# Numerical Evaluation of a Soliton Pair with Long Range Interaction

Joachim Wabnig, Josef Resch, Dominik Theuerkauf,
Fabian Anmasser and Manfried Faber

###### Abstract

Within the model of topological particles (MTP) we determine the interaction
energy of monopole pairs, sources and sinks of a Coulombic field. The
monopoles are represented by topological solitons of finite size and mass,
described by a field without any divergences. We fix the soliton centres in
numerical calculations at varying distance. Due to the finite size of the
solitons we get deviations from the Coulomb potential at distances of a few
soliton radii. We compare the numerical results for these deviations with the
running of the coupling in perturbative QED.

##  1 Introduction

In Refs. [1, 2] we proposed a dynamical model for monopoles without any
singularities, the model of topological particles (MTP). Particles are
identified by topological quantum numbers, masses of particles originate in
field energy, charges show Coulombic behaviour and are quantised in units of
an elementary charge. Depending on the interpretation, charges can be either
electric or magnetic. The model and its many predictions were extensively
discussed in Refs. [2].

Magnetic monopoles were invented by Dirac in 1931 [3, 4] as quantised
singularities in the electromagnetic field. He found that their existence
would explain the quantisation of electric charge, proven in Millikan’s
experiments [5] but not explained by Maxwell theory [6]. Dirac monopoles have
two types of singularities, the Dirac string, a line-like singularity
connecting monopoles and antimonopoles, and the singularity in the centre of
the monopole, a singularity analogous to the singularity of point-like
electrons. Wu and Yang succeeded to formulate magnetic monopoles without the
line-like singularities of the Dirac strings by using either a fibre-bundle
construction with two different gauge fields [7], one for the northern and one
for the southern hemisphere of the monopole, or by a non-abelian SU(2) gauge
field in 3+1D [8, 9]. These Wu-Yang monopoles still suffer from the point-like
singularities in the centre. There are monopole solutions without any
singularity in the Georgi-Glashow model [10], the ’tHooft-Polyakov monopoles
[11, 12]. The Georgi-Glashow model, formulated in 3+1D, has 15 degrees of
freedom, an adjoint Higgs field with three degrees of freedom and an SU(2)
gauge field with $4\cdot 3=12$ field components. Only one degree of freedom is
needed for the Sine-Gordon model [13], a model in 1+1D. It is most interesting
that in addition to waves, it has kink and anti-kink solutions which interact
with each other. The kink-antikink configurations are attracting and the kink-
kink configurations repelling. The simplicity of the Sine-Gordon model
inspired Skyrme [14, 15, 16, 17] to a model in 3+1D with a scalar SU(2)-valued
field, i.e. three degrees of freedom. Stable topological solitons (Skyrmions)
emerge in that model with the properties of particles, interacting at short
range. MTP was inspired by the simplicity and physical content of the Sine-
Gordon model, it was first formulated in [1]. It has the same degrees of
freedom as the Skyrme model, but uses a different Lagrangian. Its relations to
electrodynamics and symmetry breaking were discussed in [2, 18, 19, 20].

In this article, we want to concentrate on numerical determinations of the
interaction energy for a pair of charges, represented by a soliton-antisoliton
pair. Due to the non-existence of magnetic monopoles, comparisons to nature
are possible only for electric charges and the predictions of QED.

In Sect. 2 we repeat the formulation and some basic properties of the model,
in Sect. 3 we present the numerical formulation in cylindrical coordinates and
in Sect. 4 we show first results of the calculations and compare the results
to perturbative QED.

##  2 Summary of the MTP

As described in Ref. [2] we use the SO(3) degrees of freedom of spatial
Dreibeins to describe electromagnetic phenomena. The calculations get simpler
using SU(2) matrices,

$Q(x)=\mathrm{e}^{-\mathrm{i}\alpha(x)\vec{\sigma}\vec{n}(x)}=\underbrace{\cos\alpha(x)}_{q_{0}(x)}-\mathrm{i}\vec{\sigma}\underbrace{\vec{n}\sin\alpha(x)}_{\vec{q}(x)}\in
SU(2)\cong\mathbb{S}^{3}$ (2.1)

in Minkowski space-time as field variables, where arrows indicate vectors in
the 3D algebra of su(2) with the basis vectors represented by the Pauli
matrices $\sigma_{i}$. The Lagrangian of MTP reads,

$\mathcal{L}_{\mathrm{MTP}}(x):=-\frac{\alpha_{f}\hbar
c}{4\pi}\left(\frac{1}{4}\,\vec{R}_{\mu\nu}(x)\vec{R}^{\mu\nu}(x)+\Lambda(x)\right)\quad\textrm{with}\quad\Lambda(x):=\frac{q_{0}^{6}(x)}{r_{0}^{4}},$
(2.2)

with

$\vec{R}_{\mu\nu}:=\vec{\Gamma}_{\mu}\times\vec{\Gamma}_{\nu}\quad\textrm{and}\quad\left(\partial_{\mu}Q\right)Q^{\dagger}=:-\mathrm{i}\vec{\sigma}\vec{\Gamma}_{\mu}.$
(2.3)

We get contact with nature by relating the electromagnetic field strength
tensor to the dual of the curvature tensor $\vec{R}_{\mu\nu}$,

$\vec{F}_{\mu\nu}:=-\frac{e_{0}}{4\pi\epsilon_{0}}^{\star\\!\\!}\vec{R}_{\mu\nu}.$
(2.4)

At large distances $d$ from the sources, measured in units of the soliton
scale parameter $r_{0}$, the non-abelian field strength gets abelian,

$\vec{F}_{\mu\nu}\stackrel{{\scriptstyle\footnotesize
d>>r_{0}}}{{\to}}(\vec{F}_{\mu\nu}\vec{n})\vec{n}.$ (2.5)

MTP has four different classes of topologically stable single soliton
configurations. Their representatives read,

$n_{i}(x)=\pm\frac{x^{i}}{r},\quad\alpha(x)=\frac{\pi}{2}\mp\arctan\frac{r_{0}}{r}=\begin{cases}\arctan\frac{r}{r_{0}}\\\
\pi-\arctan\frac{r}{r_{0}}\end{cases}$ (2.6)

The imaginary part of their Q-fields are schematically depicted in Table 1.

$\vec{n}=\vec{r}/r$ | $\vec{n}=-\vec{r}/r$ | -$\vec{n}=\vec{r}/r$ | $\vec{n}=\vec{r}/r$
---|---|---|---
$q_{0}\geq 0$ | $q_{0}\leq 0$ | $q_{0}\geq 0$ | $q_{0}\leq 0$
$Z=1$ | $Z=-1$ | $Z=-1$ | $Z=1$
$\mathcal{Q}=\frac{1}{2}$ | $\mathcal{Q}=\frac{1}{2}$ | $\mathcal{Q}=-\frac{1}{2}$ | $\mathcal{Q}=-\frac{1}{2}$
|  |  |
Table 1: Scheme of single soliton configurations. The fields $\vec{n}$ and
$q_{0}$ and the topological quantum numbers $Z$ and $\mathcal{Q}$ are
indicated. The diagrams show the imaginary components
$\vec{q}=\vec{n}\sin\alpha$ of the soliton field, in full red for the
hemisphere with $q_{0}>0$ and in dashed green for $q_{0}<0$.

The fields (2.6) are solutions of the non-linear Euler-Lagrange equations [2].
They differ in two quantum numbers related to charge and spin. In the minimum
of the potential the $Q$-field is purely imaginary. Therefore, the sign $Z$ of
the $\vec{n}$-field in Eq. (2.6) decides about the charge quantum number
defined by a map $\Pi_{2}(\mathbb{S}^{2})$. Field configurations are further
characterised by the number $\mathcal{Q}$ of coverings of $\mathbb{S}^{3}$, by
the map $\Pi_{3}(\mathbb{S}^{3})$. With the sign of $\mathcal{Q}$ we define an
internal chirality $\chi$ and with the absolute value of $\mathcal{Q}$ the
spin quantum number $s$,

$\mathcal{Q}=\chi\cdot s\quad\textrm{with}\quad s:=|\mathcal{Q}|.$ (2.7)

The spin quantum number of two-soliton configurations indicates that $\chi$
can be related to the sign of the magnetic spin quantum number.

The configurations within each of the four classes may differ by Poincaré
transformations. The rest mass of solitons,

$E_{0}=\frac{\alpha_{f}\hbar c}{r_{0}}\frac{\pi}{4},$ (2.8)

can be adjusted to the electron rest energy $m_{e}c_{0}^{2}=0.510\,998\,95$
MeV by choosing,

$r_{0}=2.213\,205\,16\leavevmode\nobreak\ \mathrm{fm},$ (2.9)

a scale which is of the order of the classical electron radius. The four
parameters $r_{0},c_{0},E_{0}$ and $e_{0}$ correspond to the natural scales of
the four quantities length, time, mass and charge of the SI, of the Système
international d’unités, involved in this model. Eq. (2.8) can therefore be
interpreted as a relation between $\alpha_{f}$ and $\hbar$.

Two solitons of different charge can be combined to two topological different
field configurations. Schematic diagrams for such configurations are shown in
Fig. 1.

Figure 1: Schematic diagrams depicting the imaginary part
$\vec{q}=\vec{n}\sin\alpha$ of the $Q$-field of two opposite unit charges by
arrows. The lines represent some electric flux lines. We observe that they
coincide with the lines of constant $\vec{n}$-field. The configurations are
rotational symmetric around the axis through the two charge centres. In the
red/green arrows, we encrypt also the positive/negative values of
$q_{0}=\cos\alpha$. For $q_{0}\to 0$ the arrows are getting darker or black.
The left configuration belongs to the topological quantum numbers
$\mathcal{Q}=S=0$ and the right one to $\mathcal{Q}=S=1$, where $S$ is the
total spin quantum number of the configuration. The numerical calculations
will be presented for the spin singlet case.

##  3 Dipoles on a cylindrical lattice

According to the Lagrangian 2.2 there are two contributions to the energy
density of a static dipole,

$\mathcal{H}\overset{\footnotesize(\ref{LagrangianMTP})}{=}\frac{\alpha_{f}\hbar
c}{4\pi}\left(\frac{1}{4}\,\vec{R}_{ij}\vec{R}^{ij}+\Lambda\right)=:\mathcal{H}_{\mathrm{cur}}+\mathcal{H}_{\mathrm{pot}},$
(3.1)

the electric part of the curvature energy and the potential energy. A detailed
derivation of these energy contributions of the $Q(x)$ field in cylindrical
coordinates was given in [21]. The curvature energy reads in these
coordinates,

$\displaystyle\mathcal{H}_{\mathrm{cur}}=\frac{\epsilon_{0}}{2}$
$\displaystyle\big{(}\vec{E}_{r}^{2}+\vec{E}_{\varphi}^{2}+\vec{E}_{z}^{2}\big{)}=\frac{\alpha_{f}\hbar
c}{8\pi}\frac{1}{r^{2}}\bigg{\\{}q_{r}^{2}\big{[}(\partial_{z}q_{0})^{2}+(\partial_{z}q_{r})^{2}+(\partial_{z}q_{z})^{2}\big{]}$
(3.2)
$\displaystyle+\frac{r^{2}}{q_{0}^{2}}\big{(}\partial_{r}q_{r}\partial_{z}q_{z}-\partial_{z}q_{r}\partial_{r}q_{z}\big{)}^{2}+q_{r}^{2}\big{[}(\partial_{r}q_{0})^{2}+(\partial_{r}q_{r})^{2}+(\partial_{r}q_{z})^{2}\big{]}\bigg{\\}}.$

Due to the cylindrical symmetry of the field configurations in Fig. 1 it is
sufficient to restrict the numerical integrations to a lattice in the
$rz$-plane, see Fig. 2.

Figure 2: Two solitons of opposite charge are separated by a distance $d$ on a
discrete lattice in the $rz$ plane. The total number of points is $n_{r}\times
2n_{z}$ and the lattice constant in both directions is $a$.

Inside the lattice, we minimise the energy according to the MTP. Outside, we
use Maxwell’s electrodynamics [6] to calculate the energy. The numerical
computations of dipole field configurations were built up on the algorithms
discussed in [22, 23, 24] and their accuracy was tested in [21] by applying it
to the analytical solvable monopole configuration. The coordinates in radial-
and $z$ direction are denoted by $\bar{r}\in\\{0,1,\dots,n_{r}\\}$ and
$\bar{z}\in\\{-n_{z},-n_{z}+1,\dots,\,n_{z}\\}$, respectively. The radius of a
soliton, $\bar{r}_{0}\in\mathbb{N}$, determines the length-scale and thereby
the lattice spacing,

$a\overset{\footnotesize(\ref{r0m2})}{=}\frac{2.21\mathrm{fm}}{\bar{r}_{0}}.$
(3.3)

A numerical gradient minimization algorithm is used to find the dipole
configurations, which minimizes the total energy according to Eq. (3.1) for
given distance $d=a\bar{d}$ between the opposite charges.

A good guess for the initial configuration helps to reduce the run-time of the
computation. For the real part of the initial $Q$-field we get from the
profile function of a single monopole,

$q_{0}(r,z>0)=\frac{\bar{r}_{0}}{\sqrt{\bar{r}_{0}^{2}+\bar{r}^{2}+(\bar{z}-\frac{\bar{d}}{2})^{2}}},\qquad
q_{0}(r,z\leq
0)=\frac{\bar{r}_{0}}{\sqrt{\bar{r}_{0}^{2}+\bar{r}^{2}+(\bar{z}+\frac{\bar{d}}{2})^{2}}}.$
(3.4)

The modulus of the vector part of the $Q$-field reads,

$|\vec{q}(x)|\overset{\footnotesize(\ref{FeldVariablen})}{=}\sqrt{1-q_{0}(x)^{2}}.$
(3.5)

An appropriate initial direction $\vec{n}$ of the $\vec{q}$ field we get from
the coincidence between electric field lines and lines of constant
$\vec{n}$-field which we observe in Fig. 1. The field lines are defined by the
differential equation,

$\frac{\mathrm{d}r}{E_{r}}=\frac{\mathrm{d}z}{E_{z}},$ (3.6)

with

$\displaystyle E_{r}(r,z)$
$\displaystyle=\frac{e_{0}}{4\pi\epsilon_{0}}\left(\frac{r}{(r^{2}+(z+\frac{d}{2})^{2})^{3/2}}-\frac{r}{(r^{2}+(z-\frac{d}{2})^{2})^{3/2}}\right),$
(3.7) $\displaystyle E_{z}(r,)$
$\displaystyle=\frac{e_{0}}{4\pi\epsilon_{0}}\left(\frac{z+\frac{d}{2}}{(r^{2}+(z+\frac{d}{2})^{2})^{3/2}}-\frac{z-\frac{d}{2}}{(r^{2}+(z-\frac{d}{2})^{2})^{3/2}}\right).$

After solving the differential Eq. (3.6), we find an analytical expression for
the initial polar angle of the $\vec{q}$ field,

$\theta(r,z)=\arccos\left(1+\frac{z-\frac{d}{2}}{\sqrt{r^{2}+(z-\frac{d}{2})^{2}}}-\frac{z+\frac{d}{2}}{\sqrt{r^{2}+(z+\frac{d}{2})^{2}}}\right).$
(3.8)

Up to now, only points on the lattice are included in the energy calculation.
Outside the lattice, we use Maxwell’s electrodynamics to take the curvature
energy into account. The error, for neglecting potential and tangential energy
components outside the lattice, was shown to be tiny ($<$ 0.2%) for a single
monopole, if the distance from the soliton core to the lattice boundaries is
greater than $10a$ [21]. Hence, we find the energy outside the lattice by
$H_{\mathrm{out}}=\frac{\epsilon_{0}}{2}\int_{\mathbb{R}^{3}\setminus\text{latt}}\mathrm{d}^{3}x|\vec{E}|^{2}$,
which is numerically integrated using a trapezoidal summation, with the
electric field strength given by Eq. 3.7.

The principle of energy minimisation is exploited to find the dipole field
configuration and its associated energy. The minimisation algorithm is taken
from https://de.mathworks.com/matlabcentral/fileexchange/75546-conjugate-
gradient-minimisation. It is a more dimensional conjugate gradient method to
find a local minimum of an energy functional depending on the $Q$ field
components at each lattice point, $E=E(q_{0}^{i},q_{r}^{i},q_{z}^{i})$ with
$i\in[1,2,...,n_{r}(2n_{z}+1)]$,

$E=\int_{\mathrm{lattice}}\mathrm{d}^{3}x\leavevmode\nobreak\
\mathcal{H}_{\mathrm{lat}}+H_{\mathrm{out}},$ (3.9)

where $\mathcal{H}_{\mathrm{lat}}$ is the lattice version of $\mathcal{H}$ of
Eq. (3.1).

Fixing the centre of solitons only, however, for small dipole distances $d$
annihilation of the two solitons is observed. To suppress this behaviour, it
was suggested in Ref. [21] to smooth the soliton field by adding to the energy
functional a term,

$E_{\mathrm{min}}:=E+\lambda\frac{\alpha_{f}\hbar
c}{a}\sum_{i}\left(\frac{\partial}{\partial\bar{x}_{i}}\frac{\vec{q}}{|\vec{q}|}\right)^{2},\quad
i=\\{\bar{r},\bar{z}\\}.$ (3.10)

in the minimisation process, where $\lambda=\bar{\lambda}/a^{2}$. During the
computations, it became clear that this additional term tends to suppress the
interaction between the solitons. Therefore, we could apply tiny values of
$\bar{\lambda}$ only. This was sufficient to approach smaller distances.

The parameters, used for the minimisation process are listed in table 2.

Parameter | Description | Used value
---|---|---
$i_{\mathrm{max}}$ | Max. number of iterations | 5000
$\Delta_{\mathrm{grad}}$ | Min. gradient difference for two consecutive iterations | 1e-6
$\Delta_{\mathrm{fun}}$ | Min. energy difference for two consecutive iterations | 1e-30
$\Delta_{\mathrm{com}}$ | Lower bound on the step size of the norm of the Q field | 1e-30
Table 2: Input parameters for the energy minimisation algorithm.

The lattice spacing we fix to
$a\overset{\footnotesize(\ref{Gitterkonst})}{=}r_{0}/3=0.738$ fm. This spacing
is small enough to get a reasonable approximation to the rest energy of
electrons.

For a dipole, it is not possible to use periodic boundary conditions. While
minimising the energy, we fix the soliton field $Q$ at the edge of the lattice
$\bar{r}=0,\bar{r}=n_{r},\bar{z}=\pm n_{z}$ according to the initial
configuration discussed in Sect. 3. This implies the assumption $q_{0}=0$
outside the lattice, and leads to a small error in the potential energy which
is difficult to avoid.

Nevertheless, boundary conditions have large influence on the numerical
results. For fixed lattice sizes $n_{r}\times(2n_{z}+1)$ and varying size $d$
of the dipole, we can not reach large $d$ values due to boundary effects. It
turns out to be much better to increase the lattice size with increasing $d$
keeping the distance from the boundary constant. The lattice sizes we are
using for our calculations are in r- and z-directions by 45 lattice spacings
larger than the relative distance $d$ between the charges. We use $n_{r}=45$
and $n_{z}=\frac{\bar{d}}{2}+45$ and all even distances $\bar{d}\leq 420$.

Due to the finite size of the lattice there remains a deviation of the
asymptotic energy $E_{\infty}$ of the dipole from the mass of two non-
interacting solitons, $2m_{e}c_{0}^{2}=1.021\,997\,90$ MeV. Since we fix the
centers of solitons at lattice sites, we get a weak even-odd effect on lattice
sizes for $E_{\infty}$. For the asymptotic values $E_{\infty}$ of the
calculated energies we get $1.009\,203$ MeV for even and $1.009\,198$ MeV for
odd $r$.

For the distance $\bar{d}=16$ we show some results of the minimisation process
for the spin singlet configuration. In Fig. 3 we compare the $\vec{q}$-field
in the initial and final configuration in a restricted region.

Figure 3: The modification of the imaginary part $\vec{q}(r,z)$ of the soliton
field is represented by arrows. The initial configuration is shown in blue and
the result of the minimisation process in red. Only a subregion of the
$45\times 111$-lattice is shown. Due to the finite size of solitons,
$\bar{r}_{0}=3$, the centres of the charges the length of the arrows
approaches zero as described by Eq. (2.6).

The minimisation modifies especially the field in the region between the
charges. The final values of the real part $q_{0}$ of the soliton field are
shown in Fig. 4. We observe that the distribution of $q_{0}$ values seems much
broader than the distribution of the energy density in Fig. 5 with very
pronounced peaks reflecting the self energy $\mathcal{H}$ of charges according
to Eq. (3.1).

Figure 4: $q_{0}$ distribution after minimisation for the configuration of
Fig. 3. Figure 5: Energy density after minimization for the configuration of
Fig. 3. Despite the regularisation of the self-energy of charges by the finite
soliton size $\bar{r}_{0}=3$ there exist still very pronounced peaks of the
energy density $\mathcal{H}$ according to Eq. (3.1) at the charge centres.

##  4 Comparison to and deviations from Coulomb potential

From the analytical calculations in Ref. [18] we know that our model respects
the inhomogeneous Maxwell equations. Therefore, we expect for point-like
charges the classical Coulombic behaviour at large distances $d$ between the
charges of the dipole. This reads after adjusting the asymptotic energies

$E_{\mathrm{cl}}(d)=E_{\infty}-\frac{\alpha_{f}\hbar
c_{0}}{d}\quad\textrm{with}\quad\alpha_{f}=137.036,\hbar
c_{0}=197.327\,\textrm{MeV fm}.$ (4.1)

In Fig. 6 we compare $E_{\mathrm{cl}}(d)$ to the numerical results for the
energy $E(d)$, see Eq. (3.9), of the dipole in the spin singlet state. The
left diagram depicts the comparison in the region $24\leavevmode\nobreak\
a\leq\bar{d}\leq 420\leavevmode\nobreak\ a$ in steps $\Delta\bar{d}=2\,a$.
There are no data for distances smaller than $d=24\leavevmode\nobreak\ a$ for
$a=r_{0}/3=0.7377\,$fm, since the soliton pair cannot be stabilised just by
fixing the soliton field to $Q=1$ at the positions of soliton centers. The
small deviations of the numerical values $E(d)$ from $E_{\mathrm{cl}}(d)$ at
$d<50\leavevmode\nobreak\ $fm are magnified in the right diagram of Fig. 6.

Figure 6: Comparison between the classical Coulomb potential
$E_{\mathrm{cl}}(d)$ and the numerical values of the energy $E(d)$ according
to Eq. (3.9) of a static electron-positron pair as a function of the
separation $d$ between the two charges. Due to the finite size of solitons we
observe deviations of the energy of the static dipole (red circles) from the
classical value for point-like sources (full blue line).

We describe these deviations by a $d$-dependence of the fine structure
constant

$E(d)=E_{\infty}-\frac{\alpha(d)\,\hbar c_{0}}{d}.$ (4.2)

In Fig. 7 we can nicely observe whether the asymptotic energy $E_{\infty}$ is
well chosen, as tiny variations of $E_{\infty}$ destroy the asymptotic
behaviour of $\alpha(d)$ at large $d$. The running of the coupling $\alpha(d)$
already detected in Fig. 6 is clearly seen in Fig. 7 below distances of 50 fm.

Figure 7: Influence of the choice of $E_{\infty}$ to the asymptotic behaviour
of $\alpha(d)$ according to Eq. (4.2). Observe that tiny variations of
$E_{\infty}$ lead to wrong asymptotic behaviour at large charge separations.

To be able to reach smaller values of $d$ before the attraction gets too
strong and the dipole is collapsing, Ref. [21] suggested to minimise the
energy funcional $E_{\mathrm{min}}$ of Eq. (3.10) which suppresses photonic
excitations. In the computations, see Fig. 8, it turns out that it helps to
approach shorter distances, but at distances of around 30 fm it may lead to
unphysical stiffness of the fields and to a strong reduction of $\alpha(d)$.

Figure 8: The suppression of photonic excitations by the $\lambda$-dependent
term in Eq. (3.10) allows to approach shorter distances but leads to an
unphysical reduction of $\alpha(d)$ around distances of 30 fm, as can be
clearly observed for $\lambda=0.1/a^{2}$.

This unphysical behaviour starts to be observable around
$\lambda=0.005/a^{2}$. Looking carefully at Fig. 7 we see a similiar but very
tiny additional stiffness also in the $\lambda=0$ results. It is not yet clear
whether the minimum in $\alpha(d)/\alpha_{f}$ at $d\approx 50$f̃m is a
physical effect or an error of the approximation, e.g. of the boundary
conditions. Due to its small size this effect is not really visible in Fig. 8
at $\lambda=0$.

For tiny $\lambda\leq 0.005/a^{2}$ we can get to smaller $d$-values. The
corresponding data is plotted in Fig. 9. It is interesting to compare the rise
of the coupling at small $d$ to the perturbative calculation in QED as
described in ref. [25]. For comparison we show the prediction of first order
perturbation theory and its long distance approximation which is well known as
Uehling potential.

Figure 9: The strong attraction at short distances does not allow to stabilize
the distance $d$ of the charges just by fixing the centers of solitons with
the energy functional (3.9). One can get to shorter distances using the
functional of Eq. (3.10) for the minimisation procedure. For tiny values of
$\lambda$ the potential is only weakly distorted. The rise of $\alpha(d)$ is
compared with the result of first order perturbation theory (4.3) in QED and
its long distance approximation, the Uehling potential (A.25).

In QED the interaction between charges is described by photon exchange. The
algebraic form of the photon propagator determines therefore the strength of
the interaction. The perturbative corrections to the tree level photon
propagator are divergent. Already the first order contribution, shown by the
electron-loop diagram in Fig. 10, is divergent and needs regularisation and
renormalisation.

Figure 10: Electron-loop diagram as contribution to the photon propagator at
four-momentum $q$ in first order perturbation theory.

A perfect regularisation method is dimensional regularisation. Its success is
understood from the conservation of symmetries besides scale invariance. In
the renormalisation procedure an infinite photon self-energy is subtracted. It
is adjusted in such a way that the divergence of the photon propagator is
cancelled. Invariance under the finite terms which are subtracted in this
procedure requests a modification of the electric charge at large momentum
transfers and small distances. This is usually interpreted as a modification
of the photon structure by virtual electron-positron pairs. This prediction of
perturbation theory leads to a radial dependent fine structure constant

$\alpha(r)=\alpha_{f}\Big{\\{}1+\underbrace{\frac{2\alpha_{f}}{3\pi}\int_{1}^{\infty}\mathrm{d}\gamma\frac{\mathrm{e}^{-2\mu\gamma
r}}{\gamma}\sqrt{1-\frac{1}{\gamma^{2}}}\Big{(}1+\frac{1}{2\gamma^{2}}\Big{)}}_{\Delta(r)}\Big{\\}},$
(4.3)

where $\mu$ is the reduced Compton wavelength of electrons and $\gamma$ is the
relativistic $\gamma$ factor for the mass of a moving particle. The correction
can be evaluated by numerical integration and leads to the curve
“perturbative” in Fig. 9. The approximation for large $\mu r$, the Uehling
potential, reads [25]

$\alpha(r)=\alpha_{f}\Big{\\{}1+\frac{\alpha_{f}}{\sqrt{2\pi}}\frac{\mathrm{e}^{-2\mu
r}}{(2\mu r)^{3/2}}\Big{\\}}.$ (4.4)

It is depicted for comparison in Fig. 9.

##  5 Conclusion

In the model of topological particles we can describe monopoles and their
interaction without any divergencies. Since analytic solutions of this model
are known for one-particle systems only, we have to analyse interacting
systems numerically. In this first calculation we investigate the classical
monopole-antimonopole potential in the spin zero state. This is an axial
symmetric configuration which can be studied in the rz-plane in cylindrical
coordinates. At large separation $d$ of the dipole the potential is purely
Coulombic. The stable soliton states have finite size $r_{0}$ of the order of
the classical electron radius. This leads to a running of the charge at short
distances. It is a natural question how this running compares to the running
of the coupling in QED. The increase of the coupling appears at roughly the
same length scale as predicted by perturbative QED.

In further calculations the accuracy of the calculation should be enlarged
using adaptive lattices. This would allow to diminish the influence of
boundary effects. Since these first numerical calculations are done in the
static limit one should expect major modifications for more realistic
dynamical scenarios. Especially interesting would be to insert the potential
which results from such more detailed calculation in the Schrödinger and Dirac
equation [26]. This would result in shifts of the Bohr energies which could be
compared with the Lamb shift. Further calculations should be done in the spin
one state and for the repulsive system of two equal charges where the
cylindrical symmetry is lost.

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##  Appendix A More details of perturbative calculation

The free propagator of a photon with momentum $q_{\mu}$, see the Feynman rule
(B.2) is transversal, i.e. proportional to the transversal projection operator
${}^{T}\\!P^{\mu\nu}:=g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}$. Due to the
Ward identity it remains transversal to all orders of magnitude. Summing up
the infinite geometrical serious of 1-particle-irreducible diagrams

$\mathrm{i}\Pi^{\mu\nu}=\mathrm{i}q^{2}\,\Pi(q^{2})\,^{T}\\!P^{\mu\nu}$ (A.1)

ends up in the expression

$\displaystyle\includegraphics[scale={0.4}]{bilder/diaPhotonSelbst}=\mathrm{i}D^{\prime\mu\nu}(q)=$
(A.2) $\displaystyle\hskip
28.45274pt=\mathrm{i}D^{\mu\nu}(q)+\mathrm{i}D^{\mu\kappa}(q)\,\mathrm{i}\Pi^{\kappa\lambda}(q)\,\mathrm{i}D^{\lambda\nu}(q)+\dots=-\mathrm{i}\,\frac{{}^{T}\\!P^{\mu\nu}}{q^{2}[1-\Pi(q^{2})]}$

for the photon propagator, where the self-energy $\Pi(q^{2})$ can be
determined in perturbation theory as a function of

$q^{2}=q_{0}^{2}-\vec{q}^{\;2},$ (A.3)

see Sect. B resulting in expression (B.7) for $\Pi(q^{2})$.

In QED the Coulomb interaction is described by one-photon exchange. This is
possible, since the free photon propagator appears also in the momentum
representation of the Coulomb potential

$V(\vec{q})\overset{\footnotesize(\ref{VrErgeb})}{=}-\alpha_{f}\hbar
c_{0}\,\frac{4\pi}{\vec{q}^{\;2}+\mu_{\gamma}^{2}}$ (A.4)

in the limit of vanishing photon mass

$m_{\gamma}=:\frac{\mu_{\gamma}\hbar}{c_{0}}.$ (A.5)

From the comparison of the QED expression (LABEL:propKorr) with the Fourier
transform (A.4) of the Coulomb potential (A.9) we conclude that the QED-
modification of the Coulomb potential has to be determined in the region
$q^{2}=-\vec{q}^{\;2}$, i.e.

$q_{0}=0\quad\Rightarrow\quad
q^{2}\overset{\footnotesize(\ref{4erImpulsQ})}{=}-\vec{q}^{\;2}=:-Q^{2}$ (A.6)

of the photon momentum in the off mass shell region. From Eq. (LABEL:propKorr)
we read the perturbative correction

$-\frac{1}{q^{2}}\overset{\footnotesize(\ref{region})}{=}\frac{1}{Q^{2}}\;\;\stackrel{{\scriptstyle\footnotesize(\ref{propKorr})}}{{\mapsto}}\;\;\frac{1}{Q^{2}[1-\Pi(-Q^{2})]}.$
(A.7)

Restricting the geometrical series which we have summed up in Eq.
(LABEL:propKorr) to the first term, we get the substitution

$\frac{1}{Q^{2}}\;\;\stackrel{{\scriptstyle\footnotesize(\ref{propKorr})}}{{\mapsto}}\;\;\frac{1}{Q^{2}}[1+\Pi(-Q^{2})].$
(A.8)

If we insert this perturbative modification into the Coulomb potential

$V(r):=-\frac{\alpha_{f}\hbar
c_{0}}{r}\mathrm{e}^{-\mu_{\gamma}r}=-\frac{\alpha_{f}\hbar
c_{0}}{r}\frac{1}{\mathrm{i}\pi}\int_{-\infty}^{\infty}\mathrm{d}Q\frac{Q\,\mathrm{e}^{\mathrm{i}Qr}}{Q^{2}+\mu_{\gamma}^{2}}$
(A.9)

we get the QED-correction to the potential at $\mu_{\gamma}=0$

$\delta
V(r)\overset{{}^{(\ref{VrErgeb})}}{\underset{{}^{(\ref{CoulKorr1O})}}{=}}-\frac{\alpha_{f}\hbar
c_{0}}{r}\frac{1}{\mathrm{i}\pi}\int_{-\infty}^{\infty}\mathrm{d}Q\frac{\mathrm{e}^{\mathrm{i}Qr}}{Q}\Pi(-Q^{2})\Big{\\}}.$
(A.10)

First order perturbation theory results in the photon self-energy

$\Pi(q^{2})\overset{\footnotesize(\ref{renormPi})}{=}\frac{2\alpha_{f}}{\pi}\int_{0}^{1}\mathrm{d}x\;x(1-x)\log\Big{[}1-x(1-x)\frac{q^{2}}{\mu^{2}})\Big{]}$
(A.11)

and the correction term to the potential

$\delta
V(r)\overset{{}^{(\ref{DeltaV})}}{\underset{{}^{(\ref{Einschleifen})}}{=}}\mathrm{i}\,\frac{2\alpha_{f}^{2}\hbar
c_{0}}{\pi^{2}r}\int_{-\infty}^{\infty}\mathrm{d}Q\frac{\mathrm{e}^{\mathrm{i}Qr}}{Q}\int_{0}^{1}\mathrm{d}x\,x(1-x)\log\Big{[}1+x(1-x)\,\frac{Q^{2}}{\mu^{2}})\Big{]}\Big{\\}}.$
(A.12)

Due to the antisymmetry of the integrand only the odd term, the sin-function
in the exponential, should contribute to the $Q$-integral. But due to the
strong oscillation of the sin-function the integral is difficult to evaluate.
It is much easier to evaluate the $Q$-integral in the complex plane. The pole
from the $1/Q$-term is suppressed by the vanishing logarithm at $Q=0$. But on
the imaginary $Q$-axis there are branch cuts in the regions of $Q$-values with
$|\textrm{Im }Q|\geq 2\mu$. For these $Q$-values there is a region of
$x(1-x)$-values, around the maximal value $1/4$, where the logarithm gets
negative. Physically, this is the region of positive $q^{2}=-Q^{2}$, where
annihilation takes place and the energy is concentrated in the mass of the
fused object which cannot be lower than $2\mu$. For the evaluation of the
integral in the complex plane we use the absence of poles. Therefore, the
complex integral can be shifted as much as possible to the upper half-plane
where the contributions of the two quarter circles in the first and the second
quadrant are exponentially suppressed. The branch cut has to be circumvented
by two straight paths at $\textrm{Re }Q=0$ from $\textrm{Im }Q:=q=\infty$ down
to $q=2\mu$ and back to $+\infty$. These two straight paths give only an
imaginary contribution since the two real contributions are equal and cancel
due to the opposite integration direction. The argument

$z:\overset{\footnotesize(\ref{DeltaVpert})}{=}=1+x(1-x)\,Q^{2}/\mu^{2}=1-x(1-x)\,q^{2}/\mu^{2},\quad
Q=\mathrm{i}q$ (A.13)

of the logarithm in Eq. (A.12) is negative in the interval

$x_{1,2}=\frac{1}{2}\Big{(}1\pm\sqrt{1-\frac{4\mu^{2}}{q^{2}}}\big{)}:=\frac{1}{2}(1\pm\beta)\quad\textrm{with}\quad\beta:=\sqrt{1-\frac{4\mu^{2}}{q^{2}}},$
(A.14)

which is characterised by real values of the velocity $\beta$ of a fusing
particle pair. In this region of real $\beta$-values the imaginary part of

$\log z=\log r\pm\mathrm{i}\pi\quad\textrm{for}\quad z=-r\pm\mathrm{i}0$
(A.15)

has opposite sign at the opposite sites $Q=\pm\eta+\mathrm{i}q$ of the branch
cut

$\textrm{Im }\log
z\overset{\footnotesize(\ref{LogVerzw})}{=}\pi\,\textrm{sign(Im
}z)\overset{\footnotesize(\ref{argLog})}{=}\pi\,\textrm{sign(Im
}Q^{2})\overset{\footnotesize(\ref{argLog})}{=}\pi\,\textrm{sign(Re }Q)$
(A.16)

depending on the infinitesimal real part of $Q$. The region with negative real
part of $z$ contributes therefore in the first quadrant
($Q=+\eta+\mathrm{i}q$) with

Im
$\displaystyle\int_{0}^{1}\mathrm{d}x\,x(1-x)\log\Big{[}1+x(1-x)\,\frac{Q^{2}}{\mu^{2}}\Big{]}\overset{\footnotesize(\ref{grenzenx})}{=}\pi\int_{\frac{1}{2}(1-\beta)}^{\frac{1}{2}(1+\beta)}\mathrm{d}x\;x(1-x)=$
$\displaystyle\overset{\footnotesize
u=x-1/2}{=}\pi\underbrace{\int_{-\beta/2}^{\beta/2}\mathrm{d}u\Big{(}\frac{1}{4}-u^{2}\Big{)}}_{\beta/4-\beta^{3}/12}=\pi\,\frac{\beta}{12}(3-\beta^{2})=$
(A.17)
$\displaystyle\overset{\footnotesize(\ref{grenzenx})}{=}\frac{\pi}{6}\,\sqrt{1-\frac{4\mu^{2}}{q^{2}}}\Big{(}1+\frac{2\mu^{2}}{q^{2}}\Big{)}.$

Since the straight path in the second quadrant contributes with the opposite
sign we have to insert the double of the result (A.17) in $\delta V(r)$

$\delta
V(r)\overset{{}^{(\ref{DeltaVpert})}}{\underset{{}^{(\ref{xIntegral})}}{=}}-\frac{\alpha_{f}\hbar
c_{0}}{r}\frac{2\alpha_{f}}{3\pi}\int_{2\mu}^{\infty}\mathrm{d}q\frac{\mathrm{e}^{-qr}}{q}\sqrt{1-\frac{4\mu^{2}}{q^{2}}}\Big{(}1+\frac{2\mu^{2}}{q^{2}}\Big{)}.$
(A.18)

There are two scales in the $q$-integral, the mass scale $2\mu$ and the
distance $r$. It is convenient to go to the dimensionless relativistic
momentum scale $\gamma=q/(2\mu)$

$\delta V(r)\overset{\footnotesize(\ref{DeltaVres})}{=}-\frac{\alpha_{f}\hbar
c_{0}}{r}\frac{2\alpha_{f}}{3\pi}\int_{1}^{\infty}\mathrm{d}\gamma\frac{\mathrm{e}^{-2\mu\gamma
r}}{\gamma}\sqrt{1-\frac{1}{\gamma^{2}}}\Big{(}1+\frac{1}{2\gamma^{2}}\Big{)}.$
(A.19)

The radial dependent fine structure constant reads therefore

$\alpha(r)=\overset{\footnotesize(\ref{DeltaVres})}{=}\alpha_{f}\Big{\\{}1+\underbrace{\frac{2\alpha_{f}}{3\pi}\int_{1}^{\infty}\mathrm{d}\gamma\frac{\mathrm{e}^{-2\mu\gamma
r}}{\gamma}\sqrt{1-\frac{1}{\gamma^{2}}}\Big{(}1+\frac{1}{2\gamma^{2}}\Big{)}}_{\Delta(r)}\Big{\\}}.$
(A.20)

The prefactor $\frac{2\alpha_{f}}{3\pi}$=0.00154 of the correction term

$\Delta(r):=\frac{2\alpha_{f}}{3\pi}\int_{1}^{\infty}\mathrm{d}\gamma\frac{\mathrm{e}^{-2\mu\gamma
r}}{\gamma}\sqrt{1-\frac{1}{\gamma^{2}}}\Big{(}1+\frac{1}{2\gamma^{2}}\Big{)}$
(A.21)

is rather small. From the exponential factor $\mathrm{e}^{-2\mu\gamma r}$ at
$\gamma=1$ we expect therefore sufficiently large correction terms $\Delta(r)$
for $r$-values much smaller than the reduced Compton wave length
$\frac{1}{2\mu}=193\,$fm of an electron pair, i.e. $\frac{r}{2\mu}\ll 1$. The
function $\frac{\beta}{\gamma}=\frac{1}{\gamma}\sqrt{1-\frac{1}{\gamma^{2}}}$
in the argument of the $\gamma$-integral is dominated by $\beta$ at $\gamma=1$
and slowly decreasing with $1/\gamma$ for large $\gamma$ with a maximum of 0.5
at $\gamma=\sqrt{2}$. The factor $1+\frac{1}{2\gamma^{2}}$ leads to a small
increase of the maximum and a small shift to lower $\gamma$ only. The
numerical evaluation of the correction term (A.21) is depicted in Fig. 11
together with an analytical approximation, valid for $r\gg\frac{1}{2\mu}$
only.

Figure 11: Double logarithmic plot of the correction terms (A.21) and (A.25)
to $\alpha(r)$ of Eq. (A.20) in the interesting $r$-region.

To get an approximation for large $2\mu r$ we can substitute $\gamma=t+1$ in
Eq. (A.21)

$\Delta(r)\overset{\footnotesize(\ref{Korrektur})}{=}\frac{2\alpha_{f}}{3\pi}\mathrm{e}^{-2\mu
r}\int_{0}^{\infty}\mathrm{d}t\,\frac{\mathrm{e}^{-2\mu
rt}}{t+1}\sqrt{1-\frac{1}{(t+1)^{2}}}\Big{(}1+\frac{1}{2(t+1)^{2}}\Big{)}.$
(A.22)

With the expansion

$\frac{1}{t+1}\sqrt{1-\frac{1}{(t+1)^{2}}}\Big{(}1+\frac{1}{2(t+1)^{2}}\Big{)}\simeq
3\sqrt{\frac{t}{2}}-\frac{29}{2}\left(\frac{t}{2}\right)^{3/2}$ (A.23)

we arrive at the integrals

$\Delta(r)\simeq\frac{\sqrt{2}\alpha_{f}}{\pi}\mathrm{e}^{-2\mu
r}\int_{0}^{\infty}\mathrm{d}t\,\mathrm{e}^{-2\mu
rt}\sqrt{t}=\frac{\sqrt{2}\alpha_{f}}{\pi}\frac{\mathrm{e}^{-2\mu r}}{(2\mu
r)^{3/2}}\underbrace{\int_{0}^{\infty}\mathrm{d}x\,\mathrm{e}^{-x}\sqrt{x}}_{\sqrt{\pi}/2}$
(A.24)

and the analytical approximation

$\Delta(r)\simeq\Delta_{\mathrm{anal}}(r):=\frac{\alpha_{f}}{\sqrt{2\pi}}\frac{\mathrm{e}^{-2\mu
r}}{(2\mu r)^{3/2}}$ (A.25)

given in [25], valid for $2\mu r\gg 1$ only. At $2\mu r=1$ we get
$\Delta_{\mathrm{anal}}(\frac{1}{2\mu})=0.00107$ which is by a factor of 4 too
large compared to the numerical value $\Delta(\frac{1}{2\mu})=0.00028$. A
comparison of the two corrections $\Delta(r)$ and $\Delta_{\mathrm{anal}}(r)$
is shown in a double logarithmic plot in Fig. 11.

##  Appendix B Perturbative correction

The processes, we want to describe, are electron-electron, positron-positron,
electron-positron scattering and electron-positron annihilation. They differ
in the particle energies and transferred momenta.

According to the Feynman rules
$\displaystyle\mathrm{i}\,S(q)=\mathrm{i}\frac{\not{q}+\mu}{q^{2}-\mu^{2}+\mathrm{i}\varepsilon}$
(B.1)
$\displaystyle\mathrm{i}\,D^{\mu\nu}(q)=-\mathrm{i}\frac{1}{q^{2}+\mathrm{i}\varepsilon}[g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}]$
(B.2)
$\displaystyle-\mathrm{i}e_{0}\gamma_{\mu}\quad\textrm{with}\quad\sum_{i}q_{i}=0,$
(B.3)
the rules for the loop integrals $\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}$ and
the factor $(-1)$ for fermion loops, we get the contribution of the electron-
loop of Fig. 10

$\mathrm{i}\Pi_{2}^{\mu\nu}(q):=-(-\mathrm{i}e_{0})^{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{tr}\Big{[}\gamma^{\mu}\frac{\mathrm{i}}{\not{k}-\mu}\gamma^{\nu}\frac{\mathrm{i}}{\not{k}+\not{q}-\mu}\Big{]}.$
(B.4)

to the photon propagator.

The regularisation of this divergent contribution to the photon propagator in
one-loop approximation is convenient in Euclidean space and $\epsilon=4-D$
dimensions. The factorials in the dimension $D$ are interpolated by the Gamma
function with poles at $0$ and all negative integers. The factors depending on
$\epsilon$ and their expansion read

$(4\pi)^{-D/2}\Gamma(\frac{\epsilon}{2})=\frac{1}{(4\pi)^{2}}\left[\frac{2}{\epsilon}-\gamma_{E}+\ln
4\pi+\mathcal{O}(\epsilon)\right]$ (B.5)

where the Euler-Mascheroni constant $\gamma_{E}$ appears in
$\Gamma(\frac{\epsilon}{2})=\frac{2}{\epsilon}-\gamma_{E}+\mathcal{O}(\epsilon)$.

After a long calculation, the expression for the photon self-energy
$\Pi(q^{2})$ reads in one-loop approximation [25] 111 corresponding to second
order in the electric charge

$\Pi_{2}(q^{2})=-\frac{2\alpha_{f}}{\pi}\int_{0}^{1}\mathrm{d}x\;x(1-x)\Big{[}\frac{2}{\epsilon}-\gamma_{E}+\log(4\pi)-\log(\frac{\mu^{2}}{[\mu]^{2}}-x(1-x)\frac{q^{2}}{[\mu]^{2}})\Big{]}.$
(B.6)

with a loop of an electron of mass $m$, inverse Compton wave length
$\mu=mc/\hbar$ and scale $[\mu]$. The behaviour of $\Pi(q^{2})$ for
$\epsilon\to 0$ demonstrates the well-known logarithmic infinity of the bare
electric charge. It can be removed by a subtraction of this infinity and can
be accompanied by a subtraction of arbitrary finite terms. There is no rule in
the theory what finite contribution should be subtracted. These finite terms
can only be fixed by a comparison with experiments. Since we know from
experiments the charge of electrons at rest, it is wise to adjust the finite
constant to this observation and define the electron self-energy by

$\displaystyle\Pi(q^{2}):$
$\displaystyle=\Pi_{2}(q^{2})-\Pi_{2}(0)\overset{\footnotesize(\ref{zweiteOrd})}{=}$
(B.7)
$\displaystyle=\frac{2\alpha_{f}}{\pi}\int_{0}^{1}\mathrm{d}x\;x(1-x)\log\Big{[}1-x(1-x)\frac{q^{2}}{\mu^{2}})\Big{]}.$

This expression reveals the famous dependence of the electron self-energy from
the transferred momentum, the running of the coupling. The sign of $q^{2}$
depends on the experiment and the time structure of the corresponding Feynman
diagram, whether the virtual photon line is time-like or space-like, whether
mainly energy or momentum are transferred.

In a scattering experiment, see Fig. 12, the transferred momentum is space-
like

$q^{2}=-Q^{2}.$ (B.8)

Figure 12: Electron-positron scattering.