%Plain Latex. Figures available by e-mail on request: speight@math.utexas.edu
\documentstyle{article}
%\setstretch{1.5}
%\renewcommand{\arraystretch}{1}
%\renewcommand{\baselinestretch}{1.6}
\topmargin=0in
\oddsidemargin=0.7cm
\textheight=8.5in
\textwidth=6in
\newfont{\SETT}{msbm10 scaled \magstep4}
\newfont{\SET}{msbm10 scaled \magstep3}
\newfont{\Set}{msbm10 scaled \magstep2}
\newfont{\Settoc}{msbm10 scaled \magstep1}
\newfont{\set}{msbm10}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\inx}{\int d^{2}{\rm{\bf x}}\, }
\newcommand{\dbin}{\int\, d^{2}{\rm{\bf x}}\, d^{2}{\rm{\bf x'}}\, }
\newcommand{\x}{{\rm{\bf x}}}
\newcommand{\y}{{\rm{\bf y}}}
\newcommand{\z}{{\rm{\bf z}}}
\newcommand{\rv}{{\rm{\bf r}}}
\newcommand{\jv}{{\rm{\bf j}}}
\newcommand{\A}{{\rm{\bf A}}}
\newcommand{\B}{{\rm{\bf B}}}
\newcommand{\Rv}{{\rm{\bf R}}}
\newcommand{\k}{\widehat{{\rm{\bf k}}}}
\newcommand{\zv}{{\rm{\bf 0}}}
\newcommand{\uv}{{\rm{\bf u}}}
\newcommand{\n}{\mbox{\boldmath $\eta$}}
\newcommand{\yd}{\dot{{\rm{\bf y}}}}
\newcommand{\zd}{\dot{{\rm{\bf z}}}}
\newcommand{\rd}{\dot{{\rm{\bf r}}}}
\newcommand{\xx}{|\x-\x'|}
\newcommand{\yz}{|\y-\z|}
\newcommand{\cd}{\partial}
\newcommand{\D}{\nabla}
\newcommand{\CD}{{{\bf D}}}
\newcommand{\Dy}{\D_{y}}
\newcommand{\Dz}{\D_{z}}
\newcommand{\Dr}{\D_{r}}
\newcommand{\rh}{\widehat{\rv}}
\newcommand{\yh}{\widehat{\y}}
\newcommand{\zh}{\widehat{\z}}
\newcommand{\dxy}{\delta(\x-\y)}
\newcommand{\dxz}{\delta(\x'-\z)}
\newcommand{\inxp}{\int d^{2}\x'\, }
\newcommand{\kn}{K_{0}}
\newcommand{\ko}{K_{1}}
\newcommand{\R}{\mbox{\set R}}
\newcommand{\Z}{\mbox{\set Z}}
\newcommand{\C}{\mbox{\set C}}
\newcommand{\M}{{\cal M}}
\newcommand{\Ms}{\M_{*}}
\newcommand{\hf}{\frac{1}{2}}
\newcommand{\ol}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\thet}{\widehat{\mbox{\boldmath $\theta$}}}
\newcommand{\al}{{\mbox{\boldmath $\alpha$}}}
\newcommand{\be}{{\mbox{\boldmath $\beta$}}}
\newcommand{\th}{\vartheta}
\newcommand{\prl}{\scriptscriptstyle \parallel}
\newcommand{\prp}{\scriptscriptstyle \perp}
\newcommand{\square}{\Box}
\begin{document}
\title{Long Range Intervortex Forces}
\author{J.M. Speight \\
Department of Mathematics \\
University of Texas at Austin \\
Austin, Texas 78712, U.S.A.}
\date{}
\maketitle
\begin{abstract}
A point particle approximation to the classical dynamics of well separated
vortices of
the abelian Higgs model is developed. The asymptotic static intervortex
potential is calculated and used to model type~II vortex scattering. A velocity
dependent interaction Lagrangian for critically coupled vortices is derived,
and reinterpreted geometrically to obtain a conjecture for the asymptotic form
of the metric on the two-vortex moduli space, as used in the geodesic
approximation. The scattering of critically coupled vortices is calculated.
\end{abstract}
\section{Introduction}
\label{sec:int}
Solitons find numerous applications in many branches of physics. In theories
of elementary particles, Lorentz invariance is a fundamental requirement,
one which appears to be incompatible with integrability in spacetimes of
dimension higher than $(1+1)$. Thus, it is topological solitons whose field
equations are nonintegrable which are most interesting and relevant from a
particle theory standpoint. A basic problem in the study of such objects is
to understand classical two soliton dynamics. In the absence of integrability
there is no hope of solving the initial value problem exactly. In fact,
usually the only exact solutions available are static in some inertial frame,
and so contain no dynamics at all. Often, even the static problem is
intractable. To make progress, one must resort to numerical analysis, or
devise some approximation scheme.
In this paper we study the interaction between two well separated abelian
Higgs vortices in $\R^{2+1}$ by means of a point particle approximation. The
idea is that,
viewed from afar, a static vortex looks like a solution of a linear field
theory in the presence of a singular point source at the vortex centre. As
will be shown, the appropriate point source is a composite scalar monopole
and magnetic dipole in a Klein-Gordon/Proca theory. The monopole charge and
dipole moment must be fixed numerically. If physics is to be model independent,
then the forces between well separated vortices should approach those between
the corresponding point particles in the linear theory as the separation grows.
Proceeding on this assumption, we calculate the asymptotic static two vortex
potential, previously found by Bettencourt and Rivers using a field
superposition ansatz
\cite{BetRiv}. As an application, the scattering of type II vortices is
calculated and comparison made with numerical experiment.
The asymptotic potential reproduces the familiar trichotomy of vortex
dynamics into type I, critical and type II regimes, according to the mass of
the Higgs field. In particular, at critical coupling there is no net force
between static point vortices, just as in the nonlinear field theory.
However, {\em moving} point vortices {\em do} exert forces on one another,
and we calculate these to lowest order in velocity by adapting the method
of linear retarded potentials. This involves calculating the interaction of
one moving point particle with the retarded field induced by another, the
difficulty here being that the linear theory is massive, so conventional
retarded potentials are not available to us (field disturbances do not travel
uniformly at the speed of light). The technique was first used by Manton
\cite{Man4} to find the asymptotic forces between two BPS monopoles, and
has since been applied to the Skyrme model \cite{Sch} and extreme
Reissner-Nordstrom black holes \cite{GibRub}. In all these systems, the
linear theory is massless. Having obtained an interaction Lagrangian,
we go on to solve the scattering problem.
For so-called Bogomol'nyi field theories, there is another approximation
scheme, also due to Manton \cite{M1}, wherein low energy soliton dynamics
is approximated by geodesic motion on the moduli space of static multisoliton
solutions with respect to the metric induced by the field kinetic energy
functional. This geodesic approximation was used by Samols \cite{Sam} to
study the scattering of two critically coupled vortices, and it is with these
results that we compare our scattering data. Since no static vortex solutions
are known exactly, Samols' work is necessarily partly numerical. However, the
present work allows us to conjecture an asymptotic formula for the metric
on the two vortex moduli space, reasoning as follows. Manton found that the
equations of motion of two monopoles in the point particle approximation
could be reinterpreted as the geodesic equation on the two monopole moduli
space with a Taub NUT metric \cite{Man4}, precisely the
asymptotic form of the Atiyah-Hitchin metric \cite{AtiHit2}.
Assuming the same model independence occurs for vortices, we can make a similar
reinterpretation of our point particle interaction Lagrangian, yielding new
geometric information about the two vortex moduli space.
\section{The abelian Higgs model}
\label{sec:ahm}
The abelian Higgs model \cite{Hig} is a field theory in $(2+1)$ dimensional
Minkowski
space consisting of a complex scalar field $\phi$ coupled to a $U(1)$ gauge
field $A_{\mu}$. The scalar field is given a Higgs symmetry-breaking self
interaction which allows topologically stable solitons called vortices to
exist. The action functional is
\beq
\label{1}
S=\int\, d^{3}x\, \left[\hf D_{\mu}\phi\ol{D^{\mu}\phi}
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\mu^{2}}{8}(|\phi|^{2}-1)^{2}\right]
\eeq
where
\beq
\label{2}
D_{\mu}\phi=(\cd_{\mu}+iA_{\mu})\phi
\eeq
is the gauge covariant derivative,
\beq
\label{3}
F_{\mu\nu}=\cd_{\mu}A_{\nu}-\cd_{\nu}A_{\mu}
\eeq
is the field strength tensor, and Minkowski space has the standard metric,
${\rm diag}(1,-1,-1)$. Note that the electric charge and vacuum magnitude
of the Higgs field have been normalized to unity, leaving only one parameter
$\mu,$ the Higgs mass.
The Euler-Lagrange equations derived from $S$ are
\bea
D_{\mu}D^{\mu}\phi-\hf\mu^{2}\phi(|\phi|^{2}-1) &=& 0 \nonumber \\
\label{6}
\cd_{\mu}F^{\mu\nu}+|\phi|^{2}A^{\nu}+\frac{i}{2}(\phi\cd^{\nu}\ol{\phi}
-\ol{\phi}\cd^{\nu}\phi) &=& 0,
\eea
a set of coupled, nonlinear, hyperbolic partial differential equations of
which no nontrivial solutions are known. To make progress, one breaks Lorentz
and gauge covariance, and concentrates on the kinetic and potential energy
functionals
\bea
\label{7}
T &=& \hf\inx(|\dot{\phi}|^{2}+|\dot{\A}|^{2}) \\
\label{8}
V &=& \hf\inx\left[\CD\phi\cdot\ol{\CD\phi}+F_{12}^{2}+
\frac{1}{4}\mu^{2}(|\phi|^{2}-1)^{2}\right]
\eea
in the gauge $A_{0}=0$. The variational problem with action $\int\, dt\, (T-V)$
is the same as that with action $S$, provided one imposes the $\nu=0$ field
equation (\ref{6}) as a constraint. Examining (\ref{8}) one sees that for a
configuration to have finite energy, the fields should satisfy the following
boundary conditions as $r=|\x|\rightarrow\infty$,
\bea
\label{9}
|\phi| &\rightarrow& 1 \\
\label{10}
\D\phi &\rightarrow& \zv \Rightarrow \A\rightarrow\frac{\D\phi}{i\phi}.
\eea
So from (\ref{9}), $\phi_{\infty}(\theta)=
\lim_{r\rightarrow\infty}\phi(r,\theta)$
takes values on the unit circle in $\C$ and hence is a map $S^{1}\rightarrow
S^{1}$. It need not be constant, provided $\A$ has the right form (\ref{10}).
Finite energy solutions fall into disjoint homotopy classes, each labelled
by the degree of $\phi_{\infty}$, some integer $n$. This integer, usually
called the winding number, cannot change under any continuous deformation of
$(\phi,\A)$ preserving finite energy (time evolution for example).
Static solutions with $n=1$ are called vortices. By continuity there must be
at least one point in the physical plane at which $\phi=0$, and given the
form of the potential energy density, there is a lump of energy located
at or near this point. So vortices are topologically stable lumps of energy,
solitons in the loose sense. Using Stokes' Theorem and the boundary conditions
(\ref{9},\ref{10}) it is easily seen that a winding $n$ configuration has
total magnetic flux
\beq
\label{11}
\Phi=-\inx F_{12}=2n\pi.
\eeq
A vortex may be visualized as a flux tube in $\R^{3+1}$ with translation
symmetry along the $x^{3}$ axis. It has a total flux of $2\pi$ penetrating the
physical plane.
Vortex dynamics splits into three regimes, depending on the value of $\mu^{2}$,
the most interesting mathematically being the case of critical coupling,
$\mu^{2}=1$. In this case, by means of an ingenious argument due to
Bogoml'nyi \cite{B}, one can find an optimal topological lower bound
on the energy of a configuration of given winding, and reduce the static field
equations to a pair of first order partial differential equations. Let $(\phi,
\A)$ be a static configuration with winding $n\geq0$. Since $\dot{\phi}=0$,
$\dot{\A}=\zv$, the constraint arising from the $A_{0}$ field equation is
trivially satisfied, and $T=0$, so the field variational problem reduces to
minimizing the potential energy functional within the winding $n$ sector.
Starting from the trivial inequality,
\bea
0 &\leq& \hf\inx\left\{|D_{1}\phi+iD_{2}\phi|^{2}+\left[-F_{12}+\hf(|\phi|^{2}
-1)\right]^{2}\right\} \nonumber \\
\label{11.1}
&=& V+\hf\inx\left[i\left(D_{2}\phi\ol{D_{1}\phi}-D_{1}\phi\ol{D_{2}\phi}
\right)-F_{12}(|\phi|^{2}-1)\right],
\eea
we rearrange to obtain,
\beq
\label{11.2}
V\geq\frac{i}{2}\inx \left[\cd_{1}(\phi\ol{D_{2}\phi})-\cd_{2}\ol{D_{1}\phi}
\right]-\hf\Phi,
\eeq
where $\Phi$ is the total magnetic flux, as defined above. Using Stokes'
Theorem with the boundary condition (\ref{10}) and flux quantization
(\ref{11}), this inequality becomes
\beq
\label{11.3}
V\geq n\pi,
\eeq
equality holding if and only if the Bogomol'nyi equations
\bea
(D_{1}+iD_{2})\phi &=& 0, \nonumber \\
\label{11.4}
-F_{12}+\hf(|\phi|^{2}-1) &=& 0
\eea
are satisfied. No explicit nontrivial solutions of (\ref{11.4}) are known,
but Taubes has rigorously proved the existence of such solutions \cite{Tau}.
Since they are minimals of $V$ they are static solutions of the field
equations. Hence, the mass of a $\mu^{2}=1$ vortex (its rest energy) is $\pi$,
a result which will be of use to us.
\section{Vortex asymptotics}
\label{sec:va}
The first task in the point particle approximation is to find out what a static
vortex looks like far from its core \cite{NieOle,BetRiv}. We place the vortex
at the origin, and use
plane polar coordinates. Substituting the ansatz
\bea
\phi &=& \sigma(r)e^{i\theta} \nonumber \\
\label{12}
(A_{0},A_{r},A_{\theta}) &=& (0,0,-a(r))
\eea
($\sigma$ is real) the field equations reduce to two coupled nonlinear
ordinary differential equations,
\bea
\frac{d^{2}\sigma}{dr^{2}}+\frac{1}{r}\frac{d\sigma}{dr}
-\frac{1}{r^{2}}\sigma(1-a)-\frac{1}{2}\mu^{2}\sigma(\sigma^{2}-1)&=& 0
\nonumber \\
\label{14}
\frac{d^{2}a}{dr^{2}}-\frac{1}{r}\frac{da}{dr}
+(1-a)\sigma^{2}&=& 0,
\eea
the equations for $A_{0}$ and $A_{r}$ being trivially satisfied. Regularity
demands that $\sigma(0)=a(0)=0$ while the boundary conditions
(\ref{9},\ref{10}) become
\beq
\label{15}
\lim_{r\rightarrow\infty}\sigma(r)=\lim_{r\rightarrow\infty}a(r)=1.
\eeq
Note that the ansatz has unit winding by construction. No exact solutions
of (\ref{14}) with these boundary conditions are known, but numerical solutions
suggest that both $\sigma$ are $a$ are monotonic functions of $r$, and that
the ansatz produces an isolated lump like structure.
We are interested in the asymptotic forms of $\sigma$ and $a$, and for these
explicit expressions do exist. Define the functions $\alpha$ and $\beta$ such
that
\beq
\label{16}
\sigma(r)=1+\alpha(r),\quad a(r)=1+\beta(r).
\eeq
Then (\ref{15}) implies that $\alpha$ and $\beta$ are small at large $r$,
so we substitute (\ref{16}) into (\ref{14}) and linearize in $\alpha$ and
$\beta$,
\bea
\mu^{2}\left(\frac{d^{2}\alpha}{d(\mu r)^{2}}
+\frac{1}{\mu r}\frac{d\alpha}{d(\mu r)}-\alpha\right)&=&0 \nonumber \\
\label{17}
\frac{d^{2}\,\,}{dr^{2}}\left(\frac{\beta}{r}\right)
+\frac{1}{r}\frac{d\,\,}{dr}\left(\frac{\beta}{r}\right)
-\left(1+\frac{1}{r^{2}}\right)\frac{\beta}{r} &=& 0.
\eea
These are the modified Bessel's equations of zeroth order for $\alpha$ in
$\mu r$ and first order for $\beta/r$ in $r$ respectively. Hence, at large
$r$,
\bea
\alpha &\sim& \frac{q}{2\pi}K_{0}(\mu r) \nonumber \\
\label{18}
\beta &\sim& \frac{m}{2\pi}rK_{1}(r),
\eea
where $K_{n}$ is the $n$--th modified Bessel's function of the second kind
\cite{AbrSte}. Note that $K_{1}\equiv-K_{0}'$.
Since we have linearized the field equations, the asymptotic solutions contain
unknown scale constants $q$ and $m$ which can only be fixed by solving
(\ref{14}) numerically. Rather than solving the boundary value problem
$\sigma(0)=a(0)=0$, $\sigma(\infty)=a(\infty)=1$, we solve the initial value
problem $\sigma(0)=a(0)=0$ using $\sigma'(0)$ and $a'(0)$ as shooting
parameters. In fact, this is a slight oversimplification: due to the
singularities of equations (\ref{14}) at the origin, we must shoot from
$r=r_{0}$, some small positive number, rather than $r=0$. Substituting Taylor
expansions for $\sigma$ and $a$ into (\ref{14}) we find that, near the
origin,
\bea
\sigma&=&a_{1}r+\frac{1}{4}a_{1}\left(b_{2}+\frac{\mu^{2}}{4}\right)r^{3}
+O(r^{5}) \nonumber \\
\label{19}
a&=&b_{2}r^{2}-\frac{a_{1}^{2}}{8}r^{4}+O(r^{5}).
\eea
We use $a_{1}$ and $b_{2}$ as shooting parameters, adjusting them until the
numerical solution has $\sigma(r_{\infty})\approx 1\approx a(r_{\infty})$,
where $r_{\infty}$ is some large positive number, the effective infinity.
Having generated such a numerical solution, we compare it at large $r$ to the
asymptotic forms (\ref{18}) and deduce $q$ and $m$.
The results of this procedure using a fourth order Runge-Kutta method with
$r_{0}=10^{-8}$ and $r_{\infty}=10$ for various values of $\mu^{2}$ are
presented in table 1. That $r_{\infty}$ is so small is unfortunate but
necessary: at large $r$ the field equations reduce to Bessel's equations,
which have two independent solutions, one exponentially decaying and the
other exponentially growing. We seek to pick out the former and completely
exclude the latter, an impossible task. Hence, all numerical solutions blow
up at large $r$, and even though $a_{1}$ and $b_{2}$ were tuned to six
decimal places, the Runge-Kutta algorithm could not shoot beyond $r=10$. The
results in table 1 should thus be treated with some caution, particularly
where $\mu^{2}$ is far from $1$.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
$\mu^{2}$ & $q$ & $m$ & $r_{c}$ \\ \hline
0.4 & -7.54 & -14.92 & 4.23 \\
0.6 & -8.71 & -12.61 & 3.75 \\
0.8 & -9.70 & -11.31 & 3.43 \\
0.9 & -10.14 & -10.89 & 3.22 \\
1.0 & -10.58 & -10.57 & - \\
1.1 & -10.98 & -10.31 & 2.98 \\
1.2 & -11.43 & -10.06 & 3.07 \\
1.3 & -11.80 & -9.85 & 2.96 \\
1.4 & -12.23 & -9.66 & 2.95 \\
1.6 & -13.04 & -9.34 & 2.88 \\
1.8 & -13.97 & -9.09 & 2.87 \\
2.0 & -14.50 & -8.86 & 2.72 \\ \hline
\end{tabular}
\end{center}
\caption{{\sf Numerical values of vortex scalar charge $q$ and magnetic dipole
moment $m$. The other data are the critical points
of the static intervortex potential.}}
\label{table}
\end{table}
Two points about the numerical charges are noteworthy. First, at critical
coupling ($\mu^{2}=1$), $q\approx m$. In fact, one can prove that $q\equiv m$
exactly in this case, because the $\mu^{2}=1$ vortex satisfies (\ref{11.4}),
a pair of {\em first} order field equations. Substituting the vortex ansatz
(\ref{12}) into (\ref{11.4}) one can solve for $a$ in terms of $\sigma$ and
$\sigma'$, eliminate $a$ and obtain a second order
ordinary differential equation for $\sigma$ alone which, on linearization,
reduces to Bessel's equation. The asymptotic form of $a$ can then be deduced
from that of $\sigma$. Second, $|q|$ and $|m|$ are monotonic functions of
$\mu^{2}$, $|q|$
increasing and $|m|$ decreasing. Bettencourt and Rivers \cite{BetRiv} also
find the asymptotic forms (\ref{18}), but leave their charges analogous to
$q$ and $m$ undetermined. For purposes of calculation, they make two
assumptions about the charges which, in the light of table 1, may prove
ill-justified. First, they assume that $q=m$ is approximately true away from
$\mu^{2}=1$, whereas in our results, $q/m$ varies between $0.50$ and $1.64$.
Second, they impose the condition that the magnetic flux of a vortex should
vanish at $r=0$ and deduce that $m=-2\pi$. This result may be valid for
very large $\mu^{2}$, but is certainly flawed close to $\mu^{2}=1$ (the
$\mu^{2}=1$ vortex has {\em maximum} magnetic flux at the origin, as is easily
seen from (\ref{11.4}), the Bogomol'nyi equations). So, they combine
assumptions which are
individually true only in widely disparate physical regimes.
\section{The point vortex}
\label{sec:pv}
The next task is to replicate the vortex asymptotics found above, using point
sources in the linear theory. To linearize the abelian Higgs model, we choose
gauge so that $\phi$ is real. Defining the field $\psi=1-\phi$, the vacuum is
then $\psi=0$, and the linear Lagrangian density is obtained by expanding
(\ref{1}) up to quadratic order in $\psi$ and $A_{\mu}$,
\beq
\label{20}
{\cal L}_{\rm free}=\hf\cd_{\mu}\psi\cd^{\mu}\psi-\hf\mu^{2}\psi^{2}
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\hf A_{\mu}A^{\mu}.
\eeq
Including the external source Lagrangian density,
\beq
\label{21}
{\cal L}_{\rm source}=\rho\psi-j_{\mu}A^{\mu},
\eeq
with scalar density $\rho$ and vector current $j_{\mu}$, we obtain the
following massive, inhomogeneous wave equations for $\psi$ and $A_{\mu}$,
\bea
\label{22}
(\square+\mu^{2})\psi &=& \rho \\
\label{23}
(\square+1)A_{\mu} &=& j_{\mu}+\cd_{\mu}(\cd_{\nu}A^{\nu}).
\eea
All gauge freedom has been exhausted, and there is no global $U(1)$ symmetry
of ${\cal L}_{\rm free}$ with whose Noether current we can identify
$j_{\mu}$ because $\psi$ is real. Hence there is no reason to assume that
$j_{\mu}$ is a conserved current, and we cannot set the extra ``fictitious
current'' term in the Proca equation (\ref{23}) to zero.
To make comparison with the asymptotic vortex fields, these must first be
converted to the real $\phi$ gauge. Since $\phi$ has non-zero winding, there
is no gauge transformation regular on all $\R^{2}$ which will accomplish this.
However, we only require comparison at large $r$, so for our purposes it is
sufficient that the transformation be regular on $\R^{2}\backslash\{\zv\}$.
Since a singular point source will be introduced into the linear theory, this
is from the outset regular only on $\R^{2}\backslash\{\zv\}$. So, we unwind
the static vortex (\ref{12}) with gauge transformation
$\phi\mapsto e^{-i\theta}\phi$, $A^{\mu}\mapsto A^{\mu}+\cd^{\mu}\theta$ to
obtain
\bea
\label{24}
\phi&=&\sigma(r)\sim 1+\frac{q}{2\pi}K_{0}(\mu r) \\
\label{25}
A_{\theta}&=&-a(r)+1\sim\frac{m}{2\pi}rK_{1}(r),
\eea
while $A_{r}=A_{0}=0$. It is convenient to introduce a unit vector $\k$ in a
fictitious third direction perpendicular to the physical plane, so that
the $\R^{3}$ vector product can be defined. In terms of the 2-vector field
$\A$, the unwound asymptotic behaviour is
\beq
\label{26}
\A\sim-\frac{m}{2\pi}K_{0}'(r)\thet=-\frac{m}{2\pi}\k\times\D K_{0}(r).
\eeq
We thus seek sources $\rho$ and $j_{\mu}$ such that the solutions of
(\ref{22},\ref{23}) are
\bea
\label{27}
\psi&=&\frac{q}{2\pi}K_{0}(\mu r) \\
\label{28}
(A^{0},\A)&=&\left(0,-\frac{m}{2\pi}\k\times\D\kn(r)\right).
\eea
The static Klein-Gordon equation in $(2+1)$ dimensions has Green's function
$\kn$,
\beq
\label{29}
(-\Delta+\mu^{2})\kn(\mu r)=2\pi\delta(\x).
\eeq
Substituting (\ref{27}) into (\ref{22}) and using (\ref{29}) one finds that
\beq
\label{30}
\rho=q\delta(\x).
\eeq
Similarly, substitution of (\ref{28}) into (\ref{23}) yields
\beq
\label{31}
\jv-\D(\D\cdot\jv)=-m\k\times\D\delta(\x).
\eeq
Taking the divergence of (\ref{31}) one sees that $\D\cdot\jv$ is a solution of
the {\em homogeneous} static Klein-Gordon equation, so if $\jv$ is a point
source (meaning $\jv=\zv$ except at $\x=\zv$) then $\D\cdot\jv=0$ everywhere.
Thus the unique point source satisfying (\ref{31}) is
\beq
\label{32}
\jv=-m\k\times\D\delta(\x).
\eeq
Since $A^{0}=0$ we take $j^{0}=0$.
The physical interpretation of these expressions for $\rho$ and $\jv$ is that
the point source consists of a scalar monopole of charge $q$ and a magnetic
dipole of moment $m$ perpendicular to the physical plane. Both $q$ and $m$ are
negative (see table 1). We refer to this composite point source as the point
vortex. One should note an important difference between the present endeavour
and the analogous analysis for BPS monopoles \cite{Man4}. Both systems
(abelian Higgs and Yang-Mills-Higgs) have topologically quantized magnetic
flux $\Phi$. In the monopole calculation, the Dirac monopole charge is assumed
to have precisely the topological value, while we must fix the dipole moment
of the point vortex by numerical solution of the static field equations. The
difference lies in the fact that monopoles exist in $\R^{3}$ where $\Phi$ is
defined as flux through a large spherical shell centred on the monopole, an
asymptotic quantity, whereas vortices exist in $\R^{2}$ where $\Phi$ is flux
through the whole plane, including the vortex core. Such a quantity is not
asymptotic, and so is not reflected in the properties of the point vortex.
In fact, although a vortex is a flux tube in the $\k$ direction, to reproduce
it asymptotically, one needs a dipole of moment $|m|$ in the $-\k$ direction
in the linear theory.
\section{The static intervortex potential}
\label{sec:sip}
Having found the scalar charge and magnetic dipole moment carried by a point
vortex, it is straightforward to calculate the force between two such vortices
held at rest, in the framework of the linear theory. The interaction Lagrangian
for two arbitrary (possibly time dependent) sources $(\rho_{1},j_{(1)})$ and
$(\rho_{2},j_{(2)})$ is
\beq
\label{33}
L_{\rm int}=L_{\psi}+L_{A}=\inx \rho_{1}\psi_{2}
-\inx j^{\mu}_{(1)}A_{\mu}^{(2)}
\eeq
where $(\psi_{i},A_{(i)})$ are the fields induced by source
$(\rho_{i},j_{(i)})$ according to the wave equations (\ref{22},\ref{23}). This
is found by extracting the cross terms in
$\inx({\cal L}_{\rm free}+{\cal L}_{\rm source})$ where $(\rho,j)$ is the
superposition of the two sources, and $(\psi,A)$ is a superposition of the
induced fields (using linearity). The expression (\ref{33}) looks asymmetric
under interchange of sources $1\leftrightarrow 2$, but in fact $L_{\rm int}$
is symmetric as may be shown using the wave equations (\ref{22},\ref{23}) and
integration by parts.
Now consider the case of two static point vortices, vortex 1 at $\y$ and
vortex 2 at $\z$. Then $\rho_{1}=q\dxy$, while the scalar field due to
$\rho_{2}$ is $\psi_{2}=q\kn(\mu|\x-\z|)/2\pi$. Hence,
\beq
\label{34}
L_{\psi}=\inx\frac{q^{2}}{2\pi}\dxy\kn(\mu|\x-\z|)
=\frac{q^{2}}{2\pi}\kn(\mu|\y-\z|).
\eeq
The magnetic interaction is similar: $j_{(1)}^{0}=0$, $\jv_{(1)}=
-m\k\times\D\dxy$ while $A_{(2)}^{0}=0$, $\A_{(2)}=-m\k\times\D\kn(|\x-\z|)$,
so
\bea
L_{A}&=&\inx\frac{m^{2}}{2\pi}[\k\times\D\dxy]\cdot[\k\times\D\kn(|\x-\z|)]
\nonumber \\
&=& -\frac{m^{2}}{2\pi}\Delta_{y}\kn(|\y-\z|) \nonumber \\
\label{35}
&=& -\frac{m^{2}}{2\pi}\kn(|\y-\z|)
\eea
using (\ref{29}) with $\y\neq\z$. The total interaction Lagrangian is a
function of $|\y-\z|$ only, so we interpret $-L_{\rm int}$ as the potential
energy of the interaction,
\beq
\label{36}
U=\frac{1}{2\pi}[m^{2}\kn(r)-q^{2}\kn(\mu r)]
\eeq
where $r$ is the vortex separation, that is
$\rv=r(\cos\vartheta,\sin\vartheta):=\y-\z$. This is the same potential as
found in \cite{BetRiv}, but we arrived at it via a different route.
This potential is consistent with the partition into type I, critical and
type II regimes. The central force due to $U$ is
\beq
\label{37}
-U'(r)=\frac{1}{2\pi}[m^{2}\ko(r)-\mu q^{2}\ko(\mu r)].
\eeq
If $\mu<1$, then $\ko\rightarrow 0$ at large $r$ faster than $\ko(\mu r)$,
so scalar attraction dominates over magnetic repulsion and the force is
negative, consistent with type I behaviour. If $\mu>1$, the reverse is true
and the force is positive at large $r$, consistent with type II behaviour.
At $\mu=1$, $m\equiv q$, as explained in section \ref{sec:va} so $U\equiv 0$
and there is no net force at all. This consistency at large $r$ emerges
regardless of the specific values of $m$ and $q$ away from $\mu=1$, and may
be attributed to the inverse relationship between a field's mass and its
range. At moderate $r$, the $\mu$ dependance of $q/m$ becomes important. Given
that $\ko$ is a strictly decreasing function, it is clear from (\ref{37}) that
there exists a unique critical point of $U$ for each $\mu\neq 1$ if and only
if $m/q>\sqrt{\mu}$ when $\mu<1$ and $m/q<\sqrt{\mu}$ when $\mu>1$. Our
numerical work suggests that $m/q$ easily passes these criteria. The
rightmost column of table 1 presents the approximate critical vortex separation
$r_{c}$ for each value of $\mu^{2}$. Potentials for $\mu=0.4$ (type I) and
$\mu=2.0$
(type II) are plotted in figure 1. Rebbi and Jacobs
\cite{Reb} have found approximate static intervortex potentials by numerically
minimizing the potential energy functional (\ref{8}) subject to the constraint
that $\phi$ has two simple zeros separated by a given distance $d$, for a range
of values of $d$. In the type I case, they find that the dominance of scalar
attraction over magnetic repulsion subsides as the vortex separation gets
small, but not to the extent of producing a stable equilibrium at non-zero $d$.
Similarly, they find that magnetic repulsion of type II vortices is
increasingly counteracted by scalar attraction (though they do not use this
terminology) as $d$ becomes small. It would appear that ${\cal U}$ is in broad
agreement with their results when $r>r_{c}$, but that the asymptotic
approximation breaks down for $rFrom the plot of $U(r)$ we see that all trajectories
which do not encroach on the interior region $r\Delta$ we set $U\equiv 0$ and calculate $\th_{\Delta}$ in the
free vortex approximation. The results of this algorithm are shown in
figure 3. The fit to the numerical simulations of \cite{Mye}
could be improved by adjusting the values of $q$ and $m$. Given the warning
attached to these charges in section \ref{sec:va}, this may well be justified.
However, such adjustment corrupts the deductive nature of the model,
so we prefer not to make it.
\section{Velocity dependent forces}
\label{sec:vdf}
So far all the results obtained could be found without using the point source
formalism. Introducing the point vortex provides a compact and convenient way
of organizing essentially the same calculations as arise in the superposition
ansatz approach. (The connexion between these two viewpoints is explored in
the context of the Skyrme model in \cite{Sch}.) The utility of the point source
formalism becomes more apparent when we go on to calculate velocity dependent
intervortex forces. It is not obvious how to approach such a calculation using
a superposition ansatz, but there is a large collection of analogous
calculations for point sources in linear field theory, most notably, classical
electrodynamics. Although one could calculate a velocity dependent interaction
Lagrangian at arbitrary $\mu^{2}$, there are three reasons why we choose to
consider only the case of critical coupling. First, when $\mu^{2}=1$ there are
no static intervortex forces, so the velocity dependent forces are leading,
whereas when $\mu^{2}$ is far from 1 they will likely be swamped by the static
potential. Second, the resulting mechanical system has the simplifying
property of Galilei symmetry if $\mu^{2}=1$, but not otherwise. Third, in the
absence of a static potential we can justify our
expansion {\it a posteriori}, as will be shown. The argument does not work
if $U\neq 0$, so our approximation may not be self consistent away from
critical coupling.
The first task is to
find expressions for the sources $\rho$ and $j$ for a point vortex moving on
some arbitrary trajectory $\y(t)$. We employ a quasi-adiabatic approximation:
we assume that at each point $\y(t)$, the source is a static point vortex
Lorentz boosted with velocity $\yd$. (An adiabatic approximation would be to
assume that the point vortex always has the static form, and simply translates
along the trajectory $\y(t)$.) The strategy is then to expand this moving
source in powers of $|\yd|$, truncating at order $|\yd|^{2}$, calculate the
time dependent fields it induces, truncating similarly, then use these to find
an interaction Lagrangian.
\subsection{Moving sources}
\label{subsec:ms}
We seek expressions for the scalar charge density $\rho$ and vector current
$j$ of a point vortex moving along some curve $\y(t)$ in $\R^{2}$. At time
$t=0$, let the point vortex be at $\x=\zv$, moving with velocity $\uv$.
Introduce rest frame coordinates $\eta^{\mu}$, related to laboratory
coordinates $x^{\mu}$ by a Lorentz boost (on $x$) with velocity $\uv$.
Explicitly,
\bea
\eta^{\prl} &=& \gamma(u)(x^{\prl}-ux^{0})=\gamma x^{\prl} \nonumber \\
\label{41}
\eta^{\prp} &=& x^{\prp}
\eea
at time $x^{0}=0$, where $\gamma(u)=(1-u^{2})^{-\hf}$. The
components $\eta^{\prl}$ and $\eta^{\prp}$ can be combined into a single
2-vector equation for $\n(\x)$ by decomposing $\n$ into parallel and
perpendicular components:
\bea
\n &=& \frac{(\n\cdot\uv)\uv}{u^{2}}+\left[\n-\frac{(\n\cdot\uv)\uv}{u^{2}}
\right] \nonumber \\
&=& \frac{\gamma(\x\cdot\uv)\uv}{u^{2}}+
\left[\x-\frac{(\x\cdot\uv)\uv}{u^{2}}\right] \nonumber \\
\label{42}
&=& \x+\hf(\x\cdot\uv)\uv+\ldots
\eea
where the ellipsis denotes discarded terms of order $u^{4}$ or greater (we
shall not persist in so noting these discarded terms; henceforth
an ellipsis will only be included where we have discarded further negligible
terms in deriving the given expression). Now generalize to the situation of
a point vortex located at $\y(t)$ moving with velocity $\yd(t)$. The rest
frame coordinates of a general point $\x$ on the surface $\x^{0}=t$
are
\beq
\label{43}
\n(t,\x)=\x-\y+\hf[(\x-\y)\cdot\yd]\yd
\eeq
by mapping $\x\mapsto\x-\y(t)$, $\uv\mapsto\yd(t)$ in equation (\ref{42}).
We must now transform the sources, applying Lorentz boosts with velocity
$-\yd$ to the rest frame distributions $\rho_{(0)}$ and $j_{(0)}$. Consider
first the scalar distribution, which, from previous work, in the rest frame
has the form
\beq
\label{44}
\rho_{(0)}(\n)=q\delta(\n).
\eeq
Since $\rho$ transforms as a Lorentz scalar, $\rho(x)=\rho_{(0)}(\n(\x))$ in
the laboratory frame. To find $\delta(\n)$ as a function of $\x$ one notes that
\beq
\label{45}
\inx f(\x)\delta(\n)=\int d^{2}\n\left|\frac{\cd\x}{\cd\n}\right|f(\x)
\delta(\n)
\eeq
for any function $f$,
where $|\cd\x/\cd\n|$ is the determinant of the Jacobian of the transformation
$\x\mapsto\n$. From (\ref{43}),
\bea
\frac{\cd\n}{\cd\x} &=& \mbox{\set I}+\hf\yd\otimes\yd \nonumber \\
\Rightarrow \frac{\cd\x}{\cd\n} &=& \mbox{\set I}-\hf\yd\otimes\yd+\ldots
\nonumber \\
\label{46}
\Rightarrow \left|\frac{\cd\x}{\cd\n}\right|&=&1-\hf|\yd|^{2}+\ldots
\eea
Substituting into (\ref{45}),
\bea
\inx f(\x)\delta(\n) &=& \left(1-\hf|\yd|^{2}\right)\int d^{2}\n f(\x)
\delta(\n)
\nonumber \\
&=& \left.\left(1-\hf|\yd|^{2}\right)f(\x)\right|_{\n=\zv} \nonumber \\
\label{eq:47}
&=& \left(1-\hf|\yd|^{2}\right)f(\y)
\eea
since $\n=\zv\Leftrightarrow\x=\y$. Thus,
\bea
\label{48}
\delta(\n) &=& \left(1-\hf|\yd|^{2}\right)\delta(\x-\y) \\
\label{49}
\Rightarrow \rho(\x) &=& \left(1-\hf|\yd|^{2}\right)q\delta(\x-\y).
\eea
The term scalar charge for $q$ is something of a misnomer since, as shown by
(\ref{49}) it is not a scalar. A plate of area $C$ in its rest frame,
carrying uniform scalar charge density $\rho$ has total scalar charge
$q=C\rho$.
Looked at from a boosted frame, the plate is squashed along the boost direction
by a factor $1/\gamma$ due to Lorentz-Fitzgerald contraction, so the area of
the plate in this frame is $C/\gamma\approx(1-u^{2}/2)C$. The charge
density is invariant, so the total scalar charge is $q'\approx(1-u^{2}/2)q$ in
agreement with the calculation above.
The moving magnetic dipole is rather more complicated, because $j^{\mu}$
transforms as a vector itself. The rest frame source is
\beq
\label{50}
(j_{(0)}^{0},\quad\jv_{(0)})=(0,-m\k\times\D_{\eta}\delta(\n)).
\eeq
Again, we perform a Lorentz boost on this with velocity $-\yd$, that is
\beq
\label{51}
j^{\mu}(x)=\Lambda^{\mu}_{\,\,\nu}j_{(0)}^{\nu}(\n(x)),
\eeq
where
\bea
\Lambda^{0}_{\,\,0} = \Lambda^{\prl}_{\,\,\prl} &=& \gamma \nonumber \\
\Lambda^{0}_{\,\,\prl}=\Lambda^{\prl}_{\,\,0} &=& \gamma|\yd| \nonumber \\
\label{52}
\Lambda^{\prp}_{\,\,\prp} &=& 1,
\eea
all other $\Lambda^{\mu}_{\,\,\nu}=0$. Explicitly,
\bea
\label{53}
j^{0}(x) &=& |\yd|\gamma j_{(0)}^{\prl}(\eta(x))=\yd\cdot\jv_{(0)}(\eta(x))+
\ldots \\
j^{\prl}(x) &=& \gamma j_{(0)}^{\prl}(\eta(x))=\left(1+\hf|\yd|^{2}\right)
j_{(0)}(\eta(x))+\ldots \nonumber \\
\label{54}
j^{\prp}(x) &=& j_{(0)}^{\prp}(\eta(x)).
\eea
Repeating the algebraic trick of (\ref{42}),
\beq
\label{55}
\jv(x)=\jv_{(0)}(\n(x))+\hf(\jv_{(0)}(\n(x))\cdot\yd)\yd+\ldots
\eeq
Now,
\bea
\D_{\eta}= \left(\frac{\cd\x}{\cd\n}\right)\D
&=& \left(\mbox{\set I}-\hf\yd\otimes\yd\right)\D \nonumber \\
\label{56}
&=& \D-\hf\yd(\yd\cdot\D),
\eea
so, using (\ref{48}), we find that
\bea
\jv_{(0)}(\n(x)) &=& -m\k\times\D_{\eta}\delta(\n) \nonumber \\
\label{57}
&=&-m\left(1-\hf|\yd|^{2}\right)\k\times\D\dxy
+\hf m(\k\times\yd)\yd\cdot\D\dxy+\ldots
\eea
Substituting (\ref{57}) into (\ref{53}) and (\ref{55}),
\bea
j^{0}(x)&=& m(\k\times\yd)\cdot\D\dxy, \nonumber \\
\jv(x) &=& -m\left(1-\hf|\yd|^{2}\right)\k\times\D\dxy \nonumber \\
\label{59}
& &+\hf m\left[\yd(\k\times\yd)\cdot\D\dxy+(\k\times\yd)\yd\cdot\D\dxy
\right],
\eea
the final result.
One should note that
\beq
\label{60}
\cd_{\mu}j^{\mu}=\frac{\cd j^{0}}{\cd t}+\D\cdot\jv
=m(\k\times\ddot{\y})\cdot\D\dxy,
\eeq
so that $\cd_{\mu}j^{\mu}\neq 0$ unless $\ddot{\y}=\zv$ (in which case the
rest frame is inertial and $\cd_{\mu}j^{\mu}=0$ follows from the vanishing
of $\D\cdot\jv$ for a static point vortex). If $\ddot{\y}=\zv$ the current is
conserved, so we can visualize the current density of a vortex moving with
constant velocity as an ordinary electric current. In analogy with standard
electrodynamics, we identify $j^{0}=\varrho$ as the electric charge density
of the distribution. From (\ref{59}) we see that $\varrho\neq 0$ for a
moving
magnetic dipole, but that $\varrho$ corresponds to an electric dipole of moment
$-m\k\times\yd=|m|\k\times\yd$. It is helpful to think of the magnetic dipole
at rest as a small clockwise current loop consisting of interpervading gases of
oppositely charged current carriers confined to a circle in $\R^{2}$,
travelling
in opposite senses, as in figure 4, so that the whole has
$\varrho=0$ everywhere. Consider such an object viewed in a frame in which it
is moving with constant velocity $\yd$. In the figure, the positive gas in the
upper half experiences greater Lorentz-Fitzgerald contraction than the negative
due to its velocity of circulation, and
{\it vice-versa} in the lower half. Thus the upper half acquires a net
positive charge density, while the lower half acquires an equal net negative
charge density (it is {\em charge} not charge {\em density}
which is scalar here), this charge splitting being the origin of our electric
dipole
$\varrho$. Note the agreement of orientation. Turning now to the current
density $\jv$, the first term of (\ref{59}) represents a current loop,
while the second and third represent anisotropies due to the aforementioned
charge splitting. The transport of the net positive charge (upper half) and
net negative charge (lower half) with velocity $\yd$ produces the current
represented by the second term in (\ref{59}). Also, as the loop moves along,
the charge must split in front of the loop and recombine behind it in order to
create the electric dipole. The current due to this process is represented by
the final term.
\subsection{Interaction Lagrangians}
\label{subsec:il}
In order to compute the interaction Lagrangians $L_{\psi}$ and $L_{A}$ in the
case of arbitrarily moving vortices, one must find the fields $\psi$ and $A$
induced by time varying sources $\rho$ and $j$. Were the linear theory
massless,
this would involve the use of retarded potentials, since disturbances of the
fields due to time varying sources would propagate uniformly at the speed of
light. For example, the potential induced by a moving point charge in classical
electrodynamics has been well studied, and explicit formulae can be found in
the literature \cite{LieWer}. Not surprisingly, the analogous problem in
{\em massive} electrodynamics (or scalar field theory) has not received such
attention: the only fundamental physical force transmitted by massive quanta
is the weak force, which has no r\^{o}le to play in the classical dynamics of
point particles. We handle the problem by introducing formal temporal Fourier
transforms, as follows. Let $\psi(t,\x)$ be the field induced by time-varying
source $\rho(t,\x)$, according to the inhomogeneous Klein-Gordon equation.
Define Fourier transforms $\wt{\psi}$ and $\wt{\rho}$ with variable $\omega$
dual to $t$. That is,
\beq
\label{61}
\psi(t,\x) =: \int_{-\infty}^{\infty} d\omega e^{i\omega t}\wt{\psi}(\omega,
\x),\qquad
\rho(t,\x) =: \int_{-\infty}^{\infty}d\omega e^{i\omega t}\wt{\rho}(\omega,\x).
\eeq
Then,
\bea
\label{62}
(\Box+\mu^{2})\psi &=& \rho \\
\Rightarrow (-\omega^{2}-\Delta +\mu^{2})\wt{\psi} &=& \wt{\rho} \nonumber \\
\label{63}
\Rightarrow [-\Delta +(\mu^{2}-\omega^{2})]\wt{\psi} &=& \wt{\rho},
\eea
so $\wt{\psi}(\omega,\x)$ satisfies the static inhomogeneous Klein-Gordon
equation with squared mass $\mu^{2}-\omega^{2}$ and source
$\wt{\rho}(\omega,\x)$. Equation (\ref{63}) is solved (at least formally)
by convolution of $\wt{\rho}$ with the Green's function
$K_{0}(\sqrt{\mu^{2}-\omega^{2}}|\x-\x'|)/2\pi$,
\beq
\label{64}
\wt{\psi}(\omega,\x)=\frac{1}{2\pi}\inxp
K_{0}(\sqrt{\mu^{2}-\omega^{2}}|\x-\x'|)\wt{\rho}(\omega,\x').
\eeq
Now expand the Green's function in $\omega/\mu$, truncating at order
$\omega^{2}/\mu^{2}$,
\bea
K_{0}(\sqrt{\mu^{2}-\omega^{2}}|\x-\x'|)
&=& K_{0}(\mu(1-\hf\frac{\omega^{2}}{\mu^{2}}+\cdots)|\x-\x'|) \nonumber \\
\label{65}
&=& K_{0}(\mu|\x-\x'|)-\frac{\omega^{2}}{2\mu^{2}}|\x-\x'|K_{0}'(\mu|\x-\x'|)
+\cdots
\eea
so that substitution into (\ref{64}) yields
\bea
\wt{\psi}(\omega,\x)&=&\frac{1}{2\pi}\inxp K_{0}(\mu|\x-\x'|)\wt{\rho}
(\omega,\x') \nonumber \\
\label{66}
& &-\frac{\omega^{2}}{4\pi\mu}\inxp|\x-\x'|
K_{0}'(\mu|\x-\x'|)\wt{\rho}(\omega,\x'),
\eea
whence we obtain $\psi(t,\x)$ by (\ref{61}),
\bea
\psi(t,\x) &=& \frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\,e^{i\omega t}
\inxp K_{0}(\mu|\x-\x'|)\wt{\rho}(\omega,\x') \nonumber \\
& & -\frac{1}{4\pi\mu}\int_{-\infty}^{\infty}d\omega\,\omega^{2}e^{i\omega t}
\inxp|\x-\x'|K_{0}'(\mu|\x-\x'|)\wt{\rho}(\omega,\x') \nonumber \\
&=& \frac{1}{2\pi}\inxp K_{0}(\mu|\x-\x'|)\int_{-\infty}^{\infty}d\omega\,
e^{i\omega t}\wt{\rho}(\omega,\x') \nonumber \\
& & +\frac{1}{4\pi\mu}\inxp|\x-\x'|K_{0}'(\mu|\x-\x'|)\frac{\cd^{2}}{\cd t^{2}}
\int_{-\infty}^{\infty}d\omega\,e^{i\omega t}\wt{\rho}(\omega,\x') \nonumber \\
&=& \frac{1}{2\pi}\inxp K_{0}(\mu|\x-\x'|)\rho(t,\x') \nonumber \\
\label{67}
& & +\frac{1}{4\pi\mu}\inxp|\x-\x'|K_{0}'(\mu|\x-\x'|)\ddot{\rho}(t,\x').
\eea
Note that truncating the expansion in $\omega$ is, in effect, the same as
neglecting higher time derivatives of $\rho$ in general, eventually acting on
$\z(t)$ in our application. No claim of rigour is attached to the above
Fourier transform manoeuvre. One should regard it as a convenient algebraic
short-hand for obtaining a perturbative ansatz for (\ref{62}).
Substitution of (\ref{67}) into (\ref{62}) explicitly verifies that
$\psi$ is indeed a solution, up to higher derivative terms ($d^{3}\rho/dt^{3}$
etc.).
Using this procedure, we find $\psi_{2}$, the field induced by time varying
source $\rho_{2}$. The interaction of such a field with another time varying
source $\rho_{1}$ is given by (\ref{33}),
\bea
L_{\psi} &=& \inx \rho_{1}\psi_{2} \nonumber \\
&=& \frac{1}{2\pi}\dbin K_{0}(\mu|\x-\x'|)\rho_{1}(t,\x)\rho_{2}(t,\x')
\nonumber \\
\label{68}
& & -\frac{1}{4\pi\mu}\dbin|\x-\x'|K_{0}'(\mu|\x-\x'|)\dot{\rho}_{1}(t,\x)
\dot{\rho}_{2}(t,\x'),
\eea
where a total time derivative has been discarded. The vector field calculation
is essentially identical, yielding interaction Lagrangian
\bea
L_{A} &=& -\frac{1}{2\pi}\dbin K_{0}(|\x-\x'|)j_{(1)}^{\mu}(t,\x)
J^{(2)}_{\mu}(t,\x') \nonumber \\
\label{69}
& & +\frac{1}{4\pi}\dbin|\x-\x'|K_{0}'(|\x-\x'|)\frac{\cd j_{(1)}^{\mu}}{\cd t}
(t,\x)\frac{\cd J^{(2)}_{\mu}}{\cd t}(t,\x'),
\eea
the only new feature being that one of the sources includes the ``fictitious
current'' due to non-conservation of $j_{\mu}^{(2)}$. That is, the field
equation
for $A_{\mu}$ with source $j_{\mu}$ (\ref{23}) looks like the Proca equation
(a triplet of Klein-Gordon equations) with source $j_{\mu}+\cd_{\mu}(\cd_{\nu}
j^{\nu})$, so we define the pseudocurrent
\beq
\label{70}
J^{(2)}_{\mu}:=j^{(2)}_{\mu}+\cd_{\mu}(\cd_{\nu}j_{(2)}^{\nu}),
\eeq
and it is this source which appears in the algebra analogously to $\rho_{2}$ in
the scalar calculation above. There is an apparent asymmetry in (\ref{69})--
it looks asymmetric under the interchange of sources $1\leftrightarrow 2$ --
but this is easily removed, given the form of the pseudocurrent (\ref{70}),
by integration by parts; taking the first integral in (\ref{69}), the extra
term due to the fictitious current is
\bea
& &-\frac{1}{2\pi}\left.\dbin K_{0}(|\x-\x'|)\,j_{(1)}^{\mu}(x)\,
\cd'_{\mu}(\cd'_{\nu}
j^{\nu}_{(2)}(x'))\right|_{t'=t} \nonumber \\
& & \qquad\qquad =\frac{1}{2\pi}\left.\dbin
j^{\mu}_{(1)}\,\cd'_{\mu}K_{0}\,\cd'_{\nu}j_{(2)}^{\nu}\right|_{t'=t} \nonumber \\
& & \qquad\qquad =\frac{1}{2\pi}\left.\dbin
j^{\mu}_{(1)}\,\cd_{\mu}K_{0}\,\cd'_{\nu}j_{(2)}^{\nu}\right|_{t'=t} \nonumber \\
\label{71}
& & \qquad\qquad =-\frac{1}{2\pi}\left.\dbin K_{0}(|\x-\x'|)\,\cd_{\mu}j_{(1)}^{\mu}(x)\,
\cd'_{\nu}j_{(2)}^{\nu}(x')\right|_{t'=t}
\eea
where, as usual, all boundary integrals have vanished. A similar calculation
for the second integral yields the extra term
\beq
\label{72}
\frac{1}{4\pi}\left.\dbin|x-\x'|K_{0}'(|\x-\x'|)\frac{\cd}{\cd t}
\cd_{\mu}j_{(1)}^{\mu}(x)\frac{\cd}{\cd t'}\cd'_{\nu}j_{(2)}^{\nu}(x')
\right|_{t'=t}
\eeq
Recall from (\ref{60})
that $\cd_{\mu}j^{\mu}$ is of order $|\ddot{\y}|$, so neither term
(\ref{71}) nor (\ref{72}) makes any contribution to $L_{A}$ at the
order to which we are calculating. We may thus discard them and work with the
formula
\bea
L_{A}&=&-\frac{1}{2\pi}\dbin K_{0}(|\x-\x'|)j_{(1)}^{\mu}(t,\x)
j^{(2)}_{\mu}(t,\x') \nonumber \\
\label{73}
& &+
\frac{1}{4\pi}\dbin|\x-\x'|K_{0}'(|\x-\x'|)\frac{\cd j_{(1)}^{\mu}}{\cd t}(t,\x)
\frac{\cd j^{(2)}_{\mu}}{\cd t}(t,\x').
\eea
It remains to substitute the point vortex sources (\ref{49}) and
(\ref{59}) into these expressions for $L_{\psi}$ and $L_{A}$, noting
that at critical coupling, $\mu=1$ and $m\equiv q$. A straightforward though
lengthy calculation, presented in the appendix, yields
\bea
L_{\rm int}&=& L_{A}+L_{\psi} \nonumber \\
\label{74}
&=&\frac{q^{2}}{4\pi}\left\{
\left[\frac{(\y-\z)\cdot(\yd-\zd)}{|\y-\z|}\right]^{2}-
\left[\frac{(\y-\z)\cdot(\k\times(\yd-\zd))}{|\y-\z|}\right]^{2}\right\}
\Lambda(|\y-\z|),
\eea
where
\beq
\label{75}
\Lambda(\lambda)=K_{0}(\lambda)+\frac{2K_{1}(\lambda)}{\lambda}.
\eeq
The Lagrangian is completed by adding to this the standard non-relativistic
kinetic Lagrangian for two point particles each of mass $\pi$, the rest energy
of the $\mu=1$ vortex, as found using the Bogomol'nyi argument. Introducing
the centre of mass position $\Rv:=\hf(\y+\z)$ and relative position $\rv:=
\y-\z$, the two vortex Lagrangian is
\beq
\label{76}
L=\pi|\dot{\Rv}|^{2}+\frac{\pi}{4}|\rd|^{2}+\frac{q^{2}}{4\pi}
[(\rh\cdot\rd)^{2}-(\rh\cdot(\k\times\rd))^{2}]\Lambda(|\rv|).
\eeq
Note that $L$ is Galilei invariant (a feature which does not generalize to
arbitrary coupling since in the presence of static intervortex forces the
inertia associated with $\Rv$ depends on the vortex separation $\rv$). So,
we can work in the centre of mass frame with the reduced Lagrangian, which
looks like the Lagrangian of a free particle moving geodesically on a
manifold (naively $\R^{2}$) with respect to the metric
\beq
\label{78}
g=\left(1+\frac{q^{2}}{\pi^{2}}\Lambda(r)\right)dr^{2}+
r^{2}\left(1-\frac{q^{2}}{\pi^{2}}\Lambda(r)\right)d\th^{2},
\eeq
where $r(\cos\th,\sin\th):=\rv$. This idea will be pursued in section
\ref{sec:cvs}. For the moment, we note that the equations of motion are
\beq
\label{79}
\ddot{r}^{i}+\Gamma^{i}_{jk}\dot{r}^{j}\dot{r}^{k}=0
\eeq
where $r^{i}$ are some coordinates on the manifold, and $\Gamma$ is the
Levi-Civita connexion derived from $g$, so if $r^{i}(t)$ is a solution of
(\ref{79}), then $|\ddot{\rv}|$ is of order $|\rd|^{2}$ for all $t$.
Differentiating (\ref{79}) with respect to time, one sees that for each
integer $n\geq 2$ there is a set of position dependent coefficients
$\Omega^{(n)\,\, i}_{j_{1}j_{2}\ldots j_{n}}$ such that
\beq
\frac{d^n r^{i}}{dt^{n}}=\Omega^{(n)\,\, i}_{j_{1}j_{2}\ldots j_{n}}
\dot{r}^{j_{1}}\dot{r}^{j_{2}}\cdots\dot{r}^{j_{n}},
\eeq
so that $|d^{n}\rv/dt^{n}|$ is of order $|\rd|^{n}$. This provides {\it a
posteriori}\, justification for truncating the expansion in time derivatives.
That is, although the assumption that higher time derivatives are negligible
may turn out to be bad for real vortex dynamics, it is at least self
consistent. Away from critical coupling the situation is different. There is a
$\cd U/\cd r_{i}$ term in (\ref{79}), $U$ being the static intervortex
potential (\ref{36}), and consequently the above argument does not work.
\section{Critical vortex scattering}
\label{sec:cvs}
As remarked previously, the approximate equations of motion of two critically
coupled point vortices can be interpreted as the geodesic equation on a
two-dimensional manifold (call it $\M$) with metric $g=G(r)dr^{2}+r^{2}H(r)
d\th^{2}$ where
\beq
\label{81}
G(r)=1+\frac{q^{2}}{\pi^{2}}\Lambda(r),\qquad
H(r)=1-\frac{q^{2}}{\pi^{2}}\Lambda(r).
\eeq
Vortices are not classically distinguishable particles, so $\rv$ and $-\rv$
correspond to the same configuration, and should be identified. Accordingly
$\th\in[0,\pi]$, $\th=0$ and $\th=\pi$ being identified, and $g$ is a metric
on the cone $\R^{2}/\sim$ where $\rv\sim\rv'\Leftrightarrow\rv=\pm\rv'$. The
function $\Lambda$ is a strictly positive, strictly decreasing function on
$(0,\infty)$ and $\Lambda\rightarrow\infty$ as $r\rightarrow 0$. Hence there
is one and only one value of $r$ for which the coefficient $H$ is zero. At this
radius, $r_{s}\approx 2.73$, there is a metric singularity where the
signature of $g$ flips from Euclidean ($r>r_{s}$) to Lorentzian ($rr_{s}$, it
is helpful to define a new manifold $\Ms\cong(\R^{2}\backslash D)/\sim$
where $D$ is the disc of radius $r_{d}>r_{s}$ centred on $\rv=\zv$ excluding
boundary, so that the restriction of $g$ to $\Ms$, call it $g_{*}$ is
strictly positive. The curvature of $(\Ms,g_{*})$ is
\beq
\label{82}
K=\frac{-1}{r\sqrt{GH}}\frac{d\,\,}{dr}\left(\frac{1}{\sqrt{G}}
\frac{d\,\,}{dr}(r\sqrt{H})\right).
\eeq
It is a matter of straightforward calculation, which we will not reproduce
here, to prove that $K>0$, that $K$ is a strictly decreasing function of $r$
and that $K\rightarrow 0$ as $r\rightarrow\infty$. These results are
independent of the specific value of $q$. One may therefore visualize
$(\Ms,g_{*})$ as a rounded cone with its cap cut off, the missing cap
representing the forbidden core region where the approximation breaks down.
Since $g$ is independent of $\th$, $(\Ms,g_{*})$ is rotationally symmetric.
This picture is very reminiscent of the geodesic approximation, where the low
energy dynamics of two vortices is approximated by geodesic motion on the
moduli space of static solutions. Samols \cite{Sam} regards physical space as
the complex plane and defines the position of each vortex to be $z_{1},z_{2}
\in\C$ where the Higgs field vanishes. The degree 2 moduli space is then
$\C\times\M$ where $\C$ is the space of centre of mass positions and $\M$ is
the space of relative positions, on which is induced a nontrivial metric
$g_{s}$ by the kinetic energy functional (\ref{7}). Defining relative
coordinates $(\sigma,\th)$ such that
\beq
\label{83}
\sigma e^{i\th}=\hf(z_{1}-z_{2}),
\eeq
rotation and parity symmetries are sufficient to restrict $g_{s}$ to the form
$g_{s}=f_{1}(\sigma)d\sigma^{2}+f_{2}(\sigma)\sigma^{2}d\th^{2}$. However,
using properties of the Bogomol'nyi equations, Samols was able to prove that
$g_{s}$ must be Hermitian in $\sigma e^{i\th}$, and this provides the extra
constraint that $f_{1}\equiv f_{2}$, so
\beq
\label{84}
g_{s}=F_{s}^{2}(\sigma)(d\sigma^{2}+\sigma^{2}d\th^{2}).
\eeq
Numerical computation of $F_{s}$ reveals that $(\M,g_{s})$ is a rounded cone
of strictly positive curvature.
Given this similarity, and the precedent set by Manton's rederivation of the
asymptotic Atiyah-Hitchin metric by point particle techniques \cite{Man4},
we are led to suggest that $g_{*}$ is the asymptotic form of $g_{s}$. We would
like to identify the radial coordinate $r$ with $2\sigma$, but given that
$H\neq G$, $g_{*}$ cannot be Hermitian with respect to $re^{i\th}$, so such
an identification is impossible. It follows that the vortex positions $\y$
and $\z$ do not coincide with zeros of the Higgs field. Coincidence
is recovered asymptotically, but more slowly than the asymptotic convergence
of $g_{*}$ to the trivial flat metric. In previous applications of the method
of linear retarded potentials, this has not happened: asymptotic coincidence
of coordinates is faster than asymptotic flatness of the metric. The
essentially new feature here is that the linear theory is massive, so it may
be this which is responsible. Defining soliton position is always
somewhat arbitrary because solitons can really only be considered independent
particles when infinitely remote from one another. Since we defined vortex
position in terms of the asymptotics of $\phi$, in the case of degree 2
configurations we should expect there to be a discrepency between this and the
usual definition, exponentially small in the vortex separation.
This does not disqualify $g_{*}$ from being the asymptotic form of $g_{s}$
(in fact, $4g_{s}$ since \cite{Sam} uses a different normalization): we can
always construct a radial coordinate in terms of which $g_{*}$ does take the
Hermitian form. If one defines the function
\beq
\label{85}
s(r)=s_{d}\exp\left[\int_{r_{d}}^{r}\frac{d\lambda}{\lambda}
\sqrt{\frac{G(\lambda)}{H(\lambda)}}\right],
\eeq
the transformation $r\mapsto s(r)$ is manifestly a bijection. In terms of
$(s,\th)$,
\beq
\label{86}
g_{*}=4F^{2}(s)(ds^{2}+s^{2}d\th^{2})
\eeq
where $F(s(r))=\hf r\sqrt{H(r)}/s(r)$. To identify $s$ with $\sigma$, we fix
$s_{d}$ such that $\lim_{r\rightarrow\infty}s(r)/r=1/2$. It is then easily
shown that $s(r)/r$ is a strictly increasing function on $[r_{d},\infty)$ with
minimum $s_{d}/r_{d}<1/2$ and supremum $1/2$ (these observations follow from
the fact that $G(\lambda)>H(\lambda)$ for all $\lambda>0$, and are independent
of the value of $q$). It follows that a given point $(r,\th)\in\Ms$ represents
a two-vortex configuration with inter-zero distance less than $r$, the
difference vanishing exponentially at large $r$. Note that if we choose
$r_{d}=r_{s}$ there is a $(\lambda-r_{s})^{-\hf}$ singularity in the integrand
of (\ref{85}) because $H(r_{s})=0$. Given the specific formula
(\ref{75}) for $\Lambda$, we must evaluate this integral numerically, so to
simplify matters we choose $r_{d}>r_{s}$, that is, we exclude a slightly
larger cap from $\Ms$ than is strictly necessary. Choosing $r_{d}=3$ one finds
that $s_{d}\approx 1.22$. Figure 5 presents a plot of $F$ compared with
Samols' profile function $F_{s}$, from which it seems plausible that $g_{*}$
is the asymptotic form of $g_{s}$. Since both functions are known only
numerically, there is no rigorous test of this.
The main object of this section is to model critical vortex scattering by
solving the geodesic problem on $(\Ms,g_{*})$. We could use $g_{*}$ in
Hermitian form, but this would introduce an extra layer of numerical
approximation, so it is better to work with $g_{*}$ in the original
coordinates $(r,\th)$. The scattering problem is defined in terms of asymptotic
parameters in any case: impact paramter $b$ and impact speed $v_{\infty}$,
both defined where the vortices are infinitely remote from one another and
there is no ambiguity in the term ``vortex position.'' In fact, the geometry of
geodesics is independent of initial velocity, as may be seen by rescaling $t$
in the geodesic equation (\ref{79}), so deflection angle is independent of
$v_{\infty}$. (It should be emphasized that this is a property of the
approximation, not the abelian Higgs model itself. In fact, numerical
simulations \cite{Mye} show that scattering {\em is} approximately speed
independent at critical coupling for low to moderate $v_{\infty}$, but this
breaks down at very high speeds.) So the scattering data $\Theta(b)$ provide
a physically interesting, coordinate independent characterization of the
metric structure on $\Ms$.
Without loss of generality, we can solve the initial value problem
$r(0)=r_{0}$, $\th(0)=0$, $\dot{r}=0$, $\dot{\th}=\omega_{0}$, parametrized by
$(r_{0},J)$, where $J=r^{2}H(r)\dot{\th}$ is the conserved momentum conjugate
to $\th$. It is easily shown that $\lim_{t\rightarrow\infty}\th$ is
\beq
\label{87}
\th_{\infty}=\int_{r_{0}}^{\infty}\frac{dr}{r^{2}H(r)}
\left[\frac{1}{G(r)}\left(
\frac{1}{r_{0}^{2}H(r_{0})}-\frac{1}{r^{2}H(r)}\right)\right]^{-\hf}.
\eeq
By equating energy at $r=r_{0}$ and as $r\rightarrow\infty$, one finds that
$b(r_{0})=\hf r_{0}\sqrt{H(r_{0})}$, so the absence of $J=2v_{\infty}b$ in
(\ref{87}) implies that $\th_{\infty}$ is independent of $v_{\infty}$ as
claimed. Approximate evaluation of the function $\th_{\infty}(r_{0})$ is
performed using the same algorithm as was used for type II vortex scattering,
summarized in equation (\ref{40}). Figure 6 shows $b$ plotted against
$\Theta(b(r_{0})):=\pi-2\th_{\infty}(r_{0})$, the deflection angle, in
comparison with Samols' scattering data, obtained using the geodesic
approximation. (A comparison of the geodesic approximation with numerical
simulations of the full field equations is made in \cite{Sam}.) The fit is
remarkably good for moderate to large $b$, but deteriorates as $b$ becomes
small. This is to be expected since small $b$ collisions probe the small $r$
region of $\Ms$. That $\Theta(b)$ is a decreasing function is a corollary of
the fact that the curvature is strictly positive \cite{AtiHit2}.
\section{Conclusion}
\label{sec:con}
In this paper we have presented a point source formalism for long range vortex
dynamics. We used this framework to rederive the static intervortex potential
from a new perspective and to calculate the velocity dependent interaction of
critically coupled vortices. Reinterpreting the latter geometrically led to a
conjectured formula for the asymptotic metric on the two vortex moduli space.
We solved the scattering problem for $\mu^{2}=2$ and $\mu^{2}=1$ vortices and
found reasonable agreement with numerical simulations, despite the simplicity
of the model. It is worth pointing out that, compared with other studies of
vortex dynamics \cite{Sam,Mye,Sha}, the present work required only very
lightweight numerical work. It would be straightforward to apply the method to
other situattions of interest: to derive the asymptotic forces between a static
vortex-antivortex pair, or higher winding conglomerations (in the type~I
regime), or larger collections of vortices for example.
After its many successes in Bogomol'nyi field theories (Yang-Mills-Higgs,
abelian Higgs and sigma models most notably) attempts are now being made to
generalize the geodesic approximation in the absence of a saturable Bogomol'nyi
bound. The idea is that a moduli space of physically relevant configurations is
proposed, usually on rather {\it ad hoc} grounds, and the metric and potential
on this space (restrictions of the kinetic and potential energy functionals of
the field theory) are calculated. Generically, this metric must be evaluated
numerically from first principles, that is, by calculating the
$L^{2}$ inner products of every (unordered) pair of tangent vectors,
at each configuration. This is a very intensive procedure. The computational
cost would be significantly reduced if the need for such numerical work could
be
contained within a relatively small core region of moduli space where the
solitons are close together -- if, for example, the asymptotic form of the
metric could be found analytically from a point source approximation, a
technique employed in \cite{LeeManSch} in the context of the Skyrme model
without pion mass.
The present work develops in a simple setting a possible way of doing this when
the linearized theory is massive. However, one should note that the
mismatch of moduli space coordinates encountered in the abelian Higgs model
could cause major problems if it occurs generically. We were able to construct
a coordinate transformation quite easily, but this was on a two dimensional
manifold with rotational symmetry. For higher dimensional moduli spaces
(if the field theory is defined on $\R^{3+1}$, or the solitons have
orientations
or internal degrees of freedom for example) the transformation may be far more
complicated.
\vspace{1cm}
\noindent
\Large
{\bf Appendix}
\normalsize
\vspace{0.75cm}
\noindent
In this appendix we present a detailed derivation of the velocity dependent
interaction Lagrangians $L_{\psi}$ and $L_{A}$ for critically coupled point
vortices. It is convenient to define the functions
\beq
\Upsilon(\lambda):=\lambda\kn'(\lambda),\qquad
\Lambda(\lambda):=\kn(\lambda)-\frac{2\kn'(\lambda)}{\lambda}.
\eeq
Substituting the scalar sources (\ref{49}) for vortex 1 at $\y(t)$ and vortex 2
at $\z(t)$ into $L_{\psi}$, (\ref{68}), one obtains
\bea
L_{\psi}&=&\frac{q^{2}}{2\pi}\dbin\kn(|\x'-\x|)\left[1-\hf(|\yd|^{2}+|\zd|^{2})
\right]\dxy\dxz \nonumber \\
& &-\frac{q^{2}}{4\pi}\dbin\Upsilon(\xx)\yd\cdot\Dy\dxy\zd\Dz\dxz+\cdots
\nonumber \\
\label{A1}
&=&\frac{q^{2}}{2\pi}\left[1-\hf(|\yd|^{2}+|\zd|^{2})\right]\kn(\yz)
1\frac{q^{2}}{4\pi}(\yd\cdot\Dy)(\zd\cdot\Dz)\Upsilon(\yz).
\eea
The magnetic interaction Lagrangian is considerably more complicated. First we
note that (\ref{69}) can be split up as follows,
\beq
\label{A2}
L_{A}=-\frac{1}{2\pi}(S_{1}-S_{2})+\frac{1}{4\pi}S_{3}
\eeq
where
\bea
S_{1}&=&\dbin\kn(\xx)j_{(1)}^{0}(t,\x)j_{(2)}^{0}(t,\x') \nonumber \\
S_{2}&=&\dbin\kn(\xx)\jv_{(1)}(t,\x)\cdot\jv_{(2)}(t,\x') \nonumber \\
\label{A3}
S_{3}&=&\dbin\Upsilon(\xx)
\cd_{t}j_{(1)}^{\mu}(t,\x)\cd_{t}j^{(2)}_{\mu}(t,\x').
\eea
Source 1 is given explicitly in (\ref{59}) and source 2 is identical, but
with $\y\rightarrow\z$, $\yd\rightarrow\zd$. Note that since $\mu^{2}=1$, we
set $m\equiv q$. Substituting these sources into $S_{1}$ and $S_{2}$ yields
\bea
S_{1}&=&q^{2}\dbin\kn(\xx)(\k\times\yd)\cdot\Dy\dxy(\k\times\zd)\cdot\Dz\dxz
\nonumber \\
&=&q^{2}(\k\times\yd)\cdot\Dy(\k\times\zd)\cdot\Dz\kn(\yz) \nonumber \\
\label{A4}
&=&-q^{2}\yd\cdot\zd\frac{\kn'(\yz)}{\yz}-q^{2}
\frac{(\k\times\yd)\cdot(\y-\z)(\k\times\zd)\cdot(\y-\z)}{\yz^{2}}\Lambda(\yz),
\\
S_{2}&=&q^{2}\dbin\kn(\xx) \nonumber \\
& &\left[\left(1-\frac{|\yd|^{2}}{2}\right)\k\times\Dy\dxy
-\hf\yd(\k\times\yd)\cdot\Dy\dxy
-\hf(\k\times\yd)\yd\cdot\Dy\dxy\right] \nonumber \\
& &\bullet\left[\left(1-\frac{|\zd|^{2}}{2}\right)\k\times\Dz\dxz
-\hf\zd(\k\times\zd)\cdot\Dz\dxz
-\hf(\k\times\zd)\zd\cdot\Dz\dxz\right] \nonumber \\
&=&q^{2}\dbin\kn(\xx)\left[1-\hf(|\yd|^{2}+|\zd|^{2})\right]\Dy\dxy\cdot\Dz\dxz
\nonumber \\
& &-\frac{q^{2}}{2}\dbin\kn(\xx)[(\k\times\Dy\dxy)\cdot\zd(\k\times\zd)\cdot
\Dz\dxz+(\y\leftrightarrow\z)] \nonumber \\
& &-\frac{q^{2}}{2}\dbin\kn(\xx)[(\k\times\Dy\dxy)\cdot(\k\times\zd)\zd\cdot
\Dz\dxy+(\y\leftrightarrow\z)] \nonumber \\
& &+\cdots \nonumber \\
&=&q^{2}\left[1-\hf(|\yd|^{2}+|\zd|^{2})\right]\Dy\cdot\Dz\kn(\yz) \nonumber \\
& &+\frac{q^{2}}{2}[(\k\times\zd)\cdot\Dy(\k\times\zd)\cdot\Dz\kn(\yz)
+(\y\leftrightarrow\z)] \nonumber \\
& &-\frac{q^{2}}{2}[\zd\cdot\Dy\zd\cdot\Dz\kn(\yz)
+(\y\leftrightarrow\z)] \nonumber \\
&=&-q^{2}\left[1-\hf(|\yd|^{2}+|\zd|^{2})\right]\kn(\yz) \nonumber \\
\label{A5}
& &+\frac{q^{2}}{2}\left\{\left[\left(\frac{\zd\cdot(\y-\z)}{\yz}\right)^{2}
-\left(\frac{(\k\times\zd)\cdot(\y-\z)}{\yz}\right)^{2}\right]
+[\y\leftrightarrow\z]\right\}\Lambda(\yz),
\eea
where use has been made of Bessel's equation and the formula
($\be$ is independent of $\y$)
\beq
\al\cdot\Dy\be\cdot\Dz\kn(|\y-\z|)=-\al\cdot\be\frac{\kn'(\yz)}{\yz}
-\frac{\al\cdot(\y-\z)\be\cdot(\y-\z)}{\yz^{2}}\Lambda(\yz).
\eeq
Note that $(\y\leftrightarrow\z)$ denotes repeated terms with $(\y,\yd)$
interchanged with $(\z,\zd)$. Finally, we turn to $S_{3}$:
\bea
S_{3}&=&-\dbin\Upsilon(\xx)\cd_{t}\jv_{(1)}(t,\x)\cdot\jv_{(2)}(t,\x')+\cdots
\nonumber \\
&=&-q^{2}\dbin\Upsilon(\xx)\Dy(\yd\cdot\dxy)\cdot\Dz(\zd\cdot\dxz) \nonumber \\
&=&-q^{2}(\yd\cdot\Dy)(\zd\cdot\Dz)\Dy\cdot\Dz\Upsilon(\yz) \nonumber \\
&=&q^{2}(\yd\cdot\Dy)(\zd\cdot\Dz)\Upsilon(\yz) \nonumber \\
\label{A8}
& &-2q^{2}\left[(\yd\cdot\zd)\frac{\kn'(\yz)}{\yz}+
\frac{\yd\cdot(\y-\z)\zd\cdot(\y-\z)}{\yz^{2}}\Lambda(\yz)\right].
\eea
It remains to substitute (\ref{A4}), (\ref{A5}) and (\ref{A8}) into (\ref{A2})
and collect terms. Note that the first two terms in $S_{2}$ and $\hf S_{3}$
combined are $-2\pi L_{\psi}$.
\bea
L_{A}&=&\frac{1}{2\pi}\left(S_{2}+\frac{S_{3}}{2}-S_{1}\right) \nonumber \\
&=&-L_{\psi}+\frac{q^{2}}{4\pi}\left[
\left(\frac{\zd\cdot(\y-\z)}{\yz}\right)^{2}+
\left(\frac{\yd\cdot(\y-\z)}{\yz}\right)^{2}\right. \nonumber \\
& &-\left(\frac{(\k\times\zd)\cdot(\y-\z)}{\yz}\right)^{2}
-\left(\frac{(\k\times\yd)\cdot(\y-\z)}{\yz}\right)^{2} \nonumber \\
& &\left.-2\frac{\yd\cdot(\y-\z)\zd\cdot(\y-\z)}{\yz^{2}}
+2\frac{(\k\times\yd)\cdot(\y-\z)(\k\times\zd)\cdot(\y-\z)}{\yz^{2}}
\right]\Lambda(\yz) \nonumber \\
\label{A9}
&=&-L_{\psi}+\frac{q^{2}}{4\pi}\left[
\left(\frac{(\yd-\zd)\cdot(\y-\z)}{\yz}\right)^{2}-
\left(\frac{(\k\times(\yd-\zd))\cdot(\y-\z)}{\yz}\right)^{2}
\right]\Lambda(\yz).
\eea
Hence $L_{\rm int}=L_{A}+L_{\psi}$ is the expression quoted, (\ref{76}).
\vspace{0.5cm}
\noindent
{\bf Acknowledgments:} I would like to thank Bernd Schroers for many valuable
conversations, and Trevor Samols for reproducing his data on critical vortex
scattering and the profile function. This work developed partly while I was a
research student at the University of Durham, England, financially supported
by the UK Particle Physics and Astronomy Research Council.
\begin{thebibliography}{xx}
\bibitem{BetRiv} L.M.A. Bettencourt and R.J. Rivers,
``Interactions between $U(1)$ cosmic strings: an analytical study''
{\sl Phys. Rev.} {\bf D51} (1995) 1842.
\bibitem{Man4} N.S. Manton,
``Monopole interactions at long range''
{\sl Phys. Lett.} {\bf 154B} (1985) 397 and {\sl Phys. Lett.} {\bf 157B} (1985)
475 (errata).
\bibitem{Sch} B.J. Schroers,
``Dynamics of moving and spinning Skyrmions''
{\sl Z. Phys.} {\bf C61} (1994) 479.
\bibitem{GibRub} G.W. Gibbons and P.J. Ruback,
``Motion of extreme Reissner-Nordstrom black holes in the low-velocity limit''
{\sl Phys. Rev. Lett.} {\bf 57} (1986) 1492.
\bibitem{M1} N.S. Manton,
``A remark on the scattering of BPS monopoles''
{\sl Phys. Lett.} {\bf 110B} (1982) 54.
\bibitem{Sam} T.M. Samols,
``Vortex scattering''
{\sl Commun. Math. Phys.} {\bf 145} (1992) 149.
\bibitem{AtiHit2} M.F. Atiyah and N.J. Hitchin,
{\sl The Geometry and Dynamics of Magnetic Monopoles}
(Princeton University Press, Princeton, USA, 1988).
\bibitem{Hig} P.W. Higgs,
``Spontaneous symmetry breakdown without massless bosons''
{\sl Phys. Rev.} {\bf 145} (1966) 1156.
\bibitem{B} E.B. Bogomol'nyi,
``The stability of classical solutions''
{\sl Sov. J. Nucl. Phys.} {\bf 24} (1976) 449.
\bibitem{Tau} A. Jaffe and C. Taubes,
{\sl Vortices and Monopoles}
(Birkh\"{a}user, Boston, USA, 1980).
\bibitem{NieOle} H.B. Nielsen and P. Olesen,
``Vortex-line models for dual strings''
{\sl Nucl. Phys.} {\bf B61} (1973) 45.
\bibitem{AbrSte} M. Abramowitz and I.A. Stegun (eds),
{\sl Pocketbook of Mathematical Functions}
(Verlag Harri Deutsch, Frankfurt, Germany 1984).
\bibitem{Mye} E. Myers, C. Rebbi and R. Strilka,
``Study of the interaction and scattering of vortices in the abelian Higgs
(or Ginzburg-Landau) model''
{\sl Phys. Rev.} {\bf D45} (1992) 1355.
\bibitem{Reb} L. Jacobs and C. Rebbi,
``Interaction of superconducting vortices''
{\sl Phys. Rev.} {\bf B19} (1979) 4486.
\bibitem{LieWer} J.D. Jackson,
{\sl Classical Electrodynamics}
(John Wiley and Sons, New York, USA, 1975).
\bibitem{Sha} P.A. Shah,
``Vortex scattering at near-critical coupling''
{\sl Nucl. Phys.} {\bf B429} (1994) 259.
\bibitem{LeeManSch} R.A. Leese, N.S. Manton and B.J. Schroers,
``Attractive channel Skyrmions and the deuteron''
{\sl Nucl. Phys.} {\bf B442} (1995) 228.
\end{thebibliography}
\newpage
\noindent
\large
{\bf Figure captions}
\normalsize
\vspace{1cm}
\newline Figure 1: The potential function $U(r)$ for $\mu^{2}=0.4$ and
$\mu^{2}=2.0$.
\vspace{0.5cm}
\newline Figure 2: The geometry of vortex scattering.
\vspace{0.5cm}
\newline Figure 3: The scattering of $\mu^{2}=2$ (type II) vortices:
deflection angle $\Theta$ versus impact parameter $b$ at four different
impact speeds. The solid curves were produced using the point particle
approximation, the crosses by numerical simulation of the full field
equations \cite{Mye}.
\vspace{0.5cm}
\newline Figure 4: A moving current loop. The unit vector $\k$ is directed
out of the page. The left hand picture shows the senses of circulation of
the charge carriers, while the right hand picture depicts the electric charge
density as seen in the laboratory frame.
\vspace{0.5cm}
\newline Figure 5: The metric profile function $F(s)$ of the metric $g_{*}$
in Hermitian form (solid curve) compared with Samols' \cite{Sam} numerically
determined function (dashed curve).
\vspace{0.5cm}
\newline Figure 6: Critical vortex scattering: deflection angle $\Theta$
versus impact parameter $b$. The results of the point particle approximation
(solid curve) should be compared with the results of the numerically
implemented geodesic approximation (dashed curve), which is in good agreement
with numerical simulations of the full field equations \cite{Sam}.
\end{document}