# Massive gauge particles versus Goldstone bosons in non-Hermitian non-Abelian
gauge theory

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

We investigate the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism for
non-Hermitian field theories with local non-Abelian gauge symmetry in
different regions of their parameter spaces. We demonstrate that the two
aspects of the mechanism, that is giving mass to gauge vector bosons and at
the same time preventing the existence of massless Goldstone bosons, remain to
be synchronized in all regimes characterized by a modified CPT symmetry. In
the domain of parameter space where the “would be Goldstone bosons” can be
identified the gauge vector bosons become massive and the Goldstone bosons
cease to exist. The mechanism is also in tact at the standard exceptional
points. However, at the zero exceptional points, that is when the eigenvalues
of the mass squared matrix vanish irrespective of the symmetry breaking, the
mechanism breaks down as the Goldstone bosons can not be identified and the
gauge vector bosons remain massless. This breakdown coincides with the
vanishing of the CPT inner product of symmetry breaking vacua defined on the
eigenvector space of mass squared matrix. We verify this behaviour for a
theory with SU(2) symmetry in which the complex scalar fields are taken in the
fundamental as well as in the adjoint representation.

††conference: Massive gauge particles versus Goldstone bosons

##  1 Introduction

Our main objective is to extend the Englert-Brout-Higgs-Guralnik-Hagen-Kibble
mechanism [1, 2, 3, 4], hereafter simply referred to as Higgs mechanism, to
non-Hermitian field theories with a local non-Abelian gauge symmetry using a
pseudo-Hermitian approach. We focus on the two key aspects for which the
mechanism was originally developed, that is to give mass to gauge vector
bosons and at the same time prevent the existence of massless Goldstone
bosons. When keeping the symmetry global one may adopt different starting
points for the study of Goldstone phases, such as the field content of local
operators, a scattering matrix based on a particle picture or an explicit
Lagrangian.

For instance, two dimensional conformal quantum field theories are well
understood in terms of their operator content characterized by infinite-
dimensional algebras of local conformal transformations [5]. A large class of
such theories, minimal models [6], are known to possess a finite operator
content and the treatment of unitary and non-unitary theories is formally
identical. The simplest massive non-unitary field theory consisting of only
one real scalar field describing in its ultraviolet limit the critical point
of the Ising model in a purely imaginary magnetic field, the Yang-Lee edge
singularity [7, 8], is known for a long time to correspond to the non-
Hermitian Lagrangian [9, 10]

$\mathcal{L}=\int
d^{d}x\left[\frac{1}{2}\left(\bigtriangledown\phi\right)^{2}+i(h-h_{c})\phi+\frac{1}{3}ig\phi^{3}\right].$
(1)

Exact scattering theories for two-dimensional models have also been identified
[11], that can be used to probe the ultraviolet limit most easily by employing
the thermodynamic Bethe Ansatz [12]. These techniques have also been employed
for hypothetical scattering matrices for massless Goldstone fermions
(Goldstinos) [13] and scattering matrices that reduce to them in certain
limits [14]. Despite the fact that the Mermin-Wagner theorem prevents the
validity of the Goldstone theorem in dimensions $d\leq 2$, it was argued in
[15] that for certain symmetry groups, e.g. $SO(N)$ with formally taking
$N<2$, this restriction can be circumvented so that Goldstone phases maybe be
identified in such type of non-Hermitian systems.

Rather than taking an operator content or a scattering matrix as a starting
point, one may of course also commence directly with a non-Hermitian
Lagrangian. From that perspective it is natural to try to extend techniques
and methods developed for the treatment of non-Hermitian quantum mechanics
[16, 17, 18, 19] to a quantum field theory setting. Such considerations have
been carried out for a scalar field theory with imaginary cubic self-
interaction terms [20, 21], with a Lagrangian identical to (1) but for
$h=h_{c}$ without a linear term, deformed harmonic oscillators [22], non-
Hermitian versions with a field theoretic Yukawa interaction [23, 24, 25, 26],
free fermion theories with a $\gamma_{5}$-mass term and the massive Thirring
model [27], $\mathcal{PT}$-symmetric versions of quantum electrodynamics [28,
29, 30] and other types of $\mathcal{PT}$-symmetric quantum field theories in
higher dimensions [22] than (1).

Non-Hermitian quantum field theories are also viewed as possible options to
overcome the limitations of the Standard Model of particle physics, which is
well known to fail in describing gravity and neutrino oscillations in a
consistent manner. In [31] a study was initiated to potentially resolving the
latter issue by extending the ordinary two-flavor neutrino oscillation to a
non-Hermitian PT-symmetric/pseudo Hermitian setting. A well studies simple
Hermitian extension of the Standard Model referred to as the two-Higgs-
doublet-model [32] involves a second Higgs doublet or possibly more [33]. Here
we investigate non-Hermitian variants of these models.

Here we are especially interested in complex non-Hermitian scalar field
theories and the question of how the aforementioned Goldstone phases manifest
in these theories, together with the subsequent extension to the Higgs
mechanism [1, 2, 3, 4] in Abelian and non-Abelian gauge theories. These issues
have been studied recently by various groups in different approaches, which
differ from their very onset: Given a generic action for a complex scalar
field theory of the form $\mathcal{I}=\int
d^{4}x\mathcal{L}(\phi,\phi^{\ast})$, one has two options in a Hermitian
theory to derive the equations of motion by means functional variation, either
to calculate $\delta\mathcal{I}/\delta\phi=0$ or
$\delta\mathcal{I}/\delta\phi^{\ast}=0$. Since the standard
$\mathcal{CPT}$-theorem [34] applies, the two resulting equations are the
same. In contrast, in a non-Hermitian theory one no longer has
$\mathcal{I}=\mathcal{I}^{\ast}$, so that the two equations are not only not
the same, but in addition one also has the new options
$\delta\mathcal{I}^{\ast}/\delta\phi=0$ and
$\delta\mathcal{I}^{\ast}/\delta\phi^{\ast}=0$. In the first approach, we
refer to as the ” _surface term approach_ ”, it was suggested [35, 36] to take
of only the two equations resulting from $\delta\mathcal{I}/\delta\phi=0$,
$\delta\mathcal{I}^{\ast}/\delta\phi^{\ast}=0$ and neglect the remaining two.
As the resulting equations are in general not compatible, the authors propose
to use some non-vanishing surface terms to compensate for the discrepancy. The
second approach [37] consists of taking $\delta\mathcal{I}/\delta\phi=0$ or
$\delta\mathcal{I}/\delta\phi^{\ast}=0$, and when determining the vacuum
allowing the real and imaginary parts of the complex scalar field to be also
complex. Thus in this approach the field content is doubled or re-defined.

Here we follow an approach, we refer to as the ” _pseudo-Hermitian approach_ ”
[38, 39], more aligned to the procedure pursued in non-Hermitian versions of
quantum mechanics, in which one employs so-called Dyson maps [40] to transform
a non-Hermitian Hamiltonian to a Hermitian Hamiltonian. Since the action
$\mathcal{I}$ contains a Lagrangian, rather than a Hamiltonian, we need to
first Legendre transform the complex Lagrangian $\mathcal{L}$ to a non-
Hermitian Hamiltonian $\mathcal{H}$, carry out the similarity transformation
by means of a Dyson map, while preserving equal time commutation relations, to
obtain a Hermitian Hamiltonian $\mathfrak{h}$, which we then inverse Legendre
transform to a real Lagrangian $\mathfrak{l}$

$\mathcal{L}\overset{\text{Legendre}}{\rightarrow}\mathcal{H}\overset{\text{Dyson}}{\rightarrow}\eta\mathcal{H}\eta^{-1}=\mathfrak{h}\overset{\text{Legendre}^{-1}}{\rightarrow}\mathfrak{l.}$
(2)

A consistent set of equations of motion is then obtained by functionally
varying the action $\mathfrak{s}=\int d^{4}x\mathfrak{l}(\varphi,\chi)$
involving this real Lagrangian $\mathfrak{l}$ with respect to the real field
components $\varphi$, $\chi\in\mathbb{R}$ of the complex scalar field
$\phi=1/\sqrt{2}(\varphi+i\chi)$, i.e. $\delta\mathfrak{s}/\delta\varphi=0$
and $\delta\mathfrak{s}/\delta\chi=0$. In order to perform the similarity
transformation, one needs to canonically quantize the theory first, which is
what we will do below.

In fact, this will be the only quantum aspect in our manuscript. Our
discussion here is kept classical and we will not carry out a second
quantization of the scalar fields, nor BRST quantization for the gauge fields,
etc. Supported by our analysis below and in our previous papers [38, 39], our
view here is that much insight can be gained by a classical treatment, apart
from the appeal to the quantum field theory for the canonical commutation
relations. One may of course fully analyse the theory in its quantum aspects,
but our view is that this should be performed on the transformed, i.e.
Hermitian, theory rather than directly on the non-Hermitian theory. In the
former scenario one would not expect any issues to arise as one is simply
dealing with a Hermitian system, whereas in the latter case one may encounter
a number of what we would refer to as “pseudo-problems”.

The above mentioned incompatibility of the variational principle is an example
for such a problem that already occurs on the level of the classical theory.
One may try to fix this problem directly for the non-Hermitian system or
simply consider the equivalent Hermitian system in which the problem is
entirely absent. Another well known example for a pseudo-problem in the
quantum mechanical context is for instance to take the variables $x$ and $p$,
that might appear in the definition of a non-Hermitian Hamiltonian, to be
physical observables. However, only the pseudo-Hermitian counterparts or the
Dyson mapped quantities $\eta x\eta^{-1}$ and $\eta p\eta^{-1}$ can be
observed and should be interpreted as being physical. Taking instead the
original operators $x$ and $p$ from the non-Hermitian theory one may derive
violations of the uncertainly relations and other properties that contradict
standard quantum mechanical principles. Again these problems entirely
disappear in the equivalent Hermitian system when considering the correct
variables.

Previously we have analysed the Goldstone theorem for a non-Hermitian scalar
field theory with an Abelian [38] and a non-Abelian symmetry [39]. Here the
main purpose is to investigate the extension to the Higgs mechanism. As it is
well known, in the standard Higgs mechanism the Goldstone bosons acquires a
mass so that our previous findings will inevitably have a bearing on the
investigation to be carried out here. Let us therefore recall the key finding
from [38, 39]: The main object of study has been the eigenvalue spectra of the
non-Hermitian squared mass matrix $M^{2}$, obtained by expanding around the
symmetry breaking or preserving vacua. The reality of these eigenvalues has
been guaranteed by a modified $\mathcal{CPT}$ -symmetry of the original action
$\mathcal{I}$. Hence, we distinguished in the usual fashion between
$\mathcal{CPT}$_-symmetry regime_ characterized by $M^{2}$ commuting with this
symmetry operator and its eigenstates being simultaneous eigenstates of the
symmetry operator. When the latter is not the case, one refers to that regime
as the $\mathcal{CPT}$_-spontaneously broken regime_ in which some eigenvalues
become complex conjugate pairs. The points in parameter space at which this
occurs are commonly referred to as _exceptional point_. As physical masses are
positive and real, we also require the eigenvalues of $M^{2}$ to be non-
negative. We encountered a special behaviour at the transition points when the
eigenvalues become zero, which we referred to [38, 39] as _zero exceptional
points of type I and type II_. 111The difference between these two types of
points is discussed in the appendix in form of a general discussion for key
blocks of the squared mass matrix. At the type I points the mass matrix is
non-diagonalizable and the continuous symmetry is broken, whereas at the type
II points the mass matrix can be diagonalized and the vacuum with broken
continuous symmetry re-acquires the symmetry at this point.

Using the above mentioned approaches, the Higgs mechanism was previously
studied for Abelian [37, 41] as well as non-Abelian gauge theories [36]
leading to slightly different findings. In [37] the interesting observation
made, that the mass of the gauge vector boson vanishes at the zero exceptional
point, was not confirmed in [41]. In addition, for the non-Abelian gauge
theories it was found in [36] that the Higgs mechanism even applies in the
spontaneously broken $\mathcal{CPT}$ -regime. Our aim is here to compare the
various observations made using these alternative approaches with a pseudo-
Hermitian approach, extend the studies to other models, symmetries and
representations within this framework.

Our manuscript is organised as follows: In section 2.1 we introduce first our
non-Hermitian model with scalar field taken in the fundamental representation
possessing a local $SU(2)\times U(1)$-symmetry, discuss the symmetry breaking
vacuum of this theory and subsequently the Higgs mechanism. We indicate how to
extend the model from $SU(2)$ to $SU(N)$. We repeat the same discussion in
section 2.2 for anon-Hermitian model with scalar field taken in the adjoint
representation. Our conclusions are stated in section 3. The distinction
between the two types of exceptional points is crucial and for that reason we
include in our appendix a discussion for part of the squared mass matrix that
explains the difference.

##  2 Pseudo-Hermitian approach to the Higgs mechanism

In this section we commence by investigating the same model considered in [36]
using, however, a pseudo-Hermitian method to compare our results with the
findings in [36]. We will observe that the mass spectrum of the fields in the
$SU(2)$ fundamental representation coincides with the one found in [36], but
the masses for the gauge vector bosons differ and in particular vanish at the
zero-exceptional points. We will extend this model to incorporate a
$SU(N)$-symmetry and continue to observe this phenomena also for these more
general systems. Finally we will consider a new model for which the fields are
taken in a different representation, the adjoint representation of $SU(2)$,
making similar observations.

###  2.1 A $SU(2)$-model in the fundamental representation

We start by applying the pseudo-Hermitian approach to a model with local
$SU(2)\times U(1)$-symmetry previously studied using the surface term approach
in [36]. The model corresponds to the gauged version of the one for which the
Goldstone mechanism was studied in [38]

$\mathcal{L}_{2}=\sum_{i=1}^{2}\left|D_{\mu}\phi_{i}\right|^{2}+m_{i}^{2}|\phi_{i}|^{2}-\mu^{2}(\phi_{1}^{\dagger}\phi_{2}-\phi_{2}^{\dagger}\phi_{1})-\frac{g}{4}(|\phi_{1}|^{2})^{2}-\frac{1}{4}\mathop{\mathrm{T}r}\left(F_{\mu\nu}F^{\mu\nu}\right).$
(3)

Here $g,\mu\in\mathbb{R}$, $m_{i}\in\mathbb{R}$ or $m_{i}\in i\mathbb{R}$ are
constants. When compared to [38] we have replaced here as usual the standard
derivatives $\partial_{\mu}$ by covariant derivatives
$D_{\mu}:=\partial_{\mu}-ieA_{\mu}$, involving a charge $e\in\mathbb{R}$ and
the Lie algebra valued gauge fields $A_{\mu}:=\tau^{a}A_{\mu}^{a}$. Here the
$\tau^{a}$, $a=1,2,3$, are taken to be Pauli matrices, which when re-defined
as $i(-1)^{a+1}\tau^{a}$ are the generators of $SU(2)$. We have also added the
standard Yang-Mills term comprised of the Lie algebra valued field strength
$F_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ie[A_{\mu},A_{\nu}]$.
The two complex scalar fields $\phi_{i}$ are taken to be in the representation
space of fundamental representation of $SU(2)$. The model described by
$\mathcal{L}_{2}$ admits a global continuous $U(1)$-symmetry, a local
continuous $SU(2)$-symmetry and two discrete antilinear
$\mathcal{CPT}$-symmetries as described in more detail in [38]. Crucially
$\mathcal{L}_{2}$ is not Hermitian, which at this point is simply to be
understood as not being invariant under complex conjugation. The Abelian
version of $\mathcal{L}_{2}$ was discussed in [37, 39].

As argued in [38], it is useful to decompose the complex fields into their
real components $\phi_{j}^{k}=1/\sqrt{2}(\varphi_{j}^{k}+i\chi_{j}^{k})$ with
$\varphi_{j}^{k}$, $\chi_{j}^{k}\in\mathbb{R}$. Thus simply rewriting the
complex scalar fields in equation (2.1) in terms of their real and imaginary
components we obtain the following Lagrangian

$\displaystyle\mathcal{L}_{2}=\frac{1}{2}\sum_{k=1}^{2}\sum_{j=1}^{2}\left\\{\left[\partial_{\mu}\varphi_{j}^{k}+e(A_{\mu}\chi_{j})^{k}\right]\left[\partial^{\mu}\varphi_{j}^{k}+e(A^{\mu}\chi_{j})^{k}\right]^{\ast}\right.$
(4)
$\displaystyle+\left[\partial_{\mu}\chi_{j}^{k}-e(A_{\mu}\varphi_{j})^{k}\right]\left[\partial^{\mu}\chi_{j}^{k}-e(A^{\mu}\varphi_{j})^{k}\right]^{\ast}\\!\\!\\!-2\\!\mathop{\mathrm{I}m}\left[\left[\partial_{\mu}\varphi_{j}^{k}+e(A_{\mu}\chi_{j})^{k}\right]^{\ast}\\!\left[\partial^{\mu}\chi_{j}^{k}-e(A^{\mu}\varphi_{j})^{k}\right]\right]$
$\displaystyle+\left.m_{j}^{2}\left[(\varphi_{j}^{k})^{2}+(\chi_{j}^{k})^{2}\right]-2i\mu^{2}(\varphi_{1}^{k}\chi_{2}^{k}-\chi_{1}^{k}\varphi_{2}^{k})-\frac{1}{4}F_{\mu\nu}^{k}\left(F^{k}\right)^{\mu\nu}\right\\}-\frac{g}{16}\left[\sum_{k=1}^{2}(\varphi_{1}^{k})^{2}+(\chi_{1}^{k})^{2}\right]^{2}.$

We use here the standard notation * for complex conjugation and $\dagger$ for
the simultaneous conjugation with transposition.

The $SU(2)$-symmetry manifests itself as follows: A change in the complex
scalar fields due to this symmetry is
$\delta\phi_{j}^{k}=i\alpha_{a}T_{a}^{kl}\phi_{j}^{l}$, where the generators
$T_{a}$ of the symmetry transformation are the standard Pauli matrices
$\sigma_{a}$, $a=1,2,3$. The infinitessimal changes for the real component
fields are then identified as

$\displaystyle\delta\varphi_{j}^{1}$ $\displaystyle=$
$\displaystyle-\alpha_{1}\chi_{j}^{2}+\alpha_{2}\varphi_{j}^{2}-\alpha_{3}\chi_{j}^{1},\quad\delta\chi_{j}^{1}=\alpha_{1}\varphi_{j}^{2}+\alpha_{2}\chi_{j}^{2}+\alpha_{3}\varphi_{j}^{1},$
(5) $\displaystyle\delta\varphi_{j}^{2}$ $\displaystyle=$
$\displaystyle-\alpha_{1}\chi_{j}^{1}-\alpha_{2}\varphi_{j}^{1}+\alpha_{3}\chi_{j}^{2},\quad\delta\chi_{j}^{2}=\alpha_{1}\varphi_{j}^{1}-\alpha_{2}\chi_{j}^{1}-\alpha_{3}\varphi_{j}^{2},$
(6)

which leave the above Lagrangian invariant. The discrete antilinear
$\mathcal{CPT}_{\pm}$-symmetries manifest themselves as

$\mathcal{CPT}_{\pm}:~{}\varphi_{j}^{k}(x_{\mu})\rightarrow\mp(-1)^{j}\varphi_{j}^{k}(-x_{\mu}),~{}~{}\chi_{j}^{k}(x_{\mu})\rightarrow\pm(-1)^{j}\chi_{j}^{k}(-x_{\mu}).$
(7)

A noteworthy remark is that it is straightforward to generalise the model from
a locally $SU(2)$-invariant one to a locally $SU(N)$-invariant one, by
extending the sum over $k$ from $2$ to $N$, while keeping the $U(1)$-symmetry
global. In what follow we will focus on $N=2$.

A crucial feature of $\mathcal{L}_{2}$ is that its $\mathcal{CPT}$-invariance
translates into pseudo-Hermiticity [42, 18], meaning that it can be mapped to
a Hermitian Lagrangian $\mathfrak{l}_{2}$ by means of the adjoint action of a
Dyson map $\eta$ as
$\mathfrak{l}_{{\color[rgb]{1,0,0}2}}=\eta\mathcal{L}_{2}\eta^{-1}$. This may
be achieved by the slightly modified version of the Dyson map used in [37, 38]

$\eta_{2}^{\pm}=\exp\left(\pm\sum_{i=1}^{2}\int
d^{3}x\Pi^{\varphi_{2}^{i}}(t^{\prime},\vec{x})\varphi_{2}^{i}(t^{\prime},\vec{x})+\Pi^{\chi_{2}^{i}}(t^{\prime},\vec{x})\chi_{2}^{i}(t^{\prime},\vec{x})\right).$
(8)

We denote here the time-dependence by $t^{\prime}$ to indicate that
commutators are understood as equal time commutators for the canonical momenta
$\Pi^{\varphi_{2}^{i}}=\partial_{t}\varphi_{2}^{i}$ and
$\Pi^{\chi_{2}^{i}}=\partial_{t}\chi_{2}^{i}$, $i=1,2$. satisfying
$\left[\psi_{j}^{k}(\mathbf{x},t),\Pi^{\psi_{l}^{m}}(\mathbf{y},t)\right]=i\delta_{jl}\delta_{km}\delta(\mathbf{x}-\mathbf{y})$,
$j,k,l,m=1,2$, for $\psi=\varphi,\chi$.

Hence $\eta_{2}^{\pm}$ is not to be viewed as explicitly time-dependent as
discussed in much detail for instance in [43]. The adjoint action of
$\eta_{2}^{+}$ on the individual fields maps as

$\varphi_{1}^{k}\rightarrow\varphi_{1}^{k}~{},~{}~{}\varphi_{2}^{k}\rightarrow-i\varphi_{2}^{k}~{},~{}~{}\chi_{1}^{k}\rightarrow\chi_{1}^{k}~{},~{}~{}\chi_{2}^{k}\rightarrow-i\chi_{2}^{k}~{},~{}~{}A_{\mu}\rightarrow
A_{\mu}~{},~{}~{}~{}~{}k=1,2.$ (9)

Thus, we convert the complex Lagrangian into the real Lagrangian

$\displaystyle\mathfrak{l}_{2}=\frac{1}{2}\sum_{j=1}^{2}(-1)^{j+1}\left\\{\left|\partial_{\mu}\varphi_{j}+e(A_{\mu}\chi_{j})\right|^{2}+\left|\partial_{\mu}\chi_{j}-e(A_{\mu}\varphi_{j})\right|^{2}+m_{j}^{2}\left[\varphi_{j}\cdot\varphi_{j}+\chi_{j}\cdot\chi_{j}\right]\right.~{}~{}~{}~{}~{}~{}$
(10)
$\displaystyle-\left.2\\!\mathop{\mathrm{I}m}\left[\left[\partial_{\mu}\varphi_{j}+e(A_{\mu}\chi_{j})\right]^{\ast}\\!\cdot\left[\partial^{\mu}\chi_{j}-e(A^{\mu}\varphi_{j})\right]\right]+(-1)^{j}2\mu^{2}(\varphi_{1}\cdot\chi_{2}-\chi_{1}\cdot\varphi_{2})\right\\}$
$\displaystyle-\frac{g}{16}\left[\varphi_{1}\cdot\varphi_{1}+\chi_{1}\cdot\chi_{1}\right]^{2}-\frac{1}{4}\mathop{\mathrm{T}r}\left(F_{\mu\nu}F^{\mu\nu}\right).$

We may transform here directly the Lagrangian rather than the Hamiltonian, as
suggested in (2), since the kinetic energy term is real and the complexity
only result from the potential term. Introducing the $2$ two-component fields
of the form

$\Phi^{k}:=\left(\begin{array}[]{c}\varphi_{1}^{k}\\\
\chi_{2}^{k}\end{array}\right)~{},~{}~{}\Psi^{k}:=\left(\begin{array}[]{c}\chi_{1}^{k}\\\
\varphi_{2}^{k}\end{array}\right),~{}~{}~{}\ ~{}~{}~{}k=1,2,$ (11)

we can re-write the Lagrangians $\mathcal{L}_{2}$ and $\mathfrak{l}_{2}$ more
compactly. Defining the $2\times 2$ matrices

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

the real Lagrangian $\mathfrak{l}_{2}$ acquires the form

$\displaystyle\mathfrak{l}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\left[\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi\right]^{\ast}\mathcal{I}\left[\partial^{\mu}\Phi+e\mathcal{I}A^{\mu}\Psi\right]+\left[\partial_{\mu}\Psi-e\mathcal{I}A_{\mu}\Phi\right]^{\ast}\mathcal{I}\left[\partial^{\mu}\Psi-e\mathcal{I}A^{\mu}\Phi\right]\right.$
$\displaystyle-2\mathop{\mathrm{I}m}\left[\left[\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi\right]^{\ast}\left[\partial^{\mu}\Psi-e\mathcal{I}A^{\mu}\Phi\right]\right]\left.-{\Phi}^{T}H_{+}\Phi-{\Psi}^{T}H_{-}\Psi\right\\}$
$\displaystyle-\frac{g}{16}\left({\Phi}^{T}E\Phi+{\Psi}^{T}E\Psi\right)^{2}-\frac{1}{4}\mathop{\mathrm{T}r}\left(F_{\mu\nu}F^{\mu\nu}\right).$

We have simplified here the index notation by implicitly contracting, keeping
in mind that we are summing over two separate index sets $k\in\\{1,2\\}$ and
$j\in\\{1,2\\}$. For instance, we set

$\displaystyle(IA_{\mu}\Phi)_{\alpha}^{k}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{I}_{\alpha\beta}A_{\mu}^{kj}\Phi_{\beta}^{j},$ (14)
$\displaystyle\left[\partial_{\mu}\Phi_{j}^{k}+e\left(\mathcal{I}A_{\mu}\Psi\right)_{j}^{k}\right]^{\ast}\mathcal{I}_{j\ell}\left[\partial^{\mu}\Phi_{\ell}^{k}+e\left(\mathcal{I}A^{\mu}\Psi\right)_{\ell}^{k}\right]$
$\displaystyle\rightarrow$
$\displaystyle\left[\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi\right]^{\ast}\mathcal{I}\left[\partial^{\mu}\Phi+e\mathcal{I}A^{\mu}\Psi\right],~{}~{}~{}~{}~{}~{}~{}$
(15) $\displaystyle{\Phi^{k}}^{T}H_{+}\Phi^{k}$ $\displaystyle\rightarrow$
$\displaystyle{\Phi}^{T}H_{+}\Phi.$ (16)

In this formulation we may think of the real and complex Lagrangians,
$\mathfrak{l}_{2}$ and $\mathcal{L}_{2}$, as being simply related by a kind of
Wick rotation in the field-configuration space

$\Phi^{k}\rightarrow T\Phi^{k}~{},~{}~{}\Psi^{k}\rightarrow
T\Psi^{k},~{}~{}\text{with }T:=\left(\begin{array}[]{cc}1&0\\\
0&-i\end{array}\right).$ (17)

One may of course wonder about the negative sign in the kinetic term of
$\mathfrak{l}_{2}$ and whether these lead to ghost fields. Thus we finish this
subsection with a short discussion that establishes that these signs disappear
when the Lagrangian is properly diagonalised. As this argument is the same in
the global and local theory, we set the gauge fields to zero without loss of
generality for this purpose and consider the corresponding action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x\left(\partial_{\mu}\Phi^{\top}\mathcal{I}\partial^{\mu}\Phi-\Phi^{\top}H_{\pm}\Phi\right)+S_{\text{int}},$
(18) $\displaystyle=$ $\displaystyle-\frac{1}{2}\int
d^{4}x\Phi^{\top}\mathcal{I}\left(\partial_{\mu}\partial^{\mu}+M^{2}\right)\Phi+S_{\text{int}},$
(19)

where $S_{\text{int}}$ contains all terms of higher order than $\Phi^{2}$. We
also assumed that surface terms vanish at infinity when integrating by parts
and used $\mathcal{I}^{2}=\mathbb{I}$, $M^{2}=\mathcal{I}H_{\pm}$. Next we
diagonalise the squared mass matrix as $M^{2}=T^{-1}DT$ and consider next only
the integrant of the first term in (19)

$\Phi^{\top}\mathcal{I}\left(\partial_{\mu}\partial^{\mu}+M^{2}\right)\Phi=\Phi^{\top}\mathcal{I}\left(\partial_{\mu}\partial^{\mu}+TDT^{-1}\right)\Phi=\Psi^{\top}\left(\partial_{\mu}\partial^{\mu}+D\right)\Psi,$
(20)

where we introduced the new field $\Psi:=$ $T^{\top}\mathcal{I}\Phi$ and used
$T^{-1}=T^{\top}\mathcal{I}$.

The latter relation is derived as follows: We start by defining the right and
left eigenvectors of $M^{2}$ as

$\mathcal{I}H_{\pm}v_{i}=\lambda_{i}v_{i},\qquad\text{and}\qquad(\mathcal{I}H_{\pm})^{\dagger}u_{i}=\lambda_{i}u_{i},$
(21)

respectively. Noting that $\mathcal{I}^{\dagger}=\mathcal{I}$ and
$H_{\pm}^{\dagger}=H_{\pm}$ the last relation implies that
$\mathcal{I}H_{\pm}\mathcal{I}u_{i}=\lambda_{i}\mathcal{I}u_{i}$. Therefore we
can express the right eigenvectors in terms of the left eigenvectors as
$v_{i}=\mathcal{I}u_{i}$. The matrix $T$ is made out of the column vectors of
$v_{i}$, i.e. $T=(v_{1},\ldots)$ so that $\mathcal{I}T=(u_{1},\ldots)$. Since
the left and right eigenvector form a biorthonormal basis, $v_{i}\cdot
u_{j}=\delta_{ij}$, it follows that $T^{\top}\mathcal{I}T=\mathbb{I}$ and
hence $T^{-1}=T^{\top}\mathcal{I}$.

####  2.1.1 The symmetry breaking vacuum

The vacuum solutions $\Phi_{0}^{k},\Psi_{0}^{k}$ by solving $\delta V=0$, with
$V$ denoting the potential

$V={\Phi}^{\top}H_{+}\Phi+{\Psi}^{\top}H_{-}\Psi+\frac{g}{16}\left({\Phi}^{\top}E\Phi+{\Psi}^{\top}E\Psi\right)^{2}$
(22)

amounts to solving the two equations

$\left(H_{-}+\frac{g}{4}R^{2}E\right)\Psi_{0}^{k}=0~{},~{}~{}\left(H_{+}+\frac{g}{4}R^{2}E\right)\Phi_{0}^{k}=0~{},~{}~{}~{}\
~{}~{}~{}k=1,2,$ (23)

with
$R^{2}:=\left|\left(\phi_{1}^{0}\right)^{1}\right|^{2}+\left|\left(\phi_{1}^{0}\right)^{2}\right|^{2}=\frac{1}{2}\sum_{k=1}^{2}{\Phi_{0}^{k}}^{T}E\Phi_{0}^{k}+{\Psi_{0}^{k}}^{T}E\Psi_{0}^{k}$=const.
Hence in the real component field configuration space the vacuum manifold is a
$S^{3}$-sphere with radius $R$. Consequently, we may consider the equations
(23) as two eigenvalue equations. Thus, besides the trivial $SU(2)$-invariant
vacuum $\Phi_{0}^{k}=\Psi_{0}^{k}=0$, $k=1,2$, we must have zero eigenvalues
in both equations, which is equivalent to requiring

$R^{2}=\frac{4}{gm_{2}^{2}}(\mu^{4}+m_{1}^{2}m_{2}^{2}).$ (24)

Since $R^{2}$ is positive, this equality imposes restrictions on the
parameters $g,\mu$ and the possible choices for $m_{1}\in\mathbb{R}$,
$m_{2}\in i\mathbb{R}$ or $m_{1}\in i\mathbb{R}$, $m_{2}\in\mathbb{R}$. The
corresponding vectors that satisfy equation) (23), suitably normalized with
regard to the standard inner product, are

$\Psi_{0}^{2}=N_{\Psi}\left(\begin{array}[]{c}m_{2}^{2}\\\
\mu^{2}\end{array}\right),~{}~{}\
\Phi_{0}^{2}=N_{\Phi}\left(\begin{array}[]{c}-m_{2}^{2}\\\
\mu^{2}\end{array}\right).$ (25)

Imposing now the constraint on $R^{2}$ as stated after equation (23), a
possible solution is $\Phi_{0}^{1}=\Psi_{0}^{1}=\Phi_{0}^{2}=0$ and
$\Psi_{0}^{2}$ as defined in (25) with normalization constant
$N_{\Psi}=\pm\sqrt{2}R/m_{2}^{2}$. Hence we recover the symmetry breaking
vacuum used in [39].

####  2.1.2 The Higgs mechanism

Let us now demonstrate how the gauge vector boson acquires a finite mass and
how at the same time the emergence of a Goldstone boson is prevented by the
Higgs mechanism [1, 2, 3, 4] in the $\mathcal{CPT}$-symmetric regime, at the
exceptional points and even in the spontaneously broken
$\mathcal{CPT}$-symmetric regime. The mechanism breaks down at the two types
of zero exceptional points.

Expanding the potential $V$ in (22) around the vacuum specified at the end of
subsection 2.1.1 leads to

$\displaystyle V(\Phi_{0}+\Phi,\Psi_{0}+\Psi)$ $\displaystyle=$ $\displaystyle
V(\Phi_{0},\Psi_{0})+\frac{1}{2}\Phi^{i}\left.\frac{\partial^{2}V(\Phi_{0},\Psi_{0})}{\partial\Phi^{i}\partial\Phi^{j}}\right|\Phi^{j}+\frac{1}{2}\Psi^{i}\left.\frac{\partial^{2}V(\Phi_{0},\Psi_{0})}{\partial\Psi^{i}\partial\Psi^{j}}\right|\Psi^{j}\qquad$
$\displaystyle+\Phi^{i}\left.\frac{\partial^{2}V(\Phi_{0},\Psi_{0})}{\partial\Phi^{i}\partial\Psi^{j}}\right|\Psi^{j}+\dots$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}-{\Phi^{i}}^{\top}\left(H_{+}+\frac{g}{4}R^{2}E\right)\Phi^{i}-{\Psi^{1}}^{\top}\left(H_{-}+\frac{g}{4}R^{2}E\right)\Psi^{1}$
$\displaystyle-{\Psi^{2}}^{\top}\left[H_{-}+\frac{g}{4}R^{2}E+-\frac{g}{2}(E\Psi_{0}^{2})^{2}E\right]\Psi^{2}+\dots$

As expected, multiplying the Hessians in (2.1.2) by $\mathcal{I}$ gives back
the squared mass matrix we found in [39]. The kinetic term is almost unchanged
except for the term involving $\Psi^{2}$

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi\right]^{\dagger}\mathcal{I}\left[\partial^{\mu}\Phi+e\mathcal{I}A^{\mu}\Psi\right]+\mathop{\mathrm{R}e}\left\\{\left(\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi\right)^{\dagger}\mathcal{I}\left(e\mathcal{I}A^{\mu}\Psi_{0}\right)\right\\}$
$\displaystyle-\mathop{\mathrm{I}m}\left\\{\left(\partial_{\mu}\Phi+e\mathcal{I}A_{\mu}\Psi+e\mathcal{I}A_{\mu}\Psi_{0}\right)^{\dagger}\left(\partial^{\mu}\Psi-e\mathcal{I}A^{\mu}\Phi\right)\right\\}+\frac{1}{2}e^{2}{\left(A_{\mu}\Psi_{0}\right)}^{\dagger}\mathcal{I}\left(A^{\mu}\Psi_{0}\right)$

The last term corresponds to the mass term of the gauge vector boson that we
evaluate to

$\displaystyle\frac{1}{2}e^{2}{\left(A_{\mu}\Psi_{0}\right)^{\ast}}\mathcal{I}\left(A^{\mu}\Psi_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}{\left(A_{\mu}\Psi_{0}\right)^{\ast}}_{\alpha}^{k}\mathcal{I}_{\alpha\beta}{\left(A^{\mu}\Psi_{0}\right)}_{\beta}^{k}$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}\left(A_{\mu}^{\dagger}A^{\mu}\right)^{kj}\left({\Psi_{0}}\right)_{\alpha}^{k}\mathcal{I}_{\alpha\beta}\left({\Psi_{0}}\right)_{\beta}^{j}$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}\left(A_{\mu}^{\dagger}A^{\mu}\right)^{22}\left({\Psi_{0}}\right)_{\alpha}^{2}\mathcal{I}_{\alpha\beta}\left({\Psi_{0}}\right)_{\beta}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{2}A_{\mu}^{a}A^{b\mu}(\tau^{a\dagger}\tau^{b})^{22}\frac{2R^{2}}{m_{2}^{4}}\left(m_{2}^{4}-\mu^{4}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}m_{g}^{2}A_{\mu}^{a}A^{a\mu},$ (30)

where we used the standard relation
$\tau^{a\dagger}\tau^{b}=\tau^{a}\tau^{b}=\delta_{ab}\mathbb{I}+i\varepsilon_{abc}\tau^{c}$.
Therefore we read off the mass of each of the three components of the gauge
vector boson as

$m_{g}:=\frac{\sqrt{2}eR}{m_{2}^{2}}\sqrt{m_{2}^{4}-\mu^{4}}.$ (31)

In [39] we identified the physical regions in the parameter space in which the
squared mass matrix has non-negative eigenvalues and in which the Goldstone
bosons can be identified. Let us now compare those regions with the values for
which the gauge vector boson becomes massive. We immediately see from the
expression in (31) that the gauge vector boson remains massless when
$\mu^{4}=m_{2}^{4}$ or when $R=0$, i.e. $\mu^{4}=-m_{1}^{2}m_{2}^{2}$. The two
sets of values correspond precisely to the two types of zero exceptional
points, type I and II, respectively, at which the squared mass matrix develops
zero eigenvalues $\lambda=0$. These points are distinct from standard
exceptional points where two eigenvalues coalesce and become complex
thereafter, here at $\lambda=\frac{\mu^{4}}{m_{2}^{2}}-m_{2}^{2}$. See the
appendix for a more detailed explanation about the distinction between these
types of exceptional points.

Thus the two aspects of the Higgs-mechanism, i.e. giving mass to the gauge
vector boson and at the same time preventing the existence of the Goldstone
bosons, remain to go hand in hand. In the $\mathcal{CPT}$-symmetric regime the
mechanism applies, but at the zero exceptional points the Higgs-mechanism
breaks down as the Goldstone bosons are not identifiable [39] and at the same
time the gauge vector boson remains massless. In contrast, at the exceptional
point the Goldstone bosons are identifiable [39], although in a different
manner, and the gauge vector bosons become massive.

Let us see this in detail by following [39] and replacing
$m_{i}^{2}\rightarrow c_{i}m_{i}^{2}$, with $c_{i}=\pm 1$ to account for all
possibilities in signs. We found that physical regions only exist for the two
cases $c_{1}=-$ $c_{2}=1$ and $c_{1}=-$ $c_{2}=-1$. For the two cases we may
then write

$\frac{m_{g}^{2}}{m_{1}^{2}}=c_{2}\frac{4e^{2}}{g}\frac{m_{1}^{6}}{m_{2}^{6}}\left(\frac{m_{2}^{4}}{m_{1}^{4}}-\frac{\mu^{4}}{m_{1}^{4}}\right)\left(\frac{\mu^{4}}{m_{1}^{4}}-\frac{m_{2}^{2}}{m_{1}^{2}}\right),$
(32)

noting that $m_{g}^{2}/m_{1}^{2}$ only depends on the two parameters
$m_{2}^{2}/m_{1}^{2}$ and $m_{2}^{4}/m_{1}^{4}$ similarly as the eigenspectrum
of the squared mass matrix [36, 39]. We require the right hand side of (32) to
be positive as depicted in depict in figure 1.

Figure 1: Regions, for which the gauge vector boson is massive (blue with
mesh) versus physical regions (orange) in which the would be Goldstone boson
can be identified, bounded by exceptional and zero exceptional points as
function of $(\mu^{4}/m_{1}^{4},m_{2}^{2}/m_{1}^{2})$ for the theory expanded
around the SU(2)-symmetry breaking vacuum. Left panel for $c_{1}=-c_{2}=1$ and
right panel for $c_{1}=-c_{2}=-1$. The coupling constant $g$ must be positive.

We observe in figure 1 that while the region in which the Goldstone boson can
be identified is bounded by exceptional as well as zero exceptional points,
the exceptional points lie well inside the region for which the gauge vector
boson is massive, i.e. they acquire a mass in the $\mathcal{CPT}$-symmetric
regime as well as in the spontaneously broken $\mathcal{CPT}$-symmetric
regime. In the $\mathcal{CPT}$-symmetric regime this agrees well with the
findings that at these points the “would be Goldstone boson” is prevented from
existing as a massless particle. We may think of the sign change in front of
the mass terms, $c_{i}\rightarrow-c_{i}$, that relates the left to the right
panel as a phase transition [44].

Let us now demonstrate this behaviour in detail and expand for this purpose
the Lagrangian around the symmetry broken vacuum up to second order in the
fields

$\displaystyle\mathfrak{l}_{2}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\frac{1}{2}\partial_{\mu}{\Phi^{k}}^{T}\mathcal{I}\partial^{\mu}\Phi^{k}+\frac{1}{2}\partial_{\mu}{\Psi^{k}}^{T}\mathcal{I}\partial^{\mu}\Psi^{k}-\frac{1}{2}{\Phi^{k}}^{T}\left(H_{+}+\frac{g}{4}R^{2}E\right)\Phi^{k}$
$\displaystyle-\frac{1}{2}{\Psi^{1}}^{T}\left(H_{-}+\frac{g}{4}R^{2}E\right)\Psi^{1}-\frac{1}{2}{\Psi^{2}}^{T}\left(H_{+}+\frac{g}{4}R^{2}E+\frac{g}{2}(E\Psi_{(0)}^{2})^{2}E\right)\Psi^{2}$
$\displaystyle+e\mathop{\mathrm{R}e}\left[\partial_{\mu}{\Phi}^{\dagger}(A^{\mu}\Psi_{0})\right]+e\mathop{\mathrm{I}m}\left[\left(\mathcal{I}A_{\mu}\Psi_{0}\right)^{\dagger}\partial^{\mu}\Psi\right]+\frac{1}{2}m_{g}^{2}A_{\mu}^{a}A^{a\mu}+\dots$

We recall now from [39] that the first two lines of the Lagrangian
$\mathfrak{l}_{2}$ can be diagonalized and the Goldstone bosons can be
identified in terms of the field content of the model. Furthermore, the
Goldstone modes are eigenvectors in the null space of squared mass matrices

$M_{\pm}^{2}:=\mathcal{I}\left(H_{\pm}+\frac{g}{4}R^{2}E\right)~{},$ (34)

computed above as $\Psi_{0}^{2}$ and $\mathcal{I}\Psi_{0}^{2}$, so that the
Goldstone modes are proportional to these two vectors. The explicit forms of
the Goldstone fields were found in [38] (equation (3.40) therein), denoted as
$\psi_{5}^{\text{Gb}}$, $\psi_{3}^{\text{Gb}}$and $\psi_{1}^{\text{Gb}}$,
therein. We express them here as

$G^{1}:=\frac{e}{m_{g}}\left({\Psi_{0}^{2}}\right)^{T}\Phi^{1}~{},~{}~{}G^{3}:=\frac{e}{m_{g}}\left({\Psi_{0}^{2}}\right)^{T}\Phi^{2}~{},~{}~{}G^{2}:=-\frac{e}{m_{g}}\left({\Psi_{0}^{2}}\right)^{T}\mathcal{I}\Psi^{1},$
(35)

respectively. As expected for the Higgs mechanism the number of “would be
Goldstone bosons” equals the amount of massive vector gauge bosons. The fact
that the Goldstone modes are inverse proportional to the mass of the gauge
bosons explains that they can not be identified for massless gauge bosons.
Keeping now only the Goldstone kinetic term from the first two lines of the
Lagrangian $\mathfrak{l}_{2}$ and the one involving the gauge fields in
equation (2.1.2), we obtain

$\mathfrak{l}_{2}=\sum_{a=1}^{3}\frac{1}{2}\partial_{\mu}G^{a}\partial^{\mu}G^{a}+e\mathop{\mathrm{R}e}\left[\partial_{\mu}{\Phi}^{\dagger}(A^{\mu}\Psi_{0})\right]+e\mathop{\mathrm{I}m}\left[\left(\mathcal{I}A_{\mu}\Psi_{0}\right)^{\dagger}\partial^{\mu}\Psi\right]+\frac{1}{2}m_{g}^{2}A_{\mu}^{a}A^{a\mu}+\dots$
(36)

Using the explicit representations of the Pauli matrices, the real and
imaginary parts are determined as

$\displaystyle\mathop{\mathrm{R}e}\left[\partial_{\mu}\Phi^{T}A^{\mu}\Psi_{0}\right]$
$\displaystyle=$ $\displaystyle
A_{\mu}^{a}\mathop{\mathrm{R}e}\left[\partial_{\mu}\Phi^{T}\tau^{a}\Psi_{0}\right]=A_{\mu}^{1}\partial_{\mu}\Phi^{T}\tau^{1}\Psi_{0}+A_{\mu}^{3}\partial_{\mu}\Phi^{T}\tau^{3}\Psi_{0}$
$\displaystyle=$ $\displaystyle
A_{\mu}^{1}\partial_{\mu}\left({\Phi^{1}}\right)^{T}{\Psi_{0}^{2}}-A_{\mu}^{3}\partial_{\mu}\left({\Phi^{2}}\right)^{T}{\Psi_{0}^{2}}$
$\displaystyle=$ $\displaystyle
A_{\mu}^{1}\frac{m_{g}}{e}\partial^{\mu}G^{1}-\frac{m_{g}}{e}A_{\mu}^{3}\partial^{\mu}G^{3}$
$\displaystyle\mathop{\mathrm{I}m}\left[\left(\mathcal{I}A_{\mu}\Psi_{0}\right)^{\dagger}\partial^{\mu}\Psi\right]$
$\displaystyle=$ $\displaystyle
A_{\mu}^{a}\mathop{\mathrm{I}m}\left[\Psi_{0}^{T}\tau^{a}\mathcal{I}\partial^{\mu}\Psi\right]=-i\left(A_{\mu}^{2}\Psi_{0}^{T}\tau^{2}\mathcal{I}\partial^{\mu}\Psi\right)$
$\displaystyle=$ $\displaystyle
A_{\mu}^{2}\left({\Psi_{0}^{2}}\right)^{T}\mathcal{I}\partial^{\mu}\Psi^{1}=-A_{\mu}^{2}\frac{m_{g}}{e}\partial^{\mu}G^{2}.$

Finally the Lagrangian in (36) can be simplified to

$\displaystyle\mathfrak{l}_{2}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{3}\frac{1}{2}\partial_{\mu}G^{a}\partial^{\mu}G^{a}-m_{g}A_{\mu}^{1}\partial^{\mu}G^{1}+m_{g}A_{\mu}^{2}\partial^{\mu}G^{2}-m_{g}A_{\mu}^{3}\partial^{\mu}G^{3}+\frac{1}{2}m_{g}^{2}A_{\mu}^{a}A^{a\mu}+\dots$
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{g}^{2}\left(A_{\mu}^{1}-\frac{1}{m_{g}}\partial_{\mu}G^{1}\right)^{2}+\frac{1}{2}m_{g}^{2}\left(A_{\mu}^{2}+\frac{1}{m_{g}}\partial_{\mu}G^{2}\right)^{2}+\frac{1}{2}m_{g}^{2}\left(A_{\mu}^{3}+\frac{1}{m_{g}}\partial_{\mu}G^{3}\right)^{2}+\dots$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{a=1}^{3}m_{g}^{2}B_{\mu}^{a}B^{a\mu}+\dots,$

where we defined the new vector gauge particle with component fields
$B_{\mu}^{a}:=A_{\mu}^{a}-\frac{1}{m_{g}}\partial_{\mu}G^{a}$. We may also
replace $A_{\mu}^{a}$ by $B_{\mu}^{a}$ in the field strength $F_{\mu\nu}$ so
that $A_{\mu}$ can be eliminated entirely from the Lagrangian. We see that the
Higgs-mechanism applies as long as $m_{g}\not=0$. However, at the zero
exceptional points, not only the gauge boson mass vanishes, but the Higgs
mechanism no longer applies, in the sense that we can not remove the degrees
of freedom of Goldstone bosons.

We summarize the behaviour we found in the different types of regimes in the
following table

| $\mathcal{CPT}$ | sp. broken $\mathcal{CPT}$ | EP | zero EP I | zero EP II
---|---|---|---|---|---
gauge bosons | massive | massive | massive | massless | massless
Goldstone bosons | $\exists$ | $\exists$ | $\exists$ | $\nexists$ | $\nexists$

Thus we encounter three different types of behaviour: In the
$\mathcal{CPT}$-symmetric regime, at the standard exceptional points as well
as in the spontaneously broken $\mathcal{CPT}$-symmetric regime the Higgs
mechanism applies in the usual way. However, in the latter regime other
particles in the theory become non-physical. At the zero exceptional points
the vector gauge bosons remain massless and no Goldstone bosons can be
identified in the global theory.

####  2.1.3 From $SU(2)$ to $SU(N)$

We will now follow the same line of reasoning as in the previous subsection
and generalize our model from possessing a $SU(2)$-symmetry to one with a
$SU(N)$-symmetry. For this purpose we simply replace the Pauli matrices in all
our expressions by the traceless and skew-Hermitian $N\times N$-matrices
corresponding to the $SU(N)$-generators $T^{a}$ with $a=1,\ldots,(N^{2}-1)$.
The vacua are still determined by the solutions of the eigenvalue problem (23)
with zero eigenvalue condition

$R^{2}=\frac{1}{2}\sum_{i=1}^{N}{\Phi_{0}^{i}}^{T}E\Phi_{0}^{i}+{\Psi_{0}^{i}}^{T}E\Psi_{0}^{i}=\text{constant}=\frac{4}{gm_{2}^{2}}(\mu^{4}+m_{1}^{2}m_{2}^{2}).$
(40)

The zero eigenvalue condition implies that the vacuum manifold is a
$S^{2N-1}$-sphere with radius $R$. This follows from the fact that $SU(N)$
acts on the $2N$ dimensional space spanned by $(\varphi_{1}^{0})^{i}$,
$(\chi_{1}^{0})^{i},i=1,\ldots,N$, with norm equal to $R^{2}$. On this space
$SU(N-1)$ simply permutes the fields amongst themselves, hence acting as a
stabilizer or isotropy subgroup. Thus the vacuum manifold corresponds to the
coset $SU(N)/SU(N-1)\cong S^{2N-1}$.

As we discussed in detail in [39], we may utilize the symmetry of the
Lagrangian to transform the vacua into convenient forms without changing the
eigenvalue spectrum of the mass matrix. Thus using the elements $T\in
SU(N)/SU(N-1)\subset SU(N)$ we may transform the vacuum into the form

$\Phi_{0}^{i}=0,~{}~{}~{}~{}\Psi_{0}^{i}=\frac{\sqrt{2}R}{\sqrt{N}m_{2}^{2}}\left(\begin{array}[]{c}m_{2}^{2}\\\
\mu^{2}\end{array}\right),~{}~{}~{}\ \text{for }i=1,\ldots,N,$ (41)

satisfying the constraint (40). Let us now use this $SU(N)$-symmetry breaking
vacuum to calculate the mass of the gauge vector boson. Taking the proper
$SU(N)$-algebra rather than the physicist’s version, as in the last subsection
for $SU(2)$, we also change $e\rightarrow ie$. Dropping here the kinetic term
reported in (2.1.2) and considering only the relevant term in the Lagrangian
we obtain

$\displaystyle\mathfrak{l}_{A}$ $\displaystyle:$
$\displaystyle=\frac{1}{2}e^{2}A_{\mu}^{a}A^{b\mu}\left({T^{a}}^{\dagger}T^{b}\right)_{ij}\left({\Psi_{0}}\right)_{\alpha}^{i}\mathcal{I}_{\alpha\beta}\left({\Psi_{0}}\right)_{\beta}^{j}$
$\displaystyle=$
$\displaystyle\frac{1}{N}e^{2}A_{\mu}^{a}A^{b\mu}R^{2}\left(1-\frac{\mu^{4}}{m_{2}^{4}}\right)\sum\nolimits_{i,j=1}^{N}\left({T^{a}}T^{b}\right)_{ij}.$

We evaluate the last factor using the identity
${T^{a}}T^{b}=\frac{1}{2N}\delta_{ab}\mathbb{I}_{N}+\frac{1}{2}\sum_{c=1}^{N^{2}-1}\left(f_{abc}+ig_{abc}\right)T^{c},$
where the $g_{abc}$ and $f_{abc}$ are completely symmetric and anti-symmetric
tensors, respectively. We note that
$\sum\nolimits_{i,j=1}^{N}(T^{c})_{ij}=\mathop{\mathrm{T}r}T^{c}=0$ due to the
skew-Hermitian nature of $T^{c}$ and
$\sum\nolimits_{i,j=1}^{N}(\mathbb{I}_{N})_{ij}=\mathop{\mathrm{T}r}\mathbb{I}_{N}=N$.
Thus we can diagonalize $\mathfrak{l}_{A}$, computing

$\mathfrak{l}_{A}=\frac{R^{2}}{2N}e^{2}\left(1-\frac{\mu^{4}}{m_{2}^{4}}\right)A_{\mu}^{a}A^{a\mu}=\frac{1}{2}m_{g}^{2}A_{\mu}^{a}A^{a\mu},$
(43)

from which we read off the masses $m_{g}^{(a)}$ of the $N^{2}-1$ gauge vector
bosons. We note that once again they vanish at the zero exceptional points,
but now for all $SU(N)$-models.

###  2.2 A $SU(2)$-symmetric model in adjoint representation

As we have demonstrated, the gauge vector boson becomes massive for the
$SU(N)$-symmetric model in the $\mathcal{CPT}$-symmetric regime and at the
exceptional point when the fields are taken to be in the representation space
of the fundamental representation. On the other hand the Higgs-mechanism
breaks down at the zero exceptional points. Remarkably it still applies when
the $\mathcal{CPT}$-symmetry is broken, although in that regime other
particles acquire complex masses so that the region is non-physical. Recall
that the regions in which the $\mathcal{CPT}$-symmetry holds are identified in
figure 1.

Let us now see whether we encounter a similar behaviour when the fields are
taken in adjoint representation. We consider here a slightly different non-
Hermitian $SU(2)$-invariant Lagrangian

$\displaystyle\mathcal{L}_{2}^{\text{ad}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathop{\mathrm{T}r}\left(D\phi_{1}\right)^{2}+\frac{1}{2}\mathop{\mathrm{T}r}\left(D\phi_{2}\right)^{2}-\frac{m_{1}^{2}}{2}\mathop{\mathrm{T}r}(\phi_{1}^{2})+\frac{m_{2}^{2}}{2}\mathop{\mathrm{T}r}(\phi_{2}^{2})-i\mu^{2}\mathop{\mathrm{T}r}(\phi_{1}\phi_{2})$
$\displaystyle-\frac{g}{4}\left[\mathop{\mathrm{T}r}(\phi_{1}^{2})\right]^{2}-\frac{1}{4}\mathop{\mathrm{T}r}\left(F^{2}\right),$

where as in equation (4) we take $g,\mu\in\mathbb{R}$, $m_{i}\in\mathbb{R}$ or
$m_{i}\in i\mathbb{R}$, to be constants. The two complex scalar fields are
expressed as $\phi_{i}=\phi_{i}^{a}T^{a}$, $i=1,2$ and $a=1,2,3$, where the
$T^{a}$ are the three $SU(2)$-generators in the adjoint representation that,
up to a factor of $2$, satisfy the same algebra as the Pauli spin matrices,
that is $[T^{a},T^{b}]=i\varepsilon_{abc}T^{c}$. Hence, the adjoint
representation is $\left(T^{a}\right)_{bc}=-i\varepsilon_{abc}$, i.e. to be
explicit

$T^{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\
0&i&0\end{array}\right),\quad T^{2}=\left(\begin{array}[]{ccc}0&0&i\\\
0&0&0\\\ -i&0&0\end{array}\right),\quad
T^{3}=\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\ 0&0&0\end{array}\right),$
(45)

such that $\mathop{\mathrm{T}r}(T^{a}T^{b})=2\delta^{ab}$ and therefore
$Tr(\phi^{2})=2\sum_{a=1}^{3}\phi^{a}\phi^{a}$. The $SU(2)$-symmetry in the
adjoint representation for each generator $T^{a}$ is therefore

$\phi_{j}\rightarrow e^{i\alpha T^{a}}\phi_{j}e^{-i\alpha
T^{a}}\approx\phi_{j}-\alpha\varepsilon_{abc}\phi_{j}^{b}T^{c},$ (46)

so that the infinitesimal changes to the fields $\phi_{i}^{a}$ result to

$\delta\phi_{i}^{a}=-\alpha\varepsilon_{abc}\phi_{i}^{b}.$ (47)

We will utilize this expression below.

In more a compact form the Lagrangian in (2.2) can be expressed equivalently
as

$\mathcal{L}_{2}^{\text{ad}}=D_{\mu}\phi_{i}^{a}D^{\mu}\phi_{i}^{a}-\phi_{i}^{a}M_{ij}^{2}\phi_{j}^{a}-g\left(\phi_{i}^{a}E_{ij}\phi_{j}^{a}\right)^{2}-\frac{1}{4}F_{\mu\nu}^{a}\left(F^{\mu\nu}\right)^{a},$
(48)

where repeated indices are summed over the appropriate index sets
$i,j,\mu,\nu\in\\{1,2\\}$ and $a,b\in\\{1,2,3\\}$. The matrix $M^{2}$ is
defined as

$M^{2}=\left(\begin{array}[]{cc}m_{1}^{2}&i\mu^{2}\\\
i\mu^{2}&-m_{2}^{2}\end{array}\right)~{},~{}$ (49)

and $E$ as in (12). The covariant derivative in the adjoint representation
acting on a complex field takes on the form

$(D_{\mu}\phi_{i})^{a}:=\partial_{\mu}\phi_{i}^{a}+e\varepsilon_{abc}A_{\mu}^{b}\phi_{i}^{c}$
(50)

Pursuing here a pseudo-Hermitian approach we perform a similarity
transformation on the Lagrangian in (48) with Dyson map

$\eta=\prod_{a=1}^{3}e^{\frac{\pi}{2}\int d^{3}x\Pi_{2}^{a}\phi_{2}^{a}},$
(51)

that maps the complex Lagrangian $\mathcal{L}_{2}^{\text{ad}}$ to a real
Lagrangian

$\mathfrak{l}_{2}^{\text{ad}}=(D_{\mu}\phi_{i})^{a}\mathcal{I}_{ij}(D^{\mu}\phi_{j})^{a}-\phi_{i}^{a}H_{ij}\phi_{j}^{a}-g\left(\phi_{i}^{a}E_{ij}\phi_{j}^{a}\right)^{2},$
(52)

where the matrix $H$ is defined as

$H:=\left(\begin{array}[]{cc}m_{1}^{2}&\mu^{2}\\\
\mu^{2}&m_{2}^{2}\end{array}\right)~{},~{}$ (53)

and $\mathcal{I}$ as in (12).

####  2.2.1 The $SU(2)$-symmetry preserving and breaking vacua

To find the different types of vacua $\phi^{0}$, we need to solve again
$\delta V=0$. The corresponding functional variation of the Lagrangian in (52)
leads to the three sets of equations

$\left(H+2gR^{2}E\right)\left(\phi^{0}\right)^{a}=0,~{}~{}\ \ \ a=1,2,3,$ (54)

with
$R^{2}:=\left(\phi_{i}^{0}\right)^{a}E_{ij}\left(\phi_{j}^{0}\right)^{a}$.
Next to the trivial $SU(2)$-symmetry preserving solution
$\left(\phi^{0}\right)^{a}=0$, a $SU(2)$-symmetry breaking solution is
obtained by requiring $\left(\phi^{0}\right)^{a}$ to become a vector in the
null space of the matrix $H+2gR^{2}E$, which is the case when

$\left(\phi^{0}\right)^{a}=\frac{N_{a}}{m_{2}^{2}}~{}\left(\begin{array}[]{c}m_{2}^{2}\\\
-\mu^{2}\end{array}\right),~{}~{}\text{and\quad}R^{2}=\frac{\mu^{4}-m_{1}^{2}m_{2}^{2}}{2gm_{2}^{2}},$
(55)

where the $N_{a}$ are normalization constants. Given the solution in (55), the
relation for $R^{2}$ imposes the additional constraint
$R^{2}=N_{1}^{2}+N_{2}^{2}+N_{3}^{2}$ on these constants. Expressing the Lie
algebra valued vacuum field $\phi_{i}^{0}=\left(\phi_{i}^{0}\right)^{a}T^{a}$
in the matrix form of the adjoint representation (45) we obtain

$\phi_{1}^{0}=i\left(\begin{array}[]{ccc}0&-N_{3}&N_{2}\\\ N_{3}&0&-N_{1}\\\
-N_{2}&N_{1}&0\end{array}\right)~{},~{}~{}\text{and\quad}\phi_{2}^{0}=-\frac{\mu^{2}}{m_{2}^{2}}\phi_{1}^{0}.$
(56)

We can now apply the $SU(2)$-symmetry to the vacuum state in the form

$\phi^{\text{vac}}=\left[\left(\phi_{1}^{0}\right)^{1},\left(\phi_{2}^{0}\right)^{1},\left(\phi_{1}^{0}\right)^{2},\left(\phi_{2}^{0}\right)^{2},\left(\phi_{1}^{0}\right)^{3},\left(\phi_{2}^{0}\right)^{3}\right],$
(57)

so that the infinitesimal changes $\delta\phi_{i}(\phi^{\text{vac}})$ with
(47) and (55) yield the following states for each generator

$\displaystyle v_{1}^{0}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}}{m_{2}^{2}}\left(0,0,N_{3}m_{2}^{2},-N_{3}\mu^{2},-N_{2}m_{2}^{2},N_{2}\mu^{2}\right),$
(58) $\displaystyle v_{2}^{0}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{2}}{m_{2}^{2}}\left(-N_{3}m_{2}^{2},N_{3}\mu^{2},0,0,N_{1}m_{2}^{2},-N_{1}\mu^{2}\right),$
(59) $\displaystyle v_{3}^{0}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{3}}{m_{2}^{2}}\left(N_{2}m_{2}^{2},-N_{2}\mu^{2},-N_{1}m_{2}^{2},N_{1}\mu^{2},0,0\right),$
(60)

as solutions for $\phi^{\text{vac}}$. Evidently, these states are linearly
dependent as

$\sum\nolimits_{i=1}^{3}\frac{N_{i}v_{i}^{0}}{\alpha_{i}}=0.$ (61)

According to Goldstone’s theorem the states $v_{i}^{0}$ should be eigenvectors
of the squared mass matrix with eigenvalue 0. As only two of them are linearly
independent we expect to find two massless Goldstone bosons, which in our
gauged model correspond to “would be Goldstone bosons”. Hence the
$SU(2)$-symmetry has been broken down to a $U(1)$-symmetry, so that the group
theoretical argument predicts two Goldstone bosons equal to the dimension of
the coset $SU(2)/U(1)$.

####  2.2.2 The squared mass matrix

Expanding the Lagrangian in equation (48) about the vacuum solution gives

$\mathfrak{l}_{2}^{\text{ad}}=(D_{\mu}\phi_{i})^{a}\mathcal{I}_{ij}(D_{\mu}\phi_{j})^{a}-\frac{1}{2}\phi_{i}^{a}H_{ij}^{(a)}\phi_{j}^{a}+2(D_{\mu}\phi_{i}^{0})^{a}\mathcal{I}_{ij}(D^{\mu}\phi_{j})^{a}+(D_{\mu}\phi_{i}^{0})^{a}\mathcal{I}_{ij}(D_{\mu}\phi_{j}^{0})^{a}+\mathcal{O}(\phi^{3}),$
(62)

where the last two terms originate from expanding the covariant kinetic term.
The Hessian matrix is then computed by differentiating (54) once more

$\hat{H}_{ij}^{ab}:=\frac{\partial^{2}\mathcal{V}}{\partial\phi_{i}^{a}\partial\phi_{j}^{b}}=2H_{ij}+4gR^{2}E_{ij}\delta^{ab}+8g\left(E\phi^{a}\right)_{i}\left(E\phi^{b}\right)_{j},$
(63)

from which we obtain the non-Hermitian squared mass matrix as

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\left.\frac{1}{2}\mathcal{I}\hat{H}\right|_{\phi^{\text{vac}}}$
(64) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}m_{1}^{2}+2gR^{2}+4gN_{1}^{2}&\mu^{2}&4gN_{1}N_{2}&0&4gN_{1}N_{3}&0\\\
-\mu^{2}&-m_{2}^{2}&0&0&0&0\\\
4gN_{1}N_{2}&0&m_{1}^{2}+2gR^{2}+4gN_{2}^{2}&\mu^{2}&4gN_{2}N_{3}&0\\\
0&0&-\mu^{2}&m_{2}^{2}&0&0\\\
4gN_{1}N_{3}&0&4gN_{2}N_{3}&0&m_{1}^{2}+2gR^{2}+4gN_{3}^{2}&\mu^{2}\\\
0&0&0&0&-\mu^{2}&-m_{2}^{2}\end{array}\right).$ (71)

The entries in the rows and columns of $M^{2}$ are labeled as
$(\phi_{1}^{1},\phi_{2}^{1},\phi_{1}^{2},\phi_{2}^{2},\phi_{1}^{3},\phi_{2}^{3})=:\Psi$.
The six eigenvalues $\lambda$ of $M^{2}$ are then computed to

$\lambda_{1,2}=0;\qquad\lambda_{3,4}=\frac{\mu^{4}-m_{2}^{4}}{m_{2}^{2}},\qquad\lambda_{\pm}=\kappa\pm\sqrt{2(\mu^{4}-m_{1}^{2}m_{2}^{2})+\kappa^{2}},$
(72)

with $\kappa:=3\mu^{4}/2m_{2}^{2}-m_{2}^{2}/2-m_{1}^{2}$. We can now verify
that the three vectors $v_{i}^{0}$ in (58)-(60), corresponding to the
infinitesimal changes of the vacuum (55) under the action of the
$SU(2)$-symmetry, are indeed vectors in the null space of $M^{2}$. Due to
their linear dependence we may choose two of them to be associated with the
two massless “would be Goldstone bosons”.

We note that there are zero exceptional points at $\mu^{4}=m_{2}^{4}$ when
$\lambda_{3,4}=0$, and at $\mu^{4}=m_{1}^{2}m_{2}^{2}$ when either
$\lambda_{-}=0$ or $\lambda_{+}=0$. The standard exceptional point for which
the two eigenvalues $\lambda_{-}$ and $\lambda_{+}$ coalesce occurs when
$m_{1}^{2}=3\mu^{4}/2m_{2}^{2}+m_{2}^{2}/2\pm\mu^{2}$. The Jordan normal form
for the mass squared matrix becomes

$\mathop{\mathrm{d}iag}D_{e}=(0,\lambda_{e}^{b},0,\lambda_{e}^{b},0,\lambda_{e}^{b},\Lambda),\text{~{}~{}~{}}\lambda_{e}^{b}=\frac{\mu^{4}}{m_{2}^{2}}-m_{2}^{2},~{}~{}~{}\Lambda=\left(\begin{array}[]{ll}\pm\mu^{2}-m_{2}^{2}&\pm(\alpha-\beta)\mu^{2}\\\
0&\pm\mu^{2}-m_{2}^{2}\end{array}\right),$ (73)

for some arbitrary constants $\alpha$ and $\beta$.

We notice that the eigenvalues in (72) do not depend on the choice of the
three normalization constants $N_{a}$, since all of these vacua are equivalent
as they are related by $SU(2)$-symmetry transformations. The physical regions
of the model are determined by the requirement that the eigenvalues are real
and positive. Taking now account of the possibility that $m_{i}\in\mathbb{R}$
or $m_{i}\in i\mathbb{R}$, by allowing for different signs in front of the
$m_{i}^{2}$ terms in setting $m_{i}^{2}\rightarrow c_{i}m_{i}^{2}$, we find
that the model does not possess any physical region when $c_{1}=c_{2}=\pm 1$
and physical regions when $c_{1}=-c_{2}=\pm 1$ as argued also in the previous
section.

####  2.2.3 The would be Goldstone bosons

Let us now identify the two massless Goldstone bosons $\psi_{1,2}^{\text{Gb}}$
in the different $\mathcal{PT}$ -regimes by the same procedure as previously
explained in [38, 39], with the difference that they will be made to vanish
due to the presence of the gauge bosons. In terms of the original scalar
fields in the model we identify the Goldstone bosons by evaluating

$\psi_{1,2}^{\text{Gb}}:=\sqrt{(\Psi^{T}\hat{I}U)_{1,2}(U^{-1}\Psi)_{1,2}},$
(74)

where the matrix $U$ diagonalizes the squared mass matrix by $U^{-1}M^{2}U=D$
with
$\mathop{\mathrm{d}iag}D=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{-},\lambda_{+})$
and $\mathop{\mathrm{d}iag}\hat{I}=\\{\mathcal{I},\mathcal{I},\mathcal{I}\\}$.
In the $\mathcal{PT}$ -symmetric regime the similarity transformation $U$ is
well defined by

$U:=(v_{1},v_{2},v_{3},v_{4},v_{-},v_{+}),$ (75)

where the $v_{i}$ are the eigenvectors of $M^{2}$. Up to normalizations
constants for each eigenvector, we obtain in our example the concrete
expressions

$v_{i}=\left[\left(m_{2}^{2}+\lambda_{i}\right)\tau_{i1},-\mu^{2}\tau_{i1},\left(m_{2}^{2}+\lambda_{i}\right)\tau_{i2},-\mu^{2}\tau_{i2},\left(m_{2}^{2}+\lambda_{i}\right)\tau_{i3},-\mu^{2}\tau_{i3}\right],$
(76)

with $\tau_{12}=\tau_{23}=\tau_{32}=\tau_{43}=0$,
$\tau_{33}=\tau_{42}=\tau_{\pm 1}=-\tau_{13}=-\tau_{22}=N_{1}$,
$\tau_{21}=\tau_{41}=\tau_{\pm 2}=N_{2}$ and $\tau_{11}=\tau_{31}=\tau_{\pm
3}=N_{3}$. Defining a $\mathcal{PT}$-inner product as $\left\langle
a|b\right\rangle_{\mathcal{PT}}:=a\hat{I}b$ these vectors can be
orthonormalized $\left\langle
v_{i}|v_{j}\right\rangle_{\mathcal{PT}}=\delta_{ij}$. Recall from the argument
after (2.19) that indeed $v_{i}=\mathcal{I}u_{i}$, so that $\left\langle
v_{i}|v_{j}\right\rangle_{\mathcal{PT}}=v_{i}\mathcal{I}^{2}u_{j}=\delta_{ij}$.
Here we should stress that $M^{2}$ is only the analogue to a Hamiltonian and
we are not attempting to construct an associated Hilbert space.

For convenience we take now $N_{1}=N_{2}=0$, $N_{3}=R$ and compute

$\psi_{1}^{\text{Gb}}:=\frac{m_{2}^{2}\phi_{1}^{3}+\mu^{2}\phi_{2}^{3}}{\sqrt{m_{2}^{4}-\mu^{4}}},\quad\text{and\quad}\psi_{2}^{\text{Gb}}:=\frac{m_{2}^{2}\phi_{1}^{2}+\mu^{2}\phi_{2}^{2}}{\sqrt{m_{2}^{4}-\mu^{4}}}.$
(77)

We note that $\det
U=\lambda_{3}\lambda_{4}(\lambda_{-}-\lambda_{+})\mu^{6}R^{4}$, indicating the
breakdown of these expressions at the exceptional points when
$\lambda_{-}=\lambda_{+}$, the zero exceptional point when
$\lambda_{3}=\lambda_{4}=0$ and at the trivial vacuum when $R=0$, as
previously observed in [38, 39]. However, at the exceptional point we may
still calculate the expressions for the Goldstone boson when taking into
account that in this case the two eigenvectors $v_{-}$ and $v_{+}$ become
identical. In order to obtain two linearly independent eigenvectors when the
squared mass matrix is converted into its Jordan normal form we multiply two
entries of the vector $v_{+}$ by some arbitrary constants $\alpha\neq\beta$ as
$(v_{+})_{1}\rightarrow\alpha(v_{+})_{1}$ and
$(v_{+})_{2}\rightarrow\beta(v_{+})_{2}$. With this change the matrix $U$
becomes invertible as $\det
U=\lambda_{3}\lambda_{4}(\beta-\alpha)(m_{2}^{2}+\kappa)N_{1}^{2}\mu^{6}R^{2}$.
We may now evaluate the expression in (74) obtaining the same formulae for the
Goldstone bosons as in (77). At the zero exceptional point it is not possible
to identify the Goldstone in terms of the original fields in the model.

####  2.2.4 The mass of the vector gauge boson

Finally we calculate the mass of the gauge vector bosons by expanding the
minimal coupling term in equation (52) around the symmetry breaking vacuum
(57)

$\displaystyle\left[D_{\mu}(\phi+\phi^{0})\right]^{T}\mathcal{I}\left[D^{\mu}(\phi+\phi^{0})\right]$
$\displaystyle=$
$\displaystyle(D_{\mu}\phi^{0})^{T}\mathcal{I}(D^{\mu}\phi^{0})+\dots$
$\displaystyle=$ $\displaystyle
e^{2}\left[\varepsilon_{abc}A_{\mu}^{b}\left(\phi_{i}^{0}\right)^{c}\right]\mathcal{I}_{ij}\left(\varepsilon_{ade}{A^{d}}^{\mu}\left(\phi_{j}^{0}\right)^{e}\right)+\dots$
$\displaystyle=$ $\displaystyle
e^{2}\left(A_{\mu}^{a}A^{a\mu}\left(\phi_{i}^{0}\right)^{b}\mathcal{I}_{ij}\left(\phi_{j}^{0}\right)^{b}-A_{\mu}^{a}A^{b\mu}\left(\phi_{i}^{0}\right)^{b}\mathcal{I}_{ij}\left(\phi_{j}^{0}\right)^{a}\right)+\dots,$

where we used the standard identity
$\varepsilon_{abc}\varepsilon_{ade}=\delta_{bd}\delta_{ce}-\delta_{be}\delta_{cd}$.
A convenient choice for the normalization constants $N_{i}$ that is compatible
with (55) and diagonalizes (2.2.4) is to set two constants to zero and the
remaining one to $R$. For instance, taking $N_{1}=N_{2}=0$, $N_{3}=R$ the only
nonvanishing terms in (2.2.4) are

$\displaystyle=$ $\displaystyle
e^{2}\left(A_{\mu}^{1}A^{1\mu}+A_{\mu}^{2}A^{2\mu}\right)\left(\phi_{i}^{0}\right)^{3}\mathcal{I}_{ij}\left(\phi_{j}^{0}\right)^{3},$
(79) $\displaystyle=$ $\displaystyle
e^{2}R^{2}\left(1-\frac{\mu^{4}}{m_{2}^{4}}\right)\left(A_{\mu}^{1}A^{1\mu}+A_{\mu}^{2}A^{2\mu}\right).$
(80)

Thus for $\mu^{4}\neq m_{2}^{4}$ and $R\neq 0$ we obtain two massive vector
gauge bosons $m_{g}^{(1)}$ and $m_{g}^{(2)}$, that is one for each “would be
Goldstone boson”. When $\mu^{4}=m_{2}^{4}$, that is then model is at the zero
exceptional point of type I, the gauge mass vector bosons remain massless.
This feature is compatible with our previous observations in [38, 39] and
above, that at these points the Goldstone bosons can not be identified.

We notice here that the two massive vector gauge bosons are proportional to
the $\mathcal{CPT}$ -inner product of the symmetry broken vacuum solution

$m_{\text{gauge}}^{2}\propto\left\langle
0|0\right\rangle_{\mathcal{CPT}}\propto\phi^{\text{vac}}\hat{I}\phi^{\text{vac}}\propto
R^{2}\left(1-\frac{\mu^{4}}{m_{2}^{4}}\right).$ (81)

Hence the vanishing of the mass for the vector gauge bosons at the two types
of zero exceptional points can be associated to the vanishing of the
$\mathcal{CPT}$ -inner product at these points. This is reminiscent of the
vanishing of the $\mathcal{CPT}$ -inner product at the standard exceptional
points, which is responsible for interesting phenomena such as the stopping of
light at these locations in the parameter space [45, 46]. We note, however, a
key difference between the two scenarios: While the $\mathcal{CPT}$ -inner
product in (81) is devised on the eigenvector space of squared mass matrix,
the latter is a $\mathcal{CPT}$ -inner product on the Hilbert space.

##  3 Conclusions

Employing a pseudo-Hermitian approach we found that the Higgs mechanism
applies in the usual way in the $\mathcal{CPT}$-symmetric regime by giving a
mass to the vector gauge bosons and preventing Goldstone bosons to exist,
which was also found in [37, 41] using different approaches. As in [41] we
also observed that in the spontaneously broken $\mathcal{CPT}$-symmetric
regime the vector gauge bosons become massive and the Higgs mechanism is in
tact. However, as in this regime other particles acquire complex masses it has
to be discarded as non-physical for that reason. Even though technically one
needs to treat the standard exceptional point differently from the other
regimes, the main principle of the Higgs mechanism still holds up. In contrast
to the finding in [41], we observed that the Higgs mechanism breaks down at
the zero exceptional points, which was also observed in [37]. We find the same
characteristic behaviour, i.e. the matching of the amounts of massive vector
gauge bosons and “would be Goldstone bosons”, for the complex scalar fields
taken in the fundamental as well as in the adjoint representation. The
vanishing of the mass for the vector gauge bosons coincides with the vanishing
of the $\mathcal{CPT}$ -inner product on the eigenvector space of squared mass
matrix.

Obviously there are many interesting extensions to these investigations, such
as for instance the treatment of models with a more involved field content or
different types of continuous symmetries.

Acknowledgments: We would like to thank Alessandro de Martino for useful
comments.

##  Appendix A Type I (standard) versus type II (zero) exceptional points

In this appendix we present a discussion that illustrates the difference
between the two types of exceptional points. The main distinction in their
behaviour is that a one-dimensional parameter space the
$\mathcal{PT}$-symmetry is spontaneously broken on one side of the type I
(standard) exceptional point, whereas for type II (zero) exceptional point the
$\mathcal{PT}$-symmetry is preserved on both sides. The zero exceptional point
occurs when two eigenvalues coalesce at zero, hence the name.

We consider here a ($3\times 3$)-matrix of a very generic form that occurs for
instance as a building block of the squared mass matrix in the model discussed
in [39], see equation (3.48) therein,

$H=\left(\begin{array}[]{ccc}A&W&0\\\ -W&B&-V\\\ 0&V&-C\\\ \end{array}\right)$
(82)

Here we carry out the discussion for a Hamiltonian $H$, having in mind the
analogy to the squared mass matrix. The determinant is easily computed to
$\det H=A\kappa-CW^{2}$, $\kappa:=V^{2}-BC$. In order to obtain a zero
eigenvalue, $\lambda_{0}=0$, we enforce now the determinant to vanish by
setting $A=W^{2}C/\kappa$. The other two eigenvalues then become

$\lambda_{\pm}=\frac{\kappa(B-C)+CW^{2}\pm\tau}{2\kappa}$ (83)

with $\tau=\sqrt{\kappa^{2}\left((B+C)^{2}-4V^{2}\right)+2\kappa
W^{2}\left(C(B+C)-2V^{2}\right)+C^{2}W^{4}}$.

The exceptional points are identified by simultaneously solving the two
equations

$\det\left(H-\lambda\mathbb{I}\right)=0,\qquad\text{and}\qquad\frac{d}{d\lambda}\det\left(H-\lambda\mathbb{I}\right)=0,$
(84)

for $W$ and $\lambda$, obtaining the two sets of eigenvalues

$\lambda_{\pm}^{e}=\frac{\hat{\kappa}^{2}-\sqrt{\kappa}V}{C},\lambda_{0}=0\qquad\text{and}\qquad\lambda_{-}^{0}=\lambda_{0}=0,\lambda_{+}^{0}=\frac{C^{3}+BV^{2}-2CV^{2}}{\hat{\kappa}^{2}}$
(85)

for the critical parameters

$W^{e}=\frac{\tilde{\kappa}}{C},\qquad\text{and}\qquad
W^{0}=i\frac{\kappa}{\hat{\kappa}},$ (86)

respectively. We abbreviated
$\tilde{\kappa}:=\pm\sqrt{\kappa\left(\kappa+V^{2}-C^{2}\right)\pm
2\kappa^{3/2}V}$ and $\hat{\kappa}:=\sqrt{V^{2}-C^{2}}$. The first set of
eigenvalues in (85) correspond to the standard exceptional point and the
second set to the zero exceptional point.

Next we calculate the bi-orthonormal basis from the normalized left and right
eigenvectors $u_{i}$, $v_{i}$, $i=0,\pm$, respectively, for $H$

$v_{0}=\frac{1}{\sqrt{N_{0}}}(-\kappa,CW,VW),\quad
v_{\pm}=\frac{1}{\sqrt{N_{\pm}}}(W(\hat{\kappa}^{2}-C\lambda_{\pm}),\kappa(C+\lambda_{\pm}),V\kappa),\quad
u_{i}=Uv_{i},$ (87)

with $U=diag(1,-1,1)$ and normalization constants
$N_{0}=\kappa^{2}+W^{2}\hat{\kappa}^{2}$,
$N_{\pm}=V^{2}\kappa^{2}+W^{2}[V^{2}-C(C+\lambda_{\pm})]^{2}-(C+\lambda_{\pm})^{2}\kappa^{2}$.
By construction these vectors satisfy the orthonormality relation $u_{i}\cdot
v_{j}=\delta_{ij}$.

We observe now that at the standard exceptional point the two eigenvectors for
the non-normalised ($N_{\pm}$ become zero at the exceptional points)
eigenvalues $\lambda_{\pm}^{e}$ coalesce, which distinguishes exceptional
points from standard degeneracy. The left and right eigenvectors become in
this case

$\displaystyle
v_{\pm}^{e,r}=\left(\tilde{\kappa},V\sqrt{\kappa}-\kappa,C\sqrt{\kappa}\right),\qquad$
$\displaystyle
v_{0}^{e,r}=\left(-C\kappa,C\tilde{\kappa},V\tilde{\kappa}\right),$ (88)
$\displaystyle
v_{\pm}^{e,l}=\left(\tilde{\kappa},\kappa-V\sqrt{\kappa},C\sqrt{\kappa}\right),\qquad$
$\displaystyle
v_{0}^{e,l}=\left(-C\kappa,-C\tilde{\kappa},V\tilde{\kappa}\right),$ (89)

with

$v_{\pm}^{e,l}\cdot v_{\pm}^{e,r}=0,\qquad\text{and}\qquad v_{0}^{e,l}\cdot
v_{0}^{e,r}=C^{2}\left(\kappa^{2}-\tilde{\kappa}^{2}\right)+V^{2}\tilde{\kappa}^{2}.$
(90)

Similarly, at the zero exceptional point the eigenvectors for the eigenvalues
$\lambda_{0}$ and $\lambda_{-}^{0}$ coalesce, which qualifies this point also
to be called “exceptional” in the standard terminology. In this case the left
and right eigenvectors become

$\displaystyle
v_{+}^{0,r}=\left(V\kappa,iV\hat{\kappa}(C-B),i\hat{\kappa}^{3}\right),\qquad$
$\displaystyle v^{0,r}=v_{-}^{0,r}=\left(i\hat{\kappa},C,V\right),$ (91)
$\displaystyle
v_{+}^{0,l}=\left(V\kappa,iV\hat{\kappa}(B-C),i\hat{\kappa}^{3}\right),\qquad$
$\displaystyle v^{0,l}=v_{-}^{0,r}=\left(i\hat{\kappa},-C,V\right),$ (92)

with

$v_{+}^{e,l}\cdot
v_{+}^{e,r}=\left(C^{3}+BV^{2}-2CV^{2}\right)^{2},\qquad\text{and}\qquad
v^{0,l}\cdot v^{0,r}=0.$ (93)

In order to understand the key difference between these two types of
exceptional points we consider at first the eigenvalues (83) near the critical
values in (86). Concerning the standard exceptional points we note that the
two eigenvalues become identical when $\tau\rightarrow 0$. Thus writing
$\tau/C^{2}=(W^{2}-(W^{e})^{2})(W^{2}-\tilde{W})$, with $\tilde{W}$ being the
second root of the polynomial in $W^{2}$ under the square root, it is now
clear that if we consider the eigenvalues as functions of $W^{2}$ the argument
of the square root has different signs for $W^{2}=(W^{e})^{2}+\epsilon$ and
$W^{2}=(W^{e})^{2}-\epsilon$. Hence the eigenvalues are real on one side of
the exceptional point in the $W^{2}$-parameter space and complex on the other.
In contrast none of the eigenvalues becomes complex in the neighbourhood of
the critical value $W^{0}$.

For completion we also report the Dyson map and hence the metric operator for
which the same behaviour may be observed. Using the operator that diagonalises
the non-Hermitian Hamiltonian $H$

$\eta=(v_{0},v_{+},v_{-}),\qquad\rho=\eta\eta^{\dagger},$ (94)

with determinant

$\det\eta=\frac{V\kappa}{\sqrt{N_{0}N_{+}N_{-}}}(\lambda_{-}-\lambda_{+})(\kappa^{2}+W^{2}\hat{\kappa}^{2}),$
(95)

we verify the pseudo and quasi Hermiticity relations

$\eta^{-1}H\eta=h=h^{\dagger},\qquad\rho H=H^{\dagger}\rho.$ (96)

We observe that the map break down at both exceptional points, i.e.
$\det\eta=0$ for the critical values $W^{e}$ and $W^{0}$, and on one side of
the standard exceptional point. In all other regions of the parameter space it
holds. Thus we find the same behaviour as already observed for the analysis of
the eigenvalues.

## References

  * [1] F. Englert and R. Brout,  Broken symmetry and the mass of gauge vector mesons,  Phys. Rev. Lett. 13(9), 321 (1964).
  * [2] P. W. Higgs,  Broken symmetries, massless particles and gauge fields,  Phys. Lett. 12, 132–133 (1964).
  * [3] P. W. Higgs,  Broken symmetries and the masses of gauge bosons,  Phys. Rev. Lett. 13(16), 508 (1964).
  * [4] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble,  Global conservation laws and massless particles,  Phys. Rev. Lett. 13(20), 585 (1964).
  * [5] P. Goddard and D. I. Olive,  Kac-Moody and Virasoro Algebras in Relation to Quantum Physics,  Int. J. Mod. Phys. A1, 303 (1986).
  * [6] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov,  Infinite conformal symmetry in two-dimensional quantum field theory,  Nucl. Phys. B241, 333–380 (1984).
  * [7] C.-N. Yang and T.-D. Lee,  Statistical theory of equations of state and phase transitions. I. Theory of condensation,  Phys. Rev. 87(3), 404 (1952).
  * [8] T.-D. Lee and C.-N. Yang,  Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model,  Phys. Rev. 87(3), 410 (1952).
  * [9] M. E. Fisher,  Yang-Lee edge singularity and $\phi$ 3 field theory,  Phys. Rev. Lett. 40(25), 1610 (1978).
  * [10] J. L. Cardy,  Conformal invariance and the Yang-Lee edge singularity in two dimensions,  Phys. Rev. Lett. 54(13), 1354 (1985).
  * [11] J. L. Cardy and G. Mussardo,  S-matrix of the Yang-Lee edge singularity in two dimensions,  Phys. Lett. B 225(3), 275–278 (1989).
  * [12] A. B. Zamolodchikov,  Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models,  Nucl Phys. B 342(3), 695–720 (1990).
  * [13] A. B. Zamolodchikov,  From tricritical Ising to critical Ising by thermodynamic Bethe ansatz,  Nucl. Phys. B 358(3), 524–546 (1991).
  * [14] O. A. Castro-Alvaredo, A. Fring, C. Korff, and J. L. Miramontes,  Thermodynamic Bethe ansatz of the homogeneous sine-Gordon models,  Nucl. Phys. B575, 535–560 (2000).
  * [15] J.-L. Jacobsen, N. Read, and H. Saleur,  Dense loops, supersymmetry, and Goldstone phases in two dimensions,  Phys. Rev. Lett. 90(9), 090601 (2003).
  * [16] H. B. Geyer, F. G. Scholtz, and I. Snyman,  Quasi-hermiticity and the Role of a Metric in Some Boson Hamiltonians,  Czech. J. Phys. 54, 1069–1073 (2004).
  * [17] C. M. Bender and S. Boettcher,  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,  Phys. Rev. Lett. 80, 5243–5246 (1998).
  * [18] A. Mostafazadeh,  Pseudo-Hermitian Representation of Quantum Mechanics,  Int. J. Geom. Meth. Mod. Phys. 7, 1191–1306 (2010).
  * [19] C. M. Bender, P. E. Dorey, C. Dunning, A. Fring, D. W. Hook, H. F. Jones, S. Kuzhel, G. Levai, and R. Tateo,  PT Symmetry: In Quantum and Classical Physics,  (World Scientific, Singapore) (2019).
  * [20] C. M. Bender, V. Branchina, and E. Messina,  Critical behavior of the P T-symmetric $i\phi^{3}$ quantum field theory,  Phys. Rev. D 87(8), 085029 (2013).
  * [21] A. M. Shalaby,  Vacuum structure and P T-symmetry breaking of the non-Hermetian $i\phi^{3}$ theory,  Phys. Rev. D 96(2), 025015 (2017).
  * [22] C. M. Bender, N. Hassanpour, S. Klevansky, and S. Sarkar,  PT-symmetric quantum field theory in D dimensions,  Physical Review D 98(12), 125003 (2018).
  * [23] J. Alexandre, C. M. Bender, and P. Millington,  Non-Hermitian extension of gauge theories and implications for neutrino physics,  JHEP 2015(11), 111 (2015).
  * [24] V. E. Rochev,  Hermitian vs PT-Symmetric Scalar Yukawa Model,  arXiv preprint arXiv:1512.03286 (2015).
  * [25] A. Y. Korchin and V. A. Kovalchuk,  Decay of the Higgs boson to $\tau^{-}\tau^{+}$ and non-Hermiticy of the Yukawa interaction,  Phys. Rev. D 94(7), 076003 (2016).
  * [26] L. Gouba,  The Yukawa Model in One Space-One Time Dimensions,  in _Mathematical Structures and Applications_ , pages 225–233, Springer, 2018.
  * [27] C. M. Bender, H. F. Jones, and R. J. Rivers,  Dual PT-symmetric quantum field theories,  Phys. Lett. B 625(3-4), 333–340 (2005).
  * [28] C. M. Bender and K. A. Milton,  A nonunitary version of massless quantum electrodynamics possessing a critical point,  J. of Phys. A: Math. and Gen. 32(7), L87 (1999).
  * [29] C. M. Bender, K. A. Milton, and V. M. Savage,  Solution of Schwinger-Dyson equations for PT-symmetric quantum field theory,  Phys. Rev. D 62(8), 085001 (2000).
  * [30] K. A. Milton, E. K. Abalo, P. Parashar, N. Pourtolami, and J. Wagner,  PT-symmetric quantum electrodynamics and unitarity,  Phil. Trans. of the Royal Society A: Math., Phys. and Eng. Sciences 371(1989), 20120057 (2013).
  * [31] T. Ohlsson,  Non-Hermitian neutrino oscillations in matter with PT symmetric Hamiltonians,  EPL 113, 61001 (2016).
  * [32] G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva,  Theory and phenomenology of two-Higgs-doublet models,  Physics reports, 516, 1-102 (2012).
  * [33] R.A. Flores and M. Sher,  Higgs masses in the standard, multi-Higgs and supersymmetric models,  Annals of Physics, 148, 95-134 (1983).
  * [34] J. Schwinger,  The theory of quantized fields. I,  Phys. Rev. 82(6), 914 (1951).
  * [35] J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve,  Spontaneous symmetry breaking and the Goldstone theorem in non-Hermitian field theories,  Phys. Rev. D 98, 045001 (2018).
  * [36] J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve,  Spontaneously breaking non-Abelian gauge symmetry in non-Hermitian field theories,  Phys. Rev. D 101(3), 035008 (2020).
  * [37] P. D. Mannheim,  Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories,  Phys. Rev. D 99(4), 045006 (2019).
  * [38] A. Fring and T. Taira,  Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories,  Phys. Rev. D 101(4), 045014 (2020).
  * [39] A. Fring and T. Taira,  Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories,  Nucl. Phys. B 950, 114834 (2020).
  * [40] F. J. Dyson,  Thermodynamic Behavior of an Ideal Ferromagnet,  Phys. Rev. 102, 1230–1244 (1956).
  * [41] J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve,  Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories,  Phys. Rev. D 99(7), 075024 (2019).
  * [42] A. Mostafazadeh,  Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries,  J. Math. Phys. 44, 974–989 (2003).
  * [43] T. Frith,  Time-dependence in non-Hermitian quantum systems,  arXiv:2002.01977, PhD Thesis, City, University of London (2019).
  * [44] L. D. Landau,  On the theory of phase transitions,  Ukr. J. Phys. 11, 19–32 (1937).
  * [45] T. Goldzak, A. A. Mailybaev, and N. Moiseyev,  Light stops at exceptional points,  Phys. Rev. Lett. 120(1), 013901 (2018).
  * [46] M.-A. Miri and A. Alù,  Exceptional points in optics and photonics,  Science 363(6422), eaar7709 (2019).