# Anomalous dispersion, superluminality and instabilities in
two-flavour theories with local non-Hermitian mass mixing

Maxim N. Chernodub<EMAIL_ADDRESS>Institut Denis Poisson, CNRS
UMR 7013, Université de Tours, 37200 France  Department of Physics, West
University of Timisoara, Bd. Vasile Pârvan 4, Timisoara 300223, Romania  Peter
Millington<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom

(January 11, 2024)

###### Abstract

Pseudo-Hermitian field theories possess a global continuous “similarity”
symmetry, interconnecting the theories with the same physical particle content
and an identical mass spectrum. In their regimes with real spectra, within
this family of similarity transformations, there is a map from the non-
Hermitian theory to its Hermitian similarity partner. We promote the
similarity transformation to a local symmetry, which requires the introduction
of a new vector similarity field as a connection in the similarity space of
non-Hermitian theories. In the case of non-Hermitian two-flavour scalar or
fermion mixing, and by virtue of a novel IR/UV mixing effect, the effect of
inhomogeneous non-Hermiticity then reveals itself via anomalous dispersion,
instabilities and superluminal group velocities at very high momenta, thus
setting an upper bound on the particle momentum propagating through
inhomogeneous backgrounds characterised by Lagrangians with non-Hermitian mass
matrices. Such a non-Hermitian extension of the Standard Model of particle
physics, encoded in a weak inhomogeneity of the non-Hermitian part of the
fermion mass matrix, may nevertheless provide us with a low-energy particle
spectrum consistent with experimentally observed properties.

##  I Introduction

Due to its wide applicability in experimental physics (see Refs. Zyablovsky
_et al._ (2014); Christodoulides and Yang (2018); El-Ganainy  _et al._ (2018);
Ashida  _et al._ (2021); Zhao and Feng (2018); Praveena and Senthilnathan
(2023) for reviews), pseudo-Hermitian quantum mechanics Mostafazadeh (2002a,
b, c), wherein Hermiticity of the Hamiltonian is superseded by an antilinear
symmetry such as parity–time-reversal ($\mathcal{PT}$) Bender and Boettcher
(1998); Bender  _et al._ (1999); Bender (2005) has inspired growing interest
in viable quantum field theories (QFTs) with non-Hermitian
Hamiltonians/Lagrangians. Examples include: non-Hermitian deformations of the
Dirac Lagrangian with a parity-odd, anti-Hermitian mass term Bender  _et al._
(2005); Alexandre  _et al._ (2015, 2017a, 2017b, 2018a) (see also Ref. Jones-
Smith and Mathur (2014)), theories of massive second-order fermions LeClair
and Neubert (2007); Ferro-Hernández  _et al._ (2023), scalar Alexandre  _et
al._ (2017b, 2018a, 2020a, 2020b); Sablevice and Millington (2023) and
fermionic Alexandre  _et al._ (2015); Mishra (2021); Alexandre  _et al._
(2020b) field theories with non-Hermitian mass mixing matrices, non-Hermitian
Yukawa theories Alexandre  _et al._ (2015); Alexandre and Mavromatos (2020);
Korchin and Kovalchuk (2022); Mavromatos  _et al._ (2023); Croney and Sarkar
(2023), scalar theories with complex Bender  _et al._ (2004a, b, 2012, 2013);
Shalaby (2017); Bender  _et al._ (2018); Felski  _et al._ (2021); Ai  _et al._
(2023) or wrong-sign Shalaby (2007); Shalaby and Al-Thoyaib (2010); Ai  _et
al._ (2022) self-interactions, theories exhibiting spontaneous symmetry
breaking Alexandre  _et al._ (2018b); Mannheim (2019); Alexandre  _et al._
(2019); Fring and Taira (2020a); Alexandre  _et al._ (2020c); Fring and Taira
(2020b, 2022) and topological defects Fring and Taira (2020c, 2021); Begun
_et al._ (2021, 2022); Correa  _et al._ (2022), holographic settings Areán
_et al._ (2020); Morales-Tejera and Landsteiner (2023), and non-Hermitian
Dirac materials in the context of condensed matter physics Juricic and Roy
(2023).

The attractiveness of non-Hermitian QFTs, whose spectra are nevertheless real
and whose evolution is nevertheless unitary in regions of unbroken antilinear
symmetry, stems from the unique phenomenology that they can exhibit. This
unique phenomenology, in part, originates from the existence of so-called
exceptional points, which mark the boundaries between regimes of broken and
unbroken antilinear symmetry, and which cannot be reproduced by Hermitian
theories.

In this work, we study one such unique phenomenology, which we previously
identified in the scalar QFT composed of two complex scalar fields with a non-
Hermitian mass mixing Chernodub and Millington (2022): that non-Hermitian QFTs
with local Lagrangian parameters naturally lead to the emergence of a so-
called “similarity” gauge field, momentum-dependent exceptional points, a new
type of high-energy instability and a novel IR/UV mixing. Our aim here is to
revisit these effects in scalar and fermionic theories with local non-
Hermitian mass mixing matrices and to describe the associated phenomena of
superluminal, negative or vanishing group velocities, and anomalous
dispersion. We note that time-dependent non-Hermitian quantum mechanical
Hamiltonians have attracted significant attention Fring and Tenney (2021a, b);
Fring  _et al._ (2022). The phenomena described in this work are, however,
unique to the case of relativistic field theories.

We focus on a model with two scalars and another with two fermions that have
in common a similar non-Hermitian mass mixing, and highlight essential
differences in the properties of these two models compared to one composed of
a single Dirac fermion with a parity-odd, anti-Hermitian mass term. We
consider local similarity transformations of the models and assume that the
mass matrix has the same eigenvalues at every point in spacetime, thereby
implying that the local “low-energy” measurements provide the same physical
masses regardless of the time (spatial coordinate) when (where) this
measurement is performed. Here, the term “low-energy” means that this
statement is true for energies lower than the energy of typical ultra-high
energy cosmic rays ($10^{18}$ eV), thereby encompassing most observations to
date. Similar to Ref. Chernodub and Millington (2022), we assume a spacetime
dependence of the mass matrix, which, together with the fixed mass
eigenvalues, corresponds to a spacetime-dependent rotation of the model in the
“similarity” space.

While we find that the ground state of the single Dirac fermion becomes
unstable, the low-energy modes, including the ground state, of the two-scalar
and two-fermion models with non-Hermitian mass mixing are stable. The
instability appears instead in the high-energy regime, implying potentially
interesting consequences for the propagation of ultra-high-energy cosmic rays
or for neutrino models with time- and space-varying mass matrices Dvali  _et
al._ (2023).

This article is structured as follows. We start our discussion in Sec. II with
a review and further update of the properties of the non-Hermitian model that
describes a two-component complex-valued scalar field characterised by
spatially inhomogeneous parameters. The local nature of the similarity map in
this benchmark model allows us to introduce a novel kind of gauge field,
dubbed the “similarity gauge field” in Ref. Chernodub and Millington (2022).
In Sec. III, we then follow the same strategy of introducing the similarity
gauge field to a non-Hermitian fermionic model with a single flavour, finding
that this single-flavour model gives us phenomenologically unacceptable
results. In detail, we describe the theory of a single massive fermion with an
anti-Hermitian, parity-odd mass term, involving the fifth gamma matrix. We
promote its mass parameters to local functions and show that this
straightforward generalisation leads to an unstable spectrum in the would-be
$\mathcal{PT}$ unbroken regime.

In Sec. IV, we generalise the one-fermion model of Sec. III to two fermion
flavours, following the strategy successfully tested for the two-component
scalar model in Sec. II. We promote the mass parameters to local functions in
Sec. IV.2 and derive the dispersion relations for physical excitations in Sec.
V, wherein we identify the high-momenta instabilities, anomalous dispersion
and superluminal regimes, similar to the case of the doublet scalar model
discussed at the beginning of the paper. The potential phenomenological
relevance of our construction is described in Sec. VI. Section VII outlines
our conclusions, and additional technical details are provided in the
Appendices.

##  II Two-Flavour Scalar Model

In this section, we revisit the scalar model with local mass parameters,
previously analysed in Ref. Chernodub and Millington (2022). Our aim is to
show that, aside from the novel IR/UV mixing effect and resulting
instabilities described in Ref. Chernodub and Millington (2022), certain modes
also exhibit anomalous dispersion and superluminal propagation.

The Lagrangian of the model (originally introduced in Ref. Alexandre  _et al._
(2017b)) is

$\mathcal{L}_{\Phi}=\partial_{\mu}\tilde{\Phi}^{{\dagger}}\partial^{\mu}\Phi-\tilde{\Phi}^{{\dagger}}M^{2}\Phi\;,$
(1)

where $\Phi=(\phi_{1},\phi_{2})$ and

$M^{2}=\begin{pmatrix}m_{1}^{2}&m_{5}^{2}\\\
-m_{5}^{2}&m_{2}^{2}\end{pmatrix}\neq M^{2{\dagger}}$ (2)

is pseudo-Hermitian with

$PM^{2}P=M^{2{\dagger}}\;,\qquad P=P^{-1}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\;.$ (3)

We take $m_{1}^{2}>m_{2}^{2}>0$ and $m_{5}^{2}>0$. The conjugate doublet
$\tilde{\Phi}^{{\dagger}}=(\tilde{\phi}_{1}^{{\dagger}},\tilde{\phi}_{2}^{{\dagger}})$
is defined via Sablevice and Millington (2023)

$\tilde{\Phi}^{{\dagger}}(x)=\eta^{-1}\Phi^{{\dagger}}(x_{\eta})\eta\pi\;,$
(4)

where, for example, we can take $\eta=\mathcal{P}$ to be the parity operator
and $\pi=P={\rm diag}(1,-1)$ to be the parity matrix, such that the complex
field $\phi_{1}$ transforms as a scalar and the complex field $\phi_{2}$ as a
pseudoscalar.111Since the field and its usual Hermitian conjugate evolve with
the Hamiltonian $H$ and its Hermitian conjugate $H^{{\dagger}}\neq H$,
respectively, a Lagrangian formulated in terms of these variables would lead
to inconsistent Euler–Lagrange equations Alexandre  _et al._ (2017b). The
discrepancy between the generator of time translations Chernodub and
Millington (2022) for the two fields would lead to further inconsistencies.
Most notably, such a Lagrangian would not transform properly under the
Poincaré group Sablevice and Millington (2023). This choice matches Ref.
Alexandre  _et al._ (2020a), but it has the disadvantage that the momentum
operator is no longer Hermitian and the operator $\eta$ is not sufficient for
constructing a positive-definite inner product Sablevice and Millington
(2023). Alternatively, we can take $\eta=\mathcal{P}\mathcal{A}$ and $\pi=PA$
Sablevice and Millington (2023), where $\mathcal{A}$ is the additional
discrete symmetry of the Lagrangian with

$A^{-1}M^{2}A=M^{2}\;.$ (5)

Either way, since the equations of motion of this theory are linear, the exact
choice does not impact the arguments presented in this work.

The squared mass eigenvalues are

$M_{\pm}^{2}=\frac{m_{1}^{2}+m_{2}^{2}}{2}\pm\sqrt{\left(\frac{m_{1}^{2}-m_{2}^{2}}{2}\right)^{2}-m_{5}^{4}}\;,$
(6)

and these are real when the parameter

$\zeta\equiv\frac{2m_{5}^{2}}{m_{1}^{2}-m_{2}^{2}}\leq 1\;.$ (7)

With these definitions, the matrix $A$ is given by

$A=\frac{1}{\sqrt{1-\zeta^{2}}}\begin{pmatrix}1&\zeta\\\
-\zeta&-1\end{pmatrix}\,,\quad
PA=\frac{1}{\sqrt{1-\zeta^{2}}}\begin{pmatrix}1&\zeta\\\
\zeta&1\end{pmatrix}\,.$ (8)

Note that $PAP=A^{\mathsf{T}}$ and $A^{-1}=A$.

The eigenvectors of the mass matrix are given by Alexandre  _et al._ (2017b)

$\mathbf{e}_{+}=\begin{pmatrix}\cosh\xi_{\rm H}\\\ -\sinh\xi_{\rm
H}\end{pmatrix}\;,\qquad\mathbf{e}_{-}=\begin{pmatrix}\sinh\xi_{\rm H}\\\
-\cosh\xi_{\rm H}\end{pmatrix}\;,$ (9)

where $\xi_{\rm H}=\frac{1}{2}{\rm arctanh}\,\zeta$. We can readily confirm
that these eigenvectors are orthonormal with respect to the inner product

$\displaystyle{\left\langle{\bm{a}},{\bm{b}}\right\rangle}_{{\mathcal{APT}}}={\bm{a}}^{{\mathcal{APT}}}\cdot{\bm{b}}={\bm{a}}^{*}\cdot
P\cdot A\cdot{\bm{b}}\,,$ (10)

viz.

$\displaystyle{\left\langle{\mathbf{e}}_{+},{\mathbf{e}}_{+}\right\rangle}_{{\mathcal{APT}}}$
$\displaystyle=$
$\displaystyle{\left\langle{\mathbf{e}}_{-},{\mathbf{e}}_{-}\right\rangle}_{{\mathcal{APT}}}=1\,,$
(11)
$\displaystyle{\left\langle{\mathbf{e}}_{+},{\mathbf{e}}_{-}\right\rangle}_{{\mathcal{APT}}}$
$\displaystyle=$
$\displaystyle{\left\langle{\mathbf{e}}_{-},{\mathbf{e}}_{+}\right\rangle}_{{\mathcal{APT}}}=0\,.$
(12)

Since the squared mass matrix is non-Hermitian, it is diagonalised by a
similarity transformation of the form

$M_{\rm diag}^{2}=S_{\rm H}^{-1}M^{2}S_{\rm H}\;,$ (13)

where

$S_{\rm H}=\begin{pmatrix}\cosh\xi_{\rm H}&-\sinh\xi_{\rm H}\\\ -\sinh\xi_{\rm
H}&\cosh\xi_{\rm H}\end{pmatrix}\;.$ (14)

Note that the similarity matrix (14) is related to the matrices (8) and (3) as
follows:

$\displaystyle S_{\rm H}\cdot S_{\rm H}=A\cdot P\;.$ (15)

In this way, the Hermitian Lagrangian

$\mathcal{L}_{\Phi,{\rm
diag}}=\partial_{\mu}\Phi^{{\dagger}}\partial^{\mu}\Phi-\Phi^{{\dagger}}M^{2}_{\rm
diag}\Phi$ (16)

corresponds to one of an infinite one-parameter family of isospectral
Hamiltonians related via similarity transformations effected by the
transformation

$\displaystyle\Phi$ $\displaystyle\to S\Phi\;,$ (17)
$\displaystyle\tilde{\Phi}^{{\dagger}}$
$\displaystyle\to\tilde{\Phi}^{{\dagger}}S^{-1}\;,$ (18)

with

$S=\begin{pmatrix}\cosh\xi&-\sinh\xi\\\ -\sinh\xi&\cosh\xi\end{pmatrix}\;.$
(19)

That $\tilde{\Phi}^{{\dagger}}$ is not the Hermitian conjugate of $\Phi$,
except in the mass eigenbasis, is then manifest in the observation that
$S^{-1}\neq S^{{\dagger}}$.

The central idea of Ref. Chernodub and Millington (2022) was to make the
parameters $m_{1}^{2}=m_{1}^{2}(x)$, $m_{2}^{2}=m_{2}^{2}(x)$ and
$m_{5}^{2}=m_{5}^{2}(x)$ spacetime dependent, and to generalise the global
similarity transformation $S$ to a local similarity transformation with
$\xi=\xi(x)$. It is then apparent that the kinetic term is not invariant under
this local similarity transformation, but following the minimal coupling
procedure, it is possible to restore invariance of the kinetic term by
promoting the partial derivatives to covariant derivatives involving a
“similarity gauge field” $C_{\mu}$, i.e.,

$\partial_{\mu}\to D_{\mu}=\partial_{\mu}-\mathcal{C}_{\mu}={\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,\partial_{\mu}+\sigma_{1}C_{\mu}\equiv\begin{pmatrix}\partial_{\mu}&C_{\mu}\\\
C_{\mu}&\partial_{\mu}\end{pmatrix}\,.$ (20)

Under the similarity transformation, we have

$\mathcal{C}_{\mu}\to S\mathcal{C}_{\mu}S^{-1}-S\partial_{\mu}S^{-1}\;.$ (21)

Recalling, however, that the tilde-conjugate field $\tilde{\Phi}$ transforms
with $S^{-1}$, we also require the tilde-conjugate covariant derivative

$\tilde{D}_{\mu}=\partial_{\mu}+\mathcal{C}_{\mu}={\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,\partial_{\mu}-\sigma_{1}C_{\mu}\equiv\begin{pmatrix}\partial_{\mu}&-C_{\mu}\\\
-C_{\mu}&\partial_{\mu}\end{pmatrix}\,.$ (22)

The similarity-invariant Lagrangian then takes the form

$\mathcal{L}_{\Phi,\mathcal{C}}=\tilde{D}_{\mu}\tilde{\Phi}^{{\dagger}}D^{\mu}\Phi-\tilde{\Phi}^{{\dagger}}M^{2}\Phi\;.$
(23)

We note that the diagonalisation of the coordinate-dependent mass matrix,
effected by a coordinate-dependent similarity transformation that does not
commute with the kinetic term, leads to the emergence of a non-vanishing
similarity gauge field. Since the terms depending on the similarity gauge
field are non-Hermitian, we therefore conclude that the non-Hermitian theory
with local parameters cannot be mapped to a Hermitian theory, and it will
consequently exhibit genuine non-Hermitian phenomena that a Hermitian theory
cannot reproduce.

Following Ref. Chernodub and Millington (2022), we now consider the particular
case in which the coordinate-dependent mass parameters $m_{1}^{2}$,
$m_{2}^{2}$ and $m_{5}^{2}$ lead to coordinate-independent mass eigenvalues
$M_{\pm}^{2}$. For a constant but non-vanishing similarity gauge field, the
spectrum of the theory is then governed by

$\left(p^{2}-C^{2}-m_{1}^{2}\right)\left(p^{2}-C^{2}-m_{2}^{2}\right)+4(C\cdot
p)^{2}+m_{5}^{4}=0\;,$ (24)

where we have considered solutions to the Klein-Gordon equation of the form

$\Phi(x)=\varphi(p)e^{-ip\cdot x}\;.$ (25)

We adopt the standard conventions for the four-vector product $p\cdot
x=p_{\mu}x^{\mu}=\omega t-{\bm{p}}\cdot{\bm{x}}$, where
$p^{\mu}=(\omega,{\bm{p}})$ is the four-momentum and $x^{\mu}=(t,{\bm{x}})$ is
the spacetime coordinate. We work with the mostly minus Minkowski signature.
We have also introduced the following scalar Lorentz invariants: $p^{2}\equiv
p_{\mu}p^{\mu}=\omega^{2}-{\bm{p}}^{2}$, $C\cdot p\equiv
C_{\mu}p^{\mu}=C_{0}\omega-{\bm{C}}\cdot{\bm{p}}$ and $C^{2}\equiv C\cdot
C\equiv C_{\mu}C^{\mu}\equiv C_{0}^{2}-{\bm{C}}^{2}$.

A generic dispersion relation corresponds to a solution of Eq. (24) for a
global, coordinate-independent vector $C^{\mu}$, which is given by a root of
an algebraic equation of the 4th order. While such a solution is definitely
possible to obtain in an analytical form, its rather complicated structure
makes its further analytical analysis difficult. Therefore, we proceed below
by considering two cases of strictly temporal or strictly spatial
perturbations, where analytical solutions are provided by much simpler
expressions. Moreover, these cases cover all possible variants for $C^{\mu}$,
since a timelike or spacelike vector can be made, respectively, strictly
temporal or strictly spatial with the help of Lorentz transformations.

For the case of a timelike similarity field $C^{\mu}=(C_{0},\bm{0})$, the
spectrum is given by

$\displaystyle\omega_{\pm,\bm{p}}^{2}$
$\displaystyle=\bm{p}^{2}-C_{0}^{2}+\frac{m_{1}^{2}+m_{2}^{2}}{2}$
$\displaystyle\phantom{=}\pm\sqrt{\left(\frac{m_{1}^{2}-m_{2}^{2}}{2}\right)^{2}-m_{5}^{4}-2C_{0}^{2}(m_{1}^{2}+m_{2}^{2}+2\bm{p}^{2})}\;.$
(26)

Noting that

$\displaystyle m_{1}^{2}+m_{2}^{2}$ $\displaystyle=M_{+}^{2}+M_{-}^{2}\;,$
(27) $\displaystyle\left(\frac{m_{1}^{2}-m_{2}^{2}}{2}\right)^{2}-m_{5}^{4}$
$\displaystyle=\left(\frac{M_{+}^{2}-M_{-}^{2}}{2}\right)^{2}\;,$ (28)

we can readily confirm that the energies $\omega_{\pm,\bm{p}}$ are coordinate
independent. The modification to the eigenfrequencies leads to a corresponding
modification of the positive definite inner product. This is described in
Appendix A.

For sufficiently high momenta, the argument of the square root can become
negative, such that the energy eigenvalues are real for momenta below some
critical momentum $p_{c}$, and they come in complex-conjugate pairs for
momenta above this critical momentum. In this way, the theory exhibits
momentum-dependent exceptional points occurring at the critical momentum
Chernodub and Millington (2022), given by

$p_{c}^{2}=\frac{\left(M_{+}^{2}-M_{-}^{2}\right)^{2}}{16C_{0}^{2}}-\frac{M_{+}^{2}+M_{-}^{2}}{2}\;.$
(29)

Proceeding similarly for a purely spacelike similarity field
$C^{\mu}=(0,\bm{C})$, the spectrum is given by

$\displaystyle\omega_{\pm,\bm{p}}^{2}$
$\displaystyle=\bm{p}^{2}-\bm{C}^{2}+\frac{m_{1}^{2}+m_{2}^{2}}{2}$
$\displaystyle\phantom{=}\pm\sqrt{\left(\frac{m_{1}^{2}-m_{2}^{2}}{2}\right)^{2}-m_{5}^{4}-4\left(\bm{p}\cdot\bm{C}\right)^{2}}\;.$
(30)

In this case, the instability arises for modes with a component
$\bm{p}_{\parallel}$ parallel to $\bm{C}$ of magnitude greater than the
critical momentum $p_{\parallel,c}$ given by

$p_{\parallel,c}^{2}=\frac{\left(M_{+}^{2}-M_{-}^{2}\right)^{2}}{16\bm{C}^{2}}\;.$
(31)

Having defined the critical momenta above, the timelike and spacelike cases
can be expressed in the following convenient forms

$\omega_{\pm,\bm{p}}^{2}=\bm{p}^{2}+\overline{M}^{2}-\begin{cases}C_{0}^{2}\mp
2|C_{0}|\sqrt{p_{c}^{2}-\bm{p}^{2}}&\quad\text{timelike}\\\ \bm{C}^{2}\mp
2|\bm{C}|\sqrt{p_{\parallel,c}^{2}-\bm{p}_{\parallel}^{2}}&\quad\text{spacelike}\end{cases}\;,$
(32)

where

$\overline{M}^{2}=\frac{M_{+}^{2}+M_{-}^{2}}{2}$ (33)

is the average squared mass. Taking $C_{0}=C$ and $\bm{C}_{i}=C\delta_{i3}$
(along the $z$-direction), the group velocities take the form

$(\bm{v}_{\pm,\bm{p}})_{i}=\frac{1}{\omega_{\pm,\bm{p}}}\left[\bm{p}_{i}\mp
C\begin{cases}\frac{\bm{p}_{i}}{\sqrt{p_{c}^{2}-\bm{p}^{2}}}&\quad\text{timelike}\\\
\frac{p_{3}\delta_{i3}}{\sqrt{p_{c}^{2}-{p}_{3}^{2}}}&\quad\text{spacelike}\end{cases}\right]\;.$
(34)

For modes propagating in the $z$ direction, $\bm{p}=(0,0,p)$, and the group
velocities take the same form for the timelike and spacelike cases, i.e.,

$v_{\pm,p}=\frac{p}{\omega_{\pm,p}}\left[1\mp\frac{C}{\sqrt{p_{c}^{2}-p^{2}}}\right]\;.$
(35)

There exists another set of critical momenta $p_{\rm stop}$ and
$p_{\parallel,{\rm stop}}$ at which the group velocity of the $\omega_{+}$
mode vanishes. These critical momenta are given by

$p_{(\parallel,){\rm stop}}^{2}=p_{(\parallel,)c}^{2}-C^{2}\;.$ (36)

Moreover, for modes with the momenta in the intermediate range,
$p_{(\parallel,){\rm
stop}}^{2}<\bm{p}^{2}_{(\parallel)}<p_{(\parallel,)c}^{2}$, the group velocity
of the $\omega_{+}$ mode is negative.

In standard optical materials, the index of refraction $n$ increases with
increasing frequency $\omega$ (so that $\partial n/\partial\omega>0$). This
corresponds to “normal” dispersion. The latter is responsible, e.g., for the
usual ordering of the colours in a rainbow, from red at the top to violet at
the bottom. In the case of anomalous refractive index, $\partial
n/\partial\omega<0$, the order of colours would get reversed, from violet at
the top to red at the bottom.

Since the “$\pm$”-modes possess different dispersions, they will have
different refractive indices $n_{\pm,p}$ determined with respect to the
corresponding phase velocities $u_{\pm,p}=\omega_{\pm,p}/p$ as follows:

$\displaystyle n_{\pm,p}=\frac{1}{u_{\pm,p}}\equiv\frac{p}{\omega_{\pm,p}}\,.$
(37)

(We set the velocity of light to unity, $c=1$.) The condition for the
anomalous refractive index then reads as

$\displaystyle\frac{\partial n_{\pm,p}}{\partial
p}\equiv\frac{\partial}{\partial
p}\frac{p}{\omega_{\pm,p}}<0\,,\quad\text{[anomalous dispersion]}\,.$ (38)

The dispersion of the “$+$” mode is always normal. However, the dispersion of
the “$-$” mode becomes anomalous at a scale $p_{a}$ below both $p_{c}$ and
$p_{\rm stop}$. For completeness, the scale $p_{a}$ is given by

$\displaystyle p_{a}^{2}$
$\displaystyle=2p_{c}^{2}+\overline{M}^{2}-\frac{C^{2}}{2}-\frac{\overline{M}^{4}}{2C^{2}}$
$\displaystyle\phantom{=}+\frac{\overline{M}^{2}-C^{2}}{2C^{2}}\sqrt{\left(\overline{M}^{2}-C^{2}\right)^{2}-4C^{2}p_{c}^{2}}\;,$
(39)

where we have assumed $C^{2}<M^{2}$, needed to preserve the real-valuedness of
the dispersion for “$-$” mode at the vanishing momentum, ${\bm{p}}=0$ as
follows from Eq. (32).

The classical relation between the dispersion $n$ of a medium and the group
and phase velocities

$\displaystyle v_{\mathrm{gr}}=\frac{1}{n+\omega\frac{\partial
n}{\partial\omega}}\,,\qquad\ n=\frac{1}{v_{\mathrm{ph}}}=\frac{p}{\omega}\,,$
(40)

implies that the group velocity may exceed the speed of light if the anomalous
dispersion becomes too extreme, with $n+\omega\partial n/\partial\omega<1$.
Indeed, both modes $\omega_{\pm}$ exhibit superluminal propagation for momenta
above the corresponding scales

$p^{2}_{\rm
SL,\pm}=\frac{p_{c}^{2}}{\overline{M}^{4}}\left[\overline{M}^{4}\pm
2C^{2}p_{c}\sqrt{p_{c}^{2}+\overline{M}^{2}}-C^{2}(2p_{c}^{2}+\overline{M}^{2})\right]\;,$
(41)

where $p_{\rm SL}<p_{c}$. Note that the superluminal region for the mode
$\omega_{+}$ occurs when its group velocity is negative, such that $p_{{\rm
stop}}<p_{{\rm SL},+}<p_{c}$.

In Sec. IV, we will see that a fermionic model with non-Hermitian two-flavour
mixing exhibits analogous behaviours, wherein we provide a more comprehensive
exposition of these uniquely non-Hermitian phenomena. A complete description
of the analogous phase diagram and the critical points is given for this
phenomenologically interesting model in Sec. V.

##  III One-Flavour Fermionic Model

Before considering the fermionic analogue of the two-flavour scalar model of
the previous section, we focus on a simple non-Hermitian extension of the
Dirac Lagrangian Bender  _et al._ (2005); Alexandre  _et al._ (2015, 2017a,
2017b, 2018a), which has the following form:

$\displaystyle{\cal
L}_{\psi}=\bar{\psi}\left(i\not{\partial}-m-m_{5}\gamma^{5}\right)\psi\;,$
(42)

where $m$ corresponds to the Hermitian mass term for the fermion field $\psi$,
and $m_{5}$ provides us with the anti-Hermitian mass term. Note that the
pseudo-Hermiticity of the Lagrangian necessitates a redefinition of the dual
field $\bar{\psi}$, which is therefore not the usual Dirac-conjugate
$\bar{\psi}\neq\psi^{{\dagger}}\gamma^{0}$ (see Refs. Alexandre  _et al._
(2022); Sablevice and Millington (2023), cf. Refs. LeClair and Neubert (2007);
Ferro-Hernández  _et al._ (2023) in the case of second-order fermions).222As
noted for the scalar case, we remark that the precise definition of
$\bar{\Psi}$ has no impact on the dispersion relations relevant to this work,
since the fermionic theory is linear.

The Lagrangian (42) gives the following classical equations of motion:

$\displaystyle\left(i\not{\partial}-m-m_{5}\gamma^{5}\right)\psi$
$\displaystyle=$ $\displaystyle 0\,,$ (43)
$\displaystyle\bar{\psi}\left(i\overleftarrow{\not{\partial}}+m+m_{5}\gamma^{5}\right)$
$\displaystyle=$ $\displaystyle 0\,.$ (44)

By inspection, we see that the dual spinor is defined in terms of the spinor
$\tilde{\psi}^{{\dagger}}$ and not $\psi^{{\dagger}}$, where these differ by
$m_{5}\to-m_{5}$ (see Ref. Alexandre  _et al._ (2022)).

The positive-frequency solutions of the Dirac equation

$\displaystyle\psi(x)=u(p)e^{-ip\cdot x}$ (45)

are expressed via the spinor $u(p)$,333The explicit form of the four-spinor
can be found in Ref. Alexandre  _et al._ (2022). which satisfies

$\displaystyle\left({\not{p}}-m-m_{5}\gamma^{5}\right)u(p)=0\;.$ (46)

The self-consistency of Eq. (46) requires $p^{2}=M^{2}$ and determines the
energy spectrum ($p_{0}\equiv E_{\bm{p}}$) via

$\displaystyle\omega_{\bm{p}}^{2}={\bm{p}}^{2}+M^{2}\;,$ (47)

in which the squared mass of the fermionic excitation is

$\displaystyle M^{2}=m^{2}-m_{5}^{2}\;.$ (48)

We always take $|m|\geqslant|m_{5}|$ to ensure that the mass $M$ is a real
quantity. This range of parameters corresponds to the
“${\mathcal{PT}}$-symmetric” phase, in which the theory is stable. If the
Hermitian mass $|m|$ is smaller than the non-Hermitian mass $|m_{5}|$, then
the system resides in the “${\mathcal{PT}}$-broken” phase, which is
characterised by paired complex branches of fermionic energies that make the
vacuum unstable.

This theory, in fact, belongs to a one-parameter family of similar theories.
This can be expressed conveniently by writing

$\displaystyle m+m_{5}\gamma^{5}=Me^{2\gamma^{5}\theta}\;,$ (49)

with

$\displaystyle m=M\cosh 2\theta\qquad\text{and}\qquad m_{5}=M\sinh 2\theta\;,$
(50)

where $\theta$ is a real parameter, and the mass of the fermionic excitations
$M$ is given in Eq. (48). This family is connected by a corresponding non-
unitary and non-compact similarity transformation that takes the form

$\displaystyle\psi$ $\displaystyle\to
e^{\omega_{5}\gamma^{5}}\psi\qquad\text{and}\qquad{\bar{\psi}}\to{\bar{\psi}}\,e^{\omega_{5}\gamma^{5}}\;.$
(51)

Applying the similarity transformation (51) with the constant parameter
$\omega_{5}=-\theta$ to the original non-Hermitian Lagrangian (42), we obtain
the corresponding Hermitian Hamiltonian

$\displaystyle{\cal L}_{\psi,{\rm
H}}=\bar{\psi}\left(i\not{\partial}-M\right)\psi.\;$ (52)

Notice that the kinetic term of the Dirac Lagrangian is invariant under the
similarity transformation (51).

We now assume that both the Hermitian and non-Hermitian masses in the original
Lagrangian (42) are functions of the spacetime coordinate $x^{\mu}$, i.e.,
$m=m(x)$ and $m_{5}=m_{5}(x)$, such that

$\displaystyle m(x)+m_{5}(x)\gamma^{5}=Me^{2\gamma^{5}\theta(x)}.$ (53)

In this way, the mass of the fermionic excitation (48) remains a coordinate-
independent quantity. Still, the original non-Hermitian Lagrangian (42) can no
longer be mapped to its Hermitian analogue (52), as we will see below.

In order to incorporate the coordinate dependence of the mass parameters $m$
and $m_{5}$, we need to promote the global similarity transformation (51) to a
local transformation. To this end, we introduce the new axial vector
similarity gauge field $C_{\mu}$ and promote the Lagrangian to

$\displaystyle{\cal
L}_{\psi,C}=\bar{\psi}\left(i\not{\partial}+i\not{C}\gamma^{5}-m-m_{5}\gamma^{5}\right)\psi\;.$
(54)

Under the local version of the similarity transformation in Eq. (51), the
similarity gauge field transforms as

$\displaystyle C_{\mu}$ $\displaystyle\to C_{\mu}-\partial_{\mu}\omega_{5}\;.$
(55)

The parameter $\omega_{5}$ originates from a non-compact gauge group. The
similarity gauge field $C_{\mu}$ is, therefore, a non-compact gauge field,
which, e.g., will not contain Abelian monopole-like singularities.

We now turn to the case of a constant similarity gauge field. For a purely
spacelike field $C^{\mu}=(0,{\bm{C}})$, the spectrum is

$\displaystyle\omega^{2}_{\pm,\bm{p}}$ $\displaystyle=$ $\displaystyle
m^{2}-m_{5}^{2}+{\bm{p}}^{2}-{\bm{C}}^{2}$ (56) $\displaystyle\pm
2i\sqrt{(m^{2}-m^{2}_{5}){\bm{C}}^{2}+({\bm{p}}\cdot{\bm{C}})^{2}}\;.$

While this spectrum functionally resembles the phenomenologically interesting
energy dispersion of the scalar doublet model (32), it possesses both unstable
and dissipative modes at any momentum $\bm{p}$. This statement applies even
for the $\mathcal{PT}$-broken regime with $m^{2}-m^{2}_{5}>0$. The complex-
valuedness of the energy spectrum renders this model of less interest for
particle phenomenology.

Unexpectedly, the spectrum (56) nevertheless contains extended “islands of
stability” in momentum space in the would-be $\mathcal{PT}$-broken regime for
which $m^{2}-m_{5}^{2}<0$. In this case, the spectrum always has an unstable
low-momentum region as both $\pm$ branches of the squared energy dispersion
(56) are negative at vanishing momentum, i.e., $\omega^{2}_{\pm}(\bm{p}=0)<0$.
However, higher-momentum modes become stable at higher momenta. To illustrate
these properties, we take a $\mathcal{PT}$-broken set of masses, i.e.,
$m_{5}^{2}-m^{2}=C_{3}^{2}>0$, with the vector similarity field pointing along
the $z$ axis, such that ${\bm{C}}=(0,0,C_{3})$. The stability region is given
by the overlap of the two shaded regions of Fig. 1, which correspond to the
stable domains for the $\omega_{+}$ and $\omega_{-}$ modes, respectively.

Figure 1: An example of the stability regions of the one-fermion spectrum (56)
in the plane of longitudinal $p_{\|}\equiv p_{3}$ and transverse
$p_{\perp}=\pm\sqrt{p_{1}^{2}+p_{2}^{2}}$ components of momenta for
$m_{5}^{2}-m^{2}=C_{3}^{2}>0$. The stability island of the $\omega_{+}$ mode
(shown in green) lies within a more extended region of stability for the
$\omega_{-}$ mode (shown in blue). In the uncoloured region, both modes are
unstable.

In this way, we find that this model exhibits momentum-dependent exceptional
points, occurring along the boundaries of these regions of stability, as were
found for the case of the non-Hermitian scalar field theory with local mass
parameters Chernodub and Millington (2022), reviewed in Sec. II.

For a purely timelike field $C^{\mu}=(C_{0},{\bm{0}})$, the spectrum is
instead

$\displaystyle\omega^{2}_{\pm,\bm{p}}=m^{2}-m_{5}^{2}+\left(|{\bm{p}}|\pm
iC_{0}\right)^{2}\;.$ (57)

For mass parameters in the physical $\mathcal{PT}$-unbroken regime
($m^{2}>m_{5}^{2}$), these spectra share the same property: they include four
complex eigenmodes, which are pairwise related to each other by complex
conjugation. In both cases, the energy branches at nonzero momentum,
${\bm{p}}^{2}>0$, always include both dissipative and unstable modes, which is
an unacceptable feature for a phenomenological appropriate model.

Summarising this section, we stress that the IR instabilities of the one-
flavour fermionic model equipped with the similarity gauge field seriously
limit its phenomenological viability. In the next section, however, we will
show that the two-flavour generalisation of the single-fermion model does not
exhibit the same problems at low momenta and, therefore, can be useful in the
non-Hermitian extensions of the Standard Model of particle physics.

##  IV Two-flavour fermionic model

In this section, we consider the following two-flavour generalisation of the
non-Hermitian Dirac Lagrangian:

$\displaystyle{\cal
L}_{\Psi}=\sum_{a=1}^{2}{\bar{\psi}}_{a}\left(i\not{\partial}-m_{a}\right)\psi_{a}-m_{5}\left({\bar{\psi}}_{1}\gamma^{5}\psi_{2}+{\bar{\psi}}_{2}\gamma^{5}\psi_{1}\right)\;.$
(58)

We take $m_{1}>m_{2}>0$ and $m_{5}>0$ without loss of generality. The model
(59) mimics closely its bosonic counterpart with two species Alexandre  _et
al._ (2017b); in fact, the two systems can be considered supersymmetric
partners Alexandre  _et al._ (2020b).

If we take $\psi_{1}$ to be right-chiral and $\psi_{2}$ to be left-chiral,
such that $\gamma^{5}\psi_{1}=+\psi_{1}$ and $\gamma^{5}\psi_{2}=-\psi_{2}$,
the Lagrangian can be written in the more convenient form

$\displaystyle{\cal
L}_{\Psi}=\bar{\Psi}\left(i\not{\partial}-{\hat{M}}\right)\Psi\;,$ (59)

where $\Psi=(\psi_{1},\psi_{2})^{T}$ is the doublet of fermionic fields, and
the mass matrix takes the form

$\displaystyle{\hat{M}}=\left(\begin{array}[]{rr}m_{1}&\phantom{-}m_{5}\\\
-m_{5}&m_{2}\end{array}\right)\;.$ (62)

Its eigenvalues are

$\displaystyle
M_{\pm}=\frac{1}{2}\left(m_{1}+m_{2}\pm\sqrt{(m_{1}-m_{2})^{2}-4m_{5}^{2}}\right)\;,$
(63)

and the energy dispersion relations take the standard relativistic form

$\displaystyle\omega_{\pm,\bm{p}}^{2}={\bm{p}}^{2}+M_{\pm}^{2}\;,$ (64)

implying that Eq. (63) represents the physical masses of the excitations in
the system. The ${\mathcal{PT}}$-unbroken domain of the model (59) occurs when
the mass parameters satisfy

$\displaystyle 4m_{5}^{2}<(m_{1}-m_{2})^{2}\,,$ (65)

which is illustrated in Fig. 2.444 Notice that a negative value of a fermionic
(as well as bosonic) mass does not have any physical consequences because the
fermions (bosons) with the masses $m$ and $-m$ have the same energy
dispersions (64).

Figure 2: The ${\mathcal{PT}}$-unbroken regions (shown in the green colour)
for the model (59) with the mass matrix (62) at $m_{5}\neq 0$.

Similarly to the two-scalar case, it is convenient to introduce the non-
Hermiticity parameter

$\displaystyle\zeta\equiv\frac{2m_{5}}{m_{1}-m_{2}}\;,$ (66)

which determines the deviation of the model from the Hermitian point. Exactly
at a vanishing value $\zeta=0$, the two fermionic flavours decouple, and the
model becomes Hermitian. For $0<|\zeta|<1$, the model is non-Hermitian but
resides in the domain of unbroken ${\mathcal{PT}}$ symmetry, where the mass
eigenvalues are real. For $|\zeta|>1$, the mass eigenvalues (63) are complex,
and the ${\mathcal{PT}}$-symmetry is broken. At the points $\zeta=\pm 1$, the
squared mass eigenvalues merge, and the mass matrix becomes defective. At this
exceptional point, which occurs at the boundary between the regimes of broken
and unbroken ${\mathcal{PT}}$ symmetry, the fermions become degenerate, and
the model acquires a global $U(2)$ symmetry.

###  IV.1 Global similarity transformation

In the ${\mathcal{PT}}$-unbroken regime, the two-fermion model (59) with
spacetime-independent mass matrix can be mapped, via a global similarity
transformation, to a Hermitian model. In order to demonstrate this property,
it is convenient to consider the following parametrisation of the mass matrix
(62):

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle M+m\cosh 2\varkappa\,,$
(67a) $\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle M-m\cosh
2\varkappa\,,$ (67b) $\displaystyle m_{5}$ $\displaystyle=$ $\displaystyle
m\sinh 2\varkappa\,,$ (67c)

which reduces Eq. (62) to

$\displaystyle{\hat{M}}=M\left(\begin{array}[]{rr}1&0\\\
0&1\end{array}\right)+m\left(\begin{array}[]{rr}\cosh
2\varkappa&\phantom{-}\sinh 2\varkappa\\\ -\sinh 2\varkappa&\cosh
2\varkappa\end{array}\right).$ (72)

In this parametrisation, the physical eigenmasses (63) take the following
simple form:

$\displaystyle M_{\pm}=M\pm m\,,$ (73)

where, according to Eq. (63),

$\displaystyle M=\frac{m_{1}+m_{2}}{2}\quad\text{and}\quad
m=\frac{1}{2}\sqrt{(m_{1}-m_{2})^{2}-4m_{5}^{2}}\;.$ (74)

It is important to notice that the physical spectrum (73) does not depend on
the parameter $\varkappa$ that labels the similarity degeneracy of the non-
Hermitian model. In the ${\mathcal{PT}}$-unbroken regime, the mass parameter
$m$ is a real-valued quantity, whereas, in the broken phase, $m$ acquires an
imaginary contribution. Notice that the standard non-Hermiticity parameter
$\zeta$ defined in Eq. (66) is related to the parameter $\varkappa$ as
$\zeta=\tanh 2\varkappa$. The exceptional points $\zeta=\pm 1$ correspond to
the asymptotic limits $\varkappa\to\pm\infty$, respectively.

Notice that the model (59) becomes Hermitian for purely imaginary
$\varkappa=\pm i|\varkappa|$ (with $M$ and $m$ real), since the off-diagonal
terms become
$i\left({\bar{\psi}}_{1}\psi_{2}-{\bar{\psi}}_{2}\psi_{1}\right)$.
Consequently, the mass eigenvalues

$\displaystyle M^{({\rm
Herm})}_{\pm}=\frac{1}{2}\left(m_{1}+m_{2}\pm\sqrt{(m_{1}-m_{2})^{2}+4|m_{5}|^{2}}\right)$
(75)

are real-valued for all values of the mass parameters, in contrast to their
non-Hermitian counterparts (63).

We now turn our attention to the global similarity transformation

$\displaystyle\begin{array}[]{l}{\mathrm{Global}}\\\ {\mathrm{similarity}}\\\
{\mathrm{transform}}\end{array}\\!:\quad\left\\{\begin{array}[]{lcl}\Psi&\to&{S}\Psi\\\
\bar{\Psi}&\to&\bar{\Psi}{S}^{-1}\end{array}\right.\;,$ (81)

which maps the non-Hermitian Lagrangian (59) into a Hermitian one of the form

$\displaystyle{\cal L}_{\Psi,{\rm
diag}}=\bar{\Psi}\left(i\not{\partial}-{\hat{M}}_{\rm diag}\right)\Psi\;,$
(82)

containing the diagonalised mass matrix

$\displaystyle{\hat{M}}_{\rm
diag}\equiv{\mathrm{diag}}(M_{+},M_{-})={S}^{-1}{\hat{M}}{S}\;.$ (83)

The similarity transformation is represented by a $SU(1,1)$ matrix ${S}$,
which operates in the isospace diagonal in the spinor space. It takes the
explicit form

$\displaystyle{S}\equiv{S}(\varkappa)=e^{-\varkappa\sigma_{1}}\equiv\begin{pmatrix}\phantom{-}\cosh\varkappa&-\sinh\varkappa\\\\[2.84526pt]
-\sinh\varkappa&\phantom{-}\cosh\varkappa\end{pmatrix}\,.$ (84)

Notice that for a real-valued parameter $\varkappa$ (i.e., in the
${\mathcal{PT}}$-unbroken regime), $S$ is Hermitian but not unitary, i.e.,
${S}^{\dagger}\equiv{S}$ and ${S}^{\dagger}\neq{S}^{-1}$, with

$\displaystyle{S}^{-1}\equiv{S}(-\varkappa)=e^{\varkappa\sigma_{1}}\equiv\begin{pmatrix}\cosh\varkappa&\sinh\varkappa\\\\[2.84526pt]
\sinh\varkappa&\cosh\varkappa\end{pmatrix}\,.$ (85)

The multiplication rule

$\displaystyle{S}(\varkappa_{1})\cdot{S}(\varkappa_{2})={S}(\varkappa_{1}+\varkappa_{2})\;,$
(86)

implies that ${S}$ belongs to an Abelian (Cartan) subgroup of the $SU(1,1)$
group.

The eigenvectors of the mass matrix ${\mathbf{e}}_{\pm}$, satisfying

$\displaystyle{\hat{M}}{\mathbf{e}}_{\pm}=M_{\pm}{\mathbf{e}}_{\pm}\;,$ (87)

are equivalent to their bosonic analogues Alexandre  _et al._ (2017b), see
Sec. II; namely,

$\displaystyle{\mathbf{e}}_{+}=\begin{pmatrix}\phantom{-}\cosh\varkappa\\\\[2.84526pt]
-\sinh\varkappa\end{pmatrix},\qquad{\mathbf{e}}_{-}=\begin{pmatrix}\phantom{-}\sinh\varkappa\\\\[2.84526pt]
-\cosh\varkappa\end{pmatrix}.\quad$ (88)

Given that they are the eigenvectors of a non-Hermitian matrix, these vectors
are not mutually orthogonal with respect to the ordinary Dirac inner product.
Therefore, we introduce the auxiliary matrix

$\displaystyle A=\begin{pmatrix}\phantom{-}\cosh 2\varkappa&\phantom{-}\sinh
2\varkappa\\\\[2.84526pt] -\sinh 2\varkappa&-\cosh 2\varkappa\end{pmatrix}\,,$
(89)

with $A\cdot A={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$. This allows us to identify a
$\mathcal{APT}$ inner product that mirrors the scalar case in Sec. II, where
the matrix

$\displaystyle P\equiv
P^{-1}=\sigma_{3}\equiv\begin{pmatrix}1&0\\\\[2.84526pt] 0&-1\end{pmatrix}\,,$
(90)

is used to fulfil the analogous pseudo-Hermiticity condition

$\displaystyle P\cdot{\hat{M}}\cdot P={\hat{M}}^{\sf T}$ (91)

for the skew-symmetric mass matrix (62). Here, the operator $\sf T$ denotes
the matrix transpose.

In the next section, as was done for the scalar model in Sec. II and the
single Dirac model in Sec. III, we will make the mass parameters of this two-
flavour model coordinate dependent and gauge the corresponding similarity
transformation (81).

###  IV.2 Local similarity transformation

To gauge the similarity group, we promote the global parameter $\varkappa$,
entering the similarity matrix (84), to a local, spacetime-dependent quantity,
$\varkappa=\varkappa(x)$. The similarity transformation becomes a local
transformation $S=S(x)\equiv S[\varkappa(x)]$, which requires, in analogy with
the usual gauge invariance, the appearance of a new vector matrix-valued
similarity gauge field ${\cal C}^{\mu}$.

The usual derivative is promoted to the covariant one, $\partial_{\mu}\to
D_{\mu}$, with

$\displaystyle D_{\mu}={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\,\partial_{\mu}-{\cal
C}_{\mu}\equiv{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,\partial_{\mu}+\sigma_{1}C_{\mu}\equiv\begin{pmatrix}\partial_{\mu}&C_{\mu}\\\
C_{\mu}&\partial_{\mu}\end{pmatrix}\,.$ (92)

The similarity gauge field ${\cal C}_{\mu}\equiv-\sigma_{1}C_{\mu}$ transforms
under the local gauge similarity transformation as follows:

$\displaystyle{\cal C}_{\mu}\to{S}{\cal
C}_{\mu}{S}^{-1}-{S}\partial_{\mu}{S}^{-1}\,.$ (93)

The vector field $C_{\mu}$ that enters the covariant derivative (92)
transforms under the local similarity transformation (93) as a $U(1)$ Cartan
gauge field:

$\displaystyle C_{\mu}\to C_{\mu}+\partial_{\mu}\varkappa\,,$ (94)

where we have used the relation
$S^{-1}\partial_{\mu}S=-\sigma_{1}\partial_{\mu}\varkappa$.

The similarity-gauged fermionic model (59) then acquires the following form:

$\displaystyle{\cal
L}_{\Psi,\mathcal{C}}=\bar{\Psi}\left(i\not{D}-{\hat{M}}\right)\Psi\,,$ (95)

where the superscript “$C$” indicates that the model is similarity-gauged. The
covariant derivative is given in Eq. (92), and $\not{D}=\gamma^{\mu}D_{\mu}$.
Written explicitly, the Lagrangian (95) is

$\displaystyle{\cal L}_{\Psi,\mathcal{C}}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{2}{\bar{\psi}}_{a}\left(i\not{\partial}-m_{a}\right)\psi_{a}$
(96)
$\displaystyle+{\bar{\psi}}_{1}\left(m_{5}+{\not{C}}\right)\psi_{2}-{\bar{\psi}}_{2}\left(m_{5}-{\not{C}}\right)\psi_{1}\;,$

wherein we see that the similarity gauge field is associated with the off-
diagonal terms.

The local similarity transformation can be summarised as follows

$\displaystyle\begin{array}[]{l}{\mathrm{Local}}\\\ {\mathrm{similarity}}\\\
{\mathrm{transform}}\end{array}\\!:\quad\left\\{\begin{array}[]{lcl}\Psi(x)&\to&S(x)\Psi(x)\\\
\bar{\Psi}(x)&\to&\bar{\Psi}(x)S^{-1}(x)\\\
C_{\mu}(x)&\to&C_{\mu}(x)+\partial_{\mu}\varkappa(x)\end{array}\;,\right.\qquad$
(103)

where $S(x)\equiv S[\varkappa(x)]$ is the local similarity transformation that
depends on the arbitrary similarity parameter $\varkappa=\varkappa(x)$, and
the $SU(1,1)$ matrix $S$ is given in Eq. (84). The similarity transformation
law (103) is strikingly similar to a usual electromagnetic $U(1)$ gauge
transformation — cf. Eq. (170) below — with the “similarity field” $C_{\mu}$
playing the role of the gauge field.

It is not difficult to realise that the coupling of the similarity gauge field
$C_{\mu}$ to the fermion doublet in the action (95) — or, in the explicit
form, in Eq. (96) — is given by the non-Hermitian term. This property implies
that even in the absence of the off-diagonal non-Hermitian mass, viz.
$m_{5}=0$, the action (96) corresponds to a non-Hermitian theory provided the
vector similarity field is non-vanishing, i.e., $C_{\mu}\neq 0$.

###  IV.3 Similarity gauge field and similarity current

Before delving into the consequences of the presence of the similarity gauge
field, it is appropriate to ask whether this field can be treated as an
independent gauge field with its own kinetic term or if it must be considered
as a non-dynamical background.

The existence of the similarity gauge invariance (94) of the fermionic action
implies that the similarity gauge field $C_{\mu}$ transforms as the usual
$U(1)$ gauge field (103), thus suggesting that the fermionic action (96) can
be supplemented with the gauge-invariant kinetic term for the gauge field

$\displaystyle{\cal
L}_{\Psi,\mathcal{C}}\overset{?}{\supset}-\frac{1}{4g^{2}}f_{\mu\nu}f^{\mu\nu},\qquad
f_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu},\qquad$ (104)

where $g$ is a new gauge coupling. This observation seems to support the idea
that the gauge field $C_{\mu}$ can be quantised. However, as we show below,
the appearance of the kinetic term (104) makes the theory inconsistent, thus
forcing us to abandon the idea of a dynamical similarity gauge field and
consider the field $C_{\mu}$ only as a background (classical) gauge field.

The similarity gauge field couples to the similarity current
$J^{\mathrm{(S)}}_{\mu}$ in the same way that the usual $U(1)$ photon gauge
field couples to the electric current. The similarity current is given by a
variation of the non-Hermitian matter action (95) with respect to the
similarity gauge field $C_{\mu}$ with the result

$\displaystyle
J^{\mathrm{(S)}}_{\mu}={\bar{\Psi}}\gamma^{\mu}\sigma^{1}\Psi\equiv{\bar{\psi}}_{1}\gamma^{\mu}\psi_{2}+{\bar{\psi}}_{2}\gamma^{\mu}\psi_{1}\,.$
(105)

Using the classical equations of motion for non-Hermitian fermions (95)

$\displaystyle\left(i{\vec{\not}{D}}-{\hat{M}}\right)\Psi$ $\displaystyle=$
$\displaystyle 0\,,$ (106a)
$\displaystyle{\bar{\Psi}}\left(i{{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\not{D}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\not{D}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\not{D}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\not{D}$\cr}}}}}-{\hat{M}}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (106j)

one can easily check that the similarity current, contrary to the electric
current, is not conserved; namely,

$\displaystyle\partial^{\mu}J^{\mathrm{(S)}}_{\mu}={\bar{\Psi}}\left[2im_{5}\sigma_{3}+(m_{1}-m_{2})\sigma_{2}\right]\Psi\,.$
(107)

Notice that this current is not conserved also in the Hermitian limit
$m_{5}\to 0$. The non-conservation property implies that the similarity gauge
field $C_{\mu}$ must necessarily be made non-propagating so that the kinetic
term (104) should not appear in the action. Otherwise, a variation of the
action with respect to the gauge field would produce an inconsistent Maxwell-
like equation $\partial^{\nu}f_{\mu\nu}=J^{\mathrm{(S)}}_{\mu}$ in which the
right-hand-side would have a zero divergence while the left-hand-side would
not. Notice that the conservation of the similarity current (107) is achieved
in a trivial limit when and only when the mass matrix (62) becomes diagonal
with equal eigenvalues.

Thus, the gauge similarity field $C_{\mu}$ should be considered as
parametrising some classical background, and could not provide, e.g., a
candidate dark photon (for a review, see Ref. Fabbrichesi  _et al._ (2021)).

###  IV.4 Constant physical masses and varying similarity background

Proceeding as we did for the two-flavour scalar model in Sec. II and the
single Dirac fermion in Sec. III, we consider the non-uniform non-Hermitian
mass matrix

$\displaystyle{\hat{M}}(x)=\left(\begin{array}[]{rr}m_{1}(x)&\phantom{-}m_{5}(x)\\\
-m_{5}(x)&m_{2}(x)\end{array}\right)\,.$ (110)

The eigenvalues of this mass matrix can then be obtained with the help of a
local similarity transformation:

$\displaystyle M_{\pm}(x)$ $\displaystyle=$ $\displaystyle M(x)\pm m(x)\,,$
(111) $\displaystyle{\hat{M}}_{\rm diag}(x)$ $\displaystyle=$
$\displaystyle{\mathrm{diag}}\left(M_{+}(x),M_{-}(x)\right)\,,$ (112)

wherein we have used the parametrisation (72) with $M=M(x)$, $m=m(x)$, and
$\varkappa=\varkappa(x)$.

In general, the physical masses (112) are also spacetime-inhomogeneous
quantities. Motivated by experimental and observational constraints on the
spacetime variation of fundamental parameters, we again consider the case in
which the physical masses are spacetime-independent. This is implemented
straightforwardly in the parametrisation (72) by setting the mass parameters
$M$ and $m$ to be constant quantities ($M_{0}$ and $m_{0}$, respectively),
thus endowing only the similarity scalar field $\varkappa$ with a coordinate
dependence, i.e., $\varkappa=\varkappa(x)$. The mass matrix (110) can then be
written in the following form:

$\displaystyle{\hat{M}}(x)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}M_{0}&0\\\ 0&M_{0}\end{pmatrix}$
$\displaystyle+\,m_{0}\left(\begin{array}[]{rr}\cosh
2\varkappa(x)&\phantom{-}\sinh 2\varkappa(x)\\\ -\sinh 2\varkappa(x)&-\cosh
2\varkappa(x)\end{array}\right)$ $\displaystyle\equiv$ $\displaystyle
M_{0}\,{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}+m_{0}\big{[}\sigma_{3}\cosh
2\varkappa(x)+i\sigma_{2}\sinh 2\varkappa(x)\big{]}\,.$

As we have already noticed, despite the spatial inhomogeneity of the mass
matrix (IV.4), the eigenvalues [cf. (73)]

$M_{0}^{(\pm)}=M_{0}\pm m_{0}$ (117)

are then constant quantities. Alternatively, in units of the entries of the
mass matrix (62), the parametrisation (IV.4) implies that the following mass
combinations are constant:

$\displaystyle m_{1}(x)+m_{2}(x)$ $\displaystyle=$ $\displaystyle 2M_{0}\,,$
(118) $\displaystyle{}[m_{1}(x)-m_{2}(x)]^{2}-4m_{5}^{2}(x)$ $\displaystyle=$
$\displaystyle 4m_{0}^{2}\,.$ (119)

Without loss of generality, we take $M_{0}>0$ and $m_{0}>0$.

The non-Hermitian mass matrix (IV.4) can be readily diagonalised, leading us
to the Lagrangian

$\displaystyle{\cal L}_{\Psi,\mathcal{C},{\rm
diag}}=\bar{\Psi}\left(i{\hat{\not}{D}}-{\hat{M}}_{\rm
diag}(x)\right)\Psi\equiv\bar{\Psi}{\hat{K}}\Psi\,,$ (120)

where

$\displaystyle{\hat{K}}=\begin{pmatrix}i\not{\partial}-M_{+}&-i\not{\partial}\varkappa\\\
-i\not{\partial}\varkappa&i\not{\partial}-M_{-}\end{pmatrix}\,,$ (121)

and wherein we recognise the similarity field
$C_{\mu}=\partial_{\mu}\varkappa(x)$. The structure (121) represents a non-
Hermitian operator in which the violation of Hermiticity is determined by the
magnitude of the off-diagonal terms.

In order to proceed further, it is convenient to work in a linear
approximation with a slowly varying scalar similarity parameter

$\displaystyle\varkappa(x)=\varkappa_{0}+C_{\mu}x^{\mu}\;,$ (122)

where $\varkappa_{0}$ is a background (coordinate-independent) quantity and
the linear variation of the scalar similarity field is given by the vector
$C^{\mu}$.

In terms of the masses (IV.4), the inhomogeneous background (122) corresponds
to the following mass matrix:

$\displaystyle{\hat{M}}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}M_{0}+m_{0}\cosh
2\varkappa_{0}&m_{0}\sinh 2\varkappa_{0}\\\ -m_{0}\sinh
2\varkappa_{0}&M_{0}-m_{0}\cosh 2\varkappa_{0}\end{array}\right)$ (128)
$\displaystyle+\,m_{0}\left(\begin{array}[]{rr}\sinh
2\varkappa_{0}&\phantom{-}\cosh 2\varkappa_{0}\\\ -\cosh 2\varkappa_{0}&-\sinh
2\varkappa_{0}\end{array}\right)C_{\mu}x^{\mu}+O(x^{2})\;.$

We will work mostly with a slight non-Hermitian perturbation of Hermitian
theories corresponding to $\varkappa_{0}=0$ because the constant (spacetime-
independent) non-Hermitian background can always be reduced to a Hermitian
theory by an inverse similarity transformation. In this case, the mass matrix
(LABEL:eq_mass_matrix_xt3) simplifies further to

$\displaystyle{\hat{M}}(x)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}M_{0}+m_{0}&-m_{0}\,C_{\mu}x^{\mu}\\\\[2.84526pt]
m_{0}\,C_{\mu}x^{\mu}&M_{0}-m_{0}\end{array}\right)+O(x^{2})\,.$ (132)

In the phenomenologically interesting limit of weak inhomogeneity, one gets
from Eqs. (62) and (LABEL:eq_mass_matrix_xt3) the following relation of the
mass matrix elements and the similarity gauge field:

$\displaystyle C_{\mu}=-\frac{\partial}{\partial
x^{\mu}}\frac{m_{5}}{m}+\dots\equiv-\frac{2}{M_{+}-M_{-}}\frac{\partial
m_{5}}{\partial x^{\mu}}+\dots\,.$ (133)

This is to say that we assume the background field to develop a small, weakly
inhomogeneous off-diagonal non-Hermitian mass, which, in turn, can be treated
as the emergence of a weak similarity gauge field $C_{\mu}$. This is connected
to the inhomogeneity of the off-diagonal mass via Eq. (133) with
$|C_{\mu}x^{\mu}|\ll 1$ at the length scale of the inhomogeneities of
$C_{\mu}$. Moreover, we assume that the strength of the similarity field
$C_{\mu}$ is much smaller than the mean values $M_{0}=(M_{+,0}+M_{-,0})/2$ and
the splitting $m_{0}=(M_{+,0}-M_{-,0})/2$ of the physical masses $M_{\pm,0}$
at $C_{\mu}=0$, given by Eqs. (73), (73) and (74), i.e.,

$\displaystyle|C_{0}|,\,|{\bm{C}}|\ll M_{0},\,m_{0}\,.$ (134)

The operator (121), applied to a spinor plane wave eigenstate
$\Psi(x)=e^{-ip\cdot x}\Psi_{0}$, with a constant spinor doublet $\Psi_{0}$,
becomes

$\displaystyle{\hat{K}}=\begin{pmatrix}\not{p}+M_{+}&-i{\not{C}}\\\
-i{\not{C}}&\not{p}+M_{-}\end{pmatrix}\;.$ (135)

The condition for the eigenvalues (106a) transforms to the compatibility
equation $\det{\hat{K}}=0$, which implies, in turn, the following relation:

$\displaystyle(p^{2}-M_{+}^{2}+C^{2})(p^{2}-M_{-}^{2}+C^{2})$ (136)
$\displaystyle\hskip
28.45274pt-C^{2}\bigl{[}4p^{2}-(M_{+}+M_{-})^{2}\bigr{]}+4(C\cdot p)^{2}=0\,.$

Equation (136) determines the energy dispersion relations for the fermions. In
the absence of the similarity gauge background, i.e., $C^{\mu}=0$, this
equation gives us two standard excitation branches

$\displaystyle\omega^{2}_{\pm,\bm{p}}={\bm{p}}^{2}+M_{\pm}^{2},\qquad\
{\text{for}}\quad C^{\mu}=0\,,$ (137)

where the physical real-valued masses $M_{\pm}$ are given in Eq. (63). The
positive-definite inner product for these mass eigenstates (generalising the
construction in Sec. IV.1) is described in Appendix A.

##  V Inhomogeneous similarity background: energy dispersions

As for the scalar case in Sec. II, the energy eigenvalues are roots of an
algebraic equation of the 4th order, given by Eq. (136). It is, therefore,
convenient again to consider the cases of purely timelike or purely spacelike
similarity gauge fields.

###  V.1 Temporal similarity field

A time-varying and spatially homogeneous mass matrix provides us with a
strictly temporal background similarity field $C^{\mu}=(C^{0},{\bm{0}})$, with
$C^{0}\equiv C_{0}$. Equation (136) then gives us the following energy
dispersion relations:

$\displaystyle\omega_{\pm,\bm{p}}^{2}$
$\displaystyle=M_{0}^{2}+m_{0}^{2}+{\bm{p}}^{2}-C_{0}^{2}$
$\displaystyle\phantom{=}\pm
2\sqrt{M_{0}^{2}(m_{0}^{2}-C_{0}^{2})-C_{0}^{2}{\bm{p}}^{2}}$
$\displaystyle\hskip 71.13188pt{\text{for}}\ C^{\mu}=(C^{0},{\bm{0}})\;.$
(138)

We describe the solutions in terms of the unperturbed, coordinate-independent
masses $M_{0}$ and $m_{0}$ in the absence of the background similarity field
($C^{\mu}=0$). Their relation to the unperturbed mass matrix (62) can be read
from Eq. (74).

The energy dispersion relation (V.1) has the following notable features:

#### – Mass shift.

The masses $M_{\pm,C}\equiv\omega_{\pm}({\bm{p}}=0)$ are affected by the
presence of the similarity gauge field, as is readily visible from Eq. (V.1).
Specifically,

$\displaystyle M_{\pm,C}$
$\displaystyle=\left(M_{0}^{2}+m_{0}^{2}-C_{0}^{2}\pm
2M_{0}\sqrt{m_{0}^{2}-C_{0}^{2}}\right)^{1/2}$
$\displaystyle=M_{\pm,0}\mp\frac{C_{0}^{2}}{M_{+,0}-M_{-,0}}+\dots\,,$ (139)

where the ellipsis denotes $O\bigl{(}C_{0}^{4}\bigr{)}$ terms and the
unperturbed masses are $M_{\pm,0}=M_{0}\pm m_{0}$ in consistency with Eq.
(73). In the leading, quadratic order, the weak field $C_{0}$ slightly
contributes to the shift between the $M_{\pm,C}$ masses, while leaving their
mean value unchanged.

#### – High-momentum instability.

The dispersion relation (V.1) develops an imaginary part at a certain spatial
momentum $p=|{\bm{p}}|$, which is restricted by the higher cutoff
$p_{c}^{(t)}$, defined via

$\displaystyle p>p^{(t)}_{c}=M_{0}\sqrt{\frac{m_{0}^{2}}{C_{0}^{2}}-1}\;.$
(140)

Given the weakness of the similarity gauge field (134), the limiting momentum
at which the particle propagation becomes unstable is much higher than the
mean mass of the particles, viz. $p_{\mathrm{max}}^{(t)}\gg M_{0}$.

Figure 3: The real and imaginary parts of the energy dispersions
$\omega_{\pm}$ for the timelike similarity field $C_{0}\neq 0$ and
${\bm{C}}=0$ are shown for the illustrative set of parameters $m_{0}=M_{0}/2$
and $C_{0}=M_{0}/10$. The imaginary parts, multiplied by a factor of 100 to
increase their visibility, appear at the critical momentum (140)
$p^{(t)}_{c}\simeq 4.90$, identified with the green arrow. The inset shows the
real part of the dispersions close to the critical point $p\simeq p^{(t)}_{c}$
(at the location indicated with the orange arrows).

At large momentum $p\gg p_{c}^{(t)}$, one gets the following asymptotic form
of the energy dispersion:

$\displaystyle\omega_{\pm,p}=p\pm
i|C_{0}|+\frac{M_{0}^{2}+m_{0}^{2}}{2p}+O\bigl{(}p^{-2}\bigr{)}\;.$ (141)

In the selected parameter region (134), the instability given by the imaginary
term in the dispersion (141) is very small. Examples of the dispersion
relation are illustrated in Fig. 3 for a suitable set of parameters. Notice
that, according to Eqs. (V.1) and (141), in the unstable region,
$p>p^{(t)}_{c}$, the particle with the lower mass $M_{-}$ has a diffusive
nature — the mode decays since $\mathrm{Re}\,\omega_{-}<0$ — whereas the
higher-mass $M_{+}$ particle is unstable — the mode grows because
$\mathrm{Re}\,\omega_{+}>0$.

Given the similarities in the form of their dispersion relations, the
instability of the fermion modes at high momentum is analogous to the
instability found for the two-scalar model in Sec. II, see Ref. Chernodub and
Millington (2022). Even so, from a general point of view, it is an unexpected
feature of the presence of the timelike similarity field. A similar bound to
Eq. (140) is obtained for the counterpart bosonic model Chernodub and
Millington (2022), see Sec. II. As noted earlier for both the scalar and
single Dirac fermion models, these instabilities represent momentum-dependent
exceptional points, beyond which we obtain complex conjugate pairs of
eigenfrequencies.

#### – Superluminal propagation.

In the unstable region, the group propagation velocities of both modes

$\displaystyle{\bm{v}}_{\pm,\bm{p}}=\frac{\partial\omega_{\pm,\bm{p}}}{\partial{\bm{p}}}\;,$
(142)

defined by the slope of the real part of the dispersion (141), always remain
smaller than the speed of light, i.e., $|{\bm{v}}_{\pm,\bm{p}}|<1$. However,
for high momenta of the order of $p^{(t)}_{c}$ — with the condition that they
are still lower than the instability threshold, $p<p^{(t)}_{c}$ — the velocity
of particle propagation (142), for both “$\pm$” modes, exceeds the speed of
light in a certain region of parameter space. This property is apparent from
the inset of Fig. 3, which zooms in on the dispersions (V.1) in a small region
around the critical momentum (140). The cusps originating from the square root
of the dispersions (V.1), show that the velocity of both $\omega_{\pm}$ modes
becomes singular at $p=p^{(t)}_{c}$.

The superluminal propagation is an unanticipated property of the non-Hermitian
model. This peculiar feature suggests that fast-moving particles, coupled with
other dynamical fields like the photon field, would emit Cherenkov radiation.
This radiation, in turn, would act as a decelerating force on the particles,
effectively impeding their advancement toward the point of instability for all
modes.

The group velocities for both “$\pm$” modes are illustrated in Fig. 4 in which
the parameters of Fig. 3 are adopted. For the “$-$”-mode, the superluminal
propagation arises at high momenta $p>p^{\mathrm{(t,-)}}_{\mathrm{SL}}$, where

$\displaystyle p^{{(t,\pm)}}_{\mathrm{SL}}=$
$\displaystyle\frac{M_{0}}{|C_{0}|}\frac{\sqrt{m_{0}^{2}-C_{0}^{2}}}{M_{0}^{2}+m_{0}^{2}}\biggl{(}C_{0}^{2}\left(M_{0}^{2}-m_{0}^{2}\right)+M_{0}^{4}+m_{0}^{4}$
$\displaystyle\pm
2m_{0}M_{0}\sqrt{(m_{0}^{2}-C_{0}^{2})(M_{0}^{2}+C_{0}^{2})}\biggr{)}^{1/2}\,.$
(143)

As was the case for the scalar model, the superluminality (“SL”) threshold
momentum (143) for the “$-$”-mode is of the order of, but noticeably lower
than, the critical momentum (140) at which the instability sets in. In the
lower-momentum region, $p<p^{{(t)}}_{\mathrm{SL}}$, the propagation of this
mode is characterised by subluminal velocities.

Figure 4: Velocities of the “$\pm$” modes of Fig. 3 for the timelike
similarity field. The inset shows a wider region in the momentum $p$ space
narrowed in the $v$-axis around the speed of light, $v=1$.

#### – Stopped propagation.

The “+”-mode has a richer structure. As the momentum $p$ increases, the group
velocity of this mode reaches a maximum in the subluminal region (which is
about 90% of the speed of light for our choice of parameters) before the
velocity of the particle then drops again. When the momentum reaches a
particular value,

$\displaystyle
p_{\mathrm{stop}}^{(t)}=\frac{\sqrt{M_{0}^{2}m_{0}^{2}-M_{0}^{2}C_{0}^{2}-C_{0}^{4}}}{|C_{0}|}\,,$
(144)

the “$+$”-mode stops propagating as the group velocity vanishes, i.e.,

$\displaystyle v_{+}\bigl{(}p_{\mathrm{stop}}^{(t)}\bigr{)}=0\,.$ (145)

Notice that for weak similarity fields (134), the stopping momentum (144) is
very close to the instability point (140).

As the momentum increases further, the group (negative) velocity exceeds the
speed of light and diverges. The superluminal thresholds for both “$\pm$”
modes can be expressed via the following single expression:

$\displaystyle v_{\pm}\bigl{(}p^{{(t,\pm)}}_{\mathrm{SL}}\bigr{)}=\mp 1\,,$
(146)

where the velocities $v_{\pm}$ are expressed in Eq. (142) via the
corresponding frequencies (V.1).

#### – Negative group velocity.

As the momentum increases above the “stopping point”
$p=p^{{(t,+)}}_{\mathrm{SL}}$, the “$+$”-mode starts to propagate in a
backward direction because the group velocity takes a negative value, i.e.,

$\displaystyle v_{+}(p)<0\qquad\text{for}\quad
p^{{(t,+)}}_{\mathrm{SL}}<p<p^{(t)}_{c}\,.$ (147)

This effect persists in a narrow region close to the critical momentum (140),
beyond which both “$\pm$”-modes develop a complex part.

The emergence of a negative group velocity for wave propagation is an
interesting phenomenon, which often appears in optics Brillouin (2013). The
negative group velocities, as well as the superluminal propagation, is also a
characteristic feature of media with anomalous dispersion Garrett and McCumber
(1970), which we discuss below.

#### – Anomalous dispersion.

The non-Hermitian two-fermion model in the background of a nonvanishing
similarity gauge field also features anomalous dispersion. One can readily
check that the “$+$”-mode of our dispersion relation (V.1) is always normal
with $\partial n_{+}/\partial p>0$ in the whole range of frequencies. However,
the “$-$”-mode exhibits anomalous dispersion in a wide region of the phase
diagram.

It is worth mentioning that properties such as superluminality (with the group
velocity exceeding the speed of light) and negative group velocity (which is
opposite to the wavevector) often appear in homogeneous media characterised by
anomalous dispersion (38). The consequences of anomalous dispersion on the
shape of a Gaussian light pulse as it propagates through such a medium were
studied in Ref. Garrett and McCumber (1970), where it was found that the pulse
remains of the Gaussian shape with the peak moving in space with a velocity
determined by the classical group-velocity expression (40).

Amusingly, in the anomalously dispersive medium, the classical group velocity
can become greater than the velocity of light in a vacuum or negative,
provided the medium has an absorption line near the optical frequencies of the
waves that make up the pulse. The absorption property is a characteristic of a
non-Hermitian system, which points out that the non-Hermiticity is responsible
for the unification of all three phenomena.

These would-be contradictory statements made in the classical context do not
mean that the system necessarily violates causality. The mechanism beyond the
superluminal propagation in anomalous dispersive media is associated with the
pulse shape distortion, even though in their studies, the pulse does not
appear to be visually distorted Garrett and McCumber (1970). Causality is also
not violated in such an anomalous medium since the absorption destroys the
relation between the group velocity and the velocity of energy propagation.
While we expect that a similar phenomenon can also happen in the present non-
Hermitian model, we leave a more detailed study for future work.

We finish this subsection with Fig. 5, which illustrates all the described
particularities of the propagation of the “$\pm$”-fermionic modes for two
inhomogeneity parameters corresponding to a relatively weak ($C_{0}=0.1M_{0}$)
and moderate ($C_{0}=0.5M_{0}$) similarity fields.

Figure 5: The regions in the $(p,m_{0})$ plane at $C_{0}=0.1M_{0}$ (left) and
$C_{0}=0.5M_{5}$ (middle) of: the high-momentum instability (yellow for both
modes), superluminal propagation (red for “$-$”-mode and magenta for
“$+$”-mode), anomalous dispersion of the “$-$”-mode (green), the line of
immobility of the “$+$”-mode (blue), the negative group velocity of the
“$+$”-mode (blue). A zoom-in on a region around $p=M_{0}$ for $C_{0}=0.5M_{5}$
(right) also shows the region where the “$+$”-mode becomes superluminal with
negative group velocity (magenta). The blue and yellow regions are separated
by the critical momentum (140).

###  V.2 Spatial similarity field

A strictly spatial perturbation of the mass matrix is equivalent to
introducing the similarity field $C^{\mu}=(0,{\bm{C}})$. In this case, Eq.
(136) gives us the following dispersion:

$\displaystyle\omega_{\pm,\bm{p}}^{2}=$ $\displaystyle
M_{0}^{2}+m_{0}^{2}+{\bm{p}}^{2}-{\bm{C}}^{2}$ $\displaystyle\pm
2\sqrt{m_{0}^{2}(M_{0}^{2}-{\bm{C}}^{2})-({\bm{C}}\cdot{\bm{p}})^{2}}$
$\displaystyle\hskip 71.13188pt{\text{for}}\quad C^{\mu}=(0,{\bm{C}})\,,$
(148)

which is naturally anisotropic due to the presence of the spatial vector
${\bm{C}}$. Notice again that the fermionic dispersions are functionally
similar, up to redefinitions of the masses, to their bosonic counterparts
discussed in Sec. II, see Ref. Chernodub and Millington (2022).

The resemblance of the dispersions (V.2) and Eq. (V.1) implies that the
spacelike similarity field has exactly the same features as the timelike field
considered earlier. In particular, the shift of the fermion masses in the
presence of the spatial similarity field is given by Eq. (V.1) for the
temporal similarity field with the replacements $m_{0}\leftrightarrow M_{0}$
and $C_{0}\to C_{3}$. The same statement also applies to the critical momenta
(140), (143), and (144) that characterise the upper bound on stability modes,
the superluminal momentum threshold, and the stopping momentum, respectively.

The high-momentum instability appears if the momentum of a fermion tangential
to the similarity field ${\bm{C}}$ exceeds the following threshold:

$\displaystyle|p_{3}|>p^{(s)}_{\mathrm{max}}=m_{0}\sqrt{\frac{M_{0}^{2}}{C_{3}^{2}}-1}\;.$
(149)

Here, we took ${\bm{C}}$ along the third axis, $C^{\mu}=\delta^{\mu 3}C_{3}$.
An increasing momentum in the direction normal to the similarity field
${\bm{C}}$ does not lead to instabilities.

At large momentum $p_{\|}\equiv p_{3}\gg p^{(s)}_{\mathrm{max}}$, one gets the
asymptotic form of the energy dispersion similar to Eq. (141),

$\displaystyle\omega_{\pm}(p_{\perp},p_{3})=p\pm
i|C_{3}|+\frac{M_{0}^{2}+m_{0}^{2}+p^{2}_{\perp}}{2p_{3}}+O\bigl{(}p^{-2}_{3}\bigr{)}\,,$
(150)

whereas in the limit $p_{\perp}=\sqrt{p_{1}^{2}+p_{2}^{2}}\to\infty$ (and in
the stable region $|p_{\|}|<p^{(s)}_{\mathrm{max}}$), we arrive to the
expected dispersion $\omega_{\pm}(p_{\perp},p_{3})=p+O(p^{-1})$, which does
not contain an imaginary part. The behaviour of the real and imaginary parts
of the dispersion relation (V.2) for the spacelike similarity field are
illustrated in Fig. 6.

|
---|---
Figure 6: The real (left) and imaginary (right) parts of the energy
dispersions $\omega_{\pm}$ for the spacelike similarity field
$C^{\mu}\equiv(0,0,0,C_{3})\neq 0$ for the same set of parameters as in Fig.
6. The imaginary parts develop when the longitudinal momentum $p_{\|}\equiv
p_{3}$ exceeds the critical momentum $p^{(s)}_{\mathrm{max}}$ [see Eq. (149)].
The plots are shown in the $(p_{\|},p_{\perp})$ plane, where
$p_{\perp}=\pm\sqrt{p_{1}^{2}+p_{2}^{2}}$ is the transverse momentum.

The superluminal propagation, negative group velocity and anomalous dispersion
are also evident from the similarity of the dispersions of fermions in the
background of the spatial (V.2) and temporal (V.1) similarity fields. Both of
them contain a square-root term, which leads to the characteristic cusps of
the dispersion at the critical momentum, which, in turn, implies the divergent
group velocity of the modes.

Interestingly, the dispersion relations for the two-flavour fermionic non-
Hermitian model with temporal (V.1) and spatial (V.2) similarity gauge fields
coincide with the corresponding dispersion relations for the non-Hermitian
model with the doublet scalar field (32) with obvious redefinitions of the
critical momenta. Therefore, the phase diagrams for both models share the same
features represented in Fig. 5.

Finally, it is appropriate to ask whether these unusual features of the non-
Hermitian model appear due to the non-Hermiticity or if they can also be found
in a Hermitian version of the model. The model can be turned to its Hermitian
version by making the off-diagonal term in the mass matrix (110) a purely
imaginary quantity $m_{5}\to im_{5}$ (with a real non-Hermitian mass $m_{5}$).
This change implies, according to Eq. (IV.4), that the scalar similarity
parameter $\varkappa$ also becomes a purely imaginary quantity, which also
provokes, following Eq. (122), the corresponding replacement of the similarity
gauge field, $C_{\mu}\to iA_{\mu}$, where $A_{\mu}$ is a real-valued vector
field. Thus, in the Hermitian version of the model, the dispersion relations
are given by Eqs. (V.1) and (V.2) with the replacement $C^{\mu}\to iC^{\mu}$.
One can readily see that the Hermitian version of the model has an
unremarkable monotonically rising behaviour as a function of momenta, devoid
of any exotic properties.

##  VI Instability, IR/UV mixing and phenomenology

The models described exhibit momentum-dependent exceptional points. In the
case of the two-scalar and two-fermion models, the low momentum modes reside
in the ${\mathcal{PT}}$-unbroken domain such that the low-lying physical
excitations have a real-valued spectrum and are stable. On the other hand, at
sufficiently high momenta, the modes reside in a $\mathcal{PT}$-broken regime.
The lower-mass branch $\omega_{-}$ develops a negative imaginary part,
implying that the amplitude of the mode decays. The higher-mass branch
$\omega_{+}$ develops a positive imaginary part signalling the growth of this
mode.

The spacetime inhomogeneity of the mass matrix in realistic models is expected
to result from the inhomogeneity of an underlying scalar condensate. In the
fermion case, this condensate would couple to the fermion bilinear via a
Yukawa term and contribute to the fermion mass terms. The instabilities of the
high-momentum fermion modes can, therefore, be understood as a result of the
interaction between the inhomogeneous scalar condensate and a propagating
fermionic particle: the particle scatters inelastically at the condensate
creating or absorbing scalar quanta. One can expect that the fluxes of highly
energetic fermionic particles propagating with momenta above the thresholds
(140) and (149) through the scalar condensate will result in the
homogenisation of the condensate. The translationally invariant condensate,
corresponding to the true ground state of the model, would then be the
ultimate outcome of these interactions. In our paper, we do not consider this
homogenising backreaction of the propagating fermionic modes on the non-
Hermitian background.

For phenomenologically relevant weak similarity fields (134), the stability
zones for the two-flavour fermion model, defined for the temporal (140) and
spatial (149) cases, determine the single critical momentum

$\displaystyle p_{c}=\frac{M_{0}m_{0}}{C}\;,$ (151)

where $C>0$ stands for either $C_{0}$ or $|\bm{C}|$ and we have neglected the
subleading terms in Eq. (151). The above equation implies that a weak
similarity field $C^{\mu}$, corresponding to a slowly varying mass matrix and
a low (IR) momentum, leads to an instability of the propagating modes at high
(UV) momentum. This IR/UV mixing was found in Ref. Chernodub and Millington
(2022), and it is a particular feature of these non-Hermitian field theories.

An estimation of the critical momentum (151) can be made with the help of Eq.
(133):

$\displaystyle
p_{c}=\frac{1}{8}(M_{+}+M_{-})(M_{+}-M_{-})^{2}\biggl{(}\frac{\partial
m_{5}}{\partial x^{\mu}}\biggr{)}^{-1}\,,$ (152)

where we have omitted subleading corrections of the order $O\bigl{(}(\partial
m_{5}/\partial x^{\mu})/m_{1,2}^{2}\bigr{)}$. For this, let us use the
physical lepton masses $M_{-}=m_{e}\simeq 0.5\,\mathrm{MeV}$ and
$M_{+}=m_{\mu}\simeq 105.7\,\mathrm{MeV}$. (We note that if, indeed, we were
to associate the fermion doublet with electrically charged degrees of freedom,
conservation of the electromagnetic current for the non-Hermitian theory would
require the components of the fermionic doublet to carry equal electric
charge, see Appendix B.) If we assume that the non-Hermitian mass $m_{5}$
varies by one MeV at the distance of one meter, the critical momentum (152)
then acquires the value

$\displaystyle p_{c}\simeq 1.6\times 10^{28}\,\mathrm{MeV}\,,$ (153)

For these conditions, the similarity field (133) is extremely small, with

$\displaystyle|C|\simeq 3.8\times 10^{-15}\,\mathrm{MeV}\,,$ (154)

so that the assumption of the weakness of the similarity field is satisfied
(134). In other words, for the negligibly tiny inhomogeneity of the non-
Hermitian masses encoded in the experimentally unobservable value of the
similarity field (154), the fermionic modes become unstable as soon as the
momentum of the fermionic particle exceeds the critical value (153).

It is important to stress again that it is possible to vary the non-Hermitian
mass parameters without affecting the physical masses $M_{\pm}$. Therefore,
even if the value of these parameters vary from one spacetime region to
another, the physical masses remain the same within a tiny, experimentally
inaccessible correction of the order of the strength of the similarity vector
field (154).

##  VII Conclusions

In this article, we have extended the proposal of Ref. Chernodub and
Millington (2022) to non-Hermitian fermionic theories with local Lagrangian
parameters, namely the parameters of the mass matrix. This leads automatically
to the appearance of an associated vector field, the “similarity gauge field”
$C^{\mu}$, which acts as a new connection in the space of similar non-
Hermitian theories.

We have argued that the similarity gauge field $C^{\mu}$ cannot be a dynamical
propagating field similar, e.g., to the electromagnetic gauge field. However,
the similarity field can still appear in the model as a non-dynamical,
background field.

The spacetime-dependent contributions to the mass matrix can be viewed as a
result of the inhomogeneity of a condensate of a scalar field coupled to the
fermion doublet via, e.g., a Yukawa coupling. Such contributions can be of
either Hermitian or non-Hermitian nature. Assuming a weak inhomogeneity of the
scalar background, one can show, following the bosonic case studied in detail
in Ref. Chernodub and Millington (2022), that inhomogeneity of a Hermitian
mass matrix leads to a rather trivial effect, resulting in a shift in (or
redefinition of) the fermionic masses. On the other hand, going beyond Ref.
Chernodub and Millington (2022), we have shown that the inhomogeneity of a
non-Hermitian mass mixing matrix leads to nontrivial effects both for two-
flavour scalar and two-flavour fermion models: anomalous dispersion,
superluminality and instabilities above certain high-momentum (UV) thresholds,
determined by the IR scale of the inhomogeneity. More precisely, the longer
the wavelength of the inhomogeneity, the higher the critical value of the
momentum above which the instability arises. This IR/UV mixing effect was
observed for the scalar case in Ref. Chernodub and Millington (2022). Thus,
weak variations of the mass parameters on cosmological scales would correspond
to instabilities at scales beyond current experimental or observational reach.

An unexpected feature of the two-flavour models is that the group velocity of
the particle propagation exceeds the speed of light at sufficiently high
momenta, which are of the same (high) magnitude as the critical momenta that
mark the onset of the instability. This feature implies that the particles,
depending on their couplings to other degrees of freedom, would produce
Cherenkov radiation, which would slow down the particles, thus preventing them
from reaching the instability point.

The inhomogeneous, similarity-gauged non-Hermitian models also possess, again
at sufficiently high momenta, negative group velocities, and anomalous
dispersion relations. Moreover, at a certain momentum, the group velocity of
one of the modes vanishes. All such exotic effects (apparent superluminality,
negative group velocity, and anomalous dispersion), found for the two-flavour
non-Hermitian field theories analysed in this work, have their classical
counterparts in allowed propagation through an absorbing medium Garrett and
McCumber (1970), which calls for future investigation of the parallels between
these physical systems.

Despite the non-Hermitian nature of the effect brought by inhomogeneities and
notwithstanding the exotic features of the system at high energies, the
propagating modes reside in the ${\mathcal{PT}}$-unbroken domain at low
energies. In particular, the physical excitations at low momenta have a real-
valued spectrum, implying that the non-Hermitian modifications do not manifest
in the experimentally accessible low-energy domain. We leave detailed
phenomenological studies for future work.

###### Acknowledgements.

The authors thank Esra Sablevice for useful discussions. This work was
supported by a Royal Society International Exchange [Grant No. IES\R3\203069].
The work of P.M. was supported by the Science and Technology Facilities
Council (STFC) [Grant No. ST/X00077X/1] and a United Kingdom Research and
Innovation (UKRI) Future Leaders Fellowship [Grant Nos. MR/V021974/1 and
MR/V021974/2]; and a Nottingham Research Fellowship from the University of
Nottingham. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising.

## Data Access Statement

No data were created or analysed in this study.

##  Appendix A Momentum-dependent inner products

In this Appendix, we describe the momentum-dependence of the positive-definite
$\mathcal{APT}$ inner product induced by a non-vanishing similarity gauge
field $C^{\mu}$. For simplicity, we will consider constant $C^{\mu}$.

The momentum-dependence is most straightforwardly studied at the level of the
$2\times 2$ flavour space. Therein, the effective squared Hamiltonian for both
the two-flavour scalar and fermion models takes the general form:

$H^{2}(\bm{p})=\begin{pmatrix}\bm{p}^{2}+m_{1}^{2}-C^{2}&m_{3}^{2}-im_{C}^{2}(\bm{p})\\\
-m_{3}^{2}-im_{C}^{2}(\bm{p})&\bm{p}^{2}+m_{2}^{2}-C^{2}\end{pmatrix},$ (155)

where $m_{3}^{2}$ and $m_{C}^{2}(\bm{p})$ are, in general, functions of the
similarity gauge field $C^{\mu}$. The dependence of $m_{C}^{2}(\bm{p})$ on the
three-momentum $\bm{p}$ is induced by the first-order spatial derivatives in
the equations of motion.

We proceed as in the main body by separately treating the timelike
$C^{\mu}=(C_{0},\bm{0})$ and spacelike $C^{\mu}=(0,\bm{C})$ cases and make the
following correspondences for the two scalar (upper element of the braces) and
two fermion models (lower element of the braces):

timelike case: $\displaystyle C=C_{0}\;,$ $\displaystyle
m_{1}^{2}=\begin{Bmatrix}m_{1}^{2}\\\ M_{0}^{2}+m_{0}^{2}\end{Bmatrix}\;,$
$\displaystyle m_{2}^{2}=\begin{Bmatrix}m_{2}^{2}\\\
M_{0}^{2}-m_{0}^{2}\end{Bmatrix}\;,$ $\displaystyle
m_{3}^{2}=\begin{Bmatrix}[m_{5}^{4}+2C_{0}^{2}(m_{1}^{2}+m_{2}^{2})]^{1/2}\\\
[m_{0}^{4}+M_{0}^{2}(C_{0}^{2}-4m_{0}^{2})]^{1/2}\end{Bmatrix}\;,$
$\displaystyle m_{C}^{2}(\bm{p})=2\,C_{0}|\bm{p}|\;,$ spacelike case:
$\displaystyle C=|\bm{C}|\;,$ $\displaystyle
m_{1}^{2}=\begin{Bmatrix}m_{1}^{2}\\\ M_{0}^{2}+m_{0}^{2}\end{Bmatrix}\;,$
$\displaystyle m_{2}^{2}=\begin{Bmatrix}m_{2}^{2}\\\
M_{0}^{2}-m_{0}^{2}\end{Bmatrix}\;,$ $\displaystyle
m_{3}^{2}=\begin{Bmatrix}m_{5}^{2}\\\
[m_{0}^{4}+4m_{0}^{2}(|\bm{C}|^{2}-M_{0}^{2})]^{1/2}\end{Bmatrix}\;,$
$\displaystyle m_{C}^{2}(\bm{p})=2\,\bm{C}\cdot\bm{p}\;.$

The parity matrix $P={\rm diag}(1,-1)$ is unchanged from the case of the
vanishing similarity gauge field. However, the non-Hermitian parameter is
given by

$\zeta\equiv\zeta_{C}(\bm{p})=\frac{2[m_{3}^{2}-im_{C}^{2}(\bm{p})]}{m_{1}^{2}-m_{2}^{2}}\in\mathbb{C}\;,$
(156)

and the eigenvectors are now complex, taking the form

$\displaystyle\mathbf{e}_{+}=\mathcal{N}\begin{pmatrix}\zeta\\\
-1+\sqrt{1-|\zeta|^{2}}\end{pmatrix},$ (157a)
$\displaystyle\mathbf{e}_{-}=\mathcal{N}\begin{pmatrix}-1+\sqrt{1-|\zeta|^{2}}\\\
\zeta^{*}\end{pmatrix},$ (157b)

with normalisation

$\mathcal{N}=\left[2\left(|\zeta|^{2}-1+\sqrt{1-|\zeta|^{2}}\right)\right]^{-1/2}.$
(158)

Notice that the normalisation $\mathcal{N}$ remains real-valued in the
$\mathcal{PT}$-symmetric regime. We also see that the non-Hermitian parameter
is momentum-dependent for a non-vanishing similarity gauge field, as we might
expect from the presence of the momentum-dependent exceptional points
identified in the main text.

The matrix $A$ appearing in the $\mathcal{APT}$ inner product becomes

$A=\frac{1}{\sqrt{1-|\zeta|^{2}}}\begin{pmatrix}1&\zeta\\\
-\zeta^{*}&-1\end{pmatrix}\;,$ (159)

and the product

$P\cdot A=\frac{1}{\sqrt{1-|\zeta|^{2}}}\begin{pmatrix}1&\zeta\\\
\zeta^{*}&1\end{pmatrix}$ (160)

gives a Hermitian matrix, as it should form the relevant positive-definite
inner product for this non-Hermitian theory.

It can readily be confirmed that the eigenvectors are orthonormal with respect
to this inner product:

$\displaystyle\mathbf{e}_{\pm}^{*}\cdot P\cdot A\cdot\mathbf{e}_{\pm}=1\;,$
(161a) $\displaystyle\mathbf{e}_{\pm}^{*}\cdot P\cdot
A\cdot\mathbf{e}_{\mp}=0\;.$ (161b)

Moreover, we can also confirm that

$\mathbf{e}_{\pm}^{*}\cdot P\cdot A\cdot
H^{2}(\bm{p})\cdot\mathbf{e}_{\pm}=\omega_{\pm,\bm{p}}^{2}\;.$ (162)

Naively, we might expect the momentum-dependence of the matrix $A$ to give
rise to subtleties with respect to the calculation of the group velocity.
However, we find that this is not the case, since

$\displaystyle\bm{v}_{g,\pm,\bm{p}}$
$\displaystyle=\frac{1}{2\omega_{\pm,\bm{p}}}\,\nabla_{\bm{p}}\omega_{\pm,\bm{p}}^{2}$
$\displaystyle=\frac{1}{2\omega_{\pm,\bm{p}}}\,\mathbf{e}_{\pm}^{*}\cdot
P\cdot A\cdot[\nabla_{\bm{p}}H^{2}(\bm{p})]\cdot\mathbf{e}_{\pm}\;,$ (163)

and we are justified in focusing our attention on the dispersion relation as
derived in the main body.

##  Appendix B Non-Hermiticity and the electromagnetic sector

The group of similarity transformations, either global or local ones, commutes
with the group of electromagnetic gauge transformations given by

$\displaystyle\begin{array}[]{l}{\mathrm{Electromagnetic}}\\\
U_{\mathrm{e.m.}}(1)\ {\mathrm{gauge}}\\\
{\mathrm{transformation}}\end{array}\\!:\quad\left\\{\begin{array}[]{lcl}\Psi(x)&\to&e^{ie\alpha(x)}\Psi(x)\\\
\bar{\Psi}(x)&\to&e^{-ie\alpha(x)}\bar{\Psi}(x)\\\
A_{\mu}(x)&\to&A_{\mu}(x)+\partial_{\mu}\alpha(x)\end{array}\right.\\!\\!\\!\\!,\qquad$
(170)

provided that the electric charges of both components $\psi_{1}$, and
$\psi_{2}$ of the fermion doublet $\Psi=(\psi_{1},\psi_{2})^{T}$ possess the
same electric charges, $e_{1}=e_{2}\equiv e$.

The covariant derivative (92) in the non-Hermitian model (95) can be extended
to include also the electromagnetic gauge field $A_{\mu}$ as follows:

$\displaystyle D_{\mu}={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,\partial_{\mu}+\sigma_{1}C_{\mu}\to{\cal
D}_{\mu}={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,(\partial_{\mu}-ieA_{\mu})+\sigma_{1}C_{\mu}.\qquad$ (171)

The electric current, corresponding to the variation of the action with
respect to the gauge field $A_{\mu}$, is given by the standard expression:

$\displaystyle j_{\mathrm{e.m.}}^{\mu}=e{\bar{\Psi}}\gamma^{\mu}\Psi\equiv
e\sum_{a=1}^{2}{\bar{\psi}}_{a}\gamma^{\mu}\psi_{a},\qquad[e_{1}=e_{2}=e].\qquad$
(172)

According to the classical equation of motion (106), the electric current
(172) is a classically conserved quantity both in the original (59) and the
similarity-gauged (95) versions of the non-Hermitian model, i.e.,

$\displaystyle\partial_{\mu}j_{\mathrm{e.m.}}^{\mu}=0\;.$ (173)

If the charges of the two components differ, i.e., $e_{1}\neq e_{2}$, then
either the global or local $SU(1,1)$ group of similarity transformations (84)
becomes broken explicitly by the Maxwell $U(1)$ group of the electromagnetic
gauge transformations:

$\displaystyle
U=e^{ie{\hat{I}}_{\mathrm{e.m.}}\alpha}\,,\qquad{\hat{I}}_{\mathrm{e.m.}}=\frac{e_{1}+e_{2}}{2e}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}+\frac{e_{1}-e_{2}}{2e}\sigma_{3}.\qquad$ (174)

If $e_{1}\neq e_{2}$ then the generator of the electromagnetic group
${\hat{I}}_{\mathrm{e.m.}}$ does not commute with the $\sigma_{1}$ generator
of the Cartan subgroup of the $SU(1,1)$ similarity group,
$[{\hat{I}}_{\mathrm{e.m.}},\sigma_{1}]\neq 0$, and the similarity group is
explicitly broken by the difference in the electromagnetic charges. The
covariant derivative (171) then reads as follows:

$\displaystyle{\cal D}_{\mu}={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}\,\partial_{\mu}-ie{\hat{I}}_{\mathrm{e.m.}}A_{\mu}+\sigma_{1}C_{\mu}\,.\qquad$
(175)

In order to maintain the gauge invariance of this theory, the mass $m_{5}$
should be promoted to a field (condensate), $\phi_{5}$, in the off-diagonal
mass term in Eq. (96):

$\displaystyle
m_{5}{\bar{\psi}}_{1}\psi_{2}\to{\bar{\psi}}_{1}\phi_{5}\psi_{2},\qquad
m_{5}{\bar{\psi}}_{2}\psi_{1}\to{\bar{\psi}}_{2}\phi_{5}^{*}\psi_{1}.$ (176)

The field $\phi_{5}$ should transform under $U(1)$ electromagnetic group as
follows:

$\displaystyle U_{\mathrm{e.m.}}(1),\qquad\phi_{5}\to
e^{-i(e_{1}-e_{2})\alpha}\phi_{5}\,,$ (177)

where the gauge transformation parameter $\alpha$ is the same as in Eq. (170).
The diagonal masses $m_{1}$ and $m_{2}$ remain gauge invariant. However, in
this case, the electric current, which can be read off from the expression for
the covariant derivative (175),

$\displaystyle
j_{\mathrm{e.m.}}^{\mu}=e{\bar{\Psi}}\gamma^{\mu}{\hat{I}}_{\mathrm{e.m.}}\Psi,\qquad[e_{1}\neq
e_{2}],\qquad$ (178)

becomes a non-conserved quantity:

$\displaystyle\partial_{\mu}j_{\mathrm{e.m.}}^{\mu}$ $\displaystyle=$
$\displaystyle(e_{2}-e_{1})\left[{\bar{\psi}}_{1}(m_{5}{+}{\not{C}})\psi_{2}{+}{\bar{\psi}}_{2}(m_{5}{-}{\not{C}})\psi_{1}\right]\qquad$
(179) $\displaystyle\equiv$
$\displaystyle(e_{2}-e_{1})\,{\bar{\Psi}}(m_{5}-i\sigma^{2}{\not{C}})\Psi\,.$
(180)

Notice that the source of the non-conservation, given on the right-hand side
of this expression, is a Hermitian quantity. The non-conservation of the
electric charge (180) makes phenomenologically questionable the fermionic non-
Hermitian models with unequal charges ($e_{1}\neq e_{2}$), thereby forcing us
to consider the fermionic doublets with equal charges ($e_{1}=e_{2}$).

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