Numerical Evaluation of a Soliton Pair with Long Range Interaction Joachim Wabnig Josef Resch Dominik Theuerkauf

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Numerical Evaluation of a Soliton Pair with Long

Range Interaction

Joachim Wabnig, Josef Resch, Dominik Theuerkauf,

Fabian Anmasser and Manfried Faber

October 25, 2022

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 elec-

tromagnetic 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 sin-

gularity 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 4·3 = 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. 2we repeat the formulation and some basic properties of the model, in Sect. 3we

present the numerical formulation in cylindrical coordinates and in Sect. 4we show ﬁrst results of

the calculations and compare the results to perturbative QED.

arXiv:2210.13374v1 [hep-lat] 24 Oct 2022

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)=e−iα(x)~σ~n(x)= cos α(x)

| {z }

q0(x)

−i~σ ~n sin α(x)

| {z }

~q(x)

∈SU(2) ∼

=S3(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 σi. The Lagrangian of MTP reads,

LMTP(x) := −αf~c

4π1

Rµν (x)~

Rµν (x) + Λ(x)with Λ(x) := q6

0(x)

,(2.2)

with ~

Rµν := ~

Γµ×~

Γνand (∂µQ)Q†=: −i~σ~

Γµ.(2.3)

We get contact with nature by relating the electromagnetic ﬁeld strength tensor to the dual of

the curvature tensor ~

Rµν ,

Fµν := −e0

4π0

Rµν .(2.4)

At large distances dfrom the sources, measured in units of the soliton scale parameter r0, the

non-abelian ﬁeld strength gets abelian,

Fµν

d>>r0

→(~

Fµν~n)~n. (2.5)

MTP has four diﬀerent classes of topologically stable single soliton conﬁgurations. Their rep-

resentatives read,

ni(x) = ±xi

r, α(x) = π

2∓arctan r0

r=(arctan r

π−arctan r

(2.6)

The imaginary part of their Q-ﬁelds are schematically depicted in Table 1. The ﬁelds (2.6) are

~n =~r/r ~n =−~r/r -~n =~r/r ~n =~r/r

q0≥0q0≤0q0≥0q0≤0

Z= 1 Z=−1Z=−1Z= 1

Q=1

2Q=1

2Q=−1

Table 1: Scheme of single soliton conﬁgurations. The ﬁelds ~n and q0and the topological quantum

numbers Zand Qare indicated. The diagrams show the imaginary components ~q =~n sin αof the

soliton ﬁeld, in full red for the hemisphere with q0>0and in dashed green for q0<0.

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 Zof the ~n-ﬁeld in Eq. (2.6) decides about the charge quantum number deﬁned

by a map Π2(S2). Field conﬁgurations are further characterised by the number Qof coverings of

S3, by the map Π3(S3). With the sign of Qwe deﬁne an internal chirality χand with the absolute

value of Qthe spin quantum number s,

Q=χ·swith s:= |Q|.(2.7)

The spin quantum number of two-soliton conﬁgurations indicates that χcan 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,

E0=αf~c

4,(2.8)

can be adjusted to the electron rest energy mec2

0= 0.510 998 95 MeV by choosing,

r0= 2.213 205 16 fm,(2.9)

a scale which is of the order of the classical electron radius. The four parameters r0, c0, E0and e0

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 αfand ~.

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 ~q =~n sin αof 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 ~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 q0= cos α. For q0→0the arrows are getting darker or black. The left conﬁguration

belongs to the topological quantum numbers Q=S= 0 and the right one to Q=S= 1, where Sis

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,

H(2.2)

=αf~c

4π1

Rij ~

Rij + Λ=: Hcur +Hpot,(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,

Hcur =0

2~

r+~

ϕ+~

z=αf~c

8π

r2q2

r(∂zq0)2+ (∂zqr)2+ (∂zqz)2

+r2

0∂rqr∂zqz−∂zqr∂rqz2+q2

r(∂rq0)2+ (∂rqr)2+ (∂rqz)2.

(3.2)

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NumericalEvaluationofaSolitonPairwithLongRangeInteractionJoachimWabnig,JosefResch,DominikTheuerkauf,FabianAnmasserandManfriedFaberOctober25,2022AbstractWithinthemodeloftopologicalparticles(MTP)wedeterminetheinteractionenergyofmonopolepairs,sourcesandsinksofaCoulombiceld.Themonopolesarerepresentedby...

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Numerical Evaluation of a Soliton Pair with Long Range Interaction Joachim Wabnig Josef Resch Dominik Theuerkauf

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