#  A classical complex $\phi^{4}$ scalar field in a gauge background

N. Mohammedi
Institut Denis Poisson (CNRS - UMR 7013),
Université de Tours,
Faculté des Sciences et Techniques,
Parc de Grandmont, F-37200 Tours, France. e-mail: noureddine.mohammedi@univ-
tours.fr

###### Abstract

We solve the equations of motion of a complex $\phi^{4}$ theory coupled to
some given gauge field background. The solutions are given in both cylindrical
and spherical coordinates and have finite energy.

##  1 Introduction

The equations of motion of the Abelian Higgs model have so far resisted all
attempts to solving them analytically. These equations, on the other hand,
carry some rich structures like vortices [1] and other topological entities
[2, 3]. It is, therefore, important to find some of these classical solutions
even in some very special situations.

In this note, we simplify the Abelian Higgs model by leaving out the gauge
field kinetic term. This means that the gauge field is not propagating and
merely furnish a background for the dynamical complex scalar field.

The issue now lies in conveniently choosing this gauge field background. Our
approach, in making this choice, consists in taking the complex scalar field
as a given function and then determine the gauge field. The guiding principles
are solvability of the equations of motion and a corresponding finite energy.
We will look for static solutions only.

This method of proceeding is inspired from other physical problems, especially
in quantum mechanics: Suppose that one is given the time-independent
Schrödinger equation in one dimension and asked what is the form of the
potential $V\left(x\right)$ and energy $E$ that accompany a wave function of
the type $\Psi\left(x\right)=c\,\exp\left(-\alpha^{2}x^{2}\right)\,\,\,\,?$
One, of course, finds the harmonic oscillator potential
$V\left(x\right)=2\alpha^{4}\hbar^{2}\,x^{2}/m$ and its ground state energy
$E=\alpha^{2}\hbar^{2}/m$. This programme is the essence of the Bijl-Jastrow
ansatz for many-body problems [4, 5].

We know that $\phi^{4}$-theory, on its own, possesses the well-studied kink
solution. In this note, we consider a gauged $\phi^{4}$-theory and find the
gauge field background that accompany the kink solution. We examine the issue
in cylindrical and spherical coordinates. In both case we give the expression
of the gauge field background as well as the corresponding energy.

##  2 A complex scalar field in a gauge background

The field theory we consider is given by the action

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\left(D_{\mu}\phi\right)^{\star}\left(D^{\mu}\phi\right)-\frac{\lambda}{2}\left(\phi^{\star}\phi-v^{2}\right)^{2}\right]\,\,\,.$
(2.1)

Here $A_{\mu}$ is a gauge field taken as a background in which the complex
scalar field $\phi$ evolves. The gauge covariant derivative is
$D_{\mu}=\partial_{\mu}+ieA_{\mu}$. We take $v^{2}$ and $\lambda$ to be both
positive . The space-time indices are raised and lowered with a metric
$g_{\mu\nu}$ and $g=\det{g_{\mu\nu}}$.

The equation of motion for the scalar field $\phi$ is

$\displaystyle\frac{1}{\sqrt{-g}}\,D_{\mu}\left(\sqrt{-g}\,D^{\mu}\phi\right)+{\lambda}\phi\left(\phi^{\star}\phi-v^{2}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,.\,\,\,\,\,$ (2.2)

The corresponding energy-momentum tensor is given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{-g}}\,\frac{\delta S}{\delta g^{\mu\nu}}$ (2.3)
$\displaystyle=$
$\displaystyle\left(D_{\mu}\phi\right)^{\star}\left(D_{\nu}\phi\right)+\left(D_{\nu}\phi\right)^{\star}\left(D_{\mu}\phi\right)-g_{\mu\nu}\left[\left(D_{\alpha}\phi\right)^{\star}\left(D^{\alpha}\phi\right)-\frac{\lambda}{2}\left(\phi^{\star}\phi-v^{2}\right)^{2}\right]\,\,\,.$

###  2.1 Static solution in cylindrical coordinates

We first use cylindrical coordinates
$x^{\mu}=\left(t\,,\,r\,,\,\theta\,,\,z\right)$ for which the metric is

$\displaystyle
g_{\mu\nu}={\rm{diag}}\left(1\,,\,-1\,,\,-r^{2}\,,\,-1\right)\,\,\,\,\,\,.$
(2.4)

We assume a Nielsen-Olesen [1] vortex ansätze111The vector field with index up
is
$A^{\mu}=\left[0\,,\,0\,,\,\frac{n}{e}\,\frac{h\left(r\right)}{r^{2}}\,,0\right]$.

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
v\,f\left(r\right)\,e^{i\,n\,\theta}\,\,\,\,\,\,,$ $\displaystyle A_{\mu}$
$\displaystyle=$
$\displaystyle\left[0\,,\,0\,,\,-\frac{n}{e}\,h\left(r\right)\,,0\right]\,\,\,\,\,\,,$
(2.5)

where $n$ is an integer in $\mathbb{Z}$.

With these ansätze, the equation of motion becomes

$\left\\{\frac{d^{2}f}{dr^{2}}-\lambda
v^{2}f\left(f^{2}-1\right)\right\\}+\left\\{\frac{1}{r}\,\frac{df}{dr}-n^{2}\frac{f}{r^{2}}\left(1-h\right)^{2}\right\\}=0\,\,\,\,\,\,.$
(2.6)

The energy per unit length (along the $z$ direction) is

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}d\theta\int_{0}^{\infty}dr\,\sqrt{-g}\,T_{00}\,\,\,\,\,\,$
(2.7) $\displaystyle=$ $\displaystyle
2\pi\,v^{2}\int_{0}^{\infty}dr\,r\left[\left(\frac{df}{dr}\right)^{2}+\frac{n^{2}}{r^{2}}\,f^{2}\,\left(1-h\right)^{2}+\frac{\lambda}{2}\,v^{2}\left(f^{2}-1\right)^{2}\right]\,\,\,\,\,\,.$

Here $T_{00}$ is read from the energy-momentum tensor (2.3). The equation of
motion (2.6) corresponds to the minimum of the energy functional $E$ (that is,
$\delta E=0$, with $h(r)$ fixed).

Our strategy is to set each expression between the curly brackets in (2.6) to
zero separately. The vanishing of the first bracket is solved by

$\displaystyle f\left(r\right)$ $\displaystyle=$
$\displaystyle\tanh\left(\alpha\,r\right)\,\,\,\,\,\,,\,\,\,\,\,\,\alpha^{2}=\frac{\lambda
v^{2}}{2}\,\,\,\,\,\,.$ (2.8)

This is the well-known kink of $\phi^{4}$ theory (the anti-kink corresponds to
$f\left(r\right)=-\tanh\left(\alpha\,r\right)$). Setting the second bracket to
zero leads to

$\displaystyle h\left(r\right)$ $\displaystyle=$ $\displaystyle
1\pm\,\frac{1}{n}\sqrt{\frac{2\alpha\,r}{\sinh\left(2\alpha\,r\right)}}\,\,\,\,\,\,.$
(2.9)

This determines the gauge field background.

The only non-vanishing component of the gauge field strength
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is

$\displaystyle
F_{r\,\theta}=\partial_{r}\,A_{\theta}=-\frac{n}{e}\,\frac{dh}{dr}=\mp\,\frac{1}{e}\,\frac{1}{2r}\,\sqrt{\frac{2\alpha
r}{\sinh\left(2\alpha r\right)}}\left[1-\frac{2\alpha r}{\tanh\left(2\alpha
r\right)}\right]\,\,\,\,\,\,.$ (2.10)

This is represented in figure (1).

Figure 1: The gauge field strength $F_{r\,\theta}$, (2.10), in cylindrical
coordinates for ${e}=\pm 1$ and $\alpha=1$.

The magnetic field (in a curved space-time) is defined as

$\displaystyle B_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\,\epsilon_{ijk}\,F^{jk}\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,$
$\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle\frac{F_{r\,\theta}}{r}=\mp\,\frac{1}{e}\,\frac{1}{2r^{2}}\,\sqrt{\frac{2\alpha
r}{\sinh\left(2\alpha r\right)}}\left[1-\frac{2\alpha r}{\tanh\left(2\alpha
r\right)}\right]\,\,\,\,\,\,,$ (2.11)

where the flat alternating tensor is such that
$\epsilon_{123}=\epsilon_{r\theta z}=+1$. For completness, we have sketshed in
figure (2) the function $f(r)$ and the magnetic field $B_{z}$.

Figure 2: The thick line is the kink
$f\left(r\right)=\tanh\left(\alpha\,r\right)$ while the dashed line represents
the background magnetic field $B_{z}$ for ${e}=\pm 1$ and $\alpha=1$ in
cylindrical coordinates.

Using the equation of motion (2.6) and the expression of $f(r)$, the energy
$E$ can be written as

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2\pi\,v^{2}\int_{0}^{\infty}dr\,\left[\frac{1}{2}\partial_{r}\left(r\partial_{r}\,f^{2}\right)-\frac{\lambda}{2}\,v^{2}\,r\left(f^{4}-1\right)\right]\,\,\,\,\,\,$
(2.12) $\displaystyle=$
$\displaystyle-2\pi\,v^{2}\alpha^{2}\int_{0}^{\infty}dr\,r\left[\tanh^{4}\left(\alpha
r\right)-1\right]\,\,\,\,\,\,.$

The total derivative term does not contribute. Performing the integral we
obtain

$E=\frac{2\pi}{3}\,{v^{2}}\left[\frac{1}{2}+4\ln\left(2\right)\right]\,\,\,\,\,\,.$
(2.13)

We recall that $E$ here is the energy per unit length.

###  2.2 Static solution in spherical coordinates

The same study could be repeated in spherical coordinates
$x^{\mu}=\left(t\,,\,r\,,\,\theta\,,\,\varphi\right)$ for which the metric is

$\displaystyle
g_{\mu\nu}={\rm{diag}}\left(1\,,\,-1\,,\,-r^{2}\,,\,-r^{2}\sin^{2}\left(\theta\right)\right)\,\,\,\,\,\,.$
(2.14)

The ansätze for the complex scalar field $\phi$ and the gauge field $A_{\mu}$
are

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
v\,f\left(r\right)\,e^{i\,n\,\theta}\,\,\,\,\,\,,$ $\displaystyle A_{\mu}$
$\displaystyle=$
$\displaystyle\left[0\,,\,0\,,\,-\frac{n}{e}\,,\,-\frac{1}{e}\,\sin\left(\theta\right)\,h\left(r\right)\right]\,\,\,\,\,\,.$
(2.15)

The equation of motion is now

$\left\\{\frac{d^{2}f}{dr^{2}}-\lambda
v^{2}f\left(f^{2}-1\right)\right\\}+\left\\{\frac{2}{r}\,\frac{df}{dr}-\frac{1}{r^{2}}fh^{2}\right\\}=0\,\,\,\,\,\,.$
(2.16)

Requiring each bracket to vanish results in

$\displaystyle f\left(r\right)$ $\displaystyle=$
$\displaystyle\tanh\left(\alpha\,r\right)\,\,\,\,\,\,,\,\,\,\,\,\,\alpha^{2}=\frac{\lambda
v^{2}}{2}\,\,\,\,\,\,,$ $\displaystyle h\left(r\right)$ $\displaystyle=$
$\displaystyle\pm\,{\sqrt{2}}\,\sqrt{\frac{2\alpha\,r}{\sinh\left(2\alpha\,r\right)}}\,\,\,\,\,\,.$
(2.17)

In the case at hand, the non-vanishing components of the gauge field strength
are

$\displaystyle F_{r\,\varphi}$ $\displaystyle=$
$\displaystyle\partial_{r}\,A_{\varphi}=-\frac{1}{e}\,\sin(\theta)\frac{dh}{dr}=\mp\,\frac{1}{\sqrt{2}\,e}\,\sin(\theta)\frac{1}{r}\,\sqrt{\frac{2\alpha
r}{\sinh\left(2\alpha r\right)}}\left[1-\frac{2\alpha r}{\tanh\left(2\alpha
r\right)}\right]\,\,\,\,\,\,,$ $\displaystyle F_{\theta\,\varphi}$
$\displaystyle=$
$\displaystyle\partial_{\theta}\,A_{\varphi}=\mp\,\frac{1}{e}\cos(\theta)\,h(r)=\mp\,\frac{\sqrt{2}}{e}\cos(\theta)\,\sqrt{\frac{2\alpha
r}{\sinh\left(2\alpha r\right)}}\,\,\,\,\,\,,$ (2.18)

It can be seen that the component $F_{r\,\varphi}$ reaches a maximum for
$\theta=\pi/2$ and a very small value of $r$. Away from this maximum
$F_{r\,\varphi}$ tends rapidly to zero. On the other hand,
$F_{\theta\,\varphi}$ has two extrema for $\theta=0$ and $\theta=\pi$ at a
very small value of $r$ and goes to zero as $r$ increases.

Using the expression $B_{i}=\frac{1}{2}\sqrt{-g}\,\epsilon_{ijk}\,F^{jk}$,
with $\epsilon_{r\theta\varphi}=+1$, the components of the magnetic field are

$B_{\theta}=-F_{r\,\varphi}\,\,\,\,\,\,,\,\,\,\,\,\,B_{r}=\frac{F_{\theta\,\varphi}}{r^{2}}\,\,\,\,\,\,.$
(2.19)

These components reach their extrema around $r=0$ and tend rapidly to zero as
$r$ increases.

The corresponding energy, in the whole volume, is

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta\int_{0}^{\infty}dr\,\sqrt{-g}\,T_{00}$
(2.20) $\displaystyle=$ $\displaystyle
4\pi\,v^{2}\int_{0}^{\infty}dr\,r^{2}\left[\left(\frac{df}{dr}\right)^{2}+\frac{1}{r^{2}}\,f^{2}\,h^{2}+\frac{\lambda}{2}\,v^{2}\left(f^{2}-1\right)^{2}\right]\,\,\,\,\,\,.$

The equation of motion (2.16) together with $f(r)=\tanh(\alpha r)$ yield

$\displaystyle E$ $\displaystyle=$ $\displaystyle
4\pi\,v^{2}\int_{0}^{\infty}dr\,\left[\frac{1}{2}\partial_{r}\left(r^{2}\partial_{r}\,f^{2}\right)-\frac{\lambda}{2}\,v^{2}\,r^{2}\left(f^{4}-1\right)\right]\,\,\,\,\,\,$
(2.21) $\displaystyle=$
$\displaystyle-4\pi\,v^{2}\alpha^{2}\int_{0}^{\infty}dr\,r^{2}\left[\tanh^{4}\left(\alpha
r\right)-1\right]\,\,\,\,\,\,.$

The evaluation of the integral in the expression of $E$ involves the
dilogarithm function ${\text{Li}}_{2}\left(z\right)$ and the final result is

$E=\frac{4\pi}{3}\,\frac{v^{2}}{\alpha}\left(1+\frac{\pi^{2}}{3}\right)\,\,\,\,\,\,.$
(2.22)

In reaching this result we have used the identity (see [6] for instance)

${\text{Li}}_{2}\left(z\right)=-{\text{Li}}_{2}\left(\frac{1}{z}\right)-\frac{1}{2}\,\ln^{2}\left(-z\right)-\frac{\pi^{2}}{6}\,\,\,\,\,\,$
(2.23)

together with the special values ${\text{Li}}_{2}\left(0\right)=0$ and
${\text{Li}}_{2}\left(-1\right)=-\pi^{2}/12$.

In conclusion, we have found simple static solutions to the equations of
motion of a complex scalar field moving in a gauge field background. These are
given in both cylindrical and spherical coordinates. In cylindrical
coordinates, the gauge field background could be interpreted as representing a
vortex located in the $(r-\theta)$ plane. This configuration of fields has a
finite energy per unit length. Similarly, in spherical coordinates, the
magnetic field is in the form of localised lumps in the $(r-\theta)$ and
$(\theta-\varphi)$ planes. In this case, the energy available in the whole
space is finite.

##  3 Discussion

There are various questions raised by the solutions presented in this
paper222I am very grateful to the two anonymous reviewers for their scrutiny
of the results of this note.. We will discuss the settings in cylindrical
coordinates as this is the one relevant to the Nielsen-Olesen vortices. Let us
recall briefly the situation when the gauge field $A_{\mu}$ is dynamical. In
this case the theory is the full Abelian Higgs model as given by the action

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left(D_{\mu}\phi\right)^{\star}\left(D^{\mu}\phi\right)-\frac{\lambda}{2}\left(\phi^{\star}\phi-v^{2}\right)^{2}\right]\,\,\,,$
(3.1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength of the gauge field $A_{\mu}$.

Using cylindrical coordinates together with the Nielsen-Olesen ansätze (2.5),
the equations of motion of the Abelian Higgs model become

$\displaystyle{r}\,\frac{d}{dr}\left(\frac{1}{r}\frac{dh}{dr}\right)+2e^{2}v^{2}f^{2}\left(1-h\right)$
$\displaystyle=0$ $\displaystyle\,\,\,\,\,\,,$ (3.2)
$\displaystyle\frac{1}{r}\,\frac{d}{dr}\left(r\frac{df}{dr}\right)-n^{2}\,\frac{f}{r^{2}}\left(1-h\right)^{2}-\lambda
v^{2}f\left(f^{2}-1\right)$ $\displaystyle=0$ $\displaystyle\,\,\,\,\,\,.$
(3.3)

Similarly, the energy per unit length (along the $z$ direction) of the
Nielsen-Olesen vortex is

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\infty}dr\,r\left[\frac{1}{r^{2}}\left(\frac{1}{2}\,\frac{n^{2}}{e^{2}}\,\left(\frac{dh}{dr}\right)^{2}+n^{2}v^{2}f^{2}\,\left(1-h\right)^{2}\right)+v^{2}\left(\frac{df}{dr}\right)^{2}+\frac{\lambda}{2}\,v^{4}\left(f^{2}-1\right)^{2}\right]$
(3.4) $\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\infty}dr\,\left\\{\frac{1}{4}\,\frac{n^{2}}{e^{2}}\partial_{r}\left[\frac{1}{r}\partial_{r}\left(\left(1-h\right)^{2}\right)\right]+r\left[v^{2}\left(\frac{df}{dr}\right)^{2}+\frac{\lambda}{2}\,v^{4}\left(f^{2}-1\right)^{2}\right]\right\\}\,\,\,\,\,\,.$

The above equations correspond to the minimization of $E$ with respect to both
$h\left(r\right)$ and $f\left(r\right)$. The second equality is a result of
the use of the equations of motion.

Since in our study the gauge field $A_{\mu}$ is taken to be a non-dynamical
background, equation (3.2) was not taken into account. This raises the
question of how close the expressions

$\displaystyle f\left(r\right)$ $\displaystyle=$
$\displaystyle\tanh\left(\alpha\,r\right)\,\,\,\,,$ $\displaystyle
h\left(r\right)$ $\displaystyle=$ $\displaystyle
1\pm\,\frac{1}{n}\sqrt{\frac{2\alpha\,r}{\sinh\left(2\alpha\,r\right)}}\,\,\,\,\,,\,\,\,\,\,\,\alpha^{2}=\frac{\lambda
v^{2}}{2}\,\,\,\,\,\,$ (3.5)

are to the solutions of the equations of motion of the Abelian Higgs model ?

Notice that for $n$ sufficiently large, the function $h\left(r\right)$ is
nearly equal to $1$. Indeed, $h\left(r\right)$ is a monotonic function and
varies between $1\pm\frac{1}{n}$ and $1$ for $r\in\left[0\,,\,\infty\right]$.
On the other hand, $h\left(r\right)=1$ is a solution to (3.2). Therefore, as a
first interpretation, we could think of the two functions $f\left(r\right)$
and $h\left(r\right)$ as approximate solutions to the equations of motion of
the Abelian Higgs model for large winding number $n$.

The other feature of the study carried out in this article is that the energy
in (2.13) is independent of the coupling constant $\lambda$. This leads one to
think that the system is in a critical regime very much like when the
Bogomolny bound is saturated at the critical coupling $\lambda=e^{2}$ and
where the energy is $E=2\pi|n|v^{2}$ [7]. It is tempting to speculate that the
critical regime in our analyses might correspond to the formation of an
aggregate of a large number of vortices resulting in a single giant vortex.

Clusters of vortices, which are metastable, have been experimentally observed
in condensed matter physics. One of the early giant vortices was observed in
the superfluid Helium 4He with $n\approx 400$ [8]. More recently, dense
arrangements of single Abrikosov vortices were detected in strongly confined
superconducting condensates [9]. Giant vortices (with $n$ ranging from $7$ up
to $60$) were shown to form in a rapidly rotating dilute-gas Bose-Einstein
condensate [10]. It is well-known that the Abelian Higgs model (Ginzburg-
Landau theory) provides the theoretical framework for these condensed matter
physics phenomena. Therefore, the results presented here could be of use as
far as the formation of giant vortices is concerned.

Another point regarding the solution presented in this note is that it depends
very much on the expressions between the two brackets in (2.6). In other
words, different assemblages of the terms are possible. The only thing that
one can say about this issue is that equation (2.6) gives the general
expression of the gauge field background $h(r)$ in terms of the dynamical
field $f(r)$. Indeed, we have

$\displaystyle\left(1-h\right)^{2}=\frac{r^{2}}{n^{2}f^{2}}\left[\frac{1}{2}\frac{1}{r}\,\frac{d}{dr}\left(r\frac{df^{2}}{dr}\right)-\left(\frac{df}{dr}\right)^{2}-\lambda
v^{2}f^{2}\left(f^{2}-1\right)\right]\,\,\,\,\,\,.$ (3.6)

The restrictions on the function $f(r)$ are then : $i)$ the expression between
brackets in (3.6) is always positive and $ii)$ the energy

$\displaystyle E$ $\displaystyle=$ $\displaystyle
2\pi\,v^{2}\int_{0}^{\infty}dr\,\left[\frac{1}{2}\partial_{r}\left(r\partial_{r}\,f^{2}\right)-\frac{\lambda}{2}\,v^{2}\,r\left(f^{4}-1\right)\right]$
(3.7)

is finite and positive. The choice of the function $f(r)$ (satisfying the
points $i)$ and $ii)$ mentioned above) is at the moment a matter of trial and
error (and this is how we find our solution).

One might also want to compare the expressions (3.5) to the numerical
solutions of the Abelian Higgs model. Our function $f(r)$ and the magnetic
field $B_{r}$ have profiles similar to the one found using numerical analyses,
as can be seen from their sketshes in figure (2).

Furthermore, we have injected the expressions of $f(r)$ and $h(r)$ given in
(3.5) into the gauge field equation of motion (3.2) and evaluated this
graphically. The equation of motion of the gauge field is

$\displaystyle\partial_{\mu}\left(\sqrt{-g}\,F^{\mu\nu}\right)+ie\sqrt{-g}\left[\phi\left(D^{\nu}\phi\right)^{\star}-\phi^{\star}\left(D^{\nu}\phi\right)\right]$
(3.8) $\displaystyle=$
$\displaystyle-\frac{n}{e}\frac{1}{r}\left[r\frac{d}{dr}\left(\frac{1}{r}\frac{dh}{dr}\right)+2e^{2}v^{2}f^{2}\left(1-h\right)\right]=0\,\,\,\,\,.$

Since $\left(1-h\right)$ contains a factor of $\frac{1}{n}$, the dependence on
the winding number cancels and the statement made below is independent of the
value of $n$.

We have first traded $r$ for the dimensionless coordinate $\rho=evr$. As a
consequence, the only physical coupling constant left in (3.2) and (3.3) is
$\xi=\frac{\lambda}{e^{2}}$. We then tuned $\xi$ in such a way that the graph
corresponding to the left hand side of (3.2) is close to zero. Figure (3) is
obtained for $\xi=4.688$.

Figure 3: The gauge field equation (3.2) for
$\xi=\frac{\lambda}{e^{2}}=4.688$. The horizontal axis is the rescaled
coordinate $\rho=evr$.

Therefore, one could affirm that the expressions of $f(r)$ and $h(r)$ in (3.5)
provide a relatively good approximate solution to the equations of motion of
the Abelian Higgs model for the particular value of the coupling constant
$\xi=4.688$.

Using (3.5) in (3.4), the corresponding energy is given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{2\pi}{3}\,{v^{2}}\left(\frac{\lambda}{2e^{2}}\right)+\frac{2\pi}{3}\,{v^{2}}\left[-1+4\ln\left(2\right)\right]$
(3.9) $\displaystyle=$
$\displaystyle\frac{2\pi}{3}\,{v^{2}}\left[1.344+4\ln\left(2\right)\right]\,\,\,\,\,\,.$

This is to be compared to
$E=\frac{2\pi}{3}\,{v^{2}}\left[0.5+4\ln\left(2\right)\right]$ as read from
(2.13).

In summary, the expressions of $f(r)$ and $h(r)$ given in (3.5) furnish a
quite good approximate solutions to the equations of motion of the Abelian
Higgs model either for a large winding number or for a special value of the
coupling constant $\xi=\frac{\lambda}{e^{2}}$.

Finally, we should mention that the extension of the study carried out in this
paper to non-Abelian gauge theories is currently under investigation. The
prototype of such theories is the $SU(2)$ Georgi-Glashow [11] model with a ’t
Hooft-Polyakov ansatz [12, 13] for the gauge field $A_{\mu}^{a}$ and the
scalar field $\phi^{a}$ ($a=1,2,3$). If the gauge fields are fixed backgrounds
then one has only one equation of motion to solve. The whole issue is then to
obtain a finite energy.

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world adversarial examples, therefore better go closer to the future practical
and trustworthy deep learning.

## Acknowledgments

We thank Haojie Hao and Zhengquan Sun for their efforts in constructing the
sustainable open-source project of this paper.

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##  VI Biography Section

Jiakai Wang is now a Research Scientist in Zhongguancun Laboratory, Beijing,
China. He received the Ph.D. degree in 2022 from Beihang University (Summa Cum
Laude), supervised by Prof. Wei Li and Prof. Xianglong Liu. Before that, he
obtained his BSc degree in 2018 from Beihang University (Summa Cum Laude). His
research interests are Trustworthy AI in Computer Vision (mainly) and
Multimodal Machine Learning, including Physical Adversarial Attacks and
Defense and Security of Practical AI.
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Donghua Wang is currently pursuing the Ph.D. degree with the College of
Computer Science and Technology, Zhejiang University, Hangzhou, China. He is
also with the Defense Innovation Institute, Chinese Academy of Military
Science, Beijing, China. His research interests include adversarial machine
learning, AI safety, and image processing.
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Jin Hu is currently working toward the Ph.D. degree in the School of Computer
Science and Engineering, Beihang University and Zhongguancun Laboratory. His
research interests include adversarial attack, generative modeling and
trustworthy AI.
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Siyang Wu received the B.S. degree in Information Security from the School of
Cyber Science and Technology, Beihang University, Beijing, China, in 2023. He
is currently working toward the Ph.D. degree in Electronic and Information
with the School of Computer Science and Engineering, Beihang University and
Zhongguancun Laboratory. His research interests include AI safety, adversarial
defense and object detection.
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Tingsong Jiang received the B.Sc. and Ph.D. degrees from the School of
Electronics Engineering and Computer Science, Peking University, Beijing,
China, in 2010 and 2017, respectively. He is currently an Assistant Professor
with the Defense Innovation Institute, Chinese Academy of Military Science,
Beijing. His research interests include adversarial machine learning, AI
safety, and knowledge graph.
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Wen Yao received the M.Sc. and Ph.D. degrees in aerospace engineering from the
National University of Defense Technology, Changsha, China, in 2007 and 2011,
respectively. She is currently a Professor with the Defense Innovation
Institute, Chinese Academy of Military Science, Beijing, China. Her research
interests include spacecraft systems engineering, multidisciplinary design
optimization, uncertainty-based optimization, and data-driven surrogate
modeling and evolutionary optimization.
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Aishan Liu is an Assistant Professor in the State Key Laboratory of Software
Development Environment, Department of Computer Science and Engineering at
Beihang University. He received his Ph.D. degree in 2021 from Beihang
University, supervised by Prof. Wei Li and Prof. Xianglong Liu. His research
interests are centered around AI Safety and Security, with broad interests in
the areas of Adversarial Examples, Backdoor Attacks, Interpretable Deep
Learning, Model Robustness, Fairness Testing, AI Testing and Evaluation, and
their applications in real-world scenarios.
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Xianglong Liu is a Full Professor in School of Computer Science and
Engineering at Beihang University. He received BS and Ph.D degrees under
supervision of Prof. Wei Li, and visited DVMM Lab, Columbia University as a
joint Ph.D student supervised by Prof. Shih-Fu Chang. His research interests
include fast visual computing (e.g., large-scale search/understanding) and
robust deep learning (e.g., network quantization, adversarial attack/defense,
few shot learning). He received NSFC Excellent Young Scientists Fund, and was
selected into 2019 Beijing Nova Program, MSRA StarTrack Program, and 2015 CCF
Young Talents Development Program.
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