# About the solution of the numerical instability for topological solitons
with long range interaction

Fabian Anmasser, Dominik Theuerkauf and Manfried Faber

###### Abstract

The computations of solutions of the field equations in the Model of
Topological Particles, formulated with a scalar SU(2)-field, have shown
instabilities leading to discrepancies between the numerical and analytical
solutions. We identify the origin of these deviations in misalignments of the
rotational axes corresponding to the SU(2) elements. The system of a single
soliton we use as an example to show that a constraint suppressing the wave-
like disturbances is able to lead to excellent agreement between the result of
the numerical minimisation procedure and the analytical solution.

##  1 Introduction

Topological solitons are interesting objects. Their masses are given by
integrals over the energy density, the particle number is a topological
quantum number and their interaction is a consequence of the topology. Well-
known models of this type are the Sine-Gordon model and the Skyrme model. Both
obey the laws of special relativity. The Sine-Gordon model [1] is a 1+1D model
with 1 degree of freedom and two types of solitons differing by their
chirality acting like a charge. They behave as expected for extended charged
particles. Particles of equal charge repel and of opposite charge attract each
other and annihilate. Bound states of particle pairs oscillate and are
therefore dubbed breathing modes.

Skyrme’s model [2, 3, 4] is formulated in 3+1D with the three degrees of
freedom of an SU(2) field with the interpretation of the meson field in
nuclei. It was intended as a model for particles with the strong, short range
interaction of nucleons. Skyrme’s model can not model the Coulomb interaction
by topological properties.

In [5] a model in 3+1D with the three degrees of freedom of an SO(3) field was
suggested, modelling a long range Coulomb interaction for topologically stable
solitons. It is of similar spirit as the Skyrme model, but due to the
different Lagrangian it allows for three topological quantum numbers
$\pi_{2}(S^{2})$,$\pi_{3}(S^{2})$ and $\pi_{3}(S^{3})$. The solitons of this
model can be interpreted as Dirac monopoles [6, 7] without any singularity,
without Dirac string and with a soft core. There are four types of stable
solitons differing in the two topological quantum numbers $\pi_{3}(S^{3})$ and
$\pi_{2}(S^{2})$ which can be interpreted as spin up and down, positive and
negative charge. The equations of motion can be solved analytically for the
one soliton systems. In addition, the model describes two types of Goldstone
bosons, i.e. massless excitations propagating with the speed of light [8]. In
Ref. [5] the model was published under the title “model of topological
fermions” but possibly it should rather be dubbed _model of topological
particles_ since it turned out that also the Goldstone bosons of the model are
characterised by a topological quantum number [9]. Due to the non-linearity of
the model more complicated systems have to be solved numerically. But the
numerics has suffered from numerical instabilities leading to large
uncertainties [10, 11, 12]. In this article we report about a successful
method to avoid these instabilities [13]. For the one soliton system we are
able to compare in a careful analysis the numerical outcome with the exact
analytical results. In this article we discuss the sources and the size of the
errors of the numerical evaluations. In Sect. 2 we give a short overview of
the model and the analytical solution for the one soliton system in Sect. 3.
After discussing the cylindrical and the lattice formulation in Sects. 4 and 5
we demonstrate the failure of the calculation in Sect. 6. Further, we suggest
in this section, to improve the calculations by a constraint. With the
improved numerics we then get good agreement with the analytical results and
give characteristic numbers for the achieved accuracy.

##  2 Formulation of the model

SU(2) is the double covering group of SO(3). With the Rodriguez formula, SU(2)
matrices can be expanded in a $\sin$ and a $\cos$ term, whereas the real
$3\times 3$ SO(3) rotational matrices in 3D need three terms. It is therefore
simpler to do the calculations in SU(2) than in SO(3). For smooth field
configurations in space-time there is the only difference that for every
configuration of the SO(3)-field of the model we get two SU(2) configurations,
differing by a multiplication with the non-trivial centre element of SU(2).
Taking this into account, the three degrees of freedom of the model are
formulated with a scalar field of SU(2) matrices

$Q(x)=q_{0}-\mathrm{i}\vec{q}(x)\vec{\sigma}\quad\textrm{with}\quad
q_{0}^{2}+\vec{q}^{2}=1,$ (2.0.1)

of unit quaternions, where $\vec{q}\vec{\sigma}=q_{i}\sigma_{i}$ is an element
of the su(2) algebra, with the usual Pauli matrices $\sigma_{i}$ and
Einstein’s summation convention applied.

The Lagrangian of the model reads [5]

$\mathcal{L}_{\mathrm{MTP}}=-\frac{\alpha_{f}\hbar
c}{4\pi}\left(\frac{1}{4}\vec{R}_{\mu\nu}\vec{R}^{\mu\nu}+\frac{q_{0}^{2m}}{r_{0}^{4}}\right),$
(2.0.2)

where $\alpha_{f}$ and $r_{0}$ are in principle arbitrary constants. Choosing
for $\alpha_{f}$ the value of Sommerfeld’s fine-structure constant the force
field of solitons can be compared with the Coulomb field and the size
parameter $r_{0}$ can be adjusted to the mass of the lightest fundamental,
charged particles existing in nature, to electrons. The dynamical term
proportional to $\vec{R}_{\mu\nu}\vec{R}^{\mu\nu}$ is equivalent to the Skyrme
term of the Skyrme model. It can be formulated with the connection field on
the SU(2) manifold $\vec{\Gamma}_{\mu}$ defined by

$\partial_{\mu}Q=-\mathrm{i}\vec{\Gamma}_{\mu}\vec{\sigma}Q,\quad\vec{\Gamma}_{\mu}(r,\varphi,z)\overset{\footnotesize(\ref{unitquaternions})}{=}q_{0}\partial_{\mu}\vec{q}-\vec{q}\partial_{\mu}q_{0}+\vec{q}\times\partial_{\mu}\vec{q},$
(2.0.3)

$\vec{R}_{\mu\nu}$ is an area density on the SU(2) manifold

$\vec{R}_{\mu\nu}=\vec{\Gamma}_{\mu}\times\vec{\Gamma}_{\nu}.$ (2.0.4)

This is valid in the gauge where the local coordinate systems on the SU(2)
manifold are chosen as $\sigma_{i}Q$, see Eq. (2.0.3), and where the Maurer-
Cartan equation

$\partial_{\mu}\vec{\Gamma}_{\nu}-\partial_{\nu}\vec{\Gamma}_{\mu}=2\vec{\Gamma}_{\mu}\times\vec{\Gamma}_{\nu},$
(2.0.5)

is satisfied. After a rotation of these local coordinates systems on the SU(2)
manifold one recognizes in $\vec{R}_{\mu\nu}$ the well-known form of the field
strength tensor in QCD and the curvature tensor in general relativity

$\vec{R}_{\mu\nu}=\partial_{\mu}\vec{\Gamma}_{\nu}-\partial_{\nu}\vec{\Gamma}_{\mu}-\vec{\Gamma}_{\mu}\times\vec{\Gamma}_{\nu}.$
(2.0.6)

The potential term proportional to $q_{0}^{2m}$ defines a two-fold degenerate
vacuum at $q_{0}=0$. It fixes the size and mass of solitons.

We can relate the geometry to physics introducing a dual vector potential
$\vec{C}_{\mu}$ and a dual field strength tensor ${}^{*}\vec{F}_{\mu\nu}$ by

$\vec{C}_{\mu}=-\frac{e_{0}}{4\pi\varepsilon_{0}c}\vec{\Gamma}_{\mu},\quad^{*}\vec{F}_{\mu\nu}=-\frac{e_{0}}{4\pi\varepsilon_{0}c}\vec{R}_{\mu\nu}.$
(2.0.7)

Since we are considering static cases, we have no magnetic fields,
$\vec{B}_{i}=0$ and get for the energy density, the 00 component of the
energy-momentum tensor

$\mathcal{H}=\underbrace{\frac{\epsilon_{0}}{2}\vec{E}_{i}\vec{E}_{i}}_{\mathcal{H}_{\mathrm{cur}}}+\underbrace{\frac{\alpha_{f}\hbar
c}{4\pi}\frac{q_{0}^{2m}}{r_{0}^{4}}}_{\mathcal{H}_{\mathrm{pot}}}.$ (2.0.8)

#### Scale dependencies

For stable, time independent solutions we get a condition for the stability of
the solutions. The two terms in Eq. (2.0.8) have different scale dependencies,
$x\to\lambda x$

$\displaystyle
H_{\mathrm{cur}}:=\int\mathrm{d}^{3}x\mathcal{H}_{\mathrm{cur}}\overset{\footnotesize(\ref{EneDens})}{=}\frac{\epsilon_{0}}{2}\int\mathrm{d}^{3}x\vec{E}_{i}\vec{E}_{i}\hskip
11.38109pt\quad\rightarrow\quad\frac{1}{\lambda}H_{\mathrm{cur}},$ (2.0.9)
$\displaystyle
H_{\mathrm{pot}}:=\int\mathrm{d}^{3}x\mathcal{H}_{\mathrm{pot}}\overset{\footnotesize(\ref{EneDens})}{=}\frac{\alpha_{f}\hbar
c}{4\pi}\int\mathrm{d}^{3}x\frac{q_{0}^{2m}}{r_{0}^{4}}\quad\rightarrow\quad\lambda^{3}H_{\mathrm{pot}}.$
(2.0.10)

The stability of the total energy

$H_{\mathrm{tot}}:=H_{\mathrm{cur}}+H_{\mathrm{pot}}$ (2.0.11)

infers

$\frac{\mathrm{d}}{\mathrm{d}\lambda}H_{\mathrm{tot}}\Bigr{\rvert}_{\lambda=1}\overset{\footnotesize(\ref{GesamtE})}{=}\frac{\mathrm{d}}{\mathrm{d}\lambda}H_{\mathrm{cur}}\Bigr{\rvert}_{\lambda=1}+\frac{\mathrm{d}}{\mathrm{d}\lambda}H_{\mathrm{pot}}\Bigr{\rvert}_{\lambda=1}\overset{{}^{(\ref{KrummBeitr})}}{\underset{{}^{(\ref{PotBeitr})}}{=}}-H_{\mathrm{cur}}+3H_{\mathrm{pot}}=0.$
(2.0.12)

This application of the Hobart-Derrick theorem [14, 15] gives a useful check
of the accuracy of numerical calculations. In Sect. 6 we will compare the
numerical results with

$\frac{H_{\mathrm{tot}}}{H_{\mathrm{pot}}}\overset{{}^{(\ref{GesamtE})}}{\underset{{}^{(\ref{EnergieBez})}}{=}}4.$
(2.0.13)

##  3 Solitonic solution

For static monopoles at the origin [5] the model reduces to spherical symmetry
in 3D with spherical coordinates $r,\vartheta,\varphi$ 111We would like to
emphasize that we use the arrow symbol like $\vec{n}$ for vectors in the su(2)
algebra and bold symbols like $\mathbf{r}$ for vectors in space.

$q_{0}=\cos\alpha(r),\quad\vec{q}=\vec{n}({\mathbf{r}})\sin\alpha(r),\quad
n_{i}({\mathbf{r}})=\frac{x_{i}}{r},\quad\alpha(r)\in[0,\frac{\pi}{2}],$
(3.0.1)

where $\mathbf{r}$ denotes a vector in space and $\vec{n}$ a unit vector in
the su(2) algebra. The Euler-Lagrange equation, a non-linear differential
equation

$\partial^{2}_{\rho}\cos\alpha+\frac{(1-\cos^{2}\alpha)\cos\alpha}{\rho^{2}}-m\rho^{2}\cos^{2m-1}\alpha=0\quad\textrm{with}\quad\rho:=\frac{r}{r_{0}}$
(3.0.2)

has for $m=3$ a simple solution

$\tan\alpha(r)=\rho.$ (3.0.3)

Due to the simplicity of this solution we will use further on the case $m=3$.
Corresponding to the three terms in Eq. (3.0.2) the radial energy density has
three contributions

$h(\rho)=\frac{\alpha_{f}\hbar
c}{r_{0}}\Big{[}\frac{\rho^{2}}{2(1+\rho^{2})^{2}}+\frac{\rho^{2}}{(1+\rho^{2})^{3}}+\frac{\rho^{2}}{(1+\rho^{2})^{3}}\Big{]},$
(3.0.4)

a radial field from the contributions of $\vec{R}_{\vartheta\varphi}$
approaching at large distances the Coulomb field of a point charge, a
tangential field from $\vec{R}_{r\vartheta}$ and $\vec{R}_{r\varphi}$ and a
potential contribution. They are depicted in Fig. 1. Integrating this radial
energy density over $\rho$ we get the energy for the monopole. Comparing this
energy to the rest energy of the lightest fundamental monopole existing in
nature 222The most accurate value known up to now is
$m_{e}c^{2}=0.51099895000(15)$ MeV. Hence 0.511 MeV is a very good
approximation to the experimental value. we can fix the radius $r_{0}$,
defining the size of the soliton [16]

$0$$0.1$$0.2$$0$$1$$2$$3$$4$$5$$6$$r/r_{0}$radial energy densitiesradial
contributiontangential contribtionpotential contribution Figure 1: Three
contributions to the energy densities for the soliton solution with $m=3$. The
radial contribution describes the Coulomb field, regularised by the structure
of the model.

$H_{\mathrm{mono}}=\int_{0}^{\infty}\mathrm{d}\rho\,h(\rho)\overset{\footnotesize(\ref{radialdensities})}{=}\frac{\alpha_{f}\hbar
c}{r_{0}}\frac{\pi}{4}=0.511\,\mathrm{MeV}\quad\rightarrow
r_{0}=2.21\,\mathrm{fm}.$ (3.0.5)

#### Electrodynamic limit

As long as we do not have analytical solutions for static systems with several
charges or scattering problems, we are able to treat the fields outside the
numerical integration region within classical electrodynamics only, a scenario
which we dubbed in Ref. [8] electrodynamic limit.

In this limit we assume $q_{0}=0$ and neglect therefore tangential and
potential energy contributions. To get estimates of the accuracy of the
numerical calculations, it is sufficient to determine the size of these
neglected contributions in spherical coordinates in the region
$\rho=\frac{r}{r_{0}}>\rho_{>}=\frac{r_{>}}{r_{0}}$. Their contributions to
the radial energy density are given by the integrals over the second and third
term in Eq. (3.0.4), by

$\displaystyle\frac{\alpha_{f}\hbar
c}{r_{0}}\int_{\rho_{>}}^{\infty}\frac{2\rho^{2}}{(1+\rho^{2})^{3}}\;\mathrm{d}\rho$
$\displaystyle\overset{\footnotesize(\ref{radialdensities})}{=}\frac{\alpha_{f}\hbar
c}{r_{0}}\frac{\pi}{8}\left[1-\frac{2}{\pi}\left(\arctan\;\rho_{>}+\frac{\rho_{>}(\rho_{>}^{2}-1)}{(\rho_{>}^{2}+1)^{2}}\right]\right)\longrightarrow$
(3.0.6)
$\displaystyle\overset{\footnotesize\rho_{>}\to\infty}{\longrightarrow}\frac{\alpha_{f}\hbar
c}{r_{0}}\frac{\pi}{8}\cdot\frac{16}{3\pi}\,\rho_{>}^{-3}.$

Dividing by the integral over the whole $r$-axis, $\frac{\alpha_{f}\hbar
c}{r_{0}}\frac{\pi}{8}$, results in the relative error of the sum of these
contributions

$\Delta_{\mathrm{el}}(\rho_{>}):=1-\frac{2}{\pi}\left(\arctan\;\rho_{>}+\frac{\rho_{>}(\rho_{>}^{2}-1)}{(\rho_{>}^{2}+1)^{2}}\right)\overset{\footnotesize\rho_{>}\to\infty}{\longrightarrow}\frac{16}{3\pi}\,\rho_{>}^{-3}.$
(3.0.7)

Figure 2: Relative error $\Delta_{\mathrm{el}}$ in electrodynamic limit, when
tangential and potential energy contributions are neglected (full blue line)
for $r>r_{>}$. For large $r_{>}$ this relative error decays with the third
power of $r_{>}/r_{0}$ (dashed red line), see Eq. (3.0.7).

In the further calculations we are using $r_{>}=10\leavevmode\nobreak\ r_{0}$
with an error of 0.167 %. From Fig. 2 we can see that for increasing $r_{>}$
the relative error decreases nicely with $(r_{>}/r_{0})^{-3}$.

For comparison, in the same region the first term in Eq. (3.0.4) contributes
with 12.6 %, which should not be neglected and is taken care of in the
electrodynamic limit. These contributions we are going to determine in
cylindrical coordinates.

##  4 Cylindrical formulation

The algorithm presented here [10, 12, 13], uses cylindrical coordinates to
work with, because it should be capable of computing dipoles as well.
Therefore, we will also do monopole calculations in cylindrical coordinates
$r,\varphi,z$ in order to get insight in the accuracy of the numerical
calculations. The general soliton field (2.0.1) for any configuration in
cylindrical coordinates reads

$Q(r,\varphi,z)\overset{\footnotesize(\ref{unitquaternions})}{=}q_{0}(r,z)-i\vec{\sigma}\vec{q}(r,\varphi,z),\quad\text{with}\quad\vec{q}(r,\varphi,z)=\begin{pmatrix}q_{r}(r,z)\cos\varphi\\\
q_{r}(r,z)\sin\varphi\\\ q_{z}(r,z)\end{pmatrix}.$ (4.0.1)

For the affine connection (2.0.3) we obtain

$\displaystyle\vec{\Gamma}_{r}$
$\displaystyle\overset{\footnotesize(\ref{derivativeQ})}{=}\begin{pmatrix}(q_{0}\partial_{r}q_{r}-q_{r}\partial_{r}q_{0})\cos\varphi+(q_{r}\partial_{r}q_{z}-q_{z}\partial_{r}q_{r})\sin\varphi\\\
(q_{0}\partial_{r}q_{r}-q_{r}\partial_{r}q_{0})\sin\varphi+(q_{z}\partial_{r}q_{r}-q_{r}\partial_{r}q_{z})\cos\varphi\\\
q_{0}\partial_{r}q_{z}-q_{z}\partial_{r}q_{0}\end{pmatrix}$ (4.0.2)
$\displaystyle\vec{\Gamma}_{\varphi}$
$\displaystyle\overset{\footnotesize(\ref{derivativeQ})}{=}q_{r}\begin{pmatrix}-q_{0}\sin\varphi-
q_{z}\cos\varphi\\\ q_{0}\cos\varphi-q_{z}\sin\varphi\\\ q_{r}\end{pmatrix}$
(4.0.3) $\displaystyle\vec{\Gamma}_{z}$
$\displaystyle\overset{\footnotesize(\ref{derivativeQ})}{=}\begin{pmatrix}(q_{r}\partial_{z}q_{z}-q_{z}\partial_{z}q_{r})\sin\varphi+(q_{0}\partial_{z}q_{r}-q_{r}\partial_{z}q_{0})\cos\varphi\\\
(q_{z}\partial_{z}q_{r}-q_{r}\partial_{z}q_{r})\cos\varphi+(q_{0}\partial_{z}q_{r}-q_{r}\partial_{z}q_{0})\sin\varphi\\\
q_{0}\partial_{z}q_{z}-q_{z}\partial_{z}q_{0}\end{pmatrix}.$ (4.0.4)

The components of the curvature tensor (2.0.4) read

$\displaystyle\vec{R}_{\varphi z}$
$\displaystyle\overset{\footnotesize(\ref{RSU2})}{=}q_{r}\begin{pmatrix}\partial_{z}q_{0}\sin\varphi+\partial_{z}q_{z}\cos\varphi\\\
-\partial_{z}q_{0}\cos\varphi+\partial_{z}q_{z}\sin\varphi\\\
-\partial_{z}q_{r}\end{pmatrix},$ (4.0.5) $\displaystyle\vec{R}_{zr}$
$\displaystyle\overset{\footnotesize(\ref{RSU2})}{=}\frac{\partial_{r}q_{r}\partial_{z}q_{z}-\partial_{z}q_{r}\partial_{r}q_{z}}{q_{0}}\begin{pmatrix}-q_{0}\sin\varphi-
q_{z}\cos\varphi\\\ q_{0}\cos\varphi-q_{z}\sin\varphi\\\ q_{r}\end{pmatrix},$
(4.0.6) $\displaystyle\vec{R}_{r\varphi}$
$\displaystyle\overset{\footnotesize(\ref{RSU2})}{=}q_{r}\begin{pmatrix}-\partial_{r}q_{0}\sin\varphi-\partial_{r}q_{z}\cos\varphi\\\
\partial_{r}q_{0}\cos\varphi-\partial_{r}q_{z}\sin\varphi\\\
\partial_{r}q_{r}\end{pmatrix}.$ (4.0.7)

and its squares

$\displaystyle\vec{R}_{\varphi z}^{2}$
$\displaystyle=q_{r}^{2}\big{[}(\partial_{z}q_{0})^{2}+(\partial_{z}q_{r})^{2}+(\partial_{z}q_{z})^{2}\big{]},$
(4.0.8) $\displaystyle\vec{R}_{zr}^{2}$
$\displaystyle=q_{0}^{-2}\big{(}\partial_{r}q_{r}\partial_{z}q_{z}-\partial_{z}q_{r}\partial_{r}q_{z}\big{)}^{2},$
$\displaystyle\vec{R}_{r\varphi}^{2}$
$\displaystyle=q_{r}^{2}\big{[}(\partial_{r}q_{0})^{2}+(\partial_{r}q_{r})^{2}+(\partial_{r}q_{z})^{2}\big{]}.$

With the proper conversion factor to SI units and the length scales for
cylindrical coordinates, $l_{\varphi}=r\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }l_{r}=l_{z}=1$ we adjust relation (2.0.7) between
curvature tensor and electric field strength to cylindrical coordinates

$\vec{E}_{r}=-\frac{e_{0}}{4\pi\epsilon_{0}}\frac{1}{l_{\varphi}l_{z}}\vec{R}_{\varphi
z},\quad\vec{E}_{\varphi}=\frac{e_{0}}{4\pi\epsilon_{0}}\frac{1}{l_{r}l_{z}}\vec{R}_{rz},\quad\vec{E}_{z}=-\frac{e_{0}}{4\pi\epsilon_{0}}\frac{1}{l_{r}l_{\varphi}}\vec{R}_{r\varphi}.$
(4.0.9)

This leads to the density of the curvature energy

$\begin{split}\mathcal{H}_{\mathrm{cur}}\overset{\footnotesize(\ref{EneDens})}{=}\frac{\epsilon_{0}}{2}&\big{(}\vec{E}_{r}^{2}+\vec{E}_{\varphi}^{2}+\vec{E}_{z}^{2}\big{)}\overset{{}^{(\ref{QuadR})}}{\underset{{}^{(\ref{Ezylinder})}}{=}}\frac{\alpha_{f}\hbar
c}{8\pi}\frac{1}{r^{2}}\bigg{\\{}q_{r}^{2}\big{[}(\partial_{z}q_{0})^{2}+(\partial_{z}q_{r})^{2}++(\partial_{z}q_{z})^{2}\big{]}\\\
&+\frac{r^{2}}{q_{0}^{2}}\big{(}\partial_{r}q_{r}\partial_{z}q_{z}-\partial_{z}q_{r}\partial_{r}q_{z}\big{)}^{2}+q_{r}^{2}\big{[}(\partial_{r}q_{0})^{2}+(\partial_{r}q_{r})^{2}+(\partial_{r}q_{z})^{2}\big{]}\bigg{\\}},\end{split}$
(4.0.10)

where we used the definition of the fine structure constant
$\alpha_{f}:=e_{0}^{2}/(4\pi\epsilon_{0}\hbar c)$.

##  5 Lattice computation

We consider the case of a monopole at the centre of a cylinder. Inside this
cylinder we introduce the dimensionless coordinates
$\bar{r},\bar{\varphi},\bar{z}\in\mathbb{Z}$, defined by the relations

$r=a\bar{r},\quad\varphi=\bar{\varphi},\qquad
z=a\bar{z},\quad\bar{r}\in\\{0,1,\dots,n_{r}\\},\quad\bar{z}\in\\{-n_{z},-n_{z}+1,\dots,\,n_{z}\\}$
(5.0.1)

who just number the lattice points. Due to the rotational symmetry around the
z-axis we perform the $\varphi$-integrations analytically and are left with a
two dimensional lattice with spacing $a$, often referred to as box in the
following. It is characterized with the number $n_{r}$ of points in $r$
direction and the number $n_{z}$ of points in $\pm z$ direction. Such a finite
lattice suffers from boundary effects. In the example of lattice QCD the
boundary problems are often diminished by periodic boundary conditions. In the
present model with solitons with long range Coulomb interaction periodic
boundary conditions are in general not useful. Since we know the field
configurations of charged particles from classical electrodynamics
analytically, we use them for the boundary conditions.

####  5.0.1 Outside electric energy

Figure 3: Integration areas

For a monopole with unit charge $e_{0}$ at the centre of a cylinder the
electric field strength reads within Maxwell’s theory

$\vec{E}(r)=\frac{e_{0}}{4\pi\epsilon_{0}}\frac{\vec{r}}{r^{3}}=\frac{1}{4\pi\epsilon_{0}}\frac{e_{0}}{\sqrt{(r^{2}+z^{2})^{3}}}\begin{pmatrix}r\\\
0\\\ z\end{pmatrix}.$ (5.0.2)

We have to integrate $|\vec{E}|^{2}$ over the whole volume except the
cylinder, characterised by its radius $R$ and its half length $Z$. Due to the
cylindrical symmetry, we get an integration factor of $2\pi$ and we are able
to restrict our consideration to the $rz$-plane, where the cylinder gets
projected to a rectangular box. We split the remaining two dimensional area
into three smaller regions, which are shown in figure 3. The energy outside
the box may be written as

$\displaystyle H^{\mathrm{out}}_{\mathrm{el}}$
$\displaystyle\overset{\footnotesize(\ref{EneDens})}{=}\frac{\epsilon_{0}}{2}\int_{\mathbb{R}^{3}\setminus\text{box}}\mathrm{d}^{3}x|\vec{E}|^{2}=\frac{\epsilon_{0}}{2}\int_{\mathbb{R}^{3}\setminus\text{box}}r\,\mathrm{d}r\,\mathrm{d}\varphi\,\mathrm{d}z|\vec{E}|^{2}=\pi\epsilon_{0}\int_{\mathbb{R}^{2}\setminus\text{box}}r\,\mathrm{d}r\,\mathrm{d}z|\vec{E}|^{2}=$
(5.0.3)
$\displaystyle=\pi\epsilon_{0}\bigg{\\{}\underbrace{\int_{-\infty}^{-Z}\int_{0}^{\infty}r\,\mathrm{d}r\,\mathrm{d}z}_{\textbf{{I{}}}}+\underbrace{\int_{-Z}^{Z}\int_{R}^{\infty}r\,\mathrm{d}r\,\mathrm{d}z}_{\textbf{{II{}}}}+\underbrace{\int_{Z}^{\infty}\int_{0}^{\infty}r\,\mathrm{d}r\,\mathrm{d}z}_{\textbf{{I{}}}}\bigg{\\}}|\vec{E}|^{2}=$
$\displaystyle\overset{\footnotesize(\ref{EMaxwell})}{=}\frac{\alpha_{f}\hbar
c}{4}\bigg{(}\frac{1}{Z}+\frac{1}{R}\arctan\frac{Z}{R}\bigg{)}.$

####  5.0.2 Outside potential energy

After we did the appraisal for the involved energies in- and outside a sphere
in Sect. 3, we want go on and find an analytical expression for the potential
energy outside the cylindrical box, which we neglect in the electrodynamic
limit. We integrate over $\mathcal{H}_{\mathrm{pot}}$ in Eq. (2.0.8)

$\displaystyle H_{\mathrm{pot}}^{\mathrm{out}}$
$\displaystyle\overset{\footnotesize(\ref{EneDens})}{=}\frac{\alpha_{f}\hbar
c}{4\pi
r_{0}^{4}}\int_{\mathbb{R}^{3}\setminus\text{box}}\mathrm{d}^{3}x\,q_{0}^{6}\overset{{}^{(\ref{RegularIgel})}}{\underset{{}^{(\ref{MinLoesm3})}}{=}}\frac{\alpha_{f}\hbar
c}{4\pi
r_{0}^{4}}\int_{\mathbb{R}^{3}\setminus\text{box}}r\,\mathrm{d}r\,\mathrm{d}\varphi\,\mathrm{d}z\,\frac{1}{[1+(r^{2}+z^{2})/r_{0}^{2}]^{3}}=$
(5.0.4) $\displaystyle=\frac{\alpha_{f}\hbar
c}{2r_{0}}\frac{1}{4}\bigg{\\{}\arctan\frac{1}{Z_{0}}-\frac{Z_{0}}{1+Z_{0}^{2}}+\frac{Z_{0}}{(1+R_{0}^{2})(1+R_{0}^{2}+Z_{0}^{2})}+\frac{1}{[1+R_{0}^{2}]^{3/2}}\arctan\frac{Z_{0}}{\sqrt{1+R_{0}^{2}}}\bigg{\\}},$
$\displaystyle\hskip 56.9055pt\textrm{with}\quad
Z_{0}=\frac{Z}{r_{0}},\;R_{0}=\frac{R}{r_{0}},$

where we were splitting up the integral in the exact same way as done in
(5.0.3) for the electrical energy outside the box.

If we consider a lattice with the specifications of $Z=10\,r_{0}$ and
$R=10\,r_{0}$ we get a relative error of 0.122 %. This value is a little
smaller than the error 0.167 %, which we got for a spherical volume in Eq.
(3.0.7). The potential energy outside the box contributes therefore to the
total energy with

$H_{\mathrm{pot}}^{\mathrm{out}}\approx 0.156\,\mathrm{keV},$ (5.0.5)

which is about 0.31 ‰ of the rest mass of an electron, $m_{e}c^{2}\approx
0.511\,\mathrm{MeV}$.

###  5.1 Discretization

Having gathered all energy contributions, we are now ready to go on with the
discretization, which is necessary for the numerical calculations. The soliton
field $Q$ stated in (4.0.1), with its components $q_{0}$, $q_{r}$ and $q_{z}$
is only defined on the sites of the $rz$-lattice.

#### Derivatives

To evaluate the curvature energy density $\mathcal{H}_{cur}$ (4.0.10), we need
the soliton field’s derivatives of the form $\partial_{i}q_{j}$ in $r$ and $z$
directions only. Labelling the points in each of the two directions with
integers as indicated in Eq. (5.0.1) we are using the five-point method for
the first derivative

$\frac{\mathrm{d}f}{\mathrm{d}x_{i}}\approx\frac{f(x_{i-2})-8f(x_{i-1})+8f(x_{i+1})-f(x_{i+2})}{12a},$
(5.1.1)

whenever possible. It is correct up to $a^{4}$. For boundary points only left
or right derivatives and at neighbouring points second order approximations
are used.

With these expressions, we are prepared to calculate the curvature energy
density $\mathcal{H}_{\mathrm{cur}}$ numerically.

#### Energies on the lattice

These derivatives are applied onto $\mathcal{H}_{\textrm{cur}}$ of Eq.
(4.0.10) to get the curvature energy on the lattice. After performing the
$\varphi$ integration analytically and using the dimensionless coordinates
$\bar{r}$ and $\bar{z}$ of Eq. (5.0.1) we can write

$H_{\mathrm{cur}}^{\mathrm{box}}\overset{{}^{(\ref{equ:curvatureEnergy})}}{\underset{{}^{(\ref{eqn:DimensionlessCord})}}{=}}2\pi\,a^{3}\int_{\mathrm{box}}\bar{r}\,\mathrm{d}\bar{r}\,\mathrm{d}\bar{z}\,\mathcal{H}_{\mathrm{cur}}=:\frac{\alpha_{f}\hbar
c}{a}\bar{H}_{\mathrm{cur}}^{\mathrm{box}}.$ (5.1.2)

From the potential energy density (2.0.8) we get similarly for the potential
energy on the lattice

$H_{\mathrm{pot}}^{\mathrm{box}}\overset{\footnotesize(\ref{EneDens})}{=}2\pi\frac{\alpha_{f}\hbar
c}{4\pi\bar{r}_{0}^{4}}\frac{1}{a}\int_{\mathrm{box}}\bar{r}\,\mathrm{d}\bar{r}\,\mathrm{d}\bar{z}\,q^{2m}_{0}=:\frac{\alpha_{f}\hbar
c}{a}\underbrace{\frac{1}{2\bar{r}_{0}^{4}}\int_{\mathrm{box}}\bar{r}\,\mathrm{d}\bar{r}\,\mathrm{d}\bar{z}\,q_{0}^{2m}}_{\bar{H}_{\mathrm{pot}}^{\mathrm{box}}}$
(5.1.3)

with the soliton radius

$\bar{r}_{0}:=\frac{r_{0}}{a}$ (5.1.4)

in lattice units. Therefore, the entire energy inside the box reads

$H^{\mathrm{box}}\overset{{}^{(\ref{eqn:HcuramLattice})}}{\underset{{}^{(\ref{eqn:HpotamLattice})}}{=}}\frac{\alpha_{f}\hbar
c}{a}\bigg{(}\bar{H}_{\mathrm{cur}}^{\mathrm{box}}+\bar{H}_{\mathrm{pot}}^{\mathrm{box}}\bigg{)}.$
(5.1.5)

This energy is computed on the lattice and minimised by a conjugate gradient
descent algorithm. Here, we want to emphasize again, that quantities with a
bar over them are always dimensionless as can be seen in (5.1.5), because
$\alpha_{f}$ has no dimension, $\hbar c\approx
200\,\mathrm{MeV\leavevmode\nobreak\ fm}$ and $a$ measures the distance
between two neighbouring points in fm. To enhance the accuracy of our
calculations, we decided to do a cubic interpolation of the curvature- and
potential energy density between the lattice points before integrating them.

Last, but not least we express the electrical energy outside the box
$H^{\mathrm{out}}_{\mathrm{el}}$ in terms of dimensionless units

$H_{\mathrm{el}}^{\mathrm{out}}\overset{\footnotesize(\ref{Helout})}{=}\frac{\alpha_{f}\hbar
c}{4}\bigg{(}\frac{1}{Z}+\frac{1}{R}\arctan\frac{Z}{R}\bigg{)}=\frac{\alpha_{f}\hbar
c}{a}\underbrace{\frac{1}{4}\bigg{(}\frac{1}{\bar{Z}}+\frac{1}{\bar{R}}\arctan\frac{\bar{Z}}{\bar{R}}\bigg{)}}_{\bar{H}^{\mathrm{out}}_{\mathrm{el}}}.$
(5.1.6)

Summing up the various energy contributions, the total energy
$H_{\mathrm{tot}}$, which will be minimised, reads

$H_{\mathrm{tot}}=H^{\mathrm{box}}+H^{\mathrm{out}}\overset{{}^{(\ref{eqn:energyInbox})}}{\underset{{}^{(\ref{ElEneAus})}}{=}}\frac{\alpha_{f}\hbar
c}{a}\bigg{(}\bar{H}_{\mathrm{cur}}^{\mathrm{box}}+\bar{H}_{\mathrm{pot}}^{\mathrm{box}}+\bar{H}_{\mathrm{el}}^{\mathrm{out}}\bigg{)}=\frac{\alpha_{f}\hbar
c}{a}\bar{H}_{\mathrm{tot}}.$ (5.1.7)

##  6 Results and accuracy

The presented algorithm essentially consists of two different parts, both
carrying a numerical error. The first part is to calculate the various energy
components and the second one is the energy minimisation to find the
associated configuration. We realized, that we have to be very careful to
avoid numerical instabilities, therefore we will take a look at both steps
separately in the following.

###  6.1 Precision of the energy computation

The computation of the energy for a monopole with radius $\bar{r}_{0}$ in
lattice units $a$ results in $\bar{H}_{\mathrm{tot}}$, the total energy
$H_{\mathrm{tot}}$ in units of $\frac{\alpha_{f}\hbar c}{a}$, see Eq. (5.1.7).
We can fix the lattice spacing $a$ from

$a\overset{\footnotesize(\ref{solRadLat})}{=}\frac{r_{0}}{\bar{r}_{0}}\overset{\footnotesize(\ref{eqn:analyLsg})}{=}\frac{2.21\mathrm{fm}}{\bar{r}_{0}}.$
(6.1.1)

Now we are capable of transforming the total energy on the lattice
$\bar{H}_{\mathrm{tot}}$ into common energy units

$H_{\mathrm{tot}}\overset{\footnotesize(\ref{GesamtEne})}{=}\frac{\alpha_{f}\hbar
c}{a}\bar{H}_{\mathrm{tot}}\overset{\footnotesize(\ref{GitKonst})}{=}\frac{1.44}{2.21}\,\bar{r}_{0}\,\bar{H}_{\mathrm{tot}}\,\mathrm{MeV},$
(6.1.2)

which we are able to compare with the analytical value. Beside the total
energy we compare the ratio $H_{\mathrm{tot}}/H_{\mathrm{pot}}$ with the
analytical result (2.0.13). With these two measures, we are now ready to do
some simulations and check the results for their accuracy.

Figure 4: Total energy and ratio $H_{\mathrm{tot}}/H_{\mathrm{pot}}$ for
different lattices of size $n_{r}\times 2n_{z}$ with $n_{r}=n_{z}$. Analytical
values are marked with dashed lines. The plot shows the numerical results for
a monopole configuration placed on the lattice. No minimisation at this point.

We placed the analytical solution of the monopole configuration on the lattice
and calculated the energy contributions as described in section 5.1. The
results before any minimisation are plotted in Fig. 4, where the two measures
of accuracy are plotted versus different grid sizes. Blue lines indicate the
total energy $H_{\mathrm{tot}}$ and brown lines the ratio
$H_{\mathrm{tot}}/H_{\mathrm{pot}}$. The analytical values are marked with
dashed lines. As expected, for bigger lattices the calculations become more
accurate. Below we will discuss the results after minimisation.

###  6.2 Precision of the conjugate gradient minimisation

Up to now, all we did was to place the centre of the analytical monopole
configuration at the origin of the lattice and calculate its total energy.
Carrying on, we want to test the minimising procedure333The 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 a function depending on many arguments,
$f=f(x_{1},x_{2},...,x_{n})$.. If there were no numerical errors, the
algorithm would stop after a few iterations. Every deviation from the original
configuration may be seen as an error.
In our case, the function to be minimised is the total energy
$\bar{H}_{\mathrm{tot}}$ which depends on the field components

$\bar{H}_{\mathrm{tot}}=\bar{H}_{\mathrm{tot}}\big{(}q_{0}(\bar{r},\bar{z}),q_{r}(\bar{r},\bar{z}),q_{z}(\bar{r},\bar{z})\big{)}\quad\textrm{with}\quad
q_{0}^{2}+q_{r}^{2}+q_{z}^{2}=1.$ (6.2.1)

The conjugate gradient method starts at a given point on the hypersurface of
$\bar{H}_{\mathrm{tot}}$, which in our case is associated with the analytical
configuration and iterates its way to a local minimum. After the difference of
energies of two consecutive iterations falls below a certain value, the
algorithm stops.

It turns out that the procedure fails.

In Fig. 5 we compare the two-component vectors $(q_{r},q_{z})$ in the box
before and after the minimisation and find clear differences between the
numerical and analytical minima. Further, it can be clearly seen in Fig. 6,
that after the minimisation the $q_{0}$ component showed up wave-like
discontinuities. These deviations from the analytical solution are not
physical and need to be avoided. From the analytical solution, we know that
the field should be smooth.

Figure 5: Shown are the vector components $\vec{q}=(q_{r},q_{z})$, to allow
for a comparison of the monopole configurations before (exact, blue) and after
(with error, brown) the minimisation procedure. The magnification on the left
illustrates the differences more clearly. Figure 6: $q_{0}$-field in the
$rz$-plane for the minimised monopole configuration. Observe the wave-like
discontinuities along the radial direction.

The contribution of the discontinuities to the overall energy is very tiny and
for bigger lattices they get negligible small. Nevertheless they are always
present in the computations and are dangerous for the determination of static
dipole configurations. Even fixing the monopole centres these small
fluctuations lead to tedious collapses of the monopoles as was shown in
several diploma theses [11, 12, 13].

###  6.3 Solution to the numerical instability

As shown in Fig. 5, the problem discussed in the previous section is
accompanied by misalignments of the $\vec{q}$-vectors. This fact suggests to
diminish the variations in the directions of the $\vec{q}$-field. Therefore,
we have added a third term in Eq. (6.3.1) to the energy functional on the
lattice in order to minimise the fluctuations of $\vec{n}:=\vec{q}/|\vec{q}|$
and forcing discontinuities to vanish in the minimisation procedure. The
energy functional which we minimise is therefore

$\bar{H}_{\mathrm{mod}}^{\lambda}:=\bar{H}_{\mathrm{tot}}+\underbrace{\lambda\sum_{\bar{i}}\left(\partial_{\bar{i}}\frac{\vec{q}}{|\vec{q}|}\right)^{2}}_{\bar{H}^{\lambda}}\,,\quad\bar{i}=\\{\bar{r},\bar{z}\\},$
(6.3.1)

where the order of magnitude is controlled by $\lambda$. This extra term
$\bar{H}^{\lambda}$ reminds us of the mass term for the pion in the Skyrme
model. Likewise, the physical interpretation of our new term is, that it gives
mass to photons and hence suppresses photonic excitations. Since we are
computing static snapshots, the application of this method is justified. We
are not interested in the size of $\bar{H}^{\lambda}$, we want to use it as a
constraint and to minimise among the set of lattice configurations with
minimal $\bar{H}^{\lambda}$.

In Fig. 7, we show a plot of $H_{\mathrm{tot}}$ for simulations with different
values for $\lambda$ for $10\times 2\cdot 10$ and $30\times 2\cdot 30$
lattices. As one may expect, we see that the results do not change
significantly any more, if the parameter $\lambda$ exceeds a certain value.
With this method the two-component vectors before and after the minimisation
show no visible difference as Fig. 8 demonstrates at $\lambda=100$. The
correction of the minimisation procedure by the additional term leads to a
nicely smooth distribution of $q_{0}$-values around the centre of the
monopole, see Fig. 9.

Figure 7: Plot of the total energy $H_{\mathrm{tot}}$ versus $\lambda$ for
lattices of size $n_{r}\times 2n_{z}$ with $n_{r}=n_{z}=10$ and 30. Increasing
$\lambda$, discontinuities get smoothed out in both cases very soon. By
choosing a high enough value, they vanish completely and rising it further has
no effect. The difference in the left and right scales is caused by the much
smaller accuracy for $n_{r}=10$. Figure 8: Same simulation and parameters as
in Fig. 5, but using in the minimization procedure the additional term
$\bar{H}^{\lambda}$ of Eq. (6.3.1) at $\lambda=100$. As can be seen, the
vector part of the configuration is nearly indistinguishable before and after
the minimisation. Figure 9: Same simulation and parameters as in Fig. 6, but
with the additional term $\bar{H}^{\lambda}$ at $\lambda=100$. Observe, that
the waves in radial direction have vanished.

In Fig. 10 we compare the results of the different numerical evaluations of
the total energy of the monopole for $Z=10\,r_{0}$ and $R=10\,r_{0}$ and
various lattice sizes $n_{r}\times 2n_{z}$ and $n_{r}=n_{z}$. We can clearly
see that deviations from the analytical results due to the lattice
discretization. As discussed in Sect. 6.2, the minimisation without constraint
leads to a strange shape of the field distribution but to a small effect on
the energy only, as can be seen comparing the results before and after
minimisation. With the additional constraint (6.3.1) the minimal
configurations shows the expected behaviour and the expected energy. The
results nicely agree before and after the minimisation with the constraint.
Remaining differences to the energy of the analytical configuration are due to
the chosen discretization of the energy functional and the missing terms for
the potential and tangential energy outside the box. Adding the analytical
expressions
$2H_{\mathrm{pot}}^{\mathrm{out}}\overset{\footnotesize(\ref{schaetzwertPotOut})}{=}0.312\leavevmode\nobreak\
\mathrm{keV}$ for the sum of both terms would lead to a small overshooting of
the analytical result.

Figure 10: Total energy $H_{\mathrm{tot}}$ of a monopole configuration with
$Z=10\,r_{0}$ and $R=10\,r_{0}$ and different lattice sizes $n_{r}\times
2n_{z}$ and $n_{r}=n_{z}$, determined with different methods as explained in
the text. The inset shows that the energy scale was chosen in order to show
the tiny differences in the results clearly.

A similar comparison we show for the ratio $H_{\mathrm{tot}}/H_{\mathrm{pot}}$
in Fig. 11. Due to the Hobart-Derrick theorem this value should approach $4$.

Figure 11: Same situations as Fig. 10. The ratio
$H_{\mathrm{tot}}/H_{\mathrm{pot}}$ is shown, which is $4$ for the analytical
minimum configuration.

##  7 Conclusions

Only the three field variables of a scalar SO(3)-field and an appropriate
Lagrangian with two terms are necessary to describe the behaviour of
topological solitons with long-range interaction. SU(2), the double covering
group of SO(3), is well suited for investigations of the topology of field
configurations. The known solutions of the field equations gives a nice
opportunity to investigate the problems appearing in the numerical
computations. Several attempts [11, 12, 13] to determine field distributions
of static charges in two [12, 13] and three [11] dimensions had failed due to
numerical instabilities. We have shown in this article that the origin of
these instabilities are misalignments of the rotational axes corresponding to
the scalar SU(2)-field. Suppressing these wave-like disturbances with a
corresponding constraint we were able to get results of the numerical
minimisation procedure in excellent agreement with the analytical solutions.
The application of this method to two soliton systems will in future be of
special importance. It will allow to get accurate results for the interaction
at short distances. Deviations from the Coulomb behaviour are expected and
should be compared with the the running coupling, well-known from field
theory.

## References

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  * [2] T. H. R. Skyrme.  A Nonlinear theory of strong interactions.  Proc. Roy. Soc. Lond. A, 247:260–278, 1958.
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  * [5] Manfried Faber.  A model for topological fermions.  Few Body Syst., 30:149–186, 2001.
  * [6] Paul A. M. Dirac.  Quantised singularities in the electromagnetic field.  Proc. Roy. Soc. Lond., A133:60–72, 1931.
  * [7] Paul A. M. Dirac.  The theory of magnetic poles.  Phys. Rev., 74:817–830, 1948.
  * [8] Manfried Faber and Alexander P. Kobushkin.  Electrodynamic limit in a model for charged solitons.  Phys.Rev., D69:116002, 2004.
  * [9] Manfried Faber.  Charges and Electromagnetic radiation as topological excitations.  Advances in High Energy Physics, 9340516:9, 2017.
  * [10] Joachim Wabnig.  Interaction in the Model of Topological Fermions.  Diploma thesis, Physics, Technische Universität Wien, 2001.
  * [11] Josef Resch.  Numerische Analyse an Dipolkonfigurationen im Modell topologischer Fermionen.  Diploma thesis, Physics, Technische Universität Wien, 2011.
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  * [14] R.H. Hobart.  On the instability of a class of unitary field models.  Proc. Phys. Soc. London, 82(2):201–203, 1963.
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[affiliation=1,2]OrchidChetia Phukan [affiliation=3]YashasviChaurasia
[affiliation=1,3]Arun BalajiBuduru [affiliation=1,3]RajeshSharma

# CoLLM: CoLLaborative Modeling Across Languages for Unified Multilingual
Abuse Detection

###### Abstract

In this study, we first investigate representations from paralingual Pre-
Trained model (PTMs) for Audio Abuse Detection (AAD) . Our results demonstrate
their superiority compared to other PTM representations on the ADIMA
benchmark. Furthermore, combining PTM representations enhances AAD
performance. Despite these improvements, challenges with cross-lingual
generalizability still remain, and certain languages necessitate training in
the same language. This demands individual models for different languages,
leading to scalability and maintenance issues, hindering the practical
deployment of AAD systems in linguistically diverse real-world environments.
To address this, we introduce, CoLLM, a novel framework, completely training-
free and allows seamless merging of models trained in different languages
through weight-averaging. This culminates in a unified model with competitive
AAD performance across all languages.

###### keywords:

Multilingual Abuse Detection, Cross-Lingual, Model Merging

##  1 Introduction

In today's digital era, online social media platforms, gaming communities, and
digital environments have become indispensable aspects of modern life. Within
these realms, audio plays a crucial role in facilitating real-time
communication and fostering connections among users. However, alongside the
advantages of audio communication in these settings, there is a growing
concern regarding audio-abusive content (AAC). This refers to the malicious or
inappropriate use of audio, encompassing behaviors such as verbal harassment,
hate speech, and the dissemination of harmful or offensive material. The
prevalence of AAC in digital spaces presents substantial challenges for both
users and platform administrators.

Due to the challenges induced by AAC to online safety and well-being, audio-
based abuse detection (AAD) has caught recent attention. While abuse detection
in other modalities like text [davidson2017automated, madhu2023detecting] and
visual [alcantara2020offensive, gao2020offensive] content has garnered
significant focus and development, AAD has not received comparable attention,
despite its critical importance in safeguarding digital spaces. This gap
underscores the urgent need for research and development efforts aimed at
advancing AAD technologies to effectively mitigate the harmful impacts of
audio-based abuse in online environments.

There has been growing interest in building effective AAD systems, for
example, Gupta et al. [9746718] explored various representations from VGG
trained on AudioSet and multilingual wav2vec2 models with Fully Connected
Network (FCN), GRU, and LSTM as classifiers. Further, Sharon et al.
[sharon22_interspeech] showed that AAD performance can be enchanced by fusing
audio and ASR transcribed textual representations. On the other hand, Thakran
et al. [thakran23_interspeech] showed that choosing the right acoustic cues
can lead to state-of-the-art (SOTA) performance in AAD without the need of
textual representations also reverberated by Spiesberger et al.
[spiesberger23_interspeech]. Thakran et al. explored representations from
multilingual as well as emotion recognition PTMs.

AAD has benefitted significantly from the availability of different PTM
representations, particularly those adept at capturing paralinguistic cues
effectively. However, representations from paralingual PTM [shor2022universal]
which has shown SOTA performance in related tasks to AAD such as speech
emotion recognition (SER), speaker identification (SI), and so on haven't been
explored or given less attention for AAD. Our hypothesis suggests that these
representations will better capture verbal cues like pitch, tone, and
intensity compared to representations of other PTMs, resulting in greater
effectiveness for AAD. To validate our hypothesis, we present a comprehensive
comparative study of five PTM (TRILLsson,Whisper, MMS, Unispeech-SAT,
x-vector) representations including paralingual, multilingual, monolingual as
well as speaker recognition PTM which are SOTA in respective tasks and highly
potential candidates for improved AAD.

We also conducted an investigative analysis by combining these PTM
representations to explore their potential for complementary behavior, akin to
observations in other tasks such as speech recognition
[arunkumar22b_interspeech]. Our investigation continued as we sought to
address the challenge of cross-lingual generalizability as real-world digital
environments are linguistically diverse and AAC can appear in different
languages. This is a difficult task as some languages require training on that
language for better

AAD performance and models trained in other languages will perform poorly on
them also pointed out by Spiesberger et al [spiesberger23_interspeech]. This
gives rise to a scenario where individual models may be required for different
languages, which poses a significant challenge in terms of scalability,
maintenance, and resource allocation. Maintaining separate models for each
language not only increases the complexity of the AAD system but also demands
additional computational resources.Moreover, it can lead to inconsistencies in
performance across languages and hinder the seamless integration of AAD
systems into multilingual environments. One way to solve this problem is to
train model on the combination of different languages, however, this is not
exactly a feasible solution, because training data for different languages may
not be available at the same time and with the increase in languages, the
computational requirement for training them together will increase. To
mitigate this, we propose, CoLLaborative Modeling Across Languages (CoLLM), a
novel framework, that doesn't require training at all and enables for the
seamless integration of models trained in various languages through the
process of weight-averaging, resulting in a single unified model that
demonstrates competitive performance in AAD across all languages while
comprising the same number of parameters as that of the individual languages
models. The proposed framework remains effective even in scenarios where
training data for certain languages may not be available at certain times.
Notably, CoLLM offers the added benefit of operating in Plug-in mode,
facilitating the seamless integration of models trained in various languages
alongside those trained on ADIMA. This approach ensures consistent performance
across upcoming new languages as well as previously seen languages.

To summarize, our main contributions are three folds:

  * •

A comprehensive empirical study to investigate the performance of paralingual
PTM representations for AAD.

  * •

An in-depth analysis by combining these PTM representations to examine their
complementary behavior for improved AAD performance.

  * •

CoLLM, a novel unified framework for achieving competitive performance across
different languages for AAD.

For reproducibility of our experiments and as a reference for future studies
to build upon CoLLM for collaborative AAD, we will open source the codes and
models built as part of our study after the double-blind review.

##  2 Pre-Trained Representations

In this section, we give brief of the PTM's representations used in our study.
We utilize TRILLsson [shor22_interspeech] as paralingual PTM that was derived
from SOTA paralingual conformer CAP12 [shor2022universal] through knowledge
distillation. Unlike CAP12, TRILLsson is openly accessible and nearly SOTA
performance in the NOSS benchmark. TRILLsson was trained on AudioSet and
Libri-light datasets, while CAP12 relies on the YT-U dataset, which
potentially may include multilingual data. We employ
TRILLsson111https://tfhub.dev/google/nonsemantic-speech-benchmark/trillsson4/1
accessible via TensorFlow Hub and the model returns a vector of 1024-dimension
size through aggregrating over time. We employ MMS [pratap2023scaling] and
Whisper [radford2023robust] as multilingual PTMs in our study and they have
shown SOTA in various downstream multilingual speech processing tasks. MMS was
trained in over 1400 languages in a self-supervised manner while Whisper on 96
languages in a weakly-supervised manner. For monolingual PTM, we choose WavLM
[chen2022wavlm], which has achieved SOTA on SUPERB. Additionally, we consider
x-vector [8461375], a SOTA time-delay neural network for speaker recognition
and trained in a supervised manner. We have choosen x-vector as its
representations have shown superior performance for related tasks such as SER
[chetiaphukan23_interspeech] and predicting shout intensity
[fukumori2023investigating] where capturing verbal cues such as pitch,
intensity, tone, etc. are very important.

We experiment with MMS222https://huggingface.co/facebook/mms-1b,
Whisper333https://huggingface.co/openai/whisper-base, and
WavLM444https://huggingface.co/microsoft/wavlm-base available in Huggingface.
We took x-vector555https://huggingface.co/speechbrain/spkrec-xvect-voxceleb
from speechbrain library accesible through Huggingface. The input audio sample
is first sampled to 16kHz before passing through these PTMs and
representations of 1280, 768, 512-dimensional vectors for MMS, WavLM, x-vector
are obtained respectively from the last hidden states through pooling average.
For Whisper, we extract representations of 512-dimension size from the encoder
through average-pooling and discarding the decoder.

Figure 1: CoLLM: Here, $M_{Hi}$, $M_{Be}$, $M_{Ka}$ stands for models on
Hindi, Bengali, Kannada respectively. Left (In-hand Mode): Models trained on
ADIMA languages and weight-averaging of the models' weights for a single
unified model; Right (Plug-in Mode) Model trained on external language, that
can be integrated into CoLLM and $M_{Plug-in}$ signifies the Plug-in model

##  3 CoLLM

In this section, we present, CoLLM, a novel framework for unified abuse
detection across multiple languages. The proposed framework is shown in Figure
1.

CoLLM operates under the assumption that models trained on different languages
share a common architecture. This assumption enables seamless integration and
collaboration among diverse linguistic models within the framework. CoLLM
allows the merging of the trained models through weight-averaging. Here, let
$W_{i}$ represent the model weights for language $i$, where $i=1,2,\ldots,n$
and $n$ is the total number of languages. First, we normalize the weights of
the models using L1 normalization given by $\hat{W}_{i}$.

$\hat{W}_{i}=\frac{W_{i}}{\lVert W_{i}\rVert_{1}}$ (1)

Normalizing the weights before averaging will ensure fair representation,
alignment of weight magnitudes, and stability in combining models trained in
different languages. Then, we average the normalized model weights and the
output is given by $\bar{W}$.

$\bar{W}=\sum_{i=1}^{n}\hat{W}_{i}$ (2)

This formulation ($\bar{W}$) ensures that the resulting unified model captures
the collective knowledge from models trained in various languages while
keeping the number of parameters the same as the individual models trained in
different languages.

Figure 2: Downstream Model Architectures; Here, FCN stands for Fully Connected
Network; Representation Fusion represents the model architecture for
combination of PTM representations

##  4 Experiments

Table 1: Evaluation Scores for models built on different PTM representations; PTM, Down stands for PTM representations, Downstream model respectively; X, U, T, M, W stands for x-vector, Unispeech-SAT, TRILLsson, MMS, Whisper respectively; Be, Bh, Gu, Ha, Hi, Ka, Ma, Od, Pu, Ta represents Bengali, Bhojpuri, Gujarati, Haryanvi, Hindi, Kannada, Malayalam, Odiya, Punjabi, Tamil respectively; Acc, F1 stands for Accuracy and Macro F1 respectively; Red, Blue, Purple represents the colors of the first, second, third scores respectively in descending order: The notations used in this Table holds for Table 2 and Figure 3 PTM |  Down | Be | Bh | Gu | Ha | Hi | Ka | Ma | Od | Pu | Ta
---|---|---|---|---|---|---|---|---|---|---|---
|  |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1
X |  CNN |  79.45 |  78.58 |  76.19 |  76.15 |  78.45 |  77.51 |  80.87 |  80.70 |  78.32 |  77.93 |  79.94 |  77.87 |  84.41 |  81.23 |  81.37 |  78.33 |  82.83 |  81.54 |  81.40 |  80.32
TF |  79.73 |  79.61 |  75.89 |  76.01 |  77.90 |  75.91 |  81.15 |  79.58 |  78.05 |  78.00 |  79.67 |  79.32 |  84.68 |  81.64 |  81.64 |  78.98 |  82.56 |  81.75 |  80.86 |  80.01
U |  CNN |  78.38 |  76.62 |  74.40 |  73.81 |  80.11 |  79.24 |  78.96 |  77.19 |  75.33 |  74.89 |  78.32 |  77.68 |  80.11 |  78.65 |  78.08 |  77.03 |  80.93 |  79.48 |  79.24 |  77.89
TF |  79.19 |  76.59 |  74.82 |  73.61 |  77.35 |  76.82 |  77.04 |  75.52 |  75.07 |  74.53 |  80.76 |  79.22 |  79.57 |  78.42 |  78.90 |  77.79 |  81.20 |  80.48 |  78.44 |  78.16
T |  CNN |  81.35 |  80.72 |  77.10 |  76.40 |  82.32 |  82.28 |  82.51 |  82.47 |  83.19 |  82.91 |  83.74 |  83.68 |  86.29 |  86.22 |  83.56 |  83.50 |  83.38 |  83.31 |  82.21 |  81.93
TF |  82.16 |  82.06 |  77.08 |  75.75 |  81.77 |  81.52 |  83.33 |  83.27 |  79.40 |  79.38 |  82.11 |  82.05 |  84.95 |  84.89 |  82.23 |  82.19 |  84.47 |  84.40 |  81.67 |  80.03
M |  CNN |  78.64 |  77.43 |  74.10 |  74.02 |  80.39 |  80.32 |  78.14 |  78.09 |  76.96 |  76.91 |  79.67 |  79.60 |  81.45 |  81.39 |  82.46 |  82.40 |  79.83 |  79.78 |  76.01 |  75.37
TF |  77.03 |  76.99 |  74.71 |  72.77 |  81.21 |  81.15 |  78.41 |  78.35 |  77.23 |  77.19 |  79.40 |  79.35 |  79.30 |  79.25 |  81.92 |  81.87 |  80.38 |  80.32 |  76.28 |  76.21
W |  CNN |  79.46 |  79.42 |  75.30 |  74.05 |  81.77 |  81.72 |  83.15 |  82.47 |  82.65 |  82.60 |  81.57 |  81.52 |  82.53 |  82.48 |  81.37 |  81.32 |  82.56 |  82.50 |  81.40 |  81.39
TF |  78.38 |  78.34 |  76.21 |  74.65 |  81.49 |  81.16 |  83.33 |  83.28 |  80.76 |  80.73 |  81.84 |  81.79 |  83.60 |  83.55 |  80.55 |  80.51 |  85.83 |  85.56 |  81.13 |  80.43

###  4.1 Dataset

We use ADIMA [9746718] for our experiments and it contains 11,775 expert-
annotated audio samples in ten Indic languages and spanning over 65 hours. It
is evenly distributed across languages and comprises 5,108 abusive and 6,667
non-abusive samples from 6,446 unique Users.

###  4.2 Downstream Classifier

We experiment with two downstream networks (CNN, Transformer) extensively used
in previous works [rathod23_interspeech, arunkumar22b_interspeech] for various
speech processing tasks. The model architectures are shown in Figure 2. The
extracted representations from the PTMs are passed through the networks. For
CNN model, we build a convolution block that contains 1D-CNN layer and
maxpooling layer followed by FCN. For Transformer model, we have used the same
convolution block as used in CNN model followed by a transformer encoder from
the vanilla transformer network from Vashwani et al. [vaswani2017attention].
We kept number of heads as 8 in the multi-head attention block for the
transformer encoder. For experiments, with combination of PTM representations,
we use pass individual PTM representations through a individual
Convolution/Transformer block as shown in Figure 2 and then concatenated and
passed through FCN for final classification. We don't intermix between
convolution and transformer block, if we use convolution block for one PTM
representations then we use another convolution block for the other PTM
representations, similarly, we follow for transformer block.

For FCN in Transformer model and the combined model, we kept the same number
of layers and neurons in each layer as used in the CNN model. For the
classification head, we use softmax activation function that outputs the
probabilities pertaining to each class i.e abuse vs non-abuse. We train the
models for 50 epochs using Rectified Adam as optimizer with a learning rate of
1e-3 and batch size of 32. We have also used early stopping and dropout for
preventing overfitting. We use Accuracy and F1-macro as metrics for evaluating
our models. We use Tensorflow library for implementations. We use the official
split given by [gupta2022adima] for training and testing our models.

Table 2: Evaluation Scores for models built on the combination of different PTM representations; Fusion represents the fusion of different PTM representations Fusion |  Down | Be | Bh | Gu | Ha | Hi | Ka | Ma | Od | Pu | Ta
---|---|---|---|---|---|---|---|---|---|---|---
|  |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1 |  Acc |  F1
X+T |  CNN |  81.08 |  80.27 |  77.38 |  74.11 |  80.11 |  79.01 |  82.78 |  82.05 |  80.21 |  80.21 |  82.65 |  81.84 |  87.36 |  85.75 |  83.56 |  81.28 |  84.74 |  82.01 |  87.20 |  83.01
TF |  80.54 |  79.72 |  76.79 |  73.12 |  82.93 |  82.87 |  80.87 |  80.17 |  80.76 |  80.75 |  80.75 |  77.93 |  85.48 |  81.53 |  83.01 |  82.22 |  85.01 |  85.00 |  83.82 |  83.28
X+W |  CNN |  78.64 |  78.37 |  77.97 |  77.38 |  82.32 |  79.28 |  82.78 |  81.84 |  78.59 |  77.50 |  80.75 |  76.95 |  85.75 |  81.25 |  82.46 |  81.64 |  83.65 |  82.28 |  82.74 |  82.47
TF |  77.02 |  76.04 |  77.97 |  75.11 |  82.32 |  78.88 |  84.69 |  84.53 |  81.30 |  81.28 |  81.84 |  79.85 |  86.55 |  82.79 |  82.19 |  81.47 |  85.01 |  85.01 |  81.94 |  77.28
M+T |  CNN |  80.54 |  79.82 |  77.67 |  74.27 |  81.21 |  79.83 |  82.24 |  80.05 |  81.03 |  80.99 |  83.74 |  79.70 |  86.29 |  82.72 |  83.56 |  81.82 |  85.28 |  84.74 |  82.75 |  82.74
TF |  79.46 |  78.45 |  76.49 |  72.84 |  78.45 |  74.78 |  81.42 |  81.25 |  78.32 |  78.14 |  81.57 |  78.31 |  83.87 |  78.94 |  81.92 |  81.07 |  83.65 |  83.63 |  83.56 |  79.44
M+W |  CNN |  82.97 |  81.62 |  77.67 |  76.78 |  83.14 |  78.86 |  84.97 |  83.87 |  82.38 |  81.54 |  81.03 |  80.75 |  84.40 |  83.87 |  84.65 |  82.19 |  84.74 |  84.72 |  80.05 |  79.24
TF |  78.65 |  78.12 |  77.97 |  74.96 |  81.22 |  77.44 |  82.24 |  82.07 |  79.94 |  79.92 |  82.38 |  79.79 |  84.94 |  80.73 |  81.91 |  80.88 |  84.46 |  84.43 |  83.55 |  78.49
T+U |  CNN |  82.70 |  82.06 |  78.27 |  76.19 |  82.32 |  79.00 |  81.96 |  81.42 |  80.21 |  80.20 |  82.65 |  79.55 |  86.55 |  85.75 |  82.74 |  81.64 |  84.74 |  84.46 |  83.28 |  78.50
TF |  81.08 |  80.33 |  75.60 |  72.78 |  80.39 |  76.38 |  84.43 |  84.29 |  80.22 |  80.16 |  81.30 |  78.15 |  85.48 |  81.76 |  80.82 |  79.66 |  85.01 |  84.99 |  81.94 |  77.14
W+U |  CNN |  82.43 |  80.81 |  77.97 |  75.22 |  81.49 |  80.66 |  85.24 |  85.10 |  83.73 |  83.68 |  82.38 |  80.30 |  86.55 |  82.10 |  84.11 |  82.73 |  84.19 |  83.37 |  83.01 |  81.40
TF |  78.92 |  78.08 |  78.87 |  76.17 |  82.04 |  78.26 |  85.25 |  85.11 |  80.22 |  80.21 |  81.30 |  79.47 |  84.68 |  80.44 |  82.74 |  82.04 |  85.29 |  85.28 |  80.32 |  75.82
W+T |  CNN |  82.70 |  81.72 |  80.35 |  78.27 |  83.42 |  80.38 |  85.79 |  85.60 |  82.11 |  80.75 |  81.84 |  81.02 |  87.90 |  82.27 |  84.11 |  83.83 |  86.37 |  84.19 |  83.28 |  82.74
TF |  81.08 |  80.20 |  77.98 |  74.23 |  79.83 |  76.10 |  85.25 |  85.14 |  82.93 |  82.11 |  82.11 |  78.84 |  87.63 |  87.36 |  84.11 |  83.28 |  86.33 |  85.56 |  82.21 |  77.55
Table 3: Evaluation Scores with CoLLM; For CoLLM, we have choosen the best combination of PTM representations and downstream network; TRILLsson with CNN, Whisper with Transformer from Table 1; Whisper + TRILLsson from Table 2 Feats |  Down | Be | Bh | Gu | Ha | Hi | Ka | Ma | Od | Pu | Ta
---|---|---|---|---|---|---|---|---|---|---|---
|  |  C |  D |  C |  D |  C |  D |  C |  D |  C |  D |  C |  D |  C |  D |  C |  D |  D |  C |  D |  F1
W |  TF |  76.20 |  |  73.12 |  |  79.13 |  |  77.23 |  |  77.41 |  |  78.48 |  |  80.37 |  |  79.03 |  |  80.10 |  |  79.83 |
T |  CNN |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
W+T |  CNN |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
W+T/2 |  CNN |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

###  4.3 Experimental Results

Within-language Evaluation: The evaluation results of the models trained and
evaluated on the same language with different PTM representations are given in
Table 1. First, we discuss the results obtained by training with single PTM
representations. Models trained on TRILLsson (Paralingual PTM) representations
performed the best across most of the languages which proves their capability
in capturing cues such as pitch, tone, intensity more effectively than other
PTM representations and that are very important for AAD. Models trained on
Whisper representations showed the second-best performance in most languages
compared to TRILLsson and leading in Haryanvi, Punjabi, and Tamil in
comparison to TRILLsson representations. This can be attributed to its
exposure to diverse linguistic contexts during pre-training, acquiring a broad
understanding of language structures and nuances and also effective in
capturing acoustic cues across different languages. Models trained on
x-vector, Unispeech-SAT, and MMS representations mixed performance between
them. Also, for the downstream networks, CNN is showing leading performance
comparison to transformer, this may be due to transformer network requiring
more training data. Our results with TRILLsson representations improves over
previous works leveraging single PTM representations [gupta2022adima] and also
[spiesberger23_interspeech], that used handcrafted features for AAD.

Table 2 shows the results when combining different PTM representations. We
experimented with the combinations where either TRILLsson and Whisper is
present as these two representations showed the best performance while using
individually. Combination of representations of TRILLsson and Whisper
performed the best among all the other representations showing its
complementary behavior. We also notice this behavior in some instances, like,
for example, combination of x-vector and TRILLsson led to improvement in
Gujarati then its individual representations performance and also in
combination of Whisper and Unispeech-SAT in Hindi. Modeling with combination
of representations leads to gain in performance, also, it may remain constant
or drop as seen for Kannada. However, modeling with combination of
representations comes at a computational cost due to the increase in number of
trainable parameters of the model. So, it is left to the researchers to decide
based on their availability of computational resources.

Cross-lingual Evaluation: We present the cross-lingual evaluation results in
Figure 3. We have only experimented with CNN as the network as CNN model has
shown superior performance in individual as well as when experimented with
combination of PTM representations. We notice that cross-lingual performance
is low compared to models trained and evaluated on the same language.

Results obtained with CoLLM: As digital spaces are ever-evolving and
linguistically diverse, this can lead to the availability of training data in
new languages but models built on these languages may lack the performance on
other languages, such as the langugaes on the languages in ADIMA, CoLLM by
operting in plug-in model will allow merging of the models trained on new
lanuages with the models trained on ADIMA. This will provide consistent
performance in all the languages. By other than languages in ADIMA, CoLLM, by
operating in plug-in will allow merging of models trained on the. Our results
with CoLLM also looks quite effective and follows a recent trend seen in NLP
and vision community for model merging without the requirement of extra GPU
and no increase in inference time etc.

((a))

((b))

Figure 3: Cross-lingual Evaluation Scores; Train, Test represents the language
trained on and the languages tested on; Here, only accuracy score is taken and
scores are rounded off to two decimal points

##  5 Conclusion

We show that representations extracted from paralingual PTM perform the best
in comparison to other PTM representations. This performance is attributed to
their effectiveness in capturing pitch, tone, intensity important for AAD much
more effectively than other representations. Additionally, we show that
combining PTM representations leads to further improvement in AAD performance.
Despite this, challenges persist in achieving cross-lingual generalizability,
with certain languages requiring language-specific training impeding the
practical deployment of AAD systems in linguistically diverse real-world
scenarios. To address this, we propose CoLLM, a novel framework that
facilitates the merging of models trained in different languages and results
in a single unified model exhibiting competitive AAD performance across
different languages. Our work can guide future studies, firstly, in selecting
the most suitable representations for AAD and secondly, act as a reference for
the development of collaborative framework to build unified AAD system for
linguistically diverse environments. Additionally, we believe the proposed
framework can serve as an inspiration for building a unified framework that
works on multiple languages for related applications.