# Internal force distributions in ’t Hooft-Polyakov monopole and Julia-Zee
dyon

J. Yu. Panteleeva  Institut für Theoretische Physik II, Ruhr-Universität
Bochum, D-44780 Bochum, Germany

###### Abstract

The energy-momentum tensor of the ’t Hooft-Polyakov monopole and the Julia-Zee
dyon are studied. This tensor contains important information about the
pressure and the shear force distributions which define mechanical properties
of systems. Obtaining the violation of the local stability criterion for the
magnetic monopole and dyon we decompose the EMTs into the long- and short-
range parts. This decomposition depends on the abelian field strength tensor
which can not be uniquely defined. We suggest to use the modified ’t Hooft
definition for the tensor. Finally, the long- and short-range parts of the
EMTs are computed and new equilibrium equations are obtained. Numerical values
for masses, $D$-terms and various mean square radii for the monopole and the
dyon are also computed.

This paper is dedicated to the bright memory of Maxim Polyakov.

##  I Introduction

Magnetic monopole is a hypothetical particle carrying a magnetic charge like
an electron carrying electric charge. Maxwell formulated his elegant equations
in 1865 without monopole because there was no evidence for it. Despite the
fact that the monopole is still experimentally not detected there is also no
proof of its nonexistence. In year 1974 A. Polyakov and G. ’t Hooft have shown
in Refs. Polyakov:1974ek ; tHooft:1974kcl  that magnetic monopole exists in
all Grand Unified Theories of elementary particles as a static soliton
solution of the classical equations of motion. In year 1975 A. Zee and B.
Julia generalised the ’t Hooft-Polyakov monopole by including the electric
charge, they called such particle a dyon in Ref. Julia:1975ff .

Usually the mass and related properties of a monopole are studied. In this
work we are more interested in mechanical properties of a monopole and a dyon
such as pressure, share force distributions and the mechanical stability
condition which can be obtained from the corresponding energy-momentum tensor
(EMT). Although the main purpose of the paper is to study the mechanical
properties of the monopole and dyon, we are also interested in studying the
local stability criterium of an arbitrary system with finite energy obtained
in 2016 in Ref. Perevalova:2016dln . The force distributions obtained from the
EMT could be useful in experimental search of monopoles and studying the
monopole stability can be helpful for theoretical insights into stability of
classical solutions in gauge theories.

The organisation of the paper is as follows. In Sec. II we introduce main
definitions corresponding to an EMT and discuss the stability conditions for
static and non-static EMT. In Sec. III we discuss the EMT of the ’t Hooft-
Polyakov monopole and its consequences. We also separate the EMT into the
long- and short-range parts in this section introducing the abelian strength
tensor. In Sec. IV we discuss the properties of Julia-Zee dyon. We conclude in
the last section.

##  II Stability conditions

###  II.1 EMT densities

The energy-momentum tensor $T_{\mu\nu}(x)$ can be defined as a functional
variation of the matter part of the action in curved space-time with respect
to metric $g^{\mu\nu}(x)$ as follows

$T_{\mu\nu}(x)=\left.\dfrac{2}{\sqrt{-g}}\dfrac{\delta S}{\delta
g^{\mu\nu}(x)}\right|_{g_{\mu\nu=\eta_{\mu\nu}}},$ (1)

where $\eta_{\mu\nu}$ is Minkowski tensor with the metric signature
$(1,-1,-1,-1)$.

The EMT encodes fundamental properties of a system. The $T_{00}$ component of
the EMT defines the energy density in the studied system corresponding to the
mass distribution in the static approximation. Integral of the $T_{00}(r)$
over the full space defines the full energy of a system or its mass in the
rest frame of the system

$M=\int d^{3}rT_{00}(r).$ (2)

The most interesting part of the EMT for our purpose is the $ij$-components,
which define the stress tensor. The $T_{ij}$ components can be associated,
according to Ref. Polyakov:2002yz , with the distribution of the shear forces
$s(r)$ and the elastic pressure $p(r)$ inside the system. For a spherically
symmetric system in static approximation in three dimensions the stress tensor
is decomposed in $s(r)$ and $p(r)$ as follows

$T_{ij}(\vec{r})=\left(\frac{r_{i}r_{j}}{r^{2}}-\dfrac{1}{3}\delta_{ij}\right)s(r)+\delta_{ij}p(r).$
(3)

The pressure and shear force distributions can be connected with the
gravitational $D(t)$-form factor, which is involved in the parametrisation of
the matrix element of the EMT operator, see e.g. Ref. Polyakov:2002yz . As it
follows from Ref. Polyakov:2002yz , the $D$-term (or the Druck-term) $D\equiv
D(0)$ can be obtained as

$D=-\dfrac{4M}{15}\int d^{3}rr^{2}s(r)=M\int d^{3}rr^{2}p(r),$ (4)

where $M$ is the mass of the system. The value of the $D$-term for a particle
can be measured experimentally, like its mass, however, the value is difficult
to extract from experimental data. For example, the experimental value of the
$D$-term for the proton is $D=-1.47\pm 0.06\pm 0.14$, where the first error is
the statistical uncertainty, and the second error is due to the systematic
uncertainties, see Ref. Burkert:2021ith . The value of the $D$-term for the
nucleon was first obtained theoretically in the Skyrme model in Ref.
Cebulla:2007ei , the authors got the value $D=-3.6$ and in the bag model the
value of the $D$-term for the nucleon is $D=-1.1$ Hudson:2017oul , other
values of the $D$-term together with the corresponding references can be found
in the table 2 of Ref. Dutrieux:2021nlz . The values of the $D$-term for the
monopole and the dyon are given in Secs. III and IV, respectively.

###  II.2 Static EMT

The EMT is the conserved Noether current associated with spacetime
translations, i.e. it satisfies the following condition

$\partial_{\mu}T^{\mu\nu}=0.$ (5)

For the static EMT the Eq. (5) turns to $\partial_{i}T^{i\nu}=0$. For the
parametrisation of Eq. (3) it implies that the pressure and shear force
distributions satisfy the following equation which is also called equilibrium
equation

$\dfrac{\partial p(r)}{\partial r}+\dfrac{2}{3}\dfrac{\partial s(r)}{\partial
r}+\dfrac{2}{r}s(r)=0.$ (6)

Assuming that the pressure and shear force densities decay at large distances
faster than $\sim\dfrac{1}{r^{3}}$, multiplying on $r^{3}$ and integrating
over $r$ one obtains the von Laue stability condition given in Ref. Laue

$\int d^{3}rp(r)=0,$ (7)

which is necessary for stability, but not sufficient, since it is also
satisfied for unstable systems, see discussion in Ref. Perevalova:2016dln .
This condition is satisfied for any system, whose EMT is conserved and implies
that the pressure distribution has at least one mode. Integrating the
differential equation (6), from positive value $r$ to infinity and requiring
that $s(r)$ and $p(r)$ vanish at the infinity, one obtains

$\dfrac{2}{3}s(r)+p(r)=2\int\limits_{r}^{\infty}dx\dfrac{s(x)}{x}.$ (8)

This equation describes the equilibrium of the internal forces inside a
system. The combination $\frac{2}{3}s(r)+p(r)$ describes the normal component
of the total force exhibited by the system on an infinitesimal piece of area
$dS_{i}$, which is denoted as $p_{\text{r}}(r)$:

$F^{i}(r)=T^{ij}(r)dS_{j}=\left(\frac{2}{3}s(r)+p(r)\right)dS^{i}=p_{\text{r}}(r)dS^{i}$
(9)

where $dS^{i}=dS\ r^{i}/r$. From the equilibrium equation (8) follows that for
the positive shear force distribution $s(r)$ the normal force is always
positive, for the negative $s(r)$ the normal force is always negative. In Ref.
Perevalova:2016dln  it was argued that the normal force has to be directed
outwards otherwise the system would collapse, this means that it has to be
non-negative, i.e.

$\dfrac{2}{3}s(r)+p(r)\geq 0.$ (10)

Thus, according to the Eq. (8) for negative shear force distribution the
system is definitely unstable. The local stability condition is also
necessary, but not sufficient for stability analogously to the von Laue
condition. However, due to its local character, it is stronger than the von
Laue condition.

###  II.3 EMT and external forces

The EMT conservation taking the form $\partial_{i}T^{ij}(r)=f^{j}(r)$ can be
interpreted according to Ref. Landau  as the equilibrium equation for internal
stress and external force $f_{j}$ (per unit of the volume). According to the
parametrisation in Eq. (3) the equilibrium equation gets the following form

$\dfrac{d}{dr}\left(\dfrac{2}{3}s(r)+p(r)\right)+2\dfrac{s(r)}{r}=f(r),$ (11)

where $f(r)$ is the normal component of the external force per unit of the
volume. This equation describes the balance between internal forces pushing
out from center and external force pulling inwards to the center. When the
corresponding forces are equal, the system is at equilibrium. The von Laue
stability condition for such equilibrium equation gets the following form

$\int d^{3}rp(r)=-\dfrac{1}{3}\int d^{3}r~{}rf(r).$ (12)

One can also rewrite the equilibrium equation (11) as

$p_{\text{r}}(r)+\sigma(r)=2\int\limits_{r}^{\infty}dx\dfrac{s(x)}{x},$ (13)

with $\sigma(r)=\int\limits_{r}^{\infty}dxf(x)$. The left-hand side describes
the normal component of a total force of the system acting on the
infinitesimal unit area $dS^{i}$ as it follows from below

$F^{i}(r)=(T^{ij}(r)+\delta^{ij}\sigma(r))dS_{j}=\left(\frac{2}{3}s(r)+p(r)+\int\limits_{r}^{\infty}dxf(x)\right)dS^{i}=\Big{(}p_{\text{r}}(r)+\sigma(r)\Big{)}dS^{i}.$
(14)

Then for systems affected by an external force, i.e. for systems described by
the equilibrium equation (11), the local stability criterium of Eq. (10) can
be modified as follows

$\dfrac{2}{3}s(r)+p(r)+\int\limits_{r}^{\infty}dxf(x)\geq 0.$ (15)

For a time-dependent system the conservation of EMT gives the following
equation $\partial_{i}T^{ij}=-\partial_{0}T^{0j}$, where the left-hand side
again describes internal force of the system and the right-hand side can be
interpreted as an external force.

###  II.4 Mean square energy radius and mechanical radius

For a positive energy density $T_{00}(r)$ the mean square radius of the energy
density can be introduced as

$\langle r^{2}\rangle_{E}=\dfrac{\int d^{3}rr^{2}T_{00}(r)}{\int
d^{3}rT_{00}(r)},$ (16)

which characterises the size of the system in which the energy is distributed.

According to the local stability condition, for a stable system the radial
force must be positive, so the mechanical mean square radius where the normal
force is distributed can be defined as in Ref. Polyakov:2018zvc , for a system
with the equilibrium equation (6) it takes the following form

$\langle r^{2}\rangle_{\text{mech}}=\dfrac{\int d^{3}r\
r^{2}p_{\text{r}}(r)}{\int d^{3}rp_{\text{r}}(r)},$ (17)

and for the system with the equilibrium equation (11) it takes has the form:

$\langle r^{2}\rangle_{\text{mech}}=\dfrac{\int d^{3}r\
r^{2}\left(p_{\text{r}}(r)+\sigma(r)\right)}{\int
d^{3}r\left(p_{\text{r}}(r)+\sigma(r)\right)},$ (18)

where $p_{\text{r}}(r)$ and $\sigma(r)$ are defined above.

The local stability condition has not been mathematically proven and thereby
is still questioned, see e.g. criticism in Ref. Ji:2021mfb . Moreover, as it
was mentioned in recent studies of Refs. Varma:2020crx ; Varma:2022kbv ;
Loiko:2022noq , the local stability condition is not satisfied in the presence
of the long-range forces. Additional to that, as pointed out in Refs.
Varma:2020crx ; Varma:2022kbv ; Metz:2020vxd ; Freese:2022jlu , there is
another problem for systems with the long-range contribution, namely, the
divergence of essential quantities, that describe mechanical properties of a
system, such as the $D$-term and the mean square radii corresponding to the
EMT densities.

At this point, it is important to mention, that the ’t Hooft-Polyakov monopole
is accepted to be a stable system, see arguments of Refs. Baacke:1990at ;
Gervalle:2022npx . As we will see later, the local stability condition is
violated for the monopole and the dyon. On the first glance, thereby one could
think that the criterium is not correct. However, the monopole and the dyon
involve electromagnetic interaction, which supports the idea that the local
stability condition does not apply correctly in the presence of long-range
forces.

##  III ’t Hooft-Polyakov monopole

###  III.1 Equations of motion

The Grand Unified Theories combine the electromagnetic, weak, and strong
forces into a single force. One of the first such theories was suggested by
Howard Georgi and Sheldon Glashow in 1974 in Ref. Georgi:1974sy . Later
Alexander Polyakov and Gerard ’t Hooft have independently found that magnetic
monopoles automatically appear in all Grand Unified Theories Polyakov:1974ek ;
tHooft:1974kcl . The most simple model, where magnetic monopole exists is the
gauge SU(2) Georgi-Glashow model with Higgs triplet field $\varphi^{a},\
a=1,2,3$, which belongs to adjoint representation. The corresponding gauge
invariant action of the model is

$S=\int d^{4}x\left[-\dfrac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu
a}+\dfrac{1}{2}(\mathcal{D}_{\mu}\varphi)^{a}(\mathcal{D}^{\mu}\varphi)^{a}-\dfrac{\lambda}{4}(\varphi^{a}\varphi^{a}-v^{2})^{2}\right],$
(19)

where $\lambda$ is a dimensionless coupling constant, $v^{2}$ is the squared
vacuum expectation value of the Higgs field; $\mu$ and $\nu$ are Lorenz
indices. The covariant derivative and the non-abelian field strength tensor
are defined as follows

$\begin{split}(\mathcal{D}_{\mu}\varphi)^{a}=\partial_{\mu}\varphi^{a}+g\epsilon^{abc}A^{b}_{\mu}\varphi^{c},\\\
F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c},\end{split}$
(20)

where $g$ is the gauge coupling constant and $A_{\mu}^{a}$ is the gauge vector
field. Choosing the vacuum expectation value as
$\varphi_{0}^{a}=\left(0,0,v\right)$ and considering small fluctuations of the
ground state in unitary gauge, one can show that the Georgi-Glashow model has
one massive scalar field $\eta_{3}(x)$ with mass $m_{H}=\sqrt{2\lambda}v$, one
massless vector field $A_{\mu}^{3}$ corresponding to the $U(1)$ subgroup of
$SU(2)$ gauge group and two massive vector fields $A_{\mu}^{1}$ and
$A_{\mu}^{2}$ both with masses $m_{V}=gv$.

In this model we are interested in a soliton solution, in other words, in a
static solution of classical field equations with finite energy. As we want to
study static soliton configuration, we consider the fields
$A_{i}^{a}(\vec{x})$ and $\varphi^{a}(\vec{x})$ independent of time. We also
fix zero component of a vector field through the gauge $A_{0}^{a}=0$ to have
zero electric field. Requiring the energy to be finite the following
configuration of fields can be found

$\begin{split}\varphi^{a}&=n^{a}vh(r),\\\
A^{a}_{i}&=\dfrac{1}{gr}\epsilon_{aij}n_{j}(1-F(r)),\end{split}$ (21)

with the unit vector $n^{a}=\frac{r^{a}}{r}$, the unknown profile functions
$h(r)$ and $F(r)$ and the following boundary conditions

$\begin{gathered}F(r)=0\hskip 10.0pt\text{at}\hskip
10.0ptr\rightarrow\infty,\hskip 20.0ptF(r)=1\hskip 10.0pt\text{at}\hskip
10.0ptr\rightarrow 0,\\\ h(r)=1\hskip 10.0pt\text{at}\hskip
10.0ptr\rightarrow\infty,\hskip 20.0pth(r)=0\hskip 10.0pt\text{at}\hskip
10.0ptr\rightarrow 0.\end{gathered}$ (22)

The explicit form of the profile functions can be obtained from equations of
motion which can be computed by varying the action with respect to scalar and
vector fields, which for the static case reduce to the following form

$\begin{split}\mathcal{D}_{i}F_{ij}^{a}&=g\epsilon^{abc}\varphi^{b}(\mathcal{D}_{j}\varphi)^{c},\\\
\mathcal{D}_{i}(\mathcal{D}_{i}\varphi)^{a}&=\lambda(\varphi^{b}\varphi^{b}-v^{2})\varphi^{a}.\end{split}$
(23)

In terms of the profile functions the equations transform to

$\begin{split}F^{\prime\prime}(r)-\dfrac{F(r)(F^{2}(r)-1)}{r^{2}}-g^{2}v^{2}F(r)h^{2}(r)&=0,\\\
h^{\prime\prime}(r)+2\dfrac{h^{\prime}(r)}{r}-2\dfrac{F^{2}(r)h(r)}{r^{2}}+\lambda
v^{2}h(r)\left(1-h^{2}(r)\right)&=0.\end{split}$ (24)

Rescaling the argument $r$ with dimensionless argument $\rho$ as
$\rho=\frac{r}{R_{0}}$, where $R_{0}=\frac{1}{m_{V}}=\frac{1}{gv}$ is a
typical size of the solution, and introducing the new parameter
$\beta^{2}=2\frac{\lambda}{g^{2}}=\frac{m_{H}^{2}}{m_{V}^{2}}$ we obtain the
following system of equations of motion

$\begin{split}F^{\prime\prime}(\rho)-\dfrac{F(\rho)(F^{2}(\rho)-1)}{\rho^{2}}-F(\rho)h^{2}(\rho)&=0,\\\
h^{\prime\prime}(\rho)+2\dfrac{h^{\prime}(\rho)}{\rho}-2\dfrac{F^{2}(\rho)h(\rho)}{\rho^{2}}+\dfrac{\beta^{2}}{2}h(\rho)\left(1-h^{2}(\rho)\right)&=0,\\\
\end{split}$ (25)

with the same boundary conditions as before. This system can be solved
analytically only for the limit $\beta=0$, otherwise one solves it
numerically. However, one can find approximate solutions at small and large
distances:

$\begin{split}F(\rho)&\underset{\rho\to\infty}{\simeq}C_{F}e^{-\rho}\left(1-\dfrac{1}{2\rho}+\dfrac{3}{8\rho^{2}}+O\left(\dfrac{1}{\rho^{3}}\right)\right),\hskip
8.0ptF(\rho)\underset{\rho\to
0}{\simeq}1+a\rho^{2}+\sum_{n=2}^{\infty}c_{n}\rho^{2n},\\\
h(\rho)&\underset{\rho\to\infty}{\simeq}1-C_{h}\dfrac{e^{-\beta\rho}}{\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{\beta^{2}-4}\dfrac{e^{-2\rho}}{\rho^{2}}\left(1-O\left(\dfrac{1}{\rho}\right)\right),\hskip
8.0pth(\rho)\underset{\rho\to
0}{\simeq}b\rho+\sum_{n=2}^{\infty}d_{n}\rho^{2n-1},\end{split}$ (26)

where $C_{F}$, $C_{h}$, $a$ and $b$ are free constants. Note, that for
$\beta=2$ the constant $C_{F}=0$. The coefficients $c_{n}$ and $d_{n}$ can be
expressed in terms of $a$ and $b$ as follows

$\begin{split}c_{2}=&\dfrac{1}{10}\left(3a^{2}+b^{2}\right),\hskip
5.0ptc_{3}=\dfrac{1}{70}\left(7a^{3}+6ab^{2}-\dfrac{b^{2}\beta^{2}}{4}\right),\\\
d_{2}=&\dfrac{b}{10}\left(4a-\dfrac{\beta^{2}}{2}\right),\hskip
5.0ptd_{3}=\dfrac{1}{280}\left(48a^{2}b+4b^{3}-4ab\beta^{2}+5b^{3}\beta^{2}+b\dfrac{\beta^{2}}{2}\right).\end{split}$
(27)

In Ref. Korenblit:2021nzq  another system of differential equations is
obtained and analytically solved for complex non-abelian monopole and dyon
fields.

###  III.2 EMT densities

The EMT for the ’t Hooft-Polyakov monopole can be obtained variating generally
covariant form of Georgi-Glashow action (19) with respect to the metric

$T_{\mu\nu}=-\eta_{\mu\nu}\mathcal{L}+F_{\mu\alpha}^{a}F^{\alpha a}_{\
\nu}+(\mathcal{D}_{\mu}\varphi)^{a}(\mathcal{D}_{\nu}\varphi)^{a}.$ (28)

For the static case one gets

$\begin{split}T_{00}&=\dfrac{1}{4}F_{ij}^{a}F_{ij}^{a}+\dfrac{1}{2}(\mathcal{D}_{i}\varphi)^{a}(\mathcal{D}_{i}\varphi)^{a}+\dfrac{\lambda}{4}(\varphi^{a}\varphi^{a}-v^{2})^{2},\\\
T_{ij}&=-\dfrac{1}{4}\delta_{ij}F_{km}^{a}F_{km}^{a}-F_{ik}^{a}F_{kj}^{a}-\dfrac{1}{2}\delta_{ij}(\mathcal{D}_{k}\varphi)^{a}(\mathcal{D}_{k}\varphi)^{a}+(\mathcal{D}_{i}\varphi)^{a}(\mathcal{D}_{j}\varphi)^{a}-\dfrac{\lambda}{4}\delta_{ij}(\varphi^{a}\varphi^{a}-v^{2})^{2}.\end{split}$
(29)

Using the ’t Hooft-Polyakov ansatz given in Eq. (21) and the decomposition in
Eq. (3) the EMT densities can be expressed in terms of the profile functions
$F(\rho)$ and $h(\rho)$:

$\begin{split}T_{00}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{F^{\prime
2}}{\rho^{2}}+\dfrac{(1-F^{2})^{2}}{2\rho^{4}}+\dfrac{1}{2}h^{\prime
2}+\dfrac{h^{2}F^{2}}{\rho^{2}}+\dfrac{\beta^{2}}{8}(1-h^{2})^{2}\right),\\\
s(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{F^{\prime
2}}{\rho^{2}}-\dfrac{(1-F^{2})^{2}}{\rho^{4}}+h^{\prime
2}-\dfrac{1}{\rho^{2}}F^{2}h^{2}\right),\\\
p(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{1}{3}\dfrac{F^{\prime
2}}{\rho^{2}}+\dfrac{(1-F^{2})^{2}}{6\rho^{4}}-\dfrac{1}{6}h^{\prime
2}-\dfrac{1}{3}\dfrac{F^{2}}{\rho^{2}}h^{2}-\dfrac{\beta^{2}}{8}(1-h^{2})^{2}\right).\end{split}$
(30)

The $T_{00}(\rho)$ component provides the information about spatial
distribution of the monopole mass, $p(\rho)$ and $s(\rho)$ describe the
pressure and shear force distributions inside the monopole. The first two
terms in these densities are originating from the term $\sim
F^{a}_{\mu\sigma}F^{a\sigma}_{\phantom{a\sigma}\nu}$ in action, the next two
terms are coming from
$\sim\mathcal{D_{\mu}}\varphi^{a}\mathcal{D_{\nu}}\varphi^{a}$, and the term
proportional to $\beta^{2}$ comes from the Higgs potential.

Before we discuss the numerical results for the EMT density distributions and
their properties, we compute their asymptotic behaviour at large and small
distances using the behaviour of the profile functions from Eq. (26):

$\begin{split}T_{00}(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{1}{2\rho^{4}}+\beta^{2}C_{h}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]+2C_{F}^{2}\dfrac{e^{-2\rho}}{\rho^{2}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]\right),\\\
s(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(-\dfrac{1}{\rho^{4}}+\beta^{2}C_{h}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]+C_{F}^{2}\dfrac{e^{-2\rho}}{\rho^{4}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]\right),\\\
p(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{1}{6\rho^{4}}-\dfrac{2}{3}\beta^{2}C_{h}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]-\dfrac{2C_{F}^{2}}{3}\dfrac{e^{-2\rho}}{\rho^{4}}\left[1+O\left(\dfrac{1}{\rho}\right)\right]\right),\end{split}$
(31)

and

$\begin{split}T_{00}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\left[6a^{2}+\dfrac{3b^{2}}{2}+\dfrac{\beta^{2}}{8}\right]+\rho^{2}\left[8a^{3}+6ab^{2}-\dfrac{\beta^{2}b^{2}}{2}\right]+O(\rho^{4})\right),\\\
s(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\left[-8a^{3}+2ab^{2}-\beta^{2}b^{2}\right]\dfrac{\rho^{2}}{5}+O(\rho^{4})\right),\\\
p(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\left[2a^{2}-\dfrac{b^{2}}{2}-\dfrac{\beta^{2}}{8}\right]+\dfrac{\rho^{2}}{3}\left[8a^{3}-2ab^{2}+\beta^{2}b^{2}\right]+O(\rho^{4})\right),\end{split}$
(32)

where $C_{h}$, $C_{F}$ and $a$, $b$ are free constants from the asymptotic
behaviour of profile functions in Eq. (26). From the large distances behaviour
it is clear that the power-law decay ($\sim\rho^{-4}$) is dominating, this
behaviour corresponds to the contribution of the electromagnetic
interaction.111We will call this contribution the long-range interaction. This
is not surprising if one remembers that the mass spectrum of the Lagrangian
(19) has one massless vector particle corresponding to the $U(1)$ group.

The asymptotic behaviour of the energy density is definitely positive at the
infinity as well as at the origin. This hints that we are dealing with an
usual system. The outer region of the pressure distribution has definitely
positive sign and that of the shear force distribution – negative one. Such
behaviour leads to violation of the local stability criterion (10). However,
this behaviour is related only to the electromagnetic contribution. Note, that
because of this long-range contribution both mean square radii in Eqs. (16)
and (17) as well as the Druck-term in Eq. (4) diverge.

In Fig. 1 the energy density for various values of $\beta$ is shown. Although
the energy is growing with $\beta$, according to Ref. Kirkman:1981ck  it does
not diverge for $\beta\rightarrow\infty$. The distributions of the shear
force, pressure and normal force are shown in Fig. 2. Although the pressure is
mostly negative, it changes the sign which allows the von Laue condition (7)
to be satisfied, see e.g. Fig. 6. It is also interesting that the maximum of
pressure is three times larger than the maximum of shear forces like in many
other systems studied before, see e.g Refs. Varma:2020crx ; Cebulla:2007ei ;
Perevalova:2016dln ; Polyakov:2018zvc ; Polyakov:2002yz . From the figure one
also sees that the stability condition (10) is violated everywhere and for
every choice of $\beta$, although it is proved in Refs. Baacke:1990at ;
Gervalle:2022npx  that the monopole is stable. Since the local stability
condition is not mathematically proved, we cannot conclude from its violation
that the monopole is unstable. This violation can be also explained by the
fact that the stability criterion is inapplicable for systems with the long-
range force that is present in the monopole. The authors of Refs.
Varma:2020crx ; Varma:2022kbv  also obtained the violation of the stability
criterion for stable systems like proton in the presence of long-range forces.
All these aspects motivate us to think that the stability condition in Eq.
(10) can be applied only for the systems where only the short-range
interactions are present.

a.

b.

Figure 1: a. Energy distribution as a function of $\rho=r/R_{0}=gvr$ for
various values of $\beta$.
b. The blue line: the full pressure distribution $p(\rho)$ as a function of
$\rho=r/R_{0}=gvr$ and the red line: the contribution of the Higgs potential
only to the full pressure.

Figure 2: The pressure distribution, the shear force distribution and
stability condition from Eq. (10) as functions of $\rho=r/R_{0}=gvr$ for
various values of $\beta$.

###  III.3 BPS limit

The equations of motion can be solved analytically only for the limit
$\beta=0$. The solution of these equations was first obtained in Ref.
Prasad:1975kr . In Ref. Bogomolny  Bogomolny solved these equations by
reducing the system of the second order equations to the first order by
considering the energy functional for the Georgi-Glashow model. By integrating
the energy density over the volume one gets the energy functional

$E_{\beta=0}=\int d^{3}rT_{00}^{\beta=0}(r)=\int
d^{3}r\left[\dfrac{1}{4}F_{ij}^{a}F_{ij}^{a}+\dfrac{1}{2}(\mathcal{D}_{i}\varphi)^{a}(\mathcal{D}_{i}\varphi)^{a}\right].$
(33)

Rewriting it in terms of the chromomagnetic field
$H_{i}^{a}=-\frac{1}{2}\epsilon_{ijk}F^{a}_{jk}$, Bogomolny introduced the
following inequality for the monopole energy, which is also called BPS bound

$E_{\beta=0}\geq\int d^{3}rH_{i}^{a}\mathcal{D}_{i}\varphi^{a},$ (34)

where the Bianchi identity $\left(\mathcal{D}_{i}H_{i}\right)^{a}=0$ and the
Gauss’s theorem were used. The BPS limit gives the possible minimum of the
static energy of the monopole and the equality holds for the Bogomolny
equation

$\mathcal{D}_{i}\varphi^{a}=H_{i}^{a}.$ (35)

From the definition of chromomagnetic field the following expression can be
obtained

$H_{i}^{a}H_{j}^{a}=F^{a}_{ik}F^{a}_{kj}+\dfrac{1}{2}\delta_{ij}F^{a}_{km}F^{a}_{km}.$
(36)

Using this expression and Bogomolny equation (35) it can be shown that the
spatial components of the EMT disappear:

$T_{ij}(r)=0,$ (37)

independently of the choice of the ansatz in Eq. (21).

The vanishing of the pressure and shear force distributions at the BPS limit
allows to assume that the main contribution for the stress tensor $T_{ij}$ is
given by the Higgs potential. As it can be seen from Fig. 1 the largest
contribution to the pressure distribution is indeed due to the term
originating from the Higgs potential in Eq. (19). Although the shear force
distribution does not depend explicitly on $\beta$, it is related to the
pressure distribution due to the differential equation (6). We think that in
all models where it is possible to construct the BPS limit, the corresponding
$ij$ components of EMT would vanish. We have also obtained such vanishing, for
example, for baby Skyrme model. The vanishing of the pressure and shear force
distributions for BSP limit indicates that the ’t Hooft-Polyakov monopole in
this limit has the isotropic matter distribution Polyakov:2018zvc .

###  III.4 Abelian field strength tensor and electromagnetic EMT

In the previous subsection we obtained the EMT in non-abelian $SU(2)$ gauge
theory, however, the theory has one massless vector field corresponding to the
abelian $U(1)$ subgroup. This massless vector field is responsible for the
electromagnetic contribution to the EMT and for the divergence of such
mechanical properties of the monopole as the $D$-term, mean square energy and
mechanical radii. We will subtract this long-range contribution from the EMT
and study the remaining short-range structure. For this we have to determine
the $U(1)$ abelian field strength tensor. Let us denote the abelian field
strength tensor as $\mathcal{F}_{\mu\nu}$. Since in the unitary gauge the
massless vector field of the theory is the third component of the vector field
$A_{3}^{\mu}$ (see Sec.III.1), the expression for the abelian field strength
tensor must be $SU(2)$ gauge invariant and coincide with
$\mathcal{F}_{\mu\nu}=\partial_{\mu}A^{3}_{\nu}-\partial_{\nu}A_{\mu}^{3}$ in
the unitary gauge tHooft:1974kcl . We introduce a general definition of the
abelian field strength tensor without any requirements on $\phi$ as

$\mathcal{F}_{\mu\nu}=\phi^{a}F^{a}_{\mu\nu}-\dfrac{c_{1}}{g}\varepsilon^{abc}\phi^{a}\mathcal{D}_{\mu}\phi^{b}\mathcal{D}_{\nu}\phi^{c}.$
(38)

The constant $c_{1}$ is not fixed by the above requirements for the abelian
field strength tensor $\mathcal{F}_{\mu\nu}$, however, it is needed to define
the magnetic charge density in such a way that it coincides with the
topological charge density. G. ’t Hooft suggested in Ref. tHooft:1974kcl  the
tensor with the unit vector
$\phi^{a}=\hat{\varphi}^{a}=\frac{\varphi^{a}}{|\varphi|}$ and $c_{1}=1$,
Faddeev offered in Ref. Faddeev  the tensor with
$\phi^{a}=\dfrac{\varphi^{a}}{v}$ and $c_{1}=0$, where $\varphi^{a}$ is the
ansatz from Eq. (21) and Boulware in Ref. Boulware:1976tv  suggested the
tensor with $\phi^{a}=\hat{\varphi}^{a}=\frac{\varphi^{a}}{|\varphi|}$ and
$c_{1}=0$. Since the under-lying theory combines the long- and short-range
forces it is not possible uniquely defined the $U(1)$ field strength tensor to
separate the short-range interaction from the long one.

For the general choice of the abelian field strength tensor the magnetic
charge density gets the following form

$\begin{gathered}\rho_{M}=-\dfrac{1}{2}\varepsilon_{ijk}\partial_{i}\mathcal{F}_{jk}=\\\
=\dfrac{1}{2}\varepsilon_{ijp}(\mathcal{D}_{p}\phi)^{a}F^{a}_{ij}(c_{1}\vec{\phi}^{2}-1)+\dfrac{c_{1}}{2g}\varepsilon^{abc}\varepsilon_{ijp}(\mathcal{D}_{i}\phi)^{a}(\mathcal{D}_{j}\phi)^{b}(\mathcal{D}_{p}\phi)^{c}-\dfrac{c_{1}}{2}\varepsilon_{ijp}\phi^{a}F^{a}_{ij}\dfrac{1}{2}\partial_{p}\vec{\phi}^{2}.\end{gathered}$
(39)

One can show that for the ’t Hooft’s definition of the abelian field strength
tensor the magnetic and topological charge densities coincide and describe
single point-like particle with the magnetic charge $\frac{4\pi}{g}$ at the
origin, see Ref. tHooft:1974kcl . Such monopole is called Dirac’s monopole. In
Ref. Dirac:1931kp  Dirac derived the quantisation of the magnetic charge as
$Q_{M}=Q_{T}/g$, where $Q_{M}$ is a magnetic charge and $Q_{T}$ is a
topological charge.

In contrast to the ’t Hooft’s definition of the abelian field strength tensor
the Faddeev’s and Boulware’s definitions describe the magnetic charge density
smoothly without singularities at the origin. The topological and magnetic
charge densities are not equal in these cases, but the quantisation of the
magnetic charge is satisfied. It is also interesting to notice that the
magnetic charge density in Eq. (39) for the Faddeev’s and and Boulware’s
definitions of the abelian strength tensor coincides with the energy density
in BSP limit in Eq. (34) up to some dimensional normalisation factor. For the
positive magnetic charge distribution the mean square radius can be determined

$\langle r^{2}\rangle_{M}=\dfrac{\int d^{3}rr^{2}\rho_{M}(r)}{\int
d^{3}r\rho_{M}(r)}.$ (40)

We define the electromagnetic EMT analogously to the EMT in electrodynamic as

$T^{C}_{\mu\nu}=\mathcal{F}_{\mu}^{~{}\alpha}\mathcal{F}_{\alpha\nu}+\dfrac{1}{4}\eta_{\mu\nu}\mathcal{F}^{\alpha\beta}\mathcal{F}_{\alpha\beta},$
(41)

where $\mathcal{F}_{\mu\nu}$ is the $U(1)$ field strength tensor. As we have
discussed $\mathcal{F}_{\mu\nu}$ is not uniquely defined. Since we are
interesting in spatial structure of the monopole it makes no sense to define
$T^{C}_{\mu\nu}$ according to the ’t Hooft’s definition of the
$\mathcal{F}_{\mu\nu}$ because of singularities at the origin. Faddeev’s and
Boulware’s smooth definitions of the abelian strength tensor
$\mathcal{F}_{\mu\nu}$ could be a better candidate to define the
$T^{C}_{\mu\nu}$, however, we suggest another definition of the $U(1)$ field
strength tensor

$\mathcal{F}_{\mu\nu}=\dfrac{\varphi^{a}}{v}F^{a}_{\mu\nu}-\dfrac{1}{gv^{3}}\varepsilon^{abc}\varphi^{a}\mathcal{D}_{\mu}\varphi^{b}\mathcal{D}_{\nu}\varphi^{c},$
(42)

which is also smooth as the Faddeev’s and Boulware’s definitions and it
coincides with the ’t Hooft’s definition at long distances. For this
definition of the abelian strength tensor the electromagnetic EMT
distributions have the following form

$\begin{split}T_{00}^{C}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\dfrac{Q^{2}(\rho)}{2\rho^{4}},\\\
p^{C}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\dfrac{Q^{2}(\rho)}{6\rho^{4}},\\\
s^{C}(\rho)&=-\dfrac{1}{R_{0}^{4}g^{2}}\dfrac{Q^{2}(\rho)}{\rho^{4}},\end{split}$
(43)

where $Q(r)=h(r)\left[1-F^{2}(r)\left(1-h(r)^{2}\right)\right]$. The function
$Q(r)$ is directly related to the magnetic charge density through

$\rho_{M}(\rho)=\dfrac{1}{gR_{0}^{3}}\dfrac{1}{\rho^{2}}\dfrac{d}{d\rho}Q(\rho).$
(44)

The electromagnetic EMT densities are shown in Fig. 3. The corresponding
asymptotic behaviour can be found in App. 68.

Figure 3: The electromagnetic pressure, share force and normal force
distributions as functions of $\rho=r/R_{0}=gvr$ from Eq. (43) for various
values of $\beta$.

Note, that since the full spatial part of the EMT in BSP limit where $\beta=0$
vanishes, see Sec. III.3, and the electromagnetic EMT does not depend on
$\beta$ (and thereby does not vanish in this limit) the short-range part of
the EMT in Eq. (45) has to be equal to the long-range part. It means that the
scalar interaction in BSP limit becomes long-range interacting.

###  III.5 Short-range part of EMT and its consequences

We denote the short-range part of the EMT as $T_{\mu\nu}^{SR}$ and define it
as follows

$\displaystyle T^{SR}_{\mu\nu}=T_{\mu\nu}-T^{C}_{\mu\nu},$ (45)

where the full EMT $T_{\mu\nu}$ is obtained in Eq. (30) and electromagnetic
EMT – in Eq. (43). After some simple algebraic calculation the following
expressions for the short-range part of the EMT densities can be found

$\begin{split}T_{00}^{SR}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{(}\dfrac{F^{\prime
2}}{\rho^{2}}+\dfrac{(1-h^{2})(1-F^{2})^{2}}{2\rho^{4}}+\dfrac{1}{2}h^{\prime
2}+\dfrac{h^{2}F^{2}}{\rho^{2}}\left(1-\dfrac{h^{2}}{\rho^{2}}\left[1-F^{2}\left(1-\dfrac{h^{2}}{2}\right)\right]\right)+\dfrac{\beta^{2}}{8}(1-h^{2})^{2}\Bigg{)},\\\
p^{SR}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{(}\dfrac{1}{3}\dfrac{F^{\prime
2}}{\rho^{2}}+\dfrac{(1-h^{2})(1-F^{2})^{2}}{6\rho^{4}}-\dfrac{1}{6}h^{\prime
2}-\dfrac{1}{3}\dfrac{F^{2}h^{2}}{\rho^{2}}\left(1+\dfrac{h^{2}}{\rho^{2}}\left[1-F^{2}\left(1-\dfrac{h^{2}}{2}\right)\right]\right)-\dfrac{\beta^{2}}{8}(1-h^{2})^{2}\Bigg{)},\\\
s^{SR}(\rho)&=\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{(}\dfrac{F^{\prime
2}}{\rho^{2}}-\dfrac{(1-h^{2})(1-F^{2})^{2}}{\rho^{4}}+h^{\prime
2}-\dfrac{F^{2}h^{2}}{\rho^{2}}\left(1-2\dfrac{h^{2}}{\rho^{2}}\left[1-F^{2}\left(1-\dfrac{h^{2}}{2}\right)\right]\right)\Bigg{)}.\end{split}$
(46)

The corresponding asymptotic behaviour can be found in App. 69. The short-
range part of the pressure and shear force distributions are presented in Fig.
4. The short-range part of the pressure distribution is negative and the
short-range part of the shear force distribution is positive for every choice
of $\beta$ in considered range of $r$. The total normal force distribution is
positive for any value of $\beta$, which satisfies the stability criterium
(15). It is also interesting to notice that the pressure distribution of the
monopole does not change drastically after the subtraction of the long-range
part, however, the shear force distribution does.

Figure 4: Short-range part of the pressure, shear force and total normal force
distributions from Eq. (46) as a function of $\rho=r/R_{0}=gvr$ for various
values of $\beta$. Note, that the total normal force is defined as in Eq.
(15).

As we have already discussed in the first section, the static EMT according to
the Noether theorem must be conserved. The conservation of EMT couples
pressure and shear force dinsities by the differential equation (6). However,
after the decomposition of the EMT into the short- and long-range parts, the
separate EMTs are not conserved and the corresponding short- and long-range
parts of the pressure and of the shear force densities couple due to the new
equilibrium equations:

$\begin{split}&\dfrac{d}{dr}\left(\dfrac{2}{3}s^{C}(r)+p^{C}(r)\right)+2\dfrac{s^{C}(r)}{r}=-\dfrac{Q_{M}(r)\rho_{M}(r)}{4\pi
r^{2}},\\\
&\dfrac{d}{dr}\left(\dfrac{2}{3}s^{SR}(r)+p^{SR}(r)\right)+2\dfrac{s^{SR}(r)}{r}=\dfrac{Q_{M}(r)\rho_{M}(r)}{4\pi
r^{2}},\end{split}$ (47)

where
$Q_{M}(r)=\int\limits_{|\vec{x}|<r}d^{3}x\rho_{M}(\vec{x})=\dfrac{4\pi}{g}Q(r)$
is magnetic charge contained in a sphere of radius $r$. So the right-hand side
describes ”Coulomb force” of the magnetically charged sphere acting on the
magnetic charge density.222Modulus of the magnetic field of the magnetically
charged sphere is $\mathcal{H}=\dfrac{Q_{M}(r)}{4\pi r^{2}}$. Thereby the
first equation is the equation of magnetostatic equilibrium between the
”Coulomb stress” pushing the monopole outwards and the magnetic ”Coulomb
force” pulling the monopole inward to the center. In contrast, the second
equation describes the balance between the ”short-range stress” pulling the
monopole inward to the center and the repulsive magnetic ”Coulomb force”
pushing the monopole outward. We notice, that since the right-hand sides of
the equilibrium equations are associated with the magnetic charge density, it
is not possible to define the long- and short-range parts of the EMT using the
ambiguity of the $\mathcal{F}_{\mu\nu}$ in such a way, that the decomposed
EMTs would be conserved separately, i.e. the right-hand sides would vanish,
except the case with zero magnetic charge density.

As we discussed, the $D$-term of the monopole diverges because of the long-
range contribution. After subtracting the long-range part from the EMT, the
corresponding $D$-term still diverges for small values of $\beta$. We
illustrate this divergence in the following. The $D$-term is defined through
the shear force distribution as in Eq. (4). Since for small values of
$\beta\ll 1$ the main contribution to the short-range part of the shear force
density is provided by the asymptotic behaviour at large-distances, see App.
69, the $D$-term diverges for the large enough $R$ as

$D\simeq-\dfrac{16\pi
M_{SR}R_{0}}{15g^{2}}\int\limits_{R}^{\infty}d\rho\rho^{4}\left(C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}-2C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}\right)\sim-\int\limits_{R}^{1/\beta}d\rho\left(\beta^{2}\rho^{2}-\dfrac{1}{\rho}\right)\sim-\dfrac{1}{\beta}-\ln\beta.$
(48)

The mean square magnetic charge radius in Eq. (40), the mean square energy
radius of the short-range part in Eq. (16) and the mean square mechanical
radius in Eq. (18) also diverge for small values of $\beta$, see App. B.

In the end of this section we present numerical results for masses, $D$-terms
and mean square radii of the ’t Hooft-Polyakov monopole, see Tab. 1. We use
the following notations: full mass of the monopole $M$ is computed with help
of the energy density from Eq. (30), electromagnetic contribution to the full
mass $M_{C}$ from Eq. (43), the short-range contribution to the full mass
$M_{SH}$ is computed with help of the expression in Eq. (46). The values of
$D$-term have been calculated with help of the short-range part of the shear
force distribution in Eq. (46) according to formula (4), the mean square
radius of the magnetic charge density is computed due to the definition in Eq.
(40), the mean square radius of the short-range part of the energy
distribution is computed according to the definition in Eq. (16) and the
short-range part of the mechanical square radius – according to Eq. (18). It
is remarkable that for the small values of $\beta$ the main contribution to
the monopole mass gives the short range part and for the large values of
$\beta\gtrsim 5$ – the long range part. It would be interesting to find the
analytic form of the pressure and shear force densities depending on the
parameter $\beta$ like it was done in Refs. Forgacs:2005vx ; Gardner:1982fk ;
Kirkman:1981ck  for the static energy.

$\beta$ | $M\dfrac{R_{0}g^{2}}{4\pi}$ | $M_{C}\dfrac{R_{0}g^{2}}{4\pi}$ | $M_{SR}\dfrac{R_{0}g^{2}}{4\pi}$ | $-\dfrac{15g^{4}}{64\pi^{2}}D$ | $\dfrac{\langle r^{2}\rangle_{M}}{R_{0}^{2}}$ | $\dfrac{\langle r^{2}\rangle_{E}}{R_{0}^{2}}$ | $\dfrac{\langle r^{2}\rangle_{\text{mech}}}{R_{0}^{2}}$
---|---|---|---|---|---|---|---
$\varepsilon$ | 0.993 | 0.153 | 0.841 | $\infty$ | $\infty$ | $\infty$ | $\infty$
0.1 | 1.036 | 0.187 | 0.849 | 7.909 | 20.517 | 12.157 | 20.255
0.5 | 1.133 | 0.267 | 0.865 | 1.022 | 4.287 | 3.710 | 3.379
1 | 1.204 | 0.336 | 0.869 | 0.43 | 2.168 | 2.436 | 1.545
5 | 1.438 | 0.794 | 0.644 | 0.293 | 0.373 | 1.602 | 0.445
8 | 1.457 | 1.084 | 0.373 | 0.213 | 0.205 | 2.328 | 0.307
10 | 1.461 | 1.289 | 0.172 | 0.108 | 0.151 | 4.479 | 0.254
Table 1: Numerical values of $D$-terms, various radii and different
contributions to the mass for varied choice of $\beta$, where $\varepsilon\ll
1$. The estimation of the accuracy of our numerical method is about 95 percent
for $0\leq\beta\leq 1$ and about 85 percent for $\beta>1$.

##  IV Julia-Zee dyon

###  IV.1 Equation of motion

In the previous section we considered monopole solution carrying magnetic
charge. In this part we will consider a dyon, that is a hypothetical particle
carrying both electric and magnetic charges, such particle was first suggested
by Julian Schwinger in 1969 in Ref. Schwinger:1969ib . The dyon solution in
Georgi-Glashow model was first obtained in 1975 by Anthony Zee and Bernard
Julia in Ref. Julia:1975ff . We will use the same Georgi-Glashow model as we
did in the first part with the same action (19). However, we will not fix the
zero component of the vector field as we did it for the monopole case,
$A^{a}_{0}(\vec{x})\neq 0$. We are again interested in the static soliton
solution with finite energy. Thereby, we choose the same spherically-symmetric
ansatz as we chose for the ’t Hooft-Polyakov monopole in Eq. (21) with the
same boundary condition as in Eq. (22). In contrast to monopole case, for the
dyon the $A_{0}^{a}$ is non-zero and one chooses the ansatz

$\displaystyle A^{a}_{0}$ $\displaystyle=\dfrac{J(r)}{gr}\dfrac{r^{a}}{r},$
(49)

which follows from requiring the energy to be finite and definition of the
abelian field strength tensor as in Eq. (42). The boundary conditions for the
function $J(r)$ are

$\begin{split}J(r)&\underset{r\to 0}{\simeq}0,\\\
J(r)&\underset{r\to\infty}{\simeq}-\dfrac{Q_{D}}{Q_{M}}+mgr,\end{split}$ (50)

where $Q_{D}$ is an electric charge of a dyon, $Q_{M}=\dfrac{4\pi}{g}$ is a
monopole charge and $m$ is a free constant, that has dimension of mass.

From the variation of the action follow the equations of motion

$\begin{split}&\mathcal{D}_{0}F^{a}_{00}-\mathcal{D}_{i}F^{a}_{i0}+g\epsilon^{abc}\varphi^{b}\mathcal{D}_{0}\varphi^{c}=0,\\\
&\mathcal{D}_{0}F^{a}_{0k}-\mathcal{D}_{i}F^{a}_{ik}-g\epsilon^{abc}\varphi^{c}\mathcal{D}_{k}\varphi^{b}=0,\\\
&\mathcal{D}_{0}\mathcal{D}_{0}\varphi^{a}-\mathcal{D}_{i}\mathcal{D}_{i}\varphi^{a}+\lambda(\varphi^{b}\varphi^{b}-v^{2})\varphi^{a}=0.\end{split}$
(51)

The equations for the profile functions in dimensionless variable $\rho=gvr$
are

$\begin{split}&F^{\prime\prime}\rho^{2}=\left(F^{2}-1\right)F-\left(J^{2}-\rho^{2}h^{2}\right)F,\\\
&h^{\prime\prime}\rho^{2}+2h^{\prime}\rho=2F^{2}h+\dfrac{\beta^{2}}{2}h\left(h^{2}-1\right)\rho^{2},\\\
&J^{\prime\prime}\rho^{2}=2JF^{2}.\end{split}$ (52)

Note, that the boundary condition for $J(\rho)$ also changes:
$J(\rho)\underset{\rho\to\infty}{\simeq}-\dfrac{Q_{D}}{Q_{M}}+C\rho$, where
$C=m/v$ is dimensionless free constant.

The approximate solution for small and large distances can be found:

$\begin{split}F(\rho)&\underset{\rho\to
0}{\simeq}1+a\rho^{2}+\sum_{n=2}^{\infty}a_{2n}\rho^{2n},\hskip
8.0ptF(\rho)\underset{\rho\to\infty}{\simeq}C_{F}e^{-\sqrt{1-C^{2}}\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
h(\rho)&\underset{\rho\to
0}{\simeq}b\rho+\sum_{n=1}^{\infty}b_{2n+1}\rho^{2n+1},\hskip
8.0pth(\rho)\underset{\rho\to\infty}{\simeq}1-C_{h}\dfrac{e^{-\beta\rho}}{\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{\beta^{2}+4C^{2}-4}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
\tilde{J}(\rho)&\underset{\rho\to
0}{\simeq}c\rho+\sum_{n=1}c_{2n+1}\rho^{2n+1},\hskip
8.0pt\tilde{J}(\rho)\underset{\rho\to\infty}{\simeq}C-\dfrac{Q_{D}}{Q_{M}}\dfrac{1}{\rho}+\dfrac{CC_{F}^{2}}{2(1-C^{2})}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right).\end{split}$
(53)

The $a,\ b,\ c$ are free constants, all other constants for the small $r$
behaviour can be expressed in terms of these constants, for example,

$a_{4}=\dfrac{1}{10}\left(3a^{2}+b^{2}-c^{2}\right),\hskip
10.0ptb_{3}=\dfrac{1}{10}\left(4ab-\dfrac{b\beta^{2}}{2}\right),\hskip
10.0ptc_{3}=\dfrac{2ac}{5}.$ (54)

Since we searched the asymptotic behaviour at large distances for real and
falling functions we define a new function $J(\rho)=\tilde{J}(\rho)\rho$ and
find that the constant $C$ has restricted region: $0\leq C<1$. From numerical
analysis of the solutions we obtain that the charge ratio is restricted in the
following range $0\leq\dfrac{Q_{D}}{Q_{M}}\leq 1$. More detailed analysis can
be found in Ref. Nishino:2019lqm , where also the dependence of the parameter
$C$ on the charge ratio $\dfrac{Q_{D}}{Q_{M}}$ is found.

###  IV.2 EMT densities

The EMT for the Julia-Zee dyon can be obtained analogically to the ’t Hooft-
Polyakov monopole by variating generally covariant form of the Georgi-Glashow
action (19) with respect to the metric and it can be decomposed into magnetic
and electric parts:

$\begin{split}T_{00}(r)&=T_{00}^{M}(r)+T_{00}^{E}(r),\\\
T_{ij}(r)&=T_{ij}^{M}(r)+T_{ij}^{E}(r),\end{split}$ (55)

where the magnetic part equals to the EMT of monopole which we have already
computed in the previous section, see Eqs. (29), (30) and (31). The electric
part of EMT is

$\begin{split}T_{00}^{E}(r)&=\dfrac{1}{2}\left(\mathcal{D}_{0}\varphi^{a}\mathcal{D}_{0}\varphi^{a}+F^{a}_{0i}F^{a}_{0i}\right),\\\
T_{ij}^{E}(r)&=\dfrac{1}{2}\left[\delta_{ij}\left(F^{a}_{0k}F^{a}_{0k}+\mathcal{D}_{0}\varphi^{a}\mathcal{D}_{0}\varphi^{a}\right)-2F^{a}_{i0}F^{a}_{j0}\right].\end{split}$
(56)

Note, that same as for the monopole case the spatial components of the EMT for
the dyon vanish in BSP limit, where $\beta=0$. For electric part of the energy
density, pressure and shear force distributions of the dyon in dimensionless
variable $\rho$ the following expressions are obtained

$\begin{split}&T_{00}^{E}(\rho)=\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{\tilde{J}^{\prime
2}}{2}+\dfrac{\tilde{J}^{2}F^{2}}{\rho^{2}}\right),\\\
p^{E}(\rho)=\dfrac{1}{R_{0}^{4}g^{2}}&\left(\dfrac{\tilde{J}^{\prime
2}}{6}+\dfrac{\tilde{J}^{2}F^{2}}{3\rho^{2}}\right),\hskip
15.0pts^{E}(\rho)=\dfrac{1}{R_{0}^{4}g^{2}}\left(-\tilde{J}^{\prime
2}+\dfrac{\tilde{J}^{2}F^{2}}{\rho^{2}}\right).\end{split}$ (57)

With the help of asymptotic behaviour of profile functions in Eq. (53) the
asymptotic behaviour of the EMT for the dyon can be obtained

$\begin{split}&T_{00}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{2}\left(1+\left(\dfrac{Q_{D}}{Q_{M}}\right)^{2}\right)\dfrac{1}{\rho^{4}}+2C_{F}^{2}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
&p(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{6}\left(1+\left(\dfrac{Q_{D}}{Q_{M}}\right)^{2}\right)\dfrac{1}{\rho^{4}}-\dfrac{2}{3}\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+O\left(\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{3}}\right)\Bigg{]},\\\
&s(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}-\left(1+\left(\dfrac{Q_{D}}{Q_{M}}\right)^{2}\right)\dfrac{1}{\rho^{4}}+\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+O\left(\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{3}}\right)\Bigg{]}.\end{split}$
(58)

From this behaviour one sees that the long-range contribution presented in the
dyon is even stronger than in the monopole case, see Eq. (31). In App. A.2 one
can find the asymptotic behaviour of the EMT near the origin as well as
behaviour of the electric part of the EMT only. Again already from the
asymptotic behaviour it is clear that the stability condition of Eq. (10) is
violated. In Fig. 5 the full pressure and shear force distributions of the
dyon are presented. The shear force and pressure distributions have very
similar form as it was in the monopole case, see Fig. 2. These distributions
are mostly negative for every choice of parameters. However, pressure
distribution changes its sing, what allows the von Laue condition (7) to be
satisfied, see e.g. Fig. 6. It is also interesting to notice that the main
contribution to the dyon energy comes from the monopole part as it can be seen
in Fig. 7. Moreover, for the growing $\beta$ this contribution is increasing.

Figure 5: The mechanical properties of dyons from Eq. (55) as functions of
$\rho=r/R_{0}=gvr$ for varied value of $\beta$ and charge ratio
$\dfrac{Q_{D}}{Q_{M}}$.

Figure 6: The pressure distributions of the monopole
$\left(\dfrac{Q_{D}}{Q_{M}}=0\right)$ and the dyon for various values of
$\beta$ and charge ratio $\dfrac{Q_{D}}{Q_{M}}$. The pressure distribution has
one mode for any choice of the parameters.

Figure 7: Comparison of the full energy densities of dyon and the
contributions of the monopole part only for various values of $\beta$ and
charge ratio $\dfrac{Q_{D}}{Q_{M}}$.

###  IV.3 Electromagnetic and short-range parts of the EMT

According to the definition of the electromagnetic EMT in Eq. (41) the long-
range part of the EMT for dyon is

$\begin{split}T_{00}^{C}&=\dfrac{1}{4}\mathcal{F}_{ij}\mathcal{F}_{ij}+\dfrac{1}{2}\mathcal{F}_{0i}\mathcal{F}_{0i},\\\
T_{ij}^{C}&=-\dfrac{1}{4}\delta_{ij}\mathcal{F}_{mn}\mathcal{F}_{mn}-\mathcal{F}_{ik}\mathcal{F}_{kj}+\dfrac{1}{2}\delta_{ij}\mathcal{F}_{0k}\mathcal{F}_{0k}-\mathcal{F}_{i0}\mathcal{F}_{j0}.\end{split}$
(59)

Using the modified ’t Hooft abelian field strength tensor in Eq. (42) one gets
the following expressions for the electromagnetic EMT densities of a dyon

$\begin{split}T_{00}^{C}(\rho)&=\dfrac{1}{2}\dfrac{1}{g^{2}R_{0}^{4}}\dfrac{1}{\rho^{4}}\left(Q^{2}(\rho)+\tilde{Q}^{2}(\rho)\right),\\\
p^{C}(\rho)&=\dfrac{1}{6}\dfrac{1}{g^{2}R_{0}^{4}}\dfrac{1}{\rho^{4}}\left(Q^{2}(\rho)+\tilde{Q}^{2}(\rho)\right),\\\
s^{C}(\rho)&=-\dfrac{1}{g^{2}R_{0}^{4}}\dfrac{1}{\rho^{4}}\left(Q^{2}(\rho)+\tilde{Q}^{2}(\rho)\right),\end{split}$
(60)

where the function $Q$ is defined in Eq. (43) and is related to the magnetic
charge density in Eq. (44) and the function $\tilde{Q}$ is related to the
electric charge density of a dyon

$\rho_{D}(\rho)=\dfrac{1}{gR_{0}^{3}}\dfrac{1}{\rho^{2}}\dfrac{d\tilde{Q}(\rho)}{d\rho},\hskip
12.0pt\text{with}\hskip
12.0pt\tilde{Q}(\rho)=\rho^{2}h(\rho)\tilde{J}^{\prime}(\rho).$ (61)

Comparing these expressions with the expressions for the monopole case in Eq.
(43), it is clear that both have similar behaviour. Hence, it is again not
possible to define the various mean square radii or $D$-terms, because they
diverge.

We will exclude the long-range contribution given in Eq. (60) from the EMT of
dyon in Eq. (55) in the same way as we did it for the monopole case using Eq.
(45). After simple algebraic calculations we obtain the following expression
for the short-range part of the EMT of the dyon

$\begin{split}T_{00}^{\text{SR}}(\rho)&=T_{00}^{\text{M,SR}}(\rho)+\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{1}{2}\tilde{J}^{\prime
2}\left(1-h^{2}\right)+\dfrac{\tilde{J}^{2}F^{2}}{\rho^{2}}\right),\\\
p^{\text{SR}}(\rho)&=p^{\text{M,SR}}(\rho)+\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{1}{6}\tilde{J}^{\prime
2}\left(1-h^{2}\right)+\dfrac{1}{3}\dfrac{\tilde{J}^{2}F^{2}}{\rho^{2}}\right),\\\
s^{\text{SR}}(\rho)&=s^{\text{M,SR}}(\rho)-\dfrac{1}{R_{0}^{4}g^{2}}\left(\tilde{J}^{\prime
2}\left(1-h^{2}\right)-\dfrac{\tilde{J}^{2}F^{2}}{\rho^{2}}\right).\end{split}$
(62)

Here $T_{00}^{\text{M,SR}}$, $p^{\text{M,SR}}$ and $s^{\text{M,SR}}$
correspond to the short-range part of the EMT of the monopole in Eq. (46). The
final short-range distributions of mechanical properties are shown in Fig. 8.
These distributions have the same behaviour as in the monopole case.

Figure 8: The short-range mechanical properties of dyons as a functions of
$\rho=r/R_{0}=gvr$ from Eq. (62) for varied value of $\beta$ and charge ratio
$\dfrac{Q_{D}}{Q_{M}}$. Note, that the total normal force is defined as in Eq.
(15).

The equilibrium equation of the long- and short-range parts of the EMT
analogously to Eq. (47) is given as

$\begin{split}\dfrac{d}{dr}\left(\dfrac{2}{3}s^{C}(r)+p^{C}(r)\right)+2\dfrac{s^{C}(r)}{r}=-\dfrac{Q_{M}(r)\rho_{M}(r)}{4\pi
r^{2}}-\dfrac{\tilde{Q}_{D}(r)\rho_{D}(r)}{4\pi r^{2}},\\\
\dfrac{d}{dr}\left(\dfrac{2}{3}s^{SR}(r)+p^{SR}(r)\right)+2\dfrac{s^{SR}(r)}{r}=\dfrac{Q_{M}(r)\rho_{M}(r)}{4\pi
r^{2}}+\dfrac{\tilde{Q}_{D}(r)\rho_{D}(r)}{4\pi r^{2}},\end{split}$ (63)

where $Q_{M}(r)$ and $\tilde{Q}_{D}(r)$ are magnetic and electric charges
contained in the sphere of radius $r$:

$\begin{gathered}Q_{M}(r)=\smashoperator[]{\int_{|\vec{x}|<r}^{}}d^{3}x\rho_{M}(\vec{x})=\dfrac{4\pi}{g}Q(r),\hskip
15.0pt\tilde{Q}_{D}(r)=\smashoperator[]{\int_{|\vec{x}|<r}^{}}d^{3}x\rho_{D}(\vec{x})=\dfrac{4\pi}{g}\tilde{Q}(r).\end{gathered}$
(64)

The first equilibrium equation describes the balance between two types of
”Coulomb forces”, namely magnetic and electric ones, pulling the dyon inwards
to the center and the ”Coulomb stress” pushing the dyon outwards. The second
equation describes the balance between the two types of the repulsive ”Coulomb
forces” pushing the dyon outward and the ”short-range stress” pulling the dyon
inwards to the center. Same as for the monopole case it is not possible to
define the short-range part of the EMT using the ambiguity of the
$\mathcal{F}_{\mu\nu}$ in such a way that the long- and short-range parts of
the EMT are conserved separately unless the electric and magnetic charge
densities both vanish.

$\beta$ | $\frac{Q_{D}}{Q_{M}}$ | $M_{M}\frac{R_{0}g^{2}}{4\pi}$ | $M\frac{R_{0}g^{2}}{4\pi}$ | $M_{C}\frac{R_{0}g^{2}}{4\pi}$ | $M_{SR}\frac{R_{0}g^{2}}{4\pi}$ | $-\dfrac{15g^{4}}{64\pi^{2}}D$ | $\dfrac{\langle r^{2}\rangle_{E}}{R_{0}^{2}}$ | $\dfrac{\langle r^{2}\rangle_{M}}{R_{0}^{2}}$ | $\dfrac{\langle r^{2}\rangle_{D}}{R_{0}^{2}}$ | $\dfrac{\langle r^{2}\rangle_{\text{mech}}}{R_{0}^{2}}$
---|---|---|---|---|---|---|---|---|---|---
$\varepsilon$ | 0.5 | 0.998 | 1.109 | 0.168 | 0.941 | $\infty$ | $\infty$ | $\infty$ | $\infty$ | $\infty$
$\varepsilon$ | 1 | 1.050 | 1.400 | 0.207 | 1.193 | $\infty$ | $\infty$ | $\infty$ | $\infty$ | $\infty$
0.1 | 0.5 | 1.043 | 1.164 | 0.21 | 0.953 | 10.826 | 13.863 | 23 | 21.447 | 24.000
0.1 | 1 | 1.099 | 1.484 | 0.263 | 1.215 | 20.832 | 18.373 | 29.247 | 28.838 | 33.253
0.5 | 0.5 | 1.142 | 1.281 | 0.304 | 0.977 | 1.33 | 4.327 | 4.803 | 5.624 | 3.883
0.5 | 1 | 1.2 | 1.654 | 0.394 | 1.26 | 2.225 | 6.136 | 6.08 | 7.732 | 4.972
1 | 0.5 | 1.220 | 1.371 | 0.386 | 0.985 | 0.529 | 2.909 | 2.416 | 3.504 | 1.630
1 | 1 | 1.28 | 1.738 | 0.477 | 1.262 | 0.698 | 4.467 | 2.954 | 5.288 | 1.200
5 | 0.5 | 1.44 | 1.604 | 0.839 | 0.375 | 0.49 | 2.139 | 0.4 | 2.201 | 0.42
5 | 1 | 1.492 | 1.825 | 0.909 | 0.915 | 0.405 | 3.299 | 2.163 | 3.489 | 0.365
Table 2: Numerical value of $D$-terms, radii and different contributions to
the mass for various values of $\beta$ and the charge ratio
$\frac{Q_{D}}{Q_{M}}$, where $\varepsilon\ll 1$. The estimation of the
accuracy of our numerical method is about 95 percent for $0\leq\beta\leq 1$
and about 85 percent for $\beta>1$.

In table 2 the full mass of dyon is denoted as $M$, the monopole contribution
to the full dyon mass as $M_{M}$, the $M_{C}$ and $M_{SR}$ are the long- and
short-range contributions to the full mass, correspondingly, the $D$-terms are
computed with the help of the short-range part of the shear force distribution
in Eq. (62). The following quantities can be also found in the table: the
energy mean square radius $\langle r^{2}\rangle_{E}$, the magnetic and
electric charge mean square radii $\langle r^{2}\rangle_{M}$ and $\langle
r^{2}\rangle_{D}$, and the mechanical radius $\langle
r^{2}\rangle_{\text{mech}}$. It is remarkable that the $D$-term is growing
with the parameter $\frac{Q_{D}}{Q_{M}}$ and decreasing with the parameter
$\beta$. As it was already mentioned the main contribution to the dyon mass is
given by the monopole part of the dyon, moreover the same as it was for the
monopole: for small values of $\beta$ the main contribution to the mass comes
from the short range region, while for the value $\beta\gtrsim 5$ from the
long-range part. The following ratios of the radii can be noticed from the
table

$\begin{split}&\dfrac{\langle r^{2}\rangle_{M}}{\langle
r^{2}\rangle_{D}}>1,\hskip 10.0pt\text{for small $\beta$ and every
$\dfrac{Q_{D}}{Q_{M}}$},\\\ &\dfrac{\langle r^{2}\rangle_{M}}{\langle
r^{2}\rangle_{D}}<1,\hskip 10.0pt\text{for $\beta\gtrsim 0.1$ and every
$\dfrac{Q_{D}}{Q_{M}}$},\\\ &\dfrac{\langle r^{2}\rangle_{M}}{\langle
r^{2}\rangle_{E}}>1,\hskip 10.0pt\text{for $\beta<1$ and every
$\dfrac{Q_{D}}{Q_{M}}$},\\\ &\dfrac{\langle r^{2}\rangle_{M}}{\langle
r^{2}\rangle_{E}}<1,\hskip 10.0pt\text{for $\beta\geq 1$ and every
$\dfrac{Q_{D}}{Q_{M}}$ },\\\ &\dfrac{\langle r^{2}\rangle_{D}}{\langle
r^{2}\rangle_{E}}>1,\hskip 10.0pt\text{for every $\beta$ and every
$\dfrac{Q_{D}}{Q_{M}}$}.\end{split}$ (65)

##  V Summary and conclusions

In this work the EMTs of the ’t Hooft-Polyakov monopole and the Julia-Zee dyon
are studied. These EMTs contain long-range contributions which present
analogies to the ”Coulomb interaction”. The local stability condition
containing the pressure and shear force distributions is violated for both
cases, the monopole and the dyon. Thereby, the applicability of the condition
in the presence of long-range contributions is questioned. Moreover, such
important quantities which give information about mechanical properties of a
system as the $D$-term, energy and charge mean square radii of the monopole
and the dyon can not be computed due to the presence of the long-range
interaction.

To shed more light on the mechanical properties of the monopole and the dyon
and on the local stability condition we exclude the long-range contribution
from the EMT of the monopole and the dyon. The difficulty of such calculation
is that this contribution can not be uniquely defined. We suggest the modified
’t Hooft definition of the abelian field strength tensor for this purpose.
Dealing with the separate long- and short-range parts of the EMTs we obtained
the equilibrium equations which couple the pressure, shear force distributions
and the external force acting on the system. In this line we have also
modified the local mechanical stability condition for systems in the presence
of external forces. According to this condition the short-range part of the ’t
Hooft-Polyakov monopole as well as of the Julia-Zee dyon is stable for every
choice of the parameter $\beta$.

Further, in this paper numerous figures describing mechanical properties of
the ’t Hooft-Polyakov monopole and the Julia-Zee dyon can be found as well as
tables with varied contributions to the masses, $D$-terms and mean square
energy, magnetic and electric charge radii.

###### Acknowledgements.

This paper is an extension of my master thesis, which was supervised by Maxim
Polyakov in 2021, who passed away in August 2021. I dedicate this paper to his
memory. I would like to thank H. Alharazin, J. Gegelia, S. Korenblit, P.
Schweitzer and Ya. Shnir for encouraging, useful comments and numerous
discussions. This work was supported in part by DFG and NSFC through funds
provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure
in QCD” (NSFC Grant No. 11621131001, DFG Project-ID 196253076 - TRR 110).

##  Appendix A Asymptotic behaviours

We specify here the asymptotic behaviours of EMT which was not directly given
in the main body of the paper.

###  A.1 For monopole part

The function $Q(\rho)$ has the following asymptotic behaviour

$\begin{split}Q(\rho)&\underset{\rho\to\infty}{\simeq}1-C_{h}\dfrac{e^{-\beta\rho}}{\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{\beta^{2}-4}\dfrac{e^{-2\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
Q(\rho)&\underset{\rho\to
0}{\simeq}\rho^{3}\left(-2ab+b^{3}\right)+\dfrac{\rho^{5}}{20}\left(-48a^{2}b+2ab(32b^{2}+\beta^{2})-b^{3}(4+3\beta^{2})\right)+O(\rho^{7}).\\\
\end{split}$ (66)

The asymptotic behaviour of the magnetic charge density is

$\begin{split}\rho_{M}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{gR_{0}^{3}}\left[3b^{3}-6ab+\rho^{2}\left(-12a^{2}b-b^{3}+16ab^{3}+\dfrac{1}{2}ab\beta^{2}-\dfrac{3b^{3}\beta^{2}}{4}\right)+O(\rho^{4})\right],\\\
\rho_{M}(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{1}{gR_{0}^{3}}\left[C_{h}\beta\dfrac{e^{-\beta\rho}}{\rho^{3}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{4C_{F}^{2}}{\beta^{2}-4}\dfrac{e^{-2\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\right].\\\
\end{split}$ (67)

The ”Coulomb EMT” behaves asymptotically as

$\begin{gathered}T_{00}^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{2\rho^{4}}-C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{C_{h}^{2}}{2}\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{\beta^{2}-4}\dfrac{e^{-2\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
p^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{6\rho^{4}}-\dfrac{C_{h}}{3}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{C_{h}^{2}}{6}\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{3(\beta^{2}-4)}\dfrac{e^{-2\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
s^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}-\dfrac{1}{\rho^{4}}+2C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-C_{h}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{4C_{F}^{2}}{\beta^{2}-4}\dfrac{e^{-2\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
T_{00}^{C}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{\rho^{2}}{2}\left(-2ab+b^{3}\right)^{2}+\dfrac{\rho^{4}}{20}b\left(-2a+b^{2}\right)\left(-48a^{2}b+2ab(32b^{2}+\beta^{2})-b^{3}(4+3\beta^{2})\right)+O(\rho^{6})\Bigg{]},\\\
p^{C}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{\rho^{2}}{6}\left(-2ab+b^{3}\right)^{2}+\dfrac{\rho^{4}}{60}b\left(-2a+b^{2}\right)\left(-48a^{2}b+2ab(32b^{2}+\beta^{2})-b^{3}(4+3\beta^{2})\right)+O(\rho^{6})\Bigg{]},\\\
s^{C}(\rho)\underset{\rho\to
0}{\simeq}-\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\rho^{2}\left(-2ab+b^{3}\right)^{2}+\dfrac{\rho^{4}}{10}b\left(-2a+b^{2}\right)\left(-48a^{2}b+2ab(32b^{2}+\beta^{2})-b^{3}(4+3\beta^{2})\right)+O(\rho^{6})\Bigg{]}.\end{gathered}$
(68)

The asymptotic behaviour of the short-range part of the EMT is

$\begin{gathered}T_{00}^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+2C_{F}^{2}\dfrac{e^{-2\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
p^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{C_{h}}{3}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2}{3}C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{2C_{F}^{2}}{3}\dfrac{e^{-2\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
s^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}-2C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+C_{F}^{2}\dfrac{e^{-2\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
T_{00}^{SR}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\left(6a^{2}+\dfrac{3}{2}b^{2}+\dfrac{\beta^{2}}{8}\right)+\left(8a^{3}+6ab^{2}-2a^{2}b^{2}+2ab^{4}-\dfrac{b^{6}}{2}-\dfrac{\beta^{2}b^{2}}{2}\right)\rho^{2}+O(\rho^{4})\Bigg{]},\\\
p^{SR}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\left(2a^{2}-\dfrac{b^{2}}{2}-\dfrac{\beta^{2}}{8}\right)+\left(8a^{3}-2ab^{2}-2a^{2}b^{2}+2ab^{4}-\dfrac{b^{6}}{2}+\beta^{2}b^{2}\right)\dfrac{\rho^{2}}{3}+O(\rho^{4})\Bigg{]},\\\
s^{SR}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[\left(-8a^{3}+2ab^{2}+20a^{2}b^{2}-20ab^{4}+5b^{6}-\beta^{2}b\right)\dfrac{\rho^{2}}{5}+O(\rho^{4})\right].\end{gathered}$
(69)

###  A.2 For dyon part

The function $\tilde{Q}(\rho)$ has the following asymptotic behaviour

$\begin{split}\tilde{Q}(\rho)&\underset{\rho\to\infty}{\simeq}gQ_{D}-gC_{h}Q_{D}\dfrac{e^{-\beta\rho}}{\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\dfrac{CC_{F}^{2}}{\sqrt{1-C^{2}}}e^{-2\sqrt{1-C^{2}}\rho}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
\tilde{Q}(\rho)&\underset{\rho\to
0}{\simeq}\rho^{3}bc+\rho^{5}\left(\dfrac{6abc}{5}+\dfrac{c}{20}(8ab-b\beta^{2})\right)+O(\rho^{7}).\end{split}$
(70)

The asymptotic behaviour of the electric charge density is

$\begin{split}\rho_{D}(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{1}{gR_{0}^{3}}\left(gQ_{D}C_{h}\beta\dfrac{e^{-\beta\rho}}{\rho^{3}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+2CC_{F}^{2}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\right),\\\
\rho_{D}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{gR_{0}^{3}}\left(3bc+\rho^{2}\left(6abc+\dfrac{c}{4}(8ab-b\beta^{2})\right)\right)+O(\rho^{4}).\end{split}$
(71)

The asymptotic behaviour of the full EMT of the dyon near the origin is

$\begin{split}T_{00}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[6a^{2}+\dfrac{3b^{2}}{2}+\dfrac{\beta^{2}}{8}+\dfrac{3c^{2}}{2}+\rho^{2}\left(8a^{3}+6ab^{2}-\dfrac{b^{2}\beta^{2}}{2}+2ac^{2}\right)+O(\rho^{4})\right],\\\
p(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[2a^{2}-\dfrac{b^{2}}{2}-\dfrac{\beta^{2}}{8}+\dfrac{c^{2}}{2}+\dfrac{\rho^{2}}{3}\left(8a^{3}-2ab^{2}+b^{2}\beta^{2}+2ac^{2}\right)+O(\rho^{4})\right],\\\
s(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[\dfrac{\rho^{2}}{5}\left(-8a^{3}+2ab^{2}-b^{2}\beta^{2}-2ac^{2}\right)+O(\rho^{4})\right].\end{split}$
(72)

The asymptotic behaviour of the electric part of the full EMT for the dyon:

$\begin{split}T_{00}^{E}(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{g^{2}Q_{D}^{2}}{2}\dfrac{1}{\rho^{4}}+C^{2}C_{F}^{2}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
p^{E}(\rho)&\underset{\rho\to\infty}{\simeq}\dfrac{g^{2}Q_{D}^{2}}{6}\dfrac{1}{\rho^{4}}+\dfrac{C^{2}C_{F}^{2}}{3}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
s^{E}(\rho)&\underset{\rho\to\infty}{\simeq}-g^{2}Q_{D}^{2}\dfrac{1}{\rho^{4}}+C^{2}C_{F}^{2}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right),\\\
T_{00}^{E}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{3c^{2}}{2}+4ac^{2}\rho^{2}+O(\rho^{4}),\\\
p^{E}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{c^{2}}{2}+\dfrac{4}{3}ac^{2}\rho^{2}+O(\rho^{4}),\\\
s^{E}(\rho)&\underset{\rho\to
0}{\simeq}\dfrac{2}{5}ac^{2}\rho^{2}+O(\rho^{4}).\end{split}$ (73)

The asymptotic behaviour of the long-range part of the EMT is

$\begin{gathered}T_{00}^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{2}(1+g^{2}Q^{2}_{D})\dfrac{1}{\rho^{4}}-C_{h}(1+g^{2}Q_{D}^{2})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\\\
-\dfrac{CC_{F}^{2}gQ_{D}}{\sqrt{1-C^{2}}}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{C_{h}^{2}}{2}(1+g^{2}Q_{D}^{2})\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
p^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\dfrac{1}{6}(1+g^{2}Q^{2}_{D})\dfrac{1}{\rho^{4}}-\dfrac{C_{h}}{3}(1+g^{2}Q^{2}_{D})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-\\\
-\dfrac{CC_{F}^{2}gQ_{D}}{3\sqrt{1-C^{2}}}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\dfrac{C_{h}^{2}}{2}(1+g^{2}Q_{D}^{2})\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
s^{C}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}-(1+g^{2}Q^{2}_{D})\dfrac{1}{\rho^{4}}+2C_{h}(1+g^{2}Q^{2}_{D})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\\\
+\dfrac{2CC_{F}^{2}gQ_{D}}{\sqrt{1-C^{2}}}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{4}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)-C_{h}^{2}(1+g^{2}Q_{D}^{2})\dfrac{e^{-2\beta\rho}}{\rho^{6}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
T_{00}^{C}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[\rho^{4}\dfrac{1}{2}b^{2}((-2a+b^{2})^{2}+c^{2})+O(\rho^{6})\right],\\\
p^{C}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[\rho^{4}\dfrac{1}{6}b^{2}((-2a+b^{2})^{2}+c^{2})+O(\rho^{6})\right],\\\
s^{C}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left[-\rho^{4}b^{2}((-2a+b^{2})^{2}+c^{2})+O(\rho^{6})\right].\end{gathered}$
(74)

The asymptotic behaviour of the short-range part of the EMT of the dyon is as
follows:

$\begin{gathered}T_{00}^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}2C_{F}^{2}\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+C_{h}(1+g^{2}Q_{D}^{2})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
p^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}-\dfrac{2}{3}\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+O\left(\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{3}}\right)+\dfrac{C_{h}}{3}(1+g^{2}Q^{2}_{D})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
s^{SR}(\rho)\underset{\rho\to\infty}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\Bigg{[}\beta^{2}C_{F}^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)+O\left(\dfrac{e^{-2\sqrt{1-C^{2}}\rho}}{\rho^{3}}\right)-2C_{h}(1+g^{2}Q^{2}_{D})\dfrac{e^{-\beta\rho}}{\rho^{5}}\left(1+O\left(\dfrac{1}{\rho}\right)\right)\Bigg{]},\\\
T_{00}^{SR}(\rho)\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(6a^{2}+\dfrac{3b^{2}}{2}+\dfrac{\beta^{2}}{8}+\dfrac{3c^{2}}{2}+\rho^{2}\left[8a^{3}+6ab^{2}-\dfrac{b^{2}\beta^{2}}{2}+2ac^{2}\right]+O(\rho^{4})\right),\\\
p(\rho)^{SR}\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(2a^{2}-\dfrac{b^{2}}{2}-\dfrac{\beta^{2}}{8}+\dfrac{c^{2}}{2}+\dfrac{\rho^{2}}{3}\left[8a^{3}-2ab^{2}+b^{2}\beta^{2}+2ac^{2}\right]+O(\rho^{4})\right),\\\
s(\rho)^{SR}\underset{\rho\to
0}{\simeq}\dfrac{1}{R_{0}^{4}g^{2}}\left(\dfrac{\rho^{2}}{5}\left[-8a^{3}+2ab^{2}-b^{2}\beta^{2}-2ac^{2}\right]+O(\rho^{4})\right).\end{gathered}$
(75)

##  Appendix B Divergence of radii

As it follows from the Eqs. (67) and (69) the main contribution to the
magnetic charge density and to the short-range part of the energy density,
respectively, is provided by the asymptotic behaviour at large-distances. Then
the corresponding mean square radii for the small values of $\beta$ and for
the large enough $R$ diverge as

$\begin{gathered}\dfrac{\langle
r_{M}^{2}\rangle}{R_{0}^{2}}\simeq\frac{\int\limits_{R}^{\infty}d\rho\rho^{4}\left(C_{h}\beta\dfrac{e^{-\beta\rho}}{\rho^{3}}\right)}{\int\limits_{R}^{\infty}d\rho\rho^{2}\left(C_{h}\beta\dfrac{e^{-\beta\rho}}{\rho^{3}}\right)}\sim\frac{\int\limits_{R}^{1/\beta}d\rho\rho}{\int\limits_{R}^{1/\beta}d\rho\dfrac{1}{\rho}}\sim\dfrac{1}{\beta^{2}\ln\left(\frac{1}{\beta}\right)},\end{gathered}$
(76) $\begin{gathered}\dfrac{\langle
r_{E}^{2}\rangle}{R_{0}^{2}}\simeq\frac{\int\limits_{R}^{\infty}d\rho\rho^{4}\left(C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}+C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\right)}{\int\limits_{R}^{\infty}d\rho\rho^{2}\left(C_{h}\dfrac{e^{-\beta\rho}}{\rho^{5}}+C_{h}^{2}\beta^{2}\dfrac{e^{-2\beta\rho}}{\rho^{2}}\right)}\sim\frac{\int\limits_{R}^{1/\beta}d\rho\left(\dfrac{1}{\rho}+\beta^{2}\rho^{2}\right)}{\int\limits_{R}^{1/\beta}d\rho\left(\dfrac{1}{\rho^{3}}+\beta^{2}\right)}\sim\dfrac{1}{\beta^{2}}-\dfrac{\ln\beta}{\beta}.\end{gathered}$
(77)

According to the modified definition of the mechanical radius in Eq. (18), to
the external force in equilibrium equation (47) for the short-range and to the
asymptotic behaviour in Eqs. (66) and (69) the mean square mechanical radius
of the monopole for small values of $\beta$ diverges as follows

$\begin{gathered}\dfrac{\langle
r^{2}\rangle_{\text{mech}}}{R_{0}^{2}}\sim\frac{\int\limits_{R}^{1/\beta}d\rho\rho^{4}\left(\dfrac{\beta}{\rho^{4}}-\dfrac{1}{\rho^{5}}\right)}{\int\limits_{R}^{1/\beta}d\rho\rho^{2}\left(\dfrac{\beta}{\rho^{4}}-\dfrac{1}{\rho^{5}}\right)}\sim\dfrac{\ln\beta}{\beta^{2}}.\end{gathered}$
(78)

Proceeding analogously to the monopole case, we obtain for the dyon the same
divergences as in Eqs. (48), (76), (77) and (78). Namely, the $D$-term
diverges as in Eq. (48), the mean square electric and magnetic charge radii as
in Eq. (76), the mean square energy radius of the short-range part as in Eq.
(77) and the mean square mechanical radius as in Eq. (78).

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