aainstitutetext:  Mani L. Bhaumik Institute for Theoretical Physics,
Department of Physics and Astronomy, University of California Los Angeles, Los
Angeles, CA 90095, USA

# Classical Dynamics of Vortex Solitons from Perturbative Scattering
Amplitudes

Callum R. T. Jones<EMAIL_ADDRESS>
###### Abstract

We introduce a novel point-particle effective description of ANO vortex
solitons in the critical Abelian Higgs Model (AHM) in $d=2+1$ based on the
small winding expansion. Identifying the effective vortices with the
elementary quanta of a complex scalar field, relativistic vortex-vortex
scattering amplitudes are calculated as a diagrammatic, perturbative expansion
in the winding number $N$. Making use of powerful techniques recently
developed for analyzing the post-Minkowskian two-body problem in general
relativity, we efficiently extract the contribution to the loop integrals from
the classical potential region, with the resulting velocity expansion
subsequently resummed to all orders. The main result of this paper is an
analytic expression for the classical, vortex-vortex potential at
$\mathcal{O}\left(N^{2}\right)$, or one-loop, with exact velocity dependence.
By truncating the resulting effective Hamiltonian at
$\mathcal{O}\left(p^{2}\right)$ we derive an analytic, perturbative expression
for the metric on the 2-vortex moduli space. Finally, the emergence of the
critical AHM from the classical limit of the $\mathcal{N}=2$ supersymmetric
AHM, and the resulting constraints on the point-particle EFT is described in
detail using an on-shell superspace construction for BPS states in $d=2+1$.

##  1 Introduction

Vortices and vortex-strings are topological solitons of central importance
across a wide variety of physical systems, from superconductors
ABRIKOSOV1957199  to QCD flux tubes Nielsen:1973cs  to cosmic strings
Kibble:1976sj . In spite of decades of study, there are very few analytic
results concerning the detailed properties of vortices. The isolated static
vortex solution is unknown in closed form, even in the BPS limit
deVega:1976xbp ; Taubes:1979tm , and instead must be obtained approximately by
numerically solving a difficult non-linear boundary value problem
deVega:1976xbp ; Speight:1996px ; Ohashi:2015yta . Almost all quantitative
results concerning the dynamical properties of vortex-vortex interactions are
obtained from lattice simulations Moriarty:1988fx ; Shellard:1988zx ;
Myers:1991yh , with a few exceptions Samols:1991ne ; Speight:1996px .
Numerical results, even very accurate ones, may obscure analytic features of
interest; it is therefore of some importance to develop complimentary analytic
perturbative methods Ohashi:2015yta ; Penin:2020cxj ; Penin:2021xgr .

In this paper we will describe an effective field theory approach to vortex-
vortex interactions based on the small winding approximation introduced in
Ohashi:2015yta . By defining an analytic continuation from integer winding
number $N$ to general real values, we can set up a perturbative expansion
around the limit $N\rightarrow 0$. As demonstrated in Ohashi:2015yta  there is
strong numerical evidence that (after a suitable Padé resummation) this
expansion gives quantitatively accurate results for the physical value $N=1$.
As discussed in more detail in Section 3.1, in the small winding limit there
is an additional separation of scales:

$\text{Compton wavelength}\hskip 14.22636pt\ll\hskip 14.22636pt\text{core
size}\hskip 14.22636pt\ll\hskip 14.22636pt\text{interaction length}.$

In $d=2+1$, vortex solitons behave in this limit like classical point-
particles interacting through the exchange of virtual mediators over a finite
distance. The resulting, manifestly relativistic, effective description can
then be used to calculate a variety of physical observables, such as vortex-
vortex scattering amplitudes, using the standard Feynman diagram expansion. We
then proceed by expanding in the limit of small relative velocity and
construct a further non-relativistic effective description in which the
mediator modes are integrated out generating an effective instantaneous
potential. The main result of this paper is (4.3) an explicit analytic
expression for the vortex-vortex potential at $\mathcal{O}\left(N^{2}\right)$
but with exact velocity dependence. All calculations in this paper are made in
the so-called critical limit, defined as the boundary between the type-I and
type-II regions, in which the static vortex-vortex forces cancel
deVega:1976xbp ; Manton:2002wb .

Some important technical methods used in this paper are drawn from recent
exciting progress concerning the post-Minkowskian two-body problem in general
relativity using scattering amplitudes Cheung:2018wkq ; Bjerrum-Bohr:2018xdl ;
Bern:2019crd . In that problem, compact astrophysical bodies, black holes and
neutron stars, are modelled initially as the elementary excitations of a
massive quantum field interacting through the exchange of virtual gravitons
Neill:2013wsa . The loop integrands of the corresponding quantum scattering
amplitudes can then be constructed using a combination of powerful, modern on-
shell methods including generalized unitarity Bern:1994zx ; Bern:1994cg  and
the double-copy Kawai:1985xq ; Bern:2008qj ; Bern:2010ue . Finite size effects
Cheung:2020sdj ; Bern:2020uwk  and spin degrees-of-freedom Vaidya:2014kza ;
Bern:2020buy  can be incorporated in a systematic way. One of the most
important innovations has been an understanding of how to efficiently extract
the contributions relevant for classical observables by expanding the
amplitudes before integration using the method of regions Beneke:1997zp . In
particular, this allows for the evaluation of the classical part of the
scattering amplitude several orders beyond what is possible in the full
quantum problem Bern:2019nnu ; Bern:2021dqo ; Bern:2021yeh . In this paper we
will adopt this framework, suitably modified for a gapped system in $2+1$
dimensions, to describe the point-particle vortex effective theory.

The outline of this paper is as follows. In Section 2 we review some basic
properties of the Abelian Higgs model, the so-called critical limit, and the
static vortex soliton solution. In Section 3 the relativistic point-particle
EFT is defined and in Section 3.4 the cubic vortex-mediator interaction Wilson
coefficients are calculated by the matching of probe amplitudes. In Section
3.3 the constraints of a hidden supersymmetry are determined using an on-shell
superspace and found to agree with the proble calculation. In Section 3.5 a
scheme for calculating the non-linear vortex solution from scattering
amplitudes is developed and found to be consistent with the known perturbative
solution up to one-loop. In Section 4 the non-relativistic EFT is described
and in Sections 3.4 and 3.5, used to calculate the vortex-vortex potential.
Subsequently, in Section 4.4 from the potential is obtained a perturbative
approximation for the metric on the 2-vortex moduli space. Further technical
details on the velocity resummation of soft integrals and the Fourier
transforms of certain loop integrals are deferred to Appendices C and D
respectively.

##  2 Static ANO Vortex

###  2.1 Review: Critical Abelian Higgs Model and the ANO Vortex

In the context of this paper the full theory or UV completion, is the
classical Abelian Higgs model (AHM) in $d=2+1$, defined in our conventions by
the action

$S_{\text{AHM}}[\phi,A_{\mu}]=\int\text{d}^{3}x\left[-\frac{1}{4}F_{\mu\nu}^{2}+|D_{\mu}\phi|^{2}-\frac{\mu^{2}}{8}\left(|\phi|^{2}-v^{2}\right)^{2}\right],$
(1)

where $D_{\mu}=\partial_{\mu}+ieA_{\mu}$. For $v^{2}>0$ the $U(1)$ gauge
symmetry is spontaneously broken, the model develops a mass gap and the
corresponding potential falls off exponentially at large distances. Expanding
around the degenerate vacuum

$\phi(x)=\left(v+\frac{\sigma(x)}{\sqrt{2}}\right)e^{i\pi(x)/v},$ (2)

the physical degrees of freedom and their interactions become manifest in
unitary gauge

$\displaystyle S_{\text{AHM}}\left[\sigma,A_{\mu}\right]$
$\displaystyle=\int\text{d}^{3}x\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+e^{2}v^{2}A_{\mu}A^{\mu}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}-\frac{\mu^{2}v^{2}}{4}\sigma^{2}\right.$
$\displaystyle\hskip 56.9055pt\left.+\sqrt{2}e^{2}v\sigma
A_{\mu}A^{\mu}+\frac{e^{2}}{2}\sigma^{2}A_{\mu}A^{\mu}-\frac{\mu^{2}v}{4\sqrt{2}}\sigma^{3}-\frac{\mu^{2}}{32}\sigma^{4}\right].$
(3)

In this form we can read off the perturbative spectrum in the broken phase, it
consists of a massive vector boson $A_{\mu}$, a massive photon, and a massive
real scalar boson $\sigma$, a Higgs boson, with corresponding masses

$m_{\sigma}=\frac{v\mu}{\sqrt{2}},\hskip 28.45274ptm_{A}=\sqrt{2}ev.$ (4)

In addition to these elementary particle degrees of freedom, the AHM also
admits a spectrum of particle-like topological soliton solutions called
Abrikosov-Nielson-Oleson (ANO) vortices Nielsen:1973cs ; ABRIKOSOV1957199 .
The static vortex solution is defined as a time-independent, finite energy
solution of the classical equations of motion. The solutions are labelled by a
toplogical charge, the so-called winding number, defined as the degree of the
map

$\phi_{\infty}:S_{\infty}^{1}\rightarrow U(1)\cong S^{1},$ (5)

from spatial infinity to the degenerate vacuum manifold. Equivalently, the ANO
vortex is charged under the magnetic 1-form symmetry corresponding to the
conserved 2-form current $\tilde{F}_{\mu\nu}$; the winding number $N$ can be
defined in a gauge invariant way as an integral of the magnetic flux

$N\equiv-\frac{e}{2\pi}\int\text{d}^{2}\mathbf{x}\;F_{12}(\mathbf{x}).$ (6)

At long-distances, the static force between a pair of identical vortices is
attractive in the type-I regime ($\mu<2e$ or $m_{\sigma}<m_{A}$) and repulsive
in the type-II regime ($\mu>2e$ or $m_{\sigma}>m_{A}$). In general, the form
of the classical solution and quantitative information about the dynamics of
vortex-vortex interactions can only be determined numerically Moriarty:1988fx
; Shellard:1988zx ; Myers:1991yh  or analytically to leading asymptotic order
Nielsen:1973cs ; Speight:1996px .

The situation is a little simpler in the so-called critical limit
corresponding in our conventions to the case $\mu=2e$ or

$m_{\sigma}=m_{A}\equiv m.$ (7)

In this limit the mass of the vortex can be calculated from a functional of
the Bogomolny form Bogomolny:1975de ; PhysRevLett.35.760

$M=\int\text{d}^{2}\mathbf{x}\;\left[\frac{1}{4}\left(F_{12}-e\left(|\phi|^{2}-v^{2}\right)\right)^{2}+\frac{1}{2}|D_{1}\phi+iD_{2}\phi|^{2}\right]+\pi
v^{2}N,$ (8)

with the critical or BPS vortex solution defined as the solution of the first
order BPS equations

$F_{12}=e\left(|\phi|^{2}-v^{2}\right),\hskip
28.45274ptD_{1}\phi+iD_{2}\phi=0,$ (9)

and the mass of the winding number $N$ solution given by

$M=\pi v^{2}N.$ (10)

It can be shown that the BPS equations admit static multi-soliton solutions
and therefore the static force between identical, critical vortices exactly
vanishes Taubes:1979tm .

Many of these simplifying features of the critical AHM can be understood as a
consequence of the fact that the model is a consistent truncation of an
$\mathcal{N}=2$ supersymmetric version of the AHM deVega:1976xbp ;
Edelstein:1993bb . This perspective and the consequences for our point-
particle EFT construction are described in detail in Section 3.3. For the rest
of the paper we will restrict our attention to the critical case, leaving the
generalization to the non-critical case to future work.

###  2.2 Perturbative Solution of the BPS Equations

Even in the critical limit an analytic expression for the static vortex
solution is unknown in closed form. For our purposes in this paper, it is
sufficient to derive a perturbative solution for the BPS equations by a small
modification of the approach described in deVega:1976xbp . We begin with an
Ansatz for the winding number $N$ vortex in temporal gauge

$\phi(\mathbf{x})=\rho(r)e^{iN\theta},\hskip
14.22636ptA_{i}(\mathbf{x})=\frac{\epsilon_{ij}x_{j}}{r^{2}}A(r),\hskip
14.22636ptA_{0}(\mathbf{x})=0.$ (11)

Inserting this Ansatz into the BPS equations (9) gives a coupled system of
first-order differential equations for the profile functions

$\displaystyle\frac{A^{\prime}(r)}{r}+e\left(\rho(r)^{2}-v^{2}\right)=0,\hskip
14.22636pt\rho^{\prime}(r)+\frac{\rho(r)}{r}\left(eA(r)-N\right)=0.$ (12)

We can solve the first of these equations,

$\rho(r)=\left(v^{2}-\frac{1}{e}\frac{A^{\prime}(r)}{r}\right)^{1/2},$ (13)

where the branch of the square root is fixed by the boundary condition
$\rho\rightarrow v$ as $r\rightarrow\infty$. Inserting this into the second
BPS equation

$A^{\prime\prime}(r)+\frac{2eA^{\prime}(r)A(r)}{r}-\left(2N+1\right)\frac{A^{\prime}(r)}{r}-2e^{2}v^{2}A(r)+2Nev^{2}=0.$
(14)

To solve this differential equation we need appropriate boundary conditions.
We impose that the solution is smooth near $r\sim 0$, this requires
$A(0)=A^{\prime}(0)=0$. Similarly, asymptotically expanding (14) we find that
an asymptotically decaying solution is consistent only if $A(r)\rightarrow
N/e$ as $r\rightarrow\infty$. Introducing dimensionless variables

$a(\xi)\equiv eA(r)-N,\hskip 14.22636pt\xi=\sqrt{2}evr,$ (15)

determining the static vortex solution is equivalent to solving the following
second-order, non-linear boundary value problem

$\left(\frac{d^{2}}{d\xi^{2}}-\frac{1}{\xi}\frac{d}{d\xi}-1\right)a(\xi)=-\frac{2a(\xi)}{\xi}\frac{da(\xi)}{d\xi},\hskip
28.45274pta(0)=-N,\hskip 28.45274pta(\xi\rightarrow\infty)=0.$ (16)

To proceed we follow the approach of deVega:1976xbp  and rewrite this in the
form of an integral equation. A Green’s function for the linear differential
operator appearing on the left-hand-side

$\left(\frac{d^{2}}{d\xi^{2}}-\frac{1}{\xi}\frac{d}{d\xi}-1\right)G(\xi,\xi^{\prime})=\delta\left(\xi-\xi^{\prime}\right),$
(17)

is found to be

$G(\xi,\xi^{\prime})=\begin{cases}-\xi
K_{1}(\xi)I_{1}(\xi^{\prime})&\xi^{\prime}<\xi\\\ -\xi
I_{1}(\xi)K_{1}(\xi^{\prime})&\xi^{\prime}>\xi\end{cases},$ (18)

where $K_{\nu}$ and $I_{\nu}$ are modified Bessel functions. Imposing the
boundary conditions in (16) we derive the following non-linear integral
equation

$\displaystyle a(\xi)$ $\displaystyle=-N\xi K_{1}(\xi)+2\xi
K_{1}(\xi)\int_{0}^{\xi}\text{d}\xi^{\prime}\;I_{1}(\xi^{\prime})\left[\frac{a(\xi^{\prime})}{\xi^{\prime}}\frac{da(\xi^{\prime})}{d\xi^{\prime}}\right]$
$\displaystyle\hskip 34.1433pt+2\xi
I_{1}(\xi)\int_{\xi}^{\infty}\text{d}\xi^{\prime}\;K_{1}(\xi^{\prime})\left[\frac{a(\xi^{\prime})}{\xi^{\prime}}\frac{da(\xi^{\prime})}{d\xi^{\prime}}\right].$
(19)

To perturbatively solve this equation we make use of the small winding
expansion Ohashi:2015yta . The idea is to continue the winding number from the
physical quantized values $N\in\mathds{Z}$ to arbitrary real values
$N\in\mathds{R}$ and perturbatively expand around the limit $N\rightarrow 0$.
We will postpone a discussion of the physical interpretation of this expansion
to Section 3.1. On the right-hand-side of (2.2) we see that the first term is
$\mathcal{O}\left(N\right)$, while the remaining terms are quadratic in
$a(\xi)$ and therefore $\mathcal{O}\left(N^{2}\right)$. So by iterating this
integral equation we derive a formal series expansion for the solution

$\displaystyle a(\xi)$
$\displaystyle=-\left[N+\frac{\pi}{3\sqrt{3}}N^{2}\right]\xi K_{1}(\xi)$
$\displaystyle\hskip 14.22636pt+2N^{2}\xi
K_{1}(\xi)\int_{\xi}^{\infty}\text{d}\xi^{\prime}\;\xi^{\prime}I_{1}(\xi^{\prime})K_{0}(\xi^{\prime})K_{1}(\xi^{\prime})-2N^{2}\xi
I_{1}(\xi)\int_{\xi}^{\infty}\text{d}\xi^{\prime}\;\xi^{\prime}K_{0}(\xi^{\prime})\left[K_{1}(\xi^{\prime})\right]^{2}$
$\displaystyle\hskip 14.22636pt+\mathcal{O}\left(N^{3}\right).$ (20)

It is instructive to compare (2.2) with the asymptotic expansion around the
limit $\xi\rightarrow\infty$ derived in deVega:1976xbp  which takes the form

$a(\xi)=-Z_{N}\xi K_{1}(\xi)+\mathcal{O}\left(e^{-2\xi}\right),$

and extract an analytic perturbative expansion for the coefficient $Z_{N}$.
The first few orders have a simple form

$Z_{N}=N+\frac{\pi}{3\sqrt{3}}N^{2}+\frac{\pi^{2}}{108}N^{3}+\mathcal{O}\left(N^{4}\right).$
(21)

The usefulness of this expansion for physical vortices depends on the radius
of convergence of the series, in particular whether or not it gives a good
approximation for small integer values. Remarkably, it was shown in
Ohashi:2015yta  that for $N=1$ the small winding series (21), truncated at
$\mathcal{O}\left(N^{6}\right)$, quantitatively reproduces independent
numerical calculations of $Z_{1}\approx 1.707864...$ with a numerical
discrepancy of $\mathcal{O}\left(10^{-4}\right)$. With the use of suitable
(global) Padé approximants this numerical discrepancy is further reduced to
$\mathcal{O}\left(10^{-6}\right)$.

Inserting (2.2) into the Ansatz (11) and transforming to unitary gauge we find
the following perturbative expansion of the static vortex solution

$\displaystyle\sigma(\mathbf{x})$ $\displaystyle=\sqrt{\frac{2M}{\pi
N}}\left[-\left(N+\frac{\pi}{3\sqrt{3}}N^{2}\right)K_{0}(mr)-\frac{N^{2}}{2}\left[K_{0}(mr)\right]^{2}\right.$
$\displaystyle\hskip
99.58464pt+2N^{2}K_{0}(mr)\int_{mr}^{\infty}\text{d}\xi\;\xi
I_{1}(\xi)K_{0}(\xi)K_{1}(\xi)$ $\displaystyle\hskip
99.58464pt+\left.2N^{2}I_{0}(mr)\int_{mr}^{\infty}\text{d}\xi\;\xi
K_{0}(\xi)\left[K_{1}(\xi)\right]^{2}\right]+\mathcal{O}\left(N^{5/2}\right)$
$\displaystyle A_{i}(\mathbf{x})$
$\displaystyle=\frac{1}{m}\sqrt{\frac{2M}{\pi
N}}\frac{\epsilon_{ij}x_{j}}{r^{2}}\left[-\left(N+\frac{\pi}{3\sqrt{3}}N^{2}\right)mrK_{1}(mr)\right.$
$\displaystyle\hskip
99.58464pt+\left.2N^{2}mrK_{1}(mr)\int_{mr}^{\infty}\text{d}\xi\;\xi
I_{1}(\xi)K_{0}(\xi)K_{1}(\xi)\right.$ $\displaystyle\hskip
99.58464pt\left.-2N^{2}mrI_{1}(mr)\int_{mr}^{\infty}\text{d}\xi\;\xi
K_{0}(\xi)\left[K_{1}(\xi)\right]^{2}\right]+\mathcal{O}\left(N^{5/2}\right).$
(22)

In Section 3.5 we will rederive these expressions using Feynman diagrams.

##  3 Relativistic Effective Field Theory

###  3.1 Physical Scales and the Small Winding Expansion

There are three relevant length scales in the problem: the quantum size of the
vortex given by the Compton wavelength

$R_{\text{Compton}}\sim\frac{1}{M},$ (23)

the characteristic interaction length between the vortices set by the mediator
masses

$R_{\text{interaction}}\sim\frac{1}{m},$ (24)

and the classical or core size of the vortex soliton111 In Ohashi:2015yta  a
precise definition of the core size is given as
$R_{\text{core}}\equiv\sqrt{2\times\frac{\int\text{d}^{2}\mathbf{x}|\mathbf{x}|^{2}\mathcal{H}(\mathbf{x})}{\int\text{d}^{2}\mathbf{x}\mathcal{H}(\mathbf{x})}},$
(25) where $\mathcal{H}(\mathbf{x})$ is the energy density of the soliton, and
the scaling behaviour (26) derived in detail.

$R_{\text{core}}\sim\frac{\sqrt{N}}{m}.$ (26)

The $\sqrt{N}$-scaling of the core size is equivalent to the statement that
the density of magnetic flux per unit area in the core is roughly constant as
a function of $N$ Penin:2020cxj ; Penin:2021xgr . For physical vortices,
$N\in\mathds{Z}$ and there is no parametric separation of scales between the
interaction length and the core size. Physically, this is equivalent to the
statement that in an effective description, valid at distances $r\gg
R_{\text{core}}$ where the vortex appears as a point-particle, there are only
contact interactions. Conversely, at the scale $r\sim R_{\text{interaction}}$
where the vortices interact by exchanging mediator particles, they are not
well described as point-particles.

In this paper, following Ohashi:2015yta , we will proceed by continuing from
quantized integer values of the winding number $N$ to arbitrary real values
and expand around the limit $N\rightarrow 0$. In this regime we instead have a
hierarchy of scales of the form

$R_{\text{core}}\ll R_{\text{interaction}},$ (27)

this means we can try and construct an EFT description at the scale $r\sim
R_{\text{interaction}}$, where the vortices behave as point-like particles
interacting by exchanging virtual mediator particles. Furthermore, we are
interested in the regime in which the vortices can be treated as semi-
classical solitons rather than elementary quantum particles, this means we are
also assuming

$R_{\text{Compton}}\ll R_{\text{core}}.$ (28)

Imposing both (27) and (28) simultaneously means that at the scale $r\sim
R_{\text{interaction}}$ the vortices have an effective description as
classical point-like particles of mass $M$ interacting by exchanging mediator
particles of mass $m$ and we have a double hierarchy of the form

$\frac{1}{M}\hskip 5.69054pt\ll\hskip 5.69054pt\frac{\sqrt{N}}{m}\hskip
5.69054pt\ll\hskip 5.69054pt\frac{1}{m}\hskip 5.69054pt\sim\hskip 5.69054ptr.$
(29)

The systematics of implementing this double expansion to construct an EFT is
described in the following section. In practice we implement this expansion in
two steps: first expanding in $N\ll 1$, corresponding to the usual Feynman
diagrammatic loop expansion, and then subsequently expanding in
$\frac{m}{M}\ll 1$ corresponding to a classical or $\hbar$-expansion.

###  3.2 Definition of the REFT

The approach taken to constructing an EFT description of vortices in regime
(29) is inspired by recent works on the post-Minkowskian expansion of the two-
body problem in classical general relativity Cheung:2018wkq ; Bern:2019crd .
We begin by constructing a relativistic field theory in which the classical
point-particle vortices are identified as the quanta of a complex scalar
field.222The “point-particle” effective field theory described in this section
is strongly reminiscent of the well established notion of particle-vortex
duality Peskin:1977kp ; Dasgupta:1981zz ; Karch:2016sxi ; Seiberg:2016gmd .
The key difference is the additional separation of scales (27). In the REFT
the vortex field $\Phi$ self-interacts over a finite distance $r\sim m^{-1}$
by exchanging mediator particles, whereas in the dual XY-model description,
vortex-vortex interactions correspond to contact terms of the form
$|\Phi|^{4}$ as appropriate for an effective description valid at distances
$r\gg m^{-1}$. It would be very interesting to understand the connection
between these two descriptions in more detail. To distinguish this
construction from the non-relativistic effective description given in Section
4.1, in which we additionally expand in small velocities, we will refer to
this as the relativistic EFT or REFT. The general form the of the effective
action is

$S_{\text{REFT}}\left[\Phi,\sigma,A_{\mu}\right]=S_{\text{AHM}}\left[\sigma,A_{\mu}\right]+S_{\text{vortex}}\left[\Phi,\sigma,A_{\mu}\right].$
(30)

It is convenient to rewrite the Lagrangian parameters of the AHM (2.1) in
terms of the directly relevant mass scales and the winding number

$e=m\sqrt{\frac{\pi N}{2M}},\hskip 28.45274ptv=\sqrt{\frac{M}{\pi N}},\hskip
28.45274pt\mu=m\sqrt{\frac{2\pi N}{M}}.$ (31)

In terms of which the AHM action takes the form

$\displaystyle S_{\text{AHM}}[\sigma,A_{\mu}]$
$\displaystyle=\int\text{d}^{3}x\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}A_{\mu}A^{\mu}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}-\frac{1}{2}m^{2}\sigma^{2}+\sqrt{\frac{\pi}{2}}M^{3/2}\left(\frac{m}{M}\right)^{2}N^{1/2}\sigma
A_{\mu}A^{\mu}\right.$ $\displaystyle\hskip
48.36967pt\left.-\sqrt{\frac{\pi}{8}}M^{3/2}\left(\frac{m}{M}\right)^{2}N^{1/2}\sigma^{3}+\frac{\pi}{4}M\left(\frac{m}{M}\right)^{2}N\sigma^{2}A_{\mu}A^{\mu}-\frac{\pi}{16}M\left(\frac{m}{M}\right)^{2}N\sigma^{4}\right],$
(32)

for reference the Feynman rules associated with this action are given in
Appendix B. The term denoted $S_{\text{vortex}}$ contains the vortex kinetic
term

$S^{(2)}_{\text{vortex}}[\Phi]=\int\text{d}^{3}x\biggr{[}|\partial_{\mu}\Phi|^{2}-M^{2}|\Phi|^{2}\biggr{]},$
(33)

as well as the interactions between the vortex and the mediator particles.
These are a priori unknown and must be determined by matching an observable
with full theory. In addition to Lorentz invariance, the REFT is invariant
under a global $U(1)$ symmetry, $\Phi\rightarrow e^{i\alpha}\Phi$, the
corresponding Noether charge is identified with the topological winding number
charge in the full theory. We begin with cubic vortex-mediator interactions,
which we expect will dominate at large distances. Up to field redefinition and
total derivatives there are three possible interactions

$\displaystyle S^{(3)}_{\text{vortex}}\left[\sigma,A_{\mu},\Phi\right]$
$\displaystyle=\int\text{d}^{3}x\left[g_{s}M^{3/2}N^{1/2}\sigma|\Phi|^{2}\right.$
$\displaystyle\hskip
51.21495pt+ig_{e}M^{1/2}N^{1/2}A_{\mu}\left(\Phi\partial^{\mu}\Phi^{*}-\Phi^{*}\partial^{\mu}\Phi\right)$
$\displaystyle\hskip
51.21495pt+\left.ig_{m}M^{-1/2}N^{1/2}\left(\frac{m}{M}\right)^{-1}\epsilon^{\mu\nu\rho}F_{\mu\nu}\left(\Phi\partial_{\rho}\Phi^{*}-\Phi^{*}\partial_{\rho}\Phi\right)\right].$
(34)

As we will see in Sections 3.4 and 3.5, the cubic vortex-mediator interactions
completely determine the static structure of the vortex soliton. As a
consequence these interactions should be invariant under the same symmetries
as the solution (11). We will therefore impose that the cubic interactions are
invariant under parity acting as

$\displaystyle
A_{\mu}(x)\xrightarrow[]{\text{P}}-{\mathcal{P}_{\mu}}^{\nu}A_{\nu}(\mathcal{P}\cdot
x),\hskip 14.22636pt\sigma(x)\xrightarrow[]{\text{P}}\sigma(\mathcal{P}\cdot
x),\hskip 14.22636pt\Phi(x)\xrightarrow[]{\text{P}}\Phi(\mathcal{P}\cdot x),$
(35)

where ${\mathcal{P}_{\mu}}^{\nu}\equiv\text{diag}\left(1,-1,1\right)$. This is
a symmetry of (3.2) only if the “electric” coupling vanishes, $g_{e}=0$.333
This interaction can be generated by adding a Chern-Simons term to the AHM,
which explicitly breaks parity invariance, and would source the $A_{r}$
component of the static soliton solution Paul:1986ix .

In (3.2) we have chosen a convenient normalization for the Wilson coefficients
that ensures uniform scaling of interactions in the soft region defined in
Section 4.1. Furthermore we have assumed that the cubic vortex-mediator
coupling scales as $\sim N^{1/2}$. Combined, with the explicit $N$-scaling of
the interactions in (3.2), this ensures that the $N$-expansion aligns with the
usual diagrammatic loop expansion.

We expect that the REFT must also contain higher-multiplicity interactions
between the vortex and $n$-mediator fields with $n\geq 2$. These interactions
do not contribute to the static structure, rather they encode the response of
the finite size soliton to external perturbations, in the gravitational
context such Wilson coefficients are usually called Love numbers Bini:2020flp
; Cheung:2020sdj ; Bern:2020uwk . In Section 4.3 we will see that certain
quartic or “seagull” interactions must necessarily be non-zero to ensure the
expected cancellation of static vortex-vortex forces in the critical limit.
Otherwise we will leave a determination of such finite size effects to future
work.

Finally, as discussed briefly in Section 2.1 the critical limit of the AHM in
$d=2+1$ corresponds to a consistent truncation of a model with $\mathcal{N}=2$
supersymmetry deVega:1976xbp ; Edelstein:1993bb . Even though this larger
model contains additional degrees of freedom that are eventually decoupled,
the enlarged symmetry implies non-trivial constraints on the self-interactions
between the states that remain. Importantly, this residual supersymmetry
should also constrain the, a priori unknown, interactions between the mediator
fields and the vortices in the point-particle REFT. Especially at higher-
multiplicity, this hidden symmetry will greatly reduce the number of Wilson
coefficients that need to be matched between full theory and the REFT. In the
following subsection we will determine these constraints at cubic order by
explicitly constructing the corresponding superamplitudes.

###  3.3 Supersymmetry Constraints

We begin by constructing an on-shell representation of the $\mathcal{N}=2$
supersymmetry algebra; it is convenient to write this in a form where the two
real valued supercharges are combined into a single complex valued supercharge

$\\{Q_{a},Q^{\dagger}_{b}\\}=2\sigma_{ab}^{\mu}P_{\mu}-2iZ\epsilon_{ab},\hskip
14.22636pt\\{Q_{a},Q_{b}\\}=0,\hskip
14.22636pt\\{Q_{a}^{\dagger},Q_{b}^{\dagger}\\}=0,$ (36)

where we have included a central charge $Z$ corresponding to the winding
number Edelstein:1993bb . We would like to write down a representation of this
algebra on the space of (outgoing) one-particle states with momentum $p^{\mu}$
and mass $m>0$. For this purpose it is convenient to introduce massive spinor
variables $\lambda_{a}$ and $\tilde{\lambda}_{a}$ defined in Appendix A, and
form the little group covariant supercharges

$q=\frac{1}{\sqrt{2}m}\lambda^{a}Q_{a},\hskip
14.22636pt\tilde{q}=\frac{1}{\sqrt{2}m}\tilde{\lambda}^{a}Q_{a},\hskip
14.22636ptq^{\dagger}=\frac{1}{\sqrt{2}m}\tilde{\lambda}^{a}Q^{\dagger}_{a},\hskip
14.22636pt\tilde{q}^{\dagger}=\frac{1}{\sqrt{2}m}\lambda^{a}Q_{a}^{\dagger},$
(37)

satisfying the simpler algebra

$\\{q,q^{\dagger}\\}=1-\frac{Z}{m},\hskip
28.45274pt\\{\tilde{q},\tilde{q}^{\dagger}\\}=1+\frac{Z}{m},$ (38)

with all other anti-commutators vanishing. In this form it is simple to write
down a representation of this algebra as multiplicative and differential
operators on the space of Grassmann polynomials in two variables $\eta$ and
$\tilde{\eta}$

$\displaystyle q=\left(1-\frac{Z}{m}\right)^{1/2}\eta,\hskip
14.22636ptq^{\dagger}=\left(1-\frac{Z}{m}\right)^{1/2}\frac{\partial}{\partial\eta},$
$\displaystyle\tilde{q}=\left(1+\frac{Z}{m}\right)^{1/2}\tilde{\eta},\hskip
14.22636pt\tilde{q}^{\dagger}=\left(1+\frac{Z}{m}\right)^{1/2}\frac{\partial}{\partial\tilde{\eta}}.$
(39)

Translating back to the original spinor supercharges we have a representation
of the superalgebra

$\displaystyle Q_{a}$
$\displaystyle=i\sqrt{2}\left[\left(1-\frac{Z}{m}\right)^{1/2}\tilde{\lambda}_{a}\eta-\left(1+\frac{Z}{m}\right)^{1/2}\lambda_{a}\tilde{\eta}\right]$
$\displaystyle Q^{\dagger}_{a}$
$\displaystyle=-i\sqrt{2}\left[\left(1-\frac{Z}{m}\right)^{1/2}\lambda_{a}\frac{\partial}{\partial\eta}-\left(1+\frac{Z}{m}\right)^{1/2}\tilde{\lambda}_{a}\frac{\partial}{\partial\tilde{\eta}}\right].$
(40)

In this form we see that there are two special cases, if $Z=+m$ or $Z=-m$ the
dependence on $\eta$ or $\tilde{\eta}$ respectively drops out. These
correspond to the BPS and anti-BPS limits in which the supermultiplets are
shortened.

The spectrum of the $\mathcal{N}=2$ AHM (in the broken phase) consists of a
pair of long multiplets containing the mediator particles. In this on-shell
superspace formalism these correspond to the on-shell superfields444See
Elvang:2011fx ; Agarwal:2013tpa  for a general discussion of on-shell
supersymmetry methods for scattering amplitudes with less than maximal
supersymmetry.

$\Gamma=\psi^{+}+\varphi\eta+\gamma^{+}\tilde{\eta}+\tilde{\psi}^{+}\eta\tilde{\eta},\hskip
28.45274pt\overline{\Gamma}=\psi^{-}+\gamma^{-}\eta+\overline{\varphi}\tilde{\eta}+\tilde{\psi}^{-}\eta\tilde{\eta},$
(41)

related to each other by a parity transformation, but to themselves under CPT.
In this context the central charge $Z$ is identified with the winding number
of the vortex solitons, and so for the mediator particles $Z=0$ or

$\displaystyle
Q_{a}=i\sqrt{2}\left[\tilde{\lambda}_{a}\eta-\lambda_{a}\tilde{\eta}\right],\hskip
28.45274ptQ^{\dagger}_{a}=-i\sqrt{2}\left[\lambda_{a}\frac{\partial}{\partial\eta}-\tilde{\lambda}_{a}\frac{\partial}{\partial\tilde{\eta}}\right].$
(42)

It is interesting to compare this spectrum to the non-supersymmetric AHM
(2.1). In addition to the fermions $\psi^{\pm}$ and $\tilde{\psi}^{\pm}$, to
form the necessary long supermultiplets we have had to introduce two real
scalar degrees of freedom $\varphi$ and $\overline{\varphi}$, while the non-
supersymmetric model contains only a single real scalar Higgs boson $\sigma$.
It therefore must be the case that in an appropriate classical limit, one of
the scalar degrees of freedom in the $\mathcal{N}=2$ AHM consistently
decouples. By considering the action of parity it is straightforward to
identify

$\sigma=\frac{1}{\sqrt{2}}\left(\varphi+\overline{\varphi}\right),\hskip
28.45274pt\tilde{\sigma}=\frac{1}{\sqrt{2}}\left(\varphi-\overline{\varphi}\right),$
(43)

where the decoupling orthogonal linear combination $\tilde{\sigma}$ is a
pseudo-scalar.

In the $\mathcal{N}=2$ version of the REFT we introduce a BPS and anti-BPS
multiplet corresponding to the vortex and anti-vortex respectively
Intriligator:2013lca . For the vortex multiplet with $Z=M$ the supercharges
are

$Q_{a}=-2i\lambda_{a}\tilde{\eta},\hskip
28.45274ptQ_{a}^{\dagger}=2i\tilde{\lambda}_{a}\frac{\partial}{\partial\tilde{\eta}},$
(44)

and the on-shell superfield

$\Sigma=\Phi+\tilde{\eta}\Psi^{+}.$ (45)

For the anti-vortex multiplet with $Z=-M$ the supercharges are

$Q_{a}=2i\tilde{\lambda}_{a}\eta,\hskip
28.45274ptQ^{\dagger}_{a}=-2i\lambda_{a}\frac{\partial}{\partial\eta},$ (46)

and the on-shell superfield

$\overline{\Sigma}=\tilde{\Psi}^{+}+\eta\overline{\Phi}.$ (47)

From (45) and (47) we observe that the spin of the fermionic components of
both the vortex and anti-vortex superfields are the same. This means that the
supersymmetrization of the REFT must violate parity, which changes the sign of
the spin.

#### Cubic mediator interactions

As a first application we will reproduce the specific relative tuning of the
cubic interactions in the critical limit of the AHM (3.2). The most general
cubic superamplitudes we can write down have the form

$\displaystyle\mathcal{M}_{3}\left(\Gamma_{1},\Gamma_{2},\Gamma_{3}\right)$
$\displaystyle=c_{\Gamma\Gamma\Gamma}\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Gamma\Gamma}\times\biggr{[}[23]\eta_{1}+[31]\eta_{2}+[12]\eta_{3}\biggr{]}$
$\displaystyle\mathcal{M}_{3}\left(\overline{\Gamma}_{1},\Gamma_{2},\Gamma_{3}\right)$
$\displaystyle=c_{\overline{\Gamma}\Gamma\Gamma}\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Gamma\Gamma}\times\biggr{[}[23]\tilde{\eta}_{1}+\langle
13]\eta_{2}-\langle 12]\eta_{3}\biggr{]},$ (48)

where

$\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Gamma\Gamma}\equiv\sum_{i<j}\left(\langle
ij\rangle\eta_{i}\eta_{j}+[ij]\tilde{\eta}_{i}\tilde{\eta}_{j}-\langle
ij]\eta_{i}\tilde{\eta}_{j}-\langle
ji]\eta_{j}\tilde{\eta}_{i}\right)-im\sum_{i}\eta_{i}\tilde{\eta}_{i}.$ (49)

The remaining cubic superamplitudes are assumed to be related to the above by
parity

$\displaystyle\mathcal{M}_{3}\left(\overline{\Gamma}_{1},\overline{\Gamma}_{2},\overline{\Gamma}_{3}\right)$
$\displaystyle=\mathcal{M}_{3}\left(\Gamma_{1},\Gamma_{2},\Gamma_{3}\right)\biggr{|}_{\lambda\leftrightarrow\tilde{\lambda},\;\eta\leftrightarrow\tilde{\eta}}$
$\displaystyle\mathcal{M}_{3}\left(\Gamma_{1},\overline{\Gamma}_{2},\overline{\Gamma}_{3}\right)$
$\displaystyle=\mathcal{M}_{3}\left(\overline{\Gamma}_{1},\Gamma_{2},\Gamma_{3}\right)\biggr{|}_{\lambda\leftrightarrow\tilde{\lambda},\;\eta\leftrightarrow\tilde{\eta}}.$
(50)

We find that there is a two-parameter family of parity-invariant cubic
interactions, labelled by the independent coefficients
$c_{\Gamma\Gamma\Gamma}$ and $c_{\overline{\Gamma}\Gamma\Gamma}$, with the
assumed spectrum and $\mathcal{N}=2$ supersymmetry. To recover the AHM we must
also impose that the real scalar $\tilde{\sigma}$ consistently decouples.
Projecting out the component amplitudes we find a non-trivial condition

$\mathcal{M}_{3}\left(\tilde{\sigma}_{1},\gamma_{2}^{+},\gamma_{3}^{+}\right)\propto\mathcal{M}_{3}\left(\varphi_{1},\gamma_{2}^{+},\gamma_{3}^{+}\right)-\mathcal{M}_{3}\left(\overline{\varphi}_{1},\gamma_{2}^{+},\gamma_{3}^{+}\right)=\left(c_{\Gamma\Gamma\Gamma}-c_{\overline{\Gamma}\Gamma\Gamma}\right)[23]^{2}.$
(51)

Requiring that this vanishes gives
$c_{\Gamma\Gamma\Gamma}=c_{\overline{\Gamma}\Gamma\Gamma}$. It is
straightforward to check that all other component amplitudes with a single
$\tilde{\sigma}$ vanish automatically. After imposing this condition the non-
decoupled bosonic component amplitudes are found to be

$\mathcal{M}_{3}\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)=-\frac{3}{\sqrt{2}}m^{2}c_{\Gamma\Gamma\Gamma},\hskip
14.22636pt\mathcal{M}_{3}\left(\sigma_{1},\gamma_{2},\gamma_{3}\right)=\sqrt{2}m^{2}c_{\Gamma\Gamma\Gamma}\left(\epsilon_{2}\cdot\epsilon_{3}\right).$
(52)

Comparing this with the on-shell 3-particle amplitudes calculated from the
critical AHM (3.2) we find that this reproduces exactly the expected relative
tuning of coefficients where we identify

$c_{\Gamma\Gamma\Gamma}=\sqrt{\pi}M^{-1/2}N^{1/2}.$ (53)

#### Cubic vortex-mediator interactions

Next we repeat the above analysis for cubic interactions between the vortices
and mediators. Here we find that there is a two-parameter family of
superamplitudes

$\displaystyle\mathcal{M}_{3}\left(\Gamma_{1},\Sigma_{2},\overline{\Sigma}_{3}\right)$
$\displaystyle=\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Sigma\overline{\Sigma}}\times\left[c_{\Gamma\Sigma\overline{\Sigma}}\;[13\rangle\right],$
$\displaystyle\mathcal{M}_{3}\left(\overline{\Gamma}_{1},\Sigma_{2},\overline{\Sigma}_{3}\right)$
$\displaystyle=\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Sigma\overline{\Sigma}}\times\left[c_{\overline{\Gamma}\Sigma\overline{\Sigma}}\;\langle
13\rangle\right],$ (54)

where

$\left[\delta^{(2)}\left(Q\right)\right]_{\Gamma\Sigma\overline{\Sigma}}\equiv-
im\eta_{1}\tilde{\eta}_{1}+\sqrt{2}\langle
12\rangle\eta_{1}\tilde{\eta}_{2}-\sqrt{2}\langle
13]\eta_{1}\eta_{3}-\sqrt{2}[12\rangle\tilde{\eta}_{1}\tilde{\eta}_{2}+\sqrt{2}[13]\tilde{\eta}_{1}\eta_{3}+2\langle
23]\tilde{\eta}_{2}\eta_{3}.$ (55)

Projecting out the bosonic component amplitudes we find

$\displaystyle\mathcal{M}_{3}\left(\varphi_{1},\Phi_{2},\overline{\Phi}_{3}\right)=\frac{c_{\Gamma\Sigma\overline{\Sigma}}}{2\sqrt{2}}\;m\left(m-2M\right),\hskip
28.45274pt\mathcal{M}_{3}\left(\overline{\varphi}_{1},\Phi_{2},\overline{\Phi}_{3}\right)=-\frac{c_{\overline{\Gamma}\Sigma\overline{\Sigma}}}{2\sqrt{2}}\;m\left(m+2M\right)$
$\displaystyle\mathcal{M}_{3}\left(\gamma^{+}_{1},\Phi_{2},\overline{\Phi}_{3}\right)=-\sqrt{2}c_{\Gamma\Sigma\overline{\Sigma}}\;[13]\langle
31],\hskip
32.72066pt\mathcal{M}_{3}\left(\gamma^{-}_{1},\Phi_{2},\overline{\Phi}_{3}\right)=\sqrt{2}c_{\overline{\Gamma}\Sigma\overline{\Sigma}}\;\langle
13]\langle 31\rangle.$ (56)

Here we encounter an interesting puzzle. If we insist that the cubic vortex-
photon interaction takes the “magnetic” form (3.2) then the couplings must be
related as
$c_{\Gamma\Sigma\overline{\Sigma}}=c_{\overline{\Gamma}\Sigma\overline{\Sigma}}$.
But this choice gives the following values for the cubic vortex-scalar
interactions

$\mathcal{M}_{3}\left(\sigma_{1},\Phi_{2},\overline{\Phi}_{3}\right)=-mMc_{\Gamma\Sigma\overline{\Sigma}},\hskip
28.45274pt\mathcal{M}_{3}\left(\tilde{\sigma}_{1},\Phi_{2},\overline{\Phi}_{3}\right)=\frac{1}{2}m^{2}c_{\Gamma\Sigma\overline{\Sigma}},$
(57)

so we find that the pseudoscalar $\tilde{\sigma}$ does not decouple. We do
observe however that the $\tilde{\sigma}$ coupling is suppressed by an
additional power of $\frac{m}{M}$ relative to the $\sigma$ coupling, and so it
will decouple in the classical limit; we will revisit this observation in
Section 4.2. Matching with the general form of the cubic interactions in the
REFT, this superspace analysis implies the following relation between Wilson
coefficients

$g_{s}=4g_{m}.$ (58)

In Section 3.4 we will verify this prediction by explicitly matching the REFT
with the perturbative vortex solution.

###  3.4 Matching with Full Theory: Probe Scattering

To determine the Wilson coefficients in the REFT we need to match an
observable with full theory. As we will see the cubic vortex-mediator
interactions (3.2) are fixed by matching a probe amplitude in both
descriptions. We introduce a fictitious complex scalar probe particle $\chi$
with scalar and magnetic type cubic interactions with the mediator fields

$\displaystyle
S_{\text{Probe}}[\chi,\sigma,A_{\mu}]=\int\text{d}^{3}x\biggr{[}|\partial_{\mu}\chi|^{2}-\mathfrak{m}^{2}|\chi|^{2}+\mathfrak{g}_{s}\sigma|\chi|^{2}+i\mathfrak{g}_{m}\epsilon^{\mu\nu\rho}F_{\mu\nu}\left(\chi^{*}\partial_{\rho}\chi-\chi\partial_{\rho}\chi^{*}\right)\biggr{]},$
(59)

where $\mathfrak{g}_{s,m}$ can be taken to be arbitrarily small. To calculate
a probe amplitude in full theory we begin by calculating an off-shell two-
point function in a vortex background

$\langle\chi^{*}(x_{1})\chi(x_{2})\rangle_{\overline{\sigma},\overline{A}_{\mu}}=\int\mathcal{D}\chi\mathcal{D}\sigma\mathcal{D}A_{\mu}\left[\chi^{*}(x_{1})\chi(x_{2})\right]e^{iS_{\text{AHM}}[\overline{\sigma}+\sigma,\overline{A}_{\mu}+A_{\mu}]+iS_{\text{Probe}}[\chi,\overline{\sigma}+\sigma,\overline{A}_{\mu}+A_{\mu}]},$
(60)

with the usual normalization factor implicit. We calculate this two-point
function perturbatively in $N$, where the expansion of the background, here
denoted $\overline{\sigma}$, $\overline{A}_{\mu}$, is given by iteratively
solving the BPS equations (2.2), working to leading-order in the probe
couplings or $\mathcal{O}\left(\mathfrak{g}_{s,m}\right)$. Terms in
$S_{\text{AHM}}$ that are zeroth order in the perturbation $\sigma,A_{\mu}$
(depend only on the background fields) are constants that cancel in the
normalization of the path integral so we can ignore them. Terms that are first
order in the perturbation vanish since the background fields are assumed to
satisfy the equations of motion. Therefore we need only terms that are
quadratic or higher in the perturbation.

We want to extract from this expression an on-shell 1-to-1 scattering
amplitude using the standard LSZ formula

$\displaystyle
i(2\pi)\delta\left(E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}\right)\mathcal{M}_{\chi\rightarrow\chi}^{\text{(Full)}}\left(\mathbf{p},\mathbf{q}\right)$
(61)
$\displaystyle=(-i)^{2}[p_{1}^{2}-\mathfrak{m}^{2}+i0][p_{2}^{2}-\mathfrak{m}^{2}+i0]\int\text{d}^{3}x_{1}\text{d}^{3}x_{2}e^{-i\left(p_{1}\cdot
x_{1}+p_{2}\cdot
x_{2}\right)}\langle\chi^{*}(x_{1})\chi(x_{2})\rangle_{\overline{\sigma},\overline{A}_{\mu}}\biggr{|}_{\begin{subarray}{c}p_{1}\rightarrow(E_{\mathbf{p}+\mathbf{q}},\mathbf{p}+\mathbf{q})\\\
p_{2}\rightarrow(-E_{\mathbf{p}},-\mathbf{p})\hskip 8.53581pt\end{subarray}}.$

$\times$$p$$p+q$ (a)

$l$$\times$$\times$ (b)

Figure 1: Diagram topologies for probe two-point function
$\langle\chi^{*}\chi\rangle$ with $\overline{\sigma}$ and $\overline{A}_{\mu}$
background insertions (denoted with $\times$) at: (a) tree-level and (b) one-
loop. The one-loop diagrams give a vanishing contribution in the potential
region.

At leading order, $\mathcal{O}\left(N^{1/2}\right)$, there is only a single
diagram topology given in Figure 1(a), corresponding to an insertion of the
leading-order perturbative solution (2.2). At next-to-leading order,
$\mathcal{O}\left(N^{3/2}\right)$, there are in principle two different types
of contribution; again from the tree-level diagram depicted in Figure 1(a)
with an insertion of the $\mathcal{O}\left(N^{3/2}\right)$ perturbative
solution, but also from the loop-level diagrams depicted in Figure 1(b) with
insertions of the $\mathcal{O}\left(N^{1/2}\right)$ solution. We can avoid
calculating such diagrams by expanding the corresponding loop integrals (in
both the full-theory and REFT calculations) in the classical potential region
Bern:2019crd .

To define this region we need to specify our choice of scattering kinematics;
we will work in the frame in which the soliton is at rest and therefore the
momentum transfer is purely spatial $q^{\mu}=\left(0,\mathbf{q}\right)$.
Denoting the incoming momentum as
$p^{\mu}=\left(E_{\mathbf{p}},\mathbf{p}\right)$ where
$E_{\mathbf{p}}\equiv\sqrt{\mathbf{p}^{2}+\mathfrak{m}^{2}}$, conservation of
energy between the initial and final state implies the non-linear constraint
$E_{\mathbf{p}}=E_{\mathbf{p}+\mathbf{q}}$. The loop momentum, routed as
indicated in Figure 1(b), in this non-relativistic notation is denoted
$l^{\mu}=\left(\omega,\mathbf{l}\right)$. In this calculation we are matching
the classical soliton solution to the REFT, so we expand to leading-order in
the hierarchy

$M\sim\mathbf{p}\sim\mathfrak{m}\gg\mathbf{q}\sim m.$ (62)

Here we are using the fact that the spatial momentum transfer $\mathbf{q}$ is
Fourier conjugate to the spatial separation $\mathbf{x}$ between the vortex
and the probe and performing an EFT matching calculation at the scale
$\mathbf{x}\sim R_{\text{interaction}}\sim m^{-1}$. Next we expand in the non-
relativistic limit for the probe

$\mathfrak{m}\gg\mathbf{p},$ (63)

for this matching calculation it is sufficient to truncate at leading or
static order. In principle we could calculate the relevant loop integrals
exactly and then subsequently make the above expansions, but at high loop
order this may not be feasible. Instead we make use of the method of regions
and expand the loop integrals before integration Beneke:1997zp . Moreover, we
can match the contributions from a single conveniently chosen region in both
full theory and the REFT instead of matching the complete result. In this case
we choose to match the contributions in the classical potential region defined
by the loop momentum scaling

$\left(\omega,\mathbf{l}\right)\sim\left(|\mathbf{p}||\mathbf{q}|,|\mathbf{q}|\right).$
(64)

In non-relativistic notation the loop integrals appearing in Figure 1(b) take
the form

$\int\frac{\text{d}^{d-1}\mathbf{l}}{(2\pi)^{d-1}}\frac{\text{d}\omega}{2\pi}\frac{\mathcal{N}\left(\omega,\mathbf{l}\right)}{[\omega^{2}-\mathbf{l}^{2}+i0][\omega^{2}-(\mathbf{l}+\mathbf{q})^{2}+i0]},$
(65)

expanding the integrand as (64) we find that order-by-order the resulting
$\omega$-integral is scaleless and therefore vanishes in dimensional
regularization. We therefore conclude that at one-loop only diagrams of the
form depicted in Figure 1(a) contribute in the potential region. It is
straightforward to extend this argument to higher loop order: for a loop
integral to have a non-vanishing contribution in the potential region it is
necessary that it should contain at least one probe propagator, but since we
are explicitly working at $\mathcal{O}\left(\mathfrak{g}_{s,m}\right)$ this
can never occur.

It is therefore possible to calculate the potential region contribution to the
probe amplitude in full theory at all orders in $N$, the result takes a simple
form

$\mathcal{M}_{\chi\rightarrow\chi}^{\text{(Full)}}\left(\mathbf{p},\mathbf{q}\right)\biggr{|}_{\text{potential
region}}=\mathfrak{g}_{s}\overline{\sigma}(\mathbf{q})-4i\mathfrak{g}_{m}\sqrt{\mathbf{p}^{2}+\mathfrak{m}^{2}}\epsilon_{ij}q_{i}\overline{A}_{j}(\mathbf{q})+\mathcal{O}\left(\mathfrak{g}_{s,m}^{2}\right).$
(66)

We now want to match this result with an equivalent calculation in the REFT.
We calculate a 2-to-2 scattering amplitude between a probe $\chi$ and point-
particle vortex $\Phi$ with the momentum of the incoming vortex is chosen to
be at rest $P^{\mu}=\left(M,0\right)$ and the momentum of the probe identical
to the above. Again we expand the result to leading order in (62) and (63), at
loop-level keeping only contributions from the potential region (64). There
are two tree-level Feynman diagrams that contribute to the REFT amplitude at
$\mathcal{O}\left(N^{1/2}\right)$ depicted in Figure 2. By explicit
calculation we find

$\mathcal{M}_{\chi\Phi\rightarrow\chi\Phi}^{(\text{REFT})}\left(\mathbf{p},\mathbf{q}\right)\biggr{|}_{N^{1/2}}=\frac{\mathfrak{g}_{s}g_{s}M^{3/2}N^{1/2}}{\mathbf{q}^{2}+m^{2}}-\frac{16\mathfrak{g}_{m}g_{m}M^{3/2}N^{1/2}m\sqrt{\mathbf{p}^{2}+\mathfrak{m}^{2}}}{\mathbf{q}^{2}+m^{2}}+\mathcal{O}\left(\mathfrak{g}_{s,m}^{2}\right).$
(67)

$P$$p$$q$ Figure 2: Feynman diagrams contributing to the vortex-probe
scattering amplitude at $\mathcal{O}\left(N^{1/2}\right)$, the double-line
denotes the vortex $\Phi$ and the single line the probe $\chi$.

Since the amplitudes we are matching between full theory and the REFT have
different numbers of external particles we need to account for a
(dimensionful) relative normalization factor; the explicit matching condition
we are imposing is then

$\mathcal{M}_{\chi\Phi\rightarrow\chi\Phi}^{(\text{REFT})}\left(\mathbf{p},\mathbf{q}\right)\biggr{|}_{\text{potential
region}}=2M\times\mathcal{M}_{\chi\rightarrow\chi}^{(\text{Full})}\left(\mathbf{p},\mathbf{q}\right)\biggr{|}_{\text{potential
region}}+\mathcal{O}\left(\mathfrak{g}_{s,m}^{2}\right).$ (68)

To calculate the right-hand-side we need the Fourier transform of the leading-
order contribution to the perturbative solution of the BPS equation (2.2),
this is found to be

$\overline{\sigma}\left(\mathbf{q}\right)=-\frac{2\sqrt{2}M^{1/2}N^{1/2}}{\mathbf{q}^{2}+m^{2}}+\mathcal{O}\left(N^{3/2}\right),\hskip
14.22636pt\overline{A}_{i}(\mathbf{q})=\frac{2i\sqrt{2\pi}M^{1/2}N^{1/2}m^{-1}\epsilon_{ij}q_{j}}{\mathbf{q}^{2}+m^{2}}+\mathcal{O}\left(N^{3/2}\right).$
(69)

Putting this together we find that for the matching condition to be satisfied
at $\mathcal{O}\left(N^{1/2}\right)$ fixes the cubic Wilson coefficients in
the REFT to take the values

$g_{s}=-4\sqrt{2\pi},\hskip 28.45274ptg_{m}=-\sqrt{2\pi}.$ (70)

As a non-trivial consistency check of this result we note this satisfies the
proportionality relation (58) required by the hidden $\mathcal{N}=2$
supersymmetry.

###  3.5 Classical Solitons from the S-Matrix

Above we used the matching condition (68) at leading-order in $N$ (tree-level)
to fix the cubic interactions of the REFT, but this relation should be true
also at higher loop orders. Naively we might think that matching at higher-
order in $N$ will fix further Wilson coefficients in the REFT action, in
particular those corresponding to quartic and higher interactions, but this is
not the case. By an argument identical to the one given above, any probe-
vortex diagram in the REFT containing quartic interactions between the vortex
and the mediators, and also $\mathcal{O}\left(\mathfrak{g}_{s,m}\right)$, will
give a vanishing contribution in the potential region. The problem of matching
higher-loop terms in (68) is therefore over-constrained and provides a highly
non-trivial consistency check on the REFT construction. Phrased a different
way, the cubic vortex-mediator interactions determined above, together with
the self-interactions of the mediator fields in the critical AHM, completely
determine the (perturbative) static soliton solution (2.2).

The purpose of this short section is to explain how to calculate the higher-
order corrections to the solution directly from the Feynman diagram expansion
of the REFT probe amplitudes. This calculation is similar in spirit to the
well-known method for perturbatively reconstructing the non-linear
Schwarzschild solution from Feynman diagrams or directly from gauge invariant
scattering amplitudes Duff:1973zz ; Neill:2013wsa ; Bjerrum-Bohr:2018xdl ;
Jakobsen:2020ksu ; Mougiakakos:2020laz .

To see how this works in practice we begin by inverting the explicit
expression for the full theory probe amplitude (66) to derive a simple
expression for the classical solution in terms of the REFT probe amplitude

$\displaystyle\overline{\sigma}(\mathbf{x})$
$\displaystyle=\frac{1}{2M}\int\frac{\text{d}^{2}\mathbf{q}}{(2\pi)^{2}}e^{i\mathbf{q}\cdot\mathbf{x}}\mathcal{M}_{\chi\Phi\rightarrow\chi\Phi}^{(\text{REFT})}\left(0,\mathbf{q}\right)\biggr{|}_{\mathfrak{g}_{s},\;\text{potential
region}}$ $\displaystyle\overline{A}_{i}(\mathbf{x})$
$\displaystyle=-\frac{i}{8\mathfrak{m}M}\int\frac{\text{d}^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\epsilon_{ij}q_{j}}{\mathbf{q}^{2}}e^{i\mathbf{q}\cdot\mathbf{x}}\mathcal{M}_{\chi\Phi\rightarrow\chi\Phi}^{(\text{REFT})}\left(0,\mathbf{q}\right)\biggr{|}_{\mathfrak{g}_{m},\;\text{potential
region}}.$ (71)

$q$$P$$l$$p$ Figure 3: Feynman diagrams contributing to the vortex-probe
amplitude at $\mathcal{O}\left(N^{3/2}\right)$.

At $L$-loops the only diagrams that give a non-vanishing contribution are the
so-called “fan diagrams” containing a single cubic probe coupling and $L+1$
cubic interactions between the vortex and the mediators. At one-loop or
$\mathcal{O}\left(N^{3/2}\right)$ there are four such diagrams depicted in
Figure 3. Expanding these diagrams in the classical potential region we find

$\displaystyle\mathcal{M}_{\chi\Phi\rightarrow\chi\Phi}^{(\text{REFT})}\left(0,\mathbf{q}\right)\bigg{|}_{N^{3/2},\;\text{potential
region}}$
$\displaystyle=4\sqrt{2}\pi^{3/2}N^{3/2}M^{3/2}\left(\mathfrak{g}_{s}-8\mathfrak{g}_{m}m\mathfrak{m}\right)\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][\left(\mathbf{l}+\mathbf{q}\right)^{2}+m^{2}]}$
(72) $\displaystyle\hskip
14.22636pt-8\sqrt{2}\pi^{3/2}N^{3/2}m^{2}M^{3/2}\left(\mathfrak{g}_{s}-4\mathfrak{g}_{m}m\mathfrak{m}\right)\frac{1}{\mathbf{q}^{2}+m^{2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][\left(\mathbf{l}+\mathbf{q}\right)^{2}+m^{2}]}.$

More information about the general strategy for simplification of these
expression is given in Section 4.3. Inserting these expressions into (3.5) and
making use of Fourier transforms derived in Appendix D we find that the result
exactly matches the $\mathcal{O}\left(N^{3/2}\right)$ perturbative solution of
the BPS equations (2.2).

##  4 Non-Relativistic Effective Field Theory

###  4.1 Definition of the NREFT

In this paper we will focus on conservative, two-body interactions between
identical vortices with winding number $N$. This means we focus on quartic
effective interactions generated by integrating out mediator potential modes.
Higher-multiplicity interactions, as well as dissipative effects (importantly
including radiation) correspond to further effective interactions, the
calculation of which we leave to future work.

The conservative two-body sector of the NREFT is defined by an effective
action of the form Neill:2013wsa ; Cheung:2018wkq

$\displaystyle S_{\text{NREFT}}\left[\Phi\right]$
$\displaystyle=\int\text{d}t\left[\int\frac{\text{d}^{2}\mathbf{k}}{(2\pi)^{2}}\Phi^{\dagger}(-\mathbf{k})\left(i\partial_{t}-\sqrt{\mathbf{k}^{2}+M^{2}}\right)\Phi(\mathbf{k})\right.$
$\displaystyle\hskip
42.67912pt\left.-\int\frac{\text{d}^{2}\mathbf{k}}{(2\pi)^{2}}\int\frac{\text{d}^{2}\mathbf{k}^{\prime}}{(2\pi)^{2}}V\left(\mathbf{k},\mathbf{k}^{\prime}\right)\Phi^{\dagger}(\mathbf{k}^{\prime})\Phi(\mathbf{k})\Phi^{\dagger}(-\mathbf{k}^{\prime})\Phi(-\mathbf{k})\right],$
(73)

where $V(\mathbf{k},\mathbf{k}^{\prime})$ corresponds to the effective vortex-
vortex potential.555Throughout this section we will abuse notation and use the
same symbol to denote the potential as a function of the “incoming” and
“outgoing” momenta
$V\left(\mathbf{p},\mathbf{p}^{\prime}\right)=V\left(\mathbf{p},\mathbf{p}-\mathbf{q}\right)$
as well as a function of instantaneous momentum and position
$V\left(\mathbf{p},\mathbf{x}\right)$, where $\mathbf{q}$ is the Fourier
conjugate variable to $\mathbf{x}$; it should be clear from context which is
intended. Scattering amplitudes in the NREFT are calculated using Feynman
rules derived from the effective action (4.1), as given in Appendix B. In
particular, for conservative two-body scattering the only diagrams that
contribute are iterated bubble diagrams depicted in Figure 4.

+$\mathbf{p}$$-\mathbf{p}$$\mathbf{p}^{\prime}$$-\mathbf{p}^{\prime}$+$\mathbf{k}$$-\mathbf{k}$+$\cdots$
Figure 4: Feynman diagrams contributing to the conservative two-body vortex-
vortex scattering amplitude in the NREFT.

In this section the NREFT amplitude is calculated in the center-of-mass frame
defined by the kinematics

$p_{1}=\left(E_{\mathbf{p}},\mathbf{p}\right),\hskip
14.22636ptp_{2}=\left(E_{\mathbf{p}},-\mathbf{p}\right),\hskip
14.22636ptp_{1}^{\prime}=\left(E_{\mathbf{p}},\mathbf{p}^{\prime}\right),\hskip
14.22636ptp_{2}^{\prime}=\left(E_{\mathbf{p}},-\mathbf{p}^{\prime}\right),$
(74)

where $\mathbf{p}^{\prime}=\mathbf{p}-\mathbf{q}$. After calculating the
trivial $\omega$-integrals, the $L$-loop contribution to the NREFT amplitude
is given by

$M_{\text{NREFT}}^{L\text{-loop}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=-\int_{\mathbf{k}_{1}...\mathbf{k}_{L}}V\left(\mathbf{p},\mathbf{k}_{1}\right)\Delta_{\Phi\Phi}\left(\mathbf{p},\mathbf{k}_{1}\right)\cdots\Delta_{\Phi\Phi}\left(\mathbf{k}_{L-1},\mathbf{k}_{L}\right)V\left(\mathbf{k}_{L},\mathbf{p}^{\prime}\right),$
(75)

where

$\Delta_{\Phi\Phi}\left(\mathbf{l},\mathbf{l}^{\prime}\right)=\frac{1}{2E_{\mathbf{l}}-2E_{\mathbf{l}^{\prime}}+i0},\hskip
28.45274pt\int_{\mathbf{k}}=\int\frac{\text{d}^{d-1}\mathbf{k}}{(2\pi)^{d-1}}.$
(76)

Formally this is the same as the Born series solution of the relativistic
Lippmann-Schwinger equation Lippmann:1950zz ; Cristofoli:2019neg

$M_{\text{NREFT}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=-V\left(\mathbf{p},\mathbf{p}^{\prime}\right)+\int_{\mathbf{k}}\frac{V\left(\mathbf{p},\mathbf{k}\right)M_{\text{NREFT}}\left(\mathbf{k},\mathbf{p}^{\prime}\right)}{2E_{\mathbf{p}}-2E_{\mathbf{k}}+i0},$
(77)

derived from an effective Hamiltonian

$H_{\text{NREFT}}\left(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{x}_{1},\mathbf{x}_{2}\right)=\sum_{i=1}^{2}\sqrt{\mathbf{p}_{i}^{2}+M^{2}}+V\left(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{x}_{1},\mathbf{x}_{2}\right).$
(78)

In this formalism the potential is a Wilson coefficient that is calculated by
matching with the non-relativistic expansion of the REFT. By standard
convention, the amplitudes calculated in the (N)REFT are
(non-)relativistically normalized and so the matching condition takes the form

$M_{\text{NREFT}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=\frac{\mathcal{M}_{\text{REFT}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)}{4\left(\mathbf{p}^{2}+M^{2}\right)}.$
(79)

This matching is performed order-by-order in $N$; in the NREFT this means the
potential has an expansion of the form

$V\left(\mathbf{p},\mathbf{p}^{\prime}\right)=\sum_{i=1}^{\infty}V^{(i)}\left(\mathbf{p},\mathbf{p}^{\prime}\right),$
(80)

where $V^{(i)}\sim N^{i}$. At leading-order, or $\mathcal{O}\left(N\right)$,
the potential is given straightforwardly by the first Born approximation

$V^{(1)}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=-\frac{\mathcal{M}^{(1)}_{\text{REFT}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)}{4\left(\mathbf{p}^{2}+M^{2}\right)}.$
(81)

At next-to-leading-order, or $\mathcal{O}\left(N^{2}\right)$, the potential is
given by rearranging the matching condition (79)

$V^{(2)}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=-\frac{\mathcal{M}^{(2)}_{\text{REFT}}\left(\mathbf{p},\mathbf{p}^{\prime}\right)}{4\left(\mathbf{p}^{2}+M^{2}\right)}-\int\frac{\text{d}^{d-1}\mathbf{k}}{(2\pi)^{d-2}}\frac{V^{(1)}\left(\mathbf{p},\mathbf{k}\right)V^{(1)}\left(\mathbf{k},\mathbf{p}^{\prime}\right)}{2E_{\mathbf{p}}-2E_{\mathbf{k}}+i0},$
(82)

the second term is variously called an iteration contribution or a Born
subtraction.

As in the probe calculation above, the matching is performed by expanding loop
integrals in the potential region. In practice it is convenient to expand the
REFT amplitudes in two steps following an approach introduced in Parra-
Martinez:2020dzs . In the first step we make a relativistic expansion in the
soft region defined by

$M\sim p\gg l\sim q\sim m,$ (83)

where we perform the calculation in a general reference frame. As we will see
this allows us to make use of standard Lorentz covariant integral reduction
techniques to simplify expressions in terms of a set of scalar soft integrals
with linearized vortex propagators. The second step is to then fix center-of-
mass frame kinematics (74) and expand the soft integrals non-relativistically
in the potential region

$\left(\omega,\mathbf{l}\right)\sim\left(|\mathbf{p}|,1\right).$ (84)

As described in detail in Appendix C, in all cases considered in this paper
the resulting velocity series can be resummed, giving the complete velocity
dependence of the vortex-vortex potential up to
$\mathcal{O}\left(N^{2}\right)$. For the iteration contributions to the NREFT
amplitudes we align the loop integration by defining
$\mathbf{k}=\mathbf{l}+\mathbf{p}$, and then expand in the classical potential
region (64).

Finally, there is an ambiguity in the potential defined using (77) arising
from the need to continue the amplitude away from the implicit constraint
surface of energy conservation Jones:2022aji . We need to choose a resolution
of the non-linear constraint $E_{\mathbf{p}}=E_{\mathbf{p}-\mathbf{q}}$,
different choices lead to formally distinct classical potentials. In the
Hamiltonian formalism (78) these different choices are related by some
canonical transformation on the phase space Bern:2019crd , while in the field
theoretic NREFT formalism (4.1) they correspond to field redefinitions. The
different choices of potential therefore lead to identical sets of physical
observables and so the choice is one of convenience. In this case there is a
canonical choice that leads to very simple expressions: in the center-of-mass
frame, the energy conservation constraint is equivalent to the condition

$\mathbf{p}\cdot\mathbf{q}=\frac{1}{2}\mathbf{q}^{2},$ (85)

which we can use to rewrite all appearances of $\mathbf{p}\cdot\mathbf{q}$ in
the amplitudes before matching. All potentials calculated in this paper are
defined in this way, a choice that in the post-Minkowskian literature is
usually called isotropic gauge Cheung:2018wkq ; Bjerrum-Bohr:2018xdl ;
Bern:2019crd .

###  4.2 Tree-Level Potential

We begin with the calculation of the vortex-vortex potential at tree-level or
$\mathcal{O}\left(N\right)$. In the REFT there are two Feynman diagrams that
contribute, corresponding to the exchange of the Higgs boson $\sigma$ and the
massive photon $\gamma$ as depicted in Figure 5(a). It is instructive to work
through this simple example in some detail as a warm-up for the more
complicated one-loop calculation in the following subsection. Using the
Feynman rules given in Appendix B and the result of the probe matching
calculation (70) the REFT tree amplitude can be written in the form

$p_{1}$$p_{2}$$q$$\sigma$$\gamma$ (a)

$\tilde{\sigma}$ (b)

Figure 5: Feynman diagrams contributing to $\mathcal{O}\left(N\right)$ vortex-
vortex potential. Only diagrams (a) corresponding to Higgs and photon exchange
are necessary in the classical limit. In the fully quantum mechanical
calculation we must also include diagram (b) corresponding to exchange of the
additional pseudoscalar $\tilde{\sigma}$ required by $\mathcal{N}=2$
supersymmetry to produce a cancellation of static forces.

$\mathcal{M}^{(1)}=-\frac{32\pi MN\left(M^{2}-(p_{1}\cdot
p_{2})\right)}{q^{2}-m^{2}}+\frac{8\pi MNm^{2}}{q^{2}-m^{2}}+\frac{32\pi
MN}{m^{2}}\left(p_{1}\cdot p_{2}+\frac{1}{4}\left(q^{2}+m^{2}\right)\right).$
(86)

As we will see, the first term corresponds to the long-range classical
potential, the numerator factor vanishes in the static limit as we expect for
BPS vortices. The second term also generates a long-range potential, but it is
suppressed relative to the first term by a factor
$\left(\frac{m}{M}\right)^{2}$ and therefore does not contribute to the
classical potential. The third term corresponds to a contact or short-range
interaction, in the corresponding position space potential it generates terms
proportional to $\delta^{(2)}\left(\mathbf{x}\right)$ and
$\nabla^{2}\delta^{(2)}\left(\mathbf{x}\right)$; from now on we will drop such
terms when they appear. Using (81) the classical long-range potential at
$\mathcal{O}\left(N\right)$ is given in momentum space as

$V^{(1)}\left(\mathbf{p},\mathbf{p}^{\prime}\right)=\frac{16\pi
MN\mathbf{p}^{2}}{\left[\mathbf{p}^{2}+M^{2}\right]\left[|\mathbf{p}-\mathbf{p}^{\prime}|^{2}+m^{2}\right]},$
(87)

and in position space

$V^{(1)}\left(\mathbf{p},\mathbf{x}\right)=\frac{8MN\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}K_{0}\left(mr\right).$
(88)

Before proceeding to one-loop, it is interesting to consider extending this
classical calculation to the full quantum long-range potential; this means
keeping the second term in (86). Here we find a puzzle, this term clearly does
not vanish in the static limit and so seems to violate our expectations about
forces between critical vortices. The resolution is to be found in the
analysis of Section 3.3: the cancellation of static forces for quantum
vortices follows from their identification with BPS states in the
$\mathcal{N}=2$ supersymmetric AHM. But as shown in (57) the supersymmetric
model contains an additional pseudoscalar boson $\tilde{\sigma}$ that only
decouples in the strict classical limit. In the quantum mechanical and
supersymmetric version of this calculation we should also include this degree-
of-freedom and therefore the additional exchange diagram depicted in Figure
5(b). Explicitly this generates an additional contribution to the REFT
amplitude

$\mathcal{M}^{(1)}\biggr{|}_{\mathcal{N}=2\;\text{AHM}}=\mathcal{M}^{(1)}\biggr{|}_{\text{critical
AHM}}-\frac{8\pi MNm^{2}}{q^{2}-m^{2}},$ (89)

which precisely cancels the second term in (86) and therefore restores the
expected cancellation of static forces. As a consequence, the quantum long-
range potential is identical to the classical long-range potential at this
order.

###  4.3 One-Loop Potential

In this section we present the calculation of the one-loop or
$\mathcal{O}\left(N^{2}\right)$ classical vortex-vortex potential. Below we
will calculate the triangle, (crossed-)box and iteration pieces individually
and assemble them at the end.666Various one-loop diagram toplogies are not
described in this section, such as bubble diagrams and mushroom diagrams. We
have explicitly checked that such diagrams give a vanishing contribution in
the classical potential region.

#### Triangle Diagrams

$q$$p_{1}$$l$$p_{2}$ Figure 6: Triangle diagrams contributing to the
$\mathcal{O}\left(N^{2}\right)$ vortex-vortex potential.

The most complicated part of the potential is generated by the triangle
diagrams depicted in Figure 6. Summing these graphs, expanding to leading-
order in the soft expansion (83) and subsequently reducing tensor numerators,
we find the result

$\displaystyle\mathcal{M}^{(2)}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=-\frac{128i\pi^{2}M^{2}N^{2}m^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)}{q^{2}-m^{2}}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
14.22636pt-64i\pi^{2}M^{2}N^{2}\left(M^{2}-2(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
14.22636pt+\frac{128i\pi^{2}M^{4}N^{2}}{q^{2}-m^{2}}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][p_{1}\cdot
l+i0]}.$ (90)

The first two terms are UV and IR finite and generate contributions to the
physical potential. The third term is logarithmically divergent777This
divergence arises solely from the second diagram in Figure 6, corresponding to
a pinch of one of the photon propagators. and so needs to be renormalized by
adding a vertex counterterm of the form
$\delta\mathcal{L}^{(3)}_{\text{vortex}}\supset\delta
g_{s}M^{3/2}N^{3/2}\sigma|\Phi|^{2}$. The finite part of this counterterm
corresponds to our choice of renormalization scheme, in this case there is a
unique choice dictated by the physics of the problem. We choose the finite
part such that the third term in (4.3) cancels completely, making any other
choice generates a contribution to the classical potential that does not
vanish in the static limit, as we expect for BPS vortices. From now on we will
drop such divergent scalar integrals when they appear.

Next we add the $1\leftrightarrow 2$ diagrams, these have an almost identical
contribution to the above but with $p_{1}\leftrightarrow p_{2}$ and
$q\rightarrow-q$. Actually, since both vortices are indistinguishable these
two contributions are identical in the center-of-mass frame888To see this
explicitly we temporarily rewrite the integral in non-relativistic notation
and make the change of variables $\mathbf{l}\rightarrow-\mathbf{l}$, therefore
if we add both orientations of triangle graph together we find

$\displaystyle\mathcal{M}^{(2)}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\mathcal{M}^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=-\frac{256i\pi^{2}M^{2}N^{2}m^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)}{q^{2}-m^{2}}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
14.22636pt-128i\pi^{2}M^{2}N^{2}\left(M^{2}-2(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}.$ (91)

Here we find a problem, while the first term manifestly vanishes in the static
limit the second does not. As we will see in the following, the remaining
diagrams do not fix this problem. Instead this is an indication that we have
missed a contribution from a seagull vertex. There are many possible contact
terms, parameterizing finite size corrections to the vortex and must be
calculated from a separate matching calculation. Here we use the weaker
constraint that the vortex-vortex potential should vanish in the static limit.

The simplest seagull term we can add to restore the cancellation of static
force has the form

$q$$p_{1}$$l$$p_{2}$ Figure 7: “Seagull” diagrams contributing to the
$\mathcal{O}\left(N^{2}\right)$ vortex-vortex potential, necessary for the
cancellation of static forces.

$S^{(4)}_{\text{vortex}}\left[\sigma,A_{\mu},\Phi\right]\supset\int\text{d}^{3}x\biggr{[}g_{\sigma\sigma}MN\sigma^{2}|\Phi|^{2}\biggr{]},$
(92)

which contributes two additional triangle graphs depicted in Figure 7.
Explicitly we find that we need

$g_{\sigma\sigma}=-2\pi.$ (93)

Putting this together we find the complete triangle contribution to the soft
REFT amplitude

$\displaystyle\mathcal{M}^{(2)}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\mathcal{M}^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\mathcal{M}^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\mathcal{M}^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=-\frac{256i\pi^{2}M^{2}N^{2}m^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)}{q^{2}-m^{2}}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
14.22636pt-256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}.$ (94)

The next step is to fix center-of-mass kinematics (74) and expand the
remaining integrals in the potential region (84). As shown in detail in
Appendix C the resulting expansion can be resummed, giving contributions to
the potential region with exact velocity dependence. The complete triangle
contribution to the momentum-space potential is then

$\displaystyle V^{(2)}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=-\frac{64\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}$
$\displaystyle\hskip
14.22636pt+\frac{64\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(\frac{1}{\mathbf{q}^{2}+m^{2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}\right).$
(95)

#### Box and Crossed-Box Diagrams

$p_{1}$$p_{2}$$l$$l+q$

$p_{1}$$p_{2}$$l$$l+q$

Figure 8: Box and crossed-box diagrams contributing to the
$\mathcal{O}\left(N^{2}\right)$ vortex-vortex potential.

Next we consider the contribution of the box and crossed-box diagrams depicted
in Figure 8. For the triangle diagrams calculated above we found that the
leading-order term in the soft expansion (83) of the one-loop amplitude scales
is $\mathcal{O}\left(q^{-2}\right)$; for the box diagrams however, the leading
order term is $\mathcal{O}\left(q^{-3}\right)$, so we must expand the box
diagrams to next-to-leading-order in the soft expansion. Such super-classical
terms must necessarily cancel with a corresponding iteration term to ensure a
vortex-vortex potential with a smooth classical or $\frac{m}{M}\rightarrow 0$
limit.

After reducing tensor numerators and discarding both terms without potential
region contributions and short-range contributions we find, for the box
diagrams

$\displaystyle\mathcal{M}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}$
$\displaystyle=256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)^{2}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0][p_{2}\cdot l-i0]}$ $\displaystyle\hskip
8.53581pt-128i\pi^{2}m^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)^{2}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]^{2}[p_{2}\cdot l-i0]}$ $\displaystyle\hskip
8.53581pt+128i\pi^{2}m^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)^{2}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0][p_{2}\cdot l-i0]^{2}}$ $\displaystyle\hskip
8.53581pt-256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
8.53581pt+256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{2}\cdot
l-i0]},$ (96)

and for the crossed-box diagrams

$\displaystyle\mathcal{M}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}$
$\displaystyle=256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$ $\displaystyle\hskip
14.22636pt-256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{2}\cdot
l-i0]}.$ (97)

Interestingly, all of the super-classical contributions to the soft expansion
of the crossed-box diagrams give contributions that vanish in the potential
region.999Specifically they generate integrals with two linearized vortex
propagators with the same sign $i0$ prescription. When expanded in the
potential region, the $\omega$-integrals of such terms give zero since they
are convergent at $\omega\rightarrow\infty$ and we can deform the contour in
such a way that we never cross a singularity. Various subtleties in the
evaluation of these $\omega$-integrals are discussed in Bern:2019crd .  The
non-vanishing crossed-box contributions exactly cancel the final two terms in
the box contribution, the sum of the two therefore nicely simplifies

$\displaystyle\mathcal{M}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}+\mathcal{M}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}$
$\displaystyle=256i\pi^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)^{2}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0][p_{2}\cdot l-i0]}$ $\displaystyle\hskip
14.22636pt-128i\pi^{2}m^{2}M^{2}N^{2}\left(M^{2}-(p_{1}\cdot
p_{2})\right)^{2}$ $\displaystyle\hskip
14.22636pt\times\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0][p_{2}\cdot l-i0]}\left[\frac{1}{p_{1}\cdot l+i0}-\frac{1}{p_{2}\cdot
l-i0}\right].$ (98)

We note that these contributions manifestly vanish in the static limit. In
Appendix C the expansion and resummation of these soft integrals is given. The
resulting contribution to the momentum-space potential is

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{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}$
$\displaystyle=-\frac{128\pi^{2}M^{2}N^{2}\mathbf{p}^{4}}{(\mathbf{p}^{2}+M^{2})^{3/2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}][\mathbf{p}\cdot\mathbf{l}-i0]}$
$\displaystyle\hskip
14.22636pt-\frac{64\pi^{2}M^{2}N^{2}\mathbf{p}^{4}}{(\mathbf{p}^{2}+M^{2})^{3/2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}][\mathbf{p}\cdot\mathbf{l}-i0]^{2}}$
$\displaystyle\hskip
14.22636pt+\frac{256\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(1-\frac{M}{(\mathbf{p}^{2}+M^{2})^{1/2}}\right)\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}]^{2}[(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}.$
(99)

#### Iteration Contributions

The final contribution to one-loop potential arises from the subtraction of
the iteration of the tree-level potential, explicitly

$V_{\text{iteration}}^{(2)}\equiv-\int\frac{\text{d}^{2}\mathbf{k}}{(2\pi)^{2}}\frac{V^{(1)}\left(\mathbf{p},\mathbf{k}\right)V^{(1)}\left(\mathbf{k},\mathbf{p}^{\prime}\right)}{2E_{\mathbf{p}}-2E_{\mathbf{k}}+i0}.$
(100)

Similar to the REFT amplitude contributions, this must also be expanded in the
classical potential region (64) to sub-leading order. Using the previous
result (87) and aligning the loop momenta, $\mathbf{k}=\mathbf{l}+\mathbf{p}$,
we calculate

$\displaystyle V^{(2)}_{\text{Iteration}}$
$\displaystyle=\frac{128\pi^{2}M^{2}N^{2}\mathbf{p}^{4}}{\left(\mathbf{p}^{2}+M^{2}\right)^{3/2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}][\mathbf{p}\cdot\mathbf{l}-i0]}$
$\displaystyle\hskip
14.22636pt+\frac{64\pi^{2}M^{2}N^{2}\mathbf{p}^{4}}{\left(\mathbf{p}^{2}+M^{2}\right)^{3/2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}][\mathbf{p}\cdot\mathbf{l}-i0]^{2}}$
$\displaystyle\hskip
14.22636pt+\frac{64\pi^{2}M^{2}N^{2}\mathbf{p}^{2}\left(\mathbf{p}^{2}+4M^{2}\right)}{\left(\mathbf{p}^{2}+M^{2}\right)^{5/2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}.$
(101)

Importantly, not only do we find that the super-classical terms cancel in the
sum of (4.3) and (4.3), so too do the classical terms proportional to the 2d
integrals containing linearized propagators. This is a useful consistency
check on the calculation: these terms, when integrated, generate branch cuts
corresponding to the “$s$-channel” unitarity cut separating the one-loop
vortex-vortex amplitude into a pair of tree amplitudes. Since we expect that
the potential is an analytic function of the momentum $\mathbf{p}$, such terms
must cancel when we sum all of the contributions.

#### Final Result

Combining (4.3), (4.3) and (4.3) together with the tree-level result (87) we
find the complete $\mathcal{O}\left(N^{2}\right)$ vortex-vortex potential is
given, in momentum space by

$\displaystyle V^{(1)}+V^{(2)}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V^{(2)}_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{1.50694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+V_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}+V_{\leavevmode\hbox
to4.62pt{\vbox to3.41pt{\pgfpicture\makeatletter\hbox{\hskip
0.80278pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{0.60278pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{(2)}+V^{(2)}_{\text{Iteration}}$
$\displaystyle=\frac{16\pi
NM\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\frac{1}{\mathbf{q}^{2}+m^{2}}+\frac{64\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(\frac{1}{\mathbf{q}^{2}+m^{2}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}\right)$
$\displaystyle\hskip
14.22636pt-\frac{64\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(1-\frac{M(\mathbf{p}^{2}+4M^{2})}{(\mathbf{p}^{2}+M^{2})^{3/2}}\right)\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}$
$\displaystyle\hskip
14.22636pt+\frac{256\pi^{2}MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(1-\frac{M}{(\mathbf{p}^{2}+M^{2})^{1/2}}\right)\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{m^{2}}{[\mathbf{l}^{2}+m^{2}]^{2}[(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}+\mathcal{O}\left(N^{3}\right).$
(102)

To calculate the position space potential we need to Fourier transform the
remaining 2d integrals.101010It is certainly possible to evaluate these
integrals exactly, for example the bubble integral evaluates to
$\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}=\frac{\sinh^{-1}\left(\frac{|\mathbf{q}|}{2m}\right)}{\pi|\mathbf{q}|\sqrt{\mathbf{q}^{2}+4m^{2}}},$
however, such explicit expressions make the subsequent Fourier transforms more
obscure, so we will not make use of them. This calculation is described in
detail in Appendix D, the final result is

$\displaystyle V\left(\mathbf{p},\mathbf{x}\right)$
$\displaystyle=\frac{8M\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(N+\frac{2\pi}{3\sqrt{3}}N^{2}\right)K_{0}(mr)+\frac{16M^{2}N^{2}\mathbf{p}^{2}(\mathbf{p}^{2}+4M^{2})}{(\mathbf{p}^{2}+M^{2})^{5/2}}K_{0}(mr)^{2}\hskip
28.45274pt$ $\displaystyle\hskip
14.22636pt+\frac{32MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(1-\frac{M}{(\mathbf{p}^{2}+M^{2})^{1/2}}\right)mrK_{0}(mr)K_{1}(mr)$
$\displaystyle\hskip
14.22636pt-\frac{32MN^{2}\mathbf{p}^{2}}{\mathbf{p}^{2}+M^{2}}\left(K_{0}(mr)\int_{mr}^{\infty}\text{d}\xi\;\xi\;K_{0}(\xi)K_{1}(\xi)I_{1}\left(\xi\right)\right.$
$\displaystyle\left.\hskip
99.58464pt+I_{0}\left(mr\right)\int_{mr}^{\infty}\text{d}\xi\;\xi\;K_{0}(\xi)K_{1}(\xi)^{2}\right)+\mathcal{O}\left(N^{3}\right).$
(103)

###  4.4 Moduli Space Metric

Following the well-known conjecture of Manton Manton:1981mp , it is known that
the slow-motion dynamics of critical vortices corresponds to geodesic motion
on the moduli space of vacua Ruback:1988ba ; Samols:1991ne . The 2-vortex
moduli space metric is not known in closed form, but has been calculated
numerically Shellard:1988zx ; Myers:1991yh  as well as analytically at leading
asymptotic order Manton:2002wb . In this section we will derive an approximate
perturbative expression for the moduli space metric from the general vortex-
vortex potential (4.3).

To extract the metric on the moduli space we truncate the effective
Hamiltonian (78) at $\mathcal{O}(\mathbf{p}^{2})$ and lift the result to a
general reference frame. In general this procedure is ambiguous and naively we
need to go back and recalculate the potential in a general frame. This is
actually unnecessary, it is sufficient to note that the leading momentum
dependence arises from the Lorentz invariant prefactors in the REFT amplitudes
(4.3, 4.3), which then reduce to Galilean boost invariant expressions in the
non-relativistic limit, explicitly

$\displaystyle M^{2}-p_{1}\cdot p_{2}$
$\displaystyle=-\frac{1}{2}|\mathbf{p}_{1}-\mathbf{p}_{2}|^{2}+\mathcal{O}\left(\mathbf{p}^{4}\right).$

From the Hamiltonian we calculate the Lagrangian by an inverse Legendre
transform that we evaluate perturbatively in $N$, the result is

$L(\dot{\mathbf{x}}_{1},\mathbf{x}_{1};\dot{\mathbf{x}}_{2},\mathbf{x}_{2})=\frac{1}{2}M\dot{\mathbf{x}}_{1}^{2}+\frac{1}{2}M\dot{\mathbf{x}}_{2}^{2}-\tilde{U}\left(r_{12}\right)|\dot{\mathbf{x}}_{1}-\dot{\mathbf{x}}_{2}|^{2},$
(104)

where

$\displaystyle\tilde{U}\left(r_{12}\right)$
$\displaystyle=2M\left(N+\frac{2\pi}{3\sqrt{3}}N^{2}\right)K_{0}(mr_{12})$
$\displaystyle\hskip
14.22636pt-8MN^{2}\left(K_{0}(mr_{12})\int_{mr_{12}}^{\infty}\text{d}\xi\;\xi\;K_{0}(\xi)K_{1}(\xi)I_{1}\left(\xi\right)\right.$
$\displaystyle\hskip
71.13188pt\left.+I_{0}\left(mr_{12}\right)\int_{mr_{12}}^{\infty}\text{d}\xi\;\xi\;K_{0}(\xi)K_{1}(\xi)^{2}\right)+\mathcal{O}(N^{3}).$
(105)

According to Manton we should interpret this effective Lagrangian as defining
a $0+1$d sigma model and read-off the metric on the moduli space of 2-vortices

$ds^{2}=\left(\frac{1}{2}M-\tilde{U}\left(r_{12}\right)\right)d\mathbf{x}_{1}^{2}+\left(\frac{1}{2}M-\tilde{U}\left(r_{12}\right)\right)d\mathbf{x}_{2}^{2}-2\tilde{U}\left(r_{12}\right)d\mathbf{x}_{1}\cdot
d\mathbf{x}_{2}.$ (106)

Isolating the leading asymptotic contribution,
$\mathcal{O}\left(e^{-mr_{12}}\right)$, and using the perturbative expansion
of $Z_{N}$ (21) we find complete agreement with the known result Manton:2002wb
. Additionally, it is known that the moduli space metric is Kähler
Samols:1991ne . To verify this we rewrite the metric in complex coordinates

$\displaystyle ds^{2}\equiv
2g_{\alpha\overline{\beta}}dz^{\alpha}d\overline{z}^{\overline{\beta}}=4M\left(dz^{1}d\overline{z}^{\overline{2}}+dz^{2}d\overline{z}^{\overline{1}}\right)-\tilde{U}\left(z^{1},\overline{z}^{\overline{1}}\right)dz^{1}d\overline{z}^{\overline{1}},$
(107)

where

$\displaystyle z^{1}$ $\displaystyle=(x_{1}-x_{2})+i(y_{1}-y_{2}),\hskip
14.22636pt\overline{z}^{\overline{1}}=(x_{1}-x_{2})-i(y_{1}-y_{2})$
$\displaystyle z^{2}$ $\displaystyle=(x_{1}+x_{2})+i(y_{1}+y_{2}),\hskip
14.22636pt\overline{z}^{\overline{2}}=(x_{1}+x_{2})-i(y_{1}+y_{2}),$ (108)

and $\mathbf{x}_{i}\equiv\left(x_{i},y_{i}\right)$. The Kähler property of the
metric can be straightforwardly verified from the closure of the fundamental
2-form $\Omega\equiv-2ig_{\alpha\overline{\beta}}dz^{\alpha}\wedge
d\overline{z}^{\overline{\beta}}$ Freedman:2012zz .111111In this case closure
is almost trivial and does not depend on the detailed form of
$\tilde{U}(z,\overline{z})$. Rather, once the Galilean boost invariance of the
$\mathcal{O}\left(\mathbf{p}^{2}\right)$ effective Hamiltonian is established,
the Kähler property of the resulting metric is a foregone conclusion.

##  5 Discussion

The main result of this paper is (4.3), the $\mathcal{O}\left(N^{2}\right)$
contribution to the effective vortex-vortex potential. Both this result and
the general point-particle EFT framework have passed non-trivial consistency
checks against known results in limiting cases. In particular, the one-loop
vortex-probe amplitude in the REFT (3.5) exactly matches the predicted
$\mathcal{O}\left(N^{3/2}\right)$ solution of the BPS equations (2.2) and, up
to $\mathcal{O}\left(N^{2}p^{2}\right)$ the potential (4.3) agrees with the
prediction of the moduli space approximation at leading asymptotic order
Manton:2002wb . This gives us a great deal of confidence in the validity of
this approach.

Nonetheless there are still many ways these calculations can be extended or
improved. In Section 4.3 we found that the REFT action obtained by matching
probe amplitudes, with only cubic vortex-mediator interactions, was
necessarily incomplete as it lead to non-vanishing static forces at
$\mathcal{O}\left(N^{2}\right)$. We interpreted this as the requirement that
the REFT must contain a “seagull” vertex and proposed a simple example (92)
that resolved the dilemma. We do not claim that this term is unique; a true
determination of such finite size terms should be obtained by matching with
the static response of the vortex soliton in the full theory along the lines
of Hui:2020xxx .

To obtain quantitatively accurate results for physical vortices with
$N\in\mathds{Z}$, we expect to need to resum the small winding expansion as
undertaken in Ohashi:2015yta  using a global Padé approximant. This will
certainly require extending the calculation of the vortex-vortex potential to
higher loop order. In such a calculation the velocity resummation of soft
integrals will almost certainly require the more sophisticated method of
velocity differential equations Parra-Martinez:2020dzs . At higher-orders we
also expect to encounter radiation-reaction effects. It would be particularly
interesting to adapt the KMOC framework Kosower:2018adc ; Herrmann:2021tct  in
this context to calculate the energy loss from radiation and compare with
known numerical results Myers:1991yh .

Finally, the calculations presented in this paper were restricted to the
theoretically simpler but physically unrealistic critical case. For
applications to physical systems this method should be extended to the
description of vortices in the generic, non-critical AHM. This should be
possible, at least in the weakly type-I or type-II regimes where departure
from criticality can be treated as a perturbation. We expect the same
separation of scales should occur in the small winding limit and so the same
point-particle EFT methods should apply. We leave this and other interesting
generalizations to future work.

Acknowledgements

I would like to thank Zvi Bern, Thomas Dumitrescu, Amey Gaikwad, Shruti
Paranjape and Michael Ruf for useful discussions. CRTJ is supported by the
Department of Energy under Award Number DE-SC0009937 and acknowledges the
continued support of the Mani L. Bhaumik Institute for Theoretical Physics.

##  Appendix A Conventions

In this paper we work in the mostly minus metric convention

$\eta_{\mu\nu}=\eta^{\mu\nu}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix}.$ (109)

The Levi-Civita symbol in $2+1d$ is defined by the sign convention
$\epsilon^{012}=\epsilon_{012}=+1$, while the $2d$ (spatial) Levi-Civita is
defined in the convention $\epsilon^{12}=+1$. We choose to work with the
following basis of real and symmetric Pauli matrices

$\displaystyle\sigma_{ab}^{0}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\hskip
28.45274pt\sigma_{ab}^{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\hskip
28.45274pt\sigma_{ab}^{2}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (110)

In $d=2+1$ there is only one type of spinor index which can be raised/lowered
with the spinor Levi-Civita symbol

$\epsilon^{ac}\epsilon^{bd}\sigma^{\mu}_{cd}\equiv\overline{\sigma}^{\mu ab},$
(111)

defined in the sign convention $\epsilon^{12}=-\epsilon_{12}=+1$. The Pauli
matrices satisfy the usual Clifford algebra relation

$\sigma^{\mu}_{ac}\overline{\sigma}^{\nu
cb}+\sigma^{\nu}_{ac}\overline{\sigma}^{\mu cb}=2\eta^{\mu\nu}\delta^{b}_{a}.$
(112)

Our conventions for the $\mathcal{N}=2$ super-Poincare algebra in $d=2+1$ are

$\displaystyle\\{Q_{a},Q^{\dagger}_{b}\\}$
$\displaystyle=2\sigma^{\mu}_{ab}P_{\mu}-2iZ\epsilon_{ab}$
$\displaystyle[Q_{a},M^{\mu\nu}]$
$\displaystyle=\frac{i}{4}{\left(\sigma^{\mu}\overline{\sigma}^{\nu}-\sigma^{\nu}\overline{\sigma}^{\mu}\right)_{a}}^{b}Q_{b}$
$\displaystyle[Q^{\dagger}_{a},M^{\mu\nu}]$
$\displaystyle=\frac{i}{4}{\left(\sigma^{\mu}\overline{\sigma}^{\nu}-\sigma^{\nu}\overline{\sigma}^{\mu}\right)_{a}}^{b}Q^{\dagger}_{b}$
$\displaystyle[M^{\mu\nu},P^{\rho}]$
$\displaystyle=-i\left(\eta^{\mu\rho}P^{\nu}-\eta^{\nu\rho}P^{\mu}\right)$
$\displaystyle[M^{\mu\nu},M^{\rho\sigma}]$
$\displaystyle=-i\left(\eta^{\mu\rho}M^{\nu\sigma}-\eta^{\nu\rho}M^{\mu\sigma}-\eta^{\mu\sigma}M^{\nu\rho}+\eta^{\nu\sigma}M^{\mu\rho}\right),$
(113)

with all other (anti-)commutators vanishing.

#### Massive Spinor Variables

To construct an on-shell superspace in Section 3.3 we make use of massive
spinor variables in $d=2+1$ defined by the conditions

$p_{ab}=p_{\mu}\sigma^{\mu}_{ab}=\lambda_{a}\tilde{\lambda}_{b}+\lambda_{b}\tilde{\lambda}_{a},\hskip
28.45274pt\lambda_{a}\tilde{\lambda}^{a}=im,$ (114)

where $p^{2}=m^{2}$ and indices on the spinors are raised/lowered with the 2d
Levi-Civita symbol

$\lambda^{a}=\epsilon^{ab}\lambda_{b},\hskip
28.45274pt\tilde{\lambda}^{a}=\epsilon^{ab}\tilde{\lambda}_{b}.$ (115)

There is a redundancy in this description

$\lambda_{a}\rightarrow e^{i\theta}\lambda_{a},\hskip
28.45274pt\tilde{\lambda}_{a}\rightarrow e^{-i\theta}\tilde{\lambda}_{a},$
(116)

corresponding to the action of the little group of massive particles
$SO(2)\cong U(1)$. The Lorentz invariant spin quantum number labelling little
group representations corresponds to the eigenvalue of the Pauli-Lubanski
scalar

$W=-\frac{1}{2}\epsilon_{\mu\nu\rho}P^{\mu}M^{\nu\rho}.$ (117)

We will work in a convention in which $\lambda_{a}$ carries $+1/2$ units of
spin and $\tilde{\lambda}_{a}$ carries $-1/2$ units. An explicit solution to
these conditions for real-valued momenta with $p^{0}>0$ are given by

$\lambda_{a}=\frac{1}{\sqrt{2\left(p^{0}+p^{2}\right)}}\begin{pmatrix}p^{1}-im\\\
-p^{0}-p^{2}\end{pmatrix},\hskip
14.22636pt\tilde{\lambda}_{a}=\frac{1}{\sqrt{2\left(p^{0}+p^{2}\right)}}\begin{pmatrix}p^{1}+im\\\
-p^{0}-p^{2}\end{pmatrix}.$ (118)

From these expressions we see that these spinors are interchanged under
crossing up to a possible phase factor, a consistent choice is given by

$\lambda_{-p}^{a}=i\tilde{\lambda}_{p}^{a},\hskip
14.22636pt\tilde{\lambda}_{-p}^{a}=i\lambda_{p}^{a}.$ (119)

It is convenient to define Lorentz invariant spinor brackets, we will use

$\langle pq\rangle=\tilde{\lambda}_{pa}\tilde{\lambda}_{q}^{a},\hskip
28.45274pt[pq]=\lambda_{pa}\lambda_{q}^{a},\hskip 28.45274pt\langle
pq]=\tilde{\lambda}_{pa}\lambda_{q}^{a},\hskip
28.45274pt[pq\rangle=\lambda_{pa}\tilde{\lambda}_{q}^{a}.$ (120)

Many Schouten-type identities among the spinor brackets can be derived from
the fundamental identity

$\epsilon^{ab}\epsilon^{cd}+\epsilon^{ac}\epsilon^{db}+\epsilon^{ad}\epsilon^{bc}=0.$
(121)

Finally we can derive expressions for the polarization vectors of an
(outgoing) massive spin-1 particle

$\varepsilon_{+}^{\mu}(p)=\frac{1}{\sqrt{2}m}\lambda_{a}\overline{\sigma}^{\mu
ab}\lambda_{b},\hskip
28.45274pt\varepsilon_{-}^{\mu}(p)=\frac{1}{\sqrt{2}m}\tilde{\lambda}_{a}\overline{\sigma}^{\mu
ab}\tilde{\lambda}_{b}.$ (122)

Using the above results we derive the crossing

$\varepsilon_{\pm}^{\mu}(-p)=-\varepsilon_{\mp}^{\mu}(p),$ (123)

and completeness relations

$\sum_{\pm}\varepsilon_{\pm}^{\mu}(-p)\varepsilon_{\pm}^{\nu}(p)=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{m^{2}}.$
(124)

##  Appendix B Feynman Rules

#### Critical Abelian Higgs Model

The Feynman rules for the self-interactions of the mediator particles in the
critical AHM, in unitary gauge and in the notation corresponding to (3.2) are
given by:

$k$ $=\hskip 34.1433pt\frac{i}{k^{2}-m^{2}+i0}$ $k$$\mu$$\nu$ $=\hskip
34.1433pt\frac{-i\left(\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^{2}}\right)}{k^{2}-m^{2}+i0}$
$\mu$$\nu$ $=\hskip
34.1433pti\sqrt{2\pi}M^{3/2}\left(\frac{m}{M}\right)^{2}N^{1/2}\eta^{\mu\nu}$
$=\hskip
34.1433pt-3i\sqrt{\frac{\pi}{2}}M^{3/2}\left(\frac{m}{M}\right)^{2}N^{1/2}$
$=\hskip 34.1433pt-\frac{3i}{2}M\left(\frac{m}{M}\right)^{2}N$ $\mu$$\nu$
$=\hskip 34.1433pti\pi M\left(\frac{m}{M}\right)^{2}N\eta^{\mu\nu}.$

#### Relativistic EFT

The Feynman rules for the interactions between the (point-particle) vortex and
the mediator particles corresponding to the REFT effective action (33), (3.2)
and the scalar-scalar interaction in (92) are given by:

$k$ $=\hskip 34.1433pt\frac{i}{k^{2}-M^{2}+i0}$

$=\hskip 34.1433ptig_{s}M^{3/2}N^{1/2}$

$q$$p$$\mu$ $=\hskip
34.1433pt-4g_{m}M^{-1/2}N^{1/2}\left(\frac{m}{M}\right)^{-1}\epsilon^{\mu\nu\rho}q_{\nu}p_{\rho}$

$=\hskip 34.1433pt2ig_{\sigma\sigma}MN$

The cubic vortex-mediator couplings were determined in (70) by matching with
full theory and the quartic coupling fixed in (93) by requiring the
cancellation of static forces at one-loop.

In addition we also need the Feynman rules for the probe particle defined by
the effective action (59), these are given by:

$k$ $=\hskip 34.1433pt\frac{i}{k^{2}-\mathfrak{m}^{2}+i0}$

$=\hskip 34.1433pti\mathfrak{g}_{s}$

$q$$p$$\mu$ $=\hskip
34.1433pt-4\mathfrak{g}_{m}\epsilon^{\mu\nu\rho}q_{\nu}p_{\rho}.$

#### Non-Relativistic EFT

The Feynman rules derived from the NREFT effective action (4.1) are given by:

$\left(\omega,\mathbf{k}\right)$ $=\hskip
34.1433pt\frac{i}{\omega-\sqrt{\mathbf{k}^{2}+M^{2}}+i0}$ $=\hskip 34.1433pt-
iV\left(\mathbf{k},\mathbf{k}^{\prime}\right).$
$\mathbf{k}$$-\mathbf{k}$$\mathbf{k}^{\prime}$$-\mathbf{k}^{\prime}$

##  Appendix C Velocity Resummation of Soft Integrals

#### Triangle Integral

The triangle soft integral is defined as

$\mathcal{I}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}.$ (125)

To simplify this we first note that if we make a change of variables
$l\rightarrow-l-q$ and drop a quantum term (which must vanish in the final
result since the original integral was homogeneous) we find

$-\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l-i0]}.$ (126)

This is not exactly the same as the original integral since the $i0$ in the
linearized propagator has changed sign. Using this we can rewrite the original
integral as

$\displaystyle\mathcal{I}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=\frac{1}{2}\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}]}\left(\frac{1}{p_{1}\cdot
l+i0}-\frac{1}{p_{1}\cdot l-i0}\right)$
$\displaystyle=-i\pi\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{\delta\left(p_{1}\cdot
l\right)}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}]},$ (127)

where we have used the distributional identity

$\frac{1}{x+i0}-\frac{1}{x-i0}=-2\pi i\delta(x).$ (128)

In this form we can calculate the $\omega$-integral explicitly

$\displaystyle=-i\pi\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{\text{d}\omega}{2\pi}\frac{\delta\left(E\omega-\mathbf{p}\cdot\mathbf{l}\right)}{[\omega^{2}-\mathbf{l}^{2}-m^{2}][\omega^{2}-(\mathbf{l}+\mathbf{q})^{2}-m^{2}]}$
$\displaystyle=-\frac{i}{2E}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[(\mathbf{u}\cdot\mathbf{l})^{2}-\mathbf{l}^{2}-m^{2}][(\mathbf{u}\cdot\mathbf{l})^{2}-(\mathbf{l}+\mathbf{q})^{2}-m^{2}]},$
(129)

where we have defined $\mathbf{u}=\frac{\mathbf{p}}{E}$. Next we expand the
integrand order-by-order in $\mathbf{u}$ and simplify the resulting tensor
integrals using IBP relations. It is convenient to introduce some notation,
define

$I_{\alpha,\beta}^{(n)}\equiv\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{(\mathbf{u}\cdot\mathbf{l})^{n}}{[\mathbf{l}^{2}+m^{2}]^{\alpha}[(\mathbf{l}+\mathbf{q})^{2}+m^{2}]^{\beta}}.$
(130)

By making a change of variables $\mathbf{l}\rightarrow-\mathbf{l}-\mathbf{q}$
and dropping a quantum term we can see that these satisfy the symmetry
property $I_{\alpha,\beta}^{(n)}=I_{\beta,\alpha}^{(n)}$. More non-trivially
from the IBP relation

$\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\;\mathbf{u}\cdot\frac{\partial}{\partial\mathbf{l}}\left(\frac{(\mathbf{u}\cdot\mathbf{l})^{n}}{[\mathbf{l}^{2}+m^{2}]^{\alpha}[(\mathbf{l}+\mathbf{q})^{2}+m^{2}]^{\beta}}\right)=0,$
(131)

we derive the identity

$n\mathbf{u}^{2}I_{\alpha,\beta}^{(n-1)}=2\alpha
I_{\alpha+1,\beta}^{(n+1)}+2\beta I_{\beta+1,\alpha}^{(n+1)}.$ (132)

This turns out to be all we need to simplify the above integral. It is
straightforward to calculate

$\displaystyle-\frac{i}{2E}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[(\mathbf{u}\cdot\mathbf{l})^{2}-\mathbf{l}^{2}+m^{2}][(\mathbf{u}\cdot\mathbf{l})^{2}-(\mathbf{l}+\mathbf{q})^{2}-m^{2}]}$
$\displaystyle=-\frac{i}{2E}\left(1+\frac{\mathbf{u}^{2}}{2}+\frac{3\mathbf{u}^{4}}{4}+\frac{5\mathbf{u}^{6}}{16}+\frac{35\mathbf{u}^{8}}{128}+...\right)\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}.$
(133)

This is enough to recognize the series and resum

$=-\frac{i}{2E\sqrt{1-\mathbf{u}^{2}}}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]},$
(134)

rewriting in terms of the original variables we have the resummed triangle
integral

$\displaystyle\mathcal{I}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{0.60278pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{1.50694pt}{-1.50694pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{1.50694pt}{-1.50694pt}\pgfsys@lineto{1.50694pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle=\int\frac{\text{d}^{d}l}{(2\pi)^{d}}\frac{1}{[l^{2}-m^{2}][(l+q)^{2}-m^{2}][p_{1}\cdot
l+i0]}$
$\displaystyle=-\frac{i}{2M}\int\frac{\text{d}^{2}\mathbf{l}}{(2\pi)^{2}}\frac{1}{[\mathbf{l}^{2}+m^{2}][(\mathbf{l}+\mathbf{q})^{2}+m^{2}]}.$
(135)

#### (Super-classical) Box Integral

The super-classical box soft integral is defined as

$\mathcal{I}_{\leavevmode\hbox to4.62pt{\vbox
to3.41pt{\pgfpicture\makeatletter\hbox{\hskip 0.80278pt\lower-3.21388pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{-0.60278pt}{0.0pt}\pgfsys@lineto{3.61665pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}}{{}}{}
{}{}{}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@lineto{2.41112pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-0.60278pt}{-3.01389pt}\pgfsys@lineto{3.61665pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.60278pt}{-3.01389pt}\pgfsys@lineto{0.60278pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{