# Complex topological soliton with real energy in particle physics

Takanobu Taira
Department of Mathematics, City, University of London,
Northampton Square, London EC1V 0HB, UK
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We summarise the procedure used to find the classical masses of Higgs
particle, massive gauge boson and t’Hooft-Polyakov monopole in non-Hermitian
gauge field theory. Their physical regions are explored, and the mechanism of
the real value of the monopole solution is analysed in different physical
regions.

††conference: Complex topological soliton with real energy in particle physics

##  1 Introduction

Quantum field theory is a key tool to analyse particle physics. The most
modern physical description of the fundamental particle interaction is
described by the model called the Standard Model. However, the model possesses
several problems such as incompatibility with the general relativity,
hierarchy problem, etc. Therefore, it is an active area to extend the standard
model.

Recently a growing number of research papers started exploring the non-
Hermitian extension of the Standard Model [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14]. We have contributed to this development by analysing the
Goldstone theorem [10, 8], The Higgs mechanism [9] and t’Hooft-Polyakov
monopoles [11]. The classical masses of Higgs particles, massive gauge boson
and monopoles were analysed. However, a detailed analysis of their
intersecting physical regions and the mechanism of the real value of the
energy of monopole was not explored. The main aim of this contribution is to
fill this gap.

There are two separate mechanisms that guarantee the real value of the
particle masses in question. First, the masses of Higgs particles are given by
a non-Hermitian mass matrix $M$. Assume that the matrix possess anti-linear
symmetry, which we refer to as $\mathcal{PT}$ symmetry, that satisfies
$[\mathcal{PT},M]=0$, $Mv=\lambda v$, $\mathcal{PT}v=e^{i\theta}v$, where
$\\{v,\lambda\\}$ are eigenvectors and eigenvalues of the mass matrix. From
this, it is trivial to show that the eigenvalues are real

$\displaystyle\mathcal{PT}Mv_{i}$ $\displaystyle=$
$\displaystyle\mathcal{PT}\lambda_{i}v_{i}=\lambda_{i}^{*}\mathcal{PT}v_{i}=\lambda_{i}^{*}e^{i\theta_{i}}v_{i},$
$\displaystyle\mathcal{PT}Mv_{i}$ $\displaystyle=$ $\displaystyle
M\mathcal{PT}v_{i}=Me^{i\theta_{i}}v_{i}=\lambda_{i}e^{i\theta_{i}}v_{i}.$

It was shown in [8] that this $\mathcal{PT}$ symmetry is related to the
$\mathcal{CPT}$ symmetry of the field-theoretic action.

On the other hand, the classical energy of the soliton solution is found by
inserting the solution into the Hamiltonian $E=H[\phi]=\int
d^{3}x\mathcal{H}(\phi)$. Therefore, the techniques from $\mathcal{PT}$
symmetric quantum mechanics shown above can not be applied.

We will show below that the energy of the soliton solutions are real when the
three conditions stated below holds. Therefore they are sufficient conditions
to guarantee the real value of particles in the model. However, we do not
claim that these are necessary conditions.

Let $\\{\phi_{1},\phi_{2}\\}$ be a set of distinct (or identical) solutions to
the equations of motion
$\delta\mathcal{L}/\delta\phi-\partial_{\mu}(\delta\mathcal{L}/\delta\partial_{\mu}\phi)=0$,
where $\mathcal{L}(\phi)$ is the field-theoretic Lagrangian density. The
classical energies of the solution are given by inserting the solution into
the Hamiltonian, $E_{i}=H[\phi_{i}]=\int d^{3}x\mathcal{H}(\phi_{i})$, for
$i\in\\{1,2\\}$. The classical mass of the solution $\phi_{1}$ and $\phi_{2}$
are real if there exist some anti-linear symmetry $\mathcal{CPT}$ (note that
is it not the standard $\mathcal{CPT}$ symmetry in quantum field theory) such
that three conditions are satisfied:

  1. 1.

$\mathcal{CPT}:\mathcal{H}[\phi(x)]\rightarrow\mathcal{H}[\mathcal{CPT}\phi(x)]=\mathcal{H}^{\dagger}[\phi(-x)]$.

  2. 2.

$\mathcal{CPT}:\phi_{1}(x)\rightarrow\phi_{2}(-x)$.

  3. 3.

$H[\phi_{1}]=H[\phi_{2}]$.

If two solutions are identical $\phi_{1}=\phi_{2}$, then the above condition
reduces to the reality condition of the soliton solution already derived in
[15]. Using the above three conditions, the real value of the classical mass
can easily be shown by the following argument

$\displaystyle\int d^{3}x\mathcal{H}[\mathcal{CPT}\phi(x)]$
$\displaystyle\overset{(1)}{=}$ $\displaystyle\int
d^{3}x\mathcal{H}^{\dagger}[\phi(-x)]=M_{1}^{\dagger},$
$\displaystyle\overset{(2)}{=}$ $\displaystyle\int
d^{3}x\mathcal{H}[\phi_{2}(-x)]=M_{2},$ $\displaystyle\implies
M_{1}^{\dagger}=M_{2}$ $\displaystyle\overset{(3)}{\implies}$ $\displaystyle
M_{1}^{\dagger}=M_{1},$

where numbers above the equal signs indicate the condition number.

The above analysis can be performed directly on the complex model. However,
the non-Hermitian theory is only well-defined once the inner-product is
identified. The modern way of the well-defined non-Hermitian quantum mechanics
was first realised by Frederik Scholtz, Hendrik Geyer, and Fritz Hahne in
1992, [16]. The authors used the mathematical condition on the operator called
the quasi-Hermiticity (the term was first coined in [17], but the metric was
not given) to define the positive definite inner product. The quasi-
Hermiticity is defined as a condition on the bounded linear operator of the
Hilbert space $A:\mathcal{H}\rightarrow\mathcal{H}$, which satisfies

  * (i)

$\braket{v}{\rho v}>0$ for all $\ket{v}\in\mathcal{H}$ and $\ket{v}\not=0$.

  * (ii)

$\rho A=A^{\dagger}\rho$ .

Where the bounded Hermitian linear operator
$\rho:\mathcal{H}\rightarrow\mathcal{H}$ is often called the metric operator
because the inner product is defined by the operator
$\braket{\cdot}{\cdot}_{\rho}:=\braket{\cdot}{\rho\cdot}$ restores the
Hermiticity of the operator. This result can be shown by using the
condition(ii)

$\displaystyle\braket{v}{Aw}_{\rho}\equiv\braket{v}{\rho
Aw}=\braket{v}{A^{\dagger}\rho w}=\braket{Av}{\rho w}=\braket{Av}{w}_{\rho},$

for all $\ket{v},\ket{w}\in\mathcal{H}$. Note that the quasi-Hermiticity alone
does not guarantee the real energy spectrum of the Hamiltonian. In fact, one
requires two extra conditions.

  * (iii)

The metric operator is invertible.

  * (iv)

$\rho=\eta^{\dagger}\eta$.

The operator which satisfies only conditions (ii) and (iv) is referred to as
the pseudo-Hermitian operator, which was first introduced in [18]. These extra
conditions were considered in [16] to prove that, given a set of pseudo-
Hermitian operators $\mathcal{A}=\\{A_{i}\\}$, the metric operator
$\rho_{\mathcal{A}}$ which satisfies conditions (i), (ii), (iii) and (iv) for
all operators of set $\mathcal{A}$ is uniquely determined if and only if all
operators of the set $\mathcal{A}$ are irreducible on the Hilbert space
$\mathcal{H}$. This procedure is analogous to the Dyson mapping first
introduced by Freeman Dyson [19] used in the study of nuclear reaction [20,
21, 22], which maps the non-Hermitian operator $A$ to Hermitian operator
$\eta^{-1}A\eta$ via Dyson map $\eta$. The relation between the metric
operator and the Dyson map is found by utilising the Hermiticity of the
expression $\eta^{-1}A\eta$ in the following way

$\eta^{-1}A\eta=(\eta^{-1}A\eta)^{\dagger}\implies
A\eta^{\dagger}\eta=\eta^{\dagger}\eta
A^{\dagger}\implies\eta^{\dagger}\eta=\rho.$ (1)

We will utilise this mapping to transform the non-Hermitian field-theoretic
Hamiltonian to a Hermitian Hamiltonian. This procedure will resolve the issue
of complex vacuum solution and Derrick’s scaling argument, as we will see
below. However, we note that the Dyson map used here introduces a negative
kinetic sign in the kinetic term of one of the fields, indicating the ghost
field problem. This issue is removed if one further diagonalise the
Hamiltonian. Such diagonalisation can be realised via field-redefinition or
via another Dyson map. A more detailed discussion of this is found in [23],
and a Dyson map which diagonalise the free part of the non-Hermitian
Hamiltonian is found in [12].

##  2 Methods

In this section, we will summarise the method used in [8, 9, 10, 11] to find
the masses of the Higgs particles, massive gauge particles and t’Hooft-
Polyakov monopoles in non-Hermitian gauge field theory. We note that the
explicit forms of the similarity transformation will not be discussed in this
paper as non-Hermitian and Hermitian theories are isospectral as long as the
$\mathcal{CPT}$ symmetry is preserved for Hamiltonian, Higgs particles and
monopole solution.

We begin with the non-Hermitian local $SU(2)$ gauge theory with matter fields
in the adjoint representation

$\displaystyle\mathcal{L}_{2}^{\text{ad}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\text{Tr}\left(D\phi_{1}\right)^{2}+\frac{m_{1}^{2}}{4}\text{Tr}(\phi_{1}^{2})$
$\displaystyle-i\frac{\mu^{2}}{2}\text{Tr}(\phi_{1}\phi_{2})-\frac{g}{64}\left[\text{Tr}(\phi_{1}^{2})\right]^{2}$
$\displaystyle+\frac{1}{4}\text{Tr}\left(D\phi_{2}\right)^{2}+\frac{m_{2}^{2}}{4}\text{Tr}(\phi_{2}^{2})-\frac{1}{8}\text{Tr}\left(F^{2}\right).$

Here we take $g,\mu\in\mathbb{R}$, $m_{i}\in\mathbb{R}$ and discrete values
$c_{i}\in\\{-1,1\\}$. The two fields $\\{\phi_{i}\\}_{i=1,2}$ are Hermitian
matrices $\phi_{i}(t,\vec{x})\equiv\phi_{i}^{a}(t,\vec{x})T^{a}$, where
$\phi_{i}^{a}(t,\vec{x})$ is a real-valued field. The three generators
$\\{T^{a}\\}_{a=1,2,3}$ of $SU(2)$ in the adjoint representation are defined
by three Hermitian matrices of the form $(T^{a})_{bc}=-i\epsilon_{abc}$,
satisfying the commutation relation $[T^{a},T^{b}]=i\epsilon^{abc}T^{c}$. One
can check that $Tr(T^{a}T^{b})=2\delta^{ab}$. The field strength tensor is
defined as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ie[A_{\mu},A_{\nu}]$,
where the gauge fields are $A_{\mu}=A_{\mu}^{a}T^{a}$. The partial derivative
is replaced with the covariant derivative
$(D_{\mu}\phi_{i})^{a}:=\partial_{\mu}\phi_{i}^{a}+e\varepsilon_{abc}A_{\mu}^{b}\phi_{i}^{c}$
to compensate for the local symmetry group $SU(2)$.

This action is invariant under the local $SU(2)$ transformation of the matter
fields $\phi_{i}\rightarrow
e^{i\alpha^{a}(x)T^{a}}\phi_{i}e^{-i\alpha^{a}(x)T^{a}}$ and gauge fields
$A_{\mu}\rightarrow
e^{i\alpha^{a}(x)T^{a}}A_{\mu}e^{-i\alpha^{a}(x)T^{a}}+\frac{1}{e}\partial_{\mu}\alpha^{a}(x)T^{a}$.
It is also symmetric under modified $\mathcal{CPT}$ symmetry, which transforms
two fields, $\phi_{1}$ and $\phi_{2}$ as

$\displaystyle\mathcal{CPT}$
$\displaystyle:\phi_{1}(t,\vec{x})\rightarrow\phi_{1}(-t,-\vec{x})$ (3)
$\displaystyle:\phi_{2}(t,\vec{x})\rightarrow-\phi_{2}(-t,-\vec{x})$
$\displaystyle:i\rightarrow-i.$

The equations of motion for the fields $\phi_{i}$ and $A_{\mu}$ of the
Lagrangian (2) are

$\displaystyle\left(D_{\mu}D^{\mu}\phi_{i}\right)^{a}+\frac{\delta
V}{\delta\phi^{a}_{i}}=0,$ (4) $\displaystyle
D_{\nu}F^{\nu\mu}_{a}-e\epsilon_{abc}\phi^{b}_{1}(D^{\mu}\phi)^{c}+e\epsilon_{abc}\phi^{b}_{2}(D^{\mu}\phi)^{c}=0,$

where repeated indices are summed over. We perform the similarity
transformation of the complex Lagrangian (2) by momentary resorting to a
quantum theory where we assume an equal time commutation relation between the
fields $\phi_{i}^{a}$ and their canonical momenta
$\Pi_{i}^{a}=\partial_{0}\phi_{i}^{a}$, satisfying the commutation relation
$[\phi_{i}^{a}(t,\vec{x}),\Pi_{j}^{b}(t,\vec{y})]=\delta(\vec{x}-\vec{y})\delta_{ij}\delta_{ab}$.
Using this relation, we can transform the corresponding complex Hamiltonian of
the Lagrangian (2) by

$\displaystyle H\rightarrow e^{\eta_{\pm}}He^{-\eta_{\pm}},$ (5)
$\displaystyle\eta_{\pm}=\prod_{a=1}^{3}\exp\left(\pm\frac{\pi}{2}\int
d^{3}x\Pi^{a}_{2}\phi^{a}_{2}\right),$

where $H$ is the field-theoretic Hamiltonian of our model (2), obtained via
Legendre transformation. This non-uniqueness of the metric is analogous to the
non-uniqueness of the metric and its connection to the observables in the
quantum mechanical setting discussed in [16, 24]. The adjoint action of
$\eta_{\pm}$ maps the complex action in the equation (2) into the following
real action

$\displaystyle\mathfrak{s}$ $\displaystyle=$ $\displaystyle\int
d^{4}x~{}\frac{1}{4}Tr\left(D\phi_{1}\right)^{2}-\frac{1}{4}Tr\left(D\phi_{2}\right)^{2}$
$\displaystyle+c_{1}\frac{m_{1}^{2}}{4}Tr(\phi_{1}^{2})-c_{2}\frac{m_{2}^{2}}{4}Tr(\phi_{2}^{2})$
$\displaystyle-
c_{3}\frac{\mu^{2}}{2}Tr(\phi_{1}\phi_{2})-\frac{g}{64}\left(Tr(\phi_{1}^{2})\right)^{2}-\frac{1}{8}Tr(F^{2})$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}x~{}~{}\frac{1}{4}Tr\left(D\phi_{1}\right)^{2}-\frac{1}{4}Tr\left(D\phi_{2}\right)^{2}$
$\displaystyle-V-\frac{1}{8}Tr(F^{2}).$

The parameter $c_{3}$ indicates the different similarity transformations by
taking the values $\pm 1$ for $\eta_{\pm}$, respectively.

For convenience, let us rewrite the above real action in terms of each
component of the fields $\phi_{i}^{a}$ as

$\displaystyle\mathfrak{l}_{2}^{\text{ad}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(D_{\mu}\phi_{i})^{a}\mathcal{I}_{ij}(D^{\mu}\phi_{j})^{a}+\frac{1}{2}\phi_{i}^{a}H_{ij}\phi_{j}^{a}$
$\displaystyle-\frac{g}{16}\left(\phi_{i}^{a}E_{ij}\phi_{j}^{a}\right)^{2}-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu},$

where the matrices $H,\mathcal{I}$ and $E$ are defined as

$\displaystyle H:=\left(\begin{array}[]{cc}m_{1}^{2}&-\mu^{2}\\\
-\mu^{2}&-m_{2}^{2}\end{array}\right)~{},~{}\mathcal{I}:=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (12) $\displaystyle
E:=\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right).$ (15)

###  2.1 Higgs and gauge masses

Next, we define the trivial solution of the equations of motion by solving
$\delta V=0$ and $D_{\mu}\phi_{i}=0$. Such vacuum is often referred to as
Higgs vacuum. The first equation can be simplified by choosing an Ansatz
$(\phi_{i}^{0})^{a}(t,\vec{x})=h_{i}^{0}\hat{r}^{a}(\vec{x})$ where
$\hat{r}=(x,y,z)/\sqrt{x^{2}+y^{2}+z^{2}}$ and $\\{h^{0}_{i}\\}$ are some
constants to be determined. Note that the vacuum solution has a rotational
symmetry $SO(3)$ since $\hat{r}^{a}\hat{r}^{a}=1$. Inserting this Ansatz into
the equation (2), we find

$\displaystyle V=-\frac{1}{2}h_{i}H_{ij}h_{j}+\frac{g}{16}h_{1}^{4}.$ (16)

Then the vacuum equation $\delta V=0$ is reduced to simple coupled third order
algebraic equations

$\displaystyle\frac{g}{4}(h_{1}^{0})^{3}-c_{1}m_{1}^{2}h_{1}^{0}+c_{3}\mu^{2}h_{2}^{0}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
c_{2}m_{2}^{2}h_{2}^{0}+c_{3}\mu^{2}h_{1}^{0}$ $\displaystyle=$ $\displaystyle
0,$ (17) $\displaystyle D_{\mu}\phi_{\alpha}$ $\displaystyle=$ $\displaystyle
0.$ (18)

The resulting vacuum solutions are

$\displaystyle\\!\\!\\!\\!h_{2}^{0}=-\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}h_{1}^{0},~{}(h_{1}^{0})^{2}=4\frac{c_{2}\mu^{4}+c_{1}m_{1}^{2}m_{2}^{2}}{gm_{2}^{2}}:=R^{2},~{}~{}~{}~{}$
(19)
$\displaystyle(A_{i}^{0})^{a}=-\frac{1}{er}\epsilon^{iaj}\hat{r}^{j}+\hat{r}^{a}\mathcal{A}_{i},~{}(A_{0}^{0})^{a}=0.$

The $\mathcal{A}_{i}$ are arbitrary functions of space-time. The Higgs
particle can be identified with the fundamental fields of the theory after
spontaneous symmetry breaking of the continuous symmetry $SU(2)$ by Taylor
expanding around the vacuum solution. Performing a Taylor expansion around the
Higgs vacuum and keeping a focus on the second-order terms with only matter
fields $\phi_{i}^{a}$, the real Lagrangian (2) contains the following term

$\displaystyle\mathfrak{s}$ $\displaystyle=$ $\displaystyle\int
d^{4}x~{}\frac{1}{2}\phi_{i}^{a}\left(-\partial_{\mu}\partial^{\mu}I_{ij}\delta^{ab}-\tilde{H}_{ij}^{ab}\right)\phi_{j}^{b}+\dots,$

where $\tilde{H}$ is a $6\times 6$ block diagonal Hermitian matrix. After
diagonalising the above term by redefining the fields with the eigenvectors of
the non-Hermitian mass matrix
$M_{ij}^{ab}:=I_{ik}\delta^{ac}\tilde{H}_{kj}^{cb}$, we find that the masses
of the fundamental fields after the symmetry breaking to be equal to the
eigenvalues of $M_{ij}^{ab}$ given as

$m_{0}^{2}=c_{2}\frac{\mu^{4}-m_{2}^{4}}{m_{2}^{2}}~{},~{}~{}~{}m_{\pm}^{2}=K\pm\sqrt{K^{2}+2L},$
(20)

where
$K=c_{1}m_{1}^{2}-c_{2}\frac{m_{2}^{2}}{2}+\frac{3\mu^{4}}{2c_{2}m_{2}^{2}}$
and $L=\mu^{4}+c_{1}c_{2}m_{1}^{2}m_{2}^{2}$. Notice that we only find three
non-zero eigenvalues. The redefined fields with zero masses (eigenvalues) are
called Goldstone fields, which can be absorbed into gauge fields $A_{\mu}^{a}$
by defining the new massive gauge fields. This process of giving mass to the
previously massless fields is called the Higgs mechanism. The mass of the
gauge fields can be found by expanding the kinetic term of $\phi$ around the
Higgs vacuum $\phi^{0}_{i}=h_{i}^{0}\hat{r}^{a}$. Without loss of generality,
we can choose a particular direction of the vacuum by taking
$\hat{r}=(0,0,1)^{T}$. This is possible due to the symmetry of the $SO(3)$
vacuum as discussed above. Keeping the term only quadratic in the gauge field,
we find

$\displaystyle\frac{1}{2}(D_{\mu}\phi_{i}+D_{\mu}\phi_{i}^{0})^{a}\mathcal{I}_{ij}(D_{\mu}\phi_{i}+D_{\mu}\phi_{i}^{0})^{a}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(eA_{\mu}\times\phi^{0}_{i}\right)^{a}\mathcal{I}_{ij}\left(eA_{\mu}\times\phi^{0}_{j}\right)^{a}+\dots$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}h_{i}^{0}\mathcal{I}_{ij}h^{0}_{j}\left(A^{1}_{\mu}{A^{1}}^{\mu}+A^{2}_{\mu}{A^{2}}^{\mu}\right)+\dots$
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{g}^{2}\left(A^{1}_{\mu}{A^{1}}^{\mu}+A^{2}_{\mu}{A^{2}}^{\mu}\right)+\dots,$

where the mass of the gauge field is identified to be
$m_{g}:=\sqrt{h_{i}^{0}\mathcal{I}_{ij}h^{0}_{j}}=e\frac{R\sqrt{m_{2}^{4}-\mu^{4}}}{m_{2}^{2}}$.

###  2.2 t’Hooft-Polyakov Monopole

To find the monopole solutions, let us consider the following Ansatz

$\displaystyle(\phi^{cl}_{i})^{a}(\vec{x})=h_{i}(r)\hat{r}^{a}~{},~{}(A_{i}^{cl})^{a}=\epsilon^{iaj}\hat{r}^{j}A(r)~{},$
(22) $\displaystyle(A_{0}^{cl})^{a}=0,$

where the subscript $cl$ denotes the classical solutions to the equations of
motion (4). The difference between this Ansatz (22) and the Higgs vacuum (19)
is that the quantity $h_{i}$ now depends on the spatial radius
$h_{i}=h_{i}(r)$. Here we are only considering the static Ansatz to simplify
our calculation, but one may, of course, also consider the time-dependent
solution by utilising the Lorentz symmetry of the model and performing a
Lorentz boost. According to Derrick’s scaling argument [25], for the monopole
solution to have finite energy, we require the two matter fields of the
equation (22) to approach the vacuum solutions in the equation (19) at spatial
infinity

$\displaystyle\lim_{r\rightarrow\infty}h_{1}(r)=h_{1}^{0\pm}=\pm R~{},~{}$
(23)
$\displaystyle\lim_{r\rightarrow\infty}h_{2}(r)=h_{2}^{0\pm}=\mp\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}R.$

Also, notice that at some fixed value of the radius $r$, the vacuum solutions
$\phi_{\alpha}^{0}$ and monopole solutions $\phi_{\alpha}^{cl}$ both belongs
to the 2-sphere in the field configuration space. For example, $\phi_{1}^{0}$
belongs to the 2-sphere with radius $R$ because $(\phi_{1}^{0})^{2}=R^{2}$.
Therefore, solutions $\phi^{cl}_{i}$ can be seen as a mapping between 2-sphere
in space-time (where the radius is given by the profile function $h_{i}$) to
2-sphere in field configuration space. Such mapping has a topological number
called the winding number $n\in\mathbb{Z}$, which can be explicitly realised
by redefining the unit vector $\hat{r}^{a}$ as

$\hat{r}_{n}^{a}=\left(\begin{array}[]{c}\sin(\theta)\cos(n\varphi)\\\
\sin(\theta)\sin(n\varphi)\\\ \cos(\theta)\end{array}\right).$ (24)

Therefore different $n$ represent topologically inequivalent solutions.

Since we require the monopole and vacuum solutions to smoothly deformed into
each other at spacial infinity, both solutions need to share the same winding
number. It is important to note that winding numbers of $\phi_{1}$ and
$\phi_{2}$ need to be equal to satisfy $D\phi_{1}=D\phi_{2}=0$, and therefore
we will denote the winding numbers of $\phi_{1}$ and $\phi_{2}$ as $n$
collectively. If they are not equal, we would have $D\phi_{1}=0$ but
$D\phi_{2}\not=0$. Next, let us insert our Ansatz equation (22) into the
equations of motion equation (4). We will also redefine the Ansatz for the
gauge fields to be
$A^{a}_{i}=\epsilon^{aib}\hat{r}^{b}\left(\frac{1-u(r)}{er}\right),~{}A_{0}^{a}=0$,
which are more in line with the original Ansatz given in [26, 27], compared to
equation (22). Inserting these expressions into the equations of motion
equation (4), we find

$\displaystyle
u^{{}^{\prime\prime}}(r)+\frac{u(r)\left[1-u^{2}(r)\right]}{r^{2}}$ (25)
$\displaystyle+\frac{e^{2}u(r)}{2}\left\\{h_{2}^{2}(r)-h_{1}^{2}(r)\right\\}=0,$
$\displaystyle
h_{1}^{{}^{\prime\prime}}(r)+\frac{2h_{1}^{{}^{\prime}}(r)}{r}-\frac{2h_{1}(r)u^{2}(r)}{r^{2}}$
(26)
$\displaystyle+g\left\\{-c_{1}\frac{m_{1}^{2}}{g}h_{1}(r)+c_{3}\frac{\mu^{2}}{g}h_{2}(r)+\frac{1}{4}h_{1}^{3}(r)\right\\}=0,$
$\displaystyle
h_{2}^{{}^{\prime\prime}}(r)+\frac{2h_{2}^{{}^{\prime}}(r)}{r}-\frac{2h_{2}(r)u^{2}(r)}{r^{2}}$
(27)
$\displaystyle+c_{2}m_{2}^{2}\left\\{h_{2}(r)+c_{3}\frac{\mu^{2}}{m_{2}^{2}}h_{1}(r)\right\\}=0.$

Notice that these differential equations are similar to the ones discussed in
[26, 27], but with the extra field $h_{2}$ and extra differential equation
(27). In the Hermitian model, the exact solutions to the differential
equations were found by taking the parameter limit called the BPS limit [26,
27], where parameters in the Hermitian model are taken to zero while keeping
the vacuum solution finite. Here we will follow the same procedure and take
the parameter limit where quantities in the curly brackets of equations (26)
and (27) vanish but keep the vacuum solutions equation (19) finite. We will
see in section 2.4 that we also find the approximate solutions in this limit.

###  2.3 The energy bound

Surprisingly, by utilising Derrick’s scaling argument, one can find the lower
bound of the monopole energy without the explicit form of the solution.

The energy of the monopole can be found by inserting the monopole solution
into the corresponding Hamiltonian of equation (2).

$\displaystyle\\!\\!\\!\\!\mathfrak{h}$ $\displaystyle=$ $\displaystyle\int
d^{3}x~{}~{}Tr\left(E^{2}\right)+Tr\left(B^{2}\right)$
$\displaystyle+Tr\left\\{(D_{0}\phi_{1})^{2}\right\\}+Tr\left\\{(D_{i}\phi_{1})^{2}\right\\}$
$\displaystyle-
Tr\left\\{(D_{0}\phi_{2})^{2}\right\\}-Tr\left\\{(D_{i}\phi_{2})^{2}\right\\}+V,$

where $E,B$ are ${E^{i}}_{a}={F_{a}}^{0i}$ ,
${B^{i}}_{a}=-\frac{1}{2}\epsilon^{ijk}F_{a}^{jk}$, $i,j,k\in\\{1,2,3\\}$. The
gauge is fixed to be the radiation gauge (i.e
${A_{a}}^{0}=0,\partial_{i}{A_{a}}^{i}=0$). Notice that our monopole Ansatz
equation (17) is static with no electric charge $E_{i}^{a}=0$ and therefore,
the above Hamiltonian reduces to

$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
d^{3}x~{}~{}Tr\left({B}^{2}\right)+Tr\left\\{(D_{i}\phi_{1})^{2}\right\\}$
$\displaystyle-Tr\left\\{(D_{i}\phi_{2})^{2}\right\\}+V$ $\displaystyle=$
$\displaystyle 2\int
d^{3}x~{}~{}{B_{i}}^{a}{B_{i}}^{a}+(D_{i}\phi_{1})^{a}(D_{i}\phi_{1})^{a}$
$\displaystyle-(D_{i}\phi_{2})^{a}(D_{i}\phi_{2})^{a}+\frac{1}{2}V.$

Here, we simplified our expression by dropping the superscripts
$A^{cl}_{i}\rightarrow A_{i}$ , $\phi_{\alpha}^{cl}\rightarrow\phi_{\alpha}$.
We also keep in mind that these fields depend on the winding numbers
$n\in\mathbb{Z}$. In the Hermitian model (i.e. when $\phi_{2}=0$), one can
rewrite the kinetic term as $B^{2}+D\phi^{2}=(B-D\phi)^{2}+2BD\phi$ and find
the lower bound to be $\int 2BD\phi$. Here we will follow a similar procedure
but introduce some arbitrary constant $\alpha,\beta\in\mathbb{R}$ such that
$B^{2}=\alpha^{2}B-\beta^{2}B$ where $\alpha^{2}-\beta^{2}=1$. This will allow
us to rewrite the above energy as

$\displaystyle E$ $\displaystyle=$ $\displaystyle 2\int
d^{3}x~{}~{}\alpha^{2}\left\\{{B_{i}}^{a}+\frac{1}{\alpha}(D_{i}\phi_{1})^{a}\right\\}^{2}$
$\displaystyle-\beta^{2}\left\\{{B_{i}}^{a}+\frac{1}{\beta}(D_{i}\phi_{2})^{a}\right\\}^{2}$
$\displaystyle+2\left\\{-\alpha{B_{i}}^{a}(D_{i}\phi_{1})^{a}+\beta{B_{i}}^{a}(D_{i}\phi_{2})^{a}\right\\}+\frac{1}{2}V.$

To proceed from here, we need to assume extra constraints on $\alpha$ and
$\beta$ such that the following inequalities are true

$\displaystyle\int
d^{3}x~{}\alpha^{2}\left\\{{B_{i}}^{a}+\frac{1}{\alpha}(D_{i}\phi_{1})^{a}\right\\}^{2}$
(31)
$\displaystyle-\beta^{2}\left\\{{B_{i}}^{a}+\frac{1}{\beta}(D_{i}\phi_{2})^{a}\right\\}^{2}$
$\displaystyle\geq$ $\displaystyle 0,$ $\displaystyle\int d^{3}xV$
$\displaystyle\geq$ $\displaystyle 0.$ (32)

With these constraints we can now write down the lower bound of the monopole
as

$\displaystyle E$ $\displaystyle\geq$ $\displaystyle 2\int
d^{3}x~{}\left\\{-\alpha{B_{i}}^{a}(D_{i}\phi_{1})^{a}+\beta{B_{i}}^{a}(D_{i}\phi_{2})^{a}\right\\}$
$\displaystyle=$ $\displaystyle 2\int
d^{3}x~{}-\alpha\left\\{{B_{i}}^{a}\partial_{i}\phi_{1}^{a}+e{B_{i}}^{a}\epsilon^{abc}{A_{i}}^{b}\phi_{1}^{c}\right\\}$
$\displaystyle+\beta\left\\{{B_{i}}^{a}\partial_{i}\phi_{2}^{a}+e{B_{i}}^{a}\epsilon^{abc}{A_{i}}^{b}\phi_{2}^{c}\right\\}$
$\displaystyle=$ $\displaystyle 2\int
d^{3}x~{}-\alpha\left\\{{B_{i}}^{a}\partial_{i}\phi_{1}^{a}+\left(-e\epsilon^{abc}{A_{i}}^{b}{B_{i}}^{c}\right)\phi_{1}^{a}\right\\}$
$\displaystyle+\beta\left\\{{B_{i}}^{a}\partial_{i}\phi_{2}^{a}+\left(-e\epsilon^{abc}{A_{i}}^{b}{B_{i}}^{c}\right)\phi_{1}^{a}\phi_{2}^{c}\right\\}$
$\displaystyle=$ $\displaystyle 2\int
d^{3}x~{}-\alpha\left\\{{B_{i}}^{a}\partial_{i}\phi_{1}^{a}+\partial_{i}{B_{i}}^{a}\phi_{1}^{a}\right\\}$
$\displaystyle+\beta\left\\{{B_{i}}^{a}\partial_{i}\phi_{2}^{a}+\partial_{i}{B_{i}}^{a}\phi_{1}^{a}\right\\}$
$\displaystyle=$ $\displaystyle 2\int
d^{3}x~{}-\alpha\partial_{i}\left({B_{i}}^{a}{\phi_{1}}^{a}\right)+\beta\partial_{i}\left({B_{i}}^{a}{\phi_{2}}^{a}\right)$
$\displaystyle=$
$\displaystyle\lim_{r\rightarrow\infty}\left(-2\alpha\int_{S_{r}}dS_{i}{B_{i}}^{a}{\phi_{1}}^{a}+2\beta\int_{S_{r}}dS_{i}{B_{i}}^{a}{\phi_{2}}^{a}\right),$

where in the fourth line, we used $D_{i}B_{i}^{a}=0$, which can be shown from
the Bianchi identity $D_{\mu}\epsilon^{\mu\nu\rho\sigma}F^{a}_{\rho\sigma}=0$.
The last line is obtained by using the Gauss theorem at some fixed value of
the radius $r$. Since the $\phi_{i}^{a}$ in the integrand is only defined over
the 2-sphere with a large radius, we can use the asymptotic conditions (23)
and replace the monopole solutions $\\{\phi_{\alpha}^{a},B_{i}^{a}\\}$ with
the Higgs vacuum $\\{(\phi_{\alpha}^{0})^{a},(B_{i}^{0})^{a}\\}$

$\displaystyle E$ $\displaystyle\geq$
$\displaystyle\left(-2\alpha{\phi^{0}_{1}}^{a}+2\beta{\phi^{0}_{2}}^{a}\right)\lim_{r\rightarrow\infty}\int_{S_{r}}dS_{i}(B_{i}^{0})^{a}$
$\displaystyle=$ $\displaystyle\left(\mp 2\alpha R\hat{r}^{a}_{n}\mp
2\beta\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}R\hat{r}^{a}_{n}\right)\lim_{r\rightarrow\infty}\int_{S_{r}}dS_{i}(B_{i}^{0})^{a},$

where the upper and lower signs of the above energy correspond to the upper
and lower signs of the vacuum solutions in equation (19). The explicit value
of $B_{0i}^{a}$ can be obtained by inserting the Higgs vacuum (19) into the
definition of the magnetic field

$B_{i}^{a}=-\frac{1}{2}{\epsilon_{i}}^{jk}\left(\partial_{j}A_{k}-\partial_{k}A_{j}+eA_{j}\times
A_{k}\right)^{a}.$ (35)

After a lengthy calculation, this expression can be simplified to
$B_{0i}^{a}=\hat{\phi^{0}}^{a}b_{i}=\hat{r}^{a}_{n}b_{i}$, where
$\hat{\phi^{0}}^{a}$ is a normalised solution
$\sum_{a}\hat{\phi^{0}}^{a}\hat{\phi^{0}}^{a}=1$. The $b_{i}$ is defined as

$b_{i}\equiv-\frac{1}{2}\epsilon_{ijk}\left\\{\partial^{j}\mathcal{A}^{k}-\partial^{k}\mathcal{A}^{j}+\frac{1}{e}\hat{r}_{n}\cdot\left(\partial^{j}\hat{r}_{n}\times\partial^{k}\hat{r}_{n}\right)\right\\}.$
(36)

Where $\mathcal{A}$ was defined in equation (19). Notice that integrating the
first term over the 2-sphere gives zero by Stoke’s theorem
$\int_{S}\partial\times\mathcal{A}=\int_{\partial S}\mathcal{A}=0$, where one
can show that Stoke’s theorem on the closed surface gives zero by dividing the
sphere into two open surfaces. The second term is a topological term which can
be evaluated as

$\int dS_{i}B_{i}=-\frac{4\pi n}{e}.$ (37)

The explicit calculation is in [28]. This is the magnetic charge of the
monopole solutions. Therefore integer $n$, which corresponds to the winding
number of the solution, comes from the Ansatz
$B_{i}^{a}=\hat{\phi^{0}}^{a}b_{i}$. In our case, there is an ambiguity of
whether to choose $B_{i}^{a}=\hat{\phi_{1}^{0}}^{a}b_{i}$ or
$B_{i}^{a}=\hat{\phi_{2}^{0}}^{a}b_{i}$. Now we see explicitly the reason why
we choose to keep the same integer values for solutions $\phi_{1}^{0}$ and
$\phi^{0}_{2}$. If the integer values of $\hat{r}_{n}^{a}$ in solutions
$\phi_{1}^{0}$, $\phi^{0}_{2}$ are different, then the integration
$\int_{S_{r}}dS_{i}(B_{i}^{0})^{a}$ will be different, leading to inconsistent
energy.

Finally, we find our lower bound of the monopole energy

$\displaystyle E$ $\displaystyle\geq$ $\displaystyle\mp
2R\left(\alpha+\beta\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}\right)\hat{r}^{a}_{n}\hat{r}^{a}_{n}\left(\frac{-4\pi
n}{e}\right)$ $\displaystyle=$ $\displaystyle\frac{\pm 8\pi
nR}{e}\left(\alpha+\beta\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}\right).$

Notice that we have some freedom to choose $\alpha,\beta\in\mathbb{R}$ as long
as our initial assumptions (31) are satisfied. We will see in the next section
that we can take a parameter limit of our model, which saturates the above
inequality and gives exact values to $\alpha$ and $\beta$.

###  2.4 The fourfold BPS scaling limit

Our main goal is now to solve the coupled differential equations (25)-(27).
Prasad, Sommerfield, and Bogomolny [26, 27] managed to find the exact solution
by taking the parameter limit, which simplifies the differential equations.
The multiple scaling limit is taken so that all the parameters of the model
tend to zero with some combinations of the parameter remaining finite. The
combinations are taken so that the vacuum solutions stay finite in this limit.
Inspired by this, we will take here a fourfold scaling limit

$g,m_{1},m_{2},\mu\rightarrow
0~{},~{}~{}~{}\frac{m_{1}^{2}}{g}<\infty~{},~{}~{}~{}\frac{\mu^{2}}{g}<\infty~{},~{}~{}~{}\frac{\mu^{2}}{m_{2}^{2}}<\infty.$
(39)

This will ensure that the vacuum solutions equation (19) stays finite, but
crucially the curly bracket parts in equations (26) and (27) vanish. There is
a physical motivation for this limit in which the mass ratio of the Higgs and
gauge mass are taken to be zero (i.e. $m_{\text{Higgs}}<<m_{g}$) as described
in [29]. We will see in the next section that the same type of behaviour is
present in our model, hence justifying equation (39). The resulting set of
differential equations, after taking the BPS limit, is similar to the ones
considered in [26, 27] with the slightly different quadratic term in equation
(25). It is natural to consider a similar Ansatz as given in [26, 27]

$\displaystyle u(r)$ $\displaystyle=$ $\displaystyle\frac{evr}{\sinh{(evr)}},$
(40) $\displaystyle h_{1}(r)$ $\displaystyle=$ $\displaystyle-\alpha f(r),$
(41) $\displaystyle h_{2}(r)$ $\displaystyle=$ $\displaystyle-\beta f(r),$
(42)

where $\alpha,\beta\in\mathbb{R}$ were introduced in section 2.3 and
$f(r)\equiv\left\\{v\coth\left(evr\right)-\frac{1}{er}\right\\}$. One can
check that this Ansatz indeed satisfies differential equations equation
(25)-(27) in the BPS limit. We have decided to put a prefactor $\alpha$ and
$\beta$ in front of equations (41) and (42) to satisfy the differential
equation (25). Note that if we take $\alpha=1$, we get exactly the same as
given in [26, 27], which is known to satisfy the first-order differential
equation called the Bogomolny equation $B_{i}-D_{i}\phi=0$. The Ansatz
(40)-(42) only differs from the ones given in [26, 27] by the prefactors
$\alpha$ and $\beta$, and therefore our Ansatz should satisfy the Bogomolny
equation with the appropriate scaling to cancel the prefactor in equations
(41) and (42)

$\displaystyle B_{i}^{b}+\frac{1}{\alpha}(D_{i}\phi_{1})^{b}$ $\displaystyle=$
$\displaystyle 0,$ (43) $\displaystyle
B_{i}^{b}+\frac{1}{\beta}(D_{i}\phi_{2})^{b}$ $\displaystyle=$ $\displaystyle
0,$ (44)

where $\phi_{\alpha}\equiv h_{\alpha}(r)\hat{r}_{n}$. If we compare these
equations to the terms appearing in the energy of the monopole equation (2.3),
then we can saturate the inequality in equation (2.3) by

$\displaystyle E[\phi_{1},\phi_{2}]=\frac{\pm 8\pi
nR}{e}\left(\alpha+\beta\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}\right),$ (45)

where upper and lower signs correspond to the vacuum solutions equation (19),
when taking the square root. We can calculate the explicit forms of $\alpha$
and $\beta$ by comparing the asymptotic conditions in equation (23)

$\displaystyle\lim_{r\rightarrow\infty}h_{1}^{\pm}$ $\displaystyle=$
$\displaystyle h_{1}^{0\pm}=\pm R,$ (46)
$\displaystyle\lim_{r\rightarrow\infty}h_{2}^{\pm}$ $\displaystyle=$
$\displaystyle h_{2}^{0\pm}=\mp\frac{c_{2}c_{3}\mu^{2}}{m_{2}^{2}}R,$

with the asymptotic values of equations (40)-(42)

$\displaystyle\lim_{r\rightarrow\infty}u(r)=0~{},~{}\lim_{r\rightarrow\infty}h_{1}^{\pm}(r)=-\alpha
v,$ (47) $\displaystyle\lim_{r\rightarrow\infty}h_{2}^{\pm}(r)=-\beta v.$

By Derrick’s scaling argument, the two asymptotic values (46) and (47) should
match, resulting in algebraic equations for $\alpha$ and $\beta$. Using
$\alpha^{2}-\beta^{2}=1$ and assuming $m_{2}^{4}\geq\mu^{4}$, we find the four
set of real solutions

$\alpha=\mp(\pm)\frac{m_{2}^{2}}{l}~{},~{}~{}~{}v=(\pm)\frac{Rl}{m_{2}^{2}}~{},~{}~{}~{}\beta=\pm(\pm)\frac{c_{2}c_{3}\mu^{2}}{l},$
(48)

where $l=\sqrt{m_{2}^{4}-\mu^{4}}$. The plus-minus signs in the brackets
correspond to the two possible solutions to the algebraic equation
$\alpha^{2}-\beta^{2}=1$. These need to be distinguished from the upper and
lower signs of $\alpha$ and $\beta$, which correspond to the vacuums solutions
(19). Inserting the explicit values of $\alpha$ and $\beta$ to the energy
equation (45) we find

$\displaystyle E[\phi_{1},\phi_{2}]$ $\displaystyle\equiv$
$\displaystyle(\pm)\frac{8\pi
nR}{em_{2}^{2}}\left(\frac{-m_{2}^{4}+\mu^{4}}{l}\right)$ $\displaystyle=$
$\displaystyle(\pm)\frac{-8\pi nR}{em_{2}^{2}}l,$

with corresponding solutions

$\displaystyle h_{1}^{\pm}(r)$ $\displaystyle=$
$\displaystyle\pm(\pm)\frac{m_{2}^{2}}{l}\left[\frac{Rl}{m_{2}^{2}}\text{coth}\left(\frac{eRl}{m_{2}^{2}}r\right)-\frac{1}{er}\right],$
(50) $\displaystyle h_{2}^{\pm}(r)$ $\displaystyle=$
$\displaystyle\mp(\pm)\frac{c_{2}c_{3}\mu^{2}}{l}\left[\frac{Rl}{m_{2}^{2}}\text{coth}\left(\frac{eRl}{m_{2}^{2}}r\right)-\frac{1}{er}\right].$

It is crucial to note that although it seems like there are two monopole
solutions $\\{h_{1}^{\pm},h_{2}^{\pm}\\}$, the two solutions are related non-
trivially in their asymptotic limit by the constraint
$\lim_{r\rightarrow\infty}h_{2}^{\pm}=(-c_{2}c_{3}\mu^{2}/m_{2}^{2})\lim_{r\rightarrow\infty}h_{1}^{\pm}$
given in equation (17). For example, one can not choose
$\\{h_{1}^{+},h_{2}^{-}\\}$ as a solution as this will break the asymptotic
constraint.

The solution (50) can be constrained further by imposing that the energy (2.4)
is real and positive.

$\displaystyle E[\phi_{1},\phi_{2}]>0\implies-(\pm)\frac{8\pi
nR}{em_{2}^{2}}l\implies-(\pm)n>0.$ (51)

Therefore we can ensure positive energy if $(\pm)=\text{sign}(n)$. The final
form of the monopole solution with positive energy are

$\displaystyle h_{1}^{\pm}(r)$ $\displaystyle=$
$\displaystyle\pm\text{sign}(n)\frac{m_{2}^{2}}{l}\left[\frac{Rl}{m_{2}^{2}}\text{coth}\left(\frac{eRl}{m_{2}^{2}}r\right)-\frac{1}{er}\right],$
(52) $\displaystyle h_{2}^{\pm}(r)$ $\displaystyle=$
$\displaystyle\mp\text{sign}(n)\frac{c_{2}c_{3}\mu^{2}}{l}\left[\frac{Rl}{m_{2}^{2}}\text{coth}\left(\frac{eRl}{m_{2}^{2}}r\right)-\frac{1}{er}\right].$

with energy $E=8|n|\pi lR/em_{2}^{2}$. We conclude this subsection by
observing that the above solution depends on the parameter $c_{3}$, which
takes value $\\{-1,1\\}$ depending on the choice of the similarity
transformation. Choosing a different values of $c_{3}$ also result in a
different asymptotic values (46), meaning solutions for $c_{3}=1$ and
$c_{3}=-1$ are topologically different. Since the energy is independent of
$c_{3}$, two distinct solutions share the same energy. Respecting one of the
main features of similarity transformation, which is to preserve the energy of
the transformed Hamiltonian.

In the next section, we will investigate in detail how the solution changes
and a new $\mathcal{CPT}$ symmetry emerges by changing the parameter values.

##  3 Results and Discussion

This section will investigate the behaviour of solution (52) in different
regimes of the parameter spaces. We will compare the physical regions of gauge
particles, Higgs particles and monopoles found in the previous section. We
will see that the two regions coincide, but the solutions in different regions
possess different $\mathcal{CPT}$ symmetries. Different symmetries of
solutions in different regions are not coincident, but the consequence of the
three reality conditions stated in the introduction. In fact, it is deeply
related to the real value of energy, which will be discussed extensively in
[30].

###  3.1 Higgs mass and exceptional points

Let us recall the masses of the particles and monopole

$\displaystyle
m_{0}^{2}=c_{2}\frac{\mu^{4}-m_{2}^{4}}{m_{2}^{2}}~{},~{}m_{\pm}^{2}=K\pm\sqrt{K^{2}+2L},$
(53) $\displaystyle m_{g}=e\frac{Rl}{m_{2}^{2}},M_{\text{mono}}=\frac{8|n|\pi
lR}{em_{2}^{2}}.$

where
$K=c_{1}m_{1}^{2}-c_{2}\frac{m_{2}^{2}}{2}+\frac{3\mu^{4}}{2c_{2}m_{2}^{2}}$
and $L=\mu^{4}+c_{1}c_{2}m_{1}^{2}m_{2}^{2}$. Notice that the masses do not
depend on $c_{3}$, meaning they do not depend on the similarity transformation
as expected. We also comment that in the BPS limit, we have $m_{0}=m_{\pm}=0$,
but $m_{g}$ and $M_{\pm}$ stays finite, such that the ratios
$m_{\text{Higgs}}/m_{g}$ vanish in the BPS limit. This is in line with the
Hermitian case [29], providing the physical interpretation with
$m_{Higgs}<<m_{g}$ for the BPS limit.

One may notice that when $c_{2}=1$, requiring positive mass $m_{0}^{2}>0$,
implies that $\mu^{4}-m_{2}^{4}>0$. This means the quantity
$l=\sqrt{m_{2}^{4}-\mu^{4}}$ is purely imaginary. One may then discard this
region as unphysical. However, we will see in the next section that there is a
disconnected region beyond $\mu^{4}-m_{2}^{4}>0$, which admit real energy
because $R$ also becomes purely complex. This is not coincident, and in fact,
we will see an emerging new $\mathcal{CPT}$ symmetry for the monopoles.

In the rest of the section, we will exclusively focus on the monopole and
gauge masses. The main message of this section is the emerging symmetry
responsible for the real value of the monopole masses. The requirement to make
the whole theory physical demands also to consider the intersection of the
physical regions between monopole masses and Higgs masses. As an example, we
plot all the masses of the theory in figure 1.

Figure 1: Monopole, gauge and Higgs masses plotted for
$m_{1}^{2}/g=-0.44,\mu/g=-0.14,e=2,c_{1}=-c_{2}=-1$. The solid line represents
the real part, and the dotted line represents the imaginary part of the
masses. The dotted vertical lines indicate the boundaries of the physical
regions where all the masses acquire real positive values. Figure 2: Both
panels are plotted for $X=-2,Y=-0.6,c_{1}=-c_{2}=1,n=1,e=2$. The solid line
represents the real part, and the dotted line represents the imaginary part of
the masses.

As one can see, intersection points of the physical regions of Higgs masses
and monopole/gauge masses are non-trivial. In fact, they are bounded by two
types of exceptional points. The first type is when two masses of Higgs
particles coincide and form a complex conjugate pair. Such a point is known as
an exceptional point where the mass matrix is non-diagonalisable, and the
corresponding eigenvectors coincide. The second type is when the gauge and the
monopole masses vanishes. Interestingly, this is where one of the Higgs masses
also vanish. Since the mass matrix already has a zero eigenvalue, as the
result of the spontaneous symmetry breaking, it seems the number of massless
fields is increased. However, at this point, the mass matrix is also non-
diagonalisable. Therefore one can not diagonalise the Hamiltonian to identify
the field which corresponds to the extra massless fundamental field. Therefore
this point is also an exceptional point. However, the eigenvalues do not
become complex conjugate pairs beyond this point, and as one can see from
figure 1 that one of the mass square $m_{0}^{2}$ become negative, and gauge
and monopole masses become complex but with no conjugate pair. We dub such a
point as zero exceptional point to distinguish from the standard exceptional
point.

###  3.2 Change in $\mathcal{CPT}$ symmetry and complex monopole solution

We begin by introducing the useful quantities $m_{1}^{2}/g\equiv
X,\mu^{2}/g\equiv Y,\mu^{2}/m_{2}^{2}\equiv Z$. The gauge mass, monopole mass
and monopole solutions can be rewritten in terms of these quantities

$\displaystyle m_{g}=eR\sqrt{1-Z^{2}},~{}m_{\text{mono}}=\frac{8|n|\pi
R}{e}\sqrt{1-Z^{2}},$ (54)
$\displaystyle\\!\\!\\!\\!h_{1}^{\pm}(r)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\pm\frac{\text{sign}(n)}{\sqrt{1-Z^{2}}}\big{[}R\sqrt{1-Z^{2}}\text{coth}(\hat{r})-\frac{1}{er}\big{]},$
(55) $\displaystyle\\!\\!\\!\\!h_{2}^{\pm}(r)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\mp\frac{\text{sign}(n)c_{2}c_{3}Z}{\sqrt{1-Z^{2}}}\left[R\sqrt{1-Z^{2}}\text{coth}(\hat{r})-\frac{1}{er}\right],~{}~{}~{}~{}$
(56)

where $R^{2}=4(c_{2}ZY+c_{1}X)$ and $\hat{r}=eR\sqrt{1-Z^{2}}r$. The monopole
masses are plotted against the gauge mass for fixed parameters with
$n\in\\{1,2,3,4\\}$ in figure 3 with weak and strong couplings $e=2,e=10$.
Notice that the gauge mass is smaller than any of the monopole masses for weak
coupling, but when $e$ is large enough, some of the monopole masses can become
smaller than the gauge mass. This is clear by inspecting the monopole, and
gauge mass in equation (54) and two masses coincide when $e=\sqrt{8|n|\pi}$.
Note that $n=0$ is not a monopole mass as it corresponds to the solution with
zero winding number, which is topologically equivalent to the trivial
solution.

Figure 3: Monopole and gauge masses plotted for
$X=1,Y=0.8,e=2,c_{1}=-c_{2}=1$. The solid line represents the real part, and
the dotted line represents the imaginary part of the masses. Figure 4: Both
panels are plotted for $X=1,Y=0.8,n=1,e=2$. The solid line represents the real
part, and the dotted line represents the imaginary part of the masses. Panel
(a) shows the monopole and gauge masses against $Z\geq 0$, with vertical lines
indicating the location of the boundaries of three regions. Panel (b) shows
three profile function $h_{1}(r)$ defined on each region indicated in panel
(a).

From figure 3, we also observe disconnected regions where both monopole and
gauge masses become real to purely complex. A more detailed plot of this is
shown in figure 4. Region 2 is bounded by two points with lower bound
$\mu^{2}/m_{2}^{2}=1$ corresponding to the zero exceptional point where the
vacuum manifolds stay finite (i.e. spontaneous symmetry breaking occur).
However, the Higgs mechanism fails because the Hamiltonian is non-
diagonalisable, as discussed in the previous section. The upper bounds
correspond to the point where the vacuum manifold vanishes. Therefore, the
spontaneous symmetry breaking does not occur, implying that the gauge fields
do not acquire a mass through the Higgs mechanism, resulting in a massless
gauge field. Most crucially, an interesting region (denoted by region 3 in
figure 5) reappears as one increases the value of $Z$. The profile function in
region 3 is purely complex, which signals that this may lead to complex
energies. However, as one can see from figure 4, the energy is real. The
reason for the real energy is that the conditions stated in the introduction
hold. We will specify below the $\mathcal{CPT}$ symmetry responsible for the
real value of the energy. Note that the profile function $h_{2}$ only differ
from $h_{1}$ by some factor in front. Therefore we omitted it from the plot.

Another physical region is when $c_{1}=-c_{2}=-1$. The monopole and gauge
masses for this case is plotted in figure 5. We observe almost an identical
plot from the figure 4 but with real and imaginary parts swapped. The profile
functions also respect these changes as regions 1 and 3 no longer have a
definite asymptotic value. The boundaries are unchanged, as one can see from
the figure 5.

Figure 5: Both panels are plotted for $X=1,Y=1,n=1,e=2$. The solid line
represents the real part, and the dotted line represents the imaginary part of
the masses.

Finally, there is an interesting parameter point $X=Y$ where region 2 vanishes
(see figure 6). The two boundaries $Z^{2}=1$ and $c_{2}ZY+c_{1}X=0$ coincide
when $X=Y$ and the zero exceptional point no longer exists because the
spontaneous symmetry breaking does not occur in this case.

Figure 6: Both panels are plotted for $X=1,Y=1,n=1,e=2$. The solid line
represents the real part, and the dotted line represents the imaginary part of
the masses.

Next, let us explain the real value of the energies in different regions.
First, to realise the conditions 1-3, stated in the introduction, we require
the following transformations

$\begin{array}[]{cc}h_{2}^{\pm}(r)\rightarrow-
h_{2}^{\pm}(r)~{},~{}~{}h_{1}^{\pm}(r)\rightarrow h_{1}^{\pm}(r)&\text{in
region 1}\\\ \text{No symmetry}&\text{in region 2}\\\
h_{2}^{\pm}(r)\rightarrow-\left(h_{2}^{\pm}(r)\right)^{*}~{},~{}~{}h_{1}^{\pm}(r)\rightarrow\left(h_{1}^{\pm}(r)\right)^{*}&\text{in
region 3}\end{array}.$

By using the explicit forms of the solutions (55) and (56). We can show that
the above transformations satisfy condition 2 stated in the introduction, in
regions 1

$\displaystyle h_{2}^{\pm}(r)\rightarrow-
h_{2}^{\pm}(r)=h_{2}^{\mp}(r)~{},~{}~{}h_{1}^{\pm}(r)\rightarrow
h_{1}^{\pm}(r),$ (57)

and in region 3

$\displaystyle
h_{2}^{\pm}(r)\rightarrow-\left(h_{2}^{\pm}(r)\right)^{*}=h_{2}^{\pm}~{},~{}$
(58) $\displaystyle
h_{1}^{\pm}(r)\rightarrow\left(h_{1}^{\pm}(r)\right)^{*}=h_{1}^{\mp}.$

Notice that in regions 1 and 3, the $\mathcal{CPT}$ relates two distinct
solutions in two different ways. For example, $h_{2}^{\pm}$ is mapped to
$h_{2}^{\mp}$ in region 1, but it is mapped to itself in region 3.

Finally, the condition 3 stated in the introduction is satisfied because the
energy does not depend on the $\pm$ signs of the solutions. This explains the
real energies of complex monopoles in region 3 and complex energy in region 2.
Indeed, we observe the predicted behaviour in figure 4. Region 2 is a hard
barrier between two $\mathcal{CPT}$ symmetric regions where solutions are
either real or purely imaginary. The same analysis can be carried out in the
other physical region $c_{1}=-c_{2}=-1$ where the symmetry is now

$\begin{array}[]{cc}\text{No symmetry}&\text{in region 1}\\\
\begin{subarray}{c}h_{2}^{\pm}(r)\rightarrow-\left(h_{2}^{\pm}(r)\right)^{*}\\\
h_{1}^{\pm}(r)\rightarrow\left(h_{1}^{\pm}(r)\right)^{*}\end{subarray}&\text{in
region 2}\\\ \text{No symmetry}&\text{in region 3}\end{array}.$ (59)

We have observed that one can find a well-defined monopole solution in two
disconnected regions. However, in the full theory where we include the Higgs
particles, it is only one of the regions which are considered physical. This
is because the Higgs mass $m_{0}^{2}$ is either positive or negative depending
on which side of $Z^{2}=1$ it is defined. Because two disconnected regions are
defined on either side of the zero exceptional point $Z^{2}=1$, the full
physical region restricts one from moving region 1 to region 3 by changing
$Z$. This is most clearly seen in figure 1 where the plot of $m_{0}^{2}$
(green line) becomes negative beyond the zero exceptional point. This may
imply that the purely complex monopole solution we observed is not a possible
solution of the theory. However, the purely complex solution can exist in the
full physical region. An example of this is shown in the figure 2 where we
observe that the profile function $h_{1}$ (therefore $h_{2}$) is purely
complex, and the Higgs masses, gauge mass are all real and positive.

##  4 Conclusions

We have found the t’Hooft-Polyakov monopole solution (52) in the non-Hermitian
theory by drawing an analogue from the standard procedure in the Hermitian
theory. The monopole masses were plotted with the massive gauge and Higgs
masses, where the physical region of the monopole masses coincided with that
of the gauge mass. It was also observed that there are two distinct physical
regions bounded by the zero exceptional point and the parameter limit where
the vacuum manifold becomes trivial. The profile function (radial part of the
monopole solution) is plotted in figures 4, 5, 6, where it is real and purely
complex in regions 1 and 3, respectively. Incidentally, the $\mathcal{CPT}$
symmetries of the solution are different in regions 1 and 3.

## Acknowledgement

TT is supported by EPSRC grant EP/W522351/1.

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