Content-Type: multipart/mixed; boundary="-------------0409020452110" This is a multi-part message in MIME format. ---------------0409020452110 Content-Type: text/plain; name="04-271.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-271.keywords" Luttinger Liquids, Ward Identities, Renormalization Group ---------------0409020452110 Content-Type: application/postscript; name="betaora2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="betaora2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: betaora2.dvi %%CreationDate: Thu Sep 02 11:41:13 2004 %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMBX12 CMR10 CMCSC10 CMTI10 CMMI10 CMR5 CMMI5 %%+ CMMI7 CMSY7 CMBX7 CMSY10 CMBX5 CMR7 CMEX10 MSBM10 CMSY5 CMSL10 %%+ EUFM10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 betaora2 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2004.09.02:1132 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 122 /z put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 84 /T put dup 106 /bar put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 101 /e put dup 110 /n put dup 111 /o put dup 112 /p put dup 116 /t put dup 118 /v put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMBX5 %!PS-AdobeFont-1.1: CMBX5 1.0 %%CreationDate: 1991 Aug 20 16:35:12 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 107 /k put dup 120 /x put readonly def /FontBBox{-40 -250 1483 756}readonly def /UniqueID 5000763 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 39 /similarequal put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 66 /B put dup 67 /C put dup 68 /D put dup 76 /L put dup 78 /N put dup 82 /R put dup 84 /T put dup 86 /V put dup 87 /W put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 66 /B put dup 68 /D put dup 84 /T put dup 86 /V put dup 87 /W put dup 106 /bar put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 67 /C put dup 69 /E put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 86 /V put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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EB94F5E14B7023A995D1019E4455A7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 74 /J put dup 78 /N put dup 80 /P put dup 98 /b put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 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%%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 126 /tilde put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 67 /C put dup 76 /L put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 51 /three put dup 53 /five put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 18 /grave put dup 22 /macron put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 66 /B put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 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133[492 7[434 2[553 615 922 307 2[307 3[492 3[553 97[{}9 1106.96 /CMSL10 rf /Fc 149[254 21[455 33[523 753 244 14[753 29[407 2[600{}8 553.48 /CMSY5 rf /Fd 173[799 82[{}1 1106.96 /MSBM10 rf /Fe 143[1107 6[523 523 13[615 1414 1599 5[523 1045 1168 26[738 738 738 738 8[892 892 4[646 646 876 876 4[830 830 4[584 584 815 815 5[369 8[461 461 507 507{}30 1106.96 /CMEX10 rf /Ff 129[441 7[465 1[346 3[489 441 489 8[394 6[441 251 1[251 29[680 2[251 3[441 441 441 441 441 441 441 4[680 1[346 346 17[441 18[606 3[{}24 774.872 /CMR7 rf /Fg 135[442 12[442 107[{}2 553.48 /CMBX5 rf /Fh 135[492 7[922 1[553 3[307 2[553 553 10[738 3[1093 678 1[603 1[938 3[908 1[763 7[854 583 727 11[0 3[738 1107 7[1107 1[861 5[1107 6[861 4[861 861 2[861 14[861 307 861{}29 1106.96 /CMSY10 rf /Fi 133[446 529 529 6[529 557 4[529 107[{}6 774.872 /CMBX7 rf /Fj 149[263 18[880 558 1[502 15[689 1[584 15[597 882 255 14[882 11[692 692 12[692 692 2[454 2[692{}16 774.872 /CMSY7 rf /Fk 133[423 1[502 1[443 1[334 418 411 3[547 786 1[471 366 313 518 1[432 420 461 396 389 480 6[603 3[523 1[523 1[667 697 564 1[701 844 607 1[490 4[650 1[635 4[692 454 1[263 263 23[418 558 586 1[532 1[408 513 1[518 1[444 538 525 2[422 456 403 2[465 502 575 11[{}50 774.872 /CMMI7 rf /Fl 137[400 1[312 371 372 1[409 1[488 680 2[326 295 458 2[375 2[351 17[484 1[595 3[428 12[407 1[254 254 24[491 2[466 1[377 6[465 21[{}22 553.48 /CMMI5 rf /Fm 129[377 32[223 1[223 29[569 1[223 223 5[377 377 377 377 377 4[569 1[300 300 40[{}14 553.48 /CMR5 rf /Fn 133[566 672 672 6[672 707 4[672 22[886 707 25[636 636 636 636 636 636 636 636 636 636 1[354 46[{}19 1106.96 /CMBX10 rf /Fo 133[515 1[633 1[537 634 400 519 499 1[557 1[664 1[330 576 456 381 638 528 542 515 576 479 1[585 6[756 643 917 1045 646 1[647 679 1[875 711 844 889 1074 753 1[614 1[920 870 712 817 916 791 840 830 588 1[861 553 861 307 307 18[724 4[516 689 721 693 660 1[484 633 572 631 1[547 1[646 2[520 550 2[492 573 626 708 11[{}66 1106.96 /CMMI10 rf /Fp 133[453 537 514 736 509 594 368 453 467 509 566 566 622 905 283 509 1[339 566 509 339 509 566 509 509 566 9[1106 2[792 622 807 849 751 2[993 694 1[581 427 823 856 723 751 836 792 779 823 5[339 339 4[566 1[566 1[566 2[339 396 339 2[453 453 26[651 622 679 11[{}56 1106.96 /CMTI10 rf /Fq 139[654 507 667 2[704 1[827 4[679 1[593 617 1[654 642 679 12[870 3[821 3[756 10[901 12[612 612 612 612 2[354 418 45[{}21 1106.96 /CMCSC10 rf /Fr 129[553 2[553 492 584 584 799 584 615 430 437 434 584 615 553 615 922 307 584 338 307 615 553 338 492 615 492 615 553 2[553 307 553 307 676 830 1[1138 830 830 799 615 815 861 753 861 830 1015 692 1[569 400 830 869 723 753 846 799 784 830 3[861 1[307 307 553 553 553 553 553 553 553 553 553 553 1[307 369 307 861 1[430 430 5[553 11[553 3[553 2[922 922 615 615 646 7[769 1[922 1[{}85 1106.96 /CMR10 rf /Fs 133[664 789 789 1[789 830 581 589 610 1[830 747 830 1245 415 2[415 830 747 457 682 830 664 830 726 10[1128 1[1039 1[1115 1[1021 2[1418 898 2[557 1[1175 4[1062 1128 12[747 747 747 747 2[415 1[415 44[{}38 1328.35 /CMBX12 rf /Ft 134[967 3[1018 713 723 755 967 1[917 1018 1528 509 2[509 1018 917 1[840 1018 2[891 9[1895 10[1103 8[1324 67[{}19 1594.02 /CMBX10 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 9197 1107 a Ft(W)-153 b(ard)612 b(iden)-51 b(tities)613 b(and)f(Chiral)f(anomaly)14597 2524 y(in)h(the)f(Luttinger)g(liquid)13998 5516 y Fs(G.)498 b(Benfatto,)h(V.)g(Mastropietro)7962 6934 y Fr(Dipartimen)-31 b(to)371 b(di)f(Matematica,)i(Univ)-31 b(ersit\022)-553 b(a)371 b(di)f(Roma)g(\\T)-92 b(or)370 b(V)-92 b(ergata")12981 8351 y(Via)371 b(della)f(Ricerca)f(Scien)-31 b(ti\014ca,)372 b(I-00133,)f(Roma)0 13043 y Fq(Abstra)-25 b(ct.)673 b Fp(Systems)452 b(of)g(inter)-57 b(acting)453 b(non)f(r)-57 b(elativistic)453 b(fermions)f(in)h Fo(d)408 b Fr(=)g(1)p Fp(,)466 b(as)451 b(wel)57 b(l)451 b(as)g(spin)h(chains)g(or)0 14460 y(inter)-57 b(acting)386 b(bidimensional)g(Ising)f(mo)-57 b(dels,)387 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b(terms)h(app)-57 b(e)g(aring)421 b(in)g(the)f(War)-57 b(d)419 b(identities,)428 b(due)420 b(to)g(the)g(pr)-57 b(esenc)g(e)421 b(of)f(cuto\013s)g(br)-57 b(e)g(aking)422 b(the)e(lo)-57 b(c)g(al)0 22964 y(gauge)430 b(symmetry.)609 b(The)429 b(c)-57 b(orr)g(elations)430 b(c)-57 b(an)429 b(b)-57 b(e)430 b(c)-57 b(ompute)g(d)430 b(and)f(show)f(a)h(Luttinger)h (liquid)f(b)-57 b(ehavior)430 b(char)-57 b(ac-)0 24382 y(terize)g(d)436 b(by)g(non)h(universal)f(critic)-57 b(al)437 b(indic)-57 b(es,)447 b(so)435 b(that)g(the)h(gener)-57 b(al)436 b(Luttinger)h(liquid)g(c)-57 b(onstruction)436 b(for)g(one)0 25799 y(dimensional)391 b(systems)g(is)f(c)-57 b(omplete)g(d)391 b(without)f(any)h(use)g(of)f(exact)g(solutions.)508 b(The)390 b(ultr)-57 b(aviolet)391 b(cuto\013)f(c)-57 b(an)391 b(b)-57 b(e)0 27216 y(r)g(emove)g(d)397 b(and)f(a)g(Quantum)g (Field)g(The)-57 b(ory)396 b(c)-57 b(orr)g(esp)g(onding)398 b(to)d(the)h(Thirring)h(mo)-57 b(del)396 b(is)g(also)f(c)-57 b(onstructe)g(d.)18265 31783 y Fs(1.)1328 b(In)-42 b(tro)42 b(duction)0 35215 y Fn(1.1)1314 b Fp(Luttinger)397 b(liquids.)443 36758 y Fr(A)379 b(k)-31 b(ey)380 b(notion)h(in)e(solid)i(state)f(ph) -31 b(ysics)379 b(is)g(the)h(one)f(of)h Fp(F)-85 b(ermi)406 b(liquids)p Fr(,)382 b(used)d(to)h(describ)31 b(e)379 b(systems)g(of)h Fp(inter-)0 38175 y(acting)341 b Fr(electrons)h(whic) -31 b(h,)348 b(in)341 b(spite)g(of)h(the)f(in)-31 b(teraction,)349 b(ha)-31 b(v)g(e)342 b(a)f(ph)-31 b(ysical)342 b(b)31 b(eha)-31 b(vior)342 b(qualitativ)-31 b(ely)346 b(similar)c(to)0 39592 y(the)418 b(one)f(of)h(the)g Fp(fr)-57 b(e)g(e)442 b(F)-85 b(ermi)441 b(gas)p Fr(.)637 b(In)417 b(analogy)i(with)g(F)-92 b(ermi)417 b(liquids,)431 b(the)418 b(notion)h(of)f Fp(Luttinger)441 b(liquids)418 b Fr(has)0 41010 y(b)31 b(een)331 b(more)g(recen)-31 b(tly)332 b(in)-31 b(tro)31 b(duced,)340 b(to)332 b(describ)31 b(e)331 b(systems)g(b)31 b(eha)-31 b(ving)333 b(qualitativ)-31 b(ely)335 b(as)d(the)f(Luttinger)h(mo)31 b(del,)0 42427 y(see)423 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b(en)401 b(a)e(monomial)k Fo(Q)p Fr(\()24226 55876 y(^)23999 56168 y Fo( )40 b Fr(\))400 b(in)g(the)f(v)-61 b(ariables)33696 55876 y(^)33469 56168 y Fo( )34230 55593 y Ff([)p Fk(h;)p Ff(0])p Fk(\033)34190 56502 y Fi(k)p Fk(;!)36550 56168 y Fr(,)407 b(its)400 b(v)-61 b(alue)400 b(is)f(0,)407 b(except)0 57919 y(in)327 b(the)f(case)g Fo(Q)p Fr(\()6944 57627 y(^)6717 57919 y Fo( )41 b Fr(\))308 b(=)9385 57089 y Fe(Q)10430 58251 y Fi(k)p Fj(2D)24 b Fk(;!)k Ff(=)p Fj(\006)14955 57627 y Fr(^)14729 57919 y Fo( )15490 57344 y Ff([)p Fk(h;)p Ff(0])p Fj(\000)15450 58253 y Fi(k)p Fk(;!)18187 57627 y Fr(^)17960 57919 y Fo( )18721 57344 y Ff([)p Fk(h;)p Ff(0]+)18681 58253 y Fi(k)p Fk(;!)21180 57919 y Fr(,)335 b(up)326 b(to)h(a)f(p)31 b(erm)-31 b(utation)329 b(of)e(the)f(v)-61 b(ariables.)479 b(In)326 b(this)g(case)h(the)0 59337 y(v)-61 b(alue)398 b(of)g(the)f(functional)j(is)d(determined,)405 b(b)-31 b(y)398 b(using)g(the)f(an)-31 b(ticomm)g(uting)402 b(prop)31 b(erties)397 b(of)h(the)f(v)-61 b(ariables,)406 b(b)-31 b(y)0 59862 y Fe(R)1230 60754 y Fh(D)31 b Fo( )2876 60352 y Ff([)p Fk(h;)p Ff(0])4654 60754 y Fo(Q)p Fr(\()6187 60462 y(^)5959 60754 y Fo( )41 b Fr(\))308 b(=)f(1)h(.)443 62171 y(The)302 b(lattice)j(\003)6736 62337 y Fk(N)7878 62171 y Fr(is)d(in)-31 b(tro)31 b(duced)303 b(only)h(to)f(allo)-31 b(w)305 b(us)c(to)j(p)31 b(erform)302 b(a)h(non)f(formal)i(treatmen)-31 b(t)305 b(of)e(the)f(Grassman-)0 63589 y(nian)333 b(in)-31 b(tegrals,)342 b(as)332 b(the)h(n)-31 b(um)g(b)31 b(er)333 b(of)g(Grassmannian)g(v)-61 b(ariables)333 b(is)f(\014nite,)341 b(and)333 b(ev)-31 b(en)g(tually)335 b(the)d(limit)j Fo(N)429 b Fh(!)307 b(1)0 65006 y Fr(is)369 b(tak)-31 b(en,)371 b(see)e([BM1].)443 66423 y(W)-92 b(e)368 b(also)i(de\014ne)f (the)h Fp(Gr)-57 b(assmannian)397 b(\014eld)369 b Fr(on)g(the)h (lattice)h(\003)26995 66589 y Fk(N)28203 66423 y Fr(as)13529 68942 y Fo( )14290 68485 y Ff([)p Fk(h;)p Ff(0])p Fk(\033)14250 69216 y Fi(x)p Fk(;!)16917 68942 y Fr(=)18661 68193 y(1)p 18219 68688 1438 45 v 18219 69702 a Fo(L\014)20094 67891 y Fe(X)19974 70274 y Fi(k)p Fj(2D)21997 68942 y Fo(e)22512 68485 y Fk(i\033)28 b Fi(kx)24706 68650 y Fr(^)24480 68942 y Fo( )25241 68367 y Ff([)p Fk(h;)p Ff(0])p Fk(\033)25201 69276 y Fi(k)p Fk(;!)27868 68942 y Fo(;)1292 b Fn(x)307 b Fh(2)g Fr(\003)32260 69108 y Fk(N)33407 68942 y Fo(:)11255 b Fr(\(1)p Fo(:)p Fr(1\))0 72441 y(Note)370 b(that)h Fo( )5835 71866 y Ff([)p Fk(h;)p Ff(0])p Fk(\033)5795 72581 y Fi(x)p Fk(;!)8523 72441 y Fr(is)e(an)-31 b(tip)31 b(erio)g(dic)371 b(b)31 b(oth)370 b(in)f(time)i(and)e(space)g(v)-61 b(ariables.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14784 b Fr(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 443 1107 a Fr(W)-92 b(e)368 b(de\014ne)12448 3205 y Fo(V)246 b Fr(\()p Fo( )14531 2748 y Ff([)p Fk(h;)p Ff(0])16310 3205 y Fr(\))308 b(=)g Fo(\025)19047 1699 y Fe(Z)20338 3205 y Fo(d)p Fn(x)g Fo( )22655 2630 y Ff([)p Fk(h;)p Ff(0]+)22615 3481 y Fi(x)p Fk(;)p Ff(+)25113 3205 y Fo( )25874 2630 y Ff([)p Fk(h;)p Ff(0])p Fj(\000)25834 3481 y Fi(x)p Fk(;)p Ff(+)28345 3205 y Fo( )29106 2630 y Ff([)p Fk(h;)p Ff(0]+)29066 3481 y Fi(x)p Fk(;)p Fj(\000)31564 3205 y Fo( )32325 2630 y Ff([)p Fk(h;)p Ff(0])p Fj(\000)32285 3481 y Fi(x)p Fk(;)p Fj(\000)44969 3205 y Fr(\(1)p Fo(:)p Fr(2\))0 6294 y(and)2063 10151 y Fo(P)154 b Fr(\()p Fo(d )4695 9694 y Ff([)p Fk(h;)p Ff(0])6473 10151 y Fr(\))308 b(=)g Fh(N)9451 9694 y Fj(\000)p Ff(1)10640 10151 y Fh(D)31 b Fo( )12286 9694 y Ff([)p Fk(h;)p Ff(0])14310 10151 y Fh(\001)554 b Fr(exp)17047 8258 y Fe(\()17938 10151 y Fh(\000)19374 9402 y Fr(1)p 18932 9896 1438 45 v 18932 10910 a Fo(L\014)21087 9099 y Fe(X)20687 11444 y Fk(!)28 b Ff(=)p Fj(\006)p Ff(1)23390 9099 y Fe(X)23270 11482 y Fi(k)p Fj(2D)25293 10151 y Fo(C)26084 10317 y Fk(h;)p Ff(0)27362 10151 y Fr(\()p Fn(k)p Fr(\)\()p Fh(\000)p Fo(ik)31142 10317 y Ff(0)31887 10151 y Fr(+)246 b Fo(!)40 b(k)35 b Fr(\))34991 9859 y(^)34764 10151 y Fo( )35525 9576 y Ff([)p Fk(h;)p Ff(0]+)35485 10485 y Fi(k)p Fk(;!)38210 9859 y Fr(^)37983 10151 y Fo( )38744 9576 y Ff([)p Fk(h;)p Ff(0])p Fj(\000)38704 10485 y Fi(k)p Fk(;!)41215 8258 y Fe(\))42599 10151 y Fo(;)2063 b Fr(\(1)p Fo(:)p Fr(3\))0 14473 y(with)464 b Fh(N)626 b Fr(=)5473 13643 y Fe(Q)6519 14805 y Fi(k)p Fj(2D)8413 14473 y Fr([\()p Fo(L\014)58 b Fr(\))11017 14071 y Fj(\000)p Ff(2)12207 14473 y Fr(\()p Fh(\000)p Fo(k)14109 14071 y Ff(2)14074 14748 y(0)14915 14473 y Fh(\000)308 b Fo(k)16695 14071 y Ff(2)17192 14473 y Fr(\))p Fo(C)18413 14639 y Fk(h;)p Ff(0)19691 14473 y Fr(\()p Fn(k)p Fr(\))21223 14071 y Ff(2)21721 14473 y Fr(])462 b(and)24736 13581 y Fe(R)25659 14473 y Fo(d)p Fn(x)g Fr(is)h(a)f(shorthand)h(for)g(\\)p Fo(a)g(a)38895 14639 y Ff(0)40038 13643 y Fe(P)41207 14805 y Fi(x)p Fj(2)p Ff(\003)42939 14916 y Fl(N)43705 14473 y Fr(".)773 b(The)0 15890 y(function)423 b Fo(C)5179 16056 y Fk(h;)p Ff(0)6457 15890 y Fr(\()p Fn(k)p Fr(\))g(acts)f(as)g(an)g(ultra)-31 b(violet)425 b(and)d(infrared)g(cuto\013)g(and)g(it)h(is)f(de\014ned)f (in)i(the)f(follo)-31 b(wing)425 b(w)-31 b(a)g(y)-92 b(.)0 17308 y(W)g(e)368 b(in)-31 b(tro)31 b(duce)370 b(a)g(p)31 b(ositiv)-31 b(e)371 b(n)-31 b(um)g(b)31 b(er)369 b Fo(\015)g(>)307 b Fr(1)370 b(and)f(a)h(p)31 b(ositiv)-31 b(e)370 b(function)h Fo(\037)31111 17474 y Ff(0)31607 17308 y Fr(\()p Fo(t)p Fr(\))308 b Fh(2)g Fo(C)35091 16906 y Fj(1)36029 17308 y Fr(\()p Fd(R)37259 17474 y Ff(+)37994 17308 y Fr(\))369 b(suc)-31 b(h)369 b(that)14410 21109 y Fo(\037)15103 21275 y Ff(0)15599 21109 y Fr(\()p Fo(t)p Fr(\))308 b(=)18335 19549 y Fe(\032)19350 20445 y Fr(1)1108 b(if)369 b(0)308 b Fh(\024)g Fo(t)f Fh(\024)g Fr(1)h Fo(;)19350 21774 y Fr(0)1108 b(if)369 b Fo(t)307 b Fh(\025)h Fo(\015)24474 21940 y Ff(0)25278 21774 y Fo(;)1292 b Fr(1)308 b Fo(<)f(\015)29479 21940 y Ff(0)30284 21774 y Fh(\024)g Fo(\015)369 b(;)44969 21109 y Fr(\(1)p Fo(:)p Fr(4\))0 24995 y(and)541 b(w)-31 b(e)540 b(de\014ne,)584 b(for)540 b(an)-31 b(y)541 b(in)-31 b(teger)542 b Fo(j)656 b Fh(\024)592 b Fr(0,)584 b Fo(f)20356 25161 y Fk(j)20823 24995 y Fr(\()p Fn(k)p Fr(\))593 b(=)g Fo(\037)25095 25161 y Ff(0)25591 24995 y Fr(\()p Fo(\015)26655 24594 y Fj(\000)p Fk(j)27814 24995 y Fh(j)p Fn(k)p Fh(j)p Fr(\))362 b Fh(\000)e Fo(\037)31806 25161 y Ff(0)32302 24995 y Fr(\()p Fo(\015)33366 24594 y Fj(\000)p Fk(j)45 b Ff(+1)35646 24995 y Fh(j)p Fn(k)p Fh(j)p Fr(\).)1007 b(Finally)542 b(w)-31 b(e)541 b(de\014ne)0 26413 y Fo(\037)693 26579 y Fk(h;)p Ff(0)1970 26413 y Fr(\()p Fn(k)p Fr(\))327 b(=)f([)p Fo(C)6114 26579 y Fk(h;)p Ff(0)7392 26413 y Fr(\()p Fn(k)p Fr(\)])9231 26011 y Fj(\000)p Ff(1)10746 26413 y Fr(=)11933 25583 y Fe(P)13102 25856 y Ff(0)13102 26745 y Fk(j)45 b Ff(=)p Fk(h)14950 26413 y Fo(f)15492 26579 y Fk(j)15959 26413 y Fr(\()p Fn(k)p Fr(\))380 b(so)g(that)i([)p Fo(C)22749 26579 y Fk(h;)p Ff(0)24027 26413 y Fr(\()p Fn(k)p Fr(\)])25866 26011 y Fj(\000)p Ff(1)27436 26413 y Fr(is)e(a)g(smo)31 b(oth)382 b(function)f(with)g(supp)31 b(ort)380 b(in)h(the)0 27946 y(in)-31 b(terv)-61 b(al)370 b Fh(f)p Fo(\015)5187 27544 y Fk(h)p Fj(\000)p Ff(1)7203 27946 y Fh(\024)307 b(j)p Fn(k)p Fh(j)h(\024)g Fo(\015)61 b Fh(g)p Fr(,)370 b(equal)h(to)f(1)f(in)h(the)f(in)-31 b(terv)-61 b(al)370 b Fh(f)p Fo(\015)26580 27544 y Fk(h)27463 27946 y Fh(\024)307 b(j)p Fn(k)p Fh(j)h(\024)g Fr(1)p Fh(g)p Fr(.)443 29524 y(W)-92 b(e)368 b(call)j Fo( )5140 29122 y Ff([)p Fk(h;)p Ff(0])7287 29524 y Fr(simply)g Fo( )408 b Fr(and)370 b(w)-31 b(e)370 b(in)-31 b(tro)31 b(duce)369 b(the)h Fp(gener)-57 b(ating)398 b(functional)6420 33166 y Fh(W)91 b Fr(\()p Fo(\036;)184 b(J)106 b Fr(\))310 b(=)e(log)13428 31659 y Fe(Z)14719 33166 y Fo(P)154 b Fr(\()p Fo(d )40 b Fr(\))p Fo(e)18296 32625 y Fj(\000)p Fk(V)191 b Ff(\()p Fk( )28 b Ff(\)+)21688 31878 y Fe(P)22856 33040 y Fl(!)23573 31817 y Fe(R)24462 32625 y Fk(d)p Fi(x)25452 32708 y Fr([)25759 32625 y Fk(J)26249 32736 y Fg(x)p Fl(;!)27510 32625 y Fk( )28124 32291 y Fm(+)28096 32847 y Fg(x)p Fl(;!)29358 32625 y Fk( )29972 32291 y Fc(\000)29944 32847 y Fg(x)p Fl(;!)31205 32625 y Ff(+)p Fk(\036)32417 32291 y Fm(+)32417 32847 y Fg(x)p Fl(;!)33678 32625 y Fk( )34292 32291 y Fc(\000)34264 32847 y Fg(x)p Fl(;!)35525 32625 y Ff(+)p Fk( )36819 32291 y Fm(+)36791 32847 y Fg(x)p Fl(;!)38052 32625 y Fk(\036)38584 32291 y Fc(\000)38584 32847 y Fg(x)p Fl(;!)39846 32708 y Fr(])40516 33166 y Fo(:)4146 b Fr(\(1)p Fo(:)p Fr(5\))10339 40006 y(+)2459 b(+)9232 42219 y Fh(\000)4673 b(\000)20301 39836 y Fo(!)4846 b(!)7018 46002 y @beginspecial @setspecial %%BeginDocument: vertici.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 r cerchio_pieno % /cerchio_pieno { gsave 0 360 newpath arc fill grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 30 70 50 50 freccia 50 50 70 70 freccia 30 30 50 50 freccia 50 50 70 30 freccia 130 70 150 50 freccia 150 50 170 70 freccia grestore %%EndDocument @endspecial 3675 47419 a Fr(Fig.)494 b(1:)e(Graphical)371 b(represen)-31 b(tation)370 b(of)g(the)f(in)-31 b(teraction)372 b Fo(V)245 b Fr(\()p Fo( )40 b Fr(\))370 b(and)g(the)f(densit)-31 b(y)370 b Fo( )40022 47017 y Ff(+)39982 47693 y Fi(x)p Fk(;!)41415 47419 y Fo( )42176 47017 y Fj(\000)42136 47693 y Fi(x)p Fk(;!)443 49941 y Fr(The)277 b(Grassmannian)h(v)-61 b(ariables)277 b Fo(\036)14972 49540 y Fk(\033)14972 50215 y Fi(x)p Fk(;!)16682 49941 y Fr(are)g(an)-31 b(tip)31 b(erio)g(dic)279 b(in)e Fo(x)26300 50107 y Ff(0)27073 49941 y Fr(and)g Fo(x)g Fr(and)g(an)-31 b(ticomm)g(uting)282 b(with)c(themselv)-31 b(es)0 51359 y(and)387 b Fo( )2931 50957 y Fk(\033)2891 51632 y Fi(x)p Fk(;!)4324 51359 y Fr(,)392 b(while)c(the)f(v)-61 b(ariables)388 b Fo(J)15078 51525 y Fi(x)p Fk(;!)16897 51359 y Fr(are)f(p)31 b(erio)g(dic)387 b(and)h(comm)-31 b(uting)390 b(with)e(themselv)-31 b(es)388 b(and)f(all)h(the)g(other)0 52776 y(v)-61 b(ariables.)487 b(The)349 b(Sc)-31 b(h)g(winger)351 b(functions)f(can)g(b)31 b(e)349 b(obtained)h(b)-31 b(y)350 b(functional)i(deriv)-61 b(ativ)-31 b(es)350 b(of)g(\(1.5\);)359 b(for)349 b(instance)11652 56623 y Fo(G)12522 56166 y Ff(2)p Fk(;)p Ff(1)12522 56896 y Fk(!)13724 56623 y Fr(\()p Fn(x)p Fr(;)184 b Fn(y)18 b Fo(;)184 b Fn(z)p Fr(\))309 b(=)20128 55874 y Fo(@)p 19104 56368 2696 45 v 19104 57382 a(@)61 b(J)20367 57548 y Fi(x)p Fk(;!)24294 55874 y Fo(@)24943 55472 y Ff(2)p 22066 56368 5603 45 v 22066 57454 a Fo(@)g(\036)23375 56981 y Ff(+)23375 57730 y Fi(y)14 b Fk(;)p Ff(+)24915 57454 y Fo(@)61 b(\036)26224 56981 y Fj(\000)26224 57730 y Fi(z)p Fk(;)p Ff(+)27801 56623 y Fh(W)91 b Fr(\()p Fo(\036;)184 b(J)106 b Fr(\))p Fh(j)32023 56789 y Fk(\036)p Ff(=)p Fk(J)73 b Ff(=0)35284 56623 y Fo(;)9378 b Fr(\(1)p Fo(:)p Fr(6\))5081 61931 y Fo(G)5951 61474 y Ff(4)p Fk(;)p Ff(1)5951 62205 y Fk(!)7152 61931 y Fr(\()p Fn(x)p Fr(;)184 b Fn(x)9417 62097 y Ff(1)9915 61931 y Fo(;)g Fn(x)11078 62097 y Ff(2)11575 61931 y Fo(;)g Fn(x)12738 62097 y Ff(3)13235 61931 y Fo(;)g Fn(x)14398 62097 y Ff(4)14896 61931 y Fr(\))308 b(=)17958 61182 y Fo(@)p 16935 61677 2696 45 v 16935 62691 a(@)61 b(J)18198 62857 y Fi(x)p Fk(;!)22497 61182 y Fo(@)23146 60781 y Ff(2)p 19897 61677 6348 45 v 19897 62763 a Fo(@)g(\036)21206 62289 y Ff(+)21206 62903 y Fi(x)21735 63014 y Fm(1)22166 62903 y Fk(;!)23070 62763 y Fo(@)g(\036)24379 62289 y Fj(\000)24379 62903 y Fi(x)24908 63014 y 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Fj(\000)24183 57082 y Fo(D)25099 57248 y Fj(\000)25847 57082 y Fr(\()p Fn(p)p Fr(\))27666 56802 y(^)27414 57082 y Fo(G)28284 56550 y Ff(2)p Fk(;)p Ff(1)28284 57358 y Fj(\000)29486 57082 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p Fr(\))249 b(+)35025 56802 y(^)34735 57082 y Fo(H)35745 56550 y Ff(2)p Fk(;)p Ff(1)35655 57358 y(+)36946 57082 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p Fr(\))3576 b(\(1)p Fo(:)p Fr(13\))0 59708 y(where)475 b Fo(\027)3854 59874 y Ff(+)4589 59708 y Fo(;)184 b(\027)5627 59874 y Fj(\000)6850 59708 y Fr(are)475 b Fo(O)31 b Fr(\()p Fo(\025)p Fr(\))476 b(and)f(w)-31 b(eakly)477 b(dep)31 b(enden)-31 b(t)476 b(on)f Fo(h)p Fr(,)502 b(once)476 b(w)-31 b(e)476 b(pro)-31 b(v)g(e)475 b(that)i Fo(\025)36837 59874 y Fk(j)37778 59708 y Fr(is)e(small)i (enough)f(for)0 61126 y Fo(j)371 b Fh(\025)307 b Fo(h)p Fr(,)364 b(and)5739 60846 y(^)5449 61126 y Fo(H)6459 60594 y Ff(2)p Fk(;)p Ff(1)6369 61401 y(+)7660 61126 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p 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b(explicitly)i(pro)-31 b(v)g(ed)384 b(in)f([BM2],)389 b(whic)-31 b(h)384 b(is)g(expressed)e(directly)j(in)e(terms)h(of)g(the) 0 61369 y(function)411 b(\001)5298 60837 y Ff(2)p Fk(;)p Ff(1)5298 61644 y(+)6499 61369 y Fr(.)612 b(In)409 b(the)g(follo)-31 b(wing)413 b(section)d(w)-31 b(e)410 b(shall)g(explain)g(ho)-31 b(w)410 b(a)g(similar)g(strategy)h(can)e(b)31 b(e)409 b(applied)h(to)0 62786 y(the)369 b(second)g(addend)h(of)g(\(1.11\).)0 65251 y Fn(1.5)1314 b Fp(War)-57 b(d)396 b(identities)h(and)f(the)g(se) -57 b(c)g(ond)397 b(addend)e(of)i(\(1.11\).)443 66772 y Fr(The)306 b(analysis)h(of)g(the)g(second)e(addend)i(of)f(\(1.11\))j (is)d(more)h(complex,)320 b(the)307 b(reason)f(b)31 b(eing)306 b(that)i Fn(p)e Fr(is)g(in)-31 b(tegrated)0 68189 y(instead)446 b(of)h(b)31 b(eing)446 b(\014xed)f(at)i(the)f(infrared)f(cuto\013)i (scale,)465 b(as)446 b(it)g(w)-31 b(as)447 b(the)e(case)h(for)g(the)g (\014rst)f(addend.)722 b(If)445 b(w)-31 b(e)0 69606 y(simply)400 b(compute)8341 69327 y(^)8091 69606 y Fo(G)8961 69205 y Ff(4)p Fk(;)p Ff(1)8961 69880 y Fk(!)10560 69606 y Fr(b)-31 b(y)399 b(our)g(series)f(expansion)i(and)f(w)-31 b(e)399 b(insert)g(it)g(in)g(the)g(second)g(addend)g(of)g(\(1.11\),)409 b(w)-31 b(e)0 71024 y(get)357 b(a)g("bad")h(b)31 b(ound,)359 b(just)e(b)-31 b(y)357 b(dimensional)i(reasons.)488 b(W)-92 b(e)356 b(can)g(ho)-31 b(w)g(ev)g(er)358 b(deriv)-31 b(e)357 b(a)g(W)-92 b(ard)356 b(iden)-31 b(tit)g(y)359 b(for)45116 70744 y(^)44866 71024 y Fo(G)45736 70622 y Ff(4)p Fk(;)p Ff(1)46937 71024 y Fr(,)0 72441 y(in)369 b(the)h(form)g(of)f(\(3.17\))j(b)31 b(elo)-31 b(w,)371 b(see)e(Fig.)493 b(6.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14784 b Fr(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 13135 8185 a Fr(^)12885 8465 y Fo(G)13755 7933 y Ff(4)p Fk(;)p Ff(1)13755 8741 y(+)7571 8247 y Fo(D)8487 8413 y Ff(+)9222 8247 y Fr(\()p Fn(p)p Fr(\))8678 5197 y Fn(k)9350 5363 y Ff(1)9847 5197 y Fr(+)11445 4090 y Fn(k)12117 4256 y Ff(2)12614 4090 y Fr(+)849 b Fn(k)14996 4256 y Ff(3)15492 4090 y Fh(\000)15320 5197 y Fn(k)15992 5363 y Ff(4)16734 5197 y Fh(\000)246 b Fn(p)16980 6181 y Fh(\000)14545 12669 y Fn(p)17755 8155 y Fr(=)21105 8407 y(^)20855 8687 y Fo(G)21725 8285 y Ff(4)21725 8960 y(+)19194 4643 y Fn(k)19866 4809 y Ff(1)20609 4643 y Fh(\000)g Fn(p)20301 5959 y Fr(+)23622 4643 y Fn(k)24294 4809 y Ff(2)23069 6734 y Fr(+)22515 11838 y Fn(k)23187 12004 y Ff(3)23069 10608 y Fh(\000)17977 12392 y Fn(k)18649 12558 y Ff(4)19391 12392 y Fh(\000)g Fn(p)19416 10276 y Fh(\000)25282 8394 y(\000)28854 8407 y Fr(^)28603 8687 y Fo(G)29473 8285 y Ff(4)29473 8960 y(+)26389 4643 y Fn(k)27061 4809 y Ff(1)28050 5959 y Fr(+)29157 4643 y Fn(k)29829 4809 y Ff(2)30571 4643 y Fr(+)g Fn(p)30817 6734 y Fr(+)30264 11838 y Fn(k)30936 12004 y Ff(3)30817 10608 y Fh(\000)26389 12392 y Fn(k)27061 12558 y Ff(4)27804 12392 y Fh(\000)g Fn(p)26943 10608 y Fh(\000)33031 8394 y Fr(+)36869 8185 y(^)36684 8465 y(\001)37606 7933 y Ff(4)p Fk(;)p Ff(1)37606 8741 y(+)33031 5197 y Fn(k)33703 5363 y Ff(1)34200 5197 y Fr(+)35245 4090 y Fn(k)35917 4256 y Ff(2)36414 4090 y Fr(+)848 b Fn(k)38795 4256 y Ff(3)39292 4090 y Fh(\000)39341 5197 y Fn(k)40013 5363 y Ff(4)40755 5197 y Fh(\000)246 b Fn(p)40559 6181 y Fh(\000)38345 12669 y Fn(p)18087 15159 y Fr(0)308 b(=)g Fn(k)20789 15325 y Ff(1)21531 15159 y Fr(+)246 b Fn(k)23310 15325 y Ff(3)24053 15159 y Fh(\000)f Fn(k)25831 15325 y Ff(2)26574 15159 y Fh(\000)h Fn(k)28353 15325 y Ff(4)7018 16604 y @beginspecial @setspecial %%BeginDocument: ward2.pst %! %%BoundingBox 0 0 300 130 % x1 y1 x2 y2 cambio_coordinate /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def 60 75 20 cerchio 50 85 30 100 freccia 90 100 70 85 freccia 50 110 55 90 freccia 65 90 70 110 freccia 60 30 55 60 freccia 65 60 60 30 freccia 130 75 15 cerchio 125 87 110 105 freccia 150 105 135 87 freccia 110 45 125 62 freccia 135 62 150 45 freccia 200 75 15 cerchio 195 87 180 105 freccia 220 105 205 87 freccia 180 45 195 62 freccia 205 62 220 45 freccia 275 75 20 cerchio 265 85 245 100 freccia 305 100 285 85 freccia 265 110 270 90 freccia 280 90 285 110 freccia 275 30 270 60 freccia 280 60 275 30 freccia 275 30 3 cerchio grestore %%EndDocument @endspecial 9026 18967 a Fr(Fig.)493 b(6:)g(Graphical)370 b(represen)-31 b(tation)371 b(of)e(the)h(W)-92 b(ard)368 b(iden)-31 b(tit)g(y)372 b(\(3.17\))443 29122 y(If)414 b(w)-31 b(e)415 b(insert)g(the)f(ab)31 b(o)-31 b(v)g(e)416 b(iden)-31 b(tit)g(y)417 b(in)e(the)g(second)f(addend)h(of)g(the)g (Dyson)f(equation,)428 b(w)-31 b(e)415 b(get)h(three)e(terms)0 30540 y(\(all)462 b(m)-31 b(ultiplied)463 b(b)-31 b(y)461 b Fo(\025)p Fr(\);)507 b(t)-31 b(w)g(o)462 b(of)f(them,)484 b(the)461 b(ones)f(in)-31 b(v)g(olving)463 b(the)29341 30260 y(^)29090 30540 y Fo(G)29960 30138 y Ff(4)29960 30813 y(+)31155 30540 y Fr(functions)f(\(whic)-31 b(h)462 b(are)e(of)h(the)f(form)0 31957 y Fo(g)40 b Fr(\()p Fn(k)1670 32123 y Ff(4)2167 31957 y Fr(\)\()p Fo(L\014)58 b Fr(\))4894 31555 y Fj(\000)p Ff(1)6269 31127 y Fe(P)7437 32289 y Fi(p)8484 31677 y Fr(^)8234 31957 y Fo(G)9104 31555 y Ff(4)9104 32231 y(+)9839 31957 y Fo(D)10755 32123 y Ff(+)11490 31957 y Fr(\()p Fn(p)p Fr(\))13057 31555 y Fj(\000)p Ff(1)14247 31957 y Fr(\),)421 b(admit)413 b(go)31 b(o)g(d)411 b(b)31 b(ounds,)420 b(see)410 b(\(3.33\))j(b)31 b(elo)-31 b(w.)618 b(On)410 b(the)g(con)-31 b(trary)-92 b(,)422 b(the)411 b(third)0 33599 y(term,)376 b(whic)-31 b(h)375 b(is)f(of)g(the)g(form)h Fo(g)40 b Fr(\()p Fn(k)14719 33765 y Ff(4)15215 33599 y Fr(\)\()p Fo(L\014)58 b Fr(\))17942 33197 y Fj(\000)p Ff(1)19317 32769 y Fe(P)20486 33931 y 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Ff(+)19369 39846 y Fo(D)20285 40012 y Ff(+)21271 39566 y Fr(^)21020 39846 y Fo(G)21890 39314 y Ff(4)p Fk(;)p Ff(1)21890 40121 y(+)24176 45531 y Fr(+)26943 39846 y Fo(\027)27490 40012 y Fj(\000)28237 39846 y Fo(D)29153 40012 y Fj(\000)30151 39566 y Fr(^)29901 39846 y Fo(G)30771 39314 y Ff(4)p Fk(;)p Ff(1)30771 40121 y Fj(\000)33031 45531 y Fr(+)35867 40119 y(^)35577 40399 y Fo(H)36587 39867 y Ff(4)p Fk(;)p Ff(1)36497 40675 y(+)7018 53741 y @beginspecial @setspecial %%BeginDocument: ward5.pst %! %%BoundingBox 0 0 300 130 % x1 y1 x2 y2 cambio_coordinate /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def % x1 y1 r cerchio_pieno % /cerchio_pieno { gsave 0 360 newpath arc fill grestore} def 40 75 20 cerchio 30 92 10 110 freccia 70 110 55 88 freccia 20 115 35 94 freccia 45 94 60 115 freccia 40 30 35 56 freccia 45 56 40 30 freccia 40 27 3 cerchio 120 75 20 cerchio 110 92 90 110 freccia 150 110 135 88 freccia 100 115 115 94 freccia 125 94 140 115 freccia 120 30 115 56 freccia 125 56 120 30 freccia 200 75 20 cerchio 190 92 170 110 freccia 230 110 215 88 freccia 180 115 195 94 freccia 205 94 220 115 freccia 200 30 195 56 freccia 205 56 200 30 freccia 270 75 20 cerchio 260 92 240 110 freccia 300 110 285 88 freccia 250 115 265 94 freccia 275 94 290 115 freccia 270 30 265 56 freccia 275 56 270 30 freccia 270 27 3 cerchio_pieno grestore %%EndDocument @endspecial 1994 55159 a Fr(Fig.)g(7:)g(The)369 b(correction)h(iden) -31 b(tit)g(y)372 b(for)19146 54879 y(^)18961 55159 y(\001)19883 54757 y Ff(4)p Fk(;)p Ff(1)19883 55432 y Fk(!)21084 55159 y Fr(;)e(the)f(\014lled)h(p)31 b(oin)-31 b(t)371 b(in)e(the)h(last)g (term)f(represen)-31 b(ts)368 b(\(1.14\))443 65314 y(By)499 b(com)-31 b(bining)501 b(the)e(ab)31 b(o)-31 b(v)g(e)500 b(equation)h(and)e(the)h(W)-92 b(ard)498 b(iden)-31 b(tities)501 b(for)32152 65035 y(^)31902 65314 y Fo(G)32772 64782 y Ff(4)p Fk(;)p Ff(1)32772 65590 y(+)34471 65314 y Fr(and)37004 65035 y(^)36754 65314 y Fo(G)37624 64782 y Ff(4)p Fk(;)p Ff(1)37624 65590 y Fj(\000)39323 65314 y Fr(w)-31 b(e)500 b(obtain,)533 b(after)0 66732 y(some)583 b(algebra,)638 b(an)582 b(equation)j(relating)18449 66452 y(^)18198 66732 y Fo(G)19068 66200 y Ff(4)p Fk(;)p Ff(1)19068 67007 y(+)20852 66732 y Fr(to)22669 66452 y(^)22418 66732 y Fo(G)23288 66330 y Ff(4)23288 67005 y(+)24605 66732 y Fr(and)e(functions)32247 66452 y(^)31957 66732 y Fo(H)32967 66200 y Ff(4)p Fk(;)p Ff(1)32877 67007 y(+)34750 66732 y Fr(and)37406 66452 y(^)37116 66732 y Fo(H)38126 66200 y Ff(4)p Fk(;)p Ff(1)38036 67007 y Fj(\000)39327 66732 y Fr(;)689 b(see)582 b(\(3.24\))j(b)31 b(e-)0 68149 y(lo)-31 b(w.)1008 b(Inserting)541 b(this)g(expression)f(in)h(the)f(second)h (addend)f(of)h(the)g(r.h.s.)1007 b(of)541 b(the)f(Dyson)h(equation,)585 b(w)-31 b(e)0 69566 y(get)328 b(our)f(\014nal)i(expression,)336 b(see)326 b(\(3.26\))k(b)31 b(elo)-31 b(w,)338 b(for)327 b(the)h(Dyson)f(equation,)338 b(con)-31 b(taining)331 b(sev)-31 b(eral)327 b(terms;)342 b(among)0 70984 y(them)394 b(the)f(ones)g(still)i(requiring)e(a)g(further)g(analysis)h(are)f(the)h (ones)f(in)-31 b(v)g(olving)396 b(the)d(functions)41142 70704 y(^)40852 70984 y Fo(H)41862 70452 y Ff(4)p Fk(;)p Ff(1)41772 71259 y Fj(\006)43063 70984 y Fr(,)399 b(namely)0 72401 y Fo(g)40 b Fr(\()p Fn(k)1670 72567 y Ff(4)2167 72401 y Fr(\)\()p Fo(L\014)58 b Fr(\))4894 71999 y Fj(\000)p Ff(1)6269 71571 y Fe(P)7437 72733 y Fi(p)8524 72121 y Fr(^)8234 72401 y Fo(H)9244 71869 y Ff(4)p Fk(;)p Ff(1)9154 72676 y Fj(\006)10445 72401 y Fo(D)11361 72567 y Ff(+)12096 72401 y Fr(\()p Fn(p)p Fr(\))13663 71999 y Fj(\000)p Ff(1)14853 72401 y Fr(,)369 b(represen)-31 b(ted)368 b(as)i(in)f(Fig.)493 b(8.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14784 b Fr(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 23358 5418 a Fr(^)23069 5698 y Fo(H)24079 5166 y Ff(4)p Fk(;)p Ff(1)23989 5973 y(+)20080 2983 y Fn(k)20752 3149 y Ff(1)20301 1531 y Fr(+)22404 2429 y Fn(k)23076 2595 y Ff(2)23843 1974 y Fr(+)26389 3536 y Fn(k)27061 3702 y Ff(3)26389 1531 y Fh(\000)24176 11838 y Fn(k)24848 12004 y Ff(4)22515 12822 y Fh(\000)24729 9624 y Fn(k)25401 9790 y Ff(4)26143 9624 y Fh(\000)246 b Fn(p)24729 8394 y Fh(\000)21962 9348 y Fn(p)21962 8394 y Fr(+)7018 16604 y @beginspecial @setspecial %%BeginDocument: dyson2.pst %! %%BoundingBox 0 0 300 150 % x1 y1 x2 y2 cambio_coordinate /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def /ovale {gsave % uso: r ang x y ovale translate rotate 1 .75 scale 0 0 3 2 roll 0 360 newpath arc stroke grestore} def /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def /cerchio_pieno { gsave 0 360 newpath arc fill grestore} def 25 0 150 100 ovale 135 115 125 135 freccia 165 115 175 135 freccia 150 140 150 119 freccia 150 60 135 85 freccia 145 82 150 60 freccia 150 60 165 85 freccia 150 25 150 60 tlinea 150 60 3 cerchio_pieno grestore %%EndDocument @endspecial 8874 18022 a(Fig.)493 b(8:)g(Graphical)370 b(represen)-31 b(tation)370 b(of)g(the)g(term)f(con)-31 b(taining)36449 17742 y(^)36159 18022 y Fo(H)37169 17490 y Ff(4)p Fk(;)p Ff(1)37079 18297 y Fj(\006)443 25191 y Fr(The)432 b(analysis)h(of)g(suc)-31 b(h)431 b(terms)h(is)g(done)g (in)g Fh(x)p Fr(4;)464 b(w)-31 b(e)433 b(again)g(start)g(b)-31 b(y)432 b(writing)h(for)f(them)h(a)f(Dyson)g(equation)0 26609 y(similar)484 b(to)f(\(1.11\),)514 b(in)483 b(whic)-31 b(h)484 b(the)f(analogous)i(of)e(the)f(\014rst)h(addend)f(in)h(the)g (l.h.s.)834 b(v)-61 b(anishes;)539 b(this)483 b(Dyson)0 28026 y(equation)371 b(allo)-31 b(ws)371 b(to)f(write)g(this)g(term)f (in)h(terms)f(of)h(a)f(function)27952 27746 y(~)27702 28026 y Fo(G)28572 28192 y Ff(4)29438 28026 y Fr(similar)h(to)g Fo(G)35298 28192 y Ff(4)35795 28026 y Fr(.)12250 36619 y(~)11999 36898 y Fo(G)12869 36497 y Ff(4)12869 37172 y(+)10339 33549 y Fn(k)11011 33715 y Ff(1)8678 33426 y Fr(+)13659 32995 y Fn(k)14331 33161 y Ff(2)15873 33979 y Fr(+)14213 40744 y Fn(k)14885 40910 y Ff(3)9232 39514 y Fh(\000)10339 40744 y Fn(k)11011 40910 y Ff(4)15320 39514 y Fh(\000)20855 37060 y Fr(=)34428 33770 y(^)34138 34050 y Fo(H)35148 33518 y Ff(4)p Fk(;)p Ff(1)35058 34325 y(+)31371 31335 y Fn(k)32043 31501 y Ff(1)31371 29883 y Fr(+)33363 30781 y Fn(k)34035 30947 y Ff(2)34913 30326 y Fr(+)37459 31888 y Fn(k)38131 32054 y Ff(3)37459 29883 y Fh(\000)35245 40190 y Fn(k)35917 40356 y Ff(4)33585 41174 y Fh(\000)35799 37976 y Fn(k)36471 38142 y Ff(4)37213 37976 y Fh(\000)246 b Fn(p)35799 36746 y Fh(\000)33031 37700 y Fn(p)33031 36746 y Fr(+)7018 44956 y @beginspecial @setspecial %%BeginDocument: dyson1.pst %! %%BoundingBox 0 0 300 150 % x1 y1 x2 y2 cambio_coordinate /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def /ovale {gsave % uso: r ang x y ovale translate rotate 1 .75 scale 0 0 3 2 roll 0 360 newpath arc stroke grestore} def /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def /cerchio_pieno { gsave 0 360 newpath arc fill grestore} def 50 75 20 cerchio 40 92 20 110 freccia 80 110 60 92 freccia 60 58 80 40 freccia 20 40 40 65 tlinea 40 65 3 cerchio_pieno 25 0 250 100 ovale 235 115 225 135 freccia 265 115 275 135 freccia 250 140 250 119 freccia 250 60 235 85 freccia 242 82 250 60 freccia 250 60 265 85 freccia 250 25 250 60 tlinea 250 60 3 cerchio_pieno grestore %%EndDocument @endspecial 5676 46374 a(Fig.)493 b(9:)g(Graphical)370 b(represen)-31 b(tation)371 b(of)e(the)h(Dyson)f(equation)i(for)e(the)g (correction.)443 52599 y(W)-92 b(e)277 b(can)g(study)7405 52319 y(~)7154 52599 y Fo(G)8024 52765 y Ff(4)8798 52599 y Fr(b)-31 b(y)278 b(a)g(m)-31 b(ultiscale)280 b(analysis)e(v)-31 b(ery)278 b(similar)h(to)f(the)g(one)g(for)33011 52319 y(^)32760 52599 y Fo(G)33630 52197 y Ff(4)34127 52599 y Fr(;)308 b(the)278 b(presence)f(of)h(a)f("sp)31 b(ecial")0 54016 y(v)-31 b(ertex)392 b(\(the)g(one)g(asso)31 b(ciated)393 b(to)f(the)g(\014lled)g(p)31 b(oin)-31 b(t)392 b(in)g(Fig)g(9\))g(has)g (ho)-31 b(w)g(ev)g(er)392 b(the)g(e\013ect)f(that)i(a)f(new)g(running)0 55433 y(coupling)333 b(app)31 b(ears,)340 b(asso)31 b(ciated)333 b(with)f(the)g(lo)31 b(cal)333 b(part)f(of)h(the)f(terms)f(with)i(four) f(external)h(lines)f(among)h(whic)-31 b(h)0 56851 y(one)366 b(is)g(the)h(dotted)g(line)g(in)f(Fig.)492 b(9,)367 b(to)g(whic)-31 b(h)367 b(the)f(bare)g(propagator)406 b(^)-591 b Fo(g)40 b Fr(\()p Fn(k)31735 57017 y Ff(4)32232 56851 y Fr(\))366 b(is)g(asso)31 b(ciated;)369 b(w)-31 b(e)366 b(will)i(call)g(this)0 58268 y(new)374 b(running)f(coupling)i(constan)-31 b(t)15355 57976 y(~)15308 58268 y Fo(\025)15954 58434 y Fk(j)16421 58268 y Fr(.)505 b(It)373 b(w)-31 b(ould)375 b(seem)e(that)i(w)-31 b(e)374 b(ha)-31 b(v)g(e)374 b(a)g(problem)g(more)g(di\016cult)h(than)f (our)0 59685 y(initial)432 b(one,)445 b(since)430 b(w)-31 b(e)430 b(ha)-31 b(v)g(e)430 b(no)-31 b(w)431 b(to)f(con)-31 b(trol)431 b(the)f(\015o)-31 b(w)430 b(of)g(t)-31 b(w)g(o)431 b(running)f(coupling)h(constan)-31 b(ts,)446 b Fo(\025)42716 59851 y Fk(j)43612 59685 y Fr(and)45871 59393 y(~)45824 59685 y Fo(\025)46470 59851 y Fk(j)46937 59685 y Fr(,)0 61102 y(instead)j(of)g(one.)729 b(Ho)-31 b(w)g(ev)g(er,)470 b(it)449 b(turns)f(out,)469 b(see)447 b(Lemma)j(4.2,)469 b(that)450 b Fo(\025)30553 61268 y Fk(j)31467 61102 y Fr(and)33744 60810 y(~)33698 61102 y Fo(\025)34344 61268 y Fk(j)35258 61102 y Fr(are)e(not)h(indep)31 b(enden)-31 b(t)449 b(but)0 62520 y(are)398 b(essen)-31 b(tially)400 b Fp(pr)-57 b(op)g(ortional)p Fr(;)412 b(this)398 b(follo)-31 b(ws)401 b(from)d(a)h(careful)f(analysis)h(of)g(the)f(expansion)h(for) 41187 62228 y(~)41141 62520 y Fo(\025)41787 62686 y Fk(j)42253 62520 y Fr(,)406 b(based)398 b(on)0 63937 y(the)461 b(prop)31 b(erties)461 b(of)g(the)g(function)h Fo(C)15867 64103 y Fk(!)16508 63937 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)309 b Fh(\000)e Fn(p)p Fr(\).)768 b(One)460 b(then)h(gets)h(for)31913 63657 y(~)31662 63937 y Fo(G)32532 64103 y Ff(4)33490 63937 y Fr(a)f(b)31 b(ound)461 b(v)-31 b(ery)461 b(similar)h(to)g(the)0 65354 y(one)c(for)4151 65075 y(^)3901 65354 y Fo(G)4771 65520 y Ff(4)5268 65354 y Fr(,)480 b(except)458 b(that)h Fo(\025)12751 65520 y Fk(h)13781 65354 y Fr(is)f(replaced)g(b)-31 b(y)21113 65062 y(~)21067 65354 y Fo(\025)21713 65520 y Fk(h)22743 65354 y Fr(\(but)25338 65062 y(~)25292 65354 y Fo(\025)25938 65520 y Fk(h)26969 65354 y Fr(and)457 b Fo(\025)29855 65520 y Fk(h)30886 65354 y Fr(are)h(prop)31 b(ortional\))460 b(and)e(there)f(is)h(no)0 66772 y(w)-31 b(a)g(v)g(e)398 b(function)g(renormalization)h(asso)31 b(ciated)398 b(to)f(the)f(external)i(line)f(with)g(momen)-31 b(tum)399 b Fn(k)39155 66938 y Ff(4)40047 66772 y Fr(\(to)f(suc)-31 b(h)396 b(line)h(is)0 68189 y(asso)31 b(ciated)409 b(a)e("bare")i (instead)f(of)g(a)g("dressed")f(propagator,)419 b(lik)-31 b(e)409 b(in)30697 67909 y(^)30447 68189 y Fo(G)31317 68355 y Ff(4)31814 68189 y Fr(\).)608 b(W)-92 b(e)406 b(can)i(ho)-31 b(w)g(ev)g(er)409 b(iden)-31 b(tify)409 b(t)-31 b(w)g(o)0 69606 y(class)370 b(of)h(terms)f(in)h(the)g (expansion)g(of)16798 69327 y(~)16548 69606 y Fo(G)17418 69772 y Ff(4)17915 69606 y Fr(,)g(see)e(Fig.)497 b(10,)372 b(and)e(summing)i(them)f(has)f(the)h(e\013ect)f(that)h(also)h(the)0 71024 y(external)k(line)f(with)h(momen)-31 b(tum)377 b Fn(k)15395 71190 y Ff(4)16266 71024 y Fr(is)e(dressed)e(b)-31 b(y)375 b(the)g(in)-31 b(teraction,)380 b(and)375 b(this)g(allo)-31 b(ws)377 b(us)d(to)h(complete)i(the)0 72441 y(pro)31 b(of)369 b(that)i Fo(\025)5938 72607 y Fk(h)6819 72441 y Fr(=)307 b Fo(\025)246 b Fr(+)g Fo(O)31 b Fr(\()p Fo(\025)11937 72039 y Ff(2)12434 72441 y Fr(\))369 b(for)g(an)-31 b(y)370 b Fo(h)p Fr(.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14784 b Fr(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 20301 5074 a Fh(\000)15034 5942 y Fr(~)14988 6234 y Fo(\025)15634 6400 y Fk(j)16045 6179 y Fc(\003)31592 5971 y Fo(\025)32238 6137 y Fk(j)32649 5916 y Fc(\003)33069 4304 y Fr(^)-591 b Fo(g)33599 3902 y Ff(\()p Fk(h)p Ff(\))7018 11070 y @beginspecial @setspecial %%BeginDocument: lambda.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 50 55 20 cerchio 35 69 10 80 freccia 5 55 30 55 freccia 36 39 10 30 freccia 69 55 100 55 tlinea 69 55 punto 200 55 20 cerchio 185 70 160 80 freccia 155 55 180 55 freccia 186 39 160 30 freccia 219 55 250 55 freccia 219 55 punto 260 55 10 cerchio 270 55 300 55 tlinea grestore %%EndDocument @endspecial 16150 12487 a Fr(Fig.)493 b(10:)h(The)369 b(last)h(resummation)443 14880 y(Finally)459 b(w)-31 b(e)458 b(sho)-31 b(w)459 b(in)e(the)h(App)31 b(endix)458 b(that)h(a)f(simple)h(extension)g(of)f(our)f(analysis)i(implies)g(that) g(also)g(the)0 16297 y(ultra)-31 b(violet)398 b(cuto\013)d(can)g(b)31 b(e)394 b(remo)-31 b(v)g(ed,)403 b(that)396 b(is)e(w)-31 b(e)395 b(can)g(construct)g(a)g(QFT)g(corresp)31 b(onding)395 b(to)g(the)g(Thirring)0 17714 y(mo)31 b(del.)11867 21022 y Fs(2.)1328 b(Renormalization)501 b(Group)e(analysis)0 24359 y Fn(2.1)1314 b Fp(Multisc)-57 b(ale)396 b(inte)-57 b(gr)g(ation.)443 25807 y Fr(W)-92 b(e)390 b(resume)h(the)g (Renormalization)k(Group)390 b(analysis)j(in)e([BM1])h(for)f(the)h (generating)g(function)h(\(1.5\).)561 b(The)0 27225 y(functional)491 b(in)-31 b(tegration)491 b(of)e(\(1.5\))h(can)e(b)31 b(e)488 b(p)31 b(erformed)488 b(iterativ)-31 b(ely)491 b(in)e(the)f(follo)-31 b(wing)492 b(w)-31 b(a)g(y)-92 b(.)851 b(W)-92 b(e)488 b(pro)-31 b(v)g(e)489 b(b)-31 b(y)0 28642 y(induction)264 b(that,)285 b(for)262 b(an)-31 b(y)263 b(negativ)-31 b(e)264 b Fo(j)63 b Fr(,)284 b(there)262 b(are)f(a)i(constan)-31 b(t)263 b Fo(E)26828 28808 y Fk(j)27295 28642 y Fr(,)284 b(a)262 b(p)31 b(ositiv)-31 b(e)263 b(function)37151 28362 y(~)36918 28642 y Fo(Z)37674 28808 y Fk(j)38140 28642 y Fr(\()p Fn(k)p Fr(\))g(and)f(functionals)0 30059 y Fh(V)769 29658 y Ff(\()p Fk(j)45 b Ff(\))2297 30059 y Fr(and)369 b Fh(B)5210 29658 y Ff(\()p Fk(j)45 b Ff(\))6737 30059 y Fr(suc)-31 b(h)369 b(that)6281 33097 y Fo(e)6796 32640 y Fj(W)65 b Ff(\()p Fk(\036;J)73 b Ff(\))10153 33097 y Fr(=)308 b Fo(e)11837 32640 y Fj(\000)p Fk(L\014)40 b(E)14328 32751 y Fl(j)14981 31591 y Fe(Z)16273 33097 y Fo(P)17164 33237 y Ff(~)16984 33433 y Fk(Z)17587 33544 y Fl(j)18000 33433 y Fk(;C)18898 33550 y Fl(h;j)20078 33097 y Fr(\()p Fo(d )21845 32640 y Ff([)p Fk(h;j)45 b Ff(])23595 33097 y Fr(\))p Fo(e)24540 32630 y Fj(\000V)25855 32296 y Fm(\()p Fl(j)33 b Fm(\))26868 32630 y Ff(\()27214 31916 y Fh(p)p 28137 31916 1017 45 v 28137 32630 a Fk(Z)28740 32741 y Fl(j)29153 32630 y Fk( )29767 32296 y Fm([)p Fl(h;j)g Fm(])31338 32630 y Ff(\)+)p Fj(B)32967 32296 y Fm(\()p Fl(j)g Fm(\))33980 32630 y Ff(\()34326 31916 y Fh(p)p 35248 31916 V 714 x Fk(Z)35851 32741 y Fl(j)36265 32630 y Fk( )36879 32296 y Fm([)p Fl(h;j)g Fm(])38449 32630 y Fk(;\036;J)73 b Ff(\))40656 33097 y Fo(;)4006 b Fr(\(2)p Fo(:)p Fr(1\))0 36351 y(where:)443 37799 y(1\))288 b Fo(P)2605 37939 y Ff(~)2425 38135 y Fk(Z)3028 38246 y Fl(j)3441 38134 y Fk(;C)4339 38251 y Fl(h;j)5520 37799 y Fr(\()p Fo(d )7287 37397 y Ff([)p Fk(h;j)45 b Ff(])9036 37799 y Fr(\))288 b(is)f(the)h Fp(e\013e)-57 b(ctive)322 b(Gr)-57 b(assmannian)322 b(me)-57 b(asur)g(e)322 b(at)e(sc)-57 b(ale)321 b Fo(j)63 b Fr(,)304 b(equal)289 b(to,)305 b(if)288 b Fo(Z)39411 37965 y Fk(j)40184 37799 y Fr(=)308 b(max)43413 37965 y Fi(k)44415 37519 y Fr(~)44182 37799 y Fo(Z)44938 37965 y Fk(j)45404 37799 y Fr(\()p Fn(k)p Fr(\),)5657 41963 y Fo(P)6548 42103 y Ff(~)6368 42299 y Fk(Z)6971 42410 y Fl(j)7385 42299 y Fk(;C)8283 42416 y Fl(h;j)9463 41963 y Fr(\()p Fo(d )11230 41506 y Ff([)p Fk(h;j)45 b Ff(])12979 41963 y Fr(\))308 b(=)16625 40912 y Fe(Y)14886 43338 y Fi(k)p Ff(:)p Fk(C)16301 43455 y Fl(h;j)17425 43338 y Ff(\()p Fi(k)p Ff(\))p Fk(>)p Ff(0)20456 40912 y Fe(Y)19964 43256 y Fk(!)28 b Ff(=)p Fj(\006)p Ff(1)22679 41047 y Fo(d)23482 40755 y Fr(^)23255 41047 y Fo( )24016 40472 y Ff([)p Fk(h;j)45 b Ff(]\)+)23976 41381 y Fi(k)p Fk(;!)26791 41047 y Fo(d)27593 40755 y Fr(^)27367 41047 y Fo( )28128 40472 y Ff([)p Fk(h;j)g Ff(])p Fj(\000)28088 41381 y Fi(k)p Fk(;!)p 22679 41709 7889 45 v 25170 42723 a Fh(N)26078 42889 y Fk(j)26545 42723 y Fr(\()p Fn(k)p Fr(\))30701 41963 y Fh(\001)13656 46054 y(\001)554 b 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50211 y Fr(\()p Fn(k)p Fr(\)[)p Fh(\000)p Fo(k)28973 49754 y Ff(2)28938 50485 y(0)29717 50211 y Fh(\000)246 b Fo(k)31435 49754 y Ff(2)31932 50211 y Fr(])32239 49754 y Ff(1)p Fk(=)p Ff(2)33939 50211 y Fo(;)10723 b Fr(\(2)p Fo(:)p Fr(3\))9391 53657 y Fo(C)10182 53823 y Fk(h;j)11430 53657 y Fr(\()p Fn(k)p Fr(\))12962 53200 y Fj(\000)p Ff(1)14459 53657 y Fr(=)16238 52221 y Fk(j)15644 52605 y Fe(X)15627 54988 y Fk(r)24 b Ff(=)p Fk(h)17444 53657 y Fo(f)17986 53823 y Fk(r)18476 53657 y Fr(\()p Fn(k)p Fr(\))309 b Fh(\021)e Fo(\037)22178 53823 y Fk(h;j)23425 53657 y Fr(\()p Fn(k)p Fr(\))1108 b Fo(;)1292 b(D)28580 53823 y Fk(!)29221 53657 y Fr(\()p Fn(k)p Fr(\))309 b(=)e Fh(\000)p Fo(ik)34048 53823 y Ff(0)34791 53657 y Fr(+)246 b Fo(!)40 b(k)343 b Fr(;)7116 b(\(2)p Fo(:)p Fr(4\))443 57091 y(2\))473 b(the)g Fp(e\013e)-57 b(ctive)493 b(p)-57 b(otential)492 b(on)f(sc)-57 b(ale)492 b Fo(j)63 b Fr(,)499 b Fh(V)19426 56689 y Ff(\()p Fk(j)45 b Ff(\))20585 57091 y Fr(\()p Fo( )40 b Fr(\),)500 b(is)472 b(a)h(sum)g(of)g(monomial)j(of) e(Grassmannian)f(v)-61 b(ariables)0 58508 y(m)-31 b(ultiplied)372 b(b)-31 b(y)370 b(suitable)g(k)-31 b(ernels.)493 b Fp(i.e.)397 b Fr(it)370 b(is)f(of)h(the)f(form)5629 62248 y Fh(V)6398 61791 y Ff(\()p Fk(j)45 b Ff(\))7557 62248 y Fr(\()p Fo( )40 b Fr(\))308 b(=)11047 60864 y Fj(1)10689 61196 y Fe(X)10655 63541 y Fk(n)p Ff(=1)14035 61499 y Fr(1)p 12640 61994 3343 45 v 12640 63007 a(\()p Fo(L\014)58 b Fr(\))14937 62688 y Ff(2)p Fk(n)17434 61196 y Fe(X)16481 63426 y Fg(k)16923 63592 y Fm(1)17355 63426 y Fl(;:::)o(;)p Fg(k)19066 63592 y Fm(2)p Fl(n)16433 64058 y(!)16924 64224 y Fm(1)17355 64058 y Fl(;:::)o(;!)19115 64224 y Fm(2)p Fl(n)20574 60864 y Ff(2)p Fk(n)20361 61196 y Fe(Y)20351 63554 y Fk(i)p Ff(=1)22196 61956 y Fr(^)21970 62248 y Fo( )22731 61756 y Fk(\033)23244 61867 y Fl(i)22691 62582 y Fi(k)23220 62693 y Fl(i)23571 62582 y Fk(;!)24392 62693 y Fl(i)25120 61968 y Fr(^)24797 62248 y Fo(W)25996 61673 y Ff(\()p Fk(j)45 b Ff(\))25842 62543 y(2)p 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b(oin)-31 b(ts,)371 b Fp(normal)d Fr(and)i Fp(sp)-57 b(e)g(cial)p Fr(.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 443 1107 a Fr(With)398 b(eac)-31 b(h)398 b(normal)g(endp)31 b(oin)-31 b(t)399 b Fo(v)437 b Fr(of)398 b(scale)g Fo(h)19853 1273 y Fk(v)20776 1107 y Fr(w)-31 b(e)398 b(asso)31 b(ciate)398 b(the)g(lo)31 b(cal)399 b(term)f Fh(L)35272 827 y Fr(^)35133 1107 y Fh(V)35902 705 y Ff(\()p Fk(h)36766 816 y Fl(v)37242 705 y Ff(\))37643 1107 y Fr(\()p Fo( )38834 705 y Ff([)p Fk(h;h)40384 816 y Fl(v)40859 705 y Fj(\000)p Ff(1])42298 1107 y Fr(\))g(of)g(\(2.27\))0 2524 y(and)369 b(one)h(space-time)g(p)31 b(oin)-31 b(t)371 b Fn(x)13224 2690 y Fk(v)13750 2524 y Fr(.)492 b(W)-92 b(e)368 b(shall)i(sa)-31 b(y)370 b(that)g(the)g (endp)31 b(oin)-31 b(t)370 b(is)f(of)h(t)-31 b(yp)31 b(e)370 b Fo(\025)p Fr(.)443 4297 y(There)c(are)h(t)-31 b(w)g(o)368 b(t)-31 b(yp)31 b(es)367 b(of)g(sp)31 b(ecial)367 b(endp)31 b(oin)-31 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Fr(,)414 b(w)-31 b(e)0 9966 y(shall)375 b(call)g Fo(n)5292 9565 y Fk(\036)5292 10240 y(\034)6254 9966 y Fr(and)f Fo(n)9075 9565 y Fk(J)9075 10240 y(\034)10067 9966 y Fr(the)h(n)-31 b(um)g(b)31 b(er)374 b(of)g(endp)31 b(oin)-31 b(t)375 b(of)g(t)-31 b(yp)31 b(e)374 b Fo(\036)f Fr(and)i Fo(J)106 b Fr(,)375 b(while)h Fo(n)33811 10132 y Fk(\034)34738 9966 y Fr(will)g(denote)e(the)g(n)-31 b(um)g(b)31 b(er)374 b(of)0 11384 y(normal)396 b(endp)31 b(oin)-31 b(ts.)569 b(Finally)-92 b(,)403 b Fh(T)13952 11608 y Fk(j)m(;n;n)15935 11387 y Fl(\036)16457 11608 y Fk(;n)17267 11387 y Fl(J)18252 11384 y Fr(will)397 b(denote)e(the)g(set)f(of)h(trees)f(b)31 b(elonging)397 b(to)e Fh(T)38630 11550 y Fk(j)m(;n)40253 11384 y Fr(with)h Fo(n)f Fr(normal)0 12801 y(endp)31 b(oin)-31 b(ts,)380 b Fo(n)6030 12399 y Fk(\036)6994 12801 y Fr(endp)31 b(oin)-31 b(ts)378 b(of)f(t)-31 b(yp)31 b(e)377 b Fo(\036)f Fr(and)h Fo(n)19677 12399 y Fk(J)20672 12801 y Fr(endp)31 b(oin)-31 b(ts)378 b(of)f(t)-31 b(yp)31 b(e)377 b Fo(J)106 b Fr(.)516 b(Giv)-31 b(en)377 b(a)g(v)-31 b(ertex)377 b Fo(v)40 b Fr(,)378 b(whic)-31 b(h)378 b(is)e(not)i(an)0 14218 y(endp)31 b(oin)-31 b(t,)415 b Fn(x)5636 14384 y Fk(v)6566 14218 y Fr(will)406 b(denote)f(the)g(family)i(of)e(all)g (space-time)h(p)31 b(oin)-31 b(ts)405 b(asso)31 b(ciated)406 b(with)g(one)e(of)h(the)g(endp)31 b(oin)-31 b(ts)0 15636 y(follo)g(wing)373 b Fo(v)40 b Fr(.)498 17408 y(3\))499 b(There)f(is)g(an)h(imp)31 b(ortan)-31 b(t)500 b(constrain)-31 b(t)500 b(on)f(the)f(scale)h(indices)f(of)h(the)g(endp)31 b(oin)-31 b(ts.)880 b(In)498 b(fact,)532 b(if)499 b Fo(v)537 b Fr(is)499 b(an)0 18826 y(endp)31 b(oin)-31 b(t)404 b(normal)g(or)f(of)g(t)-31 b(yp)31 b(e)403 b Fo(J)106 b Fr(,)412 b Fo(h)15719 18992 y Fk(v)16608 18826 y Fr(=)364 b Fo(h)18471 18992 y Fk(v)18942 18770 y Fc(0)19564 18826 y Fr(+)269 b(1,)412 b(if)403 b Fo(v)23591 18424 y Fj(0)24304 18826 y Fr(is)f(the)h(non)g(trivial)i(v)-31 b(ertex)404 b(immediately)i(preceding)0 20243 y Fo(v)40 b Fr(.)554 b(This)390 b(constrain)-31 b(t)392 b(tak)-31 b(es)390 b(in)-31 b(to)392 b(accoun)-31 b(t)391 b(the)f(fact)h(that)g(at)g (least)g(one)f(of)g(the)g Fo( )430 b Fr(\014elds)389 b(asso)31 b(ciated)392 b(with)f(an)0 21660 y(endp)31 b(oin)-31 b(t)387 b(normal)h(or)e(of)h(t)-31 b(yp)31 b(e)387 b Fo(J)492 b Fr(has)386 b(to)h(b)31 b(e)386 b(con)-31 b(tracted)388 b(in)e(a)h(propagator)h(of)f(scale)f Fo(h)37526 21826 y Fk(v)37997 21605 y Fc(0)38351 21660 y Fr(,)391 b(as)386 b(a)h(consequence)0 23078 y(of)370 b(our)f(de\014nitions.)443 24851 y(On)287 b(the)h(con)-31 b(trary)-92 b(,)305 b(if)288 b Fo(v)326 b Fr(is)288 b(an)g(endp)31 b(oin)-31 b(t)288 b(of)g(t)-31 b(yp)31 b(e)288 b Fo(\036)p Fr(,)304 b(w)-31 b(e)288 b(shall)g(only)h(imp)31 b(ose)288 b(the)g(condition)h(that)g Fo(h)41892 25017 y Fk(v)42725 24851 y Fh(\025)308 b Fo(h)44532 25017 y Fk(v)45003 24795 y Fc(0)45440 24851 y Fr(+)83 b(1.)0 26268 y(In)333 b(this)g(case)g(the)g(only)i Fo( )372 b Fr(\014eld)334 b(asso)31 b(ciated)334 b(with)g 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Fr(of)324 b(the)g(tree)f(a)g (set)g Fo(P)25344 34232 y Fk(v)25870 34066 y Fr(,)333 b(the)323 b Fp(external)353 b(\014elds)323 b Fr(of)h Fo(v)40 b Fr(.)476 b(The)324 b(set)f Fo(P)42515 34232 y Fk(v)43364 34066 y Fr(includes)0 35483 y(b)31 b(oth)343 b(the)f(\014eld)h(v)-61 b(ariables)343 b(of)f(t)-31 b(yp)31 b(e)343 b Fo( )382 b Fr(whic)-31 b(h)343 b(b)31 b(elong)343 b(to)g(one)g(of)g(the)f(endp)31 b(oin)-31 b(ts)343 b(follo)-31 b(wing)346 b Fo(v)382 b Fr(and)342 b(are)g(not)h(y)-31 b(et)0 36900 y(con)g(tracted)369 b(at)g(scale)f Fo(h)10083 37066 y Fk(v)10976 36900 y Fr(\(in)h(the)f(iterativ)-31 b(e)370 b(in)-31 b(tegration)370 b(pro)31 b(cedure\),)368 b(to)h(b)31 b(e)367 b(called)i Fp(normal)395 b(external)f(\014elds)p Fr(,)0 38317 y(and)312 b(those)g(whic)-31 b(h)313 b(b)31 b(elong)312 b(to)h(an)e(endp)31 b(oin)-31 b(t)313 b(normal)g(or)f(of)g (t)-31 b(yp)31 b(e)312 b Fo(J)418 b Fr(and)312 b(are)f(con)-31 b(tracted)313 b(with)g(a)f(\014eld)g(v)-61 b(ariable)0 39735 y(b)31 b(elonging)359 b(to)f(an)g(endp)31 b(oin)-31 b(t)401 b(~)-596 b Fo(v)397 b Fr(of)358 b(t)-31 b(yp)31 b(e)358 b Fo(\036)e Fr(through)i(a)g(propagator)h Fo(g)29449 39333 y Fk(Q;)p Ff(\()p Fk(h)31314 39444 y Fm(~)-418 b Fl(v)31748 39333 y Fj(\000)p Ff(1\))33282 39735 y Fr(,)360 b(to)e(b)31 b(e)357 b(called)i Fp(sp)-57 b(e)g(cial)386 b(external)0 41152 y(\014elds)369 b Fr(of)h Fo(v)40 b Fr(.)443 42925 y(These)341 b(subsets)f(m)-31 b(ust)342 b(satisfy)g(v)-61 b(arious)341 b(constrain)-31 b(ts.)485 b(First)341 b(of)h(all,)348 b(if)342 b Fo(v)380 b Fr(is)341 b(not)h(an)g(endp)31 b(oin)-31 b(t)342 b(and)f Fo(v)42803 43091 y Ff(1)43300 42925 y Fo(;)184 b(:)g(:)g(:)j(;)d(v)46295 43091 y Fk(s)46713 43202 y Fl(v)0 44342 y Fr(are)319 b(the)g Fo(s)4173 44508 y Fk(v)5018 44342 y Fr(v)-31 b(ertices)319 b(immediately)k(follo)-31 b(wing)322 b(it,)331 b(then)319 b Fo(P)24408 44508 y Fk(v)25241 44342 y Fh(\032)308 b([)27148 44508 y Fk(i)27516 44342 y Fo(P)28227 44508 y Fk(v)28670 44619 y Fl(i)29076 44342 y Fr(.)475 b(W)-92 b(e)319 b(shall)h(denote)f Fo(Q)38645 44508 y Fk(v)39088 44619 y Fl(i)39813 44342 y Fr(the)g(in)-31 b(tersection)0 45760 y(of)380 b Fo(P)1982 45926 y Fk(v)2888 45760 y Fr(and)g Fo(P)5762 45926 y Fk(v)6205 46037 y Fl(i)6610 45760 y Fr(;)385 b(this)c(de\014nition)g(implies)g(that)g Fo(P)21290 45926 y Fk(v)22141 45760 y Fr(=)325 b Fh([)24065 45926 y Fk(i)24434 45760 y Fo(Q)25309 45926 y Fk(v)25752 46037 y Fl(i)26157 45760 y Fr(.)525 b(The)380 b(subsets)f Fo(P)33828 45926 y Fk(v)34271 46037 y Fl(i)34676 45760 y Fh(n)p Fo(Q)36104 45926 y Fk(v)36547 46037 y Fl(i)36954 45760 y Fr(,)j(whose)e(union)h(will)g(b)31 b(e)0 47177 y(made,)345 b(b)-31 b(y)339 b(de\014nition,)346 b(of)339 b(the)f Fp(internal)368 b(\014elds)337 b Fr(of)i Fo(v)40 b Fr(,)344 b(ha)-31 b(v)g(e)338 b(to)h(b)31 b(e)338 b(non)g(empt)-31 b(y)-92 b(,)346 b(if)338 b Fo(s)34934 47343 y Fk(v)35767 47177 y Fo(>)308 b Fr(1,)345 b(that)339 b(is)f(if)g Fo(v)377 b Fr(is)338 b(a)h(non)0 48594 y(trivial)371 b(v)-31 b(ertex.)443 50367 y(Moreo)g(v)g(er,)416 b(if)407 b(the)g(set)f Fo(P)11133 50533 y Fk(v)11576 50644 y Fm(0)12469 50367 y Fr(con)-31 b(tains)408 b(only)f(sp)31 b(ecial)407 b(external)h(\014elds,)416 b(that)407 b(is)g(if)g Fh(j)p Fo(P)36115 50533 y Fk(v)36558 50644 y Fm(0)37045 50367 y Fh(j)370 b Fr(=)f Fo(n)39616 49965 y Fk(\036)40204 50367 y Fr(,)416 b(and)449 b(~)-595 b Fo(v)43654 50533 y Ff(0)44556 50367 y Fr(is)407 b(the)0 51784 y(v)-31 b(ertex)370 b(immediately)i(follo)-31 b(wing)373 b Fo(v)14869 51950 y Ff(0)15365 51784 y Fr(,)d(then)f Fh(j)p Fo(P)19581 51950 y Fk(v)20024 52061 y Fm(0)20511 51784 y Fh(j)308 b Fo(<)g Fh(j)p Fo(P)23352 51950 y Ff(~)-480 b Fk(v)23756 52061 y Fm(0)24243 51784 y Fh(j)p Fr(.)0 54502 y Fn(2.3)1314 b Fp(Dimensional)397 b(b)-57 b(ounds.)443 56275 y Fr(W)-92 b(e)368 b(can)i(write)7113 59937 y Fo(S)7856 59362 y Ff(\()p Fk(h)p Ff(\))7792 60338 y(2)p Fk(m)9019 60117 y Fl(\036)9539 60338 y Fk(;n)10349 60117 y Fl(J)10941 59937 y Fr(\()p Fo(\036;)184 b(J)106 b Fr(\))309 b(=)2992 63145 y(=)4554 61761 y Fj(1)4195 62093 y Fe(X)4161 64437 y Fk(n)p Ff(=0)7011 61761 y Fj(\000)p Ff(1)6778 62093 y Fe(X)6013 64476 y Fk(j)6379 64587 y Fm(0)6812 64476 y Ff(=)p Fk(h)p Fj(\000)p Ff(1)11701 62093 y Fe(X)9460 64316 y Fl(\034)64 b Fc(2T)10879 64677 y Fl(j)11205 64843 y Fm(0)11637 64677 y Fl(;n;)p Fm(2)p Fl(m)13690 64510 y(\036)14209 64677 y(;n)14951 64514 y(J)10488 65527 y Fc(j)p Fl(P)11226 65638 y(v)11626 65804 y Fm(0)12113 65527 y Fc(j)p Fm(=2)p Fl(m)13993 65360 y(\036)15859 62093 y Fe(X)16366 64418 y Fk(!)p 16366 64595 586 45 v 17642 61638 a Fe(Z)18934 63145 y Fo(d)p Fn(x)p 19510 63322 672 45 v 20366 61761 a Ff(2)p Fk(m)21593 61427 y Fl(\036)20533 62093 y Fe(Y)20523 64451 y Fk(i)p Ff(=1)22299 63145 y Fo(\036)22959 62688 y Fk(\033)23472 62799 y Fl(i)22959 63418 y Fi(x)23488 63529 y Fl(i)23838 63418 y Fk(;!)24659 63529 y Fl(i)25485 61761 y Fk(n)26032 61427 y Fl(J)25320 62093 y Fe(Y)25249 64437 y Fk(r)24 b Ff(=1)26989 63145 y Fo(J)27603 63311 y Fi(x)28132 63595 y Fm(2)p Fl(m)29189 63428 y(\036)29710 63595 y Fm(+)p Fl(r)30725 63311 y Fk(;!)31546 63595 y Fm(2)p Fl(m)32603 63428 y(\036)33123 63595 y Fm(+)p Fl(r)34193 63145 y Fo(S)34872 63369 y Ff(2)p Fk(m)36099 63148 y Fl(\036)36620 63369 y Fk(;n)37430 63148 y Fl(J)37966 63369 y Fk(;\034)42 b(;!)p 38942 63546 586 45 v 39583 63145 a Fr(\()p Fn(x)p 40013 63322 672 45 v Fr(\))308 b Fo(;)44415 62624 y Fr(\(2)p Fo(:)p Fr(31\))0 69251 y(where)404 b Fo(!)p 3236 69428 729 45 v 405 w Fr(=)366 b Fo(!)p 5557 69428 V 405 w Fr(=)f Fh(f)p Fo(!)9119 69417 y Ff(1)9616 69251 y Fo(;)184 b(:)g(:)g(:)k(;)c(!)12764 69475 y Ff(2)p Fk(m)13991 69254 y Fl(\036)14513 69475 y Ff(+)p Fk(n)15740 69254 y Fl(J)16331 69251 y Fh(g)p Fr(,)414 b Fn(x)p 17605 69428 672 45 v 365 w Fr(=)366 b Fh(f)p Fn(x)21094 69417 y Ff(1)21591 69251 y Fo(;)184 b(:)g(:)g(:)j(;)d Fn(x)24721 69475 y Ff(2)p Fk(m)25948 69254 y Fl(\036)26470 69475 y Ff(+)p Fk(n)27697 69254 y Fl(J)28288 69251 y Fh(g)404 b Fr(and)h Fo(\033)32066 69417 y Fk(i)32799 69251 y Fr(=)366 b(+)404 b(if)g Fo(i)g Fr(is)g(o)31 b(dd,)413 b Fo(\033)41440 69417 y Fk(i)42174 69251 y Fr(=)365 b Fh(\000)404 b Fr(if)h Fo(i)f Fr(is)0 70668 y(ev)-31 b(en.)443 72441 y(Let)391 b(us)g(de\014ne)7158 72149 y(\026)7111 72441 y Fo(\025)7757 72607 y Fk(j)8568 72441 y Fr(=)345 b(max)11834 72607 y Fk(k)19 b Fj(\025)p Fk(j)13666 72441 y Fh(j)p Fo(\025)14619 72607 y Fk(k)15164 72441 y Fh(j)p Fr(;)403 b(in)392 b Fh(x)p Fr(3)g(of)g([BM1])h(it)f(is)f (pro)-31 b(v)g(ed)392 b(that)h(the)f(k)-31 b(ernels)391 b(satisfy)i(the)e(follo)-31 b(wing)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 0 1107 a Fr(b)31 b(ound:)7684 1924 y Fe(Z)8975 3430 y Fo(d)p Fn(x)p 9551 3607 672 45 v Fh(j)p Fo(S)11209 3655 y Ff(2)p Fk(m)12436 3434 y Fl(\036)12957 3655 y Fk(;n)13767 3434 y Fl(J)14303 3655 y Fk(;\034)42 b(;!)p 15279 3832 586 45 v 15920 3430 a Fr(\()p Fn(x)p 16350 3607 672 45 v Fr(\))p Fh(j)309 b(\024)e Fo(L\014)428 b 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7497 y Fr(\()20310 6748 y Fo(Z)21066 6914 y Fk(h)21584 7025 y Fl(v)p 19744 7242 2938 45 v 19744 8256 a Fo(Z)20500 8422 y Fk(h)21018 8533 y Fl(v)21492 8422 y Fj(\000)p Ff(1)22814 7497 y Fr(\))23244 7040 y Fj(j)p Fk(P)24071 7151 y Fl(v)24546 7040 y Fj(j)p Fk(=)p Ff(2)25759 7497 y Fo(\015)26393 7040 y Fj(\000)p Fk(d)27546 7151 y Fl(v)28384 7497 y Fo(;)44415 5401 y Fr(\(2)p Fo(:)p Fr(32\))0 10815 y(where)424 b Fo(h)3894 10981 y Fk(i)4686 10815 y Fr(is)g(the)h(scale)f (of)h(the)f(propagator)i(linking)g(the)e Fo(i)p Fr(-th)h(endp)31 b(oin)-31 b(t)425 b(of)g(t)-31 b(yp)31 b(e)425 b Fo(\036)e Fr(to)i(the)g(tree,)42998 10523 y(\026)42987 10815 y Fo(h)43625 10981 y Fk(r)44539 10815 y Fr(is)f(the)0 12232 y(scale)370 b(of)f(the)h Fo(r)31 b Fr(-th)369 b(endp)31 b(oin)-31 b(t)371 b(of)e(t)-31 b(yp)31 b(e)370 b Fo(J)475 b Fr(and)15838 14927 y Fo(d)16414 15093 y Fk(v)17247 14927 y Fr(=)308 b Fh(\000)p Fr(2)246 b(+)g Fh(j)p Fo(P)22201 15093 y Fk(v)22727 14927 y Fh(j)p Fo(=)p Fr(2)i(+)e Fo(n)26159 14470 y Fk(J)26159 15200 y(v)27023 14927 y Fr(+)313 b(~)-620 b Fo(z)49 b Fr(\()p Fo(P)29835 15093 y Fk(v)30361 14927 y Fr(\))308 b Fo(;)13009 b Fr(\(2)p Fo(:)p Fr(33\))0 17621 y(with)12280 19807 y(~)-620 b Fo(z)49 b Fr(\()p Fo(P)13918 19973 y Fk(v)14444 19807 y Fr(\))308 b(=)16350 17914 y Fe(\()17427 18672 y Fo(z)49 b Fr(\()p Fo(P)19132 18838 y Fk(v)19657 18672 y Fr(\))1108 b(if)369 b Fo(n)22873 18270 y Fk(\036)22873 18946 y(v)23769 18672 y Fh(\024)307 b Fr(1)h Fo(;)184 b(n)26953 18270 y Fk(J)26953 18946 y(v)27880 18672 y Fr(=)308 b(0)g Fo(;)17427 20000 y Fr(1)3215 b(if)369 b Fo(n)22873 19599 y Fk(\036)22873 20274 y(v)23769 20000 y Fr(=)307 b(0)h Fo(;)184 b(n)26953 19599 y Fk(J)26953 20274 y(v)27880 20000 y Fr(=)308 b(1)g Fo(;)184 b Fh(j)p Fo(P)31419 20166 y Fk(v)31946 20000 y Fh(j)308 b Fr(=)f(2)h Fo(;)17427 21329 y Fr(0)3215 b(otherwise)44415 19807 y(\(2)p Fo(:)p Fr(34\))0 23022 y(and)369 b Fo(z)49 b Fr(\()p Fo(P)3857 23188 y Fk(v)4383 23022 y Fr(\))308 b(=)f(1)370 b(if)g Fh(j)p Fo(P)9245 23188 y Fk(v)9771 23022 y Fh(j)308 b Fr(=)f(4,)370 b Fo(z)49 b Fr(\()p Fo(P)14489 23188 y Fk(v)15015 23022 y Fr(\))308 b(=)f(2)370 b(if)f Fh(j)p Fo(P)19876 23188 y Fk(v)20402 23022 y Fh(j)308 b Fr(=)g(2)369 b(and)h(zero)f(otherwise.)443 25384 y(As)378 b(explained)h(in)g Fh(x)p Fr(5)f(of)h([BM1],)j(one)c(can)h(sum)f(o)-31 b(v)g(er)379 b(the)f(trees)g Fo(\034)503 b Fr(only)379 b(if)g Fo(d)33108 25550 y Fk(v)33956 25384 y Fo(>)323 b Fr(0.)520 b(While)379 b(it)g(is)f(not)h(true)f(in)0 26801 y(general)414 b(that)g Fo(d)6878 26967 y Fk(v)7784 26801 y Fo(>)380 b Fr(0)413 b(in)g(\(2.32\),)427 b(it)414 b(is)e(true)h(for)g(the)g(trees)f(con)-31 b(tributing)415 b(to)33959 26521 y(^)33709 26801 y Fo(G)34579 26400 y Ff(2)p Fk(;)p Ff(1)34579 27075 y Fk(!)35780 26801 y Fr(,)36762 26521 y(^)36511 26801 y Fo(G)37381 26400 y Ff(2)37381 27075 y Fk(!)38022 26801 y Fr(,)39004 26521 y(^)38753 26801 y Fo(G)39623 26400 y Ff(4)39623 27075 y(+)40771 26801 y Fr(with)f(external)0 28219 y(momen)-31 b(ta)342 b(computed)e(at)h(the)e(cuto\013)h(scale;)351 b(hence)339 b(b)-31 b(y)340 b(using)f(the)h(b)31 b(ound)339 b(\(2.32\),)349 b(one)340 b(can)g(pro)-31 b(v)g(e,)346 b(see)339 b([BM2])0 29636 y Fh(x)p Fr(3.5,)371 b(the)e(follo)-31 b(wing)373 b(theorem.)0 31998 y Fq(Theorem)418 b(2.1)1313 b Fp(Ther)-57 b(e)397 b(exists)e Fo(")15020 32164 y Ff(0)15913 31998 y Fp(such)g(that,)h(if)22154 31706 y Fr(\026)22108 31998 y Fo(\025)22754 32164 y Fk(h)23635 31998 y Fh(\024)307 b Fo(")25319 32164 y Ff(0)26212 31998 y Fp(and)396 b Fh(j)28728 31706 y Fr(\026)28669 31998 y Fn(k)p Fh(j)308 b Fr(=)g Fo(\015)31759 31596 y Fk(h)32333 31998 y Fp(,)396 b(then)14406 35305 y Fr(^)14155 35585 y Fo(G)15025 35128 y Ff(2)p Fk(;)p Ff(1)15025 35858 y Fk(!)16226 35585 y Fr(\(2)17268 35293 y(\026)17209 35585 y Fn(k)q Fo(;)18433 35293 y Fr(\026)18373 35585 y Fn(k)q Fr(\))308 b(=)f Fh(\000)23394 34819 y Fo(Z)24229 34244 y Ff(\(2\))24150 35153 y Fk(h)p 21946 35330 4919 45 v 21946 36384 a Fo(Z)22781 36002 y Ff(2)22702 36718 y Fk(h)23278 36384 y Fo(D)24194 36550 y Fk(!)24835 36384 y Fr(\()25324 36091 y(\026)25265 36384 y Fn(k)p Fr(\))26367 36064 y Ff(2)26997 35585 y Fr([1)247 b(+)f Fo(O)31 b Fr(\()30562 35293 y(\026)30516 35585 y Fo(\025)31162 35128 y Ff(2)31162 35858 y Fk(h)31736 35585 y Fr(\)])308 b Fo(;)11327 b Fr(\(2)p Fo(:)p Fr(35\))16224 39288 y(^)15974 39568 y Fo(G)16844 39111 y Ff(2)16844 39842 y Fk(!)17485 39568 y Fr(\()17974 39276 y(\026)17915 39568 y Fn(k)p Fr(\))308 b(=)22559 38819 y(1)p 20626 39313 4420 45 v 20626 40367 a Fo(Z)21382 40533 y Fk(h)21956 40367 y Fo(D)22872 40533 y Fk(!)23513 40367 y Fr(\()24002 40075 y(\026)23943 40367 y Fn(k)p Fr(\))25179 39568 y([1)247 b(+)f Fo(O)31 b Fr(\()28744 39276 y(\026)28698 39568 y Fo(\025)29344 39111 y Ff(2)29344 39842 y Fk(h)29917 39568 y Fr(\)])309 b Fo(;)13145 b Fr(\(2)p Fo(:)p Fr(36\))13482 42275 y(^)13232 42555 y Fo(G)14102 42098 y Ff(4)14102 42829 y(+)14837 42555 y Fr(\()15326 42263 y(\026)15267 42555 y Fn(k)p Fo(;)184 b Fh(\000)17350 42263 y Fr(\026)17291 42555 y Fn(k)q Fo(;)g Fh(\000)19375 42263 y Fr(\026)19316 42555 y Fn(k)q Fr(\))308 b(=)f Fo(Z)22730 42081 y Fj(\000)p Ff(2)22651 42889 y Fk(h)23919 42555 y Fh(j)24285 42263 y Fr(\026)24226 42555 y Fn(k)p Fh(j)25205 42098 y Fj(\000)p Ff(4)26394 42555 y Fr([)p Fh(\000)p Fo(\025)28208 42721 y Fk(h)29028 42555 y Fr(+)246 b Fo(O)31 b Fr(\()31486 42263 y(\026)31440 42555 y Fo(\025)32086 42098 y Ff(2)32086 42829 y Fk(h)32660 42555 y Fr(\)])308 b Fo(:)10403 b Fr(\(2)p Fo(:)p Fr(37\))443 47087 y(The)312 b(expansion)g(for)g (\(2.35\),)326 b(\(2.36\),)h(\(2.37\))314 b(in)e(terms)g(of)g(the)g (running)g(coupling)h(constan)-31 b(t)313 b Fo(\025)40242 47253 y Fk(j)41020 47087 y Fr(is)e(con)-31 b(v)g(ergen)g(t)0 48504 y Fp(if)356 b Fo(\025)1680 48670 y Fk(j)2502 48504 y Fr(is)f(small)i(enough)f(for)g(all)h Fo(j)371 b Fh(\025)307 b Fo(h)p Fr(.)488 b(This)356 b(prop)31 b(ert)-31 b(y)355 b(is)h(surely)f(true)g(if)i Fh(j)p Fo(h)p Fh(j)f Fr(is)f(at)h(most)h (of)f(order)f Fh(j)p Fo(\025)p Fh(j)43729 48102 y Fj(\000)p Ff(1)44918 48504 y Fr(,)k(but)0 49921 y(to)349 b(pro)-31 b(v)g(e)349 b(that)h(it)f(is)g(true)f(for)g(an)-31 b(y)350 b Fh(j)p Fo(h)p Fh(j)f Fr(is)f(quite)h(non)-31 b(trivial,)356 b(as)349 b(it)g(is)f(consequence)h(of)g(in)-31 b(tricate)350 b(cancellations)0 51339 y(whic)-31 b(h)495 b(are)e(presen)-31 b(t)493 b(in)h(the)g(Beta)g(function.)868 b(In)493 b(the)h(follo)-31 b(wing)497 b(section)d(w)-31 b(e)495 b(will)g(sho)-31 b(w,)526 b(b)-31 b(y)494 b(using)g(W)-92 b(ard)0 52756 y(iden)-31 b(tities)435 b(and)e(a)g(Dyson)f(equation,)451 b(that)434 b(indeed)f Fo(\025)23089 52922 y Fk(j)23988 52756 y Fr(is)g(small)h(enough)f(for)g(all)h Fo(j)477 b Fh(\025)413 b Fo(h)433 b Fp(for)454 b(any)h Fo(h)p Fr(,)449 b(that)434 b(is)0 54173 y(uniformly)371 b(in)e(the)h(infrared) f(cuto\013,)h(so)f(that)i(the)e(ab)31 b(o)-31 b(v)g(e)371 b(theorem)e(can)h(b)31 b(e)369 b(applied.)13630 58740 y Fs(3.)1329 b(V)-125 b(anishing)501 b(of)d(Beta)h(function)0 62047 y Fn(3.1)1314 b Fp(The)397 b(main)f(the)-57 b(or)g(em)443 63465 y Fr(The)369 b(main)i(result)e(of)g(this)h(pap)31 b(er)369 b(is)g(the)g(follo)-31 b(wing)373 b(theorem.)0 65827 y Fq(Theorem)405 b(3.1)1302 b Fp(The)385 b(mo)-57 b(del)385 b(\(1.5\))f(is)h(wel)57 b(l)385 b(de\014ne)-57 b(d)386 b(in)f(the)g(limit)h Fo(h)307 b Fh(!)h(\0001)p Fp(.)506 b(In)385 b(fact)g(ther)-57 b(e)385 b(ar)-57 b(e)385 b(c)-57 b(onstants)0 67244 y Fo(")516 67410 y Ff(1)1409 67244 y Fp(and)396 b Fo(c)4038 67410 y Ff(2)4930 67244 y Fp(such)g(that)f Fh(j)p Fo(\025)p Fh(j)308 b(\024)g Fo(")12964 67410 y Ff(1)13857 67244 y Fp(implies)17694 66952 y Fr(\026)17647 67244 y Fo(\025)18293 67410 y Fk(j)19067 67244 y Fh(\024)g Fo(c)20715 67410 y Ff(2)21211 67244 y Fo(")21727 67410 y Ff(1)22224 67244 y Fp(,)396 b(for)h(any)f Fo(j)371 b(<)307 b Fr(0)p Fp(.)0 69606 y Fq(Pr)-25 b(oof)456 b(-)863 b Fr(The)406 b(pro)31 b(of)406 b(of)h(Theorem)g(3.1)g(is)f (done)g(b)-31 b(y)406 b(con)-31 b(tradiction.)607 b(Assume)405 b(that)j(there)d(exists)i(a)f Fo(h)369 b Fh(\024)g Fr(0)0 71024 y(suc)-31 b(h)369 b(that)15550 72149 y(\026)15504 72441 y Fo(\025)16150 72607 y Fk(h)p Ff(+1)18152 72441 y Fh(\024)307 b Fo(c)19799 72607 y Ff(2)20296 72441 y Fo(")20812 72607 y Ff(1)21616 72441 y Fo(<)g Fh(j)p Fo(\025)23737 72607 y Fk(h)24311 72441 y Fh(j)h(\024)g Fr(2)p Fo(c)27127 72607 y Ff(2)27624 72441 y Fo(")28140 72607 y Ff(1)28944 72441 y Fh(\024)f Fo(")30628 72607 y Ff(0)31433 72441 y Fo(;)13229 b Fr(\(3)p Fo(:)p Fr(1\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 0 1107 a Fr(where)416 b Fo(")3764 1273 y Ff(0)4677 1107 y Fr(is)h(the)g(same)g(as)f(in)h(Theorem)g(2.1.) 636 b(W)-92 b(e)416 b(sho)-31 b(w)417 b(that)h(this)f(is)g(not)g(p)31 b(ossible,)429 b(if)417 b Fo(")39118 1273 y Ff(1)39615 1107 y Fo(;)184 b(c)40585 1273 y Ff(2)41499 1107 y Fr(are)416 b(suitably)0 2524 y(c)-31 b(hosen.)443 3942 y(Let)430 b(us)f(consider)g(the)h(mo)31 b(del)431 b(with)g(cuto\013)f Fo(\015)20042 3540 y Fk(h)20616 3942 y Fr(.)674 b(In)429 b(the)h(follo)-31 b(wing)433 b(sections)d(w)-31 b(e)430 b(shall)h(pro)-31 b(v)g(e)430 b(that,)447 b(in)430 b(this)0 5359 y(mo)31 b(del,)19002 6776 y Fh(j)p Fo(\025)19955 6942 y Fk(h)20775 6776 y Fh(\000)246 b Fo(\025)p Fh(j)308 b(\024)f Fo(c)24790 6942 y Ff(3)25333 6484 y Fr(\026)25287 6776 y Fo(\025)25933 6319 y Ff(2)25933 7050 y Fk(h)p Ff(+1)27934 6776 y Fo(:)16728 b Fr(\(3)p Fo(:)p Fr(2\))0 8688 y(Ho)-31 b(w)g(ev)g(er,)375 b(as)e(a)f(consequence)h(of)g(the)f (remark)h(after)g(\(2.19\),)j Fo(\025)27053 8854 y Fk(j)27892 8688 y Fr(is)c(the)h(same,)h(for)e(an)-31 b(y)373 b Fo(j)j Fh(\025)313 b Fo(h)p Fr(,)374 b(in)e(the)h(mo)31 b(del)0 10105 y(with)335 b(or)f(without)i(cuto\013;)347 b(in)334 b(fact)h Fo(g)15254 9704 y Ff(\()p Fk(j)45 b Ff(\))16412 10105 y Fr(\()p Fn(k)p Fr(\))334 b(do)31 b(es)334 b(not)h(dep)31 b(end)333 b(on)h Fo(h)g Fr(for)g Fo(j)371 b(>)307 b(h)p Fr(,)341 b(and)334 b Fo(\025)36658 10271 y Fk(h)37565 10105 y Fr(only)h(dep)31 b(ends)333 b(on)i(the)0 11523 y(propagators)278 b Fo(g)6576 11121 y Ff(\()p Fk(j)45 b Ff(\))7734 11523 y Fr(\()p Fn(k)p Fr(\))277 b(with)h Fo(j)370 b(>)308 b(h)276 b Fr(b)-31 b(y)277 b(de\014nition.)463 b(Hence,)295 b(from)277 b(\(3.2\))i(w)-31 b(e)277 b(get)g(the)g(b)31 b(ound)276 b Fh(j)p Fo(\025)39620 11689 y Fk(h)40194 11523 y Fh(j)308 b(\024)g Fo(")42494 11689 y Ff(1)43051 11523 y Fr(+)61 b Fo(c)44452 11689 y Ff(3)44948 11523 y Fo(c)45427 11121 y Ff(2)45427 11797 y(2)45924 11523 y Fo(")46440 11121 y Ff(2)46440 11797 y(1)46937 11523 y Fr(,)0 12940 y(whic)-31 b(h)370 b(is)f(in)h(con)-31 b(tradiction)372 b(with)e(\(3.1\))i(if,)e(for)f(instance,)i Fo(c)25589 13106 y Ff(2)26393 12940 y Fr(=)307 b(2)370 b(and)f Fo(")31152 13106 y Ff(1)31956 12940 y Fh(\024)308 b Fr(1)p Fo(=)p Fr(\(4)p Fo(c)35693 13106 y Ff(3)36191 12940 y Fr(\).)p 46801 13161 443 886 v 443 16719 a(Theorem)370 b(3.1)g(implies,)h(as)e(pro)-31 b(v)g(ed)370 b(in)f([BM3],)i(that)16503 18834 y Fh(j)p Fo(\014)17436 19000 y Fk(j)m(;\025)18644 18834 y Fr(\()p Fo(\025)19720 19000 y Fk(h)20294 18834 y Fo(;)184 b(::;)g(\025)22536 19000 y Fk(h)23112 18834 y Fr(\))p Fh(j)308 b(\024)g Fo(C)79 b Fh(j)p Fo(\025)27149 19000 y Fk(h)27723 18834 y Fh(j)28030 18377 y Ff(2)28527 18834 y Fo(\015)29161 18377 y Fk(\034)90 b(j)30433 18834 y Fo(;)14229 b Fr(\(3)p Fo(:)p Fr(3\))0 20949 y(a)344 b(prop)31 b(ert)-31 b(y)345 b(called)g Fp(vanishing)374 b(of)g(Beta)f(function)p Fr(,)350 b(whic)-31 b(h)346 b(implies)f(that)g(there)f(exists)h(lim)38523 21115 y Fk(j)45 b Fj(!\0001)41631 20949 y Fo(\025)42277 21115 y Fk(j)43087 20949 y Fr(and)345 b(that)0 22366 y(this)446 b(limit)h(is)e(an)h(analytic)h(function)g(of)e Fo(\025)p Fr(.)721 b(In)444 b(its)i(turn,)464 b(the)446 b(existence)g(of)f(this)h (limit)h(implies)g(that)f(there)0 23784 y(exist)370 b(the)f(limits) 11500 25941 y(lim)10834 26639 y Fk(j)45 b Fj(!\0001)13887 25941 y Fr(log)15634 25051 y Fo(Z)16469 24476 y Ff(\(2\))16390 25359 y Fk(j)g Fj(\000)p Ff(1)p 15634 25687 2356 45 v 15800 26976 a Fo(Z)16635 26401 y Ff(\(2\))16556 27284 y Fk(j)18430 25941 y Fr(=)307 b Fo(\021)20148 26107 y Ff(2)20644 25941 y Fr(\()p Fo(\025)p Fr(\))1108 b Fo(;)1996 b Fr(lim)24857 26664 y Fk(k)19 b Fj(!\0001)27987 25941 y Fr(log)29735 25193 y Fo(Z)30491 25359 y Fk(j)45 b Fj(\000)p Ff(1)p 29735 25687 V 30301 26701 a Fo(Z)31057 26867 y Fk(j)32530 25941 y Fr(=)308 b Fo(\021)40 b Fr(\()p Fo(\025)p Fr(\))307 b Fo(;)8560 b Fr(\(3)p Fo(:)p Fr(4\))0 28941 y(with)370 b Fo(\021)40 b Fr(\()p Fo(\025)p Fr(\))308 b(=)f Fo(a)6678 29107 y Ff(2)7175 28941 y Fo(\025)7821 28540 y Ff(2)8563 28941 y Fr(+)246 b Fo(O)31 b Fr(\()p Fo(\025)11621 28540 y Ff(3)12118 28941 y Fr(\))370 b(and)f Fo(\021)15620 29107 y Ff(1)16116 28941 y Fr(\()p Fo(\025)p Fr(\))309 b(=)e Fo(a)19684 29107 y Ff(2)20181 28941 y Fo(\025)20827 28540 y Ff(2)21569 28941 y Fr(+)246 b Fo(O)31 b Fr(\()p Fo(\025)24627 28540 y Ff(3)25124 28941 y Fr(\),)370 b Fo(a)26816 29107 y Ff(2)27620 28941 y Fo(>)308 b Fr(0.)0 31304 y Fn(3.2)1314 b Fp(The)397 b(Dyson)f(e)-57 b(quation.)443 32721 y Fr(Let)369 b(us)g(no)-31 b(w)370 b(pro)-31 b(v)g(e)370 b(the)f(b)31 b(ound)369 b(\(3.2\).)495 b(W)-92 b(e)368 b(de\014ne)8826 34836 y Fo(G)9696 34379 y Ff(4)p Fk(;)p Ff(1)9696 35109 y Fk(!)10897 34836 y Fr(\()p Fn(z)p Fo(;)184 b Fn(x)13056 35002 y Ff(1)13554 34836 y Fo(;)g Fn(x)14717 35002 y Ff(2)15214 34836 y Fo(;)g Fn(x)16377 35002 y Ff(3)16875 34836 y Fo(;)g Fn(x)18038 35002 y Ff(4)18535 34836 y Fr(\))308 b(=)p Fo(<)g(\032)21875 35002 y Fi(z)p Fk(;!)23225 34836 y Fr(;)184 b Fo( )24477 34362 y Fj(\000)24437 35111 y Fi(x)24966 35222 y Fm(1)25399 35111 y Fk(;)p Ff(+)26397 34836 y Fr(;)g Fo( )27649 34362 y Ff(+)27609 35111 y Fi(x)28138 35222 y Fm(2)28571 35111 y Fk(;)p Ff(+)29569 34836 y Fr(;)g Fo( )30821 34362 y Fj(\000)30781 35111 y Fi(x)31310 35222 y Fm(3)31743 35111 y Fk(;)p Fj(\000)32753 34836 y Fr(;)g Fo( )34005 34362 y Ff(+)33965 35111 y Fi(x)34494 35222 y Fm(4)34927 35111 y Fk(;)p Fj(\000)36245 34836 y Fo(>)37106 34379 y Fk(T)38111 34836 y Fo(;)6551 b Fr(\(3)p Fo(:)p Fr(5\))11287 36950 y Fo(G)12157 36493 y Ff(4)12157 37224 y(+)12892 36950 y Fr(\()p Fn(x)13994 37116 y Ff(1)14491 36950 y Fo(;)184 b Fn(x)15654 37116 y Ff(2)16152 36950 y Fo(;)g Fn(x)17315 37116 y Ff(3)17812 36950 y Fo(;)g Fn(x)18975 37116 y Ff(4)19473 36950 y Fr(\))308 b(=)p Fo(<)f( )23001 36477 y Fj(\000)22961 37226 y Fi(x)23490 37337 y Fm(1)23922 37226 y Fk(;)p Ff(+)24920 36950 y Fo( )25681 36477 y Ff(+)25641 37226 y Fi(x)26170 37337 y Fm(2)26602 37226 y Fk(;)p Ff(+)27600 36950 y Fr(;)184 b Fo( )28852 36477 y Fj(\000)28812 37226 y Fi(x)29341 37337 y Fm(3)29774 37226 y Fk(;)p Fj(\000)30784 36950 y Fo( )31545 36477 y Ff(+)31505 37226 y Fi(x)32034 37337 y Fm(4)32466 37226 y Fk(;)p Fj(\000)33784 36950 y Fo(>)34645 36493 y Fk(T)35650 36950 y Fo(;)9012 b Fr(\(3)p Fo(:)p Fr(6\))0 38862 y(where)19420 40280 y Fo(\032)19992 40446 y Fi(x)p Fk(;!)21733 40280 y Fr(=)307 b Fo( )23662 39822 y Ff(+)23622 40553 y Fi(x)p Fk(;!)25055 40280 y Fo( )25816 39822 y Fj(\000)25776 40553 y Fi(x)p Fk(;!)27516 40280 y Fo(:)17146 b Fr(\(3)p Fo(:)p Fr(7\))0 42358 y(Moreo)-31 b(v)g(er,)342 b(w)-31 b(e)334 b(shall)g(denote)h(b)-31 b(y)14576 42078 y(^)14325 42358 y Fo(G)15195 41826 y Ff(4)p Fk(;)p Ff(1)15195 42633 y(+)p Fk(!)16516 42358 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)18816 42524 y Ff(1)19314 42358 y Fo(;)g Fn(k)20477 42524 y Ff(2)20974 42358 y Fo(;)g Fn(k)22137 42524 y Ff(3)22635 42358 y Fo(;)g Fn(k)23798 42524 y Ff(4)24295 42358 y Fr(\))334 b(and)27427 42078 y(^)27176 42358 y Fo(G)28046 41956 y Ff(4)28046 42631 y(+)28781 42358 y Fr(\()p Fn(k)29883 42524 y Ff(1)30380 42358 y Fo(;)184 b Fn(k)31543 42524 y Ff(2)32041 42358 y Fo(;)g Fn(k)33204 42524 y Ff(3)33701 42358 y Fo(;)g Fn(k)34864 42524 y Ff(4)35361 42358 y Fr(\))334 b(the)g(corresp)31 b(onding)334 b(F)-92 b(ou-)0 43775 y(rier)467 b(transforms,)494 b(depriv)-31 b(ed)468 b(of)h(the)f(momen)-31 b(tum)471 b(conserv)-61 b(ation)469 b(delta.)790 b(Note)469 b(that,)495 b(as)468 b(a)g(consequence)g(of)0 45192 y(\(1.1\),)379 b(if)c(the)f Fo( )6651 44791 y Ff(+)7761 45192 y Fr(momen)-31 b(ta)376 b(are)f(in)-31 b(terpreted)375 b(as)f(\\ingoing)k(momen)-31 b(ta")377 b(in)e(the)g(usual)g(graph)f(pictures,)j(then)0 46610 y(the)422 b Fo( )2720 46208 y Fj(\000)3888 46610 y Fr(momen)-31 b(ta)424 b(are)e(\\outgoing)i(momen)-31 b(ta";)451 b(our)422 b(de\014nition)h(of)f(F)-92 b(ourier)421 b(transform)h(is)g(suc)-31 b(h)421 b(that)i(ev)-31 b(en)0 48027 y Fn(p)p Fr(,)451 b(the)434 b(momen)-31 b(tum)436 b(asso)31 b(ciated)435 b(with)g(the)f Fo(\032)g Fr(\014eld,)451 b(is)434 b(an)g(ingoing)i(momen)-31 b(tum.)689 b(Hence,)450 b(the)435 b(momen)-31 b(tum)0 49444 y(conserv)-61 b(ation)318 b(implies)h(that)f Fn(k)13046 49610 y Ff(1)13685 49444 y Fr(+)142 b Fn(k)15360 49610 y Ff(3)16164 49444 y Fr(=)307 b Fn(k)18004 49610 y Ff(2)18643 49444 y Fr(+)142 b Fn(k)20318 49610 y Ff(4)20956 49444 y Fr(+)g Fn(p)p Fr(,)328 b(in)318 b(the)f(case)g(of)30146 49164 y(^)29895 49444 y Fo(G)30765 49042 y Ff(4)p Fk(;)p Ff(1)30765 49718 y Fk(!)31966 49444 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)34266 49610 y Ff(1)34764 49444 y Fo(;)g Fn(k)35927 49610 y Ff(2)36425 49444 y Fo(;)g Fn(k)37588 49610 y Ff(3)38085 49444 y Fo(;)g Fn(k)39248 49610 y Ff(4)39746 49444 y Fr(\))317 b(and)h Fn(k)43266 49610 y Ff(1)43904 49444 y Fr(+)142 b Fn(k)45579 49610 y Ff(3)46383 49444 y Fr(=)0 50861 y Fn(k)672 51027 y Ff(2)1414 50861 y Fr(+)246 b Fn(k)3193 51027 y Ff(4)4059 50861 y Fr(in)369 b(the)h(case)f(of)11111 50582 y(^)10861 50861 y Fo(G)11731 50460 y Ff(4)11731 51135 y(+)12466 50861 y Fr(\()p Fn(k)13568 51027 y Ff(1)14065 50861 y Fo(;)184 b Fn(k)15228 51027 y Ff(2)15725 50861 y Fo(;)g Fn(k)16888 51027 y Ff(3)17386 50861 y Fo(;)g Fn(k)18549 51027 y Ff(4)19046 50861 y Fr(\).)443 52279 y(It)457 b(is)g(p)31 b(ossible)457 b(to)h(deriv)-31 b(e)457 b(a)h(Dyson)f (equation)h(whic)-31 b(h,)481 b(com)-31 b(bined)458 b(with)h(the)e(W) -92 b(ard)456 b(iden)-31 b(tit)g(y)460 b(\(4.9\))f(of)f(ref.)0 53696 y([BM3],)371 b(giv)-31 b(es)370 b(a)f(relation)i(b)31 b(et)-31 b(w)g(een)370 b Fo(G)16484 53294 y Ff(4)16981 53696 y Fr(,)g Fo(G)18528 53294 y Ff(2)19394 53696 y Fr(and)f Fo(G)22416 53294 y Ff(2)p Fk(;)p Ff(1)23617 53696 y Fr(.)443 55113 y(If)475 b Fo(Z)564 b Fr(=)4322 54222 y Fe(R)5244 55113 y Fo(P)154 b Fr(\()p Fo(d )40 b Fr(\))184 b(exp)q Fh(f\000)p Fo(V)246 b Fr(\()p Fo( )40 b Fr(\))p Fh(g)477 b Fr(and)f Fo(<)484 b Fh(\001)i Fo(>)475 b Fr(denotes)h(the)g(exp)31 b(ectation)478 b(with)e(resp)31 b(ect)475 b(to)i Fo(Z)41887 54712 y Fj(\000)p Ff(1)43259 54222 y Fe(R)44182 55113 y Fo(P)154 b Fr(\()p Fo(d )40 b Fr(\))0 56531 y(exp)p Fh(f\000)p Fo(V)246 b Fr(\()p Fo( )40 b Fr(\))p Fh(g)p Fr(,)5996 58646 y Fo(G)6866 58188 y Ff(4)6866 58919 y(+)7601 58646 y Fr(\()p Fn(x)8703 58812 y Ff(1)9200 58646 y Fo(;)184 b Fn(x)10363 58812 y Ff(2)10861 58646 y Fo(;)g Fn(x)12024 58812 y Ff(3)12521 58646 y Fo(;)g Fn(x)13684 58812 y Ff(4)14182 58646 y Fr(\))308 b(=)p Fo(<)f( )17710 58172 y Fj(\000)17670 58921 y Fi(x)18199 59032 y Fm(1)18631 58921 y Fk(;)p Ff(+)19629 58646 y Fo( )20390 58172 y Ff(+)20350 58921 y Fi(x)20879 59032 y Fm(2)21311 58921 y Fk(;)p Ff(+)22309 58646 y Fo( )23070 58172 y Fj(\000)23030 58921 y Fi(x)23559 59032 y Fm(3)23991 58921 y Fk(;)p Fj(\000)25001 58646 y Fo( )25762 58172 y Ff(+)25722 58921 y Fi(x)26251 59032 y Fm(4)26683 58921 y Fk(;)p Fj(\000)28001 58646 y Fo(>)g Fh(\000)p Fo(G)30900 58188 y Ff(2)30900 58919 y(+)31636 58646 y Fr(\()p Fn(x)32738 58812 y Ff(1)33235 58646 y Fo(;)184 b Fn(x)34398 58812 y Ff(2)34895 58646 y Fr(\))p Fo(G)36195 58188 y Ff(2)36195 58919 y Fj(\000)36943 58646 y Fr(\()p Fn(x)38045 58812 y Ff(3)38542 58646 y Fo(;)g Fn(x)39705 58812 y Ff(4)40202 58646 y Fr(\))308 b Fo(;)3722 b Fr(\(3)p Fo(:)p Fr(8\))0 60760 y(where)369 b(w)-31 b(e)370 b(used)e(the)i(fact)g(that)g Fo(<)308 b( )15776 60359 y Fj(\000)15736 61034 y Fi(x)p Fk(;!)17169 60760 y Fo( )17930 60287 y Ff(+)17890 61036 y Fi(y)14 b Fk(;)p Fj(\000)p Fk(!)20335 60760 y Fo(>)p Fr(=)308 b(0.)443 62178 y(Let)313 b Fo(g)2898 62344 y Fk(!)3539 62178 y Fr(\()p Fn(x)p Fr(\))h(b)31 b(e)312 b(the)i(free)e(propagator,)327 b(whose)313 b(F)-92 b(ourier)313 b(transform)g(is)g Fo(g)30283 62344 y Fk(!)30924 62178 y Fr(\()p Fn(k)p Fr(\))c(=)e Fo(\037)34626 62344 y Fk(h;)p Ff(0)35903 62178 y Fr(\()p Fn(k)p Fr(\))p Fo(=)p Fr(\()p Fh(\000)p Fo(ik)40236 62344 y Ff(0)40869 62178 y Fr(+)134 b Fo(!)40 b(k)35 b Fr(\).)474 b(Then,)0 63595 y(w)-31 b(e)370 b(can)f(write)h(the)g(ab)31 b(o)-31 b(v)g(e)370 b(equation)h(as)3424 66125 y Fo(G)4294 65668 y Ff(4)4294 66399 y(+)5029 66125 y Fr(\()p Fn(x)6131 66291 y Ff(1)6628 66125 y Fo(;)184 b Fn(x)7791 66291 y Ff(2)8289 66125 y Fo(;)g Fn(x)9452 66291 y Ff(3)9949 66125 y Fo(;)g Fn(x)11112 66291 y Ff(4)11610 66125 y Fr(\))308 b(=)f Fh(\000)p Fo(\025)15207 64618 y Fe(Z)16499 66125 y Fo(d)p Fn(z)g Fo(g)18476 66291 y Fj(\000)19223 66125 y Fr(\()p Fn(z)246 b Fh(\000)g Fn(x)22244 66291 y Ff(4)22741 66125 y Fr(\))308 b Fo(<)f( )25408 65652 y Fj(\000)25368 66401 y Fi(x)25897 66512 y Fm(1)26329 66401 y Fk(;)p Ff(+)27327 66125 y Fo( )28088 65652 y Ff(+)28048 66401 y Fi(x)28577 66512 y Fm(2)29009 66401 y Fk(;)p Ff(+)30007 66125 y Fo( )30768 65652 y Fj(\000)30728 66401 y Fi(x)31257 66512 y Fm(3)31689 66401 y Fk(;)p Fj(\000)32699 66125 y Fo( )33460 65652 y Ff(+)33420 66401 y Fi(z)p Fk(;)p Fj(\000)34877 66125 y Fo( )35638 65652 y Ff(+)35598 66401 y Fi(z)p Fk(;)p Ff(+)37042 66125 y Fo( )37803 65652 y Fj(\000)37763 66401 y Fi(z)p Fk(;)p Ff(+)39515 66125 y Fo(>)h Fr(+)3670 69028 y(+)246 b Fo(\025)307 b(G)6600 68571 y Ff(2)6600 69301 y(+)7335 69028 y Fr(\()p Fn(x)8437 69194 y Ff(1)8934 69028 y Fo(;)184 b Fn(x)10097 69194 y Ff(2)10595 69028 y Fr(\))11209 67521 y Fe(Z)12501 69028 y Fo(d)p Fn(z)308 b Fo(g)14479 69194 y Fj(\000)15226 69028 y Fr(\()p Fn(z)246 b Fh(\000)g Fn(x)18247 69194 y Ff(4)18744 69028 y Fr(\))307 b Fo(<)h( )21411 68554 y Fj(\000)21371 69303 y Fi(x)21900 69414 y Fm(3)22332 69303 y Fk(;)p Fj(\000)23342 69028 y Fo( )24103 68554 y Ff(+)24063 69303 y Fi(z)p Fk(;)p Fj(\000)25520 69028 y Fo( )26281 68554 y Ff(+)26241 69303 y Fi(z)p Fk(;)p Ff(+)27685 69028 y Fo( )28446 68554 y Fj(\000)28406 69303 y Fi(z)p Fk(;)p Ff(+)30158 69028 y Fo(>)p Fr(=)3732 71930 y(=)f Fh(\000)p Fo(\025)6591 70424 y Fe(Z)7883 71930 y Fo(d)p Fn(z)p Fo(g)9553 72096 y Fj(\000)p Ff(1)10741 71930 y Fr(\()p Fn(z)246 b Fh(\000)g Fn(x)13762 72096 y Ff(4)14259 71930 y Fr(\))308 b Fo(<)f Fr([)p Fo( )17233 71457 y Fj(\000)17193 72206 y Fi(x)17722 72317 y Fm(1)18155 72206 y Fk(;)p Ff(+)19153 71930 y Fo( )19914 71457 y Ff(+)19874 72206 y Fi(x)20403 72317 y Fm(2)20835 72206 y Fk(;)p Ff(+)21832 71930 y Fr(])184 b(;)g([)p Fo( )23882 71457 y Fj(\000)23842 72206 y Fi(x)24371 72317 y Fm(3)24806 72206 y Fk(;)p Fj(\000)25816 71930 y Fo( )26577 71457 y Ff(+)26537 72206 y Fi(z)p Fk(;)p Fj(\000)27994 71930 y Fo( )28755 71457 y Ff(+)28715 72206 y Fi(z)p Fk(;)p Ff(+)30159 71930 y Fo( )30920 71457 y Fj(\000)30880 72206 y Fi(z)p Fk(;)p Ff(+)32324 71930 y Fr(])308 b Fo(>)33800 71473 y Fk(T)35113 71930 y Fo(:)44969 69028 y Fr(\(3)p Fo(:)p Fr(9\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 0 1107 a Fr(F)-92 b(rom)369 b(\(3.9\))i(w)-31 b(e)370 b(get)3863 4330 y Fh(\000)246 b Fo(G)5840 3873 y Ff(4)5840 4604 y(+)6576 4330 y Fr(\()p Fn(x)7678 4496 y Ff(1)8175 4330 y Fo(;)184 b Fn(x)9338 4496 y Ff(2)9835 4330 y Fo(;)g Fn(x)10998 4496 y Ff(3)11495 4330 y Fo(;)g Fn(x)12658 4496 y Ff(4)13156 4330 y Fr(\))308 b(=)f Fo(\025)15892 2823 y Fe(Z)17184 4330 y Fo(d)p Fn(z)p Fo(g)18854 4496 y Fj(\000)19601 4330 y Fr(\()p Fn(z)246 b Fh(\000)g Fn(x)22622 4496 y Ff(4)23119 4330 y Fr(\))308 b Fo(<)f( )25786 3857 y Fj(\000)25746 4606 y Fi(x)26275 4717 y Fm(1)26707 4606 y Fk(;)p Ff(+)27705 4330 y Fr(;)184 b Fo( )28957 3857 y Ff(+)28917 4606 y Fi(x)29446 4717 y Fm(2)29879 4606 y Fk(;)p Ff(+)30877 4330 y Fr(;)g Fo(\032)31940 4496 y Fi(z)p Fk(;)p Ff(+)33693 4330 y Fo(>)34554 3873 y Fk(T)35251 4330 y Fo(<)308 b( )37181 3857 y Fj(\000)37141 4606 y Fi(x)37670 4717 y Fm(3)38102 4606 y Fk(;)p Fj(\000)39112 4330 y Fo( )39873 3857 y Ff(+)39833 4606 y Fi(z)p Fk(;)p Fj(\000)41597 4330 y Fo(>)g Fr(+)4970 7233 y(+)246 b Fo(\025)6907 5726 y Fe(Z)8199 7233 y Fo(d)p Fn(z)p Fo(g)9869 7399 y Fj(\000)10616 7233 y Fr(\()p Fn(z)g Fh(\000)g Fn(x)13637 7399 y Ff(4)14134 7233 y Fr(\))308 b Fo(<)f(\032)16612 7399 y Fi(z)p Fk(;)p Ff(+)18057 7233 y Fr(;)184 b Fo( )19309 6759 y Fj(\000)19269 7508 y Fi(x)19798 7619 y Fm(1)20231 7508 y Fk(;)p Ff(+)21229 7233 y Fr(;)g Fo( )22481 6759 y Ff(+)22441 7508 y Fi(x)22970 7619 y Fm(2)23403 7508 y Fk(;)p Ff(+)24401 7233 y Fr(;)g Fo( )25653 6759 y Fj(\000)25613 7508 y Fi(x)26142 7619 y Fm(3)26575 7508 y Fk(;)p Fj(\000)27585 7233 y Fr(;)g Fo( )28837 6759 y Ff(+)28797 7508 y Fi(z)p Fk(;)p Fj(\000)30562 7233 y Fo(>)31423 6776 y Fk(T)32428 7233 y Fr(+)11126 b(\(3)p Fo(:)p Fr(10\))4970 10136 y(+)246 b Fo(\025)6907 8629 y Fe(Z)8199 10136 y Fo(d)p Fn(z)p Fo(g)9869 10302 y Fj(\000)10616 10136 y Fr(\()p Fn(z)g Fh(\000)g Fn(x)13637 10302 y Ff(4)14134 10136 y Fr(\))308 b Fo(<)f( )16801 9662 y Fj(\000)16761 10411 y Fi(x)17290 10522 y Fm(1)17722 10411 y Fk(;)p Ff(+)18720 10136 y Fr(;)184 b Fo( )19972 9662 y Ff(+)19932 10411 y Fi(x)20461 10522 y Fm(2)20894 10411 y Fk(;)p Ff(+)21892 10136 y Fr(;)g Fo( )23144 9662 y Fj(\000)23104 10411 y Fi(x)23633 10522 y Fm(3)24066 10411 y Fk(;)p Fj(\000)25076 10136 y Fr(;)g Fo( )26328 9662 y Ff(+)26288 10411 y Fi(z)p Fk(;)p Fj(\000)28053 10136 y Fo(>)28914 9679 y Fk(T)29612 10136 y Fo(<)307 b(\032)31352 10302 y Fi(z)p Fk(;)p Ff(+)33104 10136 y Fo(>)615 b(:)0 13297 y Fr(The)381 b(last)h(addend)f(is)g(v)-61 b(anishing,)385 b(since)c Fo(<)327 b(\032)19108 13463 y Fi(z)p Fk(;!)20786 13297 y Fo(>)p Fr(=)g(0)381 b(b)-31 b(y)381 b(the)g(propagator)i(parit)-31 b(y)382 b(prop)31 b(erties.)528 b(In)380 b(terms)h(of)0 14715 y(F)-92 b(ourier)368 b(transforms,)i(w)-31 b(e)370 b(get)g(the)f Fp(Dyson)396 b(e)-57 b(quation)8684 17643 y Fh(\000)9795 17363 y Fr(^)9545 17643 y Fo(G)10415 17186 y Ff(4)10415 17916 y(+)11150 17643 y Fr(\()p Fn(k)12252 17809 y Ff(1)12749 17643 y Fo(;)184 b Fn(k)13912 17809 y Ff(2)14409 17643 y Fo(;)g Fn(k)15572 17809 y Ff(3)16070 17643 y Fo(;)g Fn(k)17233 17809 y Ff(4)17730 17643 y Fr(\))308 b(=)g Fo(\025)38 b Fr(^)-591 b Fo(g)20811 17809 y Fj(\000)21558 17643 y Fr(\()p Fn(k)22660 17809 y Ff(4)23156 17643 y Fr(\))23586 16414 y Fe(h)24360 17363 y Fr(^)24110 17643 y Fo(G)24980 17186 y Ff(2)24980 17916 y Fj(\000)25727 17643 y Fr(\()p Fn(k)26829 17809 y Ff(3)27326 17643 y Fr(\))28007 17363 y(^)27756 17643 y Fo(G)28626 17111 y Ff(2)p Fk(;)p Ff(1)28626 17918 y(+)29828 17643 y Fr(\()p Fn(k)30930 17809 y Ff(1)31673 17643 y Fh(\000)246 b Fn(k)33452 17809 y Ff(2)33948 17643 y Fo(;)184 b Fn(k)35111 17809 y Ff(1)35609 17643 y Fo(;)g Fn(k)36772 17809 y Ff(2)37269 17643 y Fr(\)+)18407 20267 y(+)20089 19518 y(1)p 19646 20013 1438 45 v 19646 21027 a Fo(L\014)21402 19216 y Fe(X)21923 21540 y Fi(p)23185 20267 y Fo(G)24055 19735 y Ff(4)p Fk(;)p Ff(1)24055 20543 y(+)25256 20267 y Fr(\()p Fn(p)p Fr(;)g Fn(k)27556 20433 y Ff(1)28054 20267 y Fo(;)g Fn(k)29217 20433 y Ff(2)29715 20267 y Fo(;)g Fn(k)30878 20433 y Ff(3)31375 20267 y Fo(;)g Fn(k)32538 20433 y Ff(4)33282 20267 y Fh(\000)246 b Fn(p)p Fr(\))35526 19038 y Fe(i)36356 20267 y Fo(;)44415 19362 y Fr(\(3)p Fo(:)p Fr(11\))0 23998 y(see)369 b(Fig.)493 b(3.)443 25415 y(Let)439 b(us)g(no)-31 b(w)440 b(supp)31 b(ose)438 b(that)i Fh(j)p Fn(k)14043 25581 y Ff(4)14540 25415 y Fh(j)425 b(\024)f Fo(\015)17191 25014 y Fk(h)17765 25415 y Fr(;)474 b(then)440 b(the)f(supp)31 b(ort)439 b(prop)31 b(erties)439 b(of)g(the)h(propagators)g(imply)h(that)0 26833 y Fh(j)p Fn(p)p Fh(j)309 b(\024)e Fo(\015)212 b Fr(+)150 b Fo(\015)5228 26431 y Fk(h)6110 26833 y Fh(\024)308 b Fr(2)p Fo(\015)61 b Fr(,)332 b(hence)321 b(w)-31 b(e)322 b(can)g(freely)f(m)-31 b(ultiply)324 b Fo(G)23914 26301 y Ff(4)p Fk(;)p Ff(1)23914 27108 y(+)25437 26833 y Fr(in)d(the)h (r.h.s.)477 b(of)322 b(\(3.11\))i(b)-31 b(y)322 b(the)f(compact)i(supp) 31 b(ort)0 28250 y(function)326 b Fo(\037)4984 28416 y Ff(0)5480 28250 y Fr(\()p Fo(\015)6544 27848 y Fj(\000)p Fk(j)7602 27959 y Fl(m)8393 28250 y Fh(j)p Fn(p)p Fh(j)p Fr(\),)336 b(with)325 b Fo(j)13719 28416 y Fk(m)14867 28250 y Fr(=)308 b([1)156 b(+)g(log)19499 28521 y Fk(\015)20252 28250 y Fr(2])g(+)g(1,)334 b Fo(\037)24172 28416 y Ff(0)24992 28250 y Fr(b)31 b(eing)325 b(de\014ned)f(as)g(in)g(\(1.4\).)480 b(It)324 b(follo)-31 b(ws)327 b(that)e(\(3.11\))0 29667 y(can)369 b(b)31 b(e)369 b(written)i(as)2690 32414 y Fh(\000)4048 32134 y Fr(^)3797 32414 y Fo(G)4667 31957 y Ff(4)4667 32687 y(+)5402 32414 y Fr(\()p Fn(k)6504 32580 y Ff(1)7001 32414 y Fo(;)184 b Fn(k)8164 32580 y Ff(2)8662 32414 y Fo(;)g Fn(k)9825 32580 y Ff(3)10322 32414 y Fo(;)g Fn(k)11485 32580 y Ff(4)11983 32414 y Fr(\))308 b(=)f Fo(\025)38 b Fr(^)-591 b Fo(g)15063 32580 y Fj(\000)15810 32414 y Fr(\()p Fn(k)16912 32580 y Ff(4)17409 32414 y Fr(\))17839 31185 y Fe(h)18613 32134 y Fr(^)18362 32414 y Fo(G)19232 31957 y Ff(2)19232 32687 y Fj(\000)19980 32414 y Fr(\()p Fn(k)21082 32580 y Ff(3)21579 32414 y Fr(\))22260 32134 y(^)22009 32414 y Fo(G)22879 31882 y Ff(2)p Fk(;)p Ff(1)22879 32689 y(+)24080 32414 y Fr(\()p Fn(k)25182 32580 y Ff(1)25925 32414 y Fh(\000)246 b Fn(k)27704 32580 y Ff(2)28201 32414 y Fo(;)184 b Fn(k)29364 32580 y Ff(1)29861 32414 y Fo(;)g Fn(k)31024 32580 y Ff(2)31521 32414 y Fr(\)+)11603 b(\(3)p Fo(:)p Fr(12\))2690 35038 y(+)4372 34289 y(1)p 3930 34784 V 3930 35797 a Fo(L\014)5685 33987 y Fe(X)6206 36311 y Fi(p)7469 35038 y Fo(\037)8162 35204 y Fk(M)9144 35038 y Fr(\()p Fn(p)p Fr(\))p Fo(G)11581 34506 y Ff(4)p Fk(;)p Ff(1)11581 35314 y(+)12783 35038 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)15083 35204 y Ff(1)15581 35038 y Fo(;)g Fn(k)16744 35204 y Ff(2)17242 35038 y Fo(;)g Fn(k)18405 35204 y Ff(3)18902 35038 y Fo(;)g Fn(k)20065 35204 y Ff(4)20809 35038 y Fh(\000)245 b Fn(p)p Fr(\))i(+)24981 34289 y(1)p 24539 34784 V 24539 35797 a Fo(L\014)26294 33987 y Fe(X)26815 36311 y Fi(p)28209 35038 y Fr(~)-684 b Fo(\037)28771 35204 y Fk(M)29753 35038 y Fr(\()p Fn(p)p Fr(\))p Fo(G)32190 34506 y Ff(4)p Fk(;)p Ff(1)32190 35314 y(+)33392 35038 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)35692 35204 y Ff(1)36190 35038 y Fo(;)g Fn(k)37353 35204 y Ff(2)37851 35038 y Fo(;)g Fn(k)39014 35204 y Ff(3)39511 35038 y Fo(;)g Fn(k)40674 35204 y Ff(4)41418 35038 y Fh(\000)245 b Fn(p)p Fr(\))43661 33809 y Fe(i)44492 35038 y Fo(;)0 38952 y Fr(where)369 b Fo(\037)3894 39118 y Fk(M)4876 38952 y Fr(\()p Fn(p)p Fr(\))h(is)f(a)h(compact)h(supp)31 b(ort)369 b(function)h(v)-61 b(anishing)370 b(for)g Fh(j)p Fn(p)p Fh(j)308 b(\025)g Fo(\015)31733 38550 y Fk(h)p Ff(+)p Fk(j)33297 38661 y Fl(m)34033 38550 y Fj(\000)p Ff(1)35590 38952 y Fr(and)17439 42728 y(~)-684 b Fo(\037)18001 42894 y Fk(M)18983 42728 y Fr(\()p Fn(p)p Fr(\))308 b(=)23457 41292 y Fk(j)23823 41403 y Fl(m)23208 41676 y Fe(X)22027 44059 y Fk(h)22545 44170 y Fl(p)23010 44059 y Ff(=)p Fk(h)p Ff(+)p Fk(j)25254 44170 y Fl(m)26173 42728 y Fo(f)26715 42894 y Fk(h)27233 43005 y Fl(p)27753 42728 y Fr(\()p Fn(p)p Fr(\))h Fo(:)14479 b Fr(\(3)p Fo(:)p Fr(13\))0 46768 y(Note)370 b(that)h(the)e(decomp)31 b(osition)372 b(of)d(the)h Fn(p)f Fr(sum)g(is)g(done)h(so)f(that)501 b(~)-684 b Fo(\037)29068 46934 y Fk(M)30050 46768 y Fr(\()p Fn(p)p Fr(\))309 b(=)e(0)370 b(if)g Fh(j)p Fn(p)p Fh(j)308 b(\024)g Fr(2)p Fo(\015)39017 46366 y Fk(h)39591 46768 y Fr(.)443 49130 y Fp(R)-57 b(emark.)519 b Fr(The)378 b(l.h.s.)519 b(of)379 b(the)f(iden)-31 b(tit)g(y)380 b(\(3.12\))g(is,)h(b)-31 b(y)378 b(\(2.37\),)383 b(of)378 b(order)f Fo(\025)31893 49296 y Fk(h)32467 49130 y Fo(\015)33101 48729 y Fj(\000)p Ff(4)p Fk(h)34808 49130 y Fo(Z)35643 48657 y Fj(\000)p Ff(2)35564 49464 y Fk(h)36831 49130 y Fr(;)383 b(if)378 b(w)-31 b(e)378 b(can)g(pro)-31 b(v)g(e)379 b(that)0 50548 y(the)482 b(l.h.s.)832 b(is)482 b(prop)31 b(ortional)484 b(to)f Fo(\025)f Fr(with)h(essen)-31 b(tially)484 b(the)e(same)h(prop)31 b(ortionalit)-31 b(y)485 b(constan)-31 b(ts,)512 b(w)-31 b(e)483 b(get)g(that)0 51965 y Fo(\025)646 52131 y Fk(h)1711 51965 y Fh(')491 b Fo(\025)p Fr(.)824 b(This)480 b(cannot)g(b)31 b(e)479 b(ac)-31 b(hiev)g(ed)481 b(if)f(w)-31 b(e)481 b(simply)f(use)f(\(2.35\),)511 b(\(2.36\))482 b(and)e(the)f(analogous)j(b)31 b(ound)480 b(for)251 53103 y(^)0 53382 y Fo(G)870 52981 y Ff(4)p Fk(;)p Ff(1)2071 53382 y Fr(,)351 b(giv)-31 b(en)347 b(in)f(Lemma)h(A1.2)h(of)e([BM3];) 355 b(for)346 b(instance,)352 b(b)-31 b(y)346 b(using)h(\(2.35\))h(and) f(\(2.36\),)353 b(w)-31 b(e)347 b(see)e(that)i(the)f(\014rst)0 54926 y(addend)412 b(in)f(the)h(r.h.s.)619 b(of)412 b(\(3.12\))i(is)d (of)h(size)f Fo(\015)19999 54524 y Fj(\000)p Ff(2)p Fk(h)21707 54926 y Fo(Z)22542 54351 y Ff(\(2\))22463 55260 y Fk(h)23730 54926 y Fo(Z)24565 54452 y Fj(\000)p Ff(2)24486 55260 y Fk(h)25753 54926 y Fo(\025)p Fr([1)275 b(+)f Fo(O)31 b Fr(\()30020 54634 y(\026)29974 54926 y Fo(\025)30620 54524 y Ff(2)30620 55239 y Fk(h)31194 54926 y Fr(\)].)620 b(W)-92 b(e)411 b(ha)-31 b(v)g(e)412 b(to)g(tak)-31 b(e)413 b(in)-31 b(to)412 b(accoun)-31 b(t)0 56343 y(some)360 b(crucial)f(cancellations)j(in)e(the)f(p)31 b(erturbativ)-31 b(e)360 b(expansion,)j(and)c(this)h(will)h(b)31 b(e)358 b(done)i(b)-31 b(y)359 b(expressing)45424 56063 y(^)45173 56343 y Fo(G)46043 55942 y Ff(2)p Fk(;)p Ff(1)0 57761 y Fr(and)2451 57481 y(^)2200 57761 y Fo(G)3070 57359 y Ff(4)p Fk(;)p Ff(1)4688 57761 y Fr(in)417 b(terms)g(of)h(other)f (functional)j(in)-31 b(tegrals)418 b(b)-31 b(y)418 b(suitable)g(W)-92 b(ard)416 b(iden)-31 b(tities)419 b(whic)-31 b(h)418 b(at)g(the)f(end)g(will)0 59178 y(allo)-31 b(w)372 b(us)c(to)i(pro)-31 b(v)g(e)370 b(\(3.2\).)0 61540 y Fn(3.3)1314 b Fp(War)-57 b(d)396 b(identities.)443 62958 y Fr(By)292 b(doing)h(in)f(\(1.5\))j (the)d(c)-31 b(hiral)293 b(Gauge)f(transformation)j Fo( )25224 62556 y Fk(\033)25184 63232 y Fi(x)p Fk(;)p Ff(+)27019 62958 y Fh(!)307 b Fo(e)28948 62556 y Fk(i\033)28 b(\013)30377 62667 y Fg(x)p Fl(;)p Fm(+)31753 62958 y Fo( )32514 62556 y Fk(\033)32474 63232 y Fi(x)p Fk(;)p Ff(+)34001 62958 y Fr(,)308 b Fo( )35377 62556 y Fk(\033)35337 63232 y Fi(x)p Fk(;)p Fj(\000)37183 62958 y Fh(!)g Fo( )39359 62556 y Fk(\033)39319 63232 y Fi(x)p Fk(;)p Fj(\000)40858 62958 y Fr(,)g(one)292 b(obtains,)0 64375 y(see)369 b([BM2],)h(the)g(W) -92 b(ard)368 b(iden)-31 b(tit)g(y)-92 b(,)372 b(see)d(Fig.)493 b(4)10483 67122 y Fo(D)11399 67288 y Ff(+)12134 67122 y Fr(\()p Fn(p)p Fr(\))p Fo(G)14571 66590 y Ff(2)p Fk(;)p Ff(1)14571 67397 y(+)15773 67122 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p Fr(\))311 b(=)c Fo(G)22015 66665 y Ff(2)22015 67395 y(+)22750 67122 y Fr(\()p Fn(q)p Fr(\))247 b Fh(\000)f Fo(G)26506 66665 y Ff(2)26506 67395 y(+)27241 67122 y Fr(\()p Fn(k)p Fr(\))h(+)f(\001)31049 66590 y Ff(2)p Fk(;)p Ff(1)31049 67397 y(+)32250 67122 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p Fr(\))311 b Fo(;)7654 b Fr(\(3)p Fo(:)p Fr(14\))0 69868 y(with)7860 71442 y(\001)8782 70910 y Ff(2)p Fk(;)p Ff(1)8782 71717 y(+)9984 71442 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fo(;)g Fn(q)p Fr(\))310 b(=)15931 70693 y(1)p 15488 71187 V 15488 72201 a Fo(\014)58 b(L)17244 70390 y Fe(X)17779 72773 y Fi(k)19027 71442 y Fo(C)19818 71608 y Ff(+)20553 71442 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))308 b Fo(<)27012 71149 y Fr(^)26786 71442 y Fo( )27547 70968 y Ff(+)27507 71776 y Fi(k)p Fk(;)p Ff(+)29260 71149 y Fr(^)29034 71442 y Fo( )29795 70968 y Fj(\000)29755 71776 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)32530 71442 y Fr(;)33249 71149 y(^)33021 71442 y Fo( )33782 70968 y Fj(\000)33742 71776 y Fi(k)p Fk(;)p Ff(+)35497 71149 y Fr(^)35270 71442 y Fo( )36031 70968 y Ff(+)35991 71717 y Fi(q)p Fk(;)p Ff(+)37825 71442 y Fo(>)38686 70985 y Fk(T)44415 71442 y Fr(\(3)p Fo(:)p Fr(15\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 0 1107 a Fr(and)8399 2524 y Fo(C)9190 2690 y Fk(!)9831 2524 y Fr(\()p Fn(k)10933 2067 y Ff(+)11669 2524 y Fo(;)184 b Fn(k)12832 2067 y Fj(\000)13580 2524 y Fr(\))308 b(=)f([)p Fo(C)16584 2690 y Fk(h;)p Ff(0)17862 2524 y Fr(\()p Fn(k)18964 2067 y Fj(\000)19712 2524 y Fr(\))246 b Fh(\000)g Fr(1])p Fo(D)23271 2690 y Fk(!)23914 2524 y Fr(\()p Fn(k)25016 2067 y Fj(\000)25763 2524 y Fr(\))h Fh(\000)f Fr([)p Fo(C)28645 2690 y Fk(h;)p Ff(0)29923 2524 y Fr(\()p Fn(k)31025 2067 y Ff(+)31760 2524 y Fr(\))h Fh(\000)f Fr(1])p Fo(D)35320 2690 y Fk(!)35962 2524 y Fr(\()p Fn(k)37064 2067 y Ff(+)37799 2524 y Fr(\))308 b Fo(:)5571 b Fr(\(3)p Fo(:)p Fr(16\))443 5602 y(In)368 b(the)i(same)f(w)-31 b(a)g(y)-92 b(,)371 b(w)-31 b(e)370 b(get)g(t)-31 b(w)g(o)371 b(other)e(W)-92 b(ard)369 b(iden)-31 b(tities)8305 8156 y Fo(D)9221 8322 y Ff(+)9956 8156 y Fr(\()p Fn(p)p Fr(\))p Fo(G)12393 7624 y Ff(4)p Fk(;)p Ff(1)12393 8431 y(+)13596 8156 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)15896 8322 y Ff(1)16394 8156 y Fo(;)g Fn(k)17557 8322 y Ff(2)18054 8156 y Fo(;)g Fn(k)19217 8322 y Ff(3)19715 8156 y Fo(;)g Fn(k)20878 8322 y Ff(4)21621 8156 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)g Fo(G)26212 7698 y Ff(4)26212 8429 y(+)26947 8156 y Fr(\()p Fn(k)28049 8322 y Ff(1)28792 8156 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)31769 8322 y Ff(2)32267 8156 y Fo(;)g Fn(k)33430 8322 y Ff(3)33927 8156 y Fo(;)g Fn(k)35090 8322 y Ff(4)35833 8156 y Fh(\000)246 b Fn(p)p Fr(\))p Fh(\000)10765 10017 y(\000)g Fo(G)12742 9560 y Ff(4)12742 10291 y(+)13477 10017 y Fr(\()p Fn(k)14579 10183 y Ff(1)15076 10017 y Fo(;)184 b Fn(k)16239 10183 y Ff(2)16983 10017 y Fr(+)245 b Fn(p)p Fo(;)184 b Fn(k)19959 10183 y Ff(3)20457 10017 y Fo(;)g Fn(k)21620 10183 y Ff(4)22364 10017 y Fh(\000)246 b Fn(p)p Fr(\))g(+)g(\001)26883 9485 y Ff(4)p Fk(;)p Ff(1)26883 10293 y(+)28392 10017 y Fo(;)44415 9041 y Fr(\(3)p Fo(:)p Fr(17\))8299 12796 y Fo(D)9215 12962 y Fj(\000)9963 12796 y Fr(\()p Fn(p)p Fr(\))p Fo(G)12400 12264 y Ff(4)p Fk(;)p Ff(1)12400 13072 y Fj(\000)13602 12796 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)15902 12962 y Ff(1)16400 12796 y Fo(;)g Fn(k)17563 12962 y Ff(2)18060 12796 y Fo(;)g Fn(k)19223 12962 y Ff(3)19721 12796 y Fo(;)g Fn(k)20884 12962 y Ff(4)21627 12796 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)g Fo(G)26218 12339 y Ff(4)26218 13070 y(+)26953 12796 y Fr(\()p Fn(k)28055 12962 y Ff(1)28552 12796 y Fo(;)184 b Fn(k)29715 12962 y Ff(2)30213 12796 y Fo(;)g Fn(k)31376 12962 y Ff(3)32119 12796 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)35096 12962 y Ff(4)35840 12796 y Fh(\000)245 b Fn(p)p Fr(\))p Fh(\000)10759 14658 y(\000)h Fo(G)12736 14201 y Ff(4)12736 14931 y(+)13471 14658 y Fr(\()p Fn(k)14573 14824 y Ff(1)15070 14658 y Fo(;)184 b Fn(k)16233 14824 y Ff(2)16730 14658 y Fo(;)g Fn(k)17893 14824 y Ff(3)18391 14658 y Fo(;)g Fn(k)19554 14824 y Ff(4)20051 14658 y Fr(\))247 b(+)f(\001)22757 14126 y Ff(4)p Fk(;)p Ff(1)22757 14933 y Fj(\000)24265 14658 y Fo(;)44415 13682 y Fr(\(3)p Fo(:)p Fr(18\))0 16903 y(where)369 b(\001)4123 16371 y Ff(4)p Fk(;)p Ff(1)4123 17179 y Fj(\006)5693 16903 y Fr(is)g(the)h(\\correction)g(term")1568 20040 y(\001)2490 19508 y Ff(4)p Fk(;)p Ff(1)2490 20315 y Fj(\006)3691 20040 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)5991 20206 y Ff(1)6489 20040 y Fo(;)g Fn(k)7652 20206 y Ff(2)8149 20040 y Fo(;)g Fn(k)9312 20206 y Ff(3)9810 20040 y Fr(\))308 b(=)12291 19291 y(1)p 11849 19785 1438 45 v 11849 20799 a Fo(\014)58 b(L)13604 18988 y Fe(X)14139 21371 y Fi(k)15388 20040 y Fo(C)16179 20206 y Fj(\006)16926 20040 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))308 b Fo(<)23385 19748 y Fr(^)23159 20040 y Fo( )23920 19566 y Ff(+)23880 20374 y Fi(k)p Fk(;)p Fj(\006)25646 19748 y Fr(^)25419 20040 y Fo( )26180 19566 y Fj(\000)26140 20374 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\006)28928 20040 y Fr(;)29646 19748 y(^)29419 20040 y Fo( )30180 19566 y Fj(\000)30140 20374 y Fi(k)30669 20485 y Fm(1)31102 20374 y Fk(;)p Ff(+)32100 20040 y Fr(;)32818 19748 y(^)32591 20040 y Fo( )33352 19566 y Ff(+)33312 20374 y Fi(k)33841 20485 y Fm(2)34274 20374 y Fk(;)p Ff(+)35271 20040 y Fr(;)35990 19748 y(^)35762 20040 y Fo( )36523 19566 y Fj(\000)36483 20374 y Fi(k)37012 20485 y Fm(3)37446 20374 y Fk(;)p Fj(\000)38456 20040 y Fr(;)39174 19748 y(^)38947 20040 y Fo( )39708 19566 y Ff(+)39668 20374 y Fi(k)40197 20485 y Fm(4)40630 20374 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)43196 20040 y Fo(>)44057 19583 y Fk(T)45369 20040 y Fo(:)44415 22423 y Fr(\(3)p Fo(:)p Fr(19\))0 25730 y Fn(3.4)945 b Fp(Counterterms)443 27147 y Fr(Eq.)971 b(\(3.17\))531 b(can)d(b)31 b(e)528 b(written,)570 b(b)-31 b(y)529 b(adding)h(and)f (subtracting)g(suitable)h(coun)-31 b(terterms)529 b Fo(\027)40137 27313 y Fj(\006)40884 27147 y Fr(,)569 b(to)529 b(b)31 b(e)528 b(\014xed)0 28565 y(prop)31 b(erly)369 b(later,)h(see)f(Fig.) 493 b(6)832 31135 y(\(1)247 b Fh(\000)f Fo(\027)3716 31301 y Ff(+)4451 31135 y Fr(\))p Fo(D)5797 31301 y Ff(+)6533 31135 y Fr(\()p Fn(p)p Fr(\))8352 30855 y(^)8100 31135 y Fo(G)8970 30603 y Ff(4)p Fk(;)p Ff(1)8970 31410 y(+)10172 31135 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)12472 31301 y Ff(1)12970 31135 y Fo(;)g Fn(k)14133 31301 y Ff(2)14631 31135 y Fo(;)g Fn(k)15794 31301 y Ff(3)16291 31135 y Fo(;)g Fn(k)17454 31301 y Ff(4)18198 31135 y Fh(\000)246 b Fn(p)p Fr(\))g Fh(\000)g Fo(\027)22342 31301 y Fj(\000)23089 31135 y Fo(D)24005 31301 y Fj(\000)24753 31135 y Fr(\()p Fn(p)p Fr(\))26572 30855 y(^)26320 31135 y Fo(G)27190 30603 y Ff(4)p Fk(;)p Ff(1)27190 31410 y Fj(\000)28392 31135 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)30692 31301 y Ff(1)31190 31135 y Fo(;)g Fn(k)32353 31301 y Ff(2)32851 31135 y Fo(;)g Fn(k)34014 31301 y Ff(3)34511 31135 y Fo(;)g Fn(k)35674 31301 y Ff(4)36418 31135 y Fh(\000)246 b Fn(p)p Fr(\))1140 33014 y(=)2559 32734 y(^)2308 33014 y Fo(G)3178 32557 y Ff(4)3178 33287 y(+)3914 33014 y Fr(\()p Fn(k)5016 33180 y Ff(1)5759 33014 y Fh(\000)f Fn(p)p Fo(;)184 b Fn(k)8735 33180 y Ff(2)9233 33014 y Fo(;)g Fn(k)10396 33180 y Ff(3)10894 33014 y Fo(;)g Fn(k)12057 33180 y Ff(4)12800 33014 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)16648 32734 y Fr(^)16398 33014 y Fo(G)17268 32557 y Ff(4)17268 33287 y(+)18003 33014 y Fr(\()p Fn(k)19105 33180 y Ff(1)19602 33014 y Fo(;)184 b Fn(k)20765 33180 y Ff(2)21508 33014 y Fr(+)246 b Fn(p)p Fo(;)184 b Fn(k)24485 33180 y Ff(3)24983 33014 y Fo(;)g Fn(k)26146 33180 y Ff(4)26889 33014 y Fh(\000)246 b Fn(p)p Fr(\))h(+)f Fo(H)31497 32482 y Ff(4)p Fk(;)p Ff(1)31407 33289 y(+)32698 33014 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)34998 33180 y Ff(1)35496 33014 y Fo(;)g Fn(k)36659 33180 y Ff(2)37156 33014 y Fo(;)g Fn(k)38319 33180 y Ff(3)38817 33014 y Fo(;)g Fn(k)39980 33180 y Ff(4)40723 33014 y Fh(\000)246 b Fn(p)p Fr(\))308 b Fo(;)44415 32020 y Fr(\(3)p Fo(:)p Fr(20\))0 35476 y(where)369 b(b)-31 b(y)369 b(de\014nition)0 38460 y Fo(H)1010 37928 y Ff(4)p Fk(;)p Ff(1)920 38736 y(+)2211 38460 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)4511 38626 y Ff(1)5009 38460 y Fo(;)g Fn(k)6172 38626 y Ff(2)6669 38460 y Fo(;)g Fn(k)7832 38626 y Ff(3)8330 38460 y Fo(;)g Fn(k)9493 38626 y Ff(4)10016 38460 y Fh(\000)25 b Fn(p)p Fr(\))308 b(=)14091 37711 y(1)p 13649 38206 V 13649 39220 a Fo(\014)58 b(L)15404 37409 y Fe(X)15939 39792 y Fi(k)17187 38460 y Fo(C)17978 38626 y Ff(+)18713 38460 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)25 b Fh(\000)g Fn(p)p Fr(\))311 b Fo(<)24732 38168 y Fr(^)24505 38460 y Fo( )25266 37987 y Ff(+)25226 38794 y Fi(k)p Fk(;)p Ff(+)26980 38168 y Fr(^)26753 38460 y Fo( )27514 37987 y Fj(\000)27474 38794 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)30249 38460 y Fr(;)30968 38168 y(^)30740 38460 y Fo( )31501 37987 y Fj(\000)31461 38794 y Fi(k)31990 38905 y Fm(1)32423 38794 y Fk(;)p Ff(+)33421 38460 y Fr(;)34140 38168 y(^)33912 38460 y Fo( )34673 37987 y Ff(+)34633 38794 y Fi(k)35162 38905 y Fm(2)35595 38794 y Fk(;)p Ff(+)36593 38460 y Fr(;)37312 38168 y(^)37084 38460 y Fo( )37845 37987 y Fj(\000)37805 38794 y Fi(k)38334 38905 y Fm(3)38767 38794 y Fk(;)p Fj(\000)39777 38460 y Fr(;)40496 38168 y(^)40268 38460 y Fo( )41029 37987 y Ff(+)40989 38794 y Fi(k)41518 38905 y Fm(4)41951 38794 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)44517 38460 y Fo(>)45378 38003 y Fk(T)46383 38460 y Fh(\000)6417 42671 y(\000)7853 41922 y Fr(1)p 7411 42417 V 7411 43430 a Fo(\014)58 b(L)9166 41620 y Fe(X)9701 44003 y Fi(k)10950 41620 y Fe(X)11456 43944 y Fk(!)12733 42671 y Fo(\027)13280 42837 y Fk(!)13921 42671 y Fo(D)14837 42837 y Fk(!)15478 42671 y Fr(\()p Fn(p)p Fr(\))309 b Fo(<)18749 42379 y Fr(^)18522 42671 y Fo( )19283 42198 y Ff(+)19243 43005 y Fi(k)p Fk(;!)20903 42379 y Fr(^)20676 42671 y Fo( )21437 42198 y Fj(\000)21397 43005 y Fi(k)p Fj(\000)p Fi(p)p Fk(;!)24078 42671 y Fr(;)24797 42379 y(^)24569 42671 y Fo( )25330 42198 y Fj(\000)25290 43005 y Fi(k)25819 43116 y Fm(1)26252 43005 y Fk(;)p Ff(+)27250 42671 y Fr(;)27969 42379 y(^)27741 42671 y Fo( )28502 42198 y Ff(+)28462 43005 y Fi(k)28991 43116 y Fm(2)29424 43005 y Fk(;)p Ff(+)30422 42671 y Fr(;)31141 42379 y(^)30913 42671 y Fo( )31674 42198 y Fj(\000)31634 43005 y Fi(k)32163 43116 y Fm(3)32596 43005 y Fk(;)p Fj(\000)33606 42671 y Fr(;)34325 42379 y(^)34097 42671 y Fo( )34858 42198 y Ff(+)34818 43005 y Fi(k)35347 43116 y Fm(4)35780 43005 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)38346 42671 y Fo(>)39207 42214 y Fk(T)40519 42671 y Fo(:)3589 b Fr(\(3)p Fo(:)p Fr(21\))443 45770 y(In)368 b(the)i(same)f(w)-31 b(a)g(y)-92 b(,)371 b(eq.)493 b(\(3.18\))372 b(can)d(b)31 b(e)369 b(written)i(as)2708 48402 y(\(1)247 b Fh(\000)f Fo(\027)5662 47945 y Fj(0)5592 48676 y(\000)6339 48402 y Fr(\))p Fo(D)7685 48568 y Fj(\000)8434 48402 y Fr(\()p Fn(p)p Fr(\))10252 48122 y(^)10001 48402 y Fo(G)10871 47870 y Ff(4)p Fk(;)p Ff(1)10871 48678 y Fj(\000)12073 48402 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)14373 48568 y Ff(1)14871 48402 y Fo(;)g Fn(k)16034 48568 y Ff(2)16531 48402 y Fo(;)g Fn(k)17694 48568 y Ff(3)18192 48402 y Fo(;)g Fn(k)19355 48568 y Ff(4)20098 48402 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)f Fo(\027)24313 47945 y Fj(0)24243 48676 y Ff(+)24978 48402 y Fo(D)25894 48568 y Ff(+)26629 48402 y Fr(\()p Fn(p)p Fr(\))28448 48122 y(^)28196 48402 y Fo(G)29066 47870 y Ff(4)p Fk(;)p Ff(1)29066 48678 y(+)30268 48402 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)32568 48568 y Ff(1)33066 48402 y Fo(;)g Fn(k)34229 48568 y Ff(2)34727 48402 y Fo(;)g Fn(k)35890 48568 y Ff(3)36387 48402 y Fo(;)g Fn(k)37550 48568 y Ff(4)38294 48402 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)3016 50281 y(=)4435 50001 y(^)4184 50281 y Fo(G)5054 49824 y Ff(4)5054 50554 y(+)5790 50281 y Fr(\()p Fn(k)6892 50447 y Ff(1)7389 50281 y Fo(;)184 b Fn(k)8552 50447 y Ff(2)9049 50281 y Fn(k)9721 50447 y Ff(3)10463 50281 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)13440 50447 y Ff(4)14184 50281 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)18032 50001 y Fr(^)17782 50281 y Fo(G)18652 49824 y Ff(4)18652 50554 y(+)19387 50281 y Fr(\()p Fn(k)20489 50447 y Ff(1)20986 50281 y Fo(;)184 b Fn(k)22149 50447 y Ff(2)22646 50281 y Fo(;)g Fn(k)23809 50447 y Ff(3)24307 50281 y Fo(;)g Fn(k)25470 50447 y Ff(4)25967 50281 y Fr(\))247 b(+)28040 50001 y(^)27751 50281 y Fo(H)28761 49749 y Ff(4)p Fk(;)p Ff(1)28671 50556 y Fj(\000)29961 50281 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)32261 50447 y Ff(1)32760 50281 y Fo(;)g Fn(k)33923 50447 y Ff(2)34420 50281 y Fo(;)g Fn(k)35583 50447 y Ff(3)36080 50281 y Fo(;)g Fn(k)37243 50447 y Ff(4)37987 50281 y Fh(\000)246 b Fn(p)p Fr(\))308 b Fo(;)44415 49288 y Fr(\(3)p Fo(:)p Fr(22\))0 52743 y(where)0 55512 y Fo(H)1010 54980 y Ff(4)p Fk(;)p Ff(1)920 55788 y Fj(\000)2211 55512 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)4511 55678 y Ff(1)5009 55512 y Fo(;)g Fn(k)6172 55678 y Ff(2)6669 55512 y Fo(;)g Fn(k)7832 55678 y Ff(3)8330 55512 y Fo(;)g Fn(k)9493 55678 y Ff(4)10007 55512 y Fh(\000)16 b Fn(p)p Fr(\))308 b(=)14072 54763 y(1)p 13630 55258 V 13630 56272 a Fo(\014)58 b(L)15385 54461 y Fe(X)15920 56844 y Fi(k)17169 55512 y Fo(C)17960 55678 y Fj(\000)18707 55512 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)16 b Fh(\000)g Fn(p)p Fr(\))310 b Fo(<)24707 55220 y Fr(^)24480 55512 y Fo( )25241 55039 y Ff(+)25201 55846 y Fi(k)p Fk(;)p Fj(\000)26967 55220 y Fr(^)26741 55512 y Fo( )27502 55039 y Fj(\000)27462 55846 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)30249 55512 y Fr(;)30968 55220 y(^)30740 55512 y Fo( )31501 55039 y Fj(\000)31461 55846 y Fi(k)31990 55957 y Fm(1)32423 55846 y Fk(;)p Ff(+)33421 55512 y Fr(;)34140 55220 y(^)33912 55512 y Fo( )34673 55039 y Ff(+)34633 55846 y Fi(k)35162 55957 y Fm(2)35595 55846 y Fk(;)p Ff(+)36593 55512 y Fr(;)37312 55220 y(^)37084 55512 y Fo( )37845 55039 y Fj(\000)37805 55846 y Fi(k)38334 55957 y Fm(3)38767 55846 y Fk(;)p Fj(\000)39777 55512 y Fr(;)40496 55220 y(^)40268 55512 y Fo( )41029 55039 y Ff(+)40989 55846 y Fi(k)41518 55957 y Fm(4)41951 55846 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)44517 55512 y Fo(>)45378 55055 y Fk(T)46383 55512 y Fh(\000)6417 59723 y(\000)7853 58974 y Fr(1)p 7411 59469 V 7411 60482 a Fo(\014)58 b(L)9166 58672 y Fe(X)9701 61055 y Fi(k)10950 58672 y Fe(X)11456 60996 y Fk(!)12733 59723 y Fo(\027)13350 59266 y Fj(0)13280 59997 y Fk(!)13921 59723 y Fo(D)14837 59889 y Fk(!)15478 59723 y Fr(\()p Fn(p)p Fr(\))309 b Fo(<)18749 59431 y Fr(^)18522 59723 y Fo( )19283 59250 y Ff(+)19243 60057 y Fi(k)p Fk(;!)20903 59431 y Fr(^)20676 59723 y Fo( )21437 59250 y Fj(\000)21397 60057 y Fi(k)p Fj(\000)p Fi(p)p Fk(;!)24078 59723 y Fr(;)24797 59431 y(^)24569 59723 y Fo( )25330 59250 y Fj(\000)25290 60057 y Fi(k)25819 60168 y Fm(1)26252 60057 y Fk(;)p Ff(+)27250 59723 y Fr(;)27969 59431 y(^)27741 59723 y Fo( )28502 59250 y Ff(+)28462 60057 y Fi(k)28991 60168 y Fm(2)29424 60057 y Fk(;)p Ff(+)30422 59723 y Fr(;)31141 59431 y(^)30913 59723 y Fo( )31674 59250 y Fj(\000)31634 60057 y Fi(k)32163 60168 y Fm(3)32596 60057 y Fk(;)p Fj(\000)33606 59723 y Fr(;)34325 59431 y(^)34097 59723 y Fo( )34858 59250 y Ff(+)34818 60057 y Fi(k)35347 60168 y Fm(4)35780 60057 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)38346 59723 y Fo(>)39207 59266 y Fk(T)40519 59723 y Fo(:)3589 b Fr(\(3)p Fo(:)p Fr(23\))0 63040 y(If)369 b(w)-31 b(e)370 b(insert)f(in)g(the)h(r.h.s.) 492 b(of)370 b(\(3.20\))i(the)d(v)-61 b(alue)370 b(of)22654 62761 y(^)22404 63040 y Fo(G)23274 62509 y Ff(4)p Fk(;)p Ff(1)23274 63316 y Fj(\000)24844 63040 y Fr(tak)-31 b(en)370 b(from)g(\(3.22\),)i(w)-31 b(e)370 b(get)827 65654 y(\(1)247 b(+)f Fo(A)p Fr(\))p Fo(D)5340 65820 y Ff(+)6076 65654 y Fr(\()p Fn(p)p Fr(\))7895 65374 y(^)7643 65654 y Fo(G)8513 65122 y Ff(4)p Fk(;)p Ff(1)8513 65929 y(+)9716 65654 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)12016 65820 y Ff(1)12514 65654 y Fo(;)g Fn(k)13677 65820 y Ff(2)14174 65654 y Fo(;)g Fn(k)15337 65820 y Ff(3)15835 65654 y Fo(;)g Fn(k)16998 65820 y Ff(4)17741 65654 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)g([)22020 65374 y(^)21769 65654 y Fo(G)22639 65197 y Ff(4)22639 65927 y(+)23374 65654 y Fr(\()p Fn(k)24476 65820 y Ff(1)25219 65654 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)28196 65820 y Ff(2)28694 65654 y Fo(;)g Fn(k)29857 65820 y Ff(3)30354 65654 y Fo(;)g Fn(k)31517 65820 y Ff(4)32261 65654 y Fh(\000)246 b Fn(p)p Fr(\))p Fh(\000)9049 b Fr(\(3)p Fo(:)p Fr(24\))1078 67123 y(^)827 67403 y Fo(G)1697 66946 y Ff(4)1697 67677 y(+)2433 67403 y Fr(\()p Fn(k)3535 67569 y Ff(1)4032 67403 y Fo(;)184 b Fn(k)5195 67569 y Ff(2)5938 67403 y Fr(+)246 b Fn(p)p Fo(;)184 b Fn(k)8915 67569 y Ff(3)9413 67403 y Fo(;)g Fn(k)10576 67569 y Ff(4)11319 67403 y Fh(\000)246 b Fn(p)p Fr(\)])h(+)f Fo(B)56 b Fr([)16677 67123 y(^)16427 67403 y Fo(G)17297 66946 y Ff(4)17297 67677 y(+)18032 67403 y Fr(\()p Fn(k)19134 67569 y Ff(1)19631 67403 y Fo(;)184 b Fn(k)20794 67569 y Ff(3)21537 67403 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)24514 67569 y Ff(4)25258 67403 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)29106 67123 y Fr(^)28856 67403 y Fo(G)29726 66946 y Ff(4)29726 67677 y(+)30461 67403 y Fr(\()p Fn(k)31563 67569 y Ff(1)32060 67403 y Fo(;)184 b Fn(k)33223 67569 y Ff(2)33720 67403 y Fo(;)g Fn(k)34883 67569 y Ff(3)35381 67403 y Fo(;)g Fn(k)36544 67569 y Ff(4)37041 67403 y Fr(\)])247 b(+)39422 67123 y(^)39132 67403 y Fo(H)40142 66871 y Ff(4)p Fk(;)p Ff(1)40052 67679 y(+)41589 67403 y Fr(+)f Fo(B)43881 67123 y Fr(^)43592 67403 y Fo(H)44602 66871 y Ff(4)p Fk(;)p Ff(1)44512 67679 y Fj(\000)46109 67403 y Fo(;)0 70016 y Fr(where)14409 71757 y Fo(A)308 b Fr(=)f Fh(\000)p Fo(\027)18123 71923 y Ff(+)19104 71757 y Fh(\000)20656 70959 y Fo(\027)21203 71125 y Fj(\000)21950 70959 y Fo(\027)22567 70557 y Fj(0)22497 71233 y Ff(+)p 20344 71502 3201 45 v 20344 72516 a Fr(1)246 b Fh(\000)g Fo(\027)22867 72135 y Fj(0)22797 72791 y(\000)24784 71757 y Fo(;)1292 b(B)363 b Fr(=)29840 71008 y Fo(\027)30387 71174 y Fj(\000)p 28887 71502 V 28887 72516 a Fr(1)246 b Fh(\000)g Fo(\027)31410 72135 y Fj(0)31340 72791 y(\000)32528 71757 y Fo(:)11580 b Fr(\(3)p Fo(:)p Fr(25\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 443 1107 a Fr(If)369 b(w)-31 b(e)369 b(insert)h(in)f(the)g(last)i(term)e(of)h(the)f(r.h.s.)493 b(of)369 b(\(3.12\))j(the)e(v)-61 b(alue)369 b(of)31009 827 y(^)30758 1107 y Fo(G)31628 575 y Ff(4)p Fk(;)p Ff(1)31628 1382 y(+)33198 1107 y Fr(tak)-31 b(en)371 b(from)e(\(3.24\),)k(w)-31 b(e)369 b(get)2795 3644 y Fh(\000)4152 3365 y Fr(^)3902 3644 y Fo(G)4772 3187 y Ff(4)4772 3918 y(+)5507 3644 y Fr(\()p Fn(k)6609 3810 y Ff(1)7106 3644 y Fo(;)184 b Fn(k)8269 3810 y Ff(2)8766 3644 y Fo(;)g Fn(k)9929 3810 y Ff(3)10427 3644 y Fo(;)g Fn(k)11590 3810 y Ff(4)12087 3644 y Fr(\))308 b(=)30729 b(\(3)p Fo(:)p Fr(26\))2856 6390 y(=)308 b Fo(\025)38 b Fr(^)-591 b Fo(g)5199 6556 y Fj(\000)5946 6390 y Fr(\()p Fn(k)7048 6556 y Ff(4)7545 6390 y Fr(\))8159 4497 y Fe(")9056 6111 y Fr(^)8805 6390 y Fo(G)9675 5933 y Ff(2)9675 6664 y Fj(\000)10423 6390 y Fr(\()p Fn(k)11525 6556 y Ff(3)12022 6390 y Fr(\))12703 6111 y(^)12452 6390 y Fo(G)13322 5858 y Ff(2)p Fk(;)p Ff(1)13322 6666 y(+)14523 6390 y Fr(\()p Fn(k)15625 6556 y Ff(1)16368 6390 y Fh(\000)246 b Fn(k)18147 6556 y Ff(2)18644 6390 y Fo(;)184 b Fn(k)19807 6556 y Ff(1)20304 6390 y Fo(;)g Fn(k)21467 6556 y Ff(2)21965 6390 y Fr(\))246 b(+)24323 5642 y(1)p 23881 6136 1438 45 v 23881 7150 a Fo(L\014)25636 5339 y Fe(X)26157 7663 y Fi(p)27420 6390 y Fo(\037)28113 6556 y Fk(M)29095 6390 y Fr(\()p Fn(p)p Fr(\))p Fo(G)31532 5858 y Ff(4)p Fk(;)p Ff(1)31532 6666 y(+)32734 6390 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)35034 6556 y Ff(1)35532 6390 y Fo(;)g Fn(k)36695 6556 y Ff(2)37193 6390 y Fo(;)g Fn(k)38356 6556 y Ff(3)38853 6390 y Fo(;)g Fn(k)40016 6556 y Ff(4)40759 6390 y Fh(\000)246 b Fn(p)p Fr(\))43003 4497 y Fe(#)43834 6390 y Fr(+)2795 10214 y(+)4035 9465 y Fo(\025)38 b Fr(^)-591 b Fo(g)5209 9631 y Fj(\000)5956 9465 y Fr(\()p Fn(k)7058 9631 y Ff(4)7554 9465 y Fr(\))p 4035 9959 3951 45 v 4211 10973 a(\(1)247 b(+)f Fo(A)p Fr(\))8693 9465 y(1)p 8251 9959 1438 45 v 8251 10973 a Fo(L\014)10006 9162 y Fe(X)10527 11487 y Fi(p)11920 10214 y Fr(~)-684 b Fo(\037)12482 10380 y Fk(M)13464 10214 y Fr(\()p Fn(p)p Fr(\))15416 9136 y(^)15164 9416 y Fo(G)16034 9014 y Ff(4)16034 9690 y(+)16771 9416 y Fr(\()p Fn(k)17873 9582 y Ff(1)18616 9416 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)21593 9582 y Ff(2)22090 9416 y Fo(;)g Fn(k)23253 9582 y Ff(3)23751 9416 y Fo(;)g Fn(k)24914 9582 y Ff(4)25657 9416 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)29505 9136 y Fr(^)29255 9416 y Fo(G)30125 9014 y Ff(4)30125 9690 y(+)30860 9416 y Fr(\()p Fn(k)31962 9582 y Ff(1)32459 9416 y Fo(;)184 b Fn(k)33622 9582 y Ff(2)34365 9416 y Fr(+)246 b Fn(p)p Fo(;)184 b Fn(k)37342 9582 y Ff(3)37840 9416 y Fo(;)g Fn(k)39003 9582 y Ff(4)39746 9416 y Fh(\000)246 b Fn(p)p Fr(\))p 15164 9959 26826 45 v 26968 10973 a Fo(D)27884 11139 y Ff(+)28620 10973 y Fr(\()p Fn(p)p Fr(\))42431 10214 y(+)2795 14037 y(+)4035 13288 y Fo(\025)38 b Fr(^)-591 b Fo(g)5209 13454 y Fj(\000)5956 13288 y Fr(\()p Fn(k)7058 13454 y Ff(4)7554 13288 y Fr(\))p 4035 13782 3951 45 v 4211 14796 a(\(1)247 b(+)f Fo(A)p Fr(\))8693 13288 y(1)p 8251 13782 1438 45 v 8251 14796 a Fo(L\014)10006 12985 y Fe(X)10527 15310 y Fi(p)11920 14037 y Fr(~)-684 b Fo(\037)12482 14203 y Fk(M)13464 14037 y Fr(\()p Fn(p)p Fr(\))15416 12960 y(^)15164 13239 y Fo(G)16034 12838 y Ff(4)16034 13513 y(+)16771 13239 y Fr(\()p Fn(k)17873 13405 y Ff(1)18370 13239 y Fo(;)184 b Fn(k)19533 13405 y Ff(2)20030 13239 y Fo(;)g Fn(k)21193 13405 y Ff(3)21937 13239 y Fh(\000)245 b Fn(p)p Fo(;)184 b Fn(k)24913 13405 y Ff(4)25657 13239 y Fh(\000)246 b Fn(p)p Fr(\))h Fh(\000)29505 12960 y Fr(^)29255 13239 y Fo(G)30125 12838 y Ff(4)30125 13513 y(+)30860 13239 y Fr(\()p Fn(k)31962 13405 y Ff(1)32459 13239 y Fo(;)184 b Fn(k)33622 13405 y Ff(2)34119 13239 y Fo(;)g Fn(k)35282 13405 y Ff(3)35780 13239 y Fo(;)g Fn(k)36943 13405 y Ff(4)37440 13239 y Fr(\))p 15164 13782 22706 45 v 24908 14796 a Fo(D)25824 14962 y Ff(+)26560 14796 y Fr(\()p Fn(p)p Fr(\))38311 14037 y(+)2795 17862 y(+)4035 17113 y Fo(\025)38 b Fr(^)-591 b Fo(g)5209 17279 y Fj(\000)5956 17113 y Fr(\()p Fn(k)7058 17279 y Ff(4)7554 17113 y Fr(\))p 4035 17608 3951 45 v 4211 18621 a(\(1)247 b(+)f Fo(A)p Fr(\))8693 17113 y(1)p 8251 17608 1438 45 v 8251 18621 a Fo(L\014)10006 16811 y Fe(X)10527 19135 y Fi(p)11920 17862 y Fr(~)-684 b Fo(\037)12482 18028 y Fk(M)13464 17862 y Fr(\()p Fn(p)p Fr(\))15455 16783 y(^)15164 17063 y Fo(H)16174 16531 y Ff(4)p Fk(;)p Ff(1)16084 17338 y(+)17376 17063 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)19676 17229 y Ff(1)20174 17063 y Fo(;)g Fn(k)21337 17229 y Ff(2)21835 17063 y Fo(;)g Fn(k)22998 17229 y Ff(3)23495 17063 y Fo(;)g Fn(k)24658 17229 y Ff(4)25402 17063 y Fh(\000)246 b Fn(p)p Fr(\))g(+)g Fo(B)30184 16783 y Fr(^)29895 17063 y Fo(H)30905 16531 y Ff(4)p Fk(;)p Ff(1)30815 17338 y Fj(\000)32105 17063 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)34405 17229 y Ff(1)34904 17063 y Fo(;)g Fn(k)36067 17229 y Ff(2)36564 17063 y Fo(;)g Fn(k)37727 17229 y Ff(3)38224 17063 y Fo(;)g Fn(k)39387 17229 y Ff(4)40131 17063 y Fh(\000)246 b Fn(p)p Fr(\))p 15164 17608 27210 45 v 27161 18621 a Fo(D)28077 18787 y Ff(+)28812 18621 y Fr(\()p Fn(p)p Fr(\))42816 17862 y Fo(:)0 21396 y Fr(Note)370 b(that)14977 22604 y(1)p 14534 23098 1438 45 v 14534 24112 a Fo(L\014)16290 22301 y Fe(X)16811 24626 y Fi(p)18204 23353 y Fr(~)-684 b Fo(\037)18766 23519 y Fk(M)19748 23353 y Fr(\()p Fn(p)p Fr(\))21700 22276 y(^)21448 22555 y Fo(G)22318 22154 y Ff(4)22318 22829 y(+)23055 22555 y Fr(\()p Fn(k)24157 22721 y Ff(1)24653 22555 y Fo(;)184 b Fn(k)25816 22721 y Ff(2)26314 22555 y Fo(;)g Fn(k)27477 22721 y Ff(3)27974 22555 y Fo(;)g Fn(k)29137 22721 y Ff(4)29635 22555 y Fr(\))p 21448 23098 8616 45 v 24148 24112 a Fo(D)25064 24278 y Ff(+)25799 24112 y Fr(\()p Fn(p)p Fr(\))30506 23353 y(=)307 b(0)h Fo(;)11573 b Fr(\(3)p Fo(:)p Fr(27\))0 26511 y(since)369 b Fo(D)3628 26677 y Ff(+)4363 26511 y Fr(\()p Fn(p)p Fr(\))i(is)e(o)31 b(dd.)492 b(Then,)370 b(b)-31 b(y)370 b(using)f(also)h(the)g(W)-92 b(ard)368 b(iden)-31 b(tit)g(y)372 b(\(3.14\),)g(w)-31 b(e)370 b(get,)g(if)10359 29048 y Fn(k)11031 29214 y Fk(i)11707 29048 y Fr(=)12935 28756 y(\026)12875 29048 y Fn(k)13547 29214 y Fk(i)15023 29048 y Fo(;)16681 28756 y Fr(\026)16622 29048 y Fn(k)17294 29214 y Ff(1)18098 29048 y Fr(=)19325 28756 y(\026)19266 29048 y Fn(k)19938 29214 y Ff(4)20742 29048 y Fr(=)308 b Fh(\000)22831 28756 y Fr(\026)22772 29048 y Fn(k)23444 29214 y Ff(2)24247 29048 y Fr(=)g Fh(\000)26336 28756 y Fr(\026)26277 29048 y Fn(k)26949 29214 y Ff(3)27753 29048 y Fr(=)28980 28756 y(\026)28921 29048 y Fn(k)1107 b Fo(;)1292 b Fh(j)32665 28756 y Fr(\026)32606 29048 y Fn(k)p Fh(j)308 b Fr(=)g Fo(\015)35696 28591 y Fk(h)36577 29048 y Fo(;)7531 b Fr(\(3)p Fo(:)p Fr(28\))0 31586 y(the)369 b(iden)-31 b(tit)g(y)2785 34750 y Fh(\000)4142 34470 y Fr(^)3892 34750 y Fo(G)4762 34293 y Ff(4)4762 35024 y(+)5497 34750 y Fr(\()5986 34458 y(\026)5927 34750 y Fn(k)6599 34916 y Ff(1)7096 34750 y Fo(;)7647 34458 y Fr(\026)7587 34750 y Fn(k)8259 34916 y Ff(2)8756 34750 y Fo(;)9307 34458 y Fr(\026)9247 34750 y Fn(k)9919 34916 y Ff(3)10417 34750 y Fo(;)10968 34458 y Fr(\026)10908 34750 y Fn(k)11580 34916 y Ff(4)12077 34750 y Fr(\))308 b(=)g Fo(\025)38 b Fr(^)-591 b Fo(g)15158 34916 y Fj(\000)15904 34750 y Fr(\()16393 34458 y(\026)16334 34750 y Fn(k)17006 34916 y Ff(4)17503 34750 y Fr(\))18185 34470 y(^)17933 34750 y Fo(G)18803 34293 y Ff(2)18803 35024 y Fj(\000)19551 34750 y Fr(\()20040 34458 y(\026)19981 34750 y Fn(k)20653 34916 y Ff(3)21150 34750 y Fr(\))21713 33953 y Fo(G)22583 33551 y Ff(2)22583 34226 y(+)23319 33953 y Fr(\()23808 33660 y(\026)23749 33953 y Fn(k)24421 34119 y Ff(2)24918 33953 y Fr(\))246 b Fh(\000)g Fo(G)27571 33551 y Ff(2)27571 34226 y(+)28307 33953 y Fr(\()28796 33660 y(\026)28737 33953 y Fn(k)29409 34119 y Ff(1)29906 33953 y Fr(\))p 21713 34496 8623 45 v 24156 35549 a Fo(D)25072 35715 y Ff(+)25807 35549 y Fr(\(2)26849 35257 y(\026)26790 35549 y Fn(k)q Fr(\))30469 34750 y(+)13085 b(\(3)p Fo(:)p Fr(29\))2785 38099 y(+)246 b Fo(\025)38 b Fr(^)-591 b Fo(g)5066 38265 y Fj(\000)5813 38099 y Fr(\()6302 37807 y(\026)6243 38099 y Fn(k)6915 38265 y Ff(4)7412 38099 y Fr(\))8093 37819 y(^)7842 38099 y Fo(G)8712 37642 y Ff(2)8712 38373 y Fj(\000)9460 38099 y Fr(\()9949 37807 y(\026)9890 38099 y Fn(k)10562 38265 y Ff(3)11058 38099 y Fr(\))11621 37300 y(\001)12543 36768 y Ff(2)p Fk(;)p Ff(1)12543 37575 y(+)13745 37300 y Fr(\(2)14787 37008 y(\026)14728 37300 y Fn(k)q Fo(;)15952 37008 y Fr(\026)15892 37300 y Fn(k)16564 37466 y Ff(1)17061 37300 y Fo(;)17612 37008 y Fr(\026)17552 37300 y Fn(k)18224 37466 y Ff(2)18722 37300 y Fr(\))p 11621 37845 7531 45 v 13518 38898 a Fo(D)14434 39064 y Ff(+)15170 38898 y Fr(\(2)16212 38606 y(\026)16153 38898 y Fn(k)p Fr(\))19531 38099 y(+)246 b Fo(\025)38 b Fr(^)-591 b Fo(g)21812 38265 y Fj(\000)22559 38099 y Fr(\()23048 37807 y(\026)22989 38099 y Fn(k)23661 38265 y Ff(4)24158 38099 y Fr(\))25163 37350 y(1)p 24721 37845 1438 45 v 24721 38859 a Fo(L\014)26476 37048 y Fe(X)26998 39372 y Fi(p)28260 38099 y Fo(\037)28953 38265 y Fk(M)29935 38099 y Fr(\()p Fn(p)p Fr(\))p Fo(G)32372 37567 y Ff(4)p Fk(;)p Ff(1)32372 38375 y(+)33574 38099 y Fr(\()p Fn(p)p Fr(;)35263 37807 y(\026)35202 38099 y Fn(k)35874 38265 y Ff(1)36372 38099 y Fo(;)36924 37807 y Fr(\026)36863 38099 y Fn(k)37535 38265 y Ff(2)38033 38099 y Fo(;)38584 37807 y Fr(\026)38524 38099 y Fn(k)39196 38265 y Ff(3)39693 38099 y Fo(;)40244 37807 y Fr(\026)40184 38099 y Fn(k)40856 38265 y Ff(4)41600 38099 y Fh(\000)246 b Fn(p)p Fr(\)+)2785 41923 y(+)4024 41174 y Fo(\025)38 b Fr(^)-591 b Fo(g)5198 41340 y Fj(\000)5945 41174 y Fr(\()6434 40882 y(\026)6375 41174 y Fn(k)7047 41340 y Ff(4)7544 41174 y Fr(\))p 4024 41668 3951 45 v 4200 42682 a(\(1)248 b(+)e Fo(A)p Fr(\))8683 41174 y(1)p 8241 41668 1438 45 v 8241 42682 a Fo(L\014)9996 40871 y Fe(X)10517 43196 y Fi(p)11910 41923 y Fr(~)-684 b Fo(\037)12472 42089 y Fk(M)13454 41923 y Fr(\()p Fn(p)p Fr(\))15406 40845 y(^)15154 41125 y Fo(G)16024 40723 y Ff(4)16024 41399 y(+)16761 41125 y Fr(\()17250 40833 y(\026)17191 41125 y Fn(k)17863 41291 y Ff(1)18606 41125 y Fh(\000)246 b Fn(p)p Fo(;)20971 40833 y Fr(\026)20911 41125 y Fn(k)21583 41291 y Ff(2)22080 41125 y Fo(;)22631 40833 y Fr(\026)22571 41125 y Fn(k)23243 41291 y Ff(3)23741 41125 y Fo(;)24292 40833 y Fr(\026)24232 41125 y Fn(k)24904 41291 y Ff(4)25647 41125 y Fh(\000)g Fn(p)p Fr(\))h Fh(\000)29495 40845 y Fr(^)29245 41125 y Fo(G)30115 40723 y Ff(4)30115 41399 y(+)30850 41125 y Fr(\()31339 40833 y(\026)31280 41125 y Fn(k)31952 41291 y Ff(1)32449 41125 y Fo(;)33000 40833 y Fr(\026)32940 41125 y Fn(k)33612 41291 y Ff(2)34355 41125 y Fr(+)f Fn(p)p Fo(;)36721 40833 y Fr(\026)36660 41125 y Fn(k)37332 41291 y Ff(3)37830 41125 y Fo(;)38381 40833 y Fr(\026)38321 41125 y Fn(k)38993 41291 y Ff(4)39736 41125 y Fh(\000)g Fn(p)p Fr(\))p 15154 41668 26826 45 v 26958 42682 a Fo(D)27874 42848 y Ff(+)28610 42682 y Fr(\()p Fn(p)p Fr(\))42421 41923 y(+)2785 45746 y(+)4024 44997 y Fo(\025)38 b Fr(^)-591 b Fo(g)5198 45163 y Fj(\000)5945 44997 y Fr(\()6434 44705 y(\026)6375 44997 y Fn(k)7047 45163 y Ff(4)7544 44997 y Fr(\))p 4024 45491 3951 45 v 4200 46505 a(\(1)248 b(+)e Fo(A)p Fr(\))8683 44997 y(1)p 8241 45491 1438 45 v 8241 46505 a Fo(L\014)9996 44694 y Fe(X)10517 47019 y Fi(p)11910 45746 y Fr(~)-684 b Fo(\037)12472 45912 y Fk(M)13454 45746 y Fr(\()p Fn(p)p Fr(\))15406 44668 y(^)15154 44948 y Fo(G)16024 44547 y Ff(4)16024 45222 y(+)16761 44948 y Fr(\()17250 44656 y(\026)17191 44948 y Fn(k)17863 45114 y Ff(1)18360 44948 y Fo(;)18911 44656 y Fr(\026)18851 44948 y Fn(k)19523 45114 y Ff(2)20020 44948 y Fo(;)20571 44656 y Fr(\026)20511 44948 y Fn(k)21183 45114 y Ff(3)21926 44948 y Fh(\000)246 b Fn(p)p Fo(;)24292 44656 y Fr(\026)24231 44948 y Fn(k)24903 45114 y Ff(4)25647 44948 y Fh(\000)g Fn(p)p Fr(\))p 15154 45491 12737 45 v 19914 46505 a Fo(D)20830 46671 y Ff(+)21565 46505 y Fr(\()p Fn(p)p Fr(\))28332 45746 y(+)2785 49571 y(+)4024 48822 y Fo(\025)38 b Fr(^)-591 b Fo(g)5198 48988 y Fj(\000)5945 48822 y Fr(\()6434 48530 y(\026)6375 48822 y Fn(k)7047 48988 y Ff(4)7544 48822 y Fr(\))p 4024 49316 3951 45 v 4200 50330 a(\(1)248 b(+)e Fo(A)p Fr(\))8683 48822 y(1)p 8241 49316 1438 45 v 8241 50330 a Fo(L\014)9996 48519 y Fe(X)10517 50844 y Fi(p)11910 49571 y Fr(~)-684 b Fo(\037)12472 49737 y Fk(M)13454 49571 y Fr(\()p Fn(p)p Fr(\))15445 48492 y(^)15154 48772 y Fo(H)16164 48240 y Ff(4)p Fk(;)p Ff(1)16074 49047 y(+)17366 48772 y Fr(\()p Fn(p)p Fr(;)19055 48479 y(\026)18994 48772 y Fn(k)19666 48938 y Ff(1)20164 48772 y Fo(;)20716 48479 y Fr(\026)20655 48772 y Fn(k)21327 48938 y Ff(2)21825 48772 y Fo(;)22376 48479 y Fr(\026)22316 48772 y Fn(k)22988 48938 y Ff(3)23485 48772 y Fo(;)24036 48479 y Fr(\026)23976 48772 y Fn(k)24648 48938 y Ff(4)25392 48772 y Fh(\000)246 b Fn(p)p Fr(\))g(+)g Fo(B)30174 48492 y Fr(^)29885 48772 y Fo(H)30895 48240 y Ff(4)p Fk(;)p Ff(1)30805 49047 y Fj(\000)32095 48772 y Fr(\()p Fn(p)p Fr(;)33784 48479 y(\026)33723 48772 y Fn(k)34395 48938 y Ff(1)34893 48772 y Fo(;)35445 48479 y Fr(\026)35384 48772 y Fn(k)36056 48938 y Ff(2)36554 48772 y Fo(;)37105 48479 y Fr(\026)37045 48772 y Fn(k)37717 48938 y Ff(3)38214 48772 y Fo(;)38765 48479 y Fr(\026)38705 48772 y Fn(k)39377 48938 y Ff(4)40121 48772 y Fh(\000)g Fn(p)p Fr(\))p 15154 49316 27210 45 v 27151 50330 a Fo(D)28067 50496 y Ff(+)28802 50330 y Fr(\()p Fn(p)p Fr(\))42806 49571 y Fo(:)443 53105 y Fr(All)535 b(the)g(terms)f(app)31 b(earing)536 b(in)f(the)f(ab)31 b(o)-31 b(v)g(e)536 b(equation)h(can)d (b)31 b(e)534 b(expressed)g(in)g(terms)h(of)g(con)-31 b(v)g(ergen)g(t)536 b(tree)0 54522 y(expansions.)493 b(The)370 b(term)f(in)h(the)f(l.h.s.)494 b(of)369 b(\(3.29\))j(is)d (giv)-31 b(en,)371 b(b)-31 b(y)370 b(\(2.37\),)i(b)-31 b(y)12835 56780 y(^)12584 57060 y Fo(G)13454 56603 y Ff(4)13454 57334 y(+)14190 57060 y Fr(\()14679 56768 y(\026)14620 57060 y Fn(k)15292 57226 y Ff(1)15789 57060 y Fo(;)16340 56768 y Fr(\026)16280 57060 y Fn(k)16952 57226 y Ff(2)17449 57060 y Fo(;)18000 56768 y Fr(\026)17940 57060 y Fn(k)18612 57226 y Ff(3)19110 57060 y Fo(;)19661 56768 y Fr(\026)19601 57060 y Fn(k)20273 57226 y Ff(4)20770 57060 y Fr(\))308 b(=)f Fo(\015)23310 56603 y Fj(\000)p Ff(4)p Fk(h)25018 57060 y Fo(Z)25853 56586 y Fj(\000)p Ff(2)25774 57394 y Fk(h)27041 57060 y Fr([)p Fh(\000)p Fo(\025)28855 57226 y Fk(h)29675 57060 y Fr(+)246 b Fo(O)31 b Fr(\()32133 56768 y(\026)32087 57060 y Fo(\025)32733 56603 y Ff(2)32733 57334 y Fk(h)33307 57060 y Fr(\)])308 b Fo(:)9756 b Fr(\(3)p Fo(:)p Fr(30\))0 59597 y(The)345 b(t)-31 b(w)g(o)346 b(terms)e(in)g(the)h(r.h.s.,)350 b(\014rst)344 b(line,)350 b(are)344 b(equal,)351 b(b)-31 b(y)345 b(\(2.36\),)352 b(to)345 b Fo(\015)30218 59196 y Fj(\000)p Ff(4)p Fk(h)31925 59597 y Fo(Z)32760 59124 y Fj(\000)p Ff(2)32681 59931 y Fk(h)33949 59597 y Fr(\()p Fo(\025)196 b Fr(+)g Fo(O)31 b Fr(\()37629 59305 y(\026)37583 59597 y Fo(\025)38229 59196 y Ff(2)38229 59911 y Fk(h)38803 59597 y Fr(\)\).)485 b(The)345 b(\014rst)f(term)0 61015 y(in)369 b(the)h(second)f(line,)h(b)-31 b(y)370 b(\(2.36\))i(and)d(eq.) 493 b(\(177\))371 b(of)f([BM2],)h(can)e(b)31 b(e)369 b(b)31 b(ounded)369 b(as)12163 62522 y Fe(\014)12163 63187 y(\014)12163 63851 y(\014)12163 64515 y(\014)12163 65179 y(\014)12532 64460 y Fo(\025)38 b Fr(^)-591 b Fo(g)13706 64626 y Fj(\000)14452 64460 y Fr(\()14941 64168 y(\026)14882 64460 y Fn(k)15554 64626 y Ff(4)16051 64460 y Fr(\))16733 64180 y(^)16481 64460 y Fo(G)17351 64003 y Ff(2)17351 64733 y Fj(\000)18099 64460 y Fr(\()18588 64168 y(\026)18529 64460 y Fn(k)19201 64626 y Ff(3)19698 64460 y Fr(\))20261 63660 y(\001)21183 63128 y Ff(2)p Fk(;)p Ff(1)21183 63936 y(+)22385 63660 y Fr(\(2)23427 63368 y(\026)23368 63660 y Fn(k)q Fo(;)24592 63368 y Fr(\026)24532 63660 y Fn(k)25204 63826 y Ff(1)25701 63660 y Fo(;)26252 63368 y Fr(\026)26192 63660 y Fn(k)26864 63826 y Ff(2)27362 63660 y Fr(\))p 20261 64205 7531 45 v 22158 65258 a Fo(D)23074 65424 y Ff(+)23809 65258 y Fr(\(2)24851 64966 y(\026)24792 65258 y Fn(k)q Fr(\))27925 62522 y Fe(\014)27925 63187 y(\014)27925 63851 y(\014)27925 64515 y(\014)27925 65179 y(\014)28601 64460 y Fh(\024)308 b Fo(C)30686 64168 y Fr(\026)30640 64460 y Fo(\025)31286 64003 y Ff(2)31286 64733 y Fk(h)31992 63711 y Fo(\015)32626 63309 y Fj(\000)p Ff(4)p Fk(h)p 31992 64205 2342 45 v 32497 65219 a Fo(Z)33332 64838 y Ff(2)33253 65553 y Fk(h)34774 64460 y Fo(;)9334 b Fr(\(3)p Fo(:)p Fr(31\))0 67904 y(while)370 b(the)g(second)f(term)g (in)h(the)f(second)g(line,)i(b)-31 b(y)369 b(eq.)493 b(\(A1.11\))372 b(of)e(ref.)492 b([BM3],)371 b(can)e(b)31 b(e)369 b(b)31 b(ounded)369 b(as)8564 69412 y Fe(\014)8564 70076 y(\014)8564 70741 y(\014)8564 71405 y(\014)8564 72069 y(\014)8933 71349 y Fo(\025)38 b Fr(^)-591 b Fo(g)10107 71515 y Fj(\000)10853 71349 y Fr(\()11342 71057 y(\026)11283 71349 y Fn(k)11955 71515 y Ff(4)12452 71349 y Fr(\))13458 70600 y(1)p 13015 71095 1438 45 v 13015 72109 a Fo(L\014)14771 70298 y Fe(X)15292 72622 y Fi(p)16554 71349 y Fo(\037)17247 71515 y Fk(M)18230 71349 y Fr(\()p Fn(p)p Fr(\))p Fo(G)20667 70817 y Ff(4)p Fk(;)p Ff(1)20667 71625 y(+)21869 71349 y Fr(\()p Fn(p)p Fr(;)23558 71057 y(\026)23497 71349 y Fn(k)24169 71515 y Ff(1)24667 71349 y Fo(;)25218 71057 y Fr(\026)25158 71349 y Fn(k)25830 71515 y Ff(2)26327 71349 y Fo(;)26879 71057 y Fr(\026)26818 71349 y Fn(k)27490 71515 y Ff(3)27988 71349 y Fo(;)28539 71057 y Fr(\026)28479 71349 y Fn(k)29151 71515 y Ff(4)29894 71349 y Fh(\000)246 b Fn(p)p Fr(\))32138 69412 y Fe(\014)32138 70076 y(\014)32138 70741 y(\014)32138 71405 y(\014)32138 72069 y(\014)32815 71349 y Fh(\024)308 b Fo(C)34900 71057 y Fr(\026)34854 71349 y Fo(\025)35500 70892 y Ff(3)35500 71623 y Fk(h)36206 70600 y Fo(\015)36840 70199 y Fj(\000)p Ff(4)p Fk(h)p 36206 71095 2342 45 v 36711 72109 a Fo(Z)37546 71727 y Ff(2)37467 72443 y Fk(h)44415 71349 y Fr(\(3)p Fo(:)p Fr(32\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 0 1107 a Fr(Moreo)-31 b(v)g(er,)370 b(b)-31 b(y)369 b(using)h(Lemma)g(A1.1)h(of)f([BM3],)g(one)g(sees)e (that)4369 2724 y Fe(\014)4369 3388 y(\014)4369 4052 y(\014)4369 4716 y(\014)4369 5381 y(\014)5313 3912 y Fr(1)p 4871 4407 1438 45 v 4871 5420 a Fo(L\014)6626 3609 y Fe(X)7148 5934 y Fi(p)8541 4661 y Fr(~)-684 b Fo(\037)9103 4827 y Fk(M)10085 4661 y Fr(\()p Fn(p)p Fr(\))12037 3584 y(^)11785 3863 y Fo(G)12655 3462 y Ff(4)12655 4137 y(+)13391 3863 y Fr(\()13880 3571 y(\026)13821 3863 y Fn(k)14493 4029 y Ff(1)15236 3863 y Fh(\000)246 b Fn(p)p Fo(;)17602 3571 y Fr(\026)17541 3863 y Fn(k)18213 4029 y Ff(2)18711 3863 y Fo(;)19262 3571 y Fr(\026)19202 3863 y Fn(k)19874 4029 y Ff(3)20371 3863 y Fo(;)20922 3571 y Fr(\026)20862 3863 y Fn(k)21534 4029 y Ff(4)22278 3863 y Fh(\000)g Fn(p)p Fr(\))g Fh(\000)26126 3584 y Fr(^)25875 3863 y Fo(G)26745 3462 y Ff(4)26745 4137 y(+)27481 3863 y Fr(\()27970 3571 y(\026)27911 3863 y Fn(k)28583 4029 y Ff(1)29080 3863 y Fo(;)29631 3571 y Fr(\026)29571 3863 y Fn(k)30243 4029 y Ff(2)30986 3863 y Fr(+)g Fn(p)p Fo(;)33351 3571 y Fr(\026)33291 3863 y Fn(k)33963 4029 y Ff(3)34461 3863 y Fo(;)35012 3571 y Fr(\026)34952 3863 y Fn(k)35624 4029 y Ff(4)36367 3863 y Fh(\000)g Fn(p)p Fr(\))p 11785 4407 26826 45 v 23589 5420 a Fo(D)24505 5586 y Ff(+)25240 5420 y Fr(\()p Fn(p)p Fr(\))39369 4661 y(+)4431 8575 y(+)6246 7827 y(1)p 5804 8321 1438 45 v 5804 9335 a Fo(L\014)7559 7524 y Fe(X)8080 9848 y Fi(p)9473 8575 y Fr(~)-684 b Fo(\037)10035 8741 y Fk(M)11017 8575 y Fr(\()p Fn(p)p Fr(\))12969 7498 y(^)12717 7778 y Fo(G)13587 7376 y Ff(4)13587 8052 y(+)14324 7778 y Fr(\()14813 7486 y(\026)14754 7778 y Fn(k)15426 7944 y Ff(1)15923 7778 y Fo(;)16474 7486 y Fr(\026)16414 7778 y Fn(k)17086 7944 y Ff(2)17583 7778 y Fo(;)18134 7486 y Fr(\026)18074 7778 y Fn(k)18746 7944 y Ff(3)19489 7778 y Fh(\000)246 b Fn(p)p Fo(;)21855 7486 y Fr(\026)21794 7778 y Fn(k)22466 7944 y Ff(4)23210 7778 y Fh(\000)g Fn(p)p Fr(\))p 12717 8321 12737 45 v 17477 9335 a Fo(D)18393 9501 y Ff(+)19128 9335 y Fr(\()p Fn(p)p Fr(\))25588 6638 y Fe(\014)25588 7302 y(\014)25588 7967 y(\014)25588 8631 y(\014)25588 9295 y(\014)26264 8575 y Fh(\024)307 b Fo(C)28348 8283 y Fr(\026)28302 8575 y Fo(\025)28948 8741 y Fk(h)29655 7827 y Fo(\015)30289 7425 y Fj(\000)p Ff(3)p Fk(h)p 29655 8321 2342 45 v 30160 9335 a Fo(Z)30995 8954 y Ff(2)30916 9669 y Fk(h)32437 8575 y Fo(:)44415 6694 y Fr(\(3)p Fo(:)p Fr(33\))0 12219 y(In)369 b(the)g(follo)-31 b(wing)373 b(sections)d(w)-31 b(e)369 b(will)i(pro)-31 b(v)g(e)370 b(the)f(follo)-31 b(wing)373 b(Lemma.)0 14581 y Fq(Lemma)486 b(3.1)1379 b Fp(Ther)-57 b(e)456 b(exists)f Fo(")13998 14747 y Ff(1)14911 14581 y Fh(\024)416 b Fo(")16704 14747 y Ff(0)17657 14581 y Fp(and)456 b(four)g Fo(\025)p Fp(-functions)h Fo(\027)28747 14747 y Ff(+)29482 14581 y Fo(;)184 b(\027)30520 14747 y Fj(\000)31268 14581 y Fo(;)g(\027)32376 14179 y Fj(0)32306 14855 y Ff(+)33041 14581 y Fo(;)g(\027)34149 14179 y Fj(0)34079 14855 y(\000)35283 14581 y Fp(of)457 b(or)-57 b(der)455 b Fo(\025)h Fp(\(uniformly)h(in)0 15998 y Fo(h)p Fp(\),)396 b(such)g(that,)f(if)8067 15706 y Fr(\026)8021 15998 y Fo(\025)8667 16164 y Fk(h)9548 15998 y Fh(\024)307 b Fo(")11232 16164 y Ff(1)11729 15998 y Fp(,)8361 17615 y Fe(\014)8361 18279 y(\014)8361 18944 y(\014)8361 19608 y(\014)8361 20272 y(\014)8730 19552 y Fo(\025)38 b Fr(^)-591 b Fo(g)9904 19718 y Fj(\000)10651 19552 y Fr(\()11140 19260 y(\026)11081 19552 y Fn(k)11753 19718 y Ff(4)12250 19552 y Fr(\))13255 18804 y(1)p 12813 19298 1438 45 v 12813 20312 a Fo(L\014)14568 18501 y Fe(X)15089 20825 y Fi(p)16483 19552 y Fr(~)-684 b Fo(\037)17045 19718 y Fk(M)18027 19552 y Fr(\()p Fn(p)p Fr(\))20018 18473 y(^)19727 18753 y Fo(H)20737 18221 y Ff(4)p Fk(;)p Ff(1)20647 19028 y Fj(\006)21939 18753 y Fr(\()p Fn(p)p Fr(;)23628 18461 y(\026)23567 18753 y Fn(k)24239 18919 y Ff(1)24737 18753 y Fo(;)25288 18461 y Fr(\026)25228 18753 y Fn(k)25900 18919 y Ff(2)26397 18753 y Fo(;)26948 18461 y Fr(\026)26888 18753 y Fn(k)27560 18919 y Ff(3)28058 18753 y Fo(;)28609 18461 y Fr(\026)28549 18753 y Fn(k)29221 18919 y Ff(4)29964 18753 y Fh(\000)246 b Fn(p)p Fr(\))p 19727 19298 12481 45 v 24359 20312 a Fo(D)25275 20478 y Ff(+)26010 20312 y Fr(\()p Fn(p)p Fr(\))32342 17615 y Fe(\014)32342 18279 y(\014)32342 18944 y(\014)32342 19608 y(\014)32342 20272 y(\014)33018 19552 y Fh(\024)308 b Fo(C)35103 19260 y Fr(\026)35057 19552 y Fo(\025)35703 19095 y Ff(2)35703 19826 y Fk(h)36409 18804 y Fo(\015)37043 18402 y Fj(\000)p Ff(4)p Fk(h)p 36409 19298 2342 45 v 36914 20312 a Fo(Z)37749 19930 y Ff(2)37670 20646 y Fk(h)44415 19552 y Fr(\(3)p Fo(:)p Fr(34\))443 24789 y(The)364 b(ab)31 b(o)-31 b(v)g(e)365 b(Lemma,)h(together)g(with)f(the)f(iden)-31 b(tit)g(y)366 b(\(3.29\),)i(follo)-31 b(wing)368 b(from)c(the)g(Dyson)g(equation)i (and)e(the)0 26207 y(W)-92 b(ard)320 b(iden)-31 b(tities,)333 b(and)322 b(the)f(previous)g(b)31 b(ounds,)331 b(pro)-31 b(v)g(ed)321 b(in)g(refs.)476 b([BM2])322 b(and)f([BM3])h(and)g(follo) -31 b(wing)324 b(from)e(the)0 27624 y(tree)369 b(expansion,)i(implies)f (\(3.2\);)i(this)e(concludes)f(the)h(pro)31 b(of)369 b(of)h(Theorem)g(3.1.)15863 30931 y Fs(4.)1328 b(Pro)42 b(of)498 b(of)g(Lemma)h(3.1)0 34238 y Fn(4.1)1314 b Fp(The)397 b(c)-57 b(orr)g(e)g(ctions.)443 35655 y Fr(W)-92 b(e)350 b(shall)i(pro)-31 b(v)g(e)352 b(\014rst)f(the)g(b)31 b(ound)351 b(\(3.34\))j(for)d Fo(H)21286 35123 y Ff(4)p Fk(;)p Ff(1)21196 35931 y(+)22487 35655 y Fr(;)358 b(the)351 b(b)31 b(ound)351 b(for)g Fo(H)31121 35123 y Ff(4)p Fk(;)p Ff(1)31031 35931 y Fj(\000)32673 35655 y Fr(is)g(done)h(essen)-31 b(tially)353 b(in)e(the)h(same)0 37073 y(w)-31 b(a)g(y)371 b(and)e(will)i(b)31 b(e)369 b(brie\015y)g(discussed)f(later.)494 b(By)369 b(using)h(\(3.21\),)i(w)-31 b(e)370 b(get)9874 40152 y(^)-591 b Fo(g)10364 40318 y Fj(\000)11111 40152 y Fr(\()p Fn(k)12213 40318 y Ff(4)12710 40152 y Fr(\))13715 39403 y(1)p 13273 39897 1438 45 v 13273 40911 a Fo(L\014)15028 39100 y Fe(X)15550 41425 y Fi(p)16943 40152 y Fr(~)-684 b Fo(\037)17505 40318 y Fk(M)18487 40152 y Fr(\()p Fn(p)p Fr(\))p Fo(D)21001 39678 y Fj(\000)p Ff(1)20970 40427 y(+)22191 40152 y Fr(\()p Fn(p)p Fr(\))24049 39872 y(^)23758 40152 y Fo(H)24768 39620 y Ff(4)p Fk(;)p Ff(1)24678 40427 y(+)25970 40152 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)28270 40318 y Ff(1)28768 40152 y Fo(;)g Fn(k)29931 40318 y Ff(2)30428 40152 y Fo(;)g Fn(k)31591 40318 y Ff(3)32089 40152 y Fo(;)g Fn(k)33252 40318 y Ff(4)33995 40152 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)711 44606 y(=)345 b(^)-591 b Fo(g)2407 44772 y Fj(\000)3154 44606 y Fr(\()p Fn(k)4256 44772 y Ff(4)4753 44606 y Fr(\))5759 43857 y(1)p 5316 44351 V 5316 45365 a Fo(L\014)7072 43554 y Fe(X)7593 45879 y Fi(p)8986 44606 y Fr(~)-684 b Fo(\037)9548 44772 y Fk(M)10530 44606 y Fr(\()p Fn(p)p Fr(\))12674 43857 y(1)p 12230 44351 V 12230 45365 a Fo(L\014)13987 43554 y Fe(X)14521 45937 y Fi(k)15903 43857 y Fo(C)16694 44023 y Ff(+)17429 43857 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))p 15903 44351 6283 45 v 17434 45365 a Fo(D)18350 45531 y Ff(+)19086 45365 y Fr(\()p Fn(p)p Fr(\))22626 44606 y Fo(<)24021 44313 y Fr(^)23794 44606 y Fo( )24555 44132 y Ff(+)24515 44939 y Fi(k)p Fk(;)p Ff(+)26269 44313 y Fr(^)26042 44606 y Fo( )26803 44132 y Fj(\000)26763 44939 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)29539 44606 y Fr(;)30257 44313 y(^)30030 44606 y Fo( )30791 44132 y Fj(\000)30751 44939 y Fi(k)31280 45050 y Fm(1)31713 44939 y Fk(;)p Ff(+)32711 44606 y Fr(;)33429 44313 y(^)33202 44606 y Fo( )33963 44132 y Ff(+)33923 44939 y Fi(k)34452 45050 y Fm(2)34885 44939 y Fk(;)p Ff(+)35882 44606 y Fr(;)36601 44313 y(^)36373 44606 y Fo( )37134 44132 y Fj(\000)37094 44939 y Fi(k)37623 45050 y Fm(3)38057 44939 y Fk(;)p Fj(\000)39067 44606 y Fr(;)39785 44313 y(^)39558 44606 y Fo( )40319 44132 y Ff(+)40279 44939 y Fi(k)40808 45050 y Fm(4)41241 44939 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)43807 44606 y Fo(>)44668 44148 y Fk(T)45673 44606 y Fr(+)1737 48561 y Fh(\000)p Fo(\027)3145 48727 y Fj(\000)3929 48561 y Fr(^)-590 b Fo(g)4420 48727 y Fj(\000)5167 48561 y Fr(\()p Fn(k)6269 48727 y Ff(4)6766 48561 y Fr(\))7771 47812 y(1)p 7329 48306 1438 45 v 7329 49320 a Fo(L\014)9084 47509 y Fe(X)9605 49834 y Fi(p)10999 48561 y Fr(~)-684 b Fo(\037)11561 48727 y Fk(M)12543 48561 y Fr(\()p Fn(p)p Fr(\))14686 47812 y(1)p 14243 48306 V 14243 49320 a Fo(L\014)15999 47509 y Fe(X)16534 49892 y Fi(k)17915 47812 y Fo(D)18831 47978 y Fj(\000)19579 47812 y Fr(\()p Fn(p)p Fr(\))p 17915 48306 3232 45 v 17921 49320 a Fo(D)18837 49486 y Ff(+)19573 49320 y Fr(\()p Fn(p)p Fr(\))21588 48561 y Fo(<)22983 48269 y Fr(^)22756 48561 y Fo( )23517 48087 y Ff(+)23477 48895 y Fi(k)p Fk(;)p Fj(\000)25243 48269 y Fr(^)25016 48561 y Fo( )25777 48087 y Fj(\000)25737 48895 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)28525 48561 y Fr(;)29243 48269 y(^)29016 48561 y Fo( )29777 48087 y Fj(\000)29737 48895 y Fi(k)30266 49006 y Fm(1)30699 48895 y Fk(;)p Ff(+)31697 48561 y Fr(;)32415 48269 y(^)32188 48561 y Fo( )32949 48087 y Ff(+)32909 48895 y Fi(k)33438 49006 y Fm(2)33871 48895 y Fk(;)p Ff(+)34869 48561 y Fr(;)35587 48269 y(^)35360 48561 y Fo( )36121 48087 y Fj(\000)36081 48895 y Fi(k)36610 49006 y Fm(3)37043 48895 y Fk(;)p Ff(+)38041 48561 y Fr(;)38759 48269 y(^)38532 48561 y Fo( )39293 48087 y Ff(+)39253 48895 y Fi(k)39782 49006 y Fm(4)40215 48895 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)42781 48561 y Fo(>)43642 48104 y Fk(T)44646 48561 y Fh(\000)2575 52399 y(\000)p Fo(\027)3983 52565 y Ff(+)4756 52399 y Fr(^)-591 b Fo(g)5246 52565 y Fj(\000)5993 52399 y Fr(\()p Fn(k)7095 52565 y Ff(4)7592 52399 y Fr(\))8598 51650 y(1)p 8155 52144 1438 45 v 8155 53158 a Fo(L\014)9911 51347 y Fe(X)10432 53672 y Fi(p)11825 52399 y Fr(~)-684 b Fo(\037)12387 52565 y Fk(M)13369 52399 y Fr(\()p Fn(p)p Fr(\))15512 51650 y(1)p 15069 52144 V 15069 53158 a Fo(L\014)16825 51347 y Fe(X)17360 53730 y Fi(k)18732 52399 y Fo(<)20127 52107 y Fr(^)19900 52399 y Fo( )20661 51925 y Ff(+)20621 52733 y Fi(k)p Fk(;)p Ff(+)22375 52107 y Fr(^)22148 52399 y Fo( )22909 51925 y Fj(\000)22869 52733 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)25645 52399 y Fr(;)26363 52107 y(^)26136 52399 y Fo( )26897 51925 y Fj(\000)26857 52733 y Fi(k)27386 52844 y Fm(1)27819 52733 y Fk(;)p Ff(+)28816 52399 y Fr(;)29535 52107 y(^)29307 52399 y Fo( )30068 51925 y Ff(+)30028 52733 y Fi(k)30557 52844 y Fm(2)30991 52733 y Fk(;)p Ff(+)31988 52399 y Fr(;)32707 52107 y(^)32479 52399 y Fo( )33240 51925 y Fj(\000)33200 52733 y Fi(k)33729 52844 y Fm(3)34162 52733 y Fk(;)p Fj(\000)35173 52399 y Fr(;)35891 52107 y(^)35664 52399 y Fo( )36425 51925 y Ff(+)36385 52733 y Fi(k)36914 52844 y Fm(4)37347 52733 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)39913 52399 y Fo(>)40774 51942 y Fk(T)42086 52399 y Fo(:)2576 b Fr(\(4)p Fo(:)p Fr(1\))0 55544 y(Let)369 b(us)g(de\014ne)10542 57312 y(~)10292 57592 y Fo(G)11162 57134 y Ff(4)11162 57865 y(+)11897 57592 y Fr(\()p Fn(k)12999 57758 y Ff(1)13496 57592 y Fo(;)184 b Fn(k)14659 57758 y Ff(2)15156 57592 y Fo(;)g Fn(k)16319 57758 y Ff(3)16817 57592 y Fo(;)g Fn(k)17980 57758 y Ff(4)18477 57592 y Fr(\))308 b(=)26123 56843 y Fo(@)26772 56441 y Ff(4)p 20649 57337 12095 45 v 20649 58423 a Fo(@)61 b(\036)21958 57949 y Ff(+)21958 58757 y Fi(k)22487 58868 y Fm(1)22919 58757 y Fk(;)p Ff(+)23917 58423 y Fo(@)g(\036)25226 57949 y Fj(\000)25226 58757 y Fi(k)25755 58868 y Fm(2)26187 58757 y Fk(;)p Ff(+)27184 58423 y Fo(@)g(\036)28493 57949 y Ff(+)28493 58757 y Fi(k)29022 58868 y Fm(3)29454 58757 y Fk(;)p Fj(\000)30464 58423 y Fo(@)g(J)31727 58589 y Fi(k)32256 58700 y Fm(4)33199 57312 y Fr(~)32877 57592 y Fo(W)34076 55654 y Fe(\014)34076 56319 y(\014)34076 56983 y(\014)34076 57647 y(\014)34076 58311 y(\014)34445 59031 y Fk(\036)p Ff(=0)36645 57592 y Fo(;)8017 b Fr(\(4)p Fo(:)p Fr(2\))0 60792 y(where)5229 63359 y(~)4907 63639 y Fo(W)461 b Fr(=)308 b(log)9196 62132 y Fe(Z)10488 63639 y Fo(P)154 b Fr(\()p Fo(d)12585 63346 y Fr(^)12359 63639 y Fo( )39 b Fr(\))p Fo(e)14064 63182 y Fj(\000)p Fk(T)15279 63293 y Fm(1)15712 63182 y Ff(\()p Fk( )28 b Ff(\)+)p Fk(\027)18142 63293 y Fm(+)18765 63182 y Fk(T)19288 63293 y Fm(+)19912 63182 y Ff(\()p Fk( )g Ff(\)+)p Fk(\027)22342 63293 y Fc(\000)22996 63182 y Fk(T)23519 63293 y Fc(\000)24174 63182 y Ff(\()p Fk( )g Ff(\))25534 63639 y Fo(e)26049 63098 y Fj(\000)p Fk(V)191 b Ff(\()27982 62894 y(^)27801 63098 y Fk( )28 b Ff(\)+)29441 62351 y Fe(P)30608 63513 y Fl(!)31325 62289 y Fe(R)32215 63098 y Fk(d)p Fi(x)p Ff([)p Fk(\036)33988 62764 y Fm(+)33988 63320 y Fg(x)p Fl(;!)35430 62894 y Ff(^)35249 63098 y Fk( )35863 62764 y Fc(\000)35835 63320 y Fg(x)p Fl(;!)37096 63098 y Ff(+)37957 62894 y(^)37776 63098 y Fk( )38390 62764 y Fm(+)38362 63320 y Fg(x)p Fl(;!)39623 63098 y Fk(\036)40155 62764 y Fc(\000)40155 63320 y Fg(x)p Fl(;!)41417 63098 y Ff(])42030 63639 y Fo(;)2632 b Fr(\(4)p Fo(:)p Fr(3\))4886 67511 y Fo(T)5533 67677 y Ff(1)6030 67511 y Fr(\()p Fo( )40 b Fr(\))308 b(=)9702 66762 y(1)p 9260 67256 1438 45 v 9260 68270 a Fo(L\014)11015 66459 y Fe(X)11537 68784 y Fi(p)12930 67511 y Fr(~)-684 b Fo(\037)13492 67677 y Fk(M)14474 67511 y Fr(\()p Fn(p)p Fr(\))16617 66762 y(1)p 16174 67256 V 16174 68270 a Fo(L\014)17930 66459 y Fe(X)18465 68843 y Fi(k)19847 66762 y Fo(C)20638 66928 y Ff(+)21373 66762 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))p 19847 67256 6283 45 v 21378 68270 a Fo(D)22294 68436 y Ff(+)23030 68270 y Fr(\()p Fn(p)p Fr(\))26262 67511 y(\()26919 67219 y(^)26692 67511 y Fo( )27453 67038 y Ff(+)27413 67845 y Fi(k)p Fk(;)p Ff(+)29167 67219 y Fr(^)28941 67511 y Fo( )29702 67038 y Fj(\000)29662 67845 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)32437 67511 y Fr(\))33094 67219 y(^)32867 67511 y Fo( )33628 67038 y Ff(+)33588 67845 y Fi(k)34117 67956 y Fm(4)34550 67845 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)37076 67231 y Fr(^)36808 67511 y Fo(J)37422 67677 y Fi(k)37951 67788 y Fm(4)38476 67511 y Fr(^)-591 b Fo(g)38966 67677 y Fj(\000)39714 67511 y Fr(\()p Fn(k)40816 67677 y Ff(4)41312 67511 y Fr(\))308 b Fo(;)2612 b Fr(\(4)p Fo(:)p Fr(4\))8134 71349 y Fo(T)8781 71515 y Ff(+)9515 71349 y Fr(\()p Fo( )40 b Fr(\))309 b(=)13188 70600 y(1)p 12746 71095 1438 45 v 12746 72109 a Fo(L\014)14501 70298 y Fe(X)15022 72622 y Fi(p)16416 71349 y Fr(~)-684 b Fo(\037)16978 71515 y Fk(M)17960 71349 y Fr(\()p Fn(p)p Fr(\))20103 70600 y(1)p 19660 71095 V 19660 72109 a Fo(L\014)21416 70298 y Fe(X)21951 72681 y Fi(k)23015 71349 y Fr(\()23672 71057 y(^)23445 71349 y Fo( )24206 70876 y Ff(+)24166 71683 y Fi(k)p Fk(;)p Ff(+)25920 71057 y Fr(^)25693 71349 y Fo( )26454 70876 y Fj(\000)26414 71683 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)29190 71349 y Fr(\))29847 71057 y(^)29620 71349 y Fo( )30381 70876 y Ff(+)30341 71683 y Fi(k)30870 71794 y Fm(4)31302 71683 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)33829 71070 y Fr(^)33561 71349 y Fo(J)34175 71515 y Fi(k)34704 71626 y Fm(4)35229 71349 y Fr(^)-591 b Fo(g)35719 71515 y Fj(\000)36466 71349 y Fr(\()p Fn(k)37568 71515 y Ff(4)38065 71349 y Fr(\))308 b Fo(;)5859 b Fr(\(4)p Fo(:)p Fr(5\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 6274 1579 a Fo(T)6921 1745 y Fj(\000)7668 1579 y Fr(\()p Fo( )40 b Fr(\))308 b(=)11341 830 y(1)p 10899 1324 1438 45 v 10899 2338 a Fo(L\014)12654 527 y Fe(X)13175 2852 y Fi(p)14568 1579 y Fr(~)-684 b Fo(\037)15130 1745 y Fk(M)16113 1579 y Fr(\()p Fn(p)p Fr(\))18256 830 y(1)p 17813 1324 V 17813 2338 a Fo(L\014)19569 527 y Fe(X)20104 2911 y Fi(k)21485 830 y Fo(D)22401 996 y Fj(\000)23149 830 y Fr(\()p Fn(p)p Fr(\))p 21485 1324 3232 45 v 21491 2338 a Fo(D)22407 2504 y Ff(+)23143 2338 y Fr(\()p Fn(p)p Fr(\))24850 1579 y(\()25507 1287 y(^)25280 1579 y Fo( )26041 1106 y Ff(+)26001 1913 y Fi(k)p Fk(;)p Fj(\000)27767 1287 y Fr(^)27540 1579 y Fo( )28301 1106 y Fj(\000)28261 1913 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)31049 1579 y Fr(\))31706 1287 y(^)31479 1579 y Fo( )32240 1106 y Ff(+)32200 1913 y Fi(k)32729 2024 y Fm(4)33162 1913 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)35688 1299 y Fr(^)35420 1579 y Fo(J)36034 1745 y Fi(k)36563 1856 y Fm(4)37088 1579 y Fr(^)-591 b Fo(g)37578 1745 y Fj(\000)38326 1579 y Fr(\()p Fn(k)39428 1745 y Ff(4)39925 1579 y Fr(\))308 b Fo(:)3999 b Fr(\(4)p Fo(:)p Fr(6\))11999 15528 y Fo(T)12646 15694 y Ff(1)23069 15528 y Fo(T)23716 15694 y Ff(+)34138 15528 y Fo(T)34785 15694 y Fj(\000)7018 14771 y @beginspecial @setspecial %%BeginDocument: verticiT.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 r cerchio_pieno % /cerchio_pieno { gsave 0 360 newpath arc fill grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 50 30 5 cerchio 50 35 50 70 3 5 onda 70 10 50 25 freccia 50 25 30 10 freccia 50 70 80 80 tlinea 50 70 20 80 freccia 150 25 150 70 3 5 onda 120 5 150 25 freccia 150 25 180 5 freccia 150 70 180 80 tlinea 150 70 120 80 freccia 250 30 5 cerchio_pieno 250 35 250 70 3 5 onda 220 5 250 25 freccia 250 25 280 5 freccia 250 70 280 80 tlinea 250 70 220 80 freccia grestore %%EndDocument @endspecial 6555 17244 a Fr(Fig.)493 b(12:)g(Graphical)370 b(represen)-31 b(tation)371 b(of)e Fo(T)25105 17410 y Ff(1)25602 17244 y Fo(;)184 b(T)26740 17410 y Ff(+)27475 17244 y Fo(;)g(T)28613 17410 y Fj(\000)29362 17244 y Fr(;)369 b(the)g(dotted)i(line)f(carries)5171 18662 y(momen)-31 b(tum)11041 18370 y(\026)10982 18662 y Fn(k)11654 18828 y Ff(4)12151 18662 y Fr(,)369 b(the)g(empt)-31 b(y)371 b(circle)e(represen)-31 b(ts)368 b Fo(C)27015 18828 y Ff(+)27750 18662 y Fr(,)i(the)f(\014lled)h(one)g Fo(D)35985 18828 y Fj(\000)36732 18662 y Fr(\()p Fn(p)p Fr(\))p Fo(=D)39768 18828 y Ff(+)40505 18662 y Fr(\()p Fn(p)p Fr(\))443 21024 y(It)f(is)g(easy)g(to)h(see)f(that)10982 20744 y(~)10731 21024 y Fo(G)11601 20622 y Ff(4)11601 21298 y(+)12706 21024 y Fr(is)g(related)h(to)g(\(4.1\))h(b)-31 b(y)369 b(an)h(iden)-31 b(tit)g(y)372 b(similar)e(to)g(\(3.9\).)495 b(In)369 b(fact)h(w)-31 b(e)370 b(can)f(write)926 23101 y Fh(\000)2283 22821 y Fr(~)2033 23101 y Fo(G)2903 22644 y Ff(4)2903 23375 y(+)3638 23101 y Fr(\()p Fn(k)4740 23267 y Ff(1)5237 23101 y Fo(;)184 b Fn(k)6400 23267 y Ff(2)6897 23101 y Fo(;)g Fn(k)8060 23267 y Ff(3)8558 23101 y Fo(;)g Fn(k)9721 23267 y Ff(4)10218 23101 y Fr(\))308 b(=)33152 b(\(4)p Fo(:)p Fr(7\))987 25489 y(=)346 b(^)-591 b Fo(g)2684 25655 y Fj(\000)3431 25489 y Fr(\()p Fn(k)4533 25655 y Ff(4)5030 25489 y Fr(\))6035 24740 y(1)p 5593 25234 1438 45 v 5593 26248 a Fo(L\014)7348 24437 y Fe(X)7870 26762 y Fi(p)9263 25489 y Fr(~)-684 b Fo(\037)9825 25655 y Fk(M)10807 25489 y Fr(\()p Fn(p)p Fr(\))12950 24740 y(1)p 12507 25234 V 12507 26248 a Fo(L\014)14263 24437 y Fe(X)14798 26820 y Fi(k)16180 24740 y Fo(C)16971 24906 y Ff(+)17706 24740 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))p 16180 25234 6283 45 v 17711 26248 a Fo(D)18627 26414 y Ff(+)19363 26248 y Fr(\()p Fn(p)p Fr(\))22903 25489 y Fo(<)307 b Fr([)24605 25197 y(^)24378 25489 y Fo( )25139 25015 y Fj(\000)25099 25823 y Fi(k)25628 25934 y Fm(1)26061 25823 y Fk(;)p Ff(+)27285 25197 y Fr(^)27059 25489 y Fo( )27820 25015 y Ff(+)27780 25823 y Fi(k)28309 25934 y Fm(2)28741 25823 y Fk(;)p Ff(+)29739 25489 y Fr(];)184 b([)31072 25197 y(^)30844 25489 y Fo( )31605 25015 y Ff(+)31565 25823 y Fi(k)p Fk(;)p Ff(+)33320 25197 y Fr(^)33093 25489 y Fo( )33854 25015 y Fj(\000)33814 25823 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)36816 25197 y Fr(^)36590 25489 y Fo( )37351 25015 y Fj(\000)37311 25823 y Fi(k)37840 25934 y Fm(3)38272 25823 y Fk(;)p Fj(\000)39509 25197 y Fr(^)39282 25489 y Fo( )40043 25015 y Ff(+)40003 25823 y Fi(k)40532 25934 y Fm(4)40964 25823 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)43223 25489 y Fr(])308 b Fo(>)44699 25032 y Fk(T)45703 25489 y Fr(+)926 29045 y Fh(\000)246 b Fo(\027)2580 29211 y Fj(\000)3365 29045 y Fr(^)-591 b Fo(g)3855 29211 y Fj(\000)4602 29045 y Fr(\()p Fn(k)5704 29211 y Ff(4)6201 29045 y Fr(\))7206 28296 y(1)p 6764 28790 1438 45 v 6764 29804 a Fo(L\014)8519 27993 y Fe(X)9041 30318 y Fi(p)10434 29045 y Fr(~)-684 b Fo(\037)10996 29211 y Fk(M)11978 29045 y Fr(\()p Fn(p)p Fr(\))14121 28296 y(1)p 13678 28790 V 13678 29804 a Fo(L\014)15434 27993 y Fe(X)15969 30376 y Fi(k)17351 28296 y Fo(D)18267 28462 y Fj(\000)19014 28296 y Fr(\()p Fn(p)p Fr(\))p 17351 28790 3232 45 v 17357 29804 a Fo(D)18273 29970 y Ff(+)19008 29804 y Fr(\()p Fn(p)p Fr(\))21023 29045 y Fo(<)307 b Fr([)22725 28753 y(^)22498 29045 y Fo( )23259 28571 y Fj(\000)23219 29379 y Fi(k)23748 29490 y Fm(1)24181 29379 y Fk(;)p Ff(+)25405 28753 y Fr(^)25179 29045 y Fo( )25940 28571 y Ff(+)25900 29379 y Fi(k)26429 29490 y Fm(2)26861 29379 y Fk(;)p Ff(+)27859 29045 y Fr(];)184 b([)29192 28753 y(^)28964 29045 y Fo( )29725 28571 y Ff(+)29685 29379 y Fi(k)p Fk(;)p Fj(\000)31452 28753 y Fr(^)31226 29045 y Fo( )31987 28571 y Fj(\000)31947 29379 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)34961 28753 y Fr(^)34734 29045 y Fo( )35495 28571 y Fj(\000)35455 29379 y Fi(k)35984 29490 y Fm(3)36416 29379 y Fk(;)p Ff(+)37641 28753 y Fr(^)37414 29045 y Fo( )38175 28571 y Ff(+)38135 29379 y Fi(k)38664 29490 y Fm(4)39096 29379 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)41355 29045 y Fr(])308 b Fo(>)42831 28588 y Fk(T)43836 29045 y Fh(\000)926 32484 y(\000)246 b Fo(\027)2580 32650 y Ff(+)3352 32484 y Fr(^)-591 b Fo(g)3842 32650 y Fj(\000)4590 32484 y Fr(\()p Fn(k)5692 32650 y Ff(4)6189 32484 y Fr(\))7194 31736 y(1)p 6752 32230 1438 45 v 6752 33244 a Fo(L\014)8507 31433 y Fe(X)9028 33757 y Fi(p)10422 32484 y Fr(~)-684 b Fo(\037)10984 32650 y Fk(M)11966 32484 y Fr(\()p Fn(p)p Fr(\))14109 31736 y(1)p 13666 32230 V 13666 33244 a Fo(L\014)15422 31433 y Fe(X)15957 33816 y Fi(k)17328 32484 y Fo(<)308 b Fr([)19031 32192 y(^)18804 32484 y Fo( )19565 32011 y Fj(\000)19525 32818 y Fi(k)20054 32929 y Fm(1)20486 32818 y Fk(;)p Ff(+)21711 32192 y Fr(^)21484 32484 y Fo( )22245 32011 y Ff(+)22205 32818 y Fi(k)22734 32929 y Fm(2)23166 32818 y Fk(;)p Ff(+)24164 32484 y Fr(];)184 b([)25498 32192 y(^)25269 32484 y Fo( )26030 32011 y Ff(+)25990 32818 y Fi(k)p Fk(;)p Ff(+)27746 32192 y Fr(^)27519 32484 y Fo( )28280 32011 y Fj(\000)28240 32818 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)31242 32192 y Fr(^)31015 32484 y Fo( )31776 32011 y Fj(\000)31736 32818 y Fi(k)32265 32929 y Fm(3)32698 32818 y Fk(;)p Fj(\000)33934 32192 y Fr(^)33708 32484 y Fo( )34469 32011 y Ff(+)34429 32818 y Fi(k)34958 32929 y Fm(4)35390 32818 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)37648 32484 y Fr(])308 b Fo(>)39124 32027 y Fk(T)40129 32484 y Fr(;)0 35558 y(moreo)-31 b(v)g(er,)371 b(if)e(w)-31 b(e)370 b(in)-31 b(tro)31 b(duce)370 b(the)f(de\014nition) 13846 38123 y Fo(\016)42 b(\032)14952 38289 y Fi(p)p Fk(;)p Ff(+)16814 38123 y Fr(=)18558 37374 y(1)p 18115 37868 V 18115 38882 a Fo(\014)58 b(L)19871 37071 y Fe(X)20406 39454 y Fi(k)21787 37374 y Fo(C)22578 37540 y Ff(+)23313 37374 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fr(\))p 21787 37868 4259 45 v 22306 38882 a Fo(D)23222 39048 y Ff(+)23958 38882 y Fr(\()p Fn(p)p Fr(\))26178 38123 y(\()26835 37831 y(^)26608 38123 y Fo( )27369 37649 y Ff(+)27329 38457 y Fi(k)p Fk(;)p Ff(+)29083 37831 y Fr(^)28856 38123 y Fo( )29617 37649 y Fj(\000)29577 38457 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)32353 38123 y Fr(\))308 b Fo(;)11571 b Fr(\(4)p Fo(:)p Fr(8\))0 41166 y(the)369 b(term)h(in)f(the)h(second)f (line)h(of)f(\(4.7\))j(can)d(b)31 b(e)369 b(rewritten)h(as)6150 43675 y(^)-591 b Fo(g)6640 43841 y Fj(\000)7387 43675 y Fr(\()p Fn(k)8489 43841 y Ff(4)8986 43675 y Fr(\))9992 42926 y(1)p 9549 43421 1438 45 v 9549 44434 a Fo(L\014)11305 42623 y Fe(X)11826 44948 y Fi(p)13219 43675 y Fr(~)-684 b Fo(\037)13781 43841 y Fk(M)14763 43675 y Fr(\()p Fn(p)p Fr(\))309 b Fo(<)f Fr([)18342 43383 y(^)18115 43675 y Fo( )18876 43202 y Fj(\000)18836 44009 y Fi(k)19365 44120 y Fm(1)19797 44009 y Fk(;)p Ff(+)21022 43383 y Fr(^)20795 43675 y Fo( )21556 43202 y Ff(+)21516 44009 y Fi(k)22045 44120 y Fm(2)22477 44009 y Fk(;)p Ff(+)23475 43675 y Fr(];)184 b([)p Fo(\016)42 b(\032)25686 43841 y Fi(p)p Fk(;)p Ff(+)27469 43383 y Fr(^)27242 43675 y Fo( )28003 43202 y Fj(\000)27963 44009 y Fi(k)28492 44120 y Fm(3)28925 44009 y Fk(;)p Fj(\000)30161 43383 y Fr(^)29935 43675 y Fo( )30696 43202 y Ff(+)30656 44009 y Fi(k)31185 44120 y Fm(4)31617 44009 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)33875 43675 y Fr(])308 b Fo(>)35351 43218 y Fk(T)36049 43675 y Fr(=)6420 47115 y(=)f Fo(g)8116 47281 y Fj(\000)8863 47115 y Fr(\()p Fn(k)9965 47281 y Ff(4)10462 47115 y Fr(\))11468 46366 y(1)p 11025 46860 V 11025 47874 a Fo(L\014)12781 46063 y Fe(X)13302 48388 y Fi(p)14695 47115 y Fr(~)-684 b Fo(\037)15257 47281 y Fk(M)16239 47115 y Fr(\()p Fn(p)p Fr(\))17990 45886 y Fe(h)18515 47115 y Fo(<)19910 46822 y Fr(^)19683 47115 y Fo( )20444 46641 y Fj(\000)20404 47448 y Fi(k)20933 47559 y Fm(1)21365 47448 y Fk(;)p Ff(+)22363 47115 y Fr(;)23082 46822 y(^)22854 47115 y Fo( )23615 46641 y Ff(+)23575 47448 y Fi(k)24104 47559 y Fm(2)24537 47448 y Fk(;)p Ff(+)25535 47115 y Fr(;)184 b Fo(\016)42 b(\032)27132 47281 y Fi(p)p Fk(;)p Ff(+)28995 47115 y Fo(>)29856 46657 y Fk(T)30861 47115 y Fo(<)32256 46822 y Fr(^)32029 47115 y Fo( )32790 46641 y Fj(\000)32750 47448 y Fi(k)33279 47559 y Fm(3)33711 47448 y Fk(;)p Fj(\000)34948 46822 y Fr(^)34722 47115 y Fo( )35483 46641 y Ff(+)35443 47448 y Fi(k)35972 47559 y Fm(4)36404 47448 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)38970 47115 y Fo(>)307 b Fr(+)3970 b(\(4)p Fo(:)p Fr(9\))6358 50365 y(+)379 b Fo(<)307 b(\016)42 b(\032)9872 50531 y Fi(p)p Fk(;)p Ff(+)11427 50365 y Fr(;)12146 50073 y(^)11918 50365 y Fo( )12679 49891 y Fj(\000)12639 50699 y Fi(k)13168 50810 y Fm(1)13601 50699 y Fk(;)p Ff(+)14599 50365 y Fr(;)15317 50073 y(^)15090 50365 y Fo( )15851 49891 y Ff(+)15811 50699 y Fi(k)16340 50810 y Fm(2)16773 50699 y Fk(;)p Ff(+)17771 50365 y Fr(;)18489 50073 y(^)18262 50365 y Fo( )19023 49891 y Fj(\000)18983 50699 y Fi(k)19512 50810 y Fm(3)19945 50699 y Fk(;)p Fj(\000)20955 50365 y Fr(;)21674 50073 y(^)21446 50365 y Fo( )22207 49891 y Ff(+)22167 50699 y Fi(k)22696 50810 y Fm(4)23129 50699 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)25695 50365 y Fo(>)26556 49908 y Fk(T)27253 49136 y Fe(i)28268 50365 y Fo(;)0 52685 y Fr(where)319 b(w)-31 b(e)320 b(used)e(the)i(fact)g(that)g Fn(p)308 b Fh(6)p Fr(=)f(0)320 b(in)f(the)h(supp)31 b(ort)319 b(of)450 b(~)-684 b Fo(\037)25652 52851 y Fk(M)26635 52685 y Fr(\()p Fn(p)p Fr(\))320 b(and)f Fo(<)308 b(\016)42 b(\032)32899 52851 y Fi(p)p Fk(;)p Ff(+)34761 52685 y Fo(>)p Fr(=)307 b(0)320 b(for)f Fn(p)308 b Fh(6)p Fr(=)f(0.)476 b(A)319 b(similar)0 54102 y(decomp)31 b(osition)414 b(can)e(b)31 b(e)411 b(done)h(for)g(the)f(other)h(t)-31 b(w)g(o)414 b(terms)d(in)h(the)g(r.h.s.)620 b(of)412 b(\(4.7\);)435 b(hence,)423 b(b)-31 b(y)412 b(using)g(\(3.21\),)0 55520 y(w)-31 b(e)370 b(get)3999 58336 y Fh(\000)5356 58056 y Fr(~)5106 58336 y Fo(G)5976 57879 y Ff(4)5976 58610 y(+)6711 58336 y Fr(\()p Fn(k)7813 58502 y Ff(1)8310 58336 y Fo(;)184 b Fn(k)9473 58502 y Ff(2)9970 58336 y Fo(;)g Fn(k)11133 58502 y Ff(3)11631 58336 y Fo(;)g Fn(k)12794 58502 y Ff(4)13291 58336 y Fr(\))308 b(=)g Fo(g)15726 58502 y Fj(\000)16473 58336 y Fr(\()p Fn(k)17575 58502 y Ff(4)18072 58336 y Fr(\))19077 57587 y(1)p 18635 58081 V 18635 59095 a Fo(L\014)20390 57284 y Fe(X)20912 59609 y Fi(p)22305 58336 y Fr(~)-684 b Fo(\037)22867 58502 y Fk(M)23849 58336 y Fr(\()p Fn(p)p Fr(\))25549 57537 y Fo(H)26559 57005 y Ff(4)p Fk(;)p Ff(1)26469 57812 y(+)27761 57537 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)30061 57703 y Ff(1)30559 57537 y Fo(;)g Fn(k)31722 57703 y Ff(2)32219 57537 y Fo(;)g Fn(k)33382 57703 y Ff(3)33880 57537 y Fo(;)g Fn(k)35043 57703 y Ff(4)35786 57537 y Fh(\000)246 b Fn(p)p Fr(\))p 25549 58081 12481 45 v 30181 59095 a Fo(D)31097 59261 y Ff(+)31832 59095 y Fr(\()p Fn(p)p Fr(\))38164 58336 y(+)3999 61586 y(+)377 b(~)-684 b Fo(\037)5799 61752 y Fk(M)6781 61586 y Fr(\()p Fn(k)7883 61752 y Ff(1)8626 61586 y Fh(\000)246 b Fn(k)10405 61752 y Ff(2)10901 61586 y Fr(\))p Fo(g)11859 61752 y Fj(\000)12607 61586 y Fr(\()p Fn(k)13709 61752 y Ff(4)14206 61586 y Fr(\))p Fo(G)15506 61129 y Ff(2)15506 61860 y Fj(\000)16254 61586 y Fr(\()p Fn(k)17356 61752 y Ff(3)17853 61586 y Fr(\))18467 60357 y Fe(h)18991 61586 y Fo(<)20386 61294 y Fr(^)20159 61586 y Fo( )20920 61113 y Fj(\000)20880 61920 y Fi(k)21409 62031 y Fm(1)21841 61920 y Fk(;)p Ff(+)22839 61586 y Fr(;)23558 61294 y(^)23330 61586 y Fo( )24091 61113 y Ff(+)24051 61920 y Fi(k)24580 62031 y Fm(2)25013 61920 y Fk(;)p Ff(+)26011 61586 y Fr(;)184 b Fo(\016)42 b(\032)27608 61752 y Fi(k)28137 61863 y Fm(1)28570 61752 y Fj(\000)p Fi(k)29791 61863 y Fm(2)30223 61752 y Fk(;)p Ff(+)31528 61586 y Fo(>)32389 61129 y Fk(T)33394 61586 y Fh(\000)10160 b Fr(\(4)p Fo(:)p Fr(10\))4070 64354 y Fh(\000)p Fo(\027)5478 64520 y Ff(+)6520 64354 y Fo(<)7915 64062 y Fr(^)7689 64354 y Fo( )8450 63880 y Fj(\000)8410 64688 y Fi(k)8939 64799 y Fm(1)9371 64688 y Fk(;)p Ff(+)10369 64354 y Fr(;)11087 64062 y(^)10860 64354 y Fo( )11621 63880 y Ff(+)11581 64688 y Fi(k)12110 64799 y Fm(2)12543 64688 y Fk(;)p Ff(+)13541 64354 y Fr(;)184 b Fo(\032)14604 64520 y Fi(k)15133 64631 y Fm(1)15566 64520 y Fj(\000)p Fi(k)16787 64631 y Fm(2)17219 64520 y Fk(;)p Ff(+)18524 64354 y Fo(>)19385 63897 y Fk(T)20390 64354 y Fh(\000)p Fo(\027)21798 64520 y Fj(\000)22678 63605 y Fo(D)23594 63771 y Fj(\000)24341 63605 y Fr(\()p Fn(k)25443 63771 y Ff(1)26186 63605 y Fh(\000)246 b Fn(k)27965 63771 y Ff(2)28462 63605 y Fr(\))p 22678 64099 6215 45 v 22684 65113 a Fo(D)23600 65279 y Ff(+)24335 65113 y Fr(\()p Fn(k)25437 65279 y Ff(1)26180 65113 y Fh(\000)g Fn(k)27959 65279 y Ff(2)28456 65113 y Fr(\))29333 64354 y Fo(<)30728 64062 y Fr(^)30501 64354 y Fo( )31262 63880 y Fj(\000)31222 64688 y Fi(k)31751 64799 y Fm(1)32183 64688 y Fk(;)p Ff(+)33181 64354 y Fr(;)33900 64062 y(^)33672 64354 y Fo( )34433 63880 y Ff(+)34393 64688 y Fi(k)34922 64799 y Fm(2)35355 64688 y Fk(;)p Ff(+)36353 64354 y Fr(;)184 b Fo(\032)37416 64520 y Fi(k)37945 64631 y Fm(1)38378 64520 y Fj(\000)p Fi(k)39599 64631 y Fm(2)40031 64520 y Fk(;)p Fj(\000)41349 64354 y Fo(>)42210 63897 y Fk(T)42907 62793 y Fe(\025)443 67116 y Fr(W)-92 b(e)440 b(no)-31 b(w)442 b(put)g Fn(k)7573 67282 y Fk(i)8368 67116 y Fr(=)9716 66823 y(\026)9657 67116 y Fn(k)10329 67282 y Fk(i)10697 67116 y Fr(,)459 b(see)441 b(\(3.28\).)711 b(Since)441 b Fh(j)20497 66823 y Fr(\026)20438 67116 y Fn(k)21110 67282 y Ff(1)21901 67116 y Fh(\000)23115 66823 y Fr(\026)23056 67116 y Fn(k)23728 67282 y Ff(2)24225 67116 y Fh(j)428 b Fr(=)f(2)p Fo(\015)27435 66714 y Fk(h)28010 67116 y Fr(,)459 b(\(3.13\))444 b(implies)e(that)574 b(~)-684 b Fo(\037)39039 67282 y Fk(M)40021 67116 y Fr(\()40510 66823 y(\026)40451 67116 y Fn(k)41123 67282 y Ff(1)41914 67116 y Fh(\000)43128 66823 y Fr(\026)43069 67116 y Fn(k)43741 67282 y Ff(2)44237 67116 y Fr(\))428 b(=)f(0;)0 68533 y(hence)369 b(w)-31 b(e)370 b(get)6355 71349 y Fh(\000)7467 71070 y Fr(~)7216 71349 y Fo(G)8086 70892 y Ff(4)8086 71623 y(+)8821 71349 y Fr(\()p Fn(k)9923 71515 y Ff(1)10420 71349 y Fo(;)184 b Fn(k)11583 71515 y Ff(2)12081 71349 y Fo(;)g Fn(k)13244 71515 y Ff(3)13741 71349 y Fo(;)g Fn(k)14904 71515 y Ff(4)15402 71349 y Fr(\))308 b(=)f Fo(g)17836 71515 y Fj(\000)18583 71349 y Fr(\()p Fn(k)19685 71515 y Ff(4)20182 71349 y Fr(\))21188 70600 y(1)p 20745 71095 1438 45 v 20745 72109 a Fo(L\014)22501 70298 y Fe(X)23022 72622 y Fi(p)24415 71349 y Fr(~)-684 b Fo(\037)24977 71515 y Fk(M)25959 71349 y Fr(\()p Fn(p)p Fr(\))27659 70550 y Fo(H)28669 70018 y Ff(4)p Fk(;)p Ff(1)28579 70825 y(+)29871 70550 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)32171 70716 y Ff(1)32669 70550 y Fo(;)g Fn(k)33832 70716 y Ff(2)34330 70550 y Fo(;)g Fn(k)35493 70716 y Ff(3)35990 70550 y Fo(;)g Fn(k)37153 70716 y Ff(4)37897 70550 y Fh(\000)246 b Fn(p)p Fr(\))p 27659 71095 12481 45 v 32291 72109 a Fo(D)33207 72275 y Ff(+)33942 72109 y Fr(\()p Fn(p)p Fr(\))40582 71349 y Fo(:)3526 b Fr(\(4)p Fo(:)p Fr(11\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 443 2993 a Fp(R)-57 b(emark.)481 b Fr(\(4.11\))336 b(sa)-31 b(ys)333 b(that)i(the)e(last)h(line)g(of)g (the)f(Dyson)g(equation)i(\(3.29\))h(can)d(b)31 b(e)333 b(written)h(as)f(a)h(functional)0 4410 y(in)-31 b(tegral)450 b(v)-31 b(ery)449 b(similar)g(to)g(the)g(one)g(for)f Fo(G)18508 4008 y Ff(4)18508 4684 y(+)19691 4410 y Fr(\(w)-31 b(e)450 b(are)e(essen)-31 b(tially)450 b(pro)31 b(ceding)449 b(as)f(in)h(the)f(deriv)-61 b(ation)450 b(of)f(the)0 5827 y(Dyson)542 b(equation,)587 b(in)542 b(the)g(opp)31 b(osite)543 b(direction\))h(except)e(that)i(the)e(in)-31 b(teraction)544 b Fo(V)788 b Fr(\(1.3\))544 b(is)d(replaced)i(b)-31 b(y)0 7245 y Fo(V)386 b Fr(+)140 b Fo(T)2680 7411 y Ff(1)3316 7245 y Fh(\000)g Fo(\027)4864 7411 y Ff(+)5598 7245 y Fo(T)6245 7411 y Ff(+)7120 7245 y Fh(\000)g Fo(\027)8668 7411 y Fj(\000)9415 7245 y Fo(T)10062 7411 y Fj(\000)10809 7245 y Fr(;)334 b(w)-31 b(e)316 b(will)i(ev)-61 b(aluate)318 b(it)f(via)g(a)f(m)-31 b(ultiscale)319 b(in)-31 b(tegration)319 b(pro)31 b(cedure)315 b(similar)i(to)g(the)g(one)0 8662 y(for)386 b Fo(G)2581 8260 y Ff(4)2581 8936 y(+)3316 8662 y Fr(,)k(and)c(in)g(the)g(expansion)h(additional)i(running)d (coupling)h(constan)-31 b(ts)387 b(will)h(app)31 b(ear;)394 b(the)386 b(expansion)h(is)0 10079 y(con)-31 b(v)g(ergen)g(t)360 b(again)h(if)e(suc)-31 b(h)359 b(new)g(running)g(couplings)h(will)g (remain)g(small)g(uniformly)g(in)f(the)g(infrared)g(cuto\013.)0 12442 y Fn(4.2)1314 b Fp(The)397 b(pr)-57 b(op)g(erties)396 b(of)g Fo(D)12513 12608 y Fk(!)13154 12442 y Fr(\()p Fn(p)p Fr(\))14721 12040 y Fj(\000)p Ff(1)15911 12442 y Fo(C)16702 12608 y Fk(!)17343 12442 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))p Fp(.)443 13859 y Fr(W)-92 b(e)351 b(shall)i(use)e(some)h(prop)31 b(erties)352 b(of)h(the)f(op)31 b(erator)352 b Fo(D)23243 14025 y Fk(!)23885 13859 y Fr(\()p Fn(p)p Fr(\))25452 13457 y Fj(\000)p Ff(1)26641 13859 y Fo(C)27432 14025 y Fk(!)28073 13859 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)213 b Fh(\000)e Fn(p)p Fr(\),)357 b(whic)-31 b(h)353 b(w)-31 b(ere)352 b(pro)-31 b(v)g(ed)352 b(in)g([BM2].)0 15276 y(Let)414 b(us)f(consider)h(\014rst)g(the)g(e\013ect)g(of)g(con)-31 b(tracting)416 b(b)31 b(oth)25148 14984 y(^)24922 15276 y Fo( )453 b Fr(\014elds)414 b(of)g Fo(\016)42 b(\032)31387 15442 y Fi(p)p Fk(;)p Ff(+)33356 15276 y Fr(on)414 b(the)g(same)g(or)g (t)-31 b(w)g(o)416 b(di\013eren)-31 b(t)0 16693 y(scales;)401 b(in)390 b(the)g(second)f(case,)396 b(w)-31 b(e)390 b(also)g(supp)31 b(ose)389 b(that)i(the)g(regularization)h(pro)31 b(cedure)389 b(\(to)i(b)31 b(e)389 b(de\014ned)h(later,)0 18111 y(in)337 b(agreemen)-31 b(t)338 b(with)g(this)f(h)-31 b(yp)31 b(othesis\))338 b(do)31 b(es)337 b(not)g(act)h(on)f(the)g(propagator)h (of)g(higher)f(scale.)482 b(Hence,)344 b(w)-31 b(e)337 b(ha)-31 b(v)g(e)0 19528 y(to)370 b(study)f(the)h(quan)-31 b(tit)g(y)12270 22725 y(\001)13192 22268 y Ff(\()p Fk(i;j)45 b Ff(\))13192 22999 y Fk(!)14927 22725 y Fr(\()p Fn(k)16029 22268 y Ff(+)16764 22725 y Fo(;)184 b Fn(k)17927 22268 y Fj(\000)18675 22725 y Fr(\))308 b(=)20714 21976 y Fo(C)21505 22142 y Fk(!)22146 21976 y Fr(\()p Fn(k)23248 21574 y Ff(+)23983 21976 y Fo(;)184 b Fn(k)25146 21574 y Fj(\000)25894 21976 y Fr(\))p 20714 22470 5611 45 v 21957 23484 a Fo(D)22873 23650 y Fk(!)23514 23484 y Fr(\()p Fn(p)p Fr(\))26495 22725 y(~)-590 b Fo(g)27026 22268 y Ff(\()p Fk(i)p Ff(\))26986 22999 y Fk(!)28086 22725 y Fr(\()p Fn(k)29188 22268 y Ff(+)29923 22725 y Fr(\))38 b(~)-591 b Fo(g)30921 22268 y Ff(\()p Fk(j)45 b Ff(\))30881 22999 y Fk(!)32079 22725 y Fr(\()p Fn(k)33181 22268 y Fj(\000)33929 22725 y Fr(\))308 b Fo(;)9441 b Fr(\(4)p Fo(:)p Fr(12\))0 25994 y(where)369 b Fn(p)308 b Fr(=)f Fn(k)6056 25592 y Ff(+)7037 25994 y Fh(\000)246 b Fn(k)8816 25592 y Fj(\000)9563 25994 y Fr(.)492 b(The)370 b(crucial)g(observ)-61 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Ff(+)18822 33020 y Fo(;)184 b Fn(k)19985 32618 y Fj(\000)20733 33020 y Fr(\))418 b(=)f(\001)23781 32445 y Ff(\()p Fk(j)m(;i)p Ff(\))23781 33149 y Fk(!)25468 33020 y Fr(\()p Fn(k)26570 32618 y Fj(\000)27317 33020 y Fo(;)184 b Fn(k)28480 32618 y Ff(+)29216 33020 y Fr(\),)452 b(w)-31 b(e)436 b(can)f(restrict)g(the)g(analysis)h(to)g(the)0 34437 y(case)369 b Fo(i)308 b Fh(\025)f Fo(j)63 b Fr(.)493 b(W)-92 b(e)368 b(de\014ne)5801 37609 y Fo(u)6435 37775 y Ff(0)6932 37609 y Fr(\()p Fn(k)p Fr(\))308 b(=)9940 36048 y Fe(\032)10955 36945 y Fr(0)5032 b(if)370 b Fh(j)p Fn(k)p Fh(j)308 b(\024)f Fr(1)10955 38273 y(1)247 b Fh(\000)f Fo(f)13404 38439 y Ff(0)13900 38273 y Fr(\()p Fn(k)p Fr(\))1108 b(if)370 b(1)308 b Fh(\024)f(j)p Fn(k)p Fh(j)22664 37609 y Fo(;)1292 b(u)24897 37775 y Fk(h)25470 37609 y Fr(\()p Fn(k)p Fr(\))309 b(=)28479 36048 y Fe(\032)29494 36999 y Fr(0)5109 b(if)369 b Fh(j)p Fn(k)p Fh(j)309 b(\025)e Fo(\015)39567 36598 y Fk(h)29494 38328 y Fr(1)246 b Fh(\000)g Fo(f)31942 38494 y Fk(h)32516 38328 y Fr(\()p Fn(k)p Fr(\))1108 b(if)369 b Fh(j)p Fn(k)p Fh(j)309 b(\024)e Fo(\015)39567 37926 y Fk(h)41135 37609 y Fo(:)2973 b Fr(\(4)p Fo(:)p Fr(14\))443 41043 y(Then)369 b(w)-31 b(e)370 b(get,)g(b)-31 b(y)370 b(using)f(\(2.19\),)k(the)c(fact)h(that) h Fo(Z)22299 41209 y Fj(\000)p Ff(1)23794 41043 y Fr(=)308 b Fo(Z)25719 41209 y Ff(0)26522 41043 y Fr(=)g(1)369 b(and)31004 40751 y(~)30766 41043 y Fo(f)31308 41209 y Fk(j)32082 41043 y Fr(=)307 b Fo(f)33792 41209 y Fk(j)34627 41043 y Fr(for)370 b Fo(j)g Fr(=)308 b(0)p Fo(;)184 b(h)p Fr(,)8438 44341 y(\001)9360 43884 y Ff(\(0)p Fk(;)p Ff(0\))9360 44615 y Fk(!)11253 44341 y Fr(\()p Fn(k)12355 43884 y Ff(+)13090 44341 y Fo(;)g Fn(k)14253 43884 y Fj(\000)15001 44341 y Fr(\))308 b(=)18327 43593 y(1)p 17041 44087 3126 45 v 17041 45101 a Fo(D)17957 45267 y Fk(!)18598 45101 y Fr(\()p Fn(p)p Fr(\))20483 42781 y Fe(\024)21460 43593 y Fo(f)22002 43759 y Ff(0)22498 43593 y Fr(\()p Fn(k)23600 43191 y Ff(+)24336 43593 y Fr(\))p 21200 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TeXDict begin 24 23 bop 443 1107 a Fr(Finally)-92 b(,)371 b(it)f(is)f(easy)g(to)h(see)f(that,)i(if)e(0)308 b Fo(>)g(i)g Fh(\025)f Fo(h)p Fr(,)14979 3747 y Fh(j)p Fr(\001)16208 3290 y Ff(\()p Fk(i;h)p Ff(\))16208 4021 y Fk(!)18050 3747 y Fr(\()p Fn(k)19152 3290 y Ff(+)19887 3747 y Fo(;)184 b Fn(k)21050 3290 y Fj(\000)21798 3747 y Fr(\))p Fh(j)309 b(\024)e Fo(C)79 b(\015)25516 3290 y Fj(\000)p Ff(\()p Fk(i)p Fj(\000)p Fk(h)p Ff(\))28613 2998 y Fo(\015)29247 2596 y Fj(\000)p Fk(h)p Fj(\000)p Fk(i)p 28613 3492 2906 45 v 28936 4506 a Fo(Z)29692 4672 y Fk(i)p Fj(\000)p Ff(1)31958 3747 y Fo(:)12150 b Fr(\(4)p Fo(:)p Fr(22\))0 6473 y(Note)437 b(that,)455 b(in)437 b(the)g(r.h.s.)694 b(of)437 b(\(4.22\),)456 b(there)436 b(is)h(apparen)-31 b(tly)438 b(a)e Fo(Z)28392 6000 y Fj(\000)p Ff(1)28313 6807 y Fk(h)p Fj(\000)p Ff(1)30456 6473 y Fr(factor)h(missing,)455 b(but)437 b(the)f(b)31 b(ound)437 b(can)0 8029 y(not)522 b(b)31 b(e)520 b(impro)-31 b(v)g(ed;)598 b(this)522 b(is)e(a)h (consequence)g(of)g(the)h(fact)f(that)28780 7750 y(~)28547 8029 y Fo(Z)29303 8195 y Fk(h)p Fj(\000)p Ff(1)31010 8029 y Fr(\()p Fn(k)p Fr(\))561 b(=)f(0)521 b(for)g Fh(j)p Fn(k)p Fh(j)561 b(\024)f Fo(\015)41346 7628 y Fk(h)p Fj(\000)p Ff(1)43053 8029 y Fr(,)f(see)520 b(eq.)0 9447 y(\(63\))404 b(of)f([BM2],)411 b(and)403 b(the)f(supp)31 b(ort)402 b(prop)31 b(erties)402 b(of)g Fo(u)22774 9613 y Fk(h)23348 9447 y Fr(\()p Fn(k)p Fr(\).)592 b(In)402 b(an)-31 b(y)402 b(case,)411 b(this)403 b(is)f(not)h(a)f(problem,)412 b(since)402 b(the)0 10864 y(dimensional)360 b(dep)31 b(endence)358 b(of)g(\001)14178 10462 y Ff(\()p Fk(i;h)p Ff(\))16378 10864 y Fr(on)h(the)f(\014eld)g(renormalization)k(constan) -31 b(ts)359 b(is)f(exactly)i Fo(Z)40745 10462 y Fj(\000)p Ff(1)41933 10864 y Fr(.)489 b(Note)359 b(also)0 12281 y(the)411 b(presence)e(in)i(the)f(b)31 b(ound)411 b(of)g(the)f(extra)h (factor)h Fo(\015)23160 11880 y Fj(\000)p Ff(\()p Fk(i)p Fj(\000)p Fk(h)p Ff(\))26123 12281 y Fr(,)421 b(with)411 b(resp)31 b(ect)410 b(to)h(the)g(dimensional)i(b)31 b(ound;)431 b(it)0 13699 y(will)371 b(allo)-31 b(w)372 b(us)c(to)i(a)-31 b(v)g(oid)371 b(renormalization)i(of)c(the)h(marginal)h(terms)e(con)-31 b(taining)372 b(\001)35754 13297 y Ff(\()p Fk(i;h)p Ff(\))37596 13699 y Fr(.)0 16061 y Fn(4.3)1314 b Fp(The)397 b(multisc)-57 b(ale)396 b(exp)-57 b(ansion)396 b(of)17000 15781 y Fr(~)16750 16061 y Fo(G)17620 15659 y Ff(4)17620 16335 y(+)18355 16061 y Fr(\()18844 15769 y(\026)18785 16061 y Fn(k)19457 16227 y Ff(1)19954 16061 y Fo(;)20505 15769 y Fr(\026)20445 16061 y Fn(k)21117 16227 y Ff(2)21614 16061 y Fo(;)22166 15769 y Fr(\026)22105 16061 y Fn(k)22777 16227 y Ff(3)23275 16061 y Fo(;)23826 15769 y Fr(\026)23766 16061 y Fn(k)24438 16227 y Ff(4)24935 16061 y Fr(\))p Fp(:)510 b(the)396 b(\014rst)g(inte)-57 b(gr)g(ation)397 b(step.)443 17478 y Fr(The)364 b(calculation)k(of)9811 17198 y(~)9560 17478 y Fo(G)10430 17077 y Ff(4)10430 17752 y(+)11530 17478 y Fr(is)c(done)g(via)i(a)e(m)-31 b(ultiscale)367 b(expansion;)g(w)-31 b(e)365 b(shall)g(concen)-31 b(trate)366 b(on)e(the)h(di\013erences)0 18896 y(with)395 b(resp)31 b(ect)392 b(to)i(that)h(describ)31 b(ed)392 b(in)i Fh(x)p Fr(2,)400 b(due)393 b(to)h(the)f(presence)g(in)g (the)h(p)31 b(oten)-31 b(tial)396 b(of)e(the)f(terms)g Fo(T)42949 19062 y Ff(1)43446 18896 y Fr(\()p Fo( )40 b Fr(\))394 b(and)0 20313 y Fo(T)647 20479 y Fj(\006)1394 20313 y Fr(\()p Fo( )40 b Fr(\).)493 b(Moreo)-31 b(v)g(er)370 b(w)-31 b(e)369 b(shall)i(supp)31 b(ose)368 b(that)i(the)g(momen)-31 b(ta)371 b Fn(k)26866 20479 y Fk(i)27604 20313 y Fr(are)e(put)g(equal)h (to)35814 20021 y(\026)35755 20313 y Fn(k)36427 20479 y Fk(i)36796 20313 y Fr(,)f(de\014ned)g(as)g(in)h(\(3.28\).)443 21730 y(Let)448 b(us)f(consider)h(the)g(\014rst)f(step)h(of)h(the)f (iterativ)-31 b(e)450 b(in)-31 b(tegration)451 b(pro)31 b(cedure,)467 b(the)448 b(in)-31 b(tegration)451 b(of)d(the)h(\014eld)0 23147 y Fo( )761 22746 y Ff(\(0\))1949 23147 y Fr(;)514 b(w)-31 b(e)466 b(shall)g(describ)31 b(e)464 b(only)i(the)g(terms)f (linear)h(in)g(the)f(external)h Fo(J)572 b Fr(\014eld,)490 b(the)465 b(only)h(ones)f(con)-31 b(tributing)0 24565 y(to)1644 24285 y(~)1393 24565 y Fo(G)2263 24163 y Ff(4)2263 24838 y(+)3408 24565 y Fr(whic)g(h)411 b(w)-31 b(ere)410 b(not)g(already)h(discussed.)613 b(W)-92 b(e)409 b(call)24692 24285 y(\026)24553 24565 y Fh(V)25322 24163 y Ff(\()p Fj(\000)p Ff(1\))27202 24565 y Fr(\()p Fo( )28393 24163 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])30865 24565 y Fr(\))h(the)f(con)-31 b(tribution)413 b(to)d(the)g(e\013ectiv)-31 b(e)0 25982 y(p)31 b(oten)-31 b(tial)372 b(of)e(suc)-31 b(h)368 b(terms)h(and)h(w) -31 b(e)370 b(write)2416 27745 y(\026)2277 28025 y Fh(V)3046 27568 y Ff(\()p Fj(\000)p Ff(1\))4927 28025 y Fr(\()p Fo( )6118 27568 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])8589 28025 y Fr(\))308 b(=)10634 27745 y(\026)10495 28025 y Fh(V)11264 27450 y Ff(\()p Fj(\000)p Ff(1\))11173 28320 y Fk(a;)p Ff(1)13145 28025 y Fr(\()p Fo( )14336 27568 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])16807 28025 y Fr(\))247 b(+)18729 27745 y(\026)18591 28025 y Fh(V)19360 27450 y Ff(\()p Fj(\000)p Ff(1\))19269 28320 y Fk(a;)p Ff(2)21240 28025 y Fr(\()p Fo( )22431 27568 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])24902 28025 y Fr(\))g(+)26824 27745 y(\026)26686 28025 y Fh(V)27455 27450 y Ff(\()p Fj(\000)p Ff(1\))27364 28359 y Fk(b;)p Ff(1)29335 28025 y Fr(\()p Fo( )30526 27568 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])32997 28025 y Fr(\))g(+)34919 27745 y(\026)34781 28025 y Fh(V)35550 27450 y Ff(\()p Fj(\000)p Ff(1\))35459 28359 y Fk(b;)p Ff(2)37430 28025 y Fr(\()p Fo( )38621 27568 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])41093 28025 y Fr(\))307 b Fo(;)2278 b Fr(\(4)p Fo(:)p Fr(23\))0 30402 y(where)3312 30122 y(\026)3173 30402 y Fh(V)3942 29827 y Ff(\()p Fj(\000)p Ff(1\))3851 30697 y Fk(a;)p Ff(1)6014 30402 y Fr(+)7204 30122 y(\026)7066 30402 y Fh(V)7835 29827 y Ff(\()p Fj(\000)p Ff(1\))7744 30697 y Fk(a;)p Ff(2)10057 30402 y Fr(is)341 b(the)h(sum)f(of)i(the)e(terms)h(in)g(whic)-31 b(h)342 b(the)g(\014eld)30385 30109 y(^)30158 30402 y Fo( )30919 29928 y Ff(+)30923 30642 y(\026)30879 30847 y Fi(k)31408 30958 y Fm(4)31840 30847 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)34440 30402 y Fr(app)31 b(earing)343 b(in)e(the)h(de\014nition)0 32264 y(of)448 b Fo(T)1986 32430 y Ff(1)2482 32264 y Fr(\()p Fo( )40 b Fr(\))448 b(or)g Fo(T)6633 32430 y Fj(\006)7380 32264 y Fr(\()p Fo( )40 b Fr(\))448 b(is)f(con)-31 b(tracted,)16630 31985 y(\026)16491 32264 y Fh(V)17260 31689 y Ff(\()p Fj(\000)p Ff(1\))17169 32560 y Fk(a;)p Ff(1)19588 32264 y Fr(and)21957 31985 y(\026)21819 32264 y Fh(V)22588 31689 y Ff(\()p Fj(\000)p Ff(1\))22497 32560 y Fk(a;)p Ff(2)24915 32264 y Fr(denoting)450 b(the)d(sum)h(o)-31 b(v)g(er)448 b(the)f(terms)h(of)g(this)f(t)-31 b(yp)31 b(e)0 33977 y(con)-31 b(taining)393 b(a)d Fo(T)6992 34143 y Ff(1)7878 33977 y Fr(or)g(a)h Fo(T)10846 34143 y Fj(\006)11982 33977 y Fr(v)-31 b(ertex,)397 b(resp)31 b(ectiv)-31 b(ely;)22184 33697 y(\026)22045 33977 y Fh(V)22814 33402 y Ff(\()p Fj(\000)p Ff(1\))22723 34311 y Fk(b;)p Ff(1)24955 33977 y Fr(+)26214 33697 y(\026)26076 33977 y Fh(V)26845 33402 y Ff(\()p Fj(\000)p Ff(1\))26754 34311 y Fk(b;)p Ff(2)29115 33977 y Fr(is)390 b(the)g(sum)g(of)h(the)f(other)g(terms,)396 b(that)391 b(is)0 35633 y(those)396 b(where)g(the)g(\014eld)10735 35341 y(^)10508 35633 y Fo( )11269 35160 y Ff(+)11273 35874 y(\026)11229 36078 y Fi(k)11758 36189 y Fm(4)12191 36078 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)14845 35633 y Fr(is)f(an)h(external)h(\014eld,)403 b(the)396 b(index)g Fo(i)352 b Fr(=)g(1)p Fo(;)184 b Fr(2)397 b(ha)-31 b(ving)398 b(the)e(same)g(meaning)h(as)0 37050 y(b)31 b(efore.)443 38468 y(Note)317 b(that)g(the)f(condition)i(\(3.28\))g(on)e(the)g (external)h(momen)-31 b(ta)318 b Fn(k)28219 38634 y Fk(i)28903 38468 y Fr(forbids)e(the)g(presence)f(of)h(v)-31 b(ertices)316 b(of)g(t)-31 b(yp)31 b(e)0 39885 y Fo(')p Fr(,)340 b(if)332 b Fo(h)307 b(<)h Fr(0,)340 b(as)331 b(w)-31 b(e)332 b(shall)h(supp)31 b(ose.)479 b(Hence,)340 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55138 y Fo(J)28109 55202 y Ff(\026)28065 55406 y Fi(k)28594 55517 y Fm(4)29267 53631 y Fe(Z)30558 55138 y Fo(d)p Fn(k)31939 54389 y Fo(C)79 b Fr(\()p Fn(k)p Fo(;)184 b Fn(p)p Fr(\))p 31939 54883 3603 45 v 32130 55897 a Fo(D)33046 56063 y Ff(+)33782 55897 y Fr(\()p Fn(p)p Fr(\))35901 54846 y(^)35674 55138 y Fo( )36435 54664 y Ff(+)36395 55472 y Fi(k)p Fk(;)p Ff(+)38149 54846 y Fr(^)37922 55138 y Fo( )38683 54664 y Fj(\000)38643 55472 y Fi(k)p Ff(+)p Fi(p)p Fk(;)p Ff(+)41713 55138 y Fo(;)2395 b Fr(\(4)p Fo(:)p Fr(24\))0 58066 y(where)288 b Fo(G)3990 57491 y Ff(\(0\))3990 58361 y(2)5179 58066 y Fr(\()p Fn(k)p Fr(\))h(is)g(a)g(suitable)h(function)g(whic)-31 b(h)289 b(can)g(b)31 b(e)288 b(expressed)g(as)g(a)h(sum)g(of)g(graphs)g (with)g(an)g(o)31 b(dd)289 b(n)-31 b(um)g(b)31 b(er)0 59628 y(of)315 b(propagators,)327 b(hence)313 b(it)i(v)-61 b(anishes)314 b(at)h Fn(k)307 b Fr(=)h(0.)474 b(This)315 b(implies)g(that)h Fo(G)30103 59053 y Ff(\(0\))30103 59923 y(2)31291 59628 y Fr(\(0\))309 b(=)f(0,)326 b(so)314 b(that)h(w)-31 b(e)315 b(can)f(regularize)0 61045 y(it)370 b(without)h(in)-31 b(tro)31 b(ducing)371 b(an)-31 b(y)370 b(running)g(coupling.)16980 64741 y Fo(G)17850 64166 y Ff(\(0\))17850 65037 y(2)7018 72441 y @beginspecial @setspecial %%BeginDocument: G20.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 150 30 5 cerchio 150 35 150 70 3 5 onda 170 10 150 25 freccia 150 25 130 10 freccia 150 70 180 80 tlinea 150 70 115 70 freccia 100 70 15 cerchio 85 70 60 70 freccia grestore %%EndDocument @endspecial 0 75098 a Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 13256 1107 a Fr(Fig.)493 b(13:)g(Graphical)370 b(represen)-31 b(tation)371 b(of)e(\(4.24\))443 3469 y(If)g(b)31 b(oth)4390 3177 y(^)4163 3469 y Fo( )4924 2996 y Ff(+)4884 3803 y Fi(k)p Fk(;)p Ff(+)6780 3469 y Fr(and)9159 3177 y(^)8933 3469 y Fo( )9694 2996 y Fj(\000)9654 3803 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)12798 3469 y Fr(in)369 b Fo(T)14736 3635 y Ff(1)15233 3469 y Fr(\()p Fo( )40 b Fr(\))370 b(are)f(con)-31 b(tracted,)371 b(w)-31 b(e)370 b(get)f(terms)h(of)f(the)h(form)14619 6122 y(~)14296 6402 y Fo(W)15495 5827 y Ff(\()p Fj(\000)p Ff(1\))15341 6697 y Fk(n)p Ff(+1)17376 6402 y Fr(\()17865 6110 y(\026)17806 6402 y Fn(k)18478 6568 y Ff(4)18975 6402 y Fo(;)184 b Fn(k)20138 6568 y Ff(1)20635 6402 y Fo(;)g(::;)g Fn(k)22903 6568 y Fk(n)23508 6402 y Fr(\))38 b(^)-591 b Fo(g)24466 6568 y Fj(\000)25214 6402 y Fr(\()25703 6110 y(\026)25644 6402 y Fn(k)26316 6568 y Ff(4)26813 6402 y Fr(\))27511 6122 y(^)27243 6402 y Fo(J)27901 6466 y Ff(\026)27857 6670 y Fi(k)28386 6781 y Fm(4)29502 5018 y Fk(n)29068 5350 y Fe(Y)29058 7708 y Fk(i)p Ff(=1)30903 6110 y Fr(^)30677 6402 y Fo( )31438 5910 y Fk(")31856 6021 y Fl(i)31398 6736 y Fi(k)31927 6847 y Fl(i)32640 6402 y Fo(;)11468 b Fr(\(4)p Fo(:)p Fr(25\))0 9207 y(where)365 b Fo(n)g Fr(is)g(an)h(o)31 b(dd)365 b(in)-31 b(teger.)492 b(W)-92 b(e)365 b(w)-31 b(an)g(t)367 b(to)f(de\014ne)f(an)g Fh(R)g Fr(op)31 b(eration)367 b(for)e(suc)-31 b(h)365 b(terms.)491 b(There)365 b(is)g(apparen)-31 b(tly)0 10624 y(a)448 b(problem,)469 b(as)448 b(the)g Fh(R)g Fr(op)31 b(eration)449 b(in)-31 b(v)g(olv)g(es)450 b(deriv)-61 b(ativ)-31 b(es)449 b(and)f(in)29308 10344 y(~)28985 10624 y Fo(W)30184 10223 y Ff(\()p Fj(\000)p Ff(1\))32513 10624 y Fr(app)31 b(ears)447 b(the)h(function)i(\001)44012 10223 y Ff(\(0)p Fk(;)p Ff(0\))46352 10624 y Fr(of)0 12042 y(the)534 b(form)f(\(4.20\))j(and)e (the)g(cuto\013)f(function)666 b(~)-684 b Fo(\037)21063 12208 y Fk(M)22045 12042 y Fr(\()p Fn(p)p Fr(\),)576 b(with)535 b(supp)31 b(ort)533 b(on)g(momen)-31 b(ta)536 b(of)e(size)f Fo(\015)42457 11640 y Fk(h)43031 12042 y Fr(.)985 b(Hence)0 13459 y(one)551 b(can)f(w)-31 b(orry)551 b(ab)31 b(out)551 b(the)g(deriv)-61 b(ativ)-31 b(es)551 b(of)g(the)f(factor)683 b(~)-684 b Fo(\037)26449 13625 y Fk(M)27431 13459 y Fr(\()p Fn(p)p Fr(\))p Fn(p)p Fo(D)30621 13625 y Ff(+)31358 13459 y Fr(\()p Fn(p)p Fr(\))32925 13057 y Fj(\000)p Ff(1)34114 13459 y Fr(.)1036 b(Ho)-31 b(w)g(ev)g(er,)598 b(as)550 b(the)g(line)h(of)0 14876 y(momen)-31 b(tum)5932 14584 y(\026)5873 14876 y Fn(k)6545 15042 y Ff(4)7328 14876 y Fh(\000)287 b Fn(p)430 b Fr(is)h(necessarily) f(at)i(scale)f(0)f(\(w)-31 b(e)432 b(are)e(considering)h(terms)g(in)f (whic)-31 b(h)432 b(it)f(is)f(con)-31 b(tracted\),)0 16294 y(then)410 b Fh(j)p Fn(p)p Fh(j)375 b(\025)g Fo(\015)6128 15892 y Fj(\000)p Ff(1)7590 16294 y Fh(\000)273 b Fo(\015)9358 15892 y Fk(h)10307 16294 y Fh(\025)374 b Fo(\015)12176 15892 y Fj(\000)p Ff(1)13365 16294 y Fo(=)p Fr(2)410 b(\(for)g Fh(j)p Fo(h)p Fh(j)g Fr(large)h(enough\),)421 b(so)409 b(that)i(w)-31 b(e)410 b(can)g(freely)f(m)-31 b(ultiply)412 b(b)-31 b(y)410 b(a)g(smo)31 b(oth)0 17711 y(cuto\013)366 b(function)498 b(\026)-684 b Fo(\037)o Fr(\()p Fn(p)p Fr(\))367 b(restricting)f Fn(p)g Fr(to)g(the)g(allo)-31 b(w)g(ed)368 b(region;)g(this)e(allo)-31 b(ws)368 b(us)d(to)h(pass)f (to)h(co)31 b(ordinate)367 b(space)0 19128 y(and)i(sho)-31 b(ws)370 b(that)g(the)g Fh(R)f Fr(op)31 b(eration)371 b(can)e(b)31 b(e)369 b(de\014ned)g(in)g(the)g(usual)h(w)-31 b(a)g(y)-92 b(.)494 b(W)-92 b(e)368 b(de\014ne)13827 21055 y Fh(L)14913 20775 y Fr(~)14590 21055 y Fo(W)15789 20480 y Ff(\()p Fj(\000)p Ff(1\))15635 21350 y(4)17670 21055 y Fr(\()18159 20763 y(\026)18100 21055 y Fn(k)18772 21221 y Ff(4)19268 21055 y Fo(;)184 b Fn(k)20431 21221 y Ff(1)20929 21055 y Fo(;)g Fn(k)22092 21221 y Ff(2)22589 21055 y Fo(;)g Fn(k)23752 21221 y Ff(3)24250 21055 y Fr(\))308 b(=)26479 20775 y(~)26156 21055 y Fo(W)27355 20480 y Ff(\()p Fj(\000)p Ff(1\))27201 21350 y(4)29236 21055 y Fr(\(0)p Fo(;)184 b(::;)g Fr(0\))312 b Fo(;)10998 b Fr(\(4)p Fo(:)p Fr(26\))13300 23126 y Fh(L)14386 22846 y Fr(~)14063 23126 y Fo(W)15262 22551 y Ff(\()p Fj(\000)p Ff(1\))15108 23421 y(2)17143 23126 y Fr(\()17632 22834 y(\026)17573 23126 y Fn(k)18245 23292 y Ff(4)18742 23126 y Fr(\))308 b(=)20971 22846 y(~)20648 23126 y Fo(W)21847 22551 y Ff(\()p Fj(\000)p Ff(1\))21693 23421 y(2)23728 23126 y Fr(\(0\))247 b(+)26554 22834 y(\026)26495 23126 y Fn(k)27167 23292 y Ff(4)27663 23126 y Fo(@)28251 23292 y Fi(k)29158 22846 y Fr(~)28835 23126 y Fo(W)30034 22551 y Ff(\()p Fj(\000)p Ff(1\))29880 23421 y(2)31915 23126 y Fr(\(0\))309 b Fo(:)10471 b Fr(\(4)p Fo(:)p Fr(27\))0 24954 y(Note)394 b(that)h(b)-31 b(y)394 b(parit)-31 b(y)395 b(the)f(\014rst)f(term)g(in)h(\(4.27\))i(is)e(v)-61 b(anishing;)407 b(this)394 b(means)g(that)h(there)e(are)g(only)i(marginal)0 26371 y(terms.)542 b(Note)386 b(also)h(that)g(the)e(lo)31 b(cal)387 b(term)f(prop)31 b(ortional)388 b(to)26148 26091 y(^)25880 26371 y Fo(J)26538 26435 y Ff(\026)26494 26639 y Fi(k)27023 26750 y Fm(4)27737 26079 y Fr(^)27510 26371 y Fo( )28271 25898 y Ff(+)28275 26612 y(\026)28231 26817 y Fi(k)28760 26928 y Fm(4)29192 26817 y Fk(;)p Fj(\000)30588 26371 y Fr(is)d(suc)-31 b(h)386 b(that)g(the)g(\014eld) 41210 26079 y(^)40983 26371 y Fo( )41744 25898 y Ff(+)41748 26612 y(\026)41704 26817 y Fi(k)42233 26928 y Fm(4)42665 26817 y Fk(;)p Fj(\000)44061 26371 y Fr(can)f(b)31 b(e)0 27789 y(con)-31 b(tracted)384 b(only)g(at)g(the)f(last)h(scale)g Fo(h)p Fr(;)390 b(hence)382 b(it)i(has)f(an)-31 b(y)384 b(in\015uence)f(on)g(the)g(in)-31 b(tegrations)386 b(of)d(all)i(the)e (scales)0 29206 y Fo(>)307 b(h)p Fr(.)12322 39645 y(~)11999 39925 y Fo(W)13198 39350 y Ff(\()p Fj(\000)p Ff(1\))13044 40220 y(4)31140 39645 y Fr(~)30817 39925 y Fo(W)32016 39350 y Ff(\()p Fj(\000)p Ff(1\))31862 40220 y(2)7018 40430 y @beginspecial @setspecial %%BeginDocument: W4W2.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 100 30 5 cerchio 100 35 100 70 3 5 onda 76 40 100 25 freccia 100 25 56 25 freccia 100 70 130 80 tlinea 100 70 76 70 freccia 50 55 30 cerchio 25 72 0 80 freccia 0 55 20 55 freccia 26 36 0 30 freccia 270 30 5 cerchio 270 35 270 70 3 5 onda 270 25 227 25 freccia 247 40 270 25 freccia 270 70 246 70 freccia 270 70 300 80 tlinea 220 55 30 cerchio 190 55 160 55 freccia grestore %%EndDocument @endspecial 10330 41847 a Fr(Fig.)493 b(14:)h(Graphical)370 b(represen)-31 b(tation)370 b(of)28556 41567 y(~)28234 41847 y Fo(W)29433 41272 y Ff(\()p Fj(\000)p Ff(1\))29279 42142 y(4)31682 41847 y Fr(and)34157 41567 y(~)33835 41847 y Fo(W)35034 41272 y Ff(\()p Fj(\000)p Ff(1\))34880 42142 y(2)443 45154 y Fr(If)332 b(only)h(one)f(among)i(the)e(\014elds) 14321 44862 y(^)14094 45154 y Fo( )14855 44680 y Ff(+)14815 45488 y Fi(k)p Fk(;)p Ff(+)16674 45154 y Fr(and)19016 44862 y(^)18789 45154 y Fo( )19550 44680 y Fj(\000)19510 45488 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Ff(+)22617 45154 y Fr(in)g Fo(T)24518 45320 y Ff(1)25015 45154 y Fr(\()p Fo( )40 b Fr(\))333 b(is)f(con)-31 b(tracted,)341 b(w)-31 b(e)333 b(note)f(\014rst)g(that)h(w)-31 b(e)333 b(cannot)0 46798 y(ha)-31 b(v)g(e)382 b(terms)f(with)h(t)-31 b(w)g(o)382 b(external)g(lines)f(\(including)22630 46518 y(^)22362 46798 y Fo(J)22976 46964 y Fi(k)23505 47075 y Fm(4)23993 46798 y Fr(\);)387 b(in)381 b(fact)h(in)f(suc)-31 b(h)381 b(a)g(case)g(there)f(is)h(an)g(external)h(line)0 48215 y(with)454 b(momen)-31 b(tum)8559 47923 y(\026)8500 48215 y Fn(k)9172 48381 y Ff(4)10121 48215 y Fr(with)454 b Fo(!)486 b Fr(=)446 b Fh(\000)453 b Fr(and)g(the)f(other)h(has)g Fo(!)486 b Fr(=)447 b(+;)494 b(this)453 b(is)g(ho)-31 b(w)g(ev)g(er)454 b(forbidden)f(b)-31 b(y)453 b(global)0 49632 y(gauge)432 b(in)-31 b(v)-61 b(ariance.)677 b(Moreo)-31 b(v)g(er,)447 b(for)430 b(the)h(same)g(reasons)f(as)g(b)31 b(efore,)446 b(w)-31 b(e)431 b(do)g(not)g(ha)-31 b(v)g(e)431 b(to)g(w)-31 b(orry)431 b(ab)31 b(out)432 b(the)0 51050 y(deriv)-61 b(ativ)-31 b(es)386 b(of)g(the)g(factor)517 b(~)-684 b Fo(\037)12608 51216 y Fk(M)13590 51050 y Fr(\()p Fn(p)p Fr(\))p Fn(p)p Fo(D)16780 51216 y Ff(+)17517 51050 y Fr(\()p Fn(p)p Fr(\))19084 50648 y Fj(\000)p Ff(1)20274 51050 y Fr(,)389 b(related)d(with)h(the)e(regularization)j(pro)31 b(cedure)385 b(of)g(the)h(terms)0 52467 y(with)370 b(four)g(external)g (lines,)g(whic)-31 b(h)370 b(ha)-31 b(v)g(e)370 b(the)g(form)3976 53302 y Fe(Z)5267 54809 y Fo(d)p Fn(k)6515 54352 y Ff(+)7477 54517 y Fr(^)7250 54809 y Fo( )8011 54335 y Ff(+)7971 55143 y Fi(k)8500 55254 y Fm(1)8932 55143 y Fk(;)p Ff(+)10157 54517 y Fr(^)9930 54809 y Fo( )10691 54335 y Fj(\000)10651 55195 y Fi(k)11180 54974 y Fc(\000)11835 55195 y Fk(;)p Ff(+)13060 54517 y Fr(^)12833 54809 y Fo( )13594 54335 y Ff(+)13554 55254 y Fi(k)14083 55033 y Fc(\000)14738 55254 y Ff(+)15462 55050 y(\026)15418 55254 y Fi(k)15947 55365 y Fm(4)16379 55254 y Fj(\000)p Fi(k)17600 55365 y Fm(1)18032 55254 y Fk(;)p Fj(\000)19080 54809 y Fr(^)-591 b Fo(g)19570 54975 y Fj(\000)20317 54809 y Fr(\()20806 54517 y(\026)20747 54809 y Fn(k)21419 54975 y Ff(4)21916 54809 y Fr(\))22614 54529 y(^)22346 54809 y Fo(J)23004 54873 y Ff(\026)22960 55077 y Fi(k)23489 55188 y Fm(4)24108 54809 y Fr(~)-684 b Fo(\037)24670 54975 y Fk(M)25652 54809 y Fr(\()p Fn(k)26754 54352 y Ff(+)27735 54809 y Fh(\000)246 b Fn(k)29514 54352 y Fj(\000)30261 54809 y Fr(\))38 b(^)-591 b Fo(g)31259 54234 y Ff(\(0\))31219 55084 y Fj(\000)32448 54809 y Fr(\()32937 54517 y(\026)32878 54809 y Fn(k)33550 54975 y Ff(4)34293 54809 y Fh(\000)246 b Fn(k)36072 54352 y Ff(+)37052 54809 y Fr(+)g Fn(k)38831 54352 y Fj(\000)39578 54809 y Fr(\))308 b Fh(\001)4037 58160 y(\001)554 b Fo(G)5768 57585 y Ff(\(0\))5768 58456 y(4)6957 58160 y Fr(\()p Fn(k)8059 57703 y Ff(+)8794 58160 y Fo(;)9346 57868 y Fr(\026)9285 58160 y Fn(k)9957 58326 y Ff(4)10455 58160 y Fo(;)184 b Fn(k)11618 58326 y Ff(1)12115 58160 y Fr(\))12729 56268 y Fe(\()13755 57361 y Fr([)p Fo(C)14853 57527 y Fk(h;)p Ff(0)16131 57361 y Fr(\()p Fn(k)17233 56959 y Fj(\000)17981 57361 y Fr(\))246 b Fh(\000)g Fr(1])p Fo(D)21540 57527 y Ff(+)22277 57361 y Fr(\()p Fn(k)23379 56959 y Fj(\000)24126 57361 y Fr(\))38 b(^)-591 b Fo(g)25124 56786 y Ff(\(0\))25084 57636 y(+)26313 57361 y Fr(\()p Fn(k)27415 56959 y Ff(+)28150 57361 y Fr(\))p 13755 57906 14826 45 v 17822 58920 a Fo(D)18738 59086 y Ff(+)19473 58920 y Fr(\()p Fn(k)20575 58600 y Ff(+)21557 58920 y Fh(\000)246 b Fn(k)23336 58600 y Fj(\000)24083 58920 y Fr(\))28959 58160 y Fh(\000)31846 57412 y Fo(u)32480 57578 y Ff(0)32976 57412 y Fr(\()p Fn(k)34078 57010 y Ff(+)34813 57412 y Fr(\))p 30199 57906 6692 45 v 30199 58920 a Fo(D)31115 59086 y Ff(+)31850 58920 y Fr(\()p Fn(k)32952 58600 y Ff(+)33934 58920 y Fh(\000)g Fn(k)35713 58600 y Fj(\000)36460 58920 y Fr(\))37023 56268 y Fe(\))38407 58160 y Fo(;)44415 56700 y Fr(\(4)p Fo(:)p Fr(28\))0 61045 y(or)369 b(the)g(similar)i(one)e(with)i(the)e (roles)g(of)h Fn(k)17882 60644 y Ff(+)18986 61045 y Fr(and)f Fn(k)21810 60644 y Fj(\000)22926 61045 y Fr(exc)-31 b(hanged.)23069 66955 y Fo(G)23939 66380 y Ff(\(0\))23939 67251 y(4)7018 72441 y @beginspecial @setspecial %%BeginDocument: G40.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 200 30 5 cerchio 200 35 200 70 3 5 onda 176 40 200 25 freccia 200 25 230 25 freccia 200 70 230 80 tlinea 200 70 176 70 freccia 150 55 30 cerchio 100 80 125 72 freccia 126 36 100 30 freccia grestore %%EndDocument @endspecial 0 75098 a Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 8671 1107 a Fr(Fig.)493 b(15:)h(Graphical)370 b(represen)-31 b(tation)370 b(of)g(a)f(single)h(addend)g(in)f(\(4.28\)) 443 3469 y(The)479 b(t)-31 b(w)g(o)481 b(terms)f(in)f(\(4.28\))j(m)-31 b(ust)480 b(b)31 b(e)479 b(treated)h(di\013eren)-31 b(tly)-92 b(,)508 b(as)479 b(concerns)g(the)g(regularization)j(pro)31 b(cedure.)0 4886 y(The)429 b(\014rst)f(term)g(is)h(suc)-31 b(h)428 b(that)h(one)g(of)g(the)g(external)g(lines)g(is)f(asso)31 b(ciated)430 b(with)g(the)e(op)31 b(erator)429 b([)p Fo(C)42539 5052 y Fk(h;)p Ff(0)43818 4886 y Fr(\()p Fn(k)44920 4485 y Fj(\000)45667 4886 y Fr(\))286 b Fh(\000)0 6304 y Fr(1])p Fo(D)1776 6470 y Ff(+)2512 6304 y Fr(\()p Fn(k)3614 5902 y Fj(\000)4362 6304 y Fr(\))p Fo(D)5708 6470 y Ff(+)6444 6304 y Fr(\()p Fn(p)p Fr(\))8011 5902 y Fj(\000)p Ff(1)9200 6304 y Fr(.)480 b(W)-92 b(e)329 b(de\014ne)g Fh(R)307 b Fr(=)h(1)330 b(for)g(suc)-31 b(h)329 b(terms;)343 b(in)330 b(fact,)339 b(when)330 b(suc)-31 b(h)329 b(external)i(line)f(is)f(con) -31 b(tracted)0 7721 y(\(and)363 b(this)f(can)g(happ)31 b(en)361 b(only)i(at)g(scale)f Fo(h)p Fr(\),)i(the)e(factor)h Fo(D)24774 7887 y Ff(+)25509 7721 y Fr(\()p Fn(k)26611 7319 y Fj(\000)27359 7721 y Fr(\))p Fo(D)28705 7887 y Ff(+)29440 7721 y Fr(\()p Fn(p)p Fr(\))31007 7319 y Fj(\000)p Ff(1)32559 7721 y Fr(pro)31 b(duces)361 b(an)h(extra)g(factor)h Fo(\015)45386 7319 y Fk(h)46322 7721 y Fr(in)0 9138 y(the)333 b(b)31 b(ound,)340 b(with)333 b(resp)31 b(ect)332 b(to)h(the)f (dimensional)j(one.)480 b(This)333 b(claim)h(simply)g(follo)-31 b(ws)334 b(b)-31 b(y)333 b(the)f(observ)-61 b(ation)334 b(that)0 10556 y Fh(j)p Fo(D)1223 10722 y Ff(+)1959 10556 y Fr(\()p Fn(p)p Fr(\))p Fh(j)320 b(\025)e Fr(1)251 b Fh(\000)f Fo(\015)7881 10154 y Fj(\000)p Ff(1)9446 10556 y Fr(as)375 b Fn(p)319 b Fr(=)f Fn(k)13688 10154 y Ff(+)14673 10556 y Fh(\000)250 b Fn(k)16456 10154 y Fj(\000)17579 10556 y Fr(and)375 b Fn(k)20409 10154 y Ff(+)21520 10556 y Fr(is)g(at)i(scale)f(0,)i(while)e Fn(k)31462 10154 y Fj(\000)32210 10556 y Fr(,)h(as)f(w)-31 b(e)376 b(said,)i(is)d(at)i (scale)f Fo(h)p Fr(.)512 b(This)0 11973 y(factor)356 b(has)f(the)g(e\013ect)f(that)i(all)g(the)f(marginal)j(terms)c(in)h (the)g(tree)g(path)g(connecting)i Fo(v)37382 12139 y Ff(0)38233 11973 y Fr(with)f(the)f(end-p)31 b(oin)-31 b(t)0 13390 y(to)370 b(whic)-31 b(h)370 b(is)f(asso)31 b(ciated)371 b(the)e Fo(T)13457 13556 y Ff(1)14322 13390 y Fr(v)-31 b(ertex)370 b(acquires)g(negativ)-31 b(e)371 b(dimension.)443 14808 y(The)467 b(second)h(term)g(in)f(\(4.28\))k(can) c(b)31 b(e)467 b(regularized)i(as)e(ab)31 b(o)-31 b(v)g(e,)494 b(b)-31 b(y)467 b(subtracting)i(the)f(v)-61 b(alue)468 b(of)g(the)g(k)-31 b(ernel)0 16225 y(computed)342 b(at)g(zero)f (external)h(momen)-31 b(ta,)350 b Fp(i.e.)398 b Fr(for)341 b Fn(k)22337 15823 y Fj(\000)23391 16225 y Fr(=)24619 15933 y(\026)24560 16225 y Fn(k)25232 16391 y Ff(4)26036 16225 y Fr(=)307 b Fn(k)27876 16391 y Ff(1)28680 16225 y Fr(=)h(0.)483 b(Note)342 b(that)h(suc)-31 b(h)341 b(lo)31 b(cal)342 b(part)g(is)f(giv)-31 b(en)0 17642 y(b)g(y)12393 18092 y Fe(Z)13684 19598 y Fo(d)p Fn(k)14932 19141 y Ff(+)15798 19598 y Fr(~)-684 b Fo(\037)16360 19764 y Fk(M)17342 19598 y Fr(\()p Fn(k)18444 19141 y Ff(+)19180 19598 y Fr(\))38 b(^)-591 b Fo(g)20178 19023 y Ff(\(0\))20138 19874 y Fj(\000)21366 19598 y Fr(\()p Fn(k)22468 19141 y Ff(+)23204 19598 y Fr(\))p Fo(G)24504 19023 y Ff(\(0\))24504 19894 y(2)25693 19598 y Fr(\()p Fn(k)26795 19141 y Ff(+)27530 19598 y Fo(;)184 b Fr(0)p Fo(;)g Fr(0\))30445 18850 y Fo(u)31079 19016 y Ff(0)31575 18850 y Fr(\()p Fn(k)32677 18448 y Ff(+)33412 18850 y Fr(\))p 30181 19344 3920 45 v 30181 20358 a Fo(D)31097 20524 y Ff(+)31836 20358 y Fr(\()p Fn(k)32938 20038 y Ff(+)33673 20358 y Fr(\))34544 19598 y Fo(;)9564 b Fr(\(4)p Fo(:)p Fr(29\))0 22197 y(and)355 b(there)f(is)h(no)g(singularit)-31 b(y)357 b(asso)31 b(ciated)356 b(with)g(the)f(factor)g Fo(D)26771 22363 y Ff(+)27507 22197 y Fr(\()p Fn(k)28609 21795 y Ff(+)29344 22197 y Fr(\))29774 21795 y Fj(\000)p Ff(1)30963 22197 y Fr(,)j(thanks)d(to)g(the)g(supp)31 b(ort)355 b(on)g(scale)g(0)0 23740 y(of)370 b(the)f(propagator)409 b(^)-591 b Fo(g)9399 23165 y Ff(\(0\))9359 24016 y Fj(\000)10587 23740 y Fr(\()p Fn(k)11689 23338 y Ff(+)12424 23740 y Fr(\).)443 25375 y(A)334 b(similar)h(\(but)g(simpler\))g(analysis)g(holds)g(for)f(the)g (terms)g(con)-31 b(tributing)336 b(to)33210 25095 y(\026)33071 25375 y Fh(V)33840 24800 y Ff(\()p Fj(\000)p Ff(1\))33749 25670 y Fk(a;)p Ff(2)35721 25375 y Fr(,)341 b(whic)-31 b(h)335 b(con)-31 b(tain)336 b(a)e(v)-31 b(ertex)0 26792 y(of)419 b(t)-31 b(yp)31 b(e)419 b Fo(T)4497 26958 y Ff(+)5650 26792 y Fr(or)f Fo(T)7702 26958 y Fj(\000)8867 26792 y Fr(and)h(are)f(of)h(order)f Fo(\025\027)18415 26958 y Fj(\006)19161 26792 y Fr(.)640 b(No)-31 b(w,)433 b(the)418 b(only)i(thing)f(to)g(analyze)h(carefully)g(is)e(the)h(p)31 b(ossible)0 28209 y(singularities)360 b(asso)31 b(ciated)359 b(with)h(the)e(factors)490 b(~)-684 b Fo(\037)20108 28375 y Fk(M)21090 28209 y Fr(\()p Fn(p)p Fr(\))359 b(and)f Fn(p)p Fo(D)26780 28375 y Ff(+)27516 28209 y Fr(\()p Fn(p)p Fr(\))29083 27808 y Fj(\000)p Ff(1)30272 28209 y Fr(.)489 b(Ho)-31 b(w)g(ev)g(er,)362 b(since)c(in)h(these)f(terms)g (the)0 29627 y(\014eld)2741 29335 y(^)2514 29627 y Fo( )3275 29153 y Ff(+)3279 29867 y(\026)3235 30072 y Fi(k)3764 30183 y Fm(4)4196 30072 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)6939 29627 y Fr(is)485 b(con)-31 b(tracted,)515 b Fh(j)p Fn(p)p Fh(j)501 b(\025)f Fo(\015)17882 29225 y Fj(\000)p Ff(1)19071 29627 y Fo(=)p Fr(2,)515 b(for)485 b Fh(j)p Fo(h)p Fh(j)g Fr(large)g(enough,)515 b(a)485 b(prop)31 b(ert)-31 b(y)485 b(already)h(used)e(b)31 b(efore;)0 31044 y(hence)369 b(the)g(regularization)j(pro)31 b(cedure)368 b(can)i(not)g(pro)31 b(duce)369 b(bad)g(dimensional)i(b)31 b(ounds.)443 33406 y(W)-92 b(e)368 b(will)j(de\014ne)436 b(~)-620 b Fo(z)8153 33572 y Fj(\000)p Ff(1)9710 33406 y Fr(and)11909 33114 y(~)11863 33406 y Fo(\025)12509 33572 y Fj(\000)p Ff(1)13697 33406 y Fr(,)369 b(so)g(that)2056 35871 y Fh(L)p Fr([)3265 35592 y(\026)3126 35871 y Fh(V)3895 35296 y Ff(\()p Fj(\000)p Ff(1\))3804 36167 y Fk(a;)p Ff(1)6023 35871 y Fr(+)7269 35592 y(\026)7130 35871 y Fh(V)7899 35296 y Ff(\()p Fj(\000)p Ff(1\))7808 36167 y Fk(a;)p Ff(2)9780 35871 y Fr(]\()p Fo( )11278 35414 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])13749 35871 y Fr(\))308 b(=)15656 34643 y Fe(h)16224 35579 y Fr(~)16178 35871 y Fo(\025)16824 36037 y Fj(\000)p Ff(1)18261 35592 y Fr(\026)18012 35871 y Fo(F)18878 35296 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])18724 36205 y Fk(\025)21349 35871 y Fr(\()p Fo( )22540 35414 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])25011 35871 y Fr(\))246 b(+)313 b(~)-620 b Fo(z)27309 36037 y Fj(\000)p Ff(1)28724 35579 y Fr(^)28498 35871 y Fo( )29259 35296 y Ff([)p Fk(h;)p Fj(\000)p Ff(1]+)29263 36112 y(\026)29219 36317 y Fi(k)29748 36428 y Fm(4)30180 36317 y Fk(;)p Fj(\000)32409 35871 y Fo(D)33325 36037 y Fj(\000)34072 35871 y Fr(\()34561 35579 y(\026)34502 35871 y Fn(k)35174 36037 y Ff(4)35671 35871 y Fr(\))36101 34643 y Fe(i)36847 35871 y Fr(^)-591 b Fo(g)37337 36037 y Fj(\000)38084 35871 y Fr(\()38573 35579 y(\026)38514 35871 y Fn(k)39186 36037 y Ff(4)39683 35871 y Fr(\))40382 35592 y(^)40113 35871 y Fo(J)40771 35935 y Ff(\026)40727 36139 y Fi(k)41256 36250 y Fm(4)42051 35871 y Fo(;)2057 b Fr(\(4)p Fo(:)p Fr(30\))0 38293 y(where)369 b(w)-31 b(e)370 b(used)e(the)i(de\014nition)2787 40410 y(\026)2539 40689 y Fo(F)3405 40114 y Ff([)p Fk(h;j)45 b Ff(])3251 41023 y Fk(\025)5153 40689 y Fr(\()p Fo( )6344 40232 y Ff([)p Fk(h;j)g 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Fn(k)37813 40855 y Ff(3)38555 40689 y Fh(\000)39722 40397 y Fr(\026)39662 40689 y Fn(k)40334 40855 y Ff(4)40831 40689 y Fr(\))308 b Fo(:)2539 b Fr(\(4)p Fo(:)p Fr(31\))443 44689 y(Note)347 b(that)g(there)f(is)h(no)f(\014rst)g(order)f(con)-31 b(tribution)349 b(to)23679 44397 y(~)23633 44689 y Fo(\025)24279 44855 y Fj(\000)p Ff(1)25467 44689 y Fr(,)i(as)346 b(follo)-31 b(ws)349 b(from)e(a)f(simple)h(calculation,)355 b(so)346 b(that)46 45814 y(~)0 46106 y Fo(\025)646 46272 y Fj(\000)p Ff(1)2190 46106 y Fr(is)356 b(of)h(order)f Fo(\025)8068 45704 y Ff(2)8920 46106 y Fr(or)h(lo)-31 b(w)g(er.)489 b(W)-92 b(e)356 b(exp)31 b(ect)356 b(indeed)h(that)g(it)g(satis\014es)g (a)f(non)h(zero)f(lo)-31 b(w)g(er)358 b(b)31 b(ound)356 b(of)h(order)e Fo(\025)46440 45704 y Ff(2)46937 46106 y Fr(,)0 47523 y(but)369 b(this)h(will)h(not)f(pla)-31 b(y)370 b(an)-31 b(y)370 b(role)g(in)f(the)h(follo)-31 b(wing.)443 49886 y(Let)349 b(us)f(consider)h(no)-31 b(w)350 b(the)f(terms)g(con)-31 b(tributing)351 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Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)6680 65363 y Fr(and)424 b Fo(F)9753 64788 y Ff(\()p Fj(\000)p Ff(1\))9599 65658 y(1)p Fk(;)p Ff(+)12057 65363 y Fr(are)h(de\014ned)f(as)g(in)h(eq.)658 b(\(132\))426 b(of)f([BM2];)454 b(they)425 b(represen)-31 b(t)423 b(the)i(terms)f(in) h(whic)-31 b(h)0 66780 y(b)31 b(oth)306 b(or)f(only)h(one)g(of)g(the)g (\014elds)f(in)h Fo(\016)42 b(\032)16318 66946 y Fi(p)p Fk(;)p Ff(+)17872 66780 y Fr(,)319 b(resp)31 b(ectiv)-31 b(ely)-92 b(,)319 b(are)305 b(con)-31 b(tracted.)473 b(Both)306 b(con)-31 b(tributions)308 b(to)e(the)f(r.h.s.)0 68364 y(of)375 b(\(4.32\))i(are)d(dimensionally)j(marginal;)i(ho)-31 b(w)g(ev)g(er,)377 b(the)e(regularization)h(of)f Fo(F)33868 67789 y Ff(\()p Fj(\000)p Ff(1\))33714 68659 y(1)p Fk(;)p Ff(+)36122 68364 y Fr(is)f(trivial,)k(as)c(it)h(is)f(of)h(the)0 69781 y(form)7184 71848 y Fo(F)8050 71273 y Ff(\()p Fj(\000)p Ff(1\))7896 72143 y(1)p Fk(;)p Ff(+)9929 71848 y Fr(\()p Fn(k)11031 71391 y Ff(+)11767 71848 y Fo(;)184 b Fn(k)12930 71391 y Fj(\000)13678 71848 y Fr(\))308 b(=)f([)16024 71048 y([)p Fo(C)17122 71214 y Fk(h;)p Ff(0)18401 71048 y Fr(\()p Fn(k)19503 70647 y Fj(\000)20250 71048 y Fr(\))247 b Fh(\000)f Fr(1])p Fo(D)23810 71214 y Ff(+)24546 71048 y Fr(\()p Fn(k)25648 70647 y Fj(\000)26396 71048 y Fr(\))38 b(^)-591 b Fo(g)27394 70473 y Ff(\(0\))27354 71324 y(+)28582 71048 y Fr(\()p Fn(k)29684 70647 y Ff(+)30420 71048 y Fr(\))246 b Fh(\000)g Fo(u)32837 71214 y Ff(0)33333 71048 y Fr(\()p Fn(k)34435 70647 y Ff(+)35171 71048 y Fr(\))p 16024 71593 19577 45 v 22467 72607 a Fo(D)23383 72773 y Ff(+)24119 72607 y Fr(\()p Fn(k)25221 72287 y Ff(+)26202 72607 y Fh(\000)g Fn(k)27981 72287 y Fj(\000)28728 72607 y Fr(\))35734 71848 y Fo(G)36604 71391 y Ff(\(2\))37793 71848 y Fr(\()p Fn(k)38895 71391 y Ff(+)39630 71848 y Fr(\))4355 b(\(4)p Fo(:)p Fr(33\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 0 1107 a Fr(or)369 b(the)g(similar)i(one,)f (obtained)h(exc)-31 b(hanging)371 b Fn(k)20179 705 y Ff(+)21283 1107 y Fr(with)f Fn(k)24476 705 y Fj(\000)25223 1107 y Fr(.)23622 7613 y(+)15873 10891 y Fo(F)16739 10316 y Ff(\()p Fj(\000)p Ff(1\))16585 11208 y(2)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)32478 10891 y Fo(F)33344 10316 y Ff(\()p Fj(\000)p Ff(1\))33190 11186 y(1)p Fk(;)p Ff(+)7018 10289 y @beginspecial @setspecial %%BeginDocument: F2F1.pst % x1 y1 x2 y2 cambio_coordinate --> l % % (x1,y1) --> (0,0) , (x2,y2) --> (l,0) , l = sqr[(x2-x1)^2+(y2-y1)^2] % /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } def % fx fy plot % % raccorda con curve di Bezier i punti di coordinate (fx[i],fy[i]), i=0,...,n % /plot { /nx fx length 1 sub def fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for stroke} def % lx ly n normonda % % disegna n sinusoidi complete, occupando un rettangolo di lati lx e ly % /normonda { /n exch def /ly exch def /lx exch def /maxang 360 n mul def /imax 18 n mul def /fx0 [ 0 20 maxang { } for ] def /fy0 [ 0 1 imax { fx0 exch get sin } for ] def /fx [ 0 1 imax { fx0 exch get maxang div lx mul } for ] def /fy [ 0 1 imax { fy0 exch get ly mul } for ] def plot} def % x1 y1 x2 y2 ly n onda % /onda { gsave /n exch def /ly exch def cambio_coordinate % st: lx ly n normonda grestore } def % x1 y1 x2 y2 freccia % /freccia { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 x2 y2 frecciafin % /frecciafin { gsave cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def % x1 y1 punto % /punto { gsave 2 0 360 newpath arc fill stroke grestore} def % x1 y1 r cerchio % /cerchio { gsave 0 360 newpath arc stroke grestore} def % x1 y1 x2 y2 tlinea % /tlinea { gsave moveto [4 4] 2 setdash lineto stroke grestore} def 80 30 5 cerchio 40 30 75 30 3 5 onda 40 30 20 50 tlinea 40 30 20 20 freccia 85 30 98 40 freccia 98 20 85 30 freccia 110 30 15 cerchio 125 30 145 40 freccia 145 20 125 30 freccia 230 30 5 cerchio 190 30 225 30 3 5 onda 190 30 170 50 tlinea 190 30 170 20 freccia 235 30 245 30 freccia 235 10 235 30 freccia 260 30 15 cerchio 275 30 295 40 freccia 295 20 275 30 freccia grestore %%EndDocument @endspecial 13256 12651 a Fr(Fig.)493 b(16:)g(Graphical)370 b(represen)-31 b(tation)371 b(of)e(\(4.33\))443 15026 y(By)412 b(the)f(o)31 b(ddness)411 b(of)h(the)g(propagator)h(in)f(the)g (momen)-31 b(tum,)425 b Fo(G)27660 14624 y Ff(\(2\))28849 15026 y Fr(\(0\))380 b(=)e(0,)423 b(hence)411 b(w)-31 b(e)412 b(can)g(regularize)g(suc)-31 b(h)0 16443 y(term)369 b(without)j(in)-31 b(tro)31 b(ducing)371 b(an)-31 b(y)370 b(lo)31 b(cal)370 b(term,)g(b)-31 b(y)370 b(simply)g(rewriting)g(it)g (as)2741 20093 y Fo(F)3607 19518 y Ff(\()p Fj(\000)p Ff(1\))3453 20388 y(1)p Fk(;)p Ff(+)5487 20093 y Fr(\()p Fn(k)6589 19636 y Ff(+)7324 20093 y Fo(;)184 b Fn(k)8487 19636 y Fj(\000)9235 20093 y Fr(\))308 b(=)g([)11582 19294 y([)p Fo(C)12680 19460 y Fk(h;)p Ff(0)13958 19294 y Fr(\()p Fn(k)15060 18892 y Fj(\000)15808 19294 y Fr(\))246 b Fh(\000)g Fr(1])p Fo(D)19367 19460 y Ff(+)20104 19294 y Fr(\()p Fn(k)21206 18892 y Fj(\000)21953 19294 y Fr(\))38 b(^)-591 b Fo(g)22951 18719 y Ff(\(0\))22911 19569 y(+)24140 19294 y Fr(\()p Fn(k)25242 18892 y Ff(+)25977 19294 y Fr(\))247 b Fh(\000)f Fo(u)28395 19460 y Ff(0)28891 19294 y Fr(\()p Fn(k)29993 18892 y Ff(+)30728 19294 y Fr(\))p 11582 19838 19577 45 v 18025 20852 a Fo(D)18941 21018 y Ff(+)19676 20852 y Fr(\()p Fn(k)20778 20533 y Ff(+)21759 20852 y Fh(\000)g Fn(k)23538 20533 y Fj(\000)24285 20852 y Fr(\))31291 20093 y([)p Fo(G)32468 19636 y Ff(\(2\))33658 20093 y Fr(\()p Fn(k)34760 19636 y Ff(+)35495 20093 y Fr(\))g Fh(\000)g Fo(G)38148 19636 y Ff(\(2\))39337 20093 y Fr(\(0\)])310 b Fo(:)2741 b Fr(\(4)p Fo(:)p Fr(34\))443 23451 y(As)369 b(sho)-31 b(wn)369 b(in)h([BM2],)h(b)-31 b(y)369 b(using)h(the)f(symmetry)h(prop)31 b(ert)-31 b(y)10094 26236 y(^)-591 b Fo(g)10624 25779 y Ff(\()p Fk(j)45 b Ff(\))10584 26510 y Fk(!)11782 26236 y Fr(\()p Fn(k)p Fr(\))308 b(=)f Fh(\000)p Fo(i!)78 b Fr(^)-591 b Fo(g)17329 25779 y Ff(\()p Fk(j)45 b Ff(\))17289 26510 y Fk(!)18487 26236 y Fr(\()p Fn(k)19589 25779 y Fj(\003)20099 26236 y Fr(\))1107 b Fo(;)1292 b Fn(k)307 b Fr(=)h(\()p Fo(k)35 b(;)184 b(k)27491 26402 y Ff(0)27989 26236 y Fr(\))p Fo(;)1293 b Fn(k)30691 25779 y Fj(\003)31507 26236 y Fr(=)308 b(\()p Fh(\000)p Fo(k)34543 26402 y Ff(0)35040 26236 y Fo(;)184 b(k)35 b Fr(\))309 b Fo(;)7227 b Fr(\(4)p Fo(:)p Fr(35\))0 29148 y Fo(F)866 28573 y Ff(\()p Fj(\000)p Ff(1\))712 29465 y(2)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)3368 29148 y Fr(can)370 b(b)31 b(e)368 b(written)j(as)7573 32556 y Fo(F)8439 32083 y Fj(\000)p Ff(1)8285 32873 y(2)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)10572 32556 y Fr(\()p Fn(k)11674 32099 y Ff(+)12409 32556 y Fo(;)184 b Fn(k)13572 32099 y Fj(\000)14320 32556 y Fr(\))308 b(=)17693 31808 y(1)p 16360 32302 3220 45 v 16360 33316 a Fo(D)17276 33482 y Ff(+)18011 33316 y Fr(\()p Fn(p)p Fr(\))19897 31660 y Fe(\002)20358 32556 y Fo(p)20915 32722 y Ff(0)21411 32556 y Fo(A)22241 32722 y Ff(0)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)24529 32556 y Fr(\()p Fn(k)25631 32099 y Ff(+)26366 32556 y Fo(;)184 b Fn(k)27529 32099 y Fj(\000)28277 32556 y Fr(\))247 b(+)f Fo(p)30618 32722 y Ff(1)31114 32556 y Fo(A)31944 32722 y Ff(1)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)34232 32556 y Fr(\()p Fn(k)35334 32099 y Ff(+)36069 32556 y Fo(;)184 b Fn(k)37232 32099 y Fj(\000)37980 32556 y Fr(\))38410 31660 y Fe(\003)39364 32556 y Fo(;)4744 b Fr(\(4)p Fo(:)p Fr(36\))0 35953 y(where)369 b Fo(A)4031 36119 y Fk(i;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)6191 35953 y Fr(\()p Fn(k)7293 35551 y Ff(+)8028 35953 y Fo(;)184 b Fn(k)9191 35551 y Fj(\000)9939 35953 y Fr(\))369 b(are)g(functions)i(suc)-31 b(h)368 b(that,)j(if)f(w)-31 b(e)370 b(de\014ne)11153 39210 y Fh(L)p Fo(F)12782 38737 y Fj(\000)p Ff(1)12628 39527 y(2)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)15223 39210 y Fr(=)17858 38461 y(1)p 16525 38956 V 16525 39969 a Fo(D)17441 40135 y Ff(+)18176 39969 y Fr(\()p Fn(p)p Fr(\))20061 39210 y([)q Fo(p)20926 39376 y Ff(0)21422 39210 y Fo(A)22252 39376 y Ff(0)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)24540 39210 y Fr(\(0)p Fo(;)184 b Fr(0\))249 b(+)d Fo(p)28910 39376 y Ff(1)29407 39210 y Fo(A)30237 39376 y Ff(1)p Fk(;)p Ff(+)p Fk(;)72 b Ff(~)-513 b Fk(!)32524 39210 y Fr(\(0)p Fo(;)184 b Fr(0\)])496 b Fo(;)8324 b Fr(\(4)p Fo(:)p Fr(37\))0 42494 y(then)12594 44223 y Fh(L)p Fo(F)14223 43749 y Fj(\000)p Ff(1)14069 44518 y(2)p Fk(;)p Ff(+)p Fk(;)p Ff(+)16758 44223 y Fr(=)308 b Fo(Z)18762 43691 y Ff(3)p Fk(;)p Ff(+)18683 44518 y Fj(\000)p Ff(1)21308 44223 y Fo(;)1291 b Fh(L)p Fo(F)24535 43749 y Fj(\000)p Ff(1)24381 44518 y(2)p Fk(;)p Ff(+)p Fk(;)p Fj(\000)27083 44223 y Fr(=)28384 43474 y Fo(D)29300 43640 y Fj(\000)30048 43474 y Fr(\()p Fn(p)p Fr(\))p 28384 43968 3232 45 v 28390 44982 a Fo(D)29306 45148 y Ff(+)30042 44982 y Fr(\()p Fn(p)p Fr(\))31749 44223 y Fo(Z)32584 43691 y Ff(3)p Fk(;)p Fj(\000)32505 44518 y(\000)p Ff(1)34343 44223 y Fo(;)9765 b Fr(\(4)p Fo(:)p Fr(38\))0 47215 y(where)347 b Fo(Z)4014 46683 y Ff(3)p Fk(;)p Ff(+)3935 47511 y Fj(\000)p Ff(1)5800 47215 y Fr(and)g Fo(Z)8765 46683 y Ff(3)p Fk(;)p Fj(\000)8686 47511 y(\000)p Ff(1)10563 47215 y Fr(are)g(suitable)h(real)f(constan) -31 b(ts.)487 b(Hence)347 b(the)g(lo)31 b(cal)348 b(part)g(of)f(the)h (marginal)h(term)e(\(4.32\))0 48633 y(is,)370 b(b)-31 b(y)369 b(de\014nition,)i(equal)g(to)14878 50090 y Fo(Z)15713 49558 y Ff(3)p Fk(;)p Ff(+)15634 50385 y Fj(\000)p Ff(1)17151 50090 y Fo(T)17798 50256 y Ff(+)18533 50090 y Fr(\()p Fo( )19724 49633 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])22195 50090 y Fr(\))247 b(+)f Fo(Z)24814 49558 y Ff(3)p Fk(;)p Fj(\000)24735 50385 y(\000)p Ff(1)26265 50090 y Fo(T)26912 50256 y Fj(\000)27659 50090 y Fr(\()p Fo( )28850 49633 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])31321 50090 y Fr(\))308 b Fo(:)12049 b Fr(\(4)p Fo(:)p Fr(39\))443 53516 y(Let)283 b(us)f(\014nally)i(consider)f(the)g(terms)f(con)-31 b(tributing)285 b(to)23609 53237 y(\026)23471 53516 y Fh(V)24240 52941 y Ff(\()p Fj(\000)p Ff(1\))24149 53850 y Fk(b;)p Ff(2)26120 53516 y Fr(,)301 b(that)284 b(is)e(those)h(where)36218 53224 y(^)35991 53516 y Fo( )36752 53043 y Ff(+)36756 53757 y(\026)36712 53962 y Fi(k)37241 54073 y Fm(4)37673 53962 y Fj(\000)p Fi(p)39259 53516 y Fr(is)g(not)g(con)-31 b(tracted)0 54934 y(and)344 b(there)f(is)g(a)h(v)-31 b(ertex)344 b(of)g(t)-31 b(yp)31 b(e)344 b Fo(T)14593 55100 y Ff(+)15671 54934 y Fr(or)f Fo(T)17648 55100 y Fj(\000)18395 54934 y Fr(.)484 b(If)343 b(ev)-31 b(en)344 b(this)g(v)-31 b(ertex)344 b(is)f(not)h(con)-31 b(tracted,)351 b(w)-31 b(e)344 b(get)g(a)f(con)-31 b(tribution)0 56351 y(similar)380 b(to)g(\(4.39\),)385 b(with)380 b Fo(\027)11606 56517 y Fj(\006)12732 56351 y Fr(in)f(place)g(of)h Fo(Z)18977 55819 y Ff(3)p Fk(;)p Fj(\006)18898 56646 y(\000)p Ff(1)20428 56351 y Fr(.)522 b(Among)380 b(the)f(terms)g(with)h(at)g(least)g(one)f (v)-31 b(ertex)379 b Fo(\025)p Fr(,)j(there)c(is,)0 57768 y(as)365 b(b)31 b(efore,)366 b(no)g(term)f(with)h(t)-31 b(w)g(o)367 b(external)f(lines;)i(hence)c(the)i(only)g(marginal)h (terms)e(ha)-31 b(v)g(e)366 b(four)f(external)i(lines)0 59186 y(and)i(can)h(b)31 b(e)369 b(written)h(in)f(the)h(form)9870 60664 y Fe(Z)11161 62171 y Fo(d)p Fn(p)131 b Fr(~)-684 b Fo(\037)13137 62337 y Fk(M)14120 62171 y Fr(\()p Fn(p)p Fr(\))15956 61891 y(^)15687 62171 y Fo(J)16301 62337 y Fi(k)16830 62448 y Fm(4)17356 62171 y Fr(^)-591 b Fo(g)17846 62337 y Fj(\000)18593 62171 y Fr(\()p Fn(k)19695 62337 y Ff(4)20192 62171 y Fr(\))20806 60664 y Fe(Z)22099 62171 y Fo(d)p Fn(k)23347 61714 y Ff(+)24266 61119 y Fe(X)24845 63486 y Ff(~)-513 b Fk(!)26276 61879 y Fr(^)26049 62171 y Fo( )26810 61698 y Ff(+)26770 62557 y Fi(k)27299 62336 y Fm(+)27924 62557 y Fk(;)72 b Ff(~)-513 b Fk(!)29054 61879 y Fr(^)28827 62171 y Fo( )29588 61698 y Fj(\000)29548 62557 y Fi(k)30077 62336 y Fm(+)30702 62557 y Fj(\000)p Fi(p)p Fk(;)72 b Ff(~)-513 b Fk(!)33161 62171 y Fh(\001)9931 65644 y(\001)10792 64083 y Fe(\024)11376 65644 y Fo(\027)11923 65810 y Ff(+)12658 65644 y Fo(G)13528 65069 y Ff(\(0\))13528 65961 y(+)p Fk(;)72 b Ff(~)-513 b Fk(!)15112 65644 y Fr(\()p Fn(k)16214 65187 y Ff(+)16949 65644 y Fo(;)184 b Fn(k)18112 65187 y Ff(+)19094 65644 y Fh(\000)246 b Fn(p)p Fr(\))g(+)g Fo(\027)23238 65810 y Fj(\000)24118 64895 y Fo(D)25034 65061 y Fj(\000)25782 64895 y Fr(\()p Fn(p)p Fr(\))p 24118 65390 V 24124 66404 a Fo(D)25040 66570 y Ff(+)25776 66404 y Fr(\()p Fn(p)p Fr(\))27483 65644 y Fo(G)28353 65069 y Ff(\(0\))28353 65961 y Fj(\000)p Fk(;)72 b Ff(~)-513 b Fk(!)29949 65644 y Fr(\()p Fn(k)31051 65187 y Ff(+)31786 65644 y Fo(;)184 b Fn(k)32949 65187 y Ff(+)33931 65644 y Fh(\000)246 b Fn(p)p Fr(\))36175 64083 y Fe(\025)37251 65644 y Fo(:)44415 63957 y Fr(\(4)p Fo(:)p Fr(40\))0 69331 y(By)317 b(using)h(the)g(symmetry)g(prop)31 b(ert)-31 b(y)317 b(\(4.35\))j(of)e(the)f(propagators,)330 b(it)318 b(is)f(easy)g(to)i(sho)-31 b(w)317 b(that)i Fo(G)40266 68756 y Ff(\(0\))40266 69606 y Fk(!)28 b(;)p Fj(\000)p Fk(!)42447 69331 y Fr(\(0)p Fo(;)184 b Fr(0\))311 b(=)c(0.)0 71024 y(Hence,)430 b(if)418 b(w)-31 b(e)417 b(regularize)i(\(4.40\))h(b)-31 b(y)417 b(subtracting)i Fo(G)23097 70449 y Ff(\(0\))23097 71341 y Fk(!)28 b(;)72 b Ff(~)-513 b Fk(!)24586 71024 y Fr(\(0)p Fo(;)184 b Fr(0\))420 b(to)e Fo(G)29734 70449 y Ff(\(0\))29734 71341 y Fk(!)28 b(;)72 b Ff(~)-513 b Fk(!)31224 71024 y Fr(\()p Fn(k)32326 70622 y Ff(+)33061 71024 y Fo(;)184 b Fn(k)34224 70622 y Ff(+)35238 71024 y Fh(\000)278 b Fn(p)p Fr(\),)430 b(w)-31 b(e)418 b(still)h(get)f(a)f(lo)31 b(cal)0 72441 y(term)369 b(of)h(the)g(form)f(\(4.39\).)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 443 1107 a Fr(By)369 b(collecting)j(all)e(the)g (lo)31 b(cal)370 b(term,)g(w)-31 b(e)370 b(can)g(write)7674 4337 y Fh(L)p Fr([)8883 4057 y(\026)8744 4337 y Fh(V)9513 3762 y Ff(\()p Fj(\000)p Ff(1\))9422 4671 y Fk(b;)p Ff(1)11641 4337 y Fr(+)12886 4057 y(\026)12748 4337 y Fh(V)13517 3762 y Ff(\()p Fj(\000)p Ff(1\))13426 4671 y Fk(b;)p Ff(2)15397 4337 y Fr(]\()p Fo( )16895 3880 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])19367 4337 y Fr(\))308 b(=)f Fo(\027)21820 4503 y Fj(\000)p Ff(1)p Fk(;)p Ff(+)23951 4337 y Fo(T)24598 4503 y Ff(+)25333 4337 y Fr(\()p Fo( )26524 3880 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])28995 4337 y Fr(\))246 b(+)g Fo(\027)31325 4503 y Fj(\000)p Ff(1)p Fk(;)p Fj(\000)33468 4337 y Fo(T)34115 4503 y Fj(\000)34862 4337 y Fr(\()p Fo( )36053 3880 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])38525 4337 y Fr(\))307 b Fo(;)4846 b Fr(\(4)p Fo(:)p Fr(41\))0 7901 y(where)369 b Fo(\027)3748 8067 y Fj(\000)p Ff(1)p Fk(;!)6092 7901 y Fr(=)308 b Fo(\027)7808 8067 y Fk(!)8694 7901 y Fr(+)246 b Fo(Z)10636 7369 y Ff(3)p Fk(;!)10557 8196 y Fj(\000)p Ff(1)12227 7901 y Fr(+)g Fo(G)14204 7326 y Ff(\(0\))14204 8031 y Fk(!)28 b(;!)15693 7901 y Fr(\(0)p Fo(;)184 b Fr(0\).)496 b(Hence)4271 10971 y(\026)4132 11250 y Fh(V)4901 10793 y Ff(\()p Fj(\000)p Ff(1\))6782 11250 y Fr(\()p Fo( )7973 10793 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])10444 11250 y Fr(\))308 b(=)f Fo(T)12997 11416 y Ff(1)13494 11250 y Fr(\()p Fo( )14685 10793 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])17156 11250 y Fr(\))246 b(+)g Fo(\027)19486 11416 y Fj(\000)p Ff(1)p Fk(;)p Ff(+)21617 11250 y Fo(T)22264 11416 y Ff(+)22999 11250 y Fr(\()p Fo( )24190 10793 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])26661 11250 y Fr(\))g(+)g Fo(\027)28991 11416 y Fj(\000)p Ff(1)p Fk(;)p Fj(\000)31134 11250 y Fo(T)31781 11416 y Fj(\000)32528 11250 y Fr(\()p Fo( )33719 10793 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])36190 11250 y Fr(\)+)4378 13354 y(+)5485 12125 y Fe(h)6054 13061 y Fr(~)6008 13354 y Fo(\025)6654 13520 y Fj(\000)p Ff(1)8090 13074 y Fr(\026)7842 13354 y Fo(F)8708 12779 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])8554 13687 y Fk(\025)11178 13354 y Fr(\()p Fo( )12369 12897 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])14840 13354 y Fr(\))h(+)312 b(~)-619 b Fo(z)17139 13520 y Fj(\000)p Ff(1)18554 13061 y Fr(^)18327 13354 y Fo( )19088 12779 y Ff([)p Fk(h;)p Fj(\000)p Ff(1]+)19092 13594 y(\026)19048 13799 y Fi(k)19577 13910 y Fm(4)20009 13799 y Fk(;)p Fj(\000)22238 13354 y Fo(D)23154 13520 y Fj(\000)23902 13354 y Fr(\()24391 13061 y(\026)24332 13354 y Fn(k)25004 13520 y Ff(4)25501 13354 y Fr(\))25931 12125 y Fe(i)26676 13354 y Fr(^)-591 b Fo(g)27166 13520 y Fj(\000)27914 13354 y Fr(\()28403 13061 y(\026)28344 13354 y Fn(k)29016 13520 y Ff(4)29512 13354 y Fr(\))30211 13074 y(^)29942 13354 y Fo(J)30600 13418 y Ff(\026)30556 13622 y Fi(k)31085 13733 y Fm(4)31819 13354 y Fr(+)33065 13074 y(\026)32926 13354 y Fh(V)33695 12779 y Ff(\()p Fj(\000)p Ff(1\))33604 13679 y Fk(R)35576 13354 y Fr(\()p Fo( )36767 12897 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])39238 13354 y Fr(\))308 b Fo(;)44415 12419 y Fr(\(4)p Fo(:)p Fr(42\))0 17153 y(where)3340 16873 y(\026)3201 17153 y Fh(V)3970 16578 y Ff(\()p Fj(\000)p Ff(1\))3879 17478 y Fk(R)5851 17153 y Fr(\()p Fo( )7042 16751 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])9513 17153 y Fr(\))369 b(is)g(the)h(sum)f(of)g (all)i(irrelev)-61 b(an)-31 b(t)370 b(terms)f(linear)h(in)f(the)h (external)g(\014eld)f Fo(J)106 b Fr(.)0 19676 y Fn(4.4)1314 b Fp(The)397 b(multisc)-57 b(ale)396 b(exp)-57 b(ansion)396 b(of)17000 19397 y Fr(~)16750 19676 y Fo(G)17620 19275 y Ff(4)17620 19950 y(+)18355 19676 y Fr(\()18844 19384 y(\026)18785 19676 y Fn(k)19457 19842 y Ff(1)19954 19676 y Fo(;)20505 19384 y Fr(\026)20445 19676 y Fn(k)21117 19842 y Ff(2)21614 19676 y Fo(;)22166 19384 y Fr(\026)22105 19676 y Fn(k)22777 19842 y Ff(3)23275 19676 y Fo(;)23826 19384 y Fr(\026)23766 19676 y Fn(k)24438 19842 y Ff(4)24935 19676 y Fr(\))p Fp(:)510 b(the)396 b(higher)g(sc)-57 b(ales)396 b(inte)-57 b(gr)g(ation.)443 21255 y Fr(The)422 b(in)-31 b(tegration)424 b(of)f(the)f(\014eld)f Fo( )14937 20853 y Ff(\()p Fj(\000)p Ff(1\))17239 21255 y Fr(is)h(done)g(in)g(a)g (similar)h(w)-31 b(a)g(y;)450 b(w)-31 b(e)422 b(shall)g(call)36286 20975 y(\026)36147 21255 y Fh(V)36916 20853 y Ff(\()p Fj(\000)p Ff(2\))38797 21255 y Fr(\()p Fo( )39988 20853 y Ff([)p Fk(h;)p Fj(\000)p Ff(2])42459 21255 y Fr(\))g(the)g(sum)0 22672 y(o)-31 b(v)g(er)368 b(all)g(terms)g(linear)g(in)f Fo(J)106 b Fr(.)492 b(As)367 b(b)31 b(efore,)368 b(the)g(condition)h (\(3.28\))h(on)d(the)h(external)g(momen)-31 b(ta)370 b Fn(k)41305 22838 y Fk(i)42040 22672 y Fr(forbids)e(the)0 24090 y(presence)g(of)i(v)-31 b(ertices)369 b(of)h(t)-31 b(yp)31 b(e)370 b Fo(')p Fr(,)f(if)h Fo(h)307 b(<)h Fh(\000)p Fr(1,)370 b(as)f(w)-31 b(e)369 b(shall)i(supp)31 b(ose.)443 25669 y(The)395 b(main)i(di\013erence)e(is)g(that)i(there)e(is)h(no)f (con)-31 b(tribution)398 b(obtained)f(b)-31 b(y)396 b(con)-31 b(tracting)398 b(b)31 b(oth)396 b(\014eld)f(v)-61 b(ariables)0 27086 y(b)31 b(elonging)317 b(to)f Fo(\016)42 b(\032)315 b Fr(in)h Fo(T)9563 27252 y Ff(1)10059 27086 y Fr(\()p Fo( )40 b Fr(\))316 b(at)g(scale)g Fh(\000)p Fr(1,)327 b(b)31 b(ecause)314 b(of)i(\(4.13\).)478 b(It)315 b(is)g(instead)h(p)31 b(ossible)315 b(to)h(get)g(marginal)i(terms)0 28503 y(with)353 b(four)e(external)i(lines)f(\(t)-31 b(w)g(o)353 b(is)f(imp)31 b(ossible\),)357 b(suc)-31 b(h)351 b(that)i(one)f(of)g(these)f (\014elds)h(is)f(con)-31 b(tracted)353 b(at)f(scale)g Fh(\000)p Fr(1.)0 29921 y(Ho)-31 b(w)g(ev)g(er,)344 b(in)336 b(this)f(case,)343 b(the)335 b(second)g(\014eld)h(v)-61 b(ariable)336 b(will)h(b)31 b(e)335 b(necessarily)g(con)-31 b(tracted)337 b(at)f(scale)g Fo(h)p Fr(,)342 b(so)335 b(that)h(w)-31 b(e)0 31338 y(can)428 b(put)g Fh(R)406 b Fr(=)f(1)429 b(for)f(suc)-31 b(h)427 b(terms;)458 b(in)428 b(fact,)445 b(the)428 b(extra)g(factor)h Fo(\015)28220 30936 y Fj(\000)p Ff(\()p Fj(\000)p Ff(1)p Fj(\000)p Fk(h)p Ff(\))32431 31338 y Fr(coming)h(from)e(the)g(b)31 b(ound)428 b(\(4.22\))0 32755 y(after)464 b(the)f(in)-31 b(tegration)467 b(of)d(the)f(last)h(scale)g(\014eld,)488 b(has)463 b(the)g(e\013ect)h(of)f(automatically)469 b(regularize)464 b(them,)488 b(and)0 34172 y(ev)-31 b(en)369 b(the)h(terms)f(con)-31 b(taining)372 b(them)e(as)f(subgraphs.)443 35751 y(The)e(terms)g(with)h (a)f Fo(T)9884 35917 y Ff(1)10747 35751 y Fr(v)-31 b(ertex,)369 b(suc)-31 b(h)366 b(that)j(the)e(\014eld)g(v)-61 b(ariables)367 b(b)31 b(elonging)369 b(to)f Fo(\016)42 b(\032)367 b Fr(are)g(not)g(con)-31 b(tracted,)370 b(can)0 37169 y(b)31 b(e)369 b(treated)h(as)f(in)g Fh(x)p Fr(4.3,)i(hence)e(do)g(not)h(need) f(a)h(regularization.)443 38747 y(It)341 b(follo)-31 b(ws)344 b(that,)k(if)342 b(the)g(irrelev)-61 b(an)-31 b(t)342 b(part)18180 38468 y(\026)18042 38747 y Fh(V)18811 38172 y Ff(\()p Fj(\000)p Ff(1\))18720 39073 y Fk(R)21033 38747 y Fr(w)-31 b(ere)341 b(absen)-31 b(t)342 b(in)g(the)f(r.h.s.)483 b(of)342 b(\(4.42\),)350 b(then)342 b(the)g(regulariza-)0 40340 y(tion)365 b(pro)31 b(cedure)363 b(w)-31 b(ould)366 b(not)f(pro)31 b(duce)363 b(an)-31 b(y)365 b(lo)31 b(cal)365 b(term)g(prop)31 b(ortional)366 b(to)32102 40060 y(\026)31853 40340 y Fo(F)32719 39765 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])32565 40673 y Fk(\025)35190 40340 y Fr(\()p Fo( )36381 39938 y Ff([)p Fk(h;)p Fj(\000)p Ff(2])38852 40340 y Fr(\),)g(starting)f (from)g(a)0 41757 y(graph)k(con)-31 b(taining)373 b(a)c Fo(T)10090 41923 y Ff(1)10955 41757 y Fr(v)-31 b(ertex.)443 43336 y(It)432 b(is)g(easy)g(to)h(see)f(that)h(all)h(other)e(terms)g (con)-31 b(taining)435 b(a)e(v)-31 b(ertex)433 b(of)f(t)-31 b(yp)31 b(e)433 b Fo(T)33185 43502 y Ff(1)34113 43336 y Fr(or)f Fo(T)36179 43502 y Fj(\006)37358 43336 y Fr(can)h(b)31 b(e)432 b(treated)h(as)f(in)0 44753 y Fh(x)p Fr(4.3.)757 b(Moreo)-31 b(v)g(er,)479 b(the)457 b(supp)31 b(ort)456 b(prop)31 b(erties)457 b(of)495 b(^)-591 b Fo(g)21603 44919 y Fj(\000)22350 44753 y Fr(\()22839 44461 y(\026)22780 44753 y Fn(k)23452 44919 y Ff(4)23949 44753 y Fr(\))457 b(immediately)j(implies)e(that)g(it)g(is)f(not)g(p)31 b(ossible)457 b(to)0 46170 y(pro)31 b(duce)453 b(a)h(graph)g(con)-31 b(tributing)456 b(to)16498 45890 y(\026)16359 46170 y Fh(V)17128 45769 y Ff(\()p Fj(\000)p Ff(2\))19009 46170 y Fr(,)475 b(con)-31 b(taining)456 b(the)521 b(~)-620 b Fo(z)27762 46336 y Fj(\000)p Ff(1)29404 46170 y Fr(v)-31 b(ertex.)746 b(Hence,)475 b(in)454 b(order)f(to)h(complete)0 47588 y(the)399 b(analysis)i(of)7559 47308 y(\026)7421 47588 y Fh(V)8190 47186 y Ff(\()p Fj(\000)p Ff(2\))10070 47588 y Fr(,)407 b(w)-31 b(e)400 b(still)h(ha)-31 b(v)g(e)400 b(to)g(consider)f(the)h(marginal)h(terms)e(con)-31 b(taining)402 b(the)40032 47295 y(~)39986 47588 y Fo(\025)40632 47754 y Fj(\000)p Ff(1)42219 47588 y Fr(v)-31 b(ertex,)408 b(for)0 49005 y(whic)-31 b(h)449 b(w)-31 b(e)449 b(simply)h(apply)f (the)g(lo)31 b(calization)452 b(pro)31 b(cedure)447 b(de\014ned)h(in)h (\(4.26\),)471 b(\(4.27\).)733 b(W)-92 b(e)448 b(shall)h(de\014ne)f(t) -31 b(w)g(o)0 50422 y(new)376 b(constan)-31 b(ts)7235 50130 y(~)7189 50422 y Fo(\025)7835 50588 y Fj(\000)p Ff(2)9399 50422 y Fr(and)442 b(~)-620 b Fo(z)12072 50588 y Fj(\000)p Ff(2)13260 50422 y Fr(,)378 b(so)d(that)17761 50130 y(~)17715 50422 y Fo(\025)18361 50588 y Fj(\000)p Ff(2)19549 50422 y Fr(\()p Fo(Z)20735 50588 y Fj(\000)p Ff(2)21923 50422 y Fr(\))22353 50020 y Ff(2)23226 50422 y Fr(is)g(the)h(co)31 b(e\016cien)-31 b(t)377 b(of)f(the)f(lo)31 b(cal)377 b(term)f(prop)31 b(ortional)377 b(to)248 51686 y(\026)0 51966 y Fo(F)866 51391 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])712 52300 y Fk(\025)3336 51966 y Fr(\()p Fo( )4527 51564 y Ff([)p Fk(h;)p Fj(\000)p Ff(2])6998 51966 y Fr(\),)467 b(while)514 b(~)-619 b Fo(z)11685 52132 y Fj(\000)p Ff(2)12873 51966 y Fo(Z)13629 52132 y Fj(\000)p Ff(2)15044 51674 y Fr(^)14817 51966 y Fo( )15578 51391 y Ff([)p Fk(h;)p Fj(\000)p Ff(2]+)15582 52207 y(\026)15538 52411 y Fi(k)16067 52522 y Fm(4)16499 52411 y Fk(;)p Fj(\000)18728 51966 y Fo(D)19644 52132 y Fj(\000)20392 51966 y Fr(\()20881 51674 y(\026)20822 51966 y Fn(k)21494 52132 y Ff(4)21991 51966 y Fr(\))38 b(^)-591 b Fo(g)22949 52132 y Fj(\000)23696 51966 y Fr(\()24185 51674 y(\026)24126 51966 y Fn(k)24798 52132 y Ff(4)25295 51966 y Fr(\))25994 51686 y(^)25725 51966 y Fo(J)26383 52030 y Ff(\026)26339 52234 y Fi(k)26868 52345 y Fm(4)27802 51966 y Fr(denotes)447 b(the)g(sum)f(of)h(all)g(lo) 31 b(cal)448 b(terms)f(with)0 53383 y(t)-31 b(w)g(o)371 b(external)f(lines)g(pro)31 b(duced)368 b(in)i(the)f(second)g(in)-31 b(tegration)372 b(step.)443 54962 y(The)479 b(ab)31 b(o)-31 b(v)g(e)480 b(pro)31 b(cedure)477 b(can)i(b)31 b(e)478 b(iterated)i(up)f(to)g(scale)g Fo(h)319 b Fr(+)g(1,)507 b(without)481 b(an)-31 b(y)479 b(imp)31 b(ortan)-31 b(t)481 b(di\013erence.)820 b(In)0 56379 y(particular,)433 b(for)419 b(all)h(marginal)h(terms)e(\(necessarily)h(with)g(four)f(external)h (lines\))f(suc)-31 b(h)419 b(that)h(one)f(of)h(the)f(\014eld)0 57796 y(v)-61 b(ariables)429 b(b)31 b(elonging)430 b(to)g Fo(\016)42 b(\032)428 b Fr(in)h Fo(T)14666 57962 y Ff(1)15162 57796 y Fr(\()p Fo( )40 b Fr(\))429 b(is)g(con)-31 b(tracted)430 b(at)f(scale)g Fo(i)406 b Fh(\025)g Fo(j)63 b Fr(,)444 b(w)-31 b(e)429 b(put)g Fh(R)406 b Fr(=)g(1.)671 b(W)-92 b(e)428 b(can)g(do)h(that,)0 59214 y(b)31 b(ecause,)388 b(in)c(this)h(case,)j(the)d(second)f(\014eld)g(v)-61 b(ariable)385 b(b)31 b(elonging)386 b(to)f Fo(\016)42 b(\032)384 b Fr(has)g(to)h(b)31 b(e)384 b(con)-31 b(tracted)386 b(at)e(scale)h Fo(h)p Fr(,)j(so)0 60631 y(that)357 b(the)e(extra)h (factor)g Fo(\015)10916 60229 y Fj(\000)p Ff(\()p Fk(i)p Fj(\000)p Fk(h)p Ff(\))14234 60631 y Fr(of)g(\(4.22\))i(has)d(the)h (e\013ect)f(of)h(automatically)k(regularize)c(their)f(con)-31 b(tribution)0 62048 y(to)370 b(the)f(tree)g(expansion)i(of)12129 61769 y(~)11878 62048 y Fo(G)12748 61647 y Ff(4)12748 62322 y(+)13484 62048 y Fr(\()13973 61756 y(\026)13914 62048 y Fn(k)14586 62214 y Ff(1)15083 62048 y Fo(;)15634 61756 y Fr(\026)15574 62048 y Fn(k)16246 62214 y Ff(2)16743 62048 y Fo(;)17294 61756 y Fr(\026)17234 62048 y Fn(k)17906 62214 y Ff(3)18403 62048 y Fo(;)18955 61756 y Fr(\026)18894 62048 y Fn(k)19566 62214 y Ff(4)20064 62048 y Fr(\),)f(to)g(b)31 b(e)368 b(describ)31 b(ed)369 b(later.)443 63627 y(Note)380 b(that,)j(as)c(in)h(the)f(case)g Fo(j)388 b Fr(=)324 b Fh(\000)p Fr(1,)382 b(there)d(is)g(no)h(problem)f(connected)h(with)h (the)e(presence)f(of)i(the)f(factors)131 65045 y(~)-684 b Fo(\037)p Fr(\()p Fn(p)p Fr(\))479 b(and)f Fo(D)5916 65211 y Fj(\000)6664 65045 y Fr(\()p Fn(p)p Fr(\))p Fo(D)9147 65211 y Ff(+)9884 65045 y Fr(\()p Fn(p)p Fr(\))11451 64643 y Fj(\000)p Ff(1)12640 65045 y Fr(.)820 b(In)478 b(fact,)507 b(if)478 b(the)h(\014eld)23760 64752 y(^)23533 65045 y Fo( )24294 64571 y Ff(+)24298 65285 y(\026)24254 65490 y Fi(k)24783 65601 y Fm(4)25215 65490 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)27952 65045 y Fr(app)31 b(earing)479 b(in)g(the)f(de\014nition)i(of)f Fo(T)43661 65211 y Ff(1)44157 65045 y Fr(\()p Fo( )40 b Fr(\))479 b(or)0 66581 y Fo(T)647 66747 y Fj(\006)1394 66581 y Fr(\()p Fo( )40 b Fr(\))263 b(is)f(con)-31 b(tracted)263 b(on)f(scale)h Fo(j)63 b Fr(,)284 b(eac)-31 b(h)262 b(momen)-31 b(tum)264 b(deriv)-61 b(ativ)-31 b(e)264 b(related)f(with)g(the)f(regularization)j(pro)31 b(cedure)0 67999 y(pro)g(duces)415 b(the)h(righ)-31 b(t)417 b Fo(\015)10011 67597 y Fj(\000)p Fk(j)11585 67999 y Fr(dimensional)h(factor,)429 b(since)415 b Fn(p)h Fr(is)g(of)g(order)f Fo(\015)31187 67597 y Fk(j)32070 67999 y Fr(and)h(the)g(deriv)-61 b(ativ)-31 b(es)416 b(of)548 b(~)-684 b Fo(\037)o Fr(\()p Fn(p)p Fr(\))417 b(are)0 69416 y(di\013eren)-31 b(t)543 b(from)g(0)g(only)h(for)f(momen)-31 b(ta)545 b(of)f(order)e Fo(\015)23040 69014 y Fk(h)23614 69416 y Fr(.)1013 b(If,)587 b(on)543 b(the)g(con)-31 b(trary)-92 b(,)587 b(the)543 b(\014eld)40102 69124 y(^)39875 69416 y Fo( )40636 68942 y Ff(+)40640 69657 y(\026)40596 69861 y Fi(k)41125 69972 y Fm(4)41557 69861 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)44358 69416 y Fr(is)g(not)0 71024 y(con)-31 b(tracted,)355 b(then)349 b(the)g(renormalization)i(pro)31 b(cedure)348 b(is)h(tuned)g(so)f(that)481 b(~)-684 b Fo(\037)p Fr(\()p Fn(p)p Fr(\))349 b(and)g Fo(D)36745 71190 y Fj(\000)37493 71024 y Fr(\()p Fn(p)p Fr(\))p Fo(D)39976 71190 y Ff(+)40713 71024 y Fr(\()p Fn(p)p Fr(\))42280 70622 y Fj(\000)p Ff(1)43818 71024 y Fr(are)f(not)0 72441 y(a\013ected)370 b(b)-31 b(y)369 b(the)g(regularization)j(pro)31 b(cedure.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 443 1107 a Fr(A)-31 b(t)370 b(step)f Fh(\000)p Fo(j)63 b Fr(,)369 b(w)-31 b(e)370 b(get)g(an)g(expression)e (of)i(the)g(form)4412 3590 y(\026)4274 3870 y Fh(V)5043 3413 y Ff(\()p Fk(j)45 b Ff(\))6201 3870 y Fr(\()p Fo( )7392 3413 y Ff([)p Fk(h;j)g Ff(])9141 3870 y Fr(\))308 b(=)g Fo(T)11695 4036 y Ff(1)12191 3870 y Fr(\()p Fo( )13382 3413 y Ff([)p Fk(h;j)45 b Ff(])15131 3870 y Fr(\))247 b(+)f Fo(\027)17462 4036 y Fk(j)m(;)p Ff(+)18823 3870 y Fo(T)19470 4036 y Ff(+)20205 3870 y Fr(\()p Fo( )21396 3413 y Ff([)p Fk(h;j)45 b Ff(])23145 3870 y Fr(\))246 b(+)g Fo(\027)25475 4036 y Fk(j)m(;)p Fj(\000)26848 3870 y Fo(T)27495 4036 y Fj(\000)28242 3870 y Fr(\()p Fo( )29433 3413 y Ff([)p Fk(h;j)45 b Ff(])31183 3870 y Fr(\)+)4520 6969 y(+)5627 4744 y Fe(2)5627 6737 y(4)6411 6677 y Fr(~)6365 6969 y Fo(\025)7011 7135 y Fk(j)7477 6969 y Fo(Z)8312 6512 y Ff(2)8233 7243 y Fk(j)9056 6689 y Fr(\026)8808 6969 y Fo(F)9674 6394 y Ff([)p Fk(h;j)g Ff(])9520 7303 y Fk(\025)11423 6969 y Fr(\()p Fo( )12614 6512 y Ff([)p Fk(h;j)g Ff(])14363 6969 y Fr(\))246 b(+)16379 5585 y Fj(\000)p Ff(1)16146 5918 y Fe(X)16244 8275 y Fk(i)p Ff(=)p Fk(j)17996 6969 y Fr(~)-620 b Fo(z)18444 7135 y Fk(i)18813 6969 y Fo(Z)19569 7135 y Fk(i)20164 6677 y Fr(^)19937 6969 y Fo( )20698 6394 y Ff([)p Fk(h;j)45 b Ff(]+)20702 7210 y(\026)20658 7414 y Fi(k)21187 7525 y 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Fh(V)29361 15113 y Ff(\()p Fk(h)p Ff(\))30627 15514 y Fr(\()p Fo( )31818 15113 y Ff(\()p Fk(h)p Ff(\))33084 15514 y Fr(\))f(and)h(an)f(arbitrary)g(n)-31 b(um)g(b)31 b(er)376 b(of)0 16932 y(terms)314 b(tak)-31 b(en)314 b(from)g(the)g(e\013ectiv)-31 b(e)315 b(p)31 b(oten)-31 b(tial)316 b Fh(V)20004 16530 y Ff(\()p Fk(h)p Ff(\))21270 16932 y Fr(\()p Fo( )22461 16530 y Ff(\()p Fk(h)p Ff(\))23726 16932 y Fr(\).)475 b(One)313 b(of)h(the)g(external)h(lines)f(is)f(asso) 31 b(ciated)315 b(with)g(the)0 18349 y(free)409 b(propagator)i Fo(g)8397 18515 y Fj(\000)9144 18349 y Fr(\()9633 18057 y(\026)9574 18349 y Fn(k)10246 18515 y Ff(4)10743 18349 y Fr(\),)420 b(the)410 b(other)g(three)f(are)g(asso)31 b(ciated)411 b(with)f(propagators)h(of)f(scale)g Fo(h)f Fr(and)g(momen)-31 b(ta)59 19474 y(\026)0 19766 y Fn(k)672 19932 y Fk(i)1040 19766 y Fr(,)370 b Fo(i)308 b Fr(=)f(1)p Fo(;)184 b Fr(2)p Fo(;)g Fr(3.)443 22152 y Fp(R)-57 b(emark.)456 b Fr(With)258 b(resp)31 b(ect)256 b(to)i(the)f(expansion)h(for)g 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(constan)-31 b(ts.)0 27821 y(A)g(t)327 b(a)f(\014rst)f(sigh)-31 b(t,)337 b(it)326 b(seems)g(that)h(no)-31 b(w)327 b(w)-31 b(e)326 b(ha)-31 b(v)g(e)327 b(a)g(problem)f(more)h(di\016cult)g(than)g (the)f(initial)j(one;)341 b(w)-31 b(e)326 b(started)0 29238 y(from)401 b(the)g(expansion)h(for)e Fo(G)12354 28837 y Ff(4)12354 29512 y(+)13090 29238 y Fr(,)408 b(whic)-31 b(h)402 b(is)e(con)-31 b(v)g(ergen)g(t)403 b(if)e(the)g(running)f (coupling)j Fo(\025)35962 29404 y Fk(j)36828 29238 y Fr(is)e(small,)410 b(and)401 b(w)-31 b(e)401 b(ha)-31 b(v)g(e)0 30656 y(reduced)367 b(the)i(problem)g(to)f(that)i(of)e(con) -31 b(trolling)372 b(the)c(\015o)-31 b(w)369 b(of)g(four)f(running)g (coupling)i(constan)-31 b(ts,)370 b Fo(\027)42932 30822 y Ff(+)p Fk(;j)44341 30656 y Fr(,)e Fo(\027)45563 30822 y Fk(j)m(;)p Fj(\000)46937 30656 y Fr(,)0 32073 y Fo(\025)646 32239 y Fk(j)1112 32073 y Fr(,)1864 31781 y(~)1817 32073 y Fo(\025)2463 32239 y Fk(j)2930 32073 y Fr(.)561 b(Ho)-31 b(w)g(ev)g(er,)400 b(w)-31 b(e)393 b(will)h(see)e(that,)400 b(under)391 b(the)i(h)-31 b(yp)31 b(othesis)27527 31781 y(\026)27481 32073 y Fo(\025)28127 32239 y Fk(h)29046 32073 y Fh(\024)346 b Fo(")30769 32239 y Ff(1)31266 32073 y Fr(,)398 b(also)393 b(the)g(\015o)-31 b(w)393 b(of)g Fo(\027)40304 32239 y Fk(j)m(;)p Ff(+)41665 32073 y Fr(,)398 b Fo(\027)42917 32239 y Fk(j)m(;)p Fj(\000)44290 32073 y Fr(,)45042 31781 y(~)44996 32073 y Fo(\025)45642 32239 y Fk(j)46500 32073 y Fr(is)0 33490 y(b)31 b(ounded;)415 b(one)399 b(uses)f(the)i(coun)-31 b(terterms)399 b Fo(\027)18509 33656 y Ff(+)19244 33490 y Fo(;)184 b(\027)20282 33656 y Fj(\000)21429 33490 y Fr(\(this)400 b(is)f(the)g(reason)h(wh)-31 b(y)399 b(w)-31 b(e)400 b(in)-31 b(tro)31 b(duced)400 b(them)g(in)f Fh(x)p Fr(3\))h(to)0 34908 y(imp)31 b(ose)363 b(that)g Fo(\027)6658 35074 y Ff(+)p Fk(;j)8067 34908 y Fo(;)184 b(\027)9105 35074 y Fk(j)m(;)p Fj(\000)10841 34908 y Fr(are)362 b(decreasing)h(and)g(v)-61 b(anishing)363 b(at)g Fo(j)371 b Fr(=)307 b Fo(h)p Fr(,)364 b(and)e(then)h(that)g(the) g(b)31 b(eta)363 b(functions)g(for)46 36033 y(~)0 36325 y Fo(\025)646 36491 y Fk(j)1481 36325 y Fr(and)370 b Fo(\025)4280 36491 y Fk(j)5115 36325 y Fr(are)f(iden)-31 b(tical)371 b(up)e(to)h(exp)31 b(onen)-31 b(tially)372 b(deca)-31 b(ying)371 b Fo(O)31 b Fr(\()p Fo(\015)27689 35923 y Fk(\034)90 b(j)28655 36325 y Fr(\))369 b(terms.)0 38711 y Fn(4.5)1314 b Fp(The)397 b(tr)-57 b(e)g(e)396 b(structur)-57 b(e)396 b(of)g(the)g(exp)-57 b(ansion.)443 40151 y Fr(In)372 b(order)g(to)i(describ)31 b(e)372 b(the)h(tree)g (expansion)h(of)21053 39872 y(\026)20915 40151 y Fh(V)21684 39750 y Ff(\()p Fk(j)45 b Ff(\))22842 40151 y Fr(\()p Fo( )24033 39750 y Ff([)p Fk(h;j)g Ff(])25782 40151 y Fr(\),)375 b Fo(j)i Fh(2)313 b Fr([)p Fo(h)249 b Fr(+)g(1)p Fo(;)184 b Fh(\000)p Fr(1],)376 b(and)36937 39872 y(~)36686 40151 y Fo(G)37556 39750 y Ff(4)37556 40425 y(+)38292 40151 y Fr(\()38781 39859 y(\026)38722 40151 y Fn(k)39394 40317 y Ff(1)39890 40151 y Fo(;)40442 39859 y Fr(\026)40381 40151 y Fn(k)41053 40317 y Ff(2)41551 40151 y Fo(;)42102 39859 y Fr(\026)42042 40151 y Fn(k)42714 40317 y Ff(3)43211 40151 y Fo(;)43762 39859 y Fr(\026)43702 40151 y Fn(k)44374 40317 y Ff(4)44872 40151 y Fr(\),)e(w)-31 b(e)0 41569 y(ha)g(v)g(e)370 b(to)g(mo)31 b(dify)371 b(the)e(tree)g(de\014nition)i (giv)-31 b(en)370 b(in)g Fh(x)p Fr(2,)g(after)g(Fig.)493 b(11,)370 b(in)f(the)h(follo)-31 b(wing)373 b(w)-31 b(a)g(y)-92 b(.)334 43954 y(1\))334 b(Instead)g(of)g(t)-31 b(w)g(o,)343 b(there)334 b(are)f(six)h(t)-31 b(yp)31 b(es)334 b(of)g(sp)31 b(ecial)335 b(endp)31 b(oin)-31 b(ts.)481 b(There)334 b(are)f(still)i(the)f(endp)31 b(oin)-31 b(ts)335 b(of)f(t)-31 b(yp)31 b(e)334 b Fo(')p Fr(,)0 45372 y(de\014ned)288 b(exactly)i(as)e(b)31 b(efore,)304 b(but)289 b(there)f(is)g(no)g(endp) 31 b(oin)-31 b(t)290 b(of)e(t)-31 b(yp)31 b(e)289 b Fo(J)106 b Fr(.)466 b(In)287 b(addition,)308 b(w)-31 b(e)288 b(ha)-31 b(v)g(e)289 b(sp)31 b(ecial)289 b(endp)31 b(oin)-31 b(ts)0 46789 y(of)370 b(t)-31 b(yp)31 b(e)369 b Fo(T)4398 46955 y Ff(1)4895 46789 y Fr(,)g Fo(T)6218 46955 y Ff(+)6953 46789 y Fr(,)h Fo(T)8277 46955 y Fj(\000)9024 46789 y Fr(,)9746 46497 y(~)9700 46789 y Fo(\025)f Fr(and)436 b(~)-620 b Fo(z)49 b Fr(,)369 b(asso)31 b(ciated)371 b(in)e(an)h(ob)-31 b(vious)370 b(w)-31 b(a)g(y)371 b(with)f(the)g (\014v)-31 b(e)369 b(lo)31 b(cal)371 b(terms)e(of)g(\(4.43\).)369 48230 y(2\))h(There)f(are)g(only)h(trees)f(with)h(one)g(and)f(only)h (one)g(sp)31 b(ecial)369 b(endp)31 b(oin)-31 b(t)371 b(of)e(t)-31 b(yp)31 b(e)370 b(di\013eren)-31 b(t)369 b(from)h Fo(')p Fr(.)294 49670 y(3\))294 b(The)h(scale)f(index)g(is)g Fh(\024)308 b Fr(+1)294 b(for)g(the)g(endp)31 b(oin)-31 b(ts)295 b(of)f(t)-31 b(yp)31 b(e)294 b Fo(T)25840 49836 y Ff(1)26337 49670 y Fr(,)309 b Fo(T)27600 49836 y Ff(+)28629 49670 y Fr(or)294 b Fo(T)30557 49836 y Fj(\000)31597 49670 y Fr(,)310 b(while)295 b(it)f(is)g Fh(\024)308 b Fr(0)294 b(for)g(the)g(endp)31 b(oin)-31 b(ts)0 51088 y(of)447 b(t)-31 b(yp)31 b(e)3952 50796 y(~)3906 51088 y Fo(\025)446 b Fr(or)512 b(~)-619 b Fo(z)48 b Fr(.)724 b(Moreo)-31 b(v)g(er,)467 b(the)446 b(scale)h(index)g(of)g(an)f(endp)31 b(oin)-31 b(t)448 b Fo(v)485 b Fr(of)447 b(t)-31 b(yp)31 b(e)447 b Fo(T)34282 51254 y Ff(1)34779 51088 y Fr(,)465 b Fo(T)36198 51254 y Ff(+)36933 51088 y Fr(,)h Fo(T)38353 51254 y Fj(\000)39546 51088 y Fr(or)41026 50796 y(~)40980 51088 y Fo(\025)446 b Fr(is)g(equal)h(to)0 52505 y Fo(h)638 52671 y Fk(v)1109 52450 y Fc(0)1709 52505 y Fr(+)246 b(1,)370 b(if)g Fo(v)5638 52103 y Fj(0)6317 52505 y Fr(is)f(the)g(non)h (trivial)h(v)-31 b(ertex)369 b(immediately)k(preceding)c Fo(v)40 b Fr(.)381 53946 y(4\))381 b(If)g(the)f(tree)h(has)g(more)g (than)g(one)g(endp)31 b(oin)-31 b(t)382 b(and)f(one)g(of)g(them)h(is)e (of)h(t)-31 b(yp)31 b(e)448 b(~)-619 b Fo(z)48 b Fr(,)384 b(the)d(v)-31 b(ertex)381 b Fo(v)41329 54112 y Ff(0)42206 53946 y Fr(of)g(the)g(tree)0 55363 y(m)-31 b(ust)370 b(ha)-31 b(v)g(e)370 b(scale)g Fo(h)e Fr(and)i(its)f(scale)h(index)g (is)f(equal)h(to)g(an)-31 b(y)370 b(v)-61 b(alue)369 b(b)31 b(et)-31 b(w)g(een)370 b Fo(h)246 b Fr(+)g(1)370 b(and)f(0.)340 56804 y(5\))342 b(Giv)-31 b(en)340 b(a)h(tree)f(with)i (one)f(endp)31 b(oin)-31 b(t)341 b Fo(v)17537 56970 y Ff(1)18374 56804 y Fr(of)g(t)-31 b(yp)31 b(e)341 b Fo(T)22715 56970 y Ff(1)23552 56804 y Fr(and)f(scale)h(index)g Fo(h)31889 56970 y Fk(v)32332 57081 y Fm(1)33126 56804 y Fr(=)308 b(+1,)347 b(the)341 b Fh(R)f Fr(op)31 b(eration)342 b(in)f(the)0 58221 y(non)f(trivial)i(v)-31 b(ertices)340 b(of)g(the)g(path)g Fh(C)404 b Fr(connecting)342 b Fo(v)22055 58387 y Ff(1)22890 58221 y Fr(to)f Fo(v)24751 58387 y Ff(0)25587 58221 y Fr(dep)31 b(ends)338 b(on)i(the)g(set)g Fo(P)35632 58387 y Ff(1)36468 58221 y Fr(of)g(external)h(lines)f(in)g(the)0 59638 y(v)-31 b(ertex)370 b Fo(v)3932 59237 y Fj(0)3892 59913 y Ff(1)4757 59638 y Fr(\(of)g(scale)g(0\))g(immediately)i (preceding)e Fo(v)22277 59804 y Ff(1)22773 59638 y Fr(.)443 61079 y(If)338 b Fo(P)2230 61245 y Ff(1)3065 61079 y Fr(con)-31 b(tains)341 b(no)e(one)g(of)g(the)g(t)-31 b(w)g(o)340 b(external)g(lines)f(of)h Fo(v)24561 61245 y Ff(1)25395 61079 y Fr(b)31 b(elonging)341 b(to)f(the)f Fo(\016)42 b(\032)338 b Fr(part)h(of)h(the)f(corresp)31 b(onding)0 62496 y Fo(T)647 62662 y Ff(1)1143 62496 y Fr(\()p Fo( )40 b Fr(\))377 b(term,)h(then)e Fh(R)f Fr(is)h(de\014ned)f (in)h(agreemen)-31 b(t)377 b(with)g(the)f(lo)31 b(calization)379 b(pro)31 b(cedure)375 b(bringing)i(to)f(\(4.38\))j(for)0 63914 y(all)359 b(v)-31 b(ertices)359 b Fo(v)346 b Fh(2)308 b(C)65 b Fr(,)360 b(suc)-31 b(h)357 b(that)i Fo(v)347 b(>)308 b(v)16228 64080 y Ff(2)16724 63914 y Fr(,)360 b Fo(v)17928 64080 y Ff(2)18782 63914 y Fr(b)31 b(eing)359 b(the)f(higher)g(v)-31 b(ertex,)361 b(p)31 b(ossibly)359 b(coinciding)h(with)f Fo(v)43224 63512 y Fj(0)43184 64188 y Ff(1)43680 63914 y Fr(,)i(whose)0 65331 y(set)369 b(of)h(external)h (lines)e(do)31 b(es)369 b(not)i(con)-31 b(tain)371 b(the)f(\014eld) 22697 65039 y(^)22470 65331 y Fo( )23231 64857 y Ff(+)23235 65572 y(\026)23191 65776 y Fi(k)23720 65887 y Fm(4)24152 65776 y Fj(\000)p Fi(p)25825 65331 y Fr(of)g Fo(T)27733 65497 y Ff(1)28230 65331 y Fr(\()p Fo( )40 b Fr(\).)494 b(F)-92 b(or)368 b(the)i(remaining)h(v)-31 b(ertices)370 b(of)g Fh(C)65 b Fr(,)369 b Fh(R)0 66748 y Fr(is)g(de\014ned)g(in)g (the)h(usual)f(w)-31 b(a)g(y)-92 b(.)443 68189 y(If)466 b Fo(P)2358 68355 y Ff(1)3320 68189 y Fr(con)-31 b(tains)468 b(b)31 b(oth)467 b(the)g(t)-31 b(w)g(o)468 b(external)f(lines)g(of)g Fo(v)23556 68355 y Ff(1)24518 68189 y Fr(b)31 b(elonging)468 b(to)f Fo(\016)42 b(\032)466 b Fr(\(hence)h(the)g(line)g(of)g(momen)-31 b(tum)59 69314 y(\026)0 69606 y Fn(k)672 69772 y Ff(4)1440 69606 y Fh(\000)271 b Fn(p)408 b Fr(can)f(not)h(b)31 b(elong)409 b(to)f Fo(P)13437 69772 y Ff(1)13933 69606 y Fr(\),)418 b(w)-31 b(e)408 b(de\014ne)f Fh(R)g Fr(in)h(agreemen)-31 b(t)408 b(with)h(the)e(remark)h(follo)-31 b(wing)411 b(\(4.24\),)420 b(up)407 b(to)0 71024 y(the)331 b(higher)g(v)-31 b(ertex)332 b Fo(v)9069 71190 y Ff(2)9872 71024 y Fo(<)308 b(v)11578 71190 y Ff(1)12074 71024 y Fr(,)339 b(where)330 b(at)i(least)g(one)f(of)g(the)g(lines)h(of)f Fo(\016)42 b(\032)331 b Fr(do)31 b(es)331 b(not)g(b)31 b(elong)332 b(to)g Fo(P)39890 71190 y Ff(1)40717 71024 y Fr(an)-31 b(ymore.)481 b(F)-92 b(or)0 72441 y Fo(v)347 b Fh(\024)307 b Fo(v)2589 72607 y Ff(2)3085 72441 y Fr(,)370 b Fh(R)f Fr(is)g(de\014ned)g(in)g(the)h(usual)f(w)-31 b(a)g(y)-92 b(.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 443 1107 a Fr(If)314 b Fo(P)2206 1273 y Ff(1)3017 1107 y Fr(con)-31 b(tains)317 b(only)e(one)g(of)h(the) f(t)-31 b(w)g(o)316 b(external)g(lines)f(of)g Fo(v)25188 1273 y Ff(1)25999 1107 y Fr(b)31 b(elonging)317 b(to)e Fo(\016)42 b(\032)315 b Fr(and)g(one)g(de\014nes)f Fo(v)41866 1273 y Ff(2)42677 1107 y Fr(as)h(b)31 b(efore,)0 2524 y Fh(R)375 b Fr(is)g(de\014ned)f(along)j Fh(C)439 b Fr(in)376 b(agreemen)-31 b(t)376 b(with)g(the)f(ob)-31 b(vious)377 b(generalization)h(of)d(\(4.34\),)380 b(for)375 b Fo(v)356 b(>)317 b(v)41473 2690 y Ff(2)41969 2524 y Fr(.)510 b(In)375 b Fo(v)44713 2690 y Ff(2)45584 2524 y Fr(one)0 3942 y(has)397 b(to)i(in)-31 b(tro)31 b(duce)398 b(a)f(new)h(lab)31 b(el)398 b(to)h(distinguish)g(t)-31 b(w)g(o)399 b(cases,)404 b(related)398 b(with)h(the)f(t)-31 b(w)g(o)399 b(di\013eren)-31 b(t)398 b(terms)f(in)h(the)0 5359 y(braces)334 b(of)h(the)f(r.h.s.)481 b(of)335 b(\(4.28\).)484 b(In)334 b(the)g(\014rst)g(case,)342 b Fh(R)307 b Fr(=)h(1)334 b(for)h(all)g Fo(v)347 b Fh(\024)307 b Fo(v)31330 5525 y Ff(2)31827 5359 y Fr(,)341 b(in)335 b(the)f(second)h(case)f Fh(R)g Fr(is)g(de\014ned)0 6776 y(in)369 b(the)h(usual)f(w)-31 b(a)g(y)-92 b(.)443 8194 y(If)347 b Fo(h)2166 8360 y Fk(v)2609 8471 y Fm(1)3404 8194 y Fh(\024)307 b Fr(0)348 b(and)g(w)-31 b(e)348 b(de\014ne)f Fo(P)13099 8360 y Ff(1)13943 8194 y Fr(and)h Fo(v)16611 8360 y Ff(2)17454 8194 y Fr(as)g(b)31 b(efore,)352 b(the)c(set)f Fo(P)26708 8360 y Ff(1)27205 8194 y Fr(,)352 b(as)347 b(w)-31 b(ell)349 b(as)f(the)g(set)f Fo(P)37064 8360 y Fk(v)37937 8194 y Fr(for)h(all)h Fo(v)d(<)308 b(v)43715 8360 y Ff(1)44211 8194 y Fr(,)352 b(m)-31 b(ust)0 9611 y(con)g(tain)401 b(at)f(least)g(one)f(of)h(the)f(t)-31 b(w)g(o)401 b(external)f(lines)g(of)f Fo(v)24036 9777 y Ff(1)24931 9611 y Fr(b)31 b(elonging)401 b(to)f Fo(\016)42 b(\032)p Fr(.)583 b(Moreo)-31 b(v)g(er,)407 b(if)400 b Fo(P)40317 9777 y Ff(1)41212 9611 y Fr(con)-31 b(tains)401 b(one)0 11028 y(of)370 b(these)f(lines,)h(then)f Fh(R)308 b Fr(=)f(1)370 b(for)f(all)h Fo(v)347 b Fh(\024)308 b Fo(v)18610 11194 y Ff(2)19106 11028 y Fr(.)344 12445 y(6\))344 b(A)g(similar,)351 b(but)344 b(simpler,)350 b(discussion)344 b(can)g(b)31 b(e)343 b(done)h(for)g(the)h(trees)e(con) -31 b(taining)347 b(an)d(endp)31 b(oin)-31 b(t)345 b(of)f(t)-31 b(yp)31 b(e)345 b Fo(T)46190 12611 y Fj(\006)46937 12445 y Fr(.)0 13863 y(W)-92 b(e)344 b(do)g(not)i(giv)-31 b(e)345 b(the)g(details,)351 b(but)345 b(only)g(stress)e(that)j(there)e(is)h (no)-31 b(w)345 b(no)g(v)-31 b(ertex)345 b Fo(v)383 b Fr(with)346 b Fh(j)p Fo(P)38517 14029 y Fk(v)39043 13863 y Fh(j)308 b Fr(=)g(2)344 b(or)h Fh(j)p Fo(P)44074 14029 y Fk(v)44600 13863 y Fh(j)308 b Fr(=)f(4,)0 15280 y(for)369 b(whic)-31 b(h)370 b Fh(R)308 b Fr(=)f(1.)0 17642 y Fn(4.6)945 b Fp(The)397 b(\015ow)e(of)h Fo(\027)9031 17808 y Fk(j)m(;)p Fj(\006)10404 17642 y Fp(.)443 19060 y Fr(The)380 b(de\014nitions)h(of) f(the)g(previous)f(sections)i(imply)g(that)g(there)e(is)h(no)g(con)-31 b(tribution)382 b(to)e Fo(\027)39110 19226 y Fk(j)m(;)p Fj(\006)40483 19060 y Fr(,)j(coming)e(from)0 20477 y(trees)464 b(with)i(a)f(sp)31 b(ecial)466 b(endp)31 b(oin)-31 b(t)465 b(of)h(t)-31 b(yp)31 b(e)18780 20185 y(~)18734 20477 y Fo(\025)464 b Fr(or)531 b(~)-619 b Fo(z)48 b Fr(.)779 b(Moreo)-31 b(v)g(er,)490 b(b)31 b(ecause)464 b(of)h(the)g(symmetry)h (\(4.35\))h(of)f(the)0 21894 y(propagators)343 b(\(see)e(remark)h (after)g(\(4.40\)\),)351 b Fo(\027)19076 22060 y Fk(j)m(;)p Ff(+)20778 21894 y Fr(gets)342 b(no)g(con)-31 b(tribution)344 b(from)e(trees)f(with)i(a)f(sp)31 b(ecial)342 b(endp)31 b(oin)-31 b(t)0 23312 y(of)490 b(t)-31 b(yp)31 b(e)489 b Fo(\027)4538 23478 y Fk(j)m(;)p Fj(\000)5912 23312 y Fr(,)519 b(and)489 b(vicev)-31 b(ersa.)854 b(Finally)-92 b(,)520 b(and)490 b(v)-31 b(ery)489 b(imp)31 b(ortan)-31 b(t,)522 b(if)490 b(a)f(tree)g(con)-31 b(tributing)491 b(to)f Fo(\027)42119 23478 y Fk(j)m(;)p Fj(\006)43982 23312 y Fr(has)f(an)0 24729 y(endp)31 b(oin)-31 b(t)370 b(of)g(t)-31 b(yp)31 b(e)370 b Fo(T)9011 24895 y Ff(1)9507 24729 y Fr(,)g(this)f(endp)31 b(oin)-31 b(t)371 b(m)-31 b(ust)369 b(ha)-31 b(v)g(e)371 b(scale)e(index)h(+1.)443 26146 y(The)f(follo)-31 b(wing)373 b(Lemma)d(has)f(an)h(imp)31 b(ortan)-31 b(t)371 b(role)f(in)f(the)h(follo)-31 b(wing.)0 28508 y Fq(Lemma)448 b(4.1)1341 b Fp(If)8308 28216 y Fr(\026)8261 28508 y Fo(\025)8907 28674 y Fk(h)9903 28508 y Fp(is)422 b(smal)57 b(l)421 b(enough)h(\()g(uniformly)h(in)g Fo(h)p Fp(\),)428 b(it)423 b(is)f(p)-57 b(ossible)422 b(to)f(cho)-57 b(ose)422 b Fo(\027)38888 28674 y Ff(+)40045 28508 y Fp(and)g Fo(\027)42768 28674 y Fj(\000)43937 28508 y Fp(so)f(that)0 29926 y Fo(\027)547 30092 y Fk(h;!)2276 29926 y Fr(=)308 b(0)396 b Fp(and)14829 31343 y Fh(j)p Fo(\027)15683 31509 y Fk(j)m(;!)16951 31343 y Fh(j)308 b(\024)f Fo(c)19213 31509 y Ff(0)19756 31051 y Fr(\026)19710 31343 y Fo(\025)20356 31509 y Fk(h)20929 31343 y Fo(\015)21563 30886 y Fk(\022)24 b(j)23607 31343 y Fo(;)1317 b(h)246 b Fr(+)g(1)308 b Fh(\024)f Fo(j)371 b Fh(\024)307 b Fr(0)h Fo(;)12001 b Fr(\(4)p Fo(:)p Fr(44\))0 33363 y Fp(wher)-57 b(e)396 b Fr(0)308 b Fo(<)f(\022)338 b(<)308 b Fr(1)p Fo(=)p Fr(4)p Fp(,)398 b Fo(c)10057 33529 y Ff(0)10949 33363 y Fp(is)e(a)g(suitable)g(c)-57 b(onstant,)397 b(indep)-57 b(endent)397 b(of)g Fo(h)p Fp(,)f(and)g Fo(\027)33379 33529 y Ff(0)p Fk(;!)35031 33363 y Fr(=)308 b Fo(\027)36747 33529 y Fk(!)37387 33363 y Fp(.)0 35725 y Fq(Pr)-25 b(oof)497 b(-)943 b Fr(The)446 b(previous)f(remarks)g(imply)i(that)f(there)f (exists)519 b(\026)-626 b Fo(")28670 35891 y Ff(1)29601 35725 y Fh(\024)508 b Fr(\026)-627 b Fo(")31412 35891 y Ff(0)31909 35725 y Fr(,)464 b(suc)-31 b(h)445 b(that,)466 b(if)39191 35433 y(\026)39145 35725 y Fo(\025)39791 35891 y Fk(h)40798 35725 y Fh(\024)508 b Fr(\026)-626 b Fo(")42610 35891 y Ff(1)43106 35725 y Fr(,)465 b(w)-31 b(e)446 b(can)0 37143 y(write)8737 38560 y Fo(\027)9284 38726 y Fk(j)45 b Fj(\000)p Ff(1)p Fk(;!)12039 38560 y Fr(=)308 b Fo(\027)13755 38726 y Fk(j)m(;!)15268 38560 y Fr(+)246 b Fo(\014)17059 38103 y Ff(\()p Fk(j)45 b Ff(\))17001 38834 y Fk(\027)t(;!)18352 38560 y Fr(\()p Fo(\025)19428 38726 y Fk(j)19895 38560 y Fo(;)184 b(\027)20933 38726 y Fk(j)m(;!)22201 38560 y Fr(;)g Fo(:)g(:)g(:)k Fr(;)c Fo(\025)25306 38726 y Ff(0)25803 38560 y Fo(;)g(\027)26841 38726 y Ff(0)p Fk(;!)28187 38560 y Fr(\))1107 b Fo(;)1292 b(h)246 b Fr(+)g(1)308 b Fh(\024)f Fo(j)371 b Fh(\024)308 b Fr(0)f Fo(;)5909 b Fr(\(4)p Fo(:)p Fr(45\))0 40580 y(with)370 b Fo(\025)3167 40746 y Ff(0)3971 40580 y Fr(=)308 b Fo(\025)p Fr(,)369 b Fo(\027)7009 40746 y Ff(0)p Fk(;!)8661 40580 y Fr(=)308 b Fo(\027)10377 40746 y Fk(!)11386 40580 y Fr(and)4082 43943 y Fo(\014)4766 43486 y Ff(\()p Fk(j)45 b Ff(\))4708 44216 y Fk(\027)t(;!)6059 43943 y Fr(\()p Fo(\025)7135 44109 y Fk(j)7602 43943 y Fo(;)184 b(\027)8640 44109 y Fk(j)m(;!)9908 43943 y Fr(;)g Fo(:)g(:)g(:)k Fr(;)c Fo(\025)13013 44109 y Ff(0)13510 43943 y Fo(;)g(\027)14548 44109 y Ff(0)p Fk(;!)15894 43943 y Fr(\))308 b(=)f Fo(\014)18484 43486 y Ff(\()p Fk(j)m(;)p Ff(1\))18426 44216 y Fk(\027)t(;!)20300 43943 y Fr(\()p Fo(\025)21376 44109 y Fk(j)21842 43943 y Fr(;)184 b Fo(:)g(:)g(:)k Fr(;)c Fo(\025)24947 44109 y Ff(0)25445 43943 y Fr(\))246 b(+)27908 42559 y Ff(0)27329 42891 y Fe(X)27228 45265 y Fk(j)27639 45044 y Fc(0)27938 45265 y Ff(=)p Fk(j)29214 43943 y Fo(\027)29761 44109 y Fk(j)30172 43887 y Fc(0)30471 44109 y Fk(;!)31532 43651 y Fr(~)31375 43943 y Fo(\014)32059 43486 y Ff(\()p Fk(j)m(;j)33442 43152 y Fc(0)33742 43486 y Ff(\))32001 44216 y Fk(\027)t(;!)34143 43943 y Fr(\()p Fo(\025)35219 44109 y Fk(j)35686 43943 y Fr(;)184 b Fo(:)g(:)g(:)k Fr(;)c Fo(\025)38791 44109 y Ff(0)39288 43943 y Fr(\))308 b Fo(:)4082 b Fr(\(4)p Fo(:)p Fr(46\))0 47409 y(Moreo)-31 b(v)g(er)369 b(giv)-31 b(en)371 b(a)e(p)31 b(ositiv)-31 b(e)371 b Fo(\022)338 b(<)307 b Fr(1)p Fo(=)p Fr(4,)371 b(there)e(are)g(constan)-31 b(ts)371 b Fo(c)27197 47575 y Ff(1)28062 47409 y Fr(and)f Fo(c)30694 47575 y Ff(2)31559 47409 y Fr(suc)-31 b(h)369 b(that)6432 49767 y Fh(j)p Fo(\014)7423 49310 y Ff(\()p Fk(j)m(;)p Ff(1\))7365 50041 y Fk(\027)t(;!)9238 49767 y Fr(\()p Fo(\025)10314 49933 y Fk(j)10781 49767 y Fr(;)184 b Fo(:)g(:)g(:)k Fr(;)c Fo(\025)13886 49933 y Ff(0)14383 49767 y Fr(\))p Fh(j)309 b(\024)e Fo(c)17076 49933 y Ff(1)17619 49475 y Fr(\026)17573 49767 y Fo(\025)18219 49933 y Fk(h)18792 49767 y Fo(\015)19426 49310 y Ff(2)p Fk(\022)24 b(j)21887 49767 y Fo(;)1292 b Fh(j)23951 49475 y Fr(~)23793 49767 y Fo(\014)24477 49310 y Ff(\()p Fk(j)m(;j)25860 48976 y Fc(0)26161 49310 y Ff(\))24419 50041 y Fk(\027)t(;!)26562 49767 y Fr(\()p Fo(\025)27638 49933 y Fk(j)28104 49767 y Fr(;)184 b Fo(:)g(:)g(:)k Fr(;)c Fo(\025)31209 49933 y Ff(0)31707 49767 y Fr(\))p Fh(j)308 b(\024)g Fo(c)34400 49933 y Ff(2)34942 49475 y Fr(\026)34896 49767 y Fo(\025)35542 49310 y Ff(2)35542 50041 y Fk(h)36116 49767 y Fo(\015)36750 49310 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(j)45 b Fj(\000)p Fk(j)39497 48976 y Fc(0)39796 49310 y Ff(\))40505 49767 y Fo(:)3603 b Fr(\(4)p Fo(:)p Fr(47\))0 52460 y(This)417 b(follo)-31 b(ws)420 b(from)d(the)g(fact)h(that)g Fo(\014)16237 51885 y Ff(\()p Fk(j)m(;)p Ff(1\))16179 52590 y Fk(\027)t(;!)18469 52460 y Fr(and)20827 52168 y(~)20669 52460 y Fo(\014)21353 51885 y Ff(\()p Fk(j)m(;j)22736 51551 y Fc(0)23037 51885 y Ff(\))21295 52590 y Fk(\027)t(;!)23854 52460 y Fr(are)f(giv)-31 b(en)418 b(b)-31 b(y)417 b(a)h(sum)e(of)i(trees)e(v)-31 b(erifying)419 b(the)e(b)31 b(ound)0 53878 y(\(2.32\))335 b(with)e Fo(d)6221 54044 y Fk(v)7055 53878 y Fo(>)307 b Fr(0,)340 b(with)333 b(at)g(least)g(an)f(end-p)31 b(oin)-31 b(t)333 b(resp)31 b(ectiv)-31 b(ely)333 b(at)g(scale)f(0)h(and)f(at)h (scale)f Fo(j)39596 53476 y Fj(0)39907 53878 y Fr(,)340 b(hence)331 b(one)i(can)0 55295 y(impro)-31 b(v)g(e)407 b(the)e(b)31 b(ound)406 b(resp)31 b(ectiv)-31 b(ely)406 b(b)-31 b(y)405 b(a)h(factor)g Fo(\015)22088 54893 y Ff(2)p Fk(\022)24 b(j)23847 55295 y Fr(and)406 b Fo(\015)26670 54893 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(j)45 b Fj(\000)p Fk(j)29417 54559 y Fc(0)29716 54893 y Ff(\))30118 55295 y Fr(.)601 b(In)404 b(the)i(follo)-31 b(wing)409 b(w)-31 b(e)406 b(shall)g(call)h(this)0 56838 y(prop)31 b(ert)-31 b(y)391 b(the)g Fp(short)415 b(memory)i(pr)-57 b(op)g(erty)p Fr(.)558 b(Note)391 b(that)i(the)e(b)31 b(ound)391 b(of)30334 56545 y(~)30176 56838 y Fo(\014)30860 56263 y Ff(\()p Fk(j)m(;j)32243 55928 y Fc(0)32543 56263 y Ff(\))30802 56967 y Fk(\027)t(;!)33335 56838 y Fr(is)g(of)h(order)38717 56545 y(\026)38671 56838 y Fo(\025)39317 56436 y Ff(2)39317 57151 y Fk(h)39890 56838 y Fr(,)397 b(instead)392 b(of)45763 56545 y(\026)45717 56838 y Fo(\025)46363 57004 y Fk(h)46937 56838 y Fr(,)0 58255 y(b)31 b(ecause)369 b(of)g(the)h(symmetry)g (\(4.35\),)i(but)e(a)f(b)31 b(ound)369 b(of)h(order)26512 57963 y(\026)26466 58255 y Fo(\025)27112 58421 y Fk(h)28054 58255 y Fr(w)-31 b(ould)371 b(b)31 b(e)368 b(su\016cien)-31 b(t.)443 59672 y(By)369 b(a)h(simple)g(iteration,)i(\(4.45\))f(can)f (also)g(b)31 b(e)369 b(written)h(in)g(the)f(form)11937 62995 y Fo(\027)12484 63161 y Fk(j)45 b Fj(\000)p Ff(1)p Fk(;!)15239 62995 y Fr(=)308 b Fo(\027)16955 63161 y Ff(0)p Fk(;!)18546 62995 y Fr(+)20332 61611 y Ff(0)19754 61943 y Fe(X)19653 64317 y Fk(j)20064 64096 y Fc(0)20363 64317 y Ff(=)p Fk(j)21638 62995 y Fo(\014)22322 62538 y Ff(\()p Fk(j)23079 62204 y Fc(0)23379 62538 y Ff(\))22264 63268 y Fk(\027)t(;!)23780 62995 y Fr(\()p Fo(\025)24856 63161 y Fk(j)25267 62939 y Fc(0)25622 62995 y Fo(;)184 b(\027)26660 63161 y Fk(j)27071 62939 y Fc(0)27372 63161 y Fk(;!)28275 62995 y Fr(;)g Fo(:)g(:)g(:)k Fr(;)c Fo(\025)31380 63161 y Ff(0)31878 62995 y Fo(;)g(\027)32916 63161 y Ff(0)p Fk(;!)34261 62995 y Fr(\))308 b Fo(:)9109 b Fr(\(4)p Fo(:)p Fr(48\))0 66551 y(W)-92 b(e)367 b(w)-31 b(an)g(t)370 b(to)e(sho)-31 b(w)368 b(that)h(it)g(is)f(p)31 b(ossible)368 b(to)g(c)-31 b(ho)31 b(ose)368 b Fo(\027)22887 66717 y Ff(0)p Fk(;!)24232 66551 y Fr(,)g(so)g(that)h Fo(\027)29209 66717 y Ff(0)p Fk(;!)30921 66551 y Fr(is)f(of)g(order)36234 66259 y(\026)36188 66551 y Fo(\025)36834 66717 y Fk(h)37774 66551 y Fr(and)g Fo(\027)40472 66717 y Fk(h;!)42202 66551 y Fr(=)307 b(0.)493 b(Since)0 67968 y(this)370 b(last)g(condition,)h(b) -31 b(y)370 b(\(4.48\),)i(is)d(equiv)-61 b(alen)-31 b(t)371 b(to)13530 71291 y Fo(\027)14077 71457 y Ff(0)p Fk(;!)15729 71291 y Fr(=)307 b Fh(\000)19087 69907 y Ff(0)18508 70239 y Fe(X)17942 72622 y Fk(j)45 b Ff(=)p Fk(h)p Ff(+1)20857 71291 y Fo(\014)21541 70834 y Ff(\()p Fk(j)g Ff(\))21483 71565 y Fk(\027)t(;!)22835 71291 y Fr(\()p Fo(\025)23911 71457 y Fk(j)24377 71291 y Fo(;)184 b(\027)25415 71457 y Fk(j)m(;!)26683 71291 y Fr(;)g Fo(:)g(:)g(:)k Fr(;)c Fo(\025)29788 71457 y Ff(0)30285 71291 y Fo(;)g(\027)31323 71457 y Ff(0)p Fk(;!)32669 71291 y Fr(\))308 b Fo(;)10701 b Fr(\(4)p Fo(:)p Fr(49\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 0 1107 a Fr(w)-31 b(e)425 b(see,)439 b(b)-31 b(y)425 b(inserting)h(\(4.49\))i(in)d(the)g(r.h.s.)660 b(of)425 b(\(4.48\),)442 b(that)426 b(w)-31 b(e)426 b(ha)-31 b(v)g(e)425 b(to)h(sho)-31 b(w)426 b(that)g(there)e(is)h(a)g(sequence)0 2524 y Fo(\027)p 0 2701 618 45 v 378 w Fr(=)307 b Fh(f)p Fo(\027)3193 2690 y Fk(j)3660 2524 y Fo(;)369 b(h)246 b Fr(+)g(1)308 b Fh(\024)g Fo(j)370 b Fh(\024)308 b Fr(0)p Fh(g)p Fr(,)370 b(suc)-31 b(h)369 b(that)h Fo(\027)17577 2690 y Ff(0)p Fk(;!)19291 2524 y Fr(is)f(of)h(order)24608 2232 y(\026)24562 2524 y Fo(\025)25208 2690 y Fk(h)26150 2524 y Fr(and)14692 6258 y Fo(\027)15239 6424 y Fk(j)16013 6258 y Fr(=)308 b Fh(\000)19536 4822 y Fk(j)18942 5206 y Fe(X)18227 7589 y Fk(j)18638 7368 y Fc(0)18938 7589 y Ff(=)p Fk(h)p Ff(+1)21441 6258 y Fo(\014)22125 5801 y Ff(\()p Fk(j)22882 5467 y Fc(0)23182 5801 y Ff(\))22067 6532 y Fk(\027)t(;!)23583 6258 y Fr(\()p Fo(\025)24659 6424 y Fk(j)25070 6203 y Fc(0)25425 6258 y Fo(;)184 b(\027)26463 6424 y Fk(j)26874 6203 y Fc(0)27230 6258 y Fo(;)g(::;)g(\025)29472 6424 y Ff(0)29971 6258 y Fo(;)g(\027)31009 6424 y Ff(0)31506 6258 y Fr(\))308 b Fo(:)11864 b Fr(\(4)p Fo(:)p Fr(50\))443 10039 y(In)465 b(order)g(to)i(pro)-31 b(v)g(e)467 b(that,)491 b(w)-31 b(e)467 b(in)-31 b(tro)31 b(duce)466 b(the)h(space)e Fa(M)25241 10205 y Fk(\022)26208 10039 y Fr(of)h(the)g(sequences)f Fo(\027)p 34689 10216 V 539 w Fr(=)k Fh(f)p Fo(\027)38205 10205 y Fk(j)38672 10039 y Fo(;)369 b(h)310 b Fr(+)h(1)469 b Fh(\024)g Fo(j)531 b Fh(\024)469 b Fr(0)p Fh(g)0 11456 y Fr(suc)-31 b(h)479 b(that)i Fh(j)p Fo(\027)5970 11622 y Fk(j)6437 11456 y Fh(j)493 b(\024)e Fo(c)9114 11164 y Fr(\026)9068 11456 y Fo(\025)9714 11622 y Fk(h)10288 11456 y Fo(\015)10922 11054 y Fk(\022)24 b(j)11834 11456 y Fr(,)508 b(for)480 b(some)g Fo(c)p Fr(;)536 b(w)-31 b(e)480 b(shall)h(think)f Fa(M)27293 11622 y Fk(\022)28274 11456 y Fr(as)f(a)h(Banac)-31 b(h)481 b(space)f(with)h(norm)f Fh(jj)p Fo(\027)p 44157 11633 V 71 w Fh(jj)45389 11622 y Fk(\022)46383 11456 y Fr(=)0 12873 y(sup)1667 13144 y Fk(h)p Ff(+1)p Fj(\024)p Fk(j)45 b Fj(\024)p Ff(0)5782 12873 y Fh(j)p Fo(\027)6636 13039 y Fk(j)7102 12873 y Fh(j)p Fo(\015)8043 12471 y Fj(\000)p Fk(\022)24 b(j)9694 12581 y Fr(\026)9648 12873 y Fo(\025)10294 12400 y Fj(\000)p Ff(1)10294 13207 y Fk(h)11482 12873 y Fr(.)492 b(W)-92 b(e)368 b(then)g(lo)31 b(ok)370 b(for)e(a)g(\014xed)g(p)31 b(oin)-31 b(t)370 b(of)e(the)h(op)31 b(erator)369 b Fn(T)307 b Fr(:)h Fa(M)37881 13039 y Fk(\022)38690 12873 y Fh(!)f Fa(M)41266 13039 y Fk(\022)42135 12873 y Fr(de\014ned)367 b(as:)13784 16711 y(\()p Fn(T)p Fo(\027)p 15100 16888 V 70 w Fr(\))16147 16877 y Fk(j)16922 16711 y Fr(=)307 b Fh(\000)20445 15275 y Fk(j)19851 15660 y Fe(X)19135 18043 y Fk(j)19546 17821 y Fc(0)19846 18043 y Ff(=)p Fk(h)p Ff(+1)22349 16711 y Fo(\014)23033 16254 y Ff(\()p Fk(j)23790 15920 y Fc(0)24090 16254 y Ff(\))22975 16985 y Fk(\027)t(;!)24491 16711 y Fr(\()p Fo(\025)25567 16877 y Fk(j)25978 16656 y Fc(0)26333 16711 y Fo(;)184 b(\027)27371 16877 y Fk(j)27782 16656 y Fc(0)28138 16711 y Fo(;)g(::;)g(\025)30380 16877 y Ff(0)30879 16711 y Fo(;)g(\027)31917 16877 y Ff(0)32415 16711 y Fr(\))308 b Fo(:)10955 b Fr(\(4)p Fo(:)p Fr(51\))443 20582 y(Note)515 b(that,)553 b(if)7357 20290 y(\026)7311 20582 y Fo(\025)7957 20748 y Fk(h)9045 20582 y Fr(is)515 b(su\016cien)-31 b(tly)516 b(small,)553 b(then)515 b Fn(T)f Fr(lea)-31 b(v)g(es)516 b(in)-31 b(v)-61 b(arian)-31 b(t)516 b(the)f(ball)h Fa(B)36882 20748 y Fk(\022)37897 20582 y Fr(of)f(radius)g Fo(c)43258 20748 y Ff(0)44304 20582 y Fr(=)550 b(2)p Fo(c)46747 20748 y Ff(1)0 21169 y Fe(P)1168 21443 y Fj(1)1168 22332 y Fk(n)p Ff(=0)3076 22000 y Fo(\015)3710 21598 y Fj(\000)p Fk(n)5358 22000 y Fr(of)353 b Fa(M)7763 22166 y Fk(\022)8264 22000 y Fr(,)j Fo(c)9406 22166 y Ff(1)10255 22000 y Fr(b)31 b(eing)353 b(the)g(constan)-31 b(t)354 b(in)f(\(4.47\).)490 b(In)352 b(fact,)357 b(b)-31 b(y)353 b(\(4.46\))i(and)e(\(4.47\),)359 b(if)353 b Fh(jj)p Fo(\027)p 40243 22177 V 71 w Fh(jj)41475 22166 y Fk(\022)42284 22000 y Fh(\024)308 b Fo(c)43932 22166 y Ff(0)44429 22000 y Fr(,)356 b(then)6259 25749 y Fh(j)p Fr(\()p Fn(T)p Fo(\027)p 7882 25926 V 71 w Fr(\))8930 25915 y Fk(j)9397 25749 y Fh(j)308 b(\024)12489 24313 y Fk(j)11895 24697 y Fe(X)11180 27080 y Fk(j)11591 26859 y Fc(0)11891 27080 y Ff(=)p Fk(h)p Ff(+1)14394 25749 y Fo(c)14873 25915 y Ff(1)15415 25457 y Fr(\026)15369 25749 y Fo(\025)16015 25915 y Fk(h)16589 25749 y Fo(\015)17223 25292 y Ff(2)p Fk(\022)24 b(j)18521 24958 y Fc(0)19122 25749 y Fr(+)21538 24313 y Fk(j)20944 24697 y Fe(X)20229 27080 y Fk(j)20640 26859 y Fc(0)20939 27080 y Ff(=)p Fk(h)p Ff(+1)24073 24365 y(0)23495 24697 y Fe(X)23442 27071 y Fk(i)p Ff(=)p Fk(j)24846 26850 y Fc(0)25330 25749 y Fo(c)25809 25915 y Ff(0)26352 25457 y Fr(\026)26306 25749 y Fo(\025)26952 25915 y Fk(h)27525 25749 y Fo(\015)28159 25292 y Fk(\022)g(i)28974 25749 y Fo(c)29453 25915 y Ff(2)29995 25457 y Fr(\026)29949 25749 y Fo(\025)30595 25292 y Ff(2)30595 26022 y Fk(h)31169 25749 y Fo(\015)31803 25292 y Ff(2)p Fk(\022)g Ff(\()p Fk(j)33447 24958 y Fc(0)33746 25292 y Fj(\000)p Fk(i)p Ff(\))35460 25749 y Fh(\024)308 b Fo(c)37108 25915 y Ff(0)37650 25457 y Fr(\026)37604 25749 y Fo(\025)38250 25915 y Fk(h)38824 25749 y Fo(\015)39458 25292 y Fk(\022)24 b(j)40678 25749 y Fo(;)3430 b Fr(\(4)p Fo(:)p Fr(52\))0 29620 y(if)370 b(2)p Fo(c)2047 29786 y Ff(2)2590 29328 y Fr(\026)2544 29620 y Fo(\025)3190 29218 y Ff(2)3190 29933 y Fk(h)3763 29620 y Fr(\()4193 28789 y Fe(P)5362 29063 y Fj(1)5362 29952 y Fk(n)p Ff(=0)7270 29620 y Fo(\015)7904 29218 y Fj(\000)p Fk(n)9199 29620 y Fr(\))9629 29218 y Ff(2)10434 29620 y Fh(\024)307 b Fr(1.)443 31037 y Fn(T)368 b Fr(is)h(a)h(also)g(a)f(con)-31 b(traction)372 b(on)e Fa(B)15206 31203 y Fk(\022)15707 31037 y Fr(,)f(if)17444 30745 y(\026)17398 31037 y Fo(\025)18044 31203 y Fk(h)18986 31037 y Fr(is)g(su\016cien)-31 b(tly)371 b(small;)g(in)f(fact,)h(if)e Fo(\027)p 33641 31214 V 71 w(;)184 b(\027)p 34750 31214 V 35368 30635 a Fj(0)35986 31037 y Fh(2)307 b Fa(M)38193 31203 y Fk(\022)38694 31037 y Fr(,)4413 34675 y Fh(j)p Fr(\()p Fn(T)p Fo(\027)70 b Fr(\))7083 34841 y Fk(j)7797 34675 y Fh(\000)246 b Fr(\()p Fn(T)p Fo(\027)p 10220 34852 V 10837 34218 a Fj(0)11148 34675 y Fr(\))11578 34841 y Fk(j)12045 34675 y Fh(j)308 b(\024)15137 33239 y Fk(j)14543 33624 y Fe(X)13828 36007 y Fk(j)14239 35785 y Fc(0)14539 36007 y Ff(=)p Fk(h)p Ff(+1)17042 34675 y Fh(j)p Fo(\014)18033 34218 y Ff(\()p Fk(j)18790 33884 y Fc(0)19090 34218 y Ff(\))17975 34949 y Fk(\027)t(;!)19492 34675 y Fr(\()p Fo(\025)20568 34841 y Fk(j)20979 34620 y Fc(0)21334 34675 y Fo(;)184 b(\027)22372 34841 y Fk(j)22783 34620 y Fc(0)23138 34675 y Fo(;)g(::;)g(\025)25380 34841 y Ff(0)25879 34675 y Fo(;)g(\027)26917 34841 y Ff(0)27415 34675 y Fr(\))246 b Fh(\000)g Fo(\014)29882 34218 y Ff(\()p Fk(j)30639 33884 y Fc(0)30939 34218 y Ff(\))29824 34949 y Fk(\027)t(;!)31341 34675 y Fr(\()p Fo(\025)32417 34841 y Fk(j)32828 34620 y Fc(0)33183 34675 y Fo(;)184 b(\027)34291 34218 y Fj(0)34221 34949 y Fk(j)34632 34728 y Fc(0)34987 34675 y Fo(;)g(::;)g(\025)37229 34841 y Ff(0)37729 34675 y Fo(;)g(\027)38837 34218 y Fj(0)38767 34949 y Ff(0)39264 34675 y Fr(\))p Fh(j)4721 38770 y(\024)7198 37334 y Fk(j)6604 37719 y Fe(X)5889 40102 y Fk(j)6300 39880 y Fc(0)6600 40102 y Ff(=)p Fk(h)p Ff(+1)9734 37387 y(0)9155 37719 y Fe(X)9103 40093 y Fk(i)p Ff(=)p Fk(j)10507 39871 y Fc(0)10991 38770 y Fh(jj)p Fo(\027)p 11605 38947 V 317 w Fh(\000)246 b Fo(\027)p 13576 38947 V 14193 38313 a Fj(0)14504 38770 y Fh(jj)15118 38936 y Fk(\022)15666 38478 y Fr(\026)15619 38770 y Fo(\025)16265 38936 y Fk(h)16839 38770 y Fo(\015)17473 38313 y Fk(\022)24 b(i)18287 38770 y Fo(c)18766 38936 y Ff(2)19309 38478 y Fr(\026)19263 38770 y Fo(\025)19909 38313 y Ff(2)19909 39044 y Fk(h)20482 38770 y Fo(\015)21116 38313 y Ff(2)p Fk(\022)g Ff(\()p Fk(j)22760 37979 y Fc(0)23060 38313 y Fj(\000)p Fk(i)p Ff(\))24774 38770 y Fh(\024)26075 38021 y Fr(1)p 26075 38516 554 45 v 26075 39530 a(2)26761 38770 y Fh(jj)p Fo(\027)p 27375 38947 618 45 v 318 w Fh(\000)246 b Fo(\027)p 29347 38947 V 29964 38313 a Fj(0)30275 38770 y Fh(jj)30889 38936 y Fk(\022)31436 38478 y Fr(\026)31390 38770 y Fo(\025)32036 38936 y Fk(h)32610 38770 y Fo(\015)33244 38313 y Fk(\022)24 b(j)34464 38770 y Fo(;)44415 36766 y Fr(\(4)p Fo(:)p Fr(53\))0 42641 y(if)415 b Fo(c)1539 42807 y Ff(2)2082 42349 y Fr(\026)2036 42641 y Fo(\025)2682 42240 y Ff(2)2682 42955 y Fk(h)3255 42641 y Fr(\()3685 41811 y Fe(P)4854 42085 y Fj(1)4854 42973 y Fk(n)p Ff(=0)6762 42641 y Fo(\015)7396 42240 y Fj(\000)p Fk(n)8691 42641 y Fr(\))9121 42240 y Ff(2)10002 42641 y Fh(\024)384 b Fr(1)p Fo(=)p Fr(2.)631 b(Hence,)427 b(b)-31 b(y)415 b(the)g(con)-31 b(traction)417 b(principle,)428 b(there)415 b(is)f(a)i(unique)f(\014xed)g(p)31 b(oin)-31 b(t)416 b Fo(\027)p 46118 42818 V 46735 42240 a Fj(\003)0 44059 y Fr(of)370 b Fn(T)e Fr(on)i Fa(B)5031 44225 y Fk(\022)5532 44059 y Fr(.)p 46801 44280 443 886 v 0 47838 a Fn(4.7)1314 b Fp(The)397 b(c)-57 b(onstants)10115 47546 y Fr(~)10069 47838 y Fo(\025)10715 48004 y Fk(j)11577 47838 y Fp(and)463 b Fr(~)-620 b Fo(z)14242 48004 y Fk(j)14709 47838 y Fp(.)443 49255 y Fr(W)-92 b(e)527 b(shall)i(no)-31 b(w)529 b(analyze)h(the)e (constan)-31 b(ts)19019 48963 y(~)18973 49255 y Fo(\025)19619 49421 y Fk(j)20613 49255 y Fr(and)595 b(~)-620 b Fo(z)23439 49421 y Fk(j)23906 49255 y Fr(,)568 b Fo(h)k Fh(\024)g Fo(j)636 b Fh(\024)573 b(\000)p Fr(1,)568 b(app)31 b(earing)529 b(in)g(the)f(expansion)h(of)251 50393 y(~)0 50673 y Fo(G)870 50271 y Ff(4)870 50946 y(+)1605 50673 y Fr(\()2094 50381 y(\026)2035 50673 y Fn(k)2707 50839 y Ff(1)3204 50673 y Fo(;)3755 50381 y Fr(\026)3695 50673 y Fn(k)4367 50839 y Ff(2)4865 50673 y Fo(;)5416 50381 y Fr(\026)5356 50673 y Fn(k)6028 50839 y Ff(3)6525 50673 y Fo(;)7076 50381 y Fr(\026)7016 50673 y Fn(k)7688 50839 y Ff(4)8185 50673 y Fr(\).)578 b(W)-92 b(e)397 b(shall)h(do)g(that)g(b)-31 b(y)398 b(comparing)h(their)f(v)-61 b(alues)397 b(with)i(the)f(v)-61 b(alues)397 b(of)h Fo(\025)40797 50839 y Fk(j)41661 50673 y Fr(and)f Fo(z)44356 50839 y Fk(j)44822 50673 y Fr(.)578 b(W)-92 b(e)0 52090 y(start)370 b(noting)h(that)f(the)f(b)31 b(eta)370 b(function)h(equation)g(for)e Fo(\025)24089 52256 y Fk(j)24924 52090 y Fr(can)h(b)31 b(e)368 b(written)j(as)15596 55595 y Fo(\025)16242 55761 y Fk(j)45 b Fj(\000)p Ff(1)18149 55595 y Fr(=)19318 54034 y Fe(\022)20832 54846 y Fo(Z)21588 55012 y Fk(j)p 20265 55341 2356 45 v 20265 56354 a Fo(Z)21021 56520 y Fk(j)g Fj(\000)p Ff(1)22753 54034 y Fe(\023)23568 54264 y Ff(2)24249 55595 y Fo(\025)24895 55761 y Fk(j)25607 55595 y Fr(+)246 b Fo(\014)27340 55761 y Fk(j)28053 55595 y Fr(+)g Fo(\014)29844 55020 y Ff(\(1\))29786 55904 y Fk(j)31340 55595 y Fo(;)12768 b Fr(\(4)p Fo(:)p Fr(54\))0 58797 y(where)376 b Fo(\014)3834 58963 y Fk(j)4676 58797 y Fr(is)h(the)f(sum)g(o)-31 b(v)g(er)377 b(the)f(lo)31 b(cal)378 b(parts)e(of)h(the)f(trees)g(with)h(at)g(least)g(t)-31 b(w)g(o)378 b(endp)31 b(oin)-31 b(ts)377 b(and)f(no)h(endp)31 b(oin)-31 b(t)0 60381 y(of)398 b(scale)g(index)g(+1,)406 b(while)399 b Fo(\014)12709 59806 y Ff(\(1\))12651 60690 y Fk(j)14295 60381 y Fr(is)f(the)g(similar)h(sum)e(o)-31 b(v)g(er)399 b(the)f(trees)f(with)i(at)f(least)h(one)e(endp)31 b(oin)-31 b(t)399 b(of)g(scale)0 61798 y(index)370 b(+1.)443 63216 y(On)f(the)g(other)h(hand)f(w)-31 b(e)370 b(can)f(write)12289 66112 y(~)12243 66404 y Fo(\025)12889 66570 y Fk(j)45 b Fj(\000)p Ff(1)14796 66404 y Fr(=)15964 64843 y Fe(\022)17478 65655 y Fo(Z)18234 65821 y Fk(j)p 16912 66149 V 16912 67163 a Fo(Z)17668 67329 y Fk(j)g Fj(\000)p Ff(1)19400 64843 y Fe(\023)20215 65073 y Ff(2)20942 66112 y Fr(~)20896 66404 y Fo(\025)21542 66570 y Fk(j)22254 66404 y Fr(+)23519 66112 y(~)23361 66404 y Fo(\014)23987 66570 y Fk(j)24699 66404 y Fr(+)25964 66112 y(~)25806 66404 y Fo(\014)26490 65829 y Ff(\(1\))26432 66713 y Fk(j)27925 66404 y Fr(+)29190 66112 y(~)29032 66404 y Fo(\014)29716 65829 y Ff(\()p Fk(T)119 b Ff(\))29658 66713 y Fk(j)31352 66404 y Fr(+)32617 66112 y(~)32459 66404 y Fo(\014)33143 65829 y Ff(\()p Fk(\027)52 b Ff(\))33085 66713 y Fk(j)34694 66404 y Fo(;)9414 b Fr(\(4)p Fo(:)p Fr(55\))0 69606 y(where)347 71024 y(1\))1836 70732 y(~)1678 71024 y Fo(\014)2304 71190 y Fk(j)3118 71024 y Fr(is)347 b(the)h(sum)f(o)-31 b(v)g(er)348 b(the)f(lo)31 b(cal)349 b(parts)e(of)h(the)g(trees)f(with)h(at)g(least)h(t)-31 b(w)g(o)349 b(endp)31 b(oin)-31 b(ts,)353 b(no)347 b(endp)31 b(oin)-31 b(t)349 b(of)f(scale)0 72441 y(index)370 b(+1)f(and)h(one)f (sp)31 b(ecial)370 b(endp)31 b(oin)-31 b(t)370 b(of)g(t)-31 b(yp)31 b(e)20962 72149 y(~)20915 72441 y Fo(\025)p Fr(.)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 358 1156 a Fr(2\))1858 864 y(~)1700 1156 y Fo(\014)2384 581 y Ff(\(1\))2326 1465 y Fk(j)3797 1156 y Fr(+)5039 864 y(~)4882 1156 y Fo(\014)5566 581 y Ff(\()p Fk(T)119 b Ff(\))5508 1465 y Fk(j)7313 1156 y Fr(is)358 b(the)h(sum)f(o)-31 b(v)g(er)358 b(the)h(trees)e(with)j(at) f(least)g(one)f(endp)31 b(oin)-31 b(t)359 b(of)g(scale)f(index)h(+1;)k (in)358 b(this)h(case,)0 2757 y(the)401 b(sp)31 b(ecial)401 b(endp)31 b(oin)-31 b(t)402 b(can)f(b)31 b(e)400 b(of)h(t)-31 b(yp)31 b(e)17676 2465 y(~)17630 2757 y Fo(\025)400 b Fr(or)g Fo(T)20710 2923 y Ff(1)21607 2757 y Fr(and,)409 b(if)401 b(it)h(is)e(of)h(t)-31 b(yp)31 b(e)401 b Fo(T)31896 2923 y Ff(1)32393 2757 y Fr(,)409 b(its)401 b(scale)g(index)g(m)-31 b(ust)401 b(b)31 b(e)400 b(equal)0 4175 y(to)370 b(+1.)3725 3883 y(~)3567 4175 y Fo(\014)4251 3600 y Ff(\(1\))4193 4483 y Fk(j)5809 4175 y Fr(and)8119 3883 y(~)7961 4175 y Fo(\014)8645 3600 y Ff(\()p Fk(T)119 b Ff(\))8587 4483 y Fk(j)10404 4175 y Fr(are,)370 b(resp)31 b(ectiv)-31 b(ely)-92 b(,)370 b(the)f(sum)g(o)-31 b(v)g(er)370 b(the)f(trees)g (with)h(the)g(endp)31 b(oin)-31 b(t)370 b(of)g(t)-31 b(yp)31 b(e)42829 3883 y(~)42783 4175 y Fo(\025)368 b Fr(or)h Fo(T)45800 4341 y Ff(1)46297 4175 y Fr(.)423 5916 y(3\))1988 5624 y(~)1830 5916 y Fo(\014)2514 5341 y Ff(\()p Fk(\027)52 b Ff(\))2456 6225 y Fk(j)4181 5916 y Fr(is)423 b(the)h(sum)f(o)-31 b(v)g(er)424 b(the)g(trees)f(with)i(at) f(least)g(t)-31 b(w)g(o)425 b(endp)31 b(oin)-31 b(ts,)439 b(whose)423 b(sp)31 b(ecial)424 b(endp)31 b(oin)-31 b(t)425 b(is)e(of)h(t)-31 b(yp)31 b(e)0 7333 y Fo(T)647 7499 y Fj(\006)1394 7333 y Fr(.)443 9711 y(A)369 b(crucial)h(role)f(in)h (this)f(pap)31 b(er)369 b(has)g(the)h(follo)-31 b(wing)373 b(Lemma.)0 12088 y Fq(Lemma)388 b(4.2)1285 b Fp(L)-57 b(et)371 b Fo(\013)312 b Fr(=)11076 11796 y(~)11030 12088 y Fo(\025)11676 12254 y Fk(h)12249 12088 y Fo(=\025)13448 12254 y Fk(h)14022 12088 y Fp(;)379 b(then)370 b(if)18270 11796 y Fr(\026)18224 12088 y Fo(\025)18870 12254 y Fk(h)19814 12088 y Fp(is)g(smal)57 b(l)369 b(enough,)376 b(ther)-57 b(e)370 b(exists)f(a)h(c)-57 b(onstant)370 b Fo(c)p Fp(,)375 b(indep)-57 b(endent)372 b(of)0 13505 y Fo(\025)p Fp(,)396 b(such)g(that)f Fh(j)p Fo(\013)t Fh(j)309 b(\024)e Fo(c)396 b Fp(and)13410 16296 y Fh(j)13763 16004 y Fr(~)13717 16296 y Fo(\025)14363 16462 y Fk(j)15075 16296 y Fh(\000)246 b Fo(\013)t(\025)17540 16462 y Fk(j)18006 16296 y Fh(j)308 b(\024)g Fo(c)20315 16004 y Fr(\026)20269 16296 y Fo(\025)20915 16462 y Fk(h)21488 16296 y Fo(\015)22122 15839 y Fk(\022)24 b(j)24166 16296 y Fo(;)1317 b(h)246 b Fr(+)g(1)308 b Fh(\024)f Fo(j)371 b Fh(\024)307 b(\000)p Fr(1)h Fo(:)10581 b Fr(\(4)p Fo(:)p Fr(56\))0 21464 y Fq(Pr)-25 b(oof)487 b(-)922 b Fr(The)436 b(main)h(p)31 b(oin)-31 b(t)436 b(is)f(the)h(remark)f(that)i(there)e(is)g(a)h(one)f(to)h(one)g(corresp) 31 b(ondence)434 b(b)31 b(et)-31 b(w)g(een)437 b(the)0 22881 y(trees)355 b(con)-31 b(tributing)358 b(to)e Fo(\014)10898 23047 y Fk(j)11719 22881 y Fr(and)g(the)g(trees)e(con)-31 b(tributing)358 b(to)26180 22589 y(~)26022 22881 y Fo(\014)26648 23047 y Fk(j)27115 22881 y Fr(.)488 b(In)355 b(fact)h(the)g(trees)e (con)-31 b(tributing)358 b(to)43771 22589 y(~)43613 22881 y Fo(\014)44239 23047 y Fk(j)45061 22881 y Fr(ha)-31 b(v)g(e)0 24299 y(only)397 b(endp)31 b(oin)-31 b(ts)398 b(of)f(t)-31 b(yp)31 b(e)396 b Fo(\025)p Fr(,)404 b(b)31 b(esides)395 b(the)i(sp)31 b(ecial)397 b(endp)31 b(oin)-31 b(t)397 b Fo(v)27297 23897 y Fj(\003)27806 24299 y Fr(,)403 b(and)397 b(the)f(external)i(\014eld)e(with)i Fo(!)393 b Fr(=)352 b Fh(\000)397 b Fr(and)0 25716 y Fo(\033)347 b Fr(=)307 b Fh(\000)h Fr(has)f(to)i(b)31 b(elong)308 b(to)h Fo(P)11998 25882 y Fk(v)12469 25661 y Fc(\003)12986 25716 y Fr(.)472 b(It)308 b(follo)-31 b(ws)310 b(that)f(w)-31 b(e)308 b(can)g(asso)31 b(ciate)308 b(uniquely)h(with)g(an)-31 b(y)309 b(tree)e(con)-31 b(tributing)310 b(to)158 26841 y(~)0 27133 y Fo(\014)626 27299 y Fk(j)1468 27133 y Fr(a)376 b(tree)g(con)-31 b(tributing)378 b(to)f Fo(\014)12920 27299 y Fk(j)13386 27133 y Fr(,)h(b)-31 b(y)376 b(simply)h (substituting)h(the)e(sp)31 b(ecial)376 b(endp)31 b(oin)-31 b(t)377 b(with)g(a)f(normal)h(endp)31 b(oin)-31 b(t,)0 28551 y(without)290 b(c)-31 b(hanging)290 b(an)-31 b(y)289 b(lab)31 b(el.)466 b(This)289 b(corresp)31 b(ondence)287 b(is)h(surjectiv)-31 b(e,)305 b(since)288 b(w)-31 b(e)288 b(ha)-31 b(v)g(e)289 b(imp)31 b(osed)289 b(the)f(condition)0 29968 y(that)393 b(the)e(trees)g(con)-31 b(tributing)394 b(to)14887 29676 y(~)14729 29968 y Fo(\014)15355 30134 y Fk(j)16212 29968 y Fr(and)e Fo(\014)19013 30134 y Fk(j)19871 29968 y Fr(do)f(not)h(ha)-31 b(v)g(e)393 b(endp)31 b(oin)-31 b(ts)392 b(of)g(scale)f(index)h(+1.)560 b(Hence,)397 b(w)-31 b(e)392 b(can)0 31385 y(write)8955 31642 y Fe(")9601 31974 y(\022)11115 32786 y Fo(Z)11871 32952 y Fk(j)p 10549 33280 2356 45 v 10549 34294 a Fo(Z)11305 34460 y Fk(j)45 b Fj(\000)p Ff(1)13037 31974 y Fe(\023)13852 32204 y Ff(2)14594 33535 y Fh(\000)246 b Fr(1)16254 31642 y Fe(#)17085 33535 y Fr(\()17561 33243 y(~)17515 33535 y Fo(\025)18161 33701 y Fk(j)18874 33535 y Fh(\000)g Fo(\013)t(\025)21339 33701 y Fk(j)21805 33535 y Fr(\))g(+)23746 33243 y(~)23588 33535 y Fo(\014)24214 33701 y Fk(j)24927 33535 y Fh(\000)g Fo(\013)t(\014)27372 33701 y Fk(j)28146 33535 y Fr(=)29547 32151 y Fj(\000)p Ff(1)29315 32483 y Fe(X)29412 34841 y Fk(i)p Ff(=)p Fk(j)31098 33535 y Fo(\014)31724 33701 y Fk(j)m(;i)32719 33535 y Fr(\()33195 33243 y(~)33149 33535 y Fo(\025)33795 33701 y Fk(i)34410 33535 y Fh(\000)g Fo(\013)t(\025)36875 33701 y Fk(i)37243 33535 y Fr(\))308 b Fo(;)6127 b Fr(\(4)p Fo(:)p Fr(57\))0 36954 y(where,)583 b(thanks)542 b(to)f(the)f(\\short)h(memory)h(prop)31 b(ert)-31 b(y")541 b(and)f(the)h(fact)g(that)h Fo(Z)34443 37120 y Fk(j)34909 36954 y Fo(=)-61 b(Z)36157 37120 y Fk(j)45 b Fj(\000)p Ff(1)38349 36954 y Fr(=)593 b(1)361 b(+)f Fo(O)31 b Fr(\()43289 36662 y(\026)43243 36954 y Fo(\025)43889 36552 y Ff(2)43889 37242 y Fk(j)44386 36954 y Fr(\),)584 b(the)0 38486 y(constan)-31 b(ts)370 b Fo(\014)5527 38652 y Fk(j)m(;i)6891 38486 y Fr(satisfy)h(the)e(b)31 b(ound)369 b Fh(j)p Fo(\014)16600 38652 y Fk(j)m(;i)17596 38486 y Fh(j)308 b(\024)f Fo(C)20295 38194 y Fr(\026)20249 38486 y Fo(\025)20895 38652 y Fk(j)21361 38486 y Fo(\015)21995 38084 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(j)45 b Fj(\000)p Fk(i)p Ff(\))25046 38486 y Fr(,)370 b(with)g Fo(\022)399 b Fr(de\014ned)369 b(as)g(in)h(Lemma)g(4.1.)443 40085 y(Among)350 b(the)e(four)h(last)g(terms)f(in)h(the)g(r.h.s.)485 b(of)349 b(\(4.55\),)356 b(the)348 b(only)i(one)e(dep)31 b(ending)349 b(on)g(the)39407 39793 y(~)39360 40085 y Fo(\025)40006 40251 y Fk(j)40821 40085 y Fr(is)42071 39793 y(~)41913 40085 y Fo(\014)42597 39510 y Ff(\(1\))42539 40393 y Fk(j)43786 40085 y Fr(,)k(whic)-31 b(h)0 41502 y(can)369 b(b)31 b(e)369 b(written)i(in)e(the)g(form)19589 43360 y(~)19431 43652 y Fo(\014)20115 43077 y Ff(\(1\))20057 43960 y Fk(j)21611 43652 y Fr(=)23013 42268 y Fj(\000)p Ff(1)22780 42600 y Fe(X)22877 44958 y Fk(i)p Ff(=)p Fk(j)24563 43652 y Fo(\014)25247 43195 y Fj(0)25189 43925 y Fk(j)m(;i)26230 43360 y Fr(~)26184 43652 y Fo(\025)26830 43818 y Fk(i)27506 43652 y Fo(;)16602 b Fr(\(4)p Fo(:)p Fr(58\))0 47090 y(the)414 b Fo(\014)2635 46688 y Fj(0)2577 47378 y Fk(j)m(;i)3985 47090 y Fr(b)31 b(eing)414 b(constan)-31 b(ts)415 b(whic)-31 b(h)415 b(satisfy)f(the)g(b)31 b(ound)414 b Fh(j)p Fo(\014)25070 46688 y Fj(0)25012 47378 y Fk(j)m(;i)26007 47090 y Fh(j)382 b(\024)g Fo(C)28855 46798 y Fr(\026)28809 47090 y Fo(\025)29455 47256 y Fk(j)29921 47090 y Fo(\015)30555 46688 y Ff(2)p Fk(\022)24 b(j)31909 47090 y Fr(,)425 b(since)413 b(they)h(are)g (related)g(to)g(trees)0 48795 y(with)381 b(an)f(endp)31 b(oin)-31 b(t)381 b(of)g(scale)f(index)h(+1.)525 b(F)-92 b(or)379 b(the)h(same)g(reasons,)j(w)-31 b(e)380 b(ha)-31 b(v)g(e)381 b(the)f(b)31 b(ounds)380 b Fh(j)39231 48503 y Fr(~)39073 48795 y Fo(\014)39757 48220 y Ff(\()p Fk(T)119 b Ff(\))39699 49104 y Fk(j)41147 48795 y Fh(j)326 b(\024)f Fo(C)43882 48503 y Fr(\026)43836 48795 y Fo(\025)44482 48961 y Fk(j)44949 48795 y Fo(\015)45583 48394 y Ff(2)p Fk(\022)24 b(j)46937 48795 y Fr(,)0 50522 y Fh(j)p Fo(\014)991 49947 y Ff(\(1\))933 50830 y Fk(j)2180 50522 y Fh(j)308 b(\024)g Fo(C)4880 50229 y Fr(\026)4834 50522 y Fo(\025)5480 50120 y Ff(2)5480 50810 y Fk(j)5977 50522 y Fo(\015)6611 50120 y Ff(2)p Fk(\022)24 b(j)7964 50522 y Fr(.)493 b(Finally)-92 b(,)371 b(b)-31 b(y)369 b(using)h(also)g(Lemma)g(4.1,)h(w)-31 b(e)370 b(see)e(that)j Fh(j)31725 50229 y Fr(~)31567 50522 y Fo(\014)32251 49947 y Ff(\()p Fk(\027)52 b Ff(\))32193 50830 y Fk(j)33494 50522 y Fh(j)308 b(\024)g Fo(C)36194 50229 y Fr(\026)36148 50522 y Fo(\025)36794 50688 y Fk(j)37306 50229 y Fr(\026)37260 50522 y Fo(\025)37906 50688 y Fk(h)38480 50522 y Fo(\015)39114 50120 y Ff(2)p Fk(\022)24 b(j)40467 50522 y Fr(.)443 51954 y(W)-92 b(e)368 b(no)-31 b(w)370 b(c)-31 b(ho)31 b(ose)370 b Fo(\013)j Fr(so)c(that)20094 53124 y(~)20048 53416 y Fo(\025)20694 53582 y Fk(h)21513 53416 y Fh(\000)246 b Fo(\013)t(\025)23978 53582 y Fk(h)24859 53416 y Fr(=)308 b(0)g Fo(;)17219 b Fr(\(4)p Fo(:)p Fr(59\))0 55654 y(and)369 b(w)-31 b(e)370 b(put)14814 57116 y Fo(x)15447 57282 y Fk(j)16221 57116 y Fr(=)17436 56824 y(~)17390 57116 y Fo(\025)18036 57282 y Fk(j)18748 57116 y Fh(\000)246 b Fo(\013)t(\025)21213 57282 y Fk(j)22786 57116 y Fo(;)1292 b(h)246 b Fr(+)g(1)308 b Fh(\024)f Fo(j)371 b Fh(\024)307 b(\000)p Fr(1)h Fo(:)11986 b Fr(\(4)p Fo(:)p Fr(60\))0 59354 y(W)-92 b(e)368 b(can)i(write)3810 63107 y Fo(x)4443 63273 y Fk(j)45 b Fj(\000)p Ff(1)6350 63107 y Fr(=)307 b Fo(x)8151 63273 y Fj(\000)p Ff(1)9585 63107 y Fr(+)11026 61724 y Fj(\000)p Ff(1)10793 62056 y Fe(X)10692 64430 y Fk(j)11103 64208 y Fc(0)11403 64430 y Ff(=)p Fk(j)12678 60882 y Fe(2)12678 62875 y(4)13701 61724 y Fj(\000)p Ff(1)13468 62056 y Fe(X)13416 64430 y Fk(i)p Ff(=)p Fk(j)14820 64208 y Fc(0)15303 63107 y Fo(\014)15929 63273 y Fk(j)16340 63052 y Fc(0)16640 63273 y Fk(;i)17271 63107 y Fo(x)17904 63273 y Fk(i)18519 63107 y Fr(+)19911 61724 y Fj(\000)p Ff(1)19678 62056 y Fe(X)19626 64430 y Fk(i)p Ff(=)p Fk(j)21030 64208 y Fc(0)21513 63107 y Fo(\014)22197 62650 y Fj(0)22139 63381 y Fk(j)22550 63160 y Fc(0)22850 63381 y Fk(;i)23481 63107 y Fr(\()p Fo(x)24544 63273 y Fk(i)25159 63107 y Fr(+)246 b Fo(\013)t(\025)27624 63273 y Fk(i)27992 63107 y Fr(\))h(+)29934 62815 y(~)29776 63107 y Fo(\014)30460 62532 y Ff(\()p Fk(T)119 b Ff(\))30402 63432 y Fk(j)30813 63211 y Fc(0)32096 63107 y Fr(+)33360 62815 y(~)33203 63107 y Fo(\014)33887 62532 y Ff(\()p Fk(\027)52 b Ff(\))33829 63432 y Fk(j)34240 63211 y Fc(0)35376 63107 y Fh(\000)246 b Fo(\013)t(\014)37879 62532 y Ff(\(1\))37821 63416 y Fk(j)39068 60882 y Fe(3)39068 62875 y(5)40298 63107 y Fo(:)3810 b Fr(\(4)p Fo(:)p Fr(61\))0 67138 y(On)369 b(the)g(other)h(hand,)g(the)f(condition)j(\(4.59\))f(implies)g(that) 5880 71168 y Fo(x)6513 71334 y Fj(\000)p Ff(1)8009 71168 y Fr(=)307 b Fh(\000)11171 69784 y Fj(\000)p Ff(1)10938 70116 y Fe(X)10222 72499 y Fk(j)10633 72278 y Fc(0)10933 72499 y Ff(=)p Fk(h)p Ff(+1)13436 68943 y Fe(2)13436 70935 y(4)14459 69784 y Fj(\000)p Ff(1)14226 70116 y Fe(X)14174 72490 y Fk(i)p Ff(=)p Fk(j)15578 72269 y Fc(0)16062 71168 y Fo(\014)16688 71334 y Fk(j)17099 71113 y Fc(0)17399 71334 y Fk(;i)18030 71168 y Fo(x)18663 71334 y Fk(i)19277 71168 y Fr(+)20669 69784 y Fj(\000)p Ff(1)20436 70116 y Fe(X)20384 72490 y Fk(i)p Ff(=)p Fk(j)21788 72269 y Fc(0)22272 71168 y Fo(\014)22956 70711 y Fj(0)22898 71442 y Fk(j)23309 71220 y Fc(0)23608 71442 y Fk(;i)24240 71168 y Fr(\()p Fo(x)25303 71334 y Fk(i)25918 71168 y Fr(+)246 b Fo(\013)t(\025)28383 71334 y Fk(i)28751 71168 y Fr(\))h(+)30692 70876 y(~)30534 71168 y Fo(\014)31218 70593 y Ff(\()p Fk(T)119 b Ff(\))31160 71493 y Fk(j)31571 71271 y Fc(0)32854 71168 y Fr(+)34119 70876 y(~)33961 71168 y Fo(\014)34645 70593 y Ff(\()p Fk(\027)52 b Ff(\))34587 71493 y Fk(j)34998 71271 y Fc(0)36134 71168 y Fh(\000)246 b Fo(\013)t(\014)38637 70593 y Ff(\(1\))38579 71476 y Fk(j)39827 68943 y Fe(3)39827 70935 y(5)41056 71168 y Fo(;)3052 b Fr(\(4)p Fo(:)p Fr(62\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 0 1107 a Fr(so)369 b(that,)i(if)f Fo(h)245 b Fr(+)h(1)308 b Fh(\024)g Fo(j)371 b Fh(\024)307 b(\000)p Fr(1,)370 b(the)f Fo(x)15725 1273 y Fk(j)16560 1107 y Fr(satisfy)i(the)e(equation)6241 5230 y Fo(x)6874 5396 y Fk(j)7648 5230 y Fr(=)307 b Fh(\000)11171 3794 y Fk(j)10577 4179 y Fe(X)9861 6562 y Fk(j)10272 6340 y Fc(0)10572 6562 y Ff(=)p Fk(h)p Ff(+1)13075 3005 y Fe(2)13075 4998 y(4)14098 3847 y Fj(\000)p Ff(1)13865 4179 y Fe(X)13813 6553 y Fk(i)p Ff(=)p Fk(j)15217 6331 y Fc(0)15701 5230 y Fo(\014)16327 5396 y Fk(j)16738 5175 y Fc(0)17038 5396 y Fk(;i)17669 5230 y Fo(x)18302 5396 y Fk(i)18916 5230 y Fr(+)20308 3847 y Fj(\000)p Ff(1)20075 4179 y Fe(X)20023 6553 y Fk(i)p Ff(=)p Fk(j)21427 6331 y Fc(0)21911 5230 y Fo(\014)22595 4773 y Fj(0)22537 5504 y Fk(j)22948 5283 y Fc(0)23247 5504 y Fk(;i)23879 5230 y Fr(\()p Fo(x)24942 5396 y Fk(i)25557 5230 y Fr(+)246 b Fo(\013)t(\025)28022 5396 y Fk(i)28390 5230 y Fr(\))h(+)30331 4938 y(~)30173 5230 y Fo(\014)30857 4655 y Ff(\()p Fk(T)119 b Ff(\))30799 5555 y Fk(j)31210 5334 y Fc(0)32493 5230 y Fr(+)33758 4938 y(~)33600 5230 y Fo(\014)34284 4655 y Ff(\()p Fk(\027)52 b Ff(\))34226 5555 y Fk(j)34637 5334 y Fc(0)35773 5230 y Fh(\000)246 b Fo(\013)t(\014)38276 4655 y Ff(\(1\))38218 5539 y Fk(j)39466 3005 y Fe(3)39466 4998 y(5)40695 5230 y Fo(:)3413 b Fr(\(4)p Fo(:)p Fr(63\))443 9346 y(W)-92 b(e)368 b(w)-31 b(an)g(t)371 b(to)f(sho)-31 b(w)370 b(that)g(equation)h(\(4.63\))h(has)d(a)h(unique)f(solution)i (satisfying)h(the)d(b)31 b(ound)17192 12190 y Fh(j)p Fo(x)18132 12356 y Fk(j)18598 12190 y Fh(j)308 b(\024)g Fo(c)20861 12356 y Ff(0)21358 12190 y Fr(\(1)246 b(+)g Fh(j)p Fo(\013)t Fh(j)25066 11898 y Fr(\026)25020 12190 y Fo(\025)25666 12356 y Fk(h)26241 12190 y Fr(\))26717 11898 y(\026)26671 12190 y Fo(\025)27317 12356 y Fk(h)27891 12190 y Fo(\015)28525 11733 y Fk(\022)24 b(j)29745 12190 y Fo(;)14363 b Fr(\(4)p Fo(:)p Fr(64\))0 15034 y(for)328 b(a)h(suitable)h(constan)-31 b(t)330 b Fo(c)11524 15200 y Ff(0)12020 15034 y Fr(,)337 b(indep)31 b(enden)-31 b(t)329 b(of)g Fo(h)p Fr(,)336 b(if)22417 14742 y(\026)22371 15034 y Fo(\025)23017 15200 y Fk(h)23919 15034 y Fr(is)328 b(small)h(enough.)480 b(Hence)328 b(w)-31 b(e)329 b(in)-31 b(tro)31 b(duce)329 b(the)g(Banac)-31 b(h)0 16689 y(space)344 b Fa(M)4095 16855 y Fk(\022)4940 16689 y Fr(of)h(sequences)e Fo(x)p 11175 16866 633 45 v 308 w Fr(=)307 b Fh(f)p Fo(x)14470 16855 y Fk(j)14937 16689 y Fo(;)184 b(h)197 b Fr(+)f(1)308 b Fh(\024)g Fo(j)370 b Fh(\024)308 b(\000)p Fr(1)p Fh(g)345 b Fr(with)g(norm)g Fh(jj)p Fo(x)p 29636 16866 V 1 w Fh(jj)30884 16855 y Fk(\022)31386 16009 y(def)31655 16689 y Fr(=)453 b(sup)34636 16959 y Fk(j)35287 16689 y Fh(j)p Fo(x)36227 16855 y Fk(j)36693 16689 y Fh(j)p Fo(\015)37634 16287 y Fj(\000)p Fk(\022)24 b(j)39285 16397 y Fr(\026)39239 16689 y Fo(\025)39885 16215 y Fj(\000)p Ff(1)39885 17023 y Fk(h)41418 16689 y Fr(and)344 b(lo)31 b(ok)346 b(for)0 18106 y(a)401 b(\014xed)f(p)31 b(oin)-31 b(t)402 b(of)f(the)f(op)31 b(erator)401 b Fn(T)360 b Fr(:)g Fa(M)17381 18272 y Fk(\022)18242 18106 y Fh(!)f Fa(M)20870 18272 y Fk(\022)21771 18106 y Fr(de\014ned)400 b(b)-31 b(y)401 b(the)g(r.h.s.)586 b(of)401 b(\(4.63\).)589 b(By)401 b(using)g(the)f(b)31 b(ounds)0 19524 y(on)400 b(the)h(v)-61 b(arious)400 b(constan)-31 b(ts)401 b(app)31 b(earing)401 b(in)g(the)f(de\014nition)i(of)e Fn(T)p Fr(,)408 b(w)-31 b(e)401 b(can)f(easily)h(pro)-31 b(v)g(e)401 b(that)g(there)f(are)g(t)-31 b(w)g(o)0 20941 y(constan)g(ts)370 b Fo(c)5380 21107 y Ff(1)6246 20941 y Fr(and)f Fo(c)8877 21107 y Ff(2)9374 20941 y Fr(,)h(suc)-31 b(h)368 b(that)9102 24753 y Fh(j)p Fr(\()p Fn(T)p Fo(x)p 10725 24930 V Fr(\))11788 24919 y Fk(j)12255 24753 y Fh(j)308 b(\024)f Fo(c)14517 24919 y Ff(1)15060 24461 y Fr(\026)15014 24753 y Fo(\025)15660 24919 y Fk(h)16233 24753 y Fr(\(1)247 b(+)f Fh(j)p Fo(\013)t Fh(j)19942 24461 y Fr(\026)19896 24753 y Fo(\025)20542 24919 y Fk(h)21117 24753 y Fr(\))p Fo(\015)22181 24296 y Fk(\022)24 b(j)23340 24753 y Fr(+)245 b Fo(c)24925 24919 y Ff(2)25468 24461 y Fr(\026)25422 24753 y Fo(\025)26068 24919 y Fk(h)28135 23317 y(j)27541 23701 y Fe(X)26826 26084 y Fk(j)27237 25863 y Fc(0)27536 26084 y Ff(=)p Fk(h)p Ff(+1)30325 23369 y Fj(\000)p Ff(1)30092 23701 y Fe(X)30040 26075 y Fk(i)p Ff(=)p Fk(j)31444 25854 y Fc(0)31927 24753 y Fo(\015)32561 24296 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(j)34205 23962 y Fc(0)34505 24296 y Fj(\000)p Fk(i)p Ff(\))35911 24753 y Fh(j)p Fo(x)36851 24919 y Fk(i)37220 24753 y Fh(j)308 b Fo(:)6273 b Fr(\(4)p Fo(:)p Fr(65\))0 28804 y(Hence,)366 b(if)f(w)-31 b(e)366 b(tak)-31 b(e)366 b Fo(c)9103 28970 y Ff(0)9907 28804 y Fr(=)307 b Fo(M)121 b(c)12749 28970 y Ff(1)13245 28804 y Fr(,)366 b Fo(M)428 b Fh(\025)308 b Fr(2,)366 b(the)f(ball)h Fa(B)22844 28970 y Fk(M)24191 28804 y Fr(of)f(radius)g Fo(c)29252 28970 y Ff(0)29748 28804 y Fr(\(1)238 b(+)g Fh(j)p Fo(\013)t Fh(j)33440 28511 y Fr(\026)33394 28804 y Fo(\025)34040 28970 y Fk(h)34614 28804 y Fr(\))365 b(in)g Fa(M)37858 28970 y Fk(\022)38723 28804 y Fr(is)g(in)-31 b(v)-61 b(arian)-31 b(t)367 b(under)0 30221 y(the)414 b(action)i(of)f Fn(T)p Fr(,)425 b(if)415 b Fo(c)9780 30387 y Ff(2)10322 29929 y Fr(\026)10276 30221 y Fo(\025)10922 30387 y Fk(h)11496 30221 y Fr(\()11926 29391 y Fe(P)13095 29664 y Fj(1)13095 30553 y Fk(n)p Ff(=0)15002 30221 y Fo(\015)15636 29819 y Fj(\000)p Fk(n)16931 30221 y Fr(\))17361 29819 y Ff(2)18241 30221 y Fh(\024)382 b Fr(1)p Fo(=)p Fr(2,)427 b(since)414 b(1)p Fo(=)p Fr(2)384 b Fh(\024)e Fr(\()p Fo(M)397 b Fh(\000)276 b Fr(1\))p Fo(=)-61 b(M)121 b Fr(.)628 b(On)414 b(the)g(other)h(hand,)426 b(under)0 31638 y(the)369 b(same)h (condition,)i Fn(T)c Fr(is)h(a)h(con)-31 b(traction)372 b(in)d(all)h Fa(M)22941 31804 y Fk(\022)23442 31638 y Fr(;)g(in)f(fact,)i(if)e Fo(x)p 28915 31815 V(;)184 b(x)p 30039 31815 V 30673 31237 a Fj(0)31291 31638 y Fh(2)307 b Fa(M)33498 31804 y Fk(\022)33999 31638 y Fr(,)369 b(then)4031 35456 y Fh(j)p Fr(\()p Fn(T)p Fo(x)p 5654 35633 V 1 w Fr(\))6718 35622 y Fk(j)7430 35456 y Fh(\000)246 b Fr(\()p Fn(T)p Fo(x)p 9853 35633 V -457 x Fj(0)10797 35456 y Fr(\))11227 35622 y Fk(j)11694 35456 y Fh(j)308 b(\024)f Fo(c)13956 35622 y Ff(2)14499 35164 y Fr(\026)14453 35456 y Fo(\025)15099 34999 y Ff(2)15099 35730 y Fk(h)15672 35456 y Fh(jj)p Fo(x)p 16286 35633 V 247 w Fh(\000)246 b Fo(x)p 18273 35633 V 18905 34999 a Fj(0)19216 35456 y Fh(jj)21324 34020 y Fk(j)20730 34405 y Fe(X)20014 36788 y Fk(j)20425 36566 y Fc(0)20726 36788 y Ff(=)p Fk(h)p Ff(+1)23514 34073 y Fj(\000)p Ff(1)23281 34405 y Fe(X)23229 36779 y Fk(i)p Ff(=)p Fk(j)24633 36557 y Fc(0)25117 35456 y Fo(\015)25751 34999 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(j)27395 34665 y Fc(0)27694 34999 y Fj(\000)p Fk(i)p Ff(\))29101 35456 y Fo(\015)29735 34999 y Fk(\022)g(i)30857 35456 y Fh(\024)32158 34707 y Fr(1)p 32158 35202 554 45 v 32158 36216 a(2)32844 35456 y Fh(jj)p Fo(x)p 33458 35633 633 45 v 247 w Fh(\000)246 b Fo(x)p 35445 35633 V -457 x Fj(0)36388 35456 y Fh(jj)37048 35164 y Fr(\026)37002 35456 y Fo(\025)37648 35622 y Fk(h)38222 35456 y Fo(\015)38856 34999 y Fk(\022)24 b(j)40076 35456 y Fo(;)4032 b Fr(\(4)p Fo(:)p Fr(66\))0 39507 y(if)439 b Fo(c)1563 39673 y Ff(2)2106 39215 y Fr(\026)2060 39507 y Fo(\025)2706 39673 y Fk(h)3279 39507 y Fr(\()3709 38677 y Fe(P)4878 38951 y Fj(1)4878 39839 y Fk(n)p Ff(=0)6786 39507 y Fo(\015)7420 39105 y Fj(\000)p Fk(n)8715 39507 y Fr(\))9145 39105 y Ff(2)10065 39507 y Fh(\024)423 b Fr(1)p Fo(=)p Fr(2.)701 b(It)439 b(follo)-31 b(ws,)459 b(b)-31 b(y)438 b(the)h(con)-31 b(traction)441 b(principle,)457 b(that)440 b(there)e(is)g(a)h(unique)g (\014xed)0 40924 y(p)31 b(oin)-31 b(t)397 b(in)f(the)g(ball)h Fa(B)9326 41090 y Fk(M)10308 40924 y Fr(,)403 b(for)396 b(an)-31 b(y)396 b Fo(M)473 b Fh(\025)352 b Fr(2,)403 b(hence)395 b(a)h(unique)h(\014xed)f(p)31 b(oin)-31 b(t)397 b(in)f Fa(M)34652 41090 y Fk(\022)35153 40924 y Fr(,)403 b(satisfying)398 b(the)e(condition)0 42342 y(\(4.64\))372 b(with)e Fo(c)6198 42508 y Ff(0)7002 42342 y Fr(=)308 b(2)p Fo(c)9203 42508 y Ff(1)9700 42342 y Fr(.)443 43792 y(T)-92 b(o)417 b(complete)g(the)g(pro)31 b(of,)429 b(w)-31 b(e)417 b(ha)-31 b(v)g(e)417 b(to)g(sho)-31 b(w)417 b(that)g Fo(\013)k Fr(can)416 b(b)31 b(e)416 b(b)31 b(ounded)416 b(uniformly)i(in)f Fo(h)p Fr(.)633 b(In)416 b(order)g(to)h(do)0 45209 y(that,)364 b(w)-31 b(e)361 b(insert)g(in)g(the)g(l.h.s.)491 b(of)361 b(\(4.62\))j(the)d(de\014nition)h(of)f Fo(x)26484 45375 y Fj(\000)p Ff(1)28033 45209 y Fr(and)g(w)-31 b(e)361 b(b)31 b(ound)361 b(the)g(r.h.s.)490 b(b)-31 b(y)361 b(using)g(\(4.64\))0 46626 y(and)369 b(\(4.65\);)k(w)-31 b(e)369 b(get)15867 48142 y Fh(j)16220 47850 y Fr(~)16174 48142 y Fo(\025)16820 48308 y Fj(\000)p Ff(1)18255 48142 y Fh(\000)246 b Fo(\013)t(\025)20720 48308 y Fj(\000)p Ff(1)21908 48142 y Fh(j)308 b(\024)g Fo(c)24171 48308 y Ff(3)24714 47850 y Fr(\026)24667 48142 y Fo(\025)25313 48308 y Fk(h)26133 48142 y Fr(+)246 b Fo(c)27719 48308 y Ff(4)28215 48142 y Fh(j)p Fo(\013)t Fh(j)29587 47850 y Fr(\026)29541 48142 y Fo(\025)30187 47685 y Ff(2)30187 48416 y Fk(h)31069 48142 y Fo(;)13039 b Fr(\(4)p Fo(:)p Fr(67\))0 50433 y(for)451 b(some)g(constan)-31 b(ts)452 b Fo(c)10093 50599 y Ff(3)11040 50433 y Fr(and)e Fo(c)13752 50599 y Ff(4)14249 50433 y Fr(.)737 b(Since)451 b Fh(j)p Fo(\025)19218 50599 y Fj(\000)p Ff(1)20406 50433 y Fh(j)444 b(\025)f Fo(c)22940 50599 y Ff(5)23437 50433 y Fh(j)p Fo(\025)p Fh(j)p Fr(,)25522 50140 y(~)25476 50433 y Fo(\025)26122 50599 y Fj(\000)p Ff(1)27753 50433 y Fh(\024)g Fo(c)29536 50599 y Ff(6)30033 50433 y Fh(j)p Fo(\025)p Fh(j)451 b Fr(and)34024 50140 y(\026)33978 50433 y Fo(\025)34624 50599 y Fk(h)35641 50433 y Fh(\024)443 b Fr(2)p Fh(j)p Fo(\025)p Fh(j)451 b Fr(b)-31 b(y)451 b(the)g(inductiv)-31 b(e)0 51850 y(h)g(yp)31 b(othesis,)370 b(w)-31 b(e)370 b(ha)-31 b(v)g(e)7459 54694 y Fh(j)p Fo(\013)t(\025)9124 54860 y Fj(\000)p Ff(1)10313 54694 y Fh(j)308 b(\024)f(j)12449 54402 y Fr(~)12403 54694 y Fo(\025)13049 54860 y Fj(\000)p Ff(1)14238 54694 y Fh(j)247 b Fr(+)f Fo(c)16378 54860 y Ff(3)16920 54402 y Fr(\026)16874 54694 y Fo(\025)17520 54860 y Fk(h)18339 54694 y Fr(+)g Fo(c)19925 54860 y Ff(4)20422 54694 y Fh(j)p Fo(\013)t Fh(j)21794 54402 y Fr(\026)21748 54694 y Fo(\025)22394 54237 y Ff(2)22394 54967 y Fk(h)23276 54694 y Fh(\))308 b(j)p Fo(\013)t Fh(j)g(\024)g Fr(\()p Fo(c)28403 54860 y Ff(6)29146 54694 y Fr(+)246 b(2)p Fo(c)31285 54860 y Ff(3)32028 54694 y Fr(+)g(2)p Fo(c)34167 54860 y Ff(4)34664 54694 y Fh(j)p Fo(\013)t Fh(j)36036 54402 y Fr(\026)35990 54694 y Fo(\025)36636 54860 y Fk(h)37210 54694 y Fr(\))p Fo(=c)38672 54860 y Ff(5)39478 54694 y Fo(;)4630 b Fr(\(4)p Fo(:)p Fr(68\))0 57538 y(so)369 b(that,)i Fh(j)p Fo(\013)t Fh(j)309 b(\024)e Fr(2\()p Fo(c)8330 57704 y Ff(6)9074 57538 y Fr(+)246 b(2)p Fo(c)11213 57704 y Ff(3)11710 57538 y Fr(\))p Fo(=c)13172 57704 y Ff(5)13669 57538 y Fr(,)370 b(if)g(4)p Fo(c)16393 57704 y Ff(4)16936 57246 y Fr(\026)16890 57538 y Fo(\025)17536 57704 y Fk(h)18417 57538 y Fh(\024)307 b Fo(c)20064 57704 y Ff(5)20561 57538 y Fr(.)p 46801 57759 443 886 v 443 61350 a Fp(R)-57 b(emark.)490 b Fr(The)360 b(ab)31 b(o)-31 b(v)g(e)360 b(Lemma)h(is)e(based)g(on)h(the)f(fact)h(that)h Fo(\025)27118 61516 y Fk(j)27943 61350 y Fr(and)30132 61058 y(~)30086 61350 y Fo(\025)30732 61516 y Fk(j)31557 61350 y Fr(ha)-31 b(v)g(e)360 b(the)g(same)g(Beta)g(function,)j(up)0 62767 y(to)350 b Fo(O)31 b Fr(\()p Fo(\015)3272 62366 y Fk(\022)24 b(j)4185 62767 y Fr(\))349 b(terms)g(\(note)h(that)g(this) f(is)g(true)g(thanks)g(to)h(our)f(c)-31 b(hoice)349 b(of)h(the)f(coun) -31 b(terterms)349 b Fo(\027)39301 62933 y Fj(\006)40048 62767 y Fr(,)354 b(whic)-31 b(h)350 b(implies)0 64185 y(that)370 b Fo(\027)2945 64351 y Fk(j)m(;)p Fj(\006)4688 64185 y Fr(are)f Fo(O)31 b Fr(\()p Fo(\015)8475 63783 y Fk(\022)24 b(j)9388 64185 y Fr(\)\).)493 b(Hence)369 b(if)h Fo(\025)15999 64351 y Fk(j)16834 64185 y Fr(is)f(small,)i(the)f (same)f(is)h(true)f(for)31025 63893 y(~)30979 64185 y Fo(\025)31625 64351 y Fk(j)32091 64185 y Fr(.)443 66580 y(W)-92 b(e)358 b(w)-31 b(an)g(t)360 b(no)-31 b(w)360 b(to)f(discuss)f(the)h(prop)31 b(erties)358 b(of)h(the)g(constan)-31 b(ts)427 b(~)-620 b Fo(z)28016 66746 y Fk(j)28482 66580 y Fr(,)362 b Fo(h)307 b Fh(\024)g Fo(j)371 b Fh(\024)307 b(\000)p Fr(1,)362 b(b)-31 b(y)359 b(comparing)h(them)g(with)0 67997 y(the)377 b(constan)-31 b(ts)377 b Fo(z)7337 68163 y Fk(j)7804 67997 y Fr(,)h(whic)-31 b(h)378 b(are)e(in)-31 b(v)g(olv)g(ed)379 b(in)d(the)h(renormalization)j(of)d(the)f(free)g (measure,)j(see)d(\(2.23\).)517 b(There)0 69414 y(is)369 b(a)h(tree)f(expansion)h(for)f(the)h Fo(z)13510 69580 y Fk(j)13976 69414 y Fr(,)f(whic)-31 b(h)370 b(can)g(b)31 b(e)368 b(written)j(as)19927 72425 y Fo(z)20442 72591 y Fk(j)21215 72425 y Fr(=)308 b Fo(\014)23010 72591 y Fk(j)23722 72425 y Fr(+)246 b Fo(\014)25513 71850 y Ff(\(1\))25455 72733 y Fk(j)27010 72425 y Fo(;)17098 b Fr(\(4)p Fo(:)p Fr(69\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 0 1156 a Fr(where)408 b Fo(\014)3866 1322 y Fk(j)4740 1156 y Fr(is)g(the)g(sum)g(o)-31 b(v)g(er)409 b(the)f(trees)f(without)k(endp)31 b(oin)-31 b(ts)408 b(of)h(scale)f(index)h(+1,)418 b(while)409 b Fo(\014)39276 581 y Ff(\(1\))39218 1465 y Fk(j)40873 1156 y Fr(is)f(the)g(sum)g(of)0 2882 y(the)369 b(others,)h(satisfying)h(the)f(b)31 b(ound)369 b Fh(j)p Fo(\014)16787 2307 y Ff(\(1\))16729 3191 y Fk(j)17976 2882 y Fh(j)308 b(\024)g Fo(C)20676 2590 y Fr(\026)20630 2882 y Fo(\025)21276 2481 y Ff(2)21276 3196 y Fk(h)21849 2882 y Fo(\015)22483 2481 y Fk(\022)24 b(j)23396 2882 y Fr(.)492 b(The)370 b(tree)f(expansion)h(of)g(the)436 b(~)-620 b Fo(z)37511 3048 y Fk(j)38346 2882 y Fr(can)370 b(b)31 b(e)368 b(written)j(as)18353 6046 y(~)-619 b Fo(z)18802 6212 y Fk(j)19575 6046 y Fr(=)20902 5754 y(~)20744 6046 y Fo(\014)21370 6212 y Fk(j)22082 6046 y Fr(+)23347 5754 y(~)23189 6046 y Fo(\014)23873 5471 y Ff(\()p Fk(\027)52 b Ff(\))23815 6354 y Fk(j)25362 6046 y Fr(+)26627 5754 y(~)26469 6046 y Fo(\014)27153 5471 y Ff(\(1\))27095 6354 y Fk(j)28650 6046 y Fo(;)15458 b Fr(\(4)p Fo(:)p Fr(70\))0 9085 y(where)3480 8793 y(~)3323 9085 y Fo(\014)3949 9251 y Fk(j)4906 9085 y Fr(is)490 b(the)h(sum)g(o)-31 b(v)g(er)491 b(the)g(trees)g(without)i(endp)31 b(oin)-31 b(ts)491 b(of)h(scale)f(index)g(+1,)522 b(suc)-31 b(h)490 b(that)i(the)f(sp)31 b(ecial)0 10668 y(endp)g(oin)-31 b(t)340 b(is)g(of)f(t)-31 b(yp)31 b(e)9403 10376 y(~)9357 10668 y Fo(\025)p Fr(,)10813 10376 y(~)10655 10668 y Fo(\014)11339 10093 y Ff(\()p Fk(\027)52 b Ff(\))11281 10977 y Fk(j)12922 10668 y Fr(is)339 b(the)g(sum)g(o)-31 b(v)g(er)340 b(the)g(trees)e(whose)i(sp)31 b(ecial)340 b(endp)31 b(oin)-31 b(t)340 b(is)f(of)h(t)-31 b(yp)31 b(e)340 b Fo(T)41849 10834 y Fj(\006)42596 10668 y Fr(,)346 b(and)45529 10376 y(~)45371 10668 y Fo(\014)46055 10093 y Ff(\(1\))45997 10977 y Fk(j)0 12085 y Fr(is)417 b(the)h(sum)f(o)-31 b(v)g(er)418 b(the)f(trees)g(with)h(at)g(least)g(an)g(endp)31 b(oin)-31 b(t)418 b(of)g(scale)g(index)f(+1)h(\(in)g(this)f(case,)430 b(if)418 b(the)f(sp)31 b(ecial)0 13503 y(endp)g(oin)-31 b(t)370 b(is)f(of)h(t)-31 b(yp)31 b(e)370 b Fo(T)10124 13669 y Ff(1)10620 13503 y Fr(,)g(its)f(scale)h(index)g(m)-31 b(ust)369 b(b)31 b(e)369 b(+1,)h(see)f(discussion)g(in)g Fh(x)p Fr(4.4\).)443 15083 y(Since)345 b(there)f(is)h(no)g(tree)f(con) -31 b(tributing)347 b(to)18678 14791 y(~)18520 15083 y Fo(\014)19204 14508 y Ff(\(1\))19146 15391 y Fk(j)20738 15083 y Fr(without)g(at)e(least)h(one)f Fo(\025)f Fr(or)33099 14791 y(~)33053 15083 y Fo(\025)g Fr(endp)31 b(oin)-31 b(t)346 b(and)f(since)g(all)g(trees)0 16500 y(con)-31 b(tributing)407 b(to)f(it)f(satisfy)h(the)f(\\short)g(memory)g(prop)31 b(ert)-31 b(y",)415 b(b)-31 b(y)405 b(using)g(Lemma)h(4.2)g(\(whic)-31 b(h)406 b(implies)g(that)0 18044 y Fh(j)353 17752 y Fr(~)307 18044 y Fo(\025)953 18210 y Fk(j)1420 18044 y Fh(j)308 b(\024)f Fo(C)4119 17752 y Fr(\026)4073 18044 y Fo(\025)4719 18210 y Fk(h)5293 18044 y Fr(\),)362 b(w)-31 b(e)361 b(get)f(the)g(b)31 b(ound)360 b Fh(j)15614 17752 y Fr(~)15456 18044 y Fo(\014)16140 17469 y Ff(\(1\))16082 18352 y Fk(j)17330 18044 y Fh(j)308 b(\024)f Fo(C)20029 17752 y Fr(\026)19983 18044 y Fo(\025)20629 18210 y Fk(h)21203 18044 y Fo(\015)21837 17642 y Fk(\022)24 b(j)22749 18044 y Fr(.)490 b(In)359 b(a)h(similar)i(manner,)g(b)-31 b(y)360 b(using)h(Lemma)g(4.1,)i(w)-31 b(e)360 b(see)0 19770 y(that)370 b Fh(j)2863 19478 y Fr(~)2705 19770 y Fo(\014)3389 19195 y Ff(\()p Fk(\027)52 b Ff(\))3331 20078 y Fk(j)4633 19770 y Fh(j)308 b(\024)g Fo(C)7333 19478 y Fr(\026)7287 19770 y Fo(\025)7933 19368 y Ff(2)7933 20083 y Fk(h)8506 19770 y Fo(\015)9140 19368 y Fk(\022)24 b(j)10053 19770 y Fr(.)443 21407 y(Let)374 b(us)f(no)-31 b(w)375 b(consider)f Fo(\014)11112 21573 y Fk(j)11953 21407 y Fr(and)14268 21115 y(~)14110 21407 y Fo(\014)14736 21573 y Fk(j)15202 21407 y Fr(.)507 b(By)374 b(an)h(argumen)-31 b(t)375 b(similar)g(to)g(that)g(used)f(in)g(the)g(pro)31 b(of)375 b(of)f(Lemma)h(4.2,)0 22825 y(w)-31 b(e)370 b(can)f(write)15967 24746 y(~)15809 25038 y Fo(\014)16435 25204 y Fk(j)17148 25038 y Fh(\000)246 b Fo(\013)t(\014)19593 25204 y Fk(j)20367 25038 y Fr(=)22231 23655 y Fj(\000)p Ff(1)21998 23987 y Fe(X)21535 26344 y Fk(i)p Ff(=)p Fk(j)45 b Ff(+1)24244 25038 y Fo(\014)24870 25204 y Fk(j)m(;i)25865 25038 y Fr(\()26341 24746 y(~)26295 25038 y Fo(\025)26941 25204 y Fk(i)27556 25038 y Fh(\000)246 b Fo(\013)t(\025)30021 25204 y Fk(i)30390 25038 y Fr(\))308 b Fo(;)12980 b Fr(\(4)p Fo(:)p Fr(71\))0 28540 y(where)369 b Fo(\013)k Fr(is)c(de\014ned)g(as)g (in)h(Lemma)g(4.2)g(and)g Fh(j)p Fo(\014)20679 28706 y Fk(j)m(;i)21674 28540 y Fh(j)308 b(\024)g Fo(C)24374 28248 y Fr(\026)24328 28540 y Fo(\025)24974 28706 y Fk(h)25547 28540 y Fo(\015)26181 28139 y Ff(2)p Fk(\022)24 b(j)27535 28540 y Fr(.)493 b(Hence,)369 b(Lemma)h(4.2)h(implies)f(that)18437 31395 y Fh(j)66 b Fr(~)-619 b Fo(z)19259 31561 y Fk(j)19972 31395 y Fh(\000)246 b Fo(\013)t(z)22306 31561 y Fk(j)22772 31395 y Fh(j)308 b(\024)g Fo(C)25472 31103 y Fr(\026)25426 31395 y Fo(\025)26072 31561 y Fk(h)26645 31395 y Fo(\015)27279 30938 y Fk(\022)24 b(j)28499 31395 y Fo(:)15609 b Fr(\(4)p Fo(:)p Fr(72\))0 35231 y Fn(4.8)1314 b Fp(The)397 b(b)-57 b(ound)396 b(of)9952 34951 y Fr(~)9702 35231 y Fo(G)10572 34830 y Ff(4)10572 35505 y(+)11307 35231 y Fr(\()11796 34939 y(\026)11737 35231 y Fn(k)12409 35397 y Ff(1)12906 35231 y Fo(;)13457 34939 y Fr(\026)13397 35231 y Fn(k)14069 35397 y Ff(2)14566 35231 y Fo(;)15117 34939 y Fr(\026)15057 35231 y Fn(k)15729 35397 y Ff(3)16227 35231 y Fo(;)16778 34939 y Fr(\026)16718 35231 y Fn(k)17390 35397 y Ff(4)17887 35231 y Fr(\))p Fp(.)443 36685 y Fr(There)487 b(are)h(v)-61 b(arious)487 b(classes)h(of)g(trees)f(con)-31 b(tributing)490 b(to)e(the)g(tree)g(expansion)g(of)36578 36405 y(~)36328 36685 y Fo(G)37198 36283 y Ff(4)37198 36959 y(+)37933 36685 y Fr(\()38422 36393 y(\026)38363 36685 y Fn(k)39035 36851 y Ff(1)39532 36685 y Fo(;)40083 36393 y Fr(\026)40023 36685 y Fn(k)40695 36851 y Ff(2)41192 36685 y Fo(;)41743 36393 y Fr(\026)41683 36685 y Fn(k)42355 36851 y Ff(3)42853 36685 y Fo(;)43404 36393 y Fr(\026)43344 36685 y Fn(k)44016 36851 y Ff(4)44513 36685 y Fr(\),)518 b(de-)0 38102 y(p)31 b(ending)393 b(on)g(the)f(t)-31 b(yp)31 b(e)393 b(of)g(the)g(sp)31 b(ecial)393 b(endp)31 b(oin)-31 b(t.)564 b(Let)393 b(us)f(consider)g (\014rst)g(the)h(family)h Fh(T)38270 38242 y Ff(~)38228 38447 y Fk(\025)39201 38102 y Fr(of)f(the)g(trees)f(with)0 39520 y(an)414 b(endp)31 b(oin)-31 b(t)414 b(of)g(t)-31 b(yp)31 b(e)10124 39227 y(~)10078 39520 y Fo(\025)p Fr(.)624 b(These)414 b(trees)e(ha)-31 b(v)g(e)415 b(the)e(same)h(structure)f(of) h(those)f(app)31 b(earing)415 b(in)e(the)h(expansion)0 40937 y(of)460 b Fo(G)2221 40535 y Ff(4)2221 41211 y(+)2956 40937 y Fr(\()3445 40645 y(\026)3386 40937 y Fn(k)4058 41103 y Ff(1)4555 40937 y Fo(;)5106 40645 y Fr(\026)5046 40937 y Fn(k)5718 41103 y Ff(2)6215 40937 y Fo(;)6766 40645 y Fr(\026)6706 40937 y Fn(k)7378 41103 y Ff(3)7876 40937 y Fo(;)8427 40645 y Fr(\026)8367 40937 y Fn(k)9039 41103 y Ff(4)9536 40937 y Fr(\),)482 b(except)460 b(for)f(the)g(fact)h (that)g(the)f(external)h(\(renormalized\))i(propagator)e(of)g(scale)f Fo(h)0 42354 y Fr(and)369 b(momen)-31 b(tum)8022 42062 y(\026)7963 42354 y Fn(k)8635 42520 y Ff(4)9499 42354 y Fr(is)369 b(substituted)g(with)h(the)f(free)f(propagator)408 b(^)-591 b Fo(g)29146 42520 y Fj(\000)29893 42354 y Fr(\()30382 42062 y(\026)30323 42354 y Fn(k)30995 42520 y Ff(4)31492 42354 y Fr(\).)493 b(It)368 b(follo)-31 b(ws,)372 b(b)-31 b(y)368 b(using)h(the)g(b)31 b(ound)0 43771 y Fh(j)353 43479 y Fr(~)307 43771 y Fo(\025)953 43937 y Fk(j)1420 43771 y Fh(j)444 b(\024)h Fo(C)4393 43479 y Fr(\026)4347 43771 y Fo(\025)4993 43937 y Fk(h)5566 43771 y Fr(,)472 b(that)453 b(a)f(tree)f(with)h Fo(n)g Fr(endp)31 b(oin)-31 b(t)452 b(is)f(b)31 b(ounded)452 b(b)-31 b(y)451 b(\()p Fo(C)29306 43479 y Fr(\026)29260 43771 y Fo(\025)29906 43937 y Fk(h)30480 43771 y Fr(\))30910 43370 y Fk(n)31513 43771 y Fo(Z)32348 43298 y Fj(\000)p Ff(1)32269 44105 y Fk(h)33537 43771 y Fo(\015)34171 43370 y Fj(\000)p Ff(4)p Fk(h)35878 43771 y Fr(,)472 b(larger)452 b(for)f(a)h(factor)g Fo(Z)46671 43937 y Fk(h)0 45189 y Fr(with)370 b(resp)31 b(ect)369 b(to)h(what)g(w)-31 b(e)370 b(need.)443 46642 y(Let)437 b(us)g(no)-31 b(w)439 b(consider)e(the)h(family)i Fh(T)16823 46808 y Ff(~)-495 b Fk(z)17716 46642 y Fr(of)438 b(the)g(trees)f(with)i(a)e(sp)31 b(ecial)439 b(endp)31 b(oin)-31 b(t)438 b(of)g(t)-31 b(yp)31 b(e)505 b(~)-620 b Fo(z)49 b Fr(.)697 b(Giv)-31 b(en)438 b(a)g(tree)0 48060 y Fo(\034)433 b Fh(2)307 b(T)2607 48200 y Ff(~)2565 48404 y Fk(\025)3145 48060 y Fr(,)370 b(w)-31 b(e)370 b(can)f(asso)31 b(ciate)370 b(with)h(it)f(the)f(class)g Fh(T)20923 48226 y Ff(~)-495 b Fk(z)32 b(;\034)22509 48060 y Fr(of)370 b(all)g Fo(\034)25916 47658 y Fj(0)26534 48060 y Fh(2)308 b(T)28225 48200 y Ff(~)28183 48404 y Fk(\025)28763 48060 y Fr(,)370 b(obtained)h(b)-31 b(y)369 b Fo(\034)494 b Fr(in)370 b(the)f(follo)-31 b(wing)373 b(w)-31 b(a)g(y:)369 49513 y(1\))370 b(w)-31 b(e)370 b(substitute)g(the)f(endp)31 b(oin)-31 b(t)371 b Fo(v)15626 49112 y Fj(\003)16503 49513 y Fr(of)e(t)-31 b(yp)31 b(e)20300 49221 y(~)20254 49513 y Fo(\025)369 b Fr(of)g Fo(\034)495 b Fr(with)370 b(an)g(endp)31 b(oin)-31 b(t)370 b(of)g(t)-31 b(yp)31 b(e)369 b Fo(\025)p Fr(;)366 50967 y(2\))f(w)-31 b(e)367 b(link)g(the)g(endp)31 b(oin)-31 b(t)368 b Fo(v)12615 50565 y Fj(\003)13489 50967 y Fr(to)g(an)f(endp)31 b(oin)-31 b(t)367 b(of)g(t)-31 b(yp)31 b(e)434 b(~)-619 b Fo(z)414 b Fr(trough)368 b(a)f(renormalized)g(propagator)i(of)e(scale)g Fo(h)p Fr(.)443 52421 y(Note)471 b(that)g Fh(T)6375 52587 y Ff(~)-495 b Fk(z)7306 52421 y Fr(=)475 b Fh([)9380 52587 y Fk(\034)90 b Fj(2T)11021 52696 y Fm(~)10977 52842 y Fl(\025)11552 52421 y Fh(T)12209 52587 y Ff(~)-495 b Fk(z)31 b(;\034)13895 52421 y Fr(and)470 b(that,)497 b(if)470 b Fo(\034)595 b Fr(has)469 b Fo(n)h Fr(endp)31 b(oin)-31 b(ts,)497 b(an)-31 b(y)470 b Fo(\034)32665 52019 y Fj(0)33451 52421 y Fh(2)475 b(T)35321 52587 y Ff(~)-495 b Fk(z)32 b(;\034)37007 52421 y Fr(has)470 b Fo(n)314 b Fr(+)f(1)470 b(endp)31 b(oin)-31 b(ts.)0 54109 y(Moreo)g(v)g(er,)411 b(since)402 b(the)g(v)-61 b(alue)402 b(of)14142 53817 y(\026)14083 54109 y Fn(k)14755 54275 y Ff(4)15653 54109 y Fr(has)g(b)31 b(e)401 b(c)-31 b(hosen)402 b(so)g(that)h Fo(f)27139 54275 y Fk(h)27713 54109 y Fr(\()28202 53817 y(\026)28143 54109 y Fn(k)28815 54275 y Ff(4)29312 54109 y Fr(\))362 b(=)g(1,)449 b(^)-591 b Fo(g)33166 53534 y Ff(\()p Fk(h)p Ff(\))33126 54385 y Fj(\000)34431 54109 y Fr(\()34920 53817 y(\026)34861 54109 y Fn(k)35533 54275 y Ff(4)36030 54109 y Fr(\))363 b(=)f Fo(Z)38881 53636 y Fj(\000)p Ff(1)38802 54443 y Fk(h)p Fj(\000)p Ff(1)40546 54109 y Fr(^)-591 b Fo(g)41036 54275 y Fj(\000)41783 54109 y Fr(\()42272 53817 y(\026)42213 54109 y Fn(k)42885 54275 y Ff(4)43382 54109 y Fr(\);)419 b(hence)0 55526 y(it)382 b(is)f(easy)h(to)g(sho)-31 b(w)381 b(that)i(the)e(sum)h(of)f(the)h(v)-61 b(alues)381 b(of)h(a)f(tree)h Fo(\034)452 b Fh(2)328 b(T)29079 55666 y Ff(~)29037 55871 y Fk(\025)29617 55526 y Fr(,)385 b(suc)-31 b(h)381 b(the)g(sp)31 b(ecial)382 b(endp)31 b(oin)-31 b(t)382 b(has)g(scale)0 56944 y(index)370 b Fo(j)3502 56542 y Fj(\003)4257 56944 y Fr(+)246 b(1,)370 b(and)f(of)h(all)g Fo(\034)12153 56542 y Fj(0)12772 56944 y Fh(2)307 b(T)14474 57110 y Ff(~)-495 b Fk(z)32 b(;\034)16060 56944 y Fr(is)369 b(obtained)i(from)e (the)h(v)-61 b(alue)369 b(of)h Fo(\034)125 b Fr(,)370 b(b)-31 b(y)369 b(substituting)39398 56652 y(~)39352 56944 y Fo(\025)39998 57110 y Fk(j)40409 56888 y Fc(\003)41296 56944 y Fr(with)16733 60779 y(\003)17502 60945 y Fk(j)17913 60724 y Fc(\003)18739 60779 y Fr(=)19953 60487 y(~)19907 60779 y Fo(\025)20553 60945 y Fk(j)20964 60724 y Fc(\003)21728 60779 y Fh(\000)246 b Fo(\025)23481 60945 y Fk(j)23892 60724 y Fc(\003)24543 59034 y Fe(P)25711 59308 y Fj(\000)p Ff(1)25711 60197 y Fk(j)45 b Ff(=)p Fk(h)27626 59865 y Fr(~)-619 b Fo(z)28075 60031 y Fk(j)28541 59865 y Fo(Z)29297 60031 y Fk(j)p 24543 60524 5221 45 v 25922 61538 a Fo(Z)26678 61704 y Fk(h)p Fj(\000)p Ff(1)30203 60779 y Fo(;)13905 b Fr(\(4)p Fo(:)p Fr(73\))0 64101 y(see)369 b(Fig.)493 b(10.)443 67445 y(On)468 b(the)i(other)f(hand,)494 b(\(4.72\))472 b(and)d(the)g(b)31 b(ound)469 b Fo(Z)22411 67611 y Fk(j)23351 67445 y Fh(\024)474 b Fo(\015)25320 67043 y Fj(\000)p Fk(C)26743 66839 y Ff(\026)26701 67043 y Fk(\025)27226 66709 y Fm(2)27226 67270 y Fl(h)27740 67043 y Fk(j)28206 67445 y Fr(,)494 b(see)469 b([BM1],)495 b(imply)471 b(that,)495 b(if)41862 67153 y(\026)41816 67445 y Fo(\025)42462 67611 y Fk(h)43504 67445 y Fr(is)469 b(small)0 68862 y(enough)13266 69907 y Fj(\000)p Ff(1)13033 70239 y Fe(X)13028 72622 y Fk(j)45 b Ff(=)p Fk(h)14821 71291 y Fh(j)67 b Fr(~)-619 b Fo(z)15644 71457 y Fk(j)16110 71291 y Fo(Z)16866 71457 y Fk(j)17578 71291 y Fh(\000)246 b Fo(\013)t(z)19912 71457 y Fk(j)20378 71291 y Fo(Z)21134 71457 y Fk(j)21601 71291 y Fh(j)308 b(\024)23622 69907 y Fj(\000)p Ff(1)23389 70239 y Fe(X)23384 72622 y Fk(j)45 b Ff(=)p Fk(h)25177 71291 y Fo(C)26093 70999 y Fr(\026)26047 71291 y Fo(\025)26693 71457 y Fk(h)27267 71291 y Fo(\015)27901 70834 y Fk(\022)24 b(j)28814 71291 y Fo(Z)29570 71457 y Fk(j)30343 71291 y Fh(\024)308 b Fo(C)32428 70999 y Fr(\026)32382 71291 y Fo(\025)33028 71457 y Fk(h)33909 71291 y Fo(:)10199 b Fr(\(4)p Fo(:)p Fr(74\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 0 1107 a Fr(It)369 b(follo)-31 b(ws,)372 b(b)-31 b(y)370 b(using)f(also)h(the)g(b)31 b(ound)369 b(\(4.56\),)j(that)13863 4266 y(\003)14632 4432 y Fk(j)15043 4211 y Fc(\003)15868 4266 y Fr(=)308 b Fo(\013)t(\025)18395 4432 y Fk(j)18806 4211 y Fc(\003)19508 2373 y Fe(")20154 4266 y Fr(1)246 b Fh(\000)22193 2522 y Fe(P)23362 2795 y Fj(\000)p Ff(1)23362 3684 y Fk(j)45 b Ff(=)p Fk(h)25210 3352 y Fo(z)25725 3518 y Fk(j)26192 3352 y Fo(Z)26948 3518 y Fk(j)p 22193 4012 5221 45 v 23572 5026 a Fo(Z)24328 5192 y Fk(h)p Fj(\000)p Ff(1)27547 2373 y Fe(#)28438 4266 y Fr(+)29678 3517 y Fo(O)31 b Fr(\()31029 3225 y(\026)30983 3517 y Fo(\025)31629 3683 y Fk(h)32203 3517 y Fr(\))p 29678 4012 2956 45 v 30491 5026 a Fo(Z)31247 5192 y Fk(h)33074 4266 y Fo(:)11034 b Fr(\(4)p Fo(:)p Fr(75\))0 7393 y(Moreo)-31 b(v)g(er,)370 b(since)f Fo(Z)8640 7559 y Fk(j)45 b Fj(\000)p Ff(1)10547 7393 y Fr(=)307 b Fo(Z)12471 7559 y Fk(j)12937 7393 y Fr(\(1)247 b(+)f Fo(z)15789 7559 y Fk(j)16255 7393 y Fr(\),)370 b(for)g Fo(j)g Fh(2)308 b Fr([)p Fh(\000)p Fr(1)p Fo(;)184 b(h)p Fr(],)371 b(and)f Fo(Z)27673 7559 y Fj(\000)p Ff(1)29168 7393 y Fr(=)308 b(1,)370 b(it)g(is)f(easy)g(to)h (c)-31 b(hec)g(k)370 b(that)18394 10615 y Fo(Z)19150 10781 y Fk(h)p Fj(\000)p Ff(1)21102 10615 y Fh(\000)22447 9232 y Fj(\000)p Ff(1)22214 9564 y Fe(X)22209 11947 y Fk(j)45 b Ff(=)p Fk(h)24003 10615 y Fo(z)24518 10781 y Fk(j)24984 10615 y Fo(Z)25740 10781 y Fk(j)26513 10615 y Fr(=)308 b(1)g Fo(:)15565 b Fr(\(4)p Fo(:)p Fr(76\))0 13951 y(This)370 b(iden)-31 b(tit)g(y)-92 b(,)371 b(Lemma)g(4.2)f(and)g (\(4.75\))h(imply)g(the)e(b)31 b(ound)20188 16809 y Fh(j)p Fr(\003)21264 16975 y Fk(j)21675 16754 y Fc(\003)22193 16809 y Fh(j)308 b(\024)f Fo(C)25081 15768 y Fr(\026)25034 16060 y Fo(\025)25680 16226 y Fk(h)p 24980 16554 1330 45 v 24980 17568 a Fo(Z)25736 17734 y Fk(h)26749 16809 y Fo(;)17359 b Fr(\(4)p Fo(:)p Fr(77\))0 19620 y(whic)-31 b(h)432 b(giv)-31 b(es)433 b(us)e(the)g(\\missing")j Fo(Z)15408 19146 y Fj(\000)p Ff(1)15329 19954 y Fk(h)17027 19620 y Fr(factor)f(for)f(the)f(sum)h(o)-31 b(v)g(er)432 b(the)f(trees)g(whose)h(sp)31 b(ecial)432 b(endp)31 b(oin)-31 b(t)433 b(is)e(of)0 21037 y(t)-31 b(yp)31 b(e)2537 20745 y(~)2491 21037 y Fo(\025)368 b Fr(or)436 b(~)-620 b Fo(z)49 b Fr(.)443 23399 y(Let)306 b(us)f(no)-31 b(w)307 b(consider)e(the)h (family)i Fh(T)15978 23565 y Fk(\027)16835 23399 y Fr(of)e(the)g(trees) f(with)i(a)g(sp)31 b(ecial)306 b(endp)31 b(oin)-31 b(t)307 b(of)f(t)-31 b(yp)31 b(e)307 b Fo(T)38144 23565 y Fj(\006)38891 23399 y Fr(.)471 b(It)306 b(is)f(easy)h(to)h(see,)0 24816 y(b)-31 b(y)269 b(using)g(Lemma)h(4.1)h(and)e(the)g(\\short)g(memory)h (prop)31 b(ert)-31 b(y",)290 b(that)270 b(the)f(sum)g(o)-31 b(v)g(er)270 b(the)f(trees)f(of)h(this)h(class)f(with)0 26234 y Fo(n)308 b Fh(\025)f Fr(0)312 b(normal)h(endp)31 b(oin)-31 b(ts)312 b(is)g(b)31 b(ounded,)323 b(for)19213 25942 y(\026)19167 26234 y Fo(\025)19813 26400 y Fk(h)20698 26234 y Fr(small)313 b(enough,)324 b(b)-31 b(y)312 b(\()p Fo(C)30437 25942 y Fr(\026)30391 26234 y Fo(\025)31037 26400 y Fk(h)31611 26234 y Fr(\))32041 25832 y Fk(n)p Ff(+1)33765 26234 y Fo(Z)34600 25760 y Fj(\000)p Ff(1)34521 26568 y Fk(h)35788 26234 y Fo(\015)36422 25832 y Fj(\000)p Ff(4)p Fk(h)38314 25404 y Fe(P)39482 25677 y Fj(\000)p Ff(1)39482 26566 y Fk(j)45 b Ff(=)p Fk(h)41331 26234 y Fo(Z)42166 25760 y Fj(\000)p Ff(2)42087 26542 y Fk(j)43354 26234 y Fo(\015)43988 25832 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(h)p Fj(\000)p Fk(j)45 b Ff(\))0 27810 y Fo(\015)634 27408 y Fk(\022)24 b(j)1854 27810 y Fh(\024)307 b Fr(\()p Fo(C)4368 27518 y Fr(\026)4322 27810 y Fo(\025)4968 27976 y Fk(h)5543 27810 y Fr(\))5973 27408 y Fk(n)p Ff(+1)7696 27810 y Fo(Z)8531 27337 y Fj(\000)p Ff(3)8452 28144 y Fk(h)9720 27810 y Fo(\015)10354 27408 y Fj(\000)p Ff(\(4)p Fj(\000)p Fk(\022)24 b Ff(\))p Fk(h)13890 27810 y Fr(,)370 b(whic)-31 b(h)370 b(is)f(ev)-31 b(en)369 b(b)31 b(etter)370 b(of)f(our)g(needs.)443 29227 y(W)-92 b(e)399 b(still)i(ha)-31 b(v)g(e)400 b(to)h(consider)e(the)h(family)i Fh(T)18832 29393 y Ff(1)19728 29227 y Fr(of)e(the)g(trees)f(with)i(a)f(sp)31 b(ecial)400 b(endp)31 b(oin)-31 b(t)401 b(of)f(t)-31 b(yp)31 b(e)400 b Fo(T)41881 29393 y Ff(1)42378 29227 y Fr(.)584 b(There)399 b(is)0 30645 y(\014rst)368 b(of)g(all)i(the)f (trivial)h(tree,)f(obtained)g(b)-31 b(y)369 b(con)-31 b(tracting)371 b(all)e(the)g Fo( )407 b Fr(lines)369 b(of)g Fo(T)33706 30811 y Ff(1)34570 30645 y Fr(on)g(scale)g Fo(h)p Fr(,)f(but)h(its)f(v)-61 b(alue)369 b(is)0 32062 y(0,)377 b(b)31 b(ecause)375 b(of)h(the)f(supp)31 b(ort)375 b(prop)31 b(erties)375 b(of)g(the)h(function)507 b(~)-684 b Fo(\037)p Fr(\()p Fn(p)p Fr(\).)511 b(Let)375 b(us)g(no)-31 b(w)376 b(consider)f(a)g(tree)g Fo(\034)443 b Fh(2)317 b(T)44220 32228 y Ff(1)45092 32062 y Fr(with)0 33479 y Fo(n)e Fh(\025)g Fr(1)374 b(endp)31 b(oin)-31 b(ts)374 b(of)h(t)-31 b(yp)31 b(e)374 b Fo(\025)p Fr(,)g(whose)g(structure)f(is) h(describ)31 b(ed)373 b(in)g(item)i(5\))g(of)f Fh(x)p Fr(4.5,)i(whic)-31 b(h)375 b(w)-31 b(e)374 b(shall)g(refer)f(to)0 34897 y(for)382 b(notation.)534 b(If)382 b(w)-31 b(e)383 b(put)f Fo(h)12047 35063 y Fk(v)12490 35174 y Fm(1)13306 34897 y Fr(=)329 b Fo(j)14952 35063 y Ff(1)15703 34897 y Fr(+)255 b(1)382 b(and)g Fo(h)20557 35063 y Fk(v)21000 35174 y Fm(2)21816 34897 y Fr(=)329 b Fo(j)23462 35063 y Ff(2)23959 34897 y Fr(,)386 b(then)c(the)g(dimensional)j(b)31 b(ound)382 b(of)g(this)h(tree)f(di\013ers)0 36314 y(from)315 b(that)h(of)f(a)f(tree)g(with)i Fo(n)136 b Fr(+)g(1)316 b(normal)g(endp)31 b(oin)-31 b(ts)315 b(con)-31 b(tributing)317 b(to)e Fo(G)31388 35912 y Ff(4)31388 36588 y(+)32123 36314 y Fr(\()32612 36022 y(\026)32553 36314 y Fn(k)33225 36480 y Ff(1)33722 36314 y Fo(;)34273 36022 y Fr(\026)34213 36314 y Fn(k)34885 36480 y Ff(2)35383 36314 y Fo(;)35934 36022 y Fr(\026)35874 36314 y Fn(k)36546 36480 y Ff(3)37043 36314 y Fo(;)37594 36022 y Fr(\026)37534 36314 y Fn(k)38206 36480 y Ff(4)38703 36314 y Fr(\))g(for)g(the)f(follo)-31 b(wing)0 37731 y(reasons:)415 39149 y(1\))416 b(there)e(is)h(a)g (factor)i Fo(Z)10870 38675 y Fj(\000)p Ff(1)10791 39483 y Fk(h)12473 39149 y Fr(missing,)427 b(b)31 b(ecause)415 b(the)g(external)h(\(renormalized\))h(propagator)g(of)f(scale)f Fo(h)g Fr(and)0 40566 y(momen)-31 b(tum)5871 40274 y(\026)5812 40566 y Fn(k)6484 40732 y Ff(4)7349 40566 y Fr(is)369 b(substituted)h(with)h(the)e(free)g(propagator)409 b(^)-591 b Fo(g)27000 40732 y Fj(\000)27747 40566 y Fr(\()28236 40274 y(\026)28177 40566 y Fn(k)28849 40732 y Ff(4)29346 40566 y Fr(\);)499 41983 y(2\))500 b(there)f(is)g(a)h(factor)g Fh(j)p Fo(\025)11493 42149 y Fk(j)11859 42260 y Fm(1)12347 41983 y Fh(j)p Fo(Z)13489 41582 y Ff(2)13410 42271 y Fk(j)13776 42382 y Fm(1)14763 41983 y Fr(missing,)533 b(b)31 b(ecause)499 b(there)g(is)g(no)h(external)g(\014eld)g (renormalization)i(in)e(the)0 43516 y Fo(T)647 43682 y Ff(1)1143 43516 y Fr(\()p Fo( )2334 43114 y Ff([)p Fk(h;j)45 b Ff(])4084 43516 y Fr(\))369 b(con)-31 b(tribution)372 b(to)12681 43236 y(\026)12543 43516 y Fh(V)13312 43114 y Ff(\()p Fk(j)45 b Ff(\))14470 43516 y Fr(\()p Fo( )15661 43114 y Ff([)p Fk(h;j)g Ff(])17410 43516 y Fr(\),)370 b(see)f(\(4.43\);)419 44933 y(3\))420 b(if)g Fo(P)3598 45099 y Ff(1)4514 44933 y Fr(con)-31 b(tains)421 b(only)f(one)g(of)g (the)f(t)-31 b(w)g(o)422 b(external)e(lines)g(of)g Fo(v)27627 45099 y Ff(1)28542 44933 y Fr(b)31 b(elonging)421 b(to)g Fo(\016)42 b(\032)p Fr(,)432 b(then)420 b(there)f(is)g(a)h(factor)0 46350 y Fo(\015)634 45948 y Fj(\000)p Ff(\()p Fk(j)2038 46059 y Fm(2)2471 45948 y Fj(\000)p Fk(h)p Ff(\))4439 46350 y Fr(missing,)361 b(b)31 b(ecause)357 b(the)g(absence)g(of)g (regularization)j(in)e(the)f(v)-31 b(ertices)357 b Fo(v)347 b Fh(\024)308 b Fo(v)36727 46516 y Ff(2)37223 46350 y Fr(,)360 b(but)d(this)g(is)g(comp)31 b(en-)0 47768 y(sated)452 b(b)-31 b(y)452 b(the)g(same)g(factor)g(arising)h(b)31 b(ecause)451 b(of)h(the)g(b)31 b(ound)451 b(\(4.22\),)476 b(see)451 b(discussion)g(after)i(\(4.28\))h(and)e(in)0 49185 y Fh(x)p Fr(4.4,)371 b(so)e(that)h(the)g(\\short)f(memory)i(prop) 31 b(ert)-31 b(y")369 b(is)g(alw)-31 b(a)g(ys)372 b(satis\014ed;)369 50602 y(4\))e(there)f(is)g(a)g(factor)i Fo(Z)10595 50129 y Fj(\000)p Ff(1)10516 50936 y Fk(h)12152 50602 y Fr(missing,)g(b)31 b(ecause)368 b(of)i(the)g(remark)f(follo)-31 b(wing)373 b(\(4.22\).)443 52019 y(It)318 b(follo)-31 b(ws)320 b(that)f(the)f(sum) g(of)g(the)g(v)-61 b(alues)318 b(of)h(all)g(trees)e Fo(\034)433 b Fh(2)307 b(T)25840 52185 y Ff(1)26654 52019 y Fr(with)319 b Fo(n)308 b Fh(\025)g Fr(1)318 b(normal)h(endp)31 b(oin)-31 b(ts,)329 b(if)42164 51727 y(\026)42118 52019 y Fo(\025)42764 52185 y Fk(h)43655 52019 y Fr(is)318 b(small)0 53437 y(enough,)370 b(is)f(b)31 b(ounded)370 b(b)-31 b(y)369 b(\()p Fo(C)12637 53145 y Fr(\026)12591 53437 y Fo(\025)13237 53603 y Fk(h)13811 53437 y Fr(\))14241 53035 y Fk(n)14844 53437 y Fo(\015)15478 53035 y Fj(\000)p Ff(4)p Fk(h)17370 52607 y Fe(P)18538 52880 y Ff(0)18538 53769 y Fk(j)18904 53880 y Fm(1)19337 53769 y Ff(=)p Fk(h)20774 53437 y Fo(Z)21609 52963 y Fj(\000)p Ff(2)21530 53745 y Fk(j)21896 53856 y Fm(1)22798 53437 y Fo(\015)23432 53035 y Ff(2)p Fk(\022)24 b Ff(\()p Fk(h)p Fj(\000)p Fk(j)26241 53146 y Fm(1)26673 53035 y Ff(\))27444 53437 y Fh(\024)307 b Fr(\()p Fo(C)29958 53145 y Fr(\026)29912 53437 y Fo(\025)30558 53603 y Fk(h)31132 53437 y Fr(\))31562 53035 y Fk(n)32165 53437 y Fo(\015)32799 53035 y Fj(\000)p Ff(4)p Fk(h)34507 53437 y Fo(Z)35342 52963 y Fj(\000)p Ff(2)35263 53771 y Fk(h)36530 53437 y Fr(.)443 54854 y(By)407 b(collecting)i(all)f(the)f (previous)g(b)31 b(ounds,)416 b(w)-31 b(e)407 b(pro)-31 b(v)g(e)408 b(that)g(the)f(b)31 b(ound)406 b(\(3.34\))k(of)d(Lemma)h (3.1)g(is)e(satis\014ed)0 56271 y(in)369 b(the)h(case)f(of)h Fo(H)7812 55739 y Ff(4)p Fk(;)p Ff(1)7722 56547 y(+)9013 56271 y Fr(.)443 58634 y Fp(R)-57 b(emark.)817 b Fr(In)477 b Fo(T)7444 58800 y Ff(1)8417 58634 y Fr(and)g(in)h(the)f(Grassmannian) h(monomials)h(m)-31 b(ultiplying)481 b Fo(\027)33686 58800 y Fk(j)m(;)p Ff(+)35047 58634 y Fo(;)184 b(\027)36085 58800 y Fk(j)m(;)p Fj(\000)37459 58634 y Fr(,)505 b(an)477 b(external)h(line)g(is)0 60051 y(alw)-31 b(a)g(ys)392 b(asso)31 b(ciated)391 b(to)f(a)g(free)g(propagator)429 b(^)-591 b Fo(g)19528 60217 y Fj(\000)20275 60051 y Fr(\()20764 59759 y(\026)20705 60051 y Fn(k)21377 60217 y Ff(4)21874 60051 y Fr(\);)401 b(this)390 b(is)g(due)f(to)i(the)f(fact)h(that,)396 b(in)390 b(deriving)h(the)f(Dyson)0 61468 y(equation)416 b(\(3.10\),)428 b(one)414 b(extracts)h(a)f(free)f(propagator.)629 b(Then)414 b(in)g(the)g(b)31 b(ounds)413 b(there)h(is)g(a)g Fo(Z)39706 61634 y Fk(h)40693 61468 y Fr(missing)h(\(suc)-31 b(h)0 62886 y(propagator)440 b(is)e(not)h(\\dressed")e(in)i(the)f(m)-31 b(ultiscale)441 b(in)-31 b(tegration)441 b(pro)31 b(cedure\),)455 b(and)439 b(at)g(the)f(end)g(the)g(crucial)0 64303 y(iden)-31 b(tit)g(y)372 b(\(4.76\))g(has)d(to)h(b)31 b(e)368 b(used)h(to)h (\\dress")e(the)i(extracted)g(propagator)h(carrying)f(momen)-31 b(tum)42567 64011 y(\026)42507 64303 y Fn(k)43179 64469 y Ff(4)43676 64303 y Fr(.)0 66665 y Fn(4.9)1314 b Fp(The)397 b(b)-57 b(ound)396 b(in)h(the)f(c)-57 b(ase)396 b(of)g Fo(H)16284 66133 y Ff(4)p Fk(;)p Ff(1)16194 66941 y Fj(\000)17485 66665 y Fr(.)443 68082 y(If)380 b(w)-31 b(e)381 b(substitute,)j(in)d (the)g(l.h.s.)527 b(of)381 b(\(4.1\))h Fo(H)19666 67550 y Ff(4)p Fk(;)p Ff(1)19576 68358 y(+)21247 68082 y Fr(with)g Fo(H)24790 67550 y Ff(4)p Fk(;)p Ff(1)24700 68358 y Fj(\000)25991 68082 y Fr(,)h(w)-31 b(e)381 b(can)g(pro)31 b(ceed)379 b(in)i(a)g(similar)h(w)-31 b(a)g(y)-92 b(.)527 b(By)381 b(using)0 69500 y(\(3.23\),)372 b(w)-31 b(e)370 b(get)9874 71349 y(^)-591 b Fo(g)10364 71515 y Fj(\000)11111 71349 y Fr(\()p Fn(k)12213 71515 y Ff(4)12710 71349 y Fr(\))13715 70600 y(1)p 13273 71095 1438 45 v 13273 72109 a Fo(L\014)15028 70298 y Fe(X)15550 72622 y Fi(p)16943 71349 y Fr(~)-684 b Fo(\037)17505 71515 y Fk(M)18487 71349 y Fr(\()p Fn(p)p Fr(\))p Fo(D)21001 70876 y Fj(\000)p Ff(1)20970 71625 y(+)22191 71349 y Fr(\()p Fn(p)p Fr(\))24049 71070 y(^)23758 71349 y Fo(H)24768 70817 y Ff(4)p Fk(;)p Ff(1)24678 71625 y Fj(\000)25970 71349 y Fr(\()p Fn(p)p Fr(;)184 b Fn(k)28270 71515 y Ff(1)28768 71349 y Fo(;)g Fn(k)29931 71515 y Ff(2)30428 71349 y Fo(;)g Fn(k)31591 71515 y Ff(3)32089 71349 y Fo(;)g Fn(k)33252 71515 y Ff(4)33995 71349 y Fh(\000)246 b Fn(p)p Fr(\))308 b(=)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 692 1579 a Fr(=)345 b(^)-590 b Fo(g)2389 1745 y Fj(\000)3136 1579 y Fr(\()p Fn(k)4238 1745 y Ff(4)4735 1579 y Fr(\))5740 830 y(1)p 5298 1324 1438 45 v 5298 2338 a Fo(L\014)7053 527 y Fe(X)7574 2852 y Fi(p)8968 1579 y Fr(~)-684 b Fo(\037)9530 1745 y Fk(M)10512 1579 y Fr(\()p Fn(p)p Fr(\))12655 830 y(1)p 12212 1324 V 12212 2338 a Fo(L\014)13968 527 y Fe(X)14503 2911 y Fi(k)15884 830 y Fo(C)16675 996 y Fj(\000)17423 830 y Fr(\()p Fn(k)p Fo(;)184 b Fn(k)247 b Fh(\000)f Fn(p)p Fr(\))p 15884 1324 6296 45 v 17422 2338 a Fo(D)18338 2504 y Ff(+)19074 2338 y Fr(\()p Fn(p)p Fr(\))22620 1579 y Fo(<)24015 1287 y Fr(^)23788 1579 y Fo( )24549 1106 y Ff(+)24509 1913 y Fi(k)p Fk(;)p Fj(\000)26275 1287 y Fr(^)26049 1579 y Fo( )26810 1106 y Fj(\000)26770 1913 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)29557 1579 y Fr(;)30276 1287 y(^)30048 1579 y Fo( )30809 1106 y Fj(\000)30769 1913 y Fi(k)31298 2024 y Fm(1)31731 1913 y Fk(;)p Ff(+)32729 1579 y Fr(;)33448 1287 y(^)33220 1579 y Fo( )33981 1106 y Ff(+)33941 1913 y Fi(k)34470 2024 y Fm(2)34903 1913 y Fk(;)p Ff(+)35901 1579 y Fr(;)36620 1287 y(^)36392 1579 y Fo( )37153 1106 y Fj(\000)37113 1913 y Fi(k)37642 2024 y Fm(3)38075 1913 y Fk(;)p Fj(\000)39085 1579 y Fr(;)39804 1287 y(^)39576 1579 y Fo( )40337 1106 y Ff(+)40297 1913 y Fi(k)40826 2024 y Fm(4)41259 1913 y Fj(\000)p Fi(p)p Fk(;)p Fj(\000)43825 1579 y Fo(>)44686 1122 y Fk(T)45691 1579 y Fr(+)1737 5581 y Fh(\000)p 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31155 y Ff(3)15584 30989 y Fo(;)g Fn(k)16747 31155 y Ff(4)17244 30989 y Fr(\))341 b(is)f(done)h(via)g(a)g (m)-31 b(ultiscale)343 b(expansion)e(essen)-31 b(tially)343 b(iden)-31 b(tical)342 b(to)g(the)0 32406 y(one)369 b(of)3541 32126 y(~)3290 32406 y Fo(G)4160 32004 y Ff(4)4160 32680 y(+)4895 32406 y Fr(\()p Fn(k)5997 32572 y Ff(1)6494 32406 y Fo(;)184 b Fn(k)7657 32572 y Ff(2)8155 32406 y Fo(;)g Fn(k)9318 32572 y Ff(3)9815 32406 y Fo(;)g Fn(k)10978 32572 y Ff(4)11476 32406 y Fr(\),)370 b(b)-31 b(y)369 b(taking)i(in)-31 b(to)371 b(accoun)-31 b(t)370 b(that)h Fo(\016)42 b(\032)27372 32572 y Fi(p)p Fk(;)p Ff(+)29295 32406 y Fr(has)369 b(to)h(b)31 b(e)369 b(substituted)h(with)13822 35515 y Fo(\016)42 b(\032)14928 35681 y Fi(p)p Fk(;)p Fj(\000)16802 35515 y Fr(=)18545 34766 y(1)p 18103 35261 1438 45 v 18103 36275 a Fo(\014)58 b(L)19858 34464 y Fe(X)20393 36847 y Fi(k)21775 34766 y Fo(C)22566 34932 y Fj(\000)23313 34766 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)p Fr(\))p 21775 35261 4271 45 v 22300 36275 a Fo(D)23216 36441 y Ff(+)23951 36275 y Fr(\()p Fn(p)p Fr(\))26178 35515 y(\()26835 35223 y(^)26608 35515 y Fo( )27369 35042 y Ff(+)27329 35849 y Fi(k)p Fk(;)p Fj(\000)29095 35223 y Fr(^)28869 35515 y Fo( )29630 35042 y Fj(\000)29590 35849 y Fi(k)p Fj(\000)p Fi(p)p Fk(;)p Fj(\000)32377 35515 y Fr(\))308 b Fo(:)10993 b Fr(\(4)p Fo(:)p Fr(82\))0 38827 y(Let)246 b(us)g(consider)g(the)g(\014rst)g(step)g(of)h(the)f (iterativ)-31 b(e)249 b(in)-31 b(tegration)249 b(pro)31 b(cedure)245 b(and)i(let)f(us)g(call)i(again)40884 38548 y(\026)40745 38827 y Fh(V)41514 38426 y Ff(\()p Fj(\000)p Ff(1\))43395 38827 y Fr(\()p Fo( )44586 38426 y Ff([)p Fk(h;)p Fj(\000)p Ff(1])47057 38827 y Fr(\))0 40245 y(the)591 b(con)-31 b(tribution)594 b(to)d(the)h(e\013ectiv)-31 b(e)591 b(p)31 b(oten)-31 b(tial)594 b(of)e(the)f(terms)g(linear)h(in)f Fo(J)106 b Fr(.)1158 b(Let)591 b(us)f(no)-31 b(w)592 b(decomp)31 b(ose)139 41382 y(\026)0 41662 y Fh(V)769 41260 y Ff(\()p Fj(\000)p Ff(1\))2650 41662 y Fr(\()p Fo( 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Fj(\000)p Ff(1)14057 27140 y(2)p Fk(;)p Fj(\000)p Fk(;)p Ff(+)16758 26845 y Fr(=)308 b Fo(Z)18762 26313 y Ff(3)p Fk(;)p Fj(\000)18683 27140 y(\000)p Ff(1)20346 26096 y Fo(D)21262 26262 y Fj(\000)22009 26096 y Fr(\()p Fn(p)p Fr(\))p 20346 26591 V 20352 27605 a Fo(D)21268 27771 y Ff(+)22003 27605 y Fr(\()p Fn(p)p Fr(\))24817 26845 y Fo(;)1292 b Fh(L)p Fo(F)28045 26372 y Fj(\000)p Ff(1)27891 27140 y(2)p Fk(;)p Fj(\000)p Fk(;)p Fj(\000)30605 26845 y Fr(=)308 b Fo(Z)32609 26313 y Ff(3)p Fk(;)p Ff(+)32530 27140 y Fj(\000)p Ff(1)34355 26845 y Fo(;)9753 b Fr(\(4)p Fo(:)p Fr(88\))0 29752 y(where)346 b Fo(Z)4013 29220 y Ff(3)p Fk(;)p Ff(+)3934 30048 y Fj(\000)p Ff(1)5798 29752 y Fr(and)h Fo(Z)8763 29220 y Ff(3)p Fk(;)p Fj(\000)8684 30048 y(\000)p Ff(1)10560 29752 y Fr(are)g(the)f(same)h (real)g(constan)-31 b(ts)347 b(app)31 b(earing)348 b(in)e(\(4.38\).)488 b(Hence,)351 b(the)c(lo)31 b(cal)348 b(part)f(of)g(the)0 31170 y(marginal)371 b(term)f(\(4.85\))i(is,)d(b)-31 b(y)370 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b(ws)434 b(that)f(the)f(tree)g(structure)f (of)h(the)g(expansion)h(is)f(the)g(same)g(as)g(that)h(describ)31 b(ed)431 b(in)h Fh(x)p Fr(4.5)h(and)f(the)0 49608 y(constan)-31 b(ts)370 b Fo(\027)5518 49207 y Fj(0)5448 49882 y Fk(!)6458 49608 y Fr(can)f(b)31 b(e)369 b(c)-31 b(hosen)369 b(again)i(so)e(that)i (the)e(b)31 b(ound)369 b(\(4.44\))j(is)d(satis\014ed)h(ev)-31 b(en)369 b(b)-31 b(y)370 b(the)f(constan)-31 b(ts)371 b Fo(\027)45616 49207 y Fj(0)45546 49897 y Fk(j)m(;!)46813 49608 y Fr(.)443 51189 y(In)287 b(the)i(analysis)g(of)f(the)h(constan) -31 b(ts)15524 50897 y(~)15478 51189 y Fo(\025)16124 51355 y Fk(j)16878 51189 y Fr(and)355 b(~)-620 b Fo(z)19464 51355 y Fk(j)20218 51189 y Fr(there)288 b(is)g(only)h(one)f (di\013erence,)305 b(concerning)288 b(the)h(b)31 b(ound)288 b(\(4.72\),)0 52607 y(whic)-31 b(h)314 b(has)f(to)h(b)31 b(e)313 b(substituted)h(with)381 b(~)-619 b Fo(z)16495 52773 y Fj(\000)p Ff(1)17817 52607 y Fh(\000)134 b Fo(\013)t(z)20039 52773 y Fj(\000)p Ff(1)21535 52607 y 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55607 y Ff(4)41571 55441 y Fr(\),)375 b(so)e(ending)0 56859 y(the)c(pro)31 b(of)370 b(of)g(Lemma)g(3.1.)2580 61111 y Fs(App)42 b(endix)498 b(A1.)1330 b(The)498 b(ultra)-42 b(violet)500 b(problem)g(and)f(the)f(Thirring)i(mo)42 b(del)443 64418 y Fr(Thanks)546 b(to)h(the)e(linearit)-31 b(y)548 b(of)f(the)f(propagator,)592 b(the)546 b(ab)31 b(o)-31 b(v)g(e)547 b(analysis)g(can)e(b)31 b(e)546 b(used)f(with)i(no) f(essen)-31 b(tial)0 65835 y(mo)31 b(di\014cations)510 b(to)e(construct)g(the)g(massless)f Fp(Thirring)524 b(mo)-57 b(del)508 b Fr(\(see)f(for)h(instance)g([Z]\),)i(b)-31 b(y)507 b(remo)-31 b(ving)510 b(the)0 67252 y(ultra)-31 b(violet)372 b(cuto\013.)494 b(W)-92 b(e)368 b(shall)i(sk)-31 b(etc)g(h)370 b(here)e(the)i(main)g(ideas;)g(the)g(details)g(will)h(b) 31 b(e)369 b(published)g(elsewhere.)443 68670 y(The)337 b(Thirring)h(mo)31 b(del)338 b(describ)31 b(es)335 b(Dirac)i(fermions)h (in)f Fo(d)307 b Fr(=)h(1)181 b(+)g(1)338 b(in)-31 b(teracting)340 b(with)e(a)f(lo)31 b(cal)339 b(curren)-31 b(t-curren)g(t)0 70087 y(in)g(teraction;)372 b(its)e(action)h(is)15751 70424 y Fe(Z)17043 71930 y Fo(d)p Fn(x)p Fr([)p Fh(\000)19686 71638 y Fr(\026)19459 71930 y Fo( )20180 72096 y Fi(x)21072 71930 y Fh(6)308 b Fo(@)61 b( )22750 72096 y Fi(x)23580 71930 y Fh(\000)24820 71182 y Fo(\025)p 24820 71676 646 45 v 24866 72690 a Fr(4)25598 71930 y Fo(J)26212 72096 y Fk(\026)26806 71930 y Fr(\()p Fn(x)p Fr(\))p Fo(J)29058 71473 y Fk(\026)29653 71930 y Fr(\()p Fn(x)p Fr(\)])12646 b(\()p Fo(A)p Fr(1)p Fo(:)p Fr(1\))0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 0 1107 a Fr(where)394 b Fh(6)350 b Fo(@)411 b Fr(=)350 b Fo(\015)6359 1273 y Ff(0)6856 1107 y Fo(@)7444 1273 y Fk(x)7946 1384 y Fm(0)8696 1107 y Fr(+)263 b Fo(\015)10393 1273 y Ff(1)10890 1107 y Fo(@)11478 1273 y Fk(x)12035 1107 y Fr(,)401 b Fn(x)350 b Fr(=)g(\()p Fo(x)16039 1273 y Ff(0)16535 1107 y Fo(;)184 b(x)p Fr(\),)19025 815 y(\026)18799 1107 y Fo( )19520 1273 y Fi(x)20454 1107 y Fr(=)350 b Fo( )22426 705 y Ff(+)22386 1381 y Fi(x)23161 1107 y Fo(\015)23734 1273 y Ff(0)24230 1107 y Fr(,)402 b Fo( )25660 1273 y Fi(x)26639 1107 y Fr(is)394 b(a)h(t)-31 b(w)g(o)396 b(comp)31 b(onen)-31 b(t)397 b(spinor)d(\014eld)h(\(to)h(not)f(b)31 b(e)0 2524 y(confused)369 b(with)i(a)e(Grassmannian)h(\014eld\),)g Fo(J)19000 2690 y Fk(\026)19594 2524 y Fr(\()p Fn(x)p Fr(\))308 b(=)22829 2232 y(\026)22603 2524 y Fo( )23324 2690 y Fi(x)23908 2524 y Fo(\015)24481 2690 y Fk(\026)25075 2524 y Fo( )25796 2690 y Fi(x)26749 2524 y Fr(and)370 b Fo(\015)29475 2690 y Ff(0)30279 2524 y Fr(=)307 b Fo(\033)32080 2690 y Ff(1)32576 2524 y Fr(,)370 b Fo(\015)33826 2690 y Ff(1)34630 2524 y Fr(=)307 b Fo(\033)36431 2690 y Ff(2)37297 2524 y Fr(are)369 b(P)-31 b(auli)371 b(matrices.)443 3978 y(The)532 b Fp(gener)-57 b(ating)547 b(functional)533 b Fr(of)f(the)g(Thirring)h(mo)31 b(del)533 b(is)f(the)g(follo)-31 b(wing)535 b(Grassmannian)e(in)-31 b(tegral)534 b(with)0 5395 y(infrared)369 b(cuto\013)h Fo(\015)7897 4994 y Fk(h)8840 5395 y Fr(and)g(ultra)-31 b(violet)372 b(cuto\013)e Fo(\015)20086 4994 y Fk(N)20926 5395 y Fr(,)f(with)i Fo(h;)184 b(N)490 b Fr(in)-31 b(tegers)370 b(and)g Fo(N)428 b(>)308 b Fr(0)4463 9097 y Fh(W)91 b Fr(\()p Fo(\036;)184 b(J)106 b Fr(\))310 b(=)d(log)11471 7590 y Fe(Z)12762 9097 y Fo(P)13473 9263 y Fk(Z)14076 9374 y Fl(N)14842 9097 y Fr(\()p Fo(d )16609 8640 y Ff([)p Fk(h;N)83 b Ff(])18732 9097 y Fr(\))184 b(exp)21222 7204 y Fe(\()22360 9097 y Fh(\000)246 b Fo(V)g Fr(\()p Fo( )25550 8640 y Ff([)p Fk(h;N)83 b Ff(])27672 9097 y Fr(\)+)4709 12860 y(+)5816 11809 y Fe(X)6323 14133 y Fk(!)7600 11354 y Fe(Z)8891 12860 y Fo(d)p Fn(x)10323 11632 y Fe(h)10846 12860 y Fo(Z)11681 12285 y Ff(\(2\))11602 13186 y Fk(N)12869 12860 y Fo(J)13483 13026 y Fi(x)p Fk(;!)14916 12860 y Fo( )15677 12403 y Ff([)p Fk(h;N)g Ff(]+)15637 13134 y Fi(x)p Fk(;!)18478 12860 y Fo( )19239 12403 y Ff([)p Fk(h;N)g Ff(])p Fj(\000)19199 13134 y Fi(x)p Fk(;!)22299 12860 y Fr(+)246 b Fo(\036)24066 12403 y Ff(+)24066 13134 y Fi(x)p Fk(;!)25498 12860 y Fo( )26259 12403 y Ff([)p Fk(h;N)83 b Ff(])p Fj(\000)26219 13134 y Fi(x)p Fk(;!)29319 12860 y Fr(+)246 b Fo( )31187 12403 y Ff([)p Fk(h;N)83 b Ff(]+)31147 13134 y Fi(x)p Fk(;!)33988 12860 y Fo(\036)34648 12403 y Ff([)p Fk(h;N)g Ff(])p Fj(\000)34648 13134 y Fi(x)p Fk(;!)37461 11632 y Fe(i)38169 10967 y(\))39368 12860 y Fo(;)44138 10979 y Fr(\()p Fo(A)p Fr(1)p Fo(:)p Fr(2\))0 16949 y(where)352 b Fo(P)3895 17115 y Fk(Z)4498 17226 y Fl(N)5264 16949 y Fr(\()p Fo(d )7031 16547 y Ff([)p Fk(h;N)83 b Ff(])9154 16949 y Fr(\))352 b(is)h(giv)-31 b(en)353 b(b)-31 b(y)353 b(\(1.3\),)358 b(with)c Fo(C)21661 17115 y Fk(h;)p Ff(0)22939 16949 y Fr(\()p Fn(k)p Fr(\))f(replaced)f(b) -31 b(y)353 b Fo(C)31488 17115 y Fk(h;N)33109 16949 y Fr(\()p Fn(k)p Fr(\))308 b(=)36117 16119 y Fe(P)37286 16393 y Fk(N)37286 17281 y(j)45 b Ff(=)p Fk(h)39135 16949 y Fo(f)39677 17115 y Fk(j)40143 16949 y Fr(\()p Fn(k)p Fr(\))353 b(and)g Fo( )44925 16374 y Ff([)p Fk(h;)p Ff(0])p Fk(\033)44885 17283 y Fi(k)p Fk(;!)0 18701 y Fr(replaced)369 b(b)-31 b(y)5907 17846 y Fh(p)p 6829 17846 1596 45 v 855 x Fo(Z)7585 18867 y Fk(N)8425 18701 y Fo( )9186 18126 y Ff([)p Fk(h;N)83 b Ff(])p Fk(\033)9146 19034 y Fi(k)p Fk(;!)11848 18701 y Fr(,)370 b(and)f Fo(V)246 b Fr(\()p Fo( )16760 18299 y Ff([)p Fk(h;N)83 b Ff(])18882 18701 y Fr(\))369 b(is)h(giv)-31 b(en)370 b(b)-31 b(y)10864 22241 y Fo(V)245 b Fr(\()p Fo( )12946 21784 y Ff([)p Fk(h;N)83 b Ff(])15069 22241 y Fr(\))308 b(=)17021 21949 y(~)16975 22241 y Fo(\025)17621 22407 y Fk(N)18645 20734 y Fe(Z)19936 22241 y Fo(d)p Fn(x)g Fo( )22253 21666 y Ff([)p Fk(h;N)83 b Ff(]+)22213 22516 y Fi(x)p Fk(;)p Ff(+)25054 22241 y Fo( )25815 21666 y Ff([)p Fk(h;N)g Ff(])p Fj(\000)25775 22516 y Fi(x)p Fk(;)p Ff(+)28629 22241 y Fo( )29390 21666 y Ff([)p Fk(h;N)g Ff(]+)29350 22516 y Fi(x)p Fk(;)p Fj(\000)32191 22241 y Fo( )32952 21666 y Ff([)p Fk(h;N)g Ff(])p Fj(\000)32912 22516 y Fi(x)p Fk(;)p Fj(\000)36073 22241 y Fr(;)7758 b(\()p Fo(A)p Fr(1)p Fo(:)p Fr(3\))0 25899 y Fo(Z)756 26065 y Fk(N)1929 25899 y Fr(is)334 b(the)h(\(bare)f(\)w)-31 b(a)g(v)g(e)336 b(function)g(renormalization,)345 b Fo(Z)24193 25324 y Ff(\(2\))24114 26224 y Fk(N)25715 25899 y Fr(is)334 b(the)g(\(bare)h(\)densit)-31 b(y)335 b(renormalization)i(and)45805 25607 y(~)45759 25899 y Fo(\025)46405 26065 y Fk(N)0 27316 y Fr(is)344 b(the)h(\(bare\))g(in)-31 b(teraction.)486 b(In)344 b(order)g(to)h(get)g(a)f(non)-31 b(trivial)347 b(limit)g(as)d Fo(N)428 b Fh(!)308 b(1)p Fr(,)349 b(it)c(is)f(con)-31 b(v)g(enien)g(t)347 b(to)e(write)45805 27024 y(~)45759 27316 y Fo(\025)46405 27482 y Fk(N)0 28860 y Fr(and)369 b Fo(Z)2987 28285 y Ff(\(2\))2908 29185 y Fk(N)4545 28860 y Fr(in)g(terms)g(of)h Fo(Z)10937 29026 y Fk(N)12145 28860 y Fr(and)g(t)-31 b(w)g(o)371 b(new)e(bare)g(constan)-31 b(ts,)371 b Fo(\025)26982 29026 y Fk(N)28190 28860 y Fr(and)f Fo(c)30822 29026 y Fk(N)31661 28860 y Fr(,)g(in)f(the)h(follo) 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b(e)376 b(small)0 37580 y(enough,)370 b(uniformly)h(in)f Fo(N)121 b Fr(.)443 39034 y(The)423 b(pro)31 b(of)424 b(of)g(this)f(claim)i(is)f (essen)-31 b(tially)425 b(a)e(corollary)i(of)f(the)f(ab)31 b(o)-31 b(v)g(e)425 b(analysis)f(for)g(the)f(infrared)h(problem.)0 40451 y(The)431 b(R)-31 b(G)429 b(analysis)j(in)f Fh(x)p Fr(2)f(can)h(b)31 b(e)430 b(rep)31 b(eated)430 b(b)-31 b(y)431 b(allo)-31 b(wing)434 b(the)c(scale)h(index)g Fo(j)493 b Fr(to)431 b(b)31 b(e)430 b(p)31 b(ositiv)-31 b(e)432 b(or)e(negativ)-31 b(e.)0 41869 y(The)383 b(W)-92 b(ard)381 b(iden)-31 b(tit)g(y)385 b(\(3.14\))g(holds)e(with)g(a)g (factor)g Fo(Z)22951 42035 y Fk(N)23791 41869 y Fo(=)-61 b(Z)25118 41294 y Ff(\(2\))25039 42194 y Fk(N)26688 41869 y Fr(m)-31 b(ultiplying)386 b Fo(G)33475 41337 y Ff(2)p Fk(;)p Ff(1)33475 42144 y(+)35058 41869 y Fr(and)d(\001)38146 41337 y Ff(2)p Fk(;)p Ff(1)38146 42144 y(+)39347 41869 y Fr(,)j(and)d(w)-31 b(e)383 b(get)g(the)0 43286 y(iden)-31 b(tit)g(y)18753 43722 y Fe(\014)18753 44386 y(\014)18753 45050 y(\014)18753 45714 y(\014)18753 46378 y(\014)19567 44893 y Fo(Z)20402 44318 y Ff(\(2\))20323 45227 y Fk(h)p 19255 45404 2648 45 v 19255 46418 a Fo(c)19734 46584 y Fk(N)20573 46418 y Fo(Z)21329 46584 y Fk(h)22281 45659 y Fh(\000)246 b Fr(1)23941 43722 y Fe(\014)23941 44386 y(\014)23941 45050 y(\014)23941 45714 y(\014)23941 46378 y(\014)24618 45659 y Fh(\024)308 b Fo(C)26703 45367 y Fr(\026)26657 45659 y Fo(\025)27303 45202 y Ff(2)27303 45932 y Fk(h)28184 45659 y Fo(:)15647 b Fr(\()p Fo(A)p Fr(1)p Fo(:)p Fr(5\))0 49194 y(In)410 b(the)h(same)f(w)-31 b(a)g(y)-92 b(,)423 b(from)411 b(the)f(Dyson)h(equation)h(\(iden)-31 b(tical)413 b(to)e(\(1.11\),)424 b(with)412 b Fo(\025)34401 49360 y Fk(N)35240 49194 y Fo(Z)35996 49360 y Fk(N)36835 49194 y Fo(=)-61 b(Z)38162 48619 y Ff(\(2\))38083 49519 y Fk(N)39761 49194 y Fr(in)411 b(place)g(of)g Fo(\025)f Fr(in)0 50611 y(the)475 b(r.h.s\),)502 b(and)475 b(pro)31 b(ceeding)475 b(as)f(in)h Fh(x)p Fr(3)g(and)g Fh(x)p Fr(4,)502 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b(requiring,)j(for)d(instance,)j(that)e Fo(Z)18730 55029 y Ff(0)19542 54863 y Fr(=)316 b(1.)508 b(In)373 b(the)i(same)f(w)-31 b(a)g(y)376 b(w)-31 b(e)374 b(can)h(\014x)f(lim)37194 55029 y Fk(N)83 b Fj(!1)39983 54863 y Fo(\025)40629 55029 y Fk(N)41843 54863 y Fr(so)374 b(that,)j(for)0 56281 y(instance,)387 b Fo(\025)5281 56447 y Ff(0)6107 56281 y Fr(=)330 b Fo(\025)p Fr(,)386 b(with)e Fo(\025)e Fr(small)i(enough;)390 b(of)384 b(course)e Fo(\025)24576 56447 y Fk(N)25745 56281 y Fr(=)329 b Fo(\025)255 b Fr(+)g Fo(O)31 b Fr(\()p Fo(\025)30903 55879 y Ff(2)31400 56281 y Fr(\).)533 b(The)383 b(c)-31 b(hoice)383 b(of)h(lim)41074 56447 y Fk(N)83 b Fj(!1)43863 56281 y Fo(c)44342 56447 y Fk(N)45564 56281 y Fr(is)383 b(a)0 57698 y(free)369 b(parameter,)h(whose)g(v)-61 b(alue)369 b(has)g(no)h(sp)31 b(ecial)370 b(role.)443 59152 y(Finally)315 b(w)-31 b(e)314 b(shall)h(discuss)e(the)h(form)g(tak)-31 b(en)315 b(from)f(W)-92 b(ard)313 b(iden)-31 b(tities)316 b(when)e(the)g(ultra)-31 b(violet)317 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y(^)31515 63900 y Fo(G)32385 63368 y Ff(4)p Fk(;)p Ff(1)32385 64175 y Fj(\000)33587 63900 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)35887 64066 y Ff(1)36385 63900 y Fo(;)g Fn(k)37548 64066 y Ff(2)38046 63900 y Fo(;)g Fn(k)39209 64066 y Ff(3)39706 63900 y Fo(;)g Fn(k)40869 64066 y Ff(4)41612 63900 y Fh(\000)246 b Fn(p)p Fr(\))282 b(\()p Fo(A)p Fr(1)p Fo(:)p Fr(6\))1757 66751 y(=)3176 66471 y(^)2925 66751 y Fo(G)3795 66294 y Ff(4)3795 67025 y(+)4530 66751 y Fr(\()p Fn(k)5632 66917 y Ff(1)6375 66751 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)9352 66917 y Ff(2)9850 66751 y Fo(;)g Fn(k)11013 66917 y Ff(3)11510 66751 y Fo(;)g Fn(k)12673 66917 y Ff(4)13417 66751 y Fh(\000)246 b Fn(p)p Fr(\))g Fh(\000)17265 66471 y Fr(^)17014 66751 y Fo(G)17884 66294 y Ff(4)17884 67025 y(+)18620 66751 y Fr(\()p Fn(k)19722 66917 y Ff(1)20219 66751 y Fo(;)184 b Fn(k)21382 66917 y Ff(2)22125 66751 y Fr(+)246 b Fn(p)p Fo(;)184 b Fn(k)25102 66917 y Ff(3)25600 66751 y Fo(;)g Fn(k)26763 66917 y Ff(4)27506 66751 y Fh(\000)246 b 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6957 y Ff(4)14386 6791 y Fh(\000)246 b Fn(p)p Fr(\))g Fh(\000)18234 6511 y Fr(^)17983 6791 y Fo(G)18853 6334 y Ff(4)18853 7064 y(+)19589 6791 y Fr(\()p Fn(k)20691 6957 y Ff(1)21188 6791 y Fo(;)184 b Fn(k)22351 6957 y Ff(2)22848 6791 y Fo(;)g Fn(k)24011 6957 y Ff(3)24508 6791 y Fo(;)g Fn(k)25671 6957 y Ff(4)26169 6791 y Fr(\))246 b(+)g Fo(Z)28708 6957 y Fk(N)29548 6791 y Fo(H)30558 6259 y Ff(4)p Fk(;)p Ff(1)30468 7066 y Fj(\000)31758 6791 y Fr(\()p Fn(p)p Fo(;)184 b Fn(k)34058 6957 y Ff(1)34557 6791 y Fo(;)g Fn(k)35720 6957 y Ff(2)36217 6791 y Fo(;)g Fn(k)37380 6957 y Ff(3)37877 6791 y Fo(;)g Fn(k)39040 6957 y Ff(4)39784 6791 y Fh(\000)246 b Fn(p)p Fr(\))308 b Fo(;)1495 b Fr(\()p Fo(A)p Fr(1)p Fo(:)p Fr(7\))0 9148 y(It)462 b(is)g(a)g(straigh)-31 b(tforw)g(ard)465 b(consequence)d(of)g (our)g(analysis)h(\(in)g(particular)g(of)g(the)f(short)g(memory)h(prop) 31 b(ert)-31 b(y)0 10565 y(w)g(e)415 b(used)e(extensiv)-31 b(ely)416 b(throughout)g(the)e(pap)31 b(er\))414 b(that)i(in)e 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Fr(\))308 b(=)f Fo(G)28825 21476 y Ff(4)28825 22207 y(+)29560 21933 y Fr(\()p Fn(k)30662 22099 y Ff(1)31405 21933 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)34382 22099 y Ff(2)34880 21933 y Fo(;)g Fn(k)36043 22099 y Ff(3)36540 21933 y Fo(;)g Fn(k)37703 22099 y Ff(4)38447 21933 y Fh(\000)246 b Fn(p)p Fr(\))p Fh(\000)2832 24900 y(\000)4190 24620 y Fr(^)3939 24900 y Fo(G)4809 24443 y Ff(4)4809 25174 y(+)5544 24900 y Fr(\()p Fn(k)6646 25066 y Ff(1)7143 24900 y Fo(;)184 b Fn(k)8306 25066 y Ff(2)9050 24900 y Fr(+)245 b Fn(p)p Fo(;)184 b Fn(k)12026 25066 y Ff(3)12524 24900 y Fo(;)g Fn(k)13687 25066 y Ff(4)14431 24900 y Fh(\000)246 b Fn(p)p Fr(\))g(+)18279 24620 y(^)18028 24900 y Fo(G)18898 24443 y Ff(4)18898 25174 y(+)19633 24900 y Fr(\()p Fn(k)20735 25066 y Ff(1)21232 24900 y Fo(;)184 b Fn(k)22395 25066 y Ff(2)22893 24900 y Fo(;)g Fn(k)24056 25066 y Ff(3)24799 24900 y Fh(\000)246 b Fn(p)p Fo(;)184 b Fn(k)27776 25066 y Ff(4)28520 24900 y Fh(\000)246 b Fn(p)p Fr(\))g Fh(\000)32368 24620 y Fr(^)32117 24900 y Fo(G)32987 24443 y Ff(4)32987 25174 y(+)33723 24900 y Fr(\()p Fn(k)34825 25066 y Ff(1)35322 24900 y Fo(;)184 b Fn(k)36485 25066 y Ff(2)36982 24900 y Fo(;)g Fn(k)38145 25066 y Ff(3)38642 24900 y Fo(;)g Fn(k)39805 25066 y Ff(4)40303 24900 y Fr(\))44138 23072 y(\()p Fo(A)p Fr(1)p Fo(:)p Fr(9\))443 27107 y(The)335 b(ab)31 b(o)-31 b(v)g(e)335 b(W)-92 b(ard)334 b(iden)-31 b(tit)g(y)337 b(is)e(iden)-31 b(tical)337 b(to)e(the)g(formal)h(one,) 342 b(obtained)336 b(b)-31 b(y)335 b(a)g Fp(total)e Fr(gauge)j (transformation,)0 28524 y(except)351 b(for)g(the)g(factor)g(\(1)210 b Fh(\000)f Fo(\027)12981 28690 y Ff(+)13924 28524 y Fh(\000)g Fo(\027)15611 28123 y Fj(0)15541 28798 y(\000)16288 28524 y Fr(\))p Fo(=c)17750 28690 y Fk(N)18941 28524 y Fr(m)-31 b(ultiplying)355 b Fo(G)25697 28123 y Ff(4)p Fk(;)p Ff(1)26898 28524 y Fr(;)i(in)351 b(other)g(w)-31 b(ords,)355 b(the)350 b(formal)j(W)-92 b(ard)349 b(iden)-31 b(tit)g(y)0 29942 y(holds)452 b(when)g(the)g(cuto\013s)g(are)f(remo)-31 b(v)g(ed,)474 b(up)451 b(to)i(a)f(\014nite)g(in)-31 b(teraction-dep)31 b(enden)-31 b(t)455 b(renormalization)f(of)f(the)0 31359 y(densit)-31 b(y)400 b(op)31 b(erator.)584 b(A)400 b(similar)g (phenomenon)h(app)31 b(ears)399 b(also)h(in)g(p)31 b(erturbativ)-31 b(e)400 b(QED)f([H])i(and)f(is)f(called)h Fp(soft)0 32902 y(br)-57 b(e)g(aking)510 b Fr(of)g(gauge)f(in)-31 b(v)-61 b(ariance.)913 b(Of)508 b(course)g(it)i(is)e(p)31 b(ossible)509 b(to)g(c)-31 b(ho)31 b(ose)509 b Fo(Z)33200 32327 y Ff(\(2\))33121 33228 y Fk(N)34897 32902 y Fr(so)g(that)h(the)e(formal)j(W)-92 b(ard)0 34320 y(iden)-31 b(tit)g(y)372 b(is)d(v)-31 b(eri\014ed,)370 b Fp(i.e.)397 b Fr(w)-31 b(e)370 b(can)f(c)-31 b(ho)31 b(ose)370 b Fo(c)18961 34486 y Fk(N)20108 34320 y Fr(=)308 b(\(1)246 b Fh(\000)g Fo(\027)24160 34486 y Ff(+)25141 34320 y Fh(\000)g Fo(\027)26865 33918 y Fj(0)26795 34593 y(\000)27542 34320 y Fr(\).)443 35737 y(On)308 b(the)h(con)-31 b(trary)-92 b(,)322 b(the)309 b(W)-92 b(ard)308 b(iden)-31 b(tities)310 b(\(3.20\))h(and)e(\(3.22\),)324 b(obtained)311 b(b)-31 b(y)308 b(a)h Fp(chir)-57 b(al)309 b Fr(gauge)h (transformation,)0 37154 y(do)508 b(not)g(tend)g(in)g(the)g(limit)i(to) e(the)g(formal)h(W)-92 b(ard)507 b(iden)-31 b(tities)510 b(\(obtained)g(b)-31 b(y)508 b(\(3.20\))i(and)e(\(3.22\))i(putting)0 38572 y Fo(\027)547 38738 y Fj(\006)1649 38572 y Fr(=)355 b Fo(\027)3482 38170 y Fj(0)3412 38845 y(\006)4514 38572 y Fr(=)g(0\);)414 b(b)31 b(eside)397 b(the)h(renormalization)j(of)d (the)g(densit)-31 b(y)399 b(op)31 b(erator,)406 b(an)398 b(extra)g(factor)h(app)31 b(ears)398 b(in)g(the)0 39989 y(iden)-31 b(tit)g(y)-92 b(,)444 b(namely)428 b Fo(G)9148 39457 y Ff(4)p Fk(;)p Ff(1)9148 40265 y Fj(\000)10776 39989 y Fr(in)f(\(3.20\))j(or)c Fo(G)17664 39457 y Ff(4)p Fk(;)p Ff(1)17664 40265 y(+)19292 39989 y Fr(in)h(\(3.22\).)668 b(This)428 b(phenomenon)g(is)e(called)i Fp(chir)-57 b(al)450 b(anomaly)p Fr(,)441 b(see)0 41406 y([Z],)371 b(and)e(is)g(is)h(presen) -31 b(t)368 b(also)i(in)g(p)31 b(erturbativ)-31 b(e)370 b(QED.)20126 44713 y Fs(References)0 47076 y Fr([A])2705 b(P)-92 b(.W.)403 b(Anderson.)592 b(The)403 b(theory)g(sup)31 b(erconductivit)-31 b(y)404 b(in)f(high)h Fo(T)31762 47242 y Fk(c)32615 47076 y Fr(cuprates,)411 b(Princeton)404 b(Univ)-31 b(ersit)g(y)4149 48493 y(Press)368 b(\(1997\))0 49711 y([Af)86 b(])2281 b(I.)385 b(A\017ec)-31 b(k:)526 b(Field)386 b(theory)g(metho)31 b(ds)386 b(and)g(quan)-31 b(tum)387 b(critical)g(phenomena.)543 b(Pro)31 b(c.)541 b(of)386 b(Les)f(Houc)-31 b(hes)4149 51128 y(summer)384 b(sc)-31 b(ho)31 b(ol)385 b(on)f(Critical)j(phenomena,)i(Random)d (Systems,)i(Gauge)d(theories,)k(North)384 b(Holland)4149 52545 y(\(1984\).)0 53763 y([BG])1882 b(G.)540 b(Benfatto,)585 b(G.)540 b(Galla)-31 b(v)g(otti:)839 b(P)-31 b(erturbation)542 b(Theory)e(of)h(the)f(F)-92 b(ermi)540 b(Surface)g(in)h(a)f(Quan)-31 b(tum)4149 55180 y(Liquid.)475 b(A)316 b(General)g(Quasiparticle)i(F) -92 b(ormalism)317 b(and)f(One-Dimensional)i(Systems.)475 b Fp(J.)347 b(Stat.)493 b(Phys.)4149 56598 y Fn(59)p Fr(,)370 b(541{664)i(\(1990\).)0 57815 y([BGPS])514 b(G.)544 b(Benfatto,)591 b(G.)545 b(Galla)-31 b(v)g(otti,)593 b(A.)545 b(Pro)31 b(cacci,)590 b(B.)545 b(Scopp)31 b(ola:)845 b(Beta)545 b(F)-92 b(unctions)545 b(and)g(Sc)-31 b(h)g(winger)4149 59232 y(F)-92 b(unctions)455 b(for)g(a)g(Man)-31 b(y)455 b(F)-92 b(ermions)455 b(System)h(in)f(One)g(Dimension.)751 b Fp(Comm.)c(Math.)e(Phys.)750 b Fn(160)p Fr(,)4149 60650 y(93{171)371 b(\(1994\).)0 61867 y([BM1])1183 b(G.)429 b(Benfatto,)448 b(V.)430 b(Mastropietro:)615 b(Renormalization)434 b(Group,)444 b(hidden)431 b(symmetries)f(and)g(appro)-31 b(xi-)4149 63285 y(mate)370 b(W)-92 b(ard)368 b(iden)-31 b(tities)372 b(in)d(the)h Fo(X)87 b(Y)245 b(Z)448 b Fr(mo)31 b(del.)494 b Fp(R)-57 b(ev.)510 b(Math.)f(Phys.)492 b Fn(13)p Fr(,)371 b(1323{1435)i(\(2001\).)0 64502 y([BM2])1183 b(G.)365 b(Benfatto,)k(V.)e(Mastropietro:)492 b(On)365 b(the)h(densit)-31 b(y-densit)g(y)368 b(critical)f(indices)f(in)h(in) -31 b(teracting)368 b(F)-92 b(ermi)4149 65920 y(systems.)492 b Fp(Comm.)510 b(Math.)f(Phys.)492 b Fn(231)p Fr(,)371 b(97{134)h(\(2002\).)0 67137 y([BM3])1183 b(G.)292 b(Benfatto,)311 b(V.)293 b(Mastropietro:)456 b(W)-92 b(ard)292 b(iden)-31 b(tities)294 b(and)f(v)-61 b(anishing)294 b(of)f(the)g(Beta)h(function) g(for)f Fo(d)308 b Fr(=)f(1)4149 68555 y(in)-31 b(teracting)371 b(F)-92 b(ermi)369 b(systems.)493 b Fp(J.)396 b(Stat.)509 b(Phys.)493 b Fn(115)p Fr(,)371 b(143{184)h(\(2004\).)0 69772 y([BoM])1183 b(F.)310 b(Bonetto,)324 b(V.)310 b(Mastropietro:)465 b(Beta)311 b(F)-92 b(unction)311 b(and)f(Anomaly)i(of)f(the)f(F)-92 b(ermi)310 b(Surface)g(for)h(a)f Fo(d)e Fr(=)f(1)4149 71190 y(System)380 b(of)g(In)-31 b(teracting)381 b(F)-92 b(ermions)380 b(in)g(a)g(P)-31 b(erio)31 b(dic)381 b(P)-31 b(oten)g(tial.)528 b Fp(Comm.)539 b(Math.)e(Phys.)524 b Fn(172)p Fr(,)384 b(57{93)4149 72607 y(\(1995\).)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 0 1107 a Fr([DL])1997 b(I.E.)380 b(Dzy)-31 b(aloshinky)-92 b(,)385 b(A.I.)380 b(Larkin:)515 b(Correlation)382 b(functions)f(for)f(a)g(one-dimensional)j(F)-92 b(ermi)379 b(system)4149 2524 y(with)370 b(long-range)h(in)-31 b(teraction)371 b(\(T)-92 b(omonaga)373 b(mo)31 b(del\).)494 b Fp(Soviet)397 b(Phys.)509 b(JETP)369 b Fn(38)p Fr(,)i(202{208)h (\(1974\).)0 3742 y([G])2666 b(T.)426 b(Giamarc)-31 b(hi.:)609 b(Quan)-31 b(tum)427 b(Ph)-31 b(ysics)427 b(in)f(one)g(dimension.)665 b Fp(International)448 b(Series)h(of)f(Mono)-57 b(gr)g(aphs)4149 5159 y(on)396 b(Physics)369 b Fn(121)p Fr(,)i(Clarendon)f(Press,)f (Oxford)g(\(2004\).)0 6377 y([GM])1651 b(G.)341 b(Gen)-31 b(tile,)348 b(V.)342 b(Mastropietro:)479 b(Renormalization)345 b(Group)c(for)g(fermions:)479 b(a)341 b(review)h(of)g(mathemat-)4149 7794 y(ical)370 b(results.)492 b Fp(Phys.)510 b(R)-57 b(ep.)493 b Fn(352)p Fr(,)371 b(273{437)h(\(2001\).)0 9012 y([H])2705 b(T.R.)407 b(Hurd:)567 b(Soft)408 b(breaking)f(of)g (Gauge)g(in)-31 b(v)-61 b(ariance)408 b(in)f(regularized)g(Quan)-31 b(tum)408 b(Electro)31 b(dynamics.)4149 10429 y Fp(Comm.)510 b(Math.)e(Phys)369 b Fn(125)p Fr(,)i(515-526)h(\(1989\).)0 11647 y([Le])2351 b(A.)372 b(Lesniewski:)499 b(E\013ectiv)-31 b(e)373 b(action)h(for)e(the)g(Y)-92 b(uk)-61 b(a)-31 b(w)g(a)374 b(2)e(quan)-31 b(tum)373 b(\014eld)g(Theory)-92 b(.)501 b Fp(Commun.)518 b(Math.)4149 13064 y(Phys.)492 b Fn(108)p Fr(,)371 b(437{467)h(\(1987\).)0 14282 y([M])2520 b(V.Mastropietro,)586 b(Small)543 b(denominators)f(and)g(anomalous)h(b) 31 b(eha)-31 b(viour)542 b(in)f(the)h(incommensurate)4149 15699 y(Hubbard-Holstein)370 b(mo)31 b(del.)p Fp(Comm.)512 b(Math.)d(Phys.)492 b Fr(201,)371 b(81{115)h(\(1999\))0 16917 y([M1])1967 b(V.Mastropietro,)298 b(Coupled)279 b(Ising)f(mo)31 b(dels)279 b(with)g(quartic)g(in)-31 b(teraction)281 b(at)d(criticalit)-31 b(y)-92 b(,)300 b Fp(Comm.)482 b(Math.)4149 18334 y(Phys)368 b Fr(244,)j(595{642)i (\(2004\))0 19552 y([MD])1674 b(W.)456 b(Metzner,)479 b(C.)457 b(Di)g(Castro:Conserv)-61 b(ation)460 b(la)-31 b(ws)457 b(and)g(correlation)i(functions)e(in)g(the)g(Luttinger)4149 20969 y(liquid.)494 b Fp(Phys.)509 b(R)-57 b(ev.)511 b(B)369 b Fn(47)p Fr(,)i(16107)g(\(1993\).)0 22187 y([ML])1828 b(D.)471 b(Mattis,)498 b(E.)472 b(Lieb:)698 b(Exact)472 b(solution)i(of)e(a)f(man)-31 b(y)473 b(fermion)f(system)g(and)g(its)g (asso)31 b(ciated)472 b(b)31 b(oson)4149 23604 y(\014eld.)492 b Fp(J.)397 b(Math.)509 b(Phys.)492 b Fn(6)p Fr(,)370 b(304{312)i(\(1965\).)0 24822 y([S])2920 b(J.)490 b(Soly)-31 b(om:)737 b(The)490 b(F)-92 b(ermi)490 b(gas)h(mo)31 b(del)491 b(of)g(one-dimensional)i(conductors.)856 b Fp(A)-57 b(dv.)844 b(in.)h(Phys.)855 b Fn(28)p Fr(,)4149 26239 y(201{303)372 b(\(1979\).)0 27457 y([T])2736 b(W.)369 b(Thirring:)493 b Fp(A)-28 b(nn.)510 b(of)396 b(Phys.)493 b Fn(3)p Fr(,)370 b(91)g(\(1958\).)0 28874 y([Z])2859 b(J.)283 b(Zinn-Justin:)451 b(Quan)-31 b(tum)286 b(\014eld)e(theory)g (and)g(critical)h(phenomena,)303 b(Oxford)284 b(publications,)303 b(\(1989\).)0 75098 y Fm(2)p Fl(=settembr)19 b(e=)p Fm(2004;)366 b(11:32)14508 b Fr(40)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0409020452110--