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\def\intk{\int \!\!\mathrm{d}^3 \G k}
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\def\la{\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigma}
\def\om{\omega}\def\Om{\Omega}
\def\AB{{_{A,B}}}\newcommand{\mlab}[1]{\label{#1}}
\def\CP{{_{C\!P}}}\def\T{{_{T}}}
\def\AB{{_{A,B}}}\def\mass{{_{1,2}}}
\def\flav{{e,\mu}}\def\1{{_{1}}}\def\2{{_{2}}}
%
\def\bp{{\bf {p}}}\def\bk{{\bf {k}}}\def\br{{\bf {r}}}\def\bx{{\bf {x}}}
\def\by{{\bf {y}}}\def\bl{{\bf {l}}}\def\bq{{\bf {q}}}\def\bj{{\bf {j}}}
\def \ak{\alpha^r_{{\bf k},e}(0)}\def \akd{\alpha^{r\dag}_{{\bf k},e}(0)}
\def\ap{\alpha^s_{{\bf p},e}(0)}\def\apd{\alpha^{s\dag}_{{\bf p},e}(0)}
\def\br{{\bf{r}}}\def\bI{{\bf{I}}}
\def\fourint{\int\!\!\!\int\!\!\!\int\!\!\!\int}\def\threeint{\int\!\!\!\int\!\!\!\int}
\def\twoint{\int\!\!\!\int}
\def\CKM{\tiny CKM}
\def\Qo{{_{Q\!,1}}}\def\Qt{{_{Q\!,2}}}\def\ko{{_{k\!,1}}}\def\kt{{_{k\!,2}}}
\newcommand{\ide}{1\hspace{-1mm}{\rm I}}
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\def\noHe0{:\;\!\!\;\!\!:H_e(0):\;\!\!\;\!\!:}
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%
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\def\pa{\partial}\def\ran{\rangle}
\def\rar{\rightarrow}\def\Rar{\Rightarrow}
\def\ri{\right}\def\ti{\tilde}
\def\we{\wedge}\def\wti{\widetilde}
%
\def\al{\alpha}\def\bt{\beta}\def\ga{\gamma}
\def\Ga{\Gamma}\def\de{\delta}\def\De{\Delta}
\def\ep{\epsilon}\def\ze{\zeta}\def\te{\theta}
\def\Te{\Theta}\def\ka{\kappa}\def\la{\lambda}
\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigma}
\def\om{\omega}\def\Om{\Omega}
%
\def\AB{{_{A,B}}}
\def\CP{{_{C\!P}}}\def\T{{_{T}}}
\def\AB{{_{A,B}}}\def\mass{{_{1,2}}}
\def\flav{{e,\mu}}\def\1{{_{1}}}\def\2{{_{2}}}
%
\def\nof{:\;\!\!\;\!\!:}
%
\def\wQ{Q}
\def\wwQ{Q}

\def\I{{_{\rm{I}}}}\def\II{{_{\rm{II}}}}
\def\A{{_{A}}}\def\B{{_{B}}}

\def\rran{\ran\!\ran}
\def\llan{\lan\!\lan}

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Chiral symmetry-breaking schemes and dynamical generation of masses
and field mixing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\author{M Blasone$^{1}$, P Jizba$^2$ and L Smaldone$^{1}$}
\address{$^1$ Dipartimento di Fisica, Universit\`a di Salerno, Via Giovanni Paolo II, 132 84084 Fisciano, Italy \& INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy}
\address{$^2$ FNSPE, Czech Technical University in Prague, B\v{r}ehov\'{a} 7, 115 19 Praha 1, Czech Republic}

\ead{blasone@sa.infn.it}
\ead{p.jizba@fjfi.cvut.cz}
\ead{lsmaldone@sa.infn.it}



\begin{abstract}
In this paper we review dynamical generation of field mixing after chiral symmetry breaking. We also study the explicit form of discrete transformations of flavor states in a two-flavor scalar model with field mixing. We find that $CPT$ symmetry is spontaneously broken on flavor vacuum because of its dynamically generated condensate structure.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The relation of flavor oscillations with $CPT$ and Lorentz and transformations has been analyzed in  modern theoretical physics literature \cite{CapCPT,Giunti:2003ku,maguejo,BLDiMLa}. Deformations of the Lorentz dispersion relations, as the ones encountered in the  treatment of neutrino oscillations in quantum field theory (QFT)~\cite{BlaVit95,qftmixing}, were explicitly studied in Ref.~\cite{maguejo}, by using the formalism developed in Ref.~\cite{MagSmo}. Analogous results can be inferred from more fundamental and exotic scenarios: In string theory, for example, the \emph{D}-foam model \cite{Ellis:2004ay} predicts deformations of the Lorentz dispersion relation of the order of some power of $E/M_P$, where $E$ is the energy of a particle and $M_P$ is the Planck mass. In this context, the scattering between open strings and $D0$-branes is modeled by an effective action which describes the dynamical generation of mixing via flavor vacuum condensates \cite{mavroman,DynMix}. This result was also studied from a non-perturbative algebraic point of view in the case of two \cite{ccpaper} and three flavors \cite{ccpaper3}.

String theory also inspired the Standard Model extension (SME) by Colladay and Kostelecky \cite{Colladay:1996iz}. In that context, $CPT$ and Lorentz violating terms were added to the Standard Model Lagrangian. Neutrino oscillations were there explicitly studied \cite{Kostelecky2004} and modified dispersion relations connected with an underlying Planck scale physics were found. Following these developments, many authors dedicated efforts to both theoretical and phenomenological implications of SME or similar models (see e.g. \cite{ColGla,Katori:2012pe,Antonelli:2018fbv}). It can be also shown   \cite{Lambiase:2017adh} that bounds on the parameters of SME can be fixed in connection with generalized uncertainty principle \cite{GUP}.  

In this paper we briefly review the dynamical mechanism of flavor vacuum condensate and field mixing generation presented in  Ref.~\cite{ccpaper}. As a consequence, Lorentz and $CPT$ symmetries are spontaneously broken. We here limit our study to the case of discrete symmetries in a simple effective model  of  two flavor scalar fields with mixing \cite{Binger:1999nj,bosons}. We explicitly show that this model presents a spontaneous symmetry breaking (SSB) of $T$ and 
$CPT$: while these are exact symmetries of the considered Lagrangian, the flavor vacuum is not left invariant by their action. To this end, the transformation properties of flavor creation and annihilation operators are explicitly derived, in the case of parity, charge conjugation and time reversal. 

The paper is organized as it follows: In Section \ref{Sec1} we review the mechanism of dynamical mixing generation described in Ref.~\cite{ccpaper}. In Section \ref{Sec2} quantization of flavor (scalar) fields is reviewed \cite{Binger:1999nj,bosons}. Discrete symmetries in the flavor representation are studied in Section \ref{Sec4}. Finally, in Section \ref{conclusions}, conclusions and perspectives are presented.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Dynamical field mixing generation} \label{Sec1}

Following Ref.\cite{ccpaper}, let us consider  a Lagrangian density $\mathcal{L}$ that is
invariant under the global \textit{chiral-flavor} group $G = SU(2)_L \times SU(2)_R \times U(1)_V $. Let the fermion field
be a flavor doublet
%
\be
\boldsymbol{\psi} \ = \
\bbm
\tilde{\psi}_1 \\ \tilde{\psi}_2
\ebm \, .
\ee
%
Under a generic chiral-group transformation $\G g$, the field $\boldsymbol{\psi}$ transforms to $\boldsymbol{\psi}'$, where~\cite{Miransky}
%
\bea
 \boldsymbol{\psi}'  =  \G g \mpsi  \ = \ \exp\lf[i \lf(\phi+\boldsymbol{\om} \cdot \frac{\boldsymbol{\si}}{2}+\boldsymbol{\om}_5 \cdot\frac{\boldsymbol{\si}}{2} \ga_5\ri)\ri] \mpsi \, . \label{chigr}
\eea
%
Here $\si_j \, , j=1,2,3$ are the Pauli matrices and $\phi, \boldsymbol{\om}$,  $\boldsymbol{\om}_5$ are real-valued
transformation parameters of $G$. Noether's theorem implies the conserved vector and axial currents
%
\bea
J^\mu \ = \ \overline{\boldsymbol{\psi}}\ga^\mu \boldsymbol{\psi} \, , \qquad \boldsymbol{J}^{\mu} \ = \ \overline{\boldsymbol{\psi}}\gamma^\mu \frac{\boldsymbol{\si}}{2} \boldsymbol{\psi}\, , \qquad \boldsymbol{J}^\mu_{5} \ = \ \overline{\boldsymbol{\psi}}\gamma^\mu \gamma_5 \frac{\boldsymbol{\si}}{2} \boldsymbol{\psi} \, ,
\eea
%
and the ensuing conserved charges
%
\bea
Q \ = \ {\intx} \, \boldsymbol{\psi}^\dagger \boldsymbol{\psi} \, , \qquad \boldsymbol{Q} \ = \ \intx \, \mpsi^\dagger  \frac{\boldsymbol{\si}}{2}  \mpsi \, , \qquad \boldsymbol{Q}_5 \ = \ \intx \, \mpsi^\dagger \frac{\boldsymbol{\si}}{2}  \ga_5 \mpsi  \, .
\eea
%
From these we recover the Lie algebra of the chiral-flavor group $G$, i.e.
%
\bea  \non
&& \label{su21} \lf[Q_i,Q_j \ri] \ = \ i \,\varepsilon_{ijk} Q_k \, , \quad
 \lf[Q_i, Q_{5,j} \ri] \ = \ i \, \varepsilon_{ijk} Q_{5,k} \, , \quad
  \lf[Q_{5,i}, Q_{5,j} \ri] \ = \ i \,\varepsilon_{ijk}{Q_{k}} \, ,\\ [1.5mm]
	\label{su23} 
&&  \lf[Q , { Q_{5,j}} \ri] \ = \ { \lf[Q , Q_{j} \ri]}  \ = \ 0 \, .
\eea
%
Here $i,j,k=1,2,3$ and $\varepsilon_{ijk}$ is the Levi-Civita pseudo-tensor.

To proceed, let us recall~\cite{Miransky} that SSB is carchterized by the existence of some
local operator(s) $\phi(x)$ so that, on the vacuum $|\Omega\ran$, it is 
%
\be \label{cssb}
\lan \lf[N_i, \phi(0) \ri] \ran \ = \  \lan \ph_i(0)\ran \  \ \equiv \ v_i \  \neq \ 0 \, ,
\ee
%
where $\lan \ldots \ran \equiv \lan \Omega | \ldots| \Omega \ran$. Here $v_i$ are the \textit{order parameters} and $N_i$ represent group generators from the quotient space $G/H$, with $H$ being the stability group. In our case $N_i$ will
be taken as $\boldsymbol{Q}$ and ${\boldsymbol{Q}}_5$ according to the SSB scheme under consideration.


By analogy with quark condensation in QCD~\cite{Miransky}, we will limit our considerations to  order parameters that are condensates of fermion-antifermion pairs.
To this end we introduce the following composite operators
%
\bea
\Phi_k \ = \  \overline{\boldsymbol{\psi}}\, \si_k \,  \mpsi \, , \qquad  \Phi^5_k \ = \  \overline{\boldsymbol{\psi}}\, \si_k \, \ga_5 \mpsi\, , \qquad  \sigma_0 \ \equiv\ \ide\, ,   \label{bil35}
\eea
%
with $k = 0,1,2,3$. For simplicity we now assume $\lan \boldsymbol{\Phi}^5\ran =0$. 

Let us now consider three specific SSB schemes $G \rightarrow H$: \\

{\bf i)} SSB sequence corresponding to a single mass generation is~\cite{Miransky, FujPap}
%
\be
SU(2)_L \times SU(2)_R \times U(1)_V \ \! \longrightarrow \ \! U(2)_V \, .
\label{SSB-scheme-1}
\ee
%
The broken-phase symmetry (which corresponds to dynamically generated mass matrix $M  =  m_0 \ide$)
is characterized by the order parameter
%
\bea
\lan \Phi_0 \ran \ = \ v_0  \ \neq 0 \, , \qquad \lan \Phi_k \ran \ = \ 0 \, , \quad  k = 1,2,3 \, . \label{diag1}
\eea
%
One can easily check that this is invariant under the residual symmetry group $H=U(2)_V$ but not under the full chiral group $G$.

{\bf ii)} As a second case we consider the SSB pattern
%
\be
SU(2)_L \times SU(2)_R \times U(1)_V \ \!\longrightarrow \ \! U(1)_V \times U(1)^3_V  \, ,
\ee
%
which is responsible for the dynamical generation of different masses $m_1,m_2$. In this case the order parameters take the form
%
\begin{eqnarray}
\lan \Phi_0 \ran \ = \  v_0  \ \neq \  0 \, ,  \qquad \lan \Phi_3 \ran  \ = \  v_3  \ \neq \  0  \, .
\label{diagm2}
\end{eqnarray}
%

{\bf iii)} Finally, we consider the SSB scheme
%
\be
SU(2)_L \times SU(2)_R \times U(1)_V \longrightarrow  U(1)_V \times U(1)^3_V \longrightarrow U(1)_V \, ,
\label{SSB_scheme_3}
\ee
%
which is responsible for the dynamical generation of field mixing.

Let us introduce
%
\bea
\Phi_{k,m} \ = \  \overline{\boldsymbol{\psi}}\, \si_k \,  \mpsi \, ,  \qquad k=1,2,3 \, ,
\eea
%
where $m$ indicates that $\mpsi$ is now a doublet of fields $\mpsi=\lf[\psi_1 \, \psi_2\ri]^T$ in the mass basis. The SSB condition now reads
%
\be \label{v1m}
\lan \Phi_{1,m} \ran \ \equiv \ v_{1,m} \ \neq \ 0 \, .
\ee
%
Hence we find \cite{ccpaper} that a \textit{necessary condition} for a dynamical generation of field mixing within chiral symmetric systems,
is the presence of  exotic  pairs in the vacuum, made up by fermions and antifermions with different masses \footnote{Generally also diagonal condensate may be present.}:
%
\be
\lan \overline{\psi}_i(x) \, \psi_j(x) \ran \neq \ 0 \, , \qquad i \ \neq \ j \, .
\ee
%
In other words, \textit{field mixing requires mixing at the level of the vacuum condensate structure}.
This conclusion is consistent with analogous results obtained in the context of QFT treatment of neutrino oscillations, in which case a \textit{flavor vacuum} has the structure of a non-trivial condensate~\cite{BlaVit95,qftmixing}. Moreover, this is an agreement with Ref.~\cite{mavroman}, where, as previously mentioned, this structure is recovered via dynamical symmetry breaking in a specific effective model generated by string-brane scattering. We remark that the above result is basically model independent (the only assumption made was the global chiral symmetry), and has a non-perturbative nature. 

In the mean-field approximation the vacuum condensate responsible for (\ref{v1m}) formally resembles the aforementioned flavor vacuum 
%
\bea \non
|0\rangle_{e,\mu}& = & \prod_{{\bf k}}\prod_{r}\Big[(1-\sin^{2}\theta\;V_{{\bf k}}^{2})- \eta^{r}\sin\theta\;\cos\theta\; V_{{\bf k}}(\alpha^{r\dag}_{{\bf k},1}\beta^{r\dag}_{-{\bf k},2}+\alpha^{r\dag}_{{\bf k},2}\beta^{r\dag}_{-{\bf k},1})  \\[2mm]
& &\hspace{-.6cm}+ \, \eta^{r}\sin^{2}\theta \;V_{{\bf k}} U_{{\bf k}}(\alpha^{r\dag}_{{\bf k},1}\beta^{r\dag}_{-{\bf k},1}-\alpha^{r\dag}_{{\bf k},2}\beta^{r\dag}_{-{\bf k},2}) +  \sin^{2}\theta \; V_{{\bf k}}^{2}\alpha^{r\dag}_{{\bf k},1}\beta^{r\dag}_{-{\bf k},2} \alpha^{r\dag}_{{\bf k},2}\beta^{r\dag}_{-{\bf k},1} \Big]|0\rangle_{1,2} \, , \label{fv}
\eea
%
where 
%
\bea \label{Vk}
&&\hspace{-.6cm} U_\G k \ = \ \mathcal{A}_\G k  \lf(1+\frac{|\G k|^2}{\lf(\om_{\G k,1}+m_1\ri)\lf(\om_{\G k,2}+m_2\ri)}\ri)\, ,\quad 
V_\G k \ = \ \mathcal{A}_\G k \lf(\frac{|\G k|}{\om_{\G k,1}+m_1}-\frac{|\G k|}{\om_{\G k,2}+m_2}\ri) ,  
%\\[2mm]
%\mathcal{A}_\G k & = & \sqrt{\lf(\frac{\om_{\G k,1}+m_1}{2 \om_{\G k,1}}\ri) \, \lf(\frac{\om_{\G k,2}+m_2}{2 \om_{\G k,2}}\ri)} \, .
\eea
%
with $\mathcal{A}_\G k =  \sqrt{\frac{\lf(\om_{\G k,1}+m_1\ri)\lf(\om_{\G k,2}+m_2\ri)}{4\om_{\G k,1}\om_{\G k,2}}}$. 

In this case \cite{ccpaper}:
%
\bea
v_{1,m} \ = \ 2\sin 2 \theta \, \intk \, \left(\frac{m_2}{\omega_{{\bf k},2}} - \frac{m_1}{\omega_{{\bf k},1}}  \right) .
\eea
%

Let us remark that such considerations are very general. Flavor vacuum was originally introduced in the context of neutrino oscillation physics~\cite{BlaVit95} but the same structure can be equally recovered by studying the bosonic case~\cite{bosons}. Therefore, just for technical simplicity, we here study the properties of flavor vacuum in this last case. However, it is understood that the origin of such condensate and the consequent $CPT$ SSB have to be retraced in a generation mechanism similar to the one described in this Section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Scalar field mixing}\label{Sec2}
Let us consider the effective Lagrange density
%
\be
\mathcal{L}(x)\ = \ \pa^\mu \boldsymbol{\ph}^\dagger_f(x) \pa_\mu \boldsymbol{\ph}_f(x) \ -\ \boldsymbol{\ph}^\dagger_f(x) \, M^2 \, \boldsymbol{\ph}_f(x) \, ,
\ee
%
where
%
\be \label{lagrflavfields}
\boldsymbol{\ph}_f(x)\ =\ \bbm \ph_A(x)  \\ \ph_B(x)  \ebm \, , \qquad M^2\ = \ \bbm m^2_A & m^2_{A B} \\ m^2_{ \, A B} & m^2_B \ebm \, ,
\ee
%
which describes the dynamics of two coupled (mixed) scalar fields that we will call flavor fields, in analogy with the terminology used in quark and neutrino physics. The Lagrange density~\eqref{lagrflavfields} can be diagonalized thanks to the following transformation:
%
\bea \label{bosmix}
\boldsymbol{\ph}_f(x)\ =\ U \,  \boldsymbol{\ph}_m(x) \, , \qquad U=\bbm \cos \te & \sin \te \\ -\sin \te & \cos \te \ebm  \, ,
\eea
%
where $\tan 2 \theta \ = \ 2 \, m^2_{AB}/\lf(m^2_B-m^2_A\ri)$. Therefore, $\mathcal{L}$ becomes
%
\be
\mathcal{L}(x)\ = \ \pa^\mu \boldsymbol{\ph}^\dagger_m(x) \,  \pa_\mu \boldsymbol{\ph}_m (x)\ -\ \boldsymbol{\ph}^\dagger_m(x) \, M^2_d \,  \boldsymbol{\ph}_m (x) \, ,
\label{II.4.a}
\ee
%
where
%
\be \label{lagrmassfields}
\boldsymbol{\ph}_m(x)\ =\ \bbm \ph_1(x) \\ \ph_2(x)  \ebm \, , \qquad M^2_d\ =\ \bbm m^2_1 & 0 \\ 0 & m^2_2 \ebm \, .
\ee
%
The Lagrange density (\ref{II.4.a}) describes two free scalar particles with masses $m_1$ and $m_2$. They can be thus expanded as:
%
\bea \non 
\ph_j(x)& = & \int \frac{\dr^3 \G k}{2\om_{\G k,j}(2 \pi)^3} \, \lf[a_{\G k,j}\, e^{-i\om_{\G k,j} t}\ + \ b^{\dagger}_{-\G k,j} \, e^{i\om_{\G k,j} t}\ri] \, e^{i\G k \cdot \G x} \, , \quad j=1,2
\eea
%
where the covariant annihilation and creation operators satisfy the following commutation relations:
%
\be
\lf[a_{\G k,i} \, , \, a^\dag_{\G p,j} \ri] \ = \ \lf[b_{\G k,i} \, , \, b^\dag_{\G p,j} \ri] \ = \ 2 \om_{\G k,i} \, (2 \pi)^3 \, \de(\G k-\G p) \,  \de_{ij} \, .
\ee
%
and annihilate the \emph{mass vacuum}:
%
\be
a_{\G k,j}|0\ran_{1,2} \ = \ b_{\G k,j}|0\ran_{1,2} \ = \ 0 \, , 
\ee
%
i.e. the ground state of the system. We now expand flavor fields in a similar way:
%
\bea \label{expph}
\ph_\si(x)\ = \ \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[a_{\G k,\si}(t)\, e^{-i\om_{\G k,\si} t}\ + \ b^{\dagger}_{-\G k,\si}(t) \, e^{i\om_{\G k,\si} t}\ri] \, e^{i\G k \cdot \G x} \, , \quad \si=A,B
\eea 
%
with $\om_{\G k,\si}= \sqrt{|\G k|^2+\mu_\si^2}$ and $\mu_\si$ are mass parameters which have to be specified. From mixing transformation \eqref{bosmix}, it follows\footnote{Here the time dependence of creation and annihilation operators indicates that flavor fields are interacting fields. Actually, this interacting model can be solved exactly, without perturbation expansion.}:
%
\be
a_{\G k,A}(t) \ = \ \intx \, e^{i \lf(\om_{\G k,A} t-\G k \cdot \G x\ri)} \, i \stackrel{\leftrightarrow}{\pa_0} \lf(\cos \theta \ph_1(x) \, + \, \sin \theta \ph_2(x)\ri) \, , 
\ee
%
and similar expressions work for the other operators. Explicitly:
%
\bea
  \left[ \begin{tabular}{c} $a_{\G k,A}$ \\ $b_{-\G k,A}^{\dagger}$
\\$a_{\G k,B}$ \\ $b_{-\G k,B}^{\dagger}$ \end{tabular} \right]
= \left[\begin{array}{cccc}
c_\theta\, \rho^{\G k}_{A 1}& c_\theta \, \lambda^{\G k}_{A 1} &
s_\theta \,\rho^{\G k}_{A 2}  &
s_\theta \,\lambda^{\G k}_{A 2}
\\  c_\theta \,\lambda^{\G k}_{A 1} & c_\theta\, \rho^{\G k}_{A 1} &  s_\theta
\,\lambda^{\G k}_{A 2} & s_\theta \,\rho^{\G k}_{A 2}
\\ - s_\theta \,\rho^{\G k}_{B 1} &-s_\theta \,\lambda^{\G k}_{B 1}& c_\theta
\,\rho^{\G k}_{B 2}
&  c_\theta \,\lambda^{\G k}_{B 2} \\  - s_\theta \,\lambda^{\G k}_{B 1} &-
s_\theta\,
\rho^{\G k}_{B 1} &  c_\theta\, \lambda^{\G k}_{B 2}& c_\theta\, \rho^{\G k}_{B 2}
\end{array}\right]
  \left[ \begin{tabular}{c} $a_{\G k,1}$ \\ $b_{-\G k,1}^{\dagger}$ \\  $a_{\G k,2}$ \\ $b_{-\G k,2}^{\dagger} $\end{tabular} \right] \, , 
 \label{4x4Bog}
\eea
%
where
$c_\theta\equiv \cos\theta$, $s_\theta\equiv \sin\theta$,
%
\be
\rho^{\G k}_{\si j} \ = \ |\rho^{\G k}_{\si j}|e^{i (\om_{\G k,\si}-\om_{\G k,j})t} \, , \qquad \la^{\G k}_{\si j} \ = \ |\la^{\G k}_{\si j}|e^{i (\om_{\G k,\si}+\om_{\G k,j})t} \,  , \quad (\si,j)=(A,1),(B,2),
\ee
%
where
%
\be
|\rho^{\G k}_{\si j}| \ = \ \ha \lf(\frac{\om_{\G k,\si}}{\om_{\G k,j}}+1\ri) \, , \qquad |\la^{\G k}_{\si j}| \ = \ \ha \lf(\frac{\om_{\G k,\si}}{\om_{\G k,j}}-1\ri) \, .
\ee
%
Note that this is a canonical transformation:
%
\be \label{flavccr}
\lf[a_{\G k,\si}(t) \, , \, a^\dag_{\G p,\rho}(t) \ri] \ = \ \lf[b_{\G k,\si}(t) \, , \, b^\dag_{\G p,\rho} (t)\ri] \ = \ 2 \om_{\G k,\si} \, (2 \pi)^3 \, \de(\G k-\G p) \,  \de_{\si \rho} \, .
\ee
%
Let us notice that we have not specified the mass parameters $\mu_\si$. The situation here is similar the one encountered in QFT in curved spacetime \cite{Curved} where one has an infinite set of creation and annihilation operators related by a Bogolyubov transformation. In our case, the Bogolyubov transformation is of the type:
%
\bea
  \left[ \begin{tabular}{c} $\tilde{a}_{\G k}$ \\ $\tilde{b}_{-\G k}^{\dagger}$
 \end{tabular} \right]
= \left[\begin{array}{cc}
 \rho^{\G k} & \lambda^{\G k}
\\  \lambda^{\G k} & \rho^{\G k} 
\end{array}\right]
  \left[ \begin{tabular}{c} $a_{\G k}$ \\ $b_{-\G k}^{\dagger}$\end{tabular} \right] \, , 
 \label{2x2Bog}
\eea
%
where
%
\be
|\rho^{\G k}| \ = \ \ha \lf(\frac{\tilde{\om}_{\G k}}{\om_{\G k}}+1\ri) \, , \qquad |\la^{\G k}_{\si}| \ = \ \ha \lf(\frac{\tilde{\om}_{\G k}}{\om_{\G k}}-1\ri) \, .
\ee
%
These are related to the usual Bogoliubov coefficients \cite{Miransky} as:
%
\be
\rho^{\G k}_B  \ = \ \rho^{\G k} \, \sqrt{\frac{\tilde{\om}_{\G k}}{\om_{\G k}}} \, , \qquad \la^{\G k}_B  \ = \ \la^{\G k} \, \sqrt{\frac{\tilde{\om}_{\G k}}{\om_{\G k}}}  \, , 
\ee
%
which satisfy the usual relation $|\rho^{\G k}_B|^2-|\la^{\G k}_B|^2=1$. In Ref. \cite{Casimiro}, it was shown that different choices of $\mu_\si$ affect the form of the Casimir force between two plates. Typical choices studied in literature are \cite{Blasone:2011zz} $\mu_A=m_1$, $\mu_B=m_2$ and $\mu_A=m_A$, $\mu_B=m_B$. 

Therefore, one can define the \emph{flavor vacuum} as the state which is annihilated by flavor annihilation operators at a fixed time $t$:
%
\be
a_{\G k,\si}(t) \, |0(t)\ran_{A,B} \ = \ b_{\G k,\si}(t) \, |0(t)\ran_{A,B} \ = \ 0 \, .
\ee
%
This is characterized by a boson-condensate structure, similar to the one encountered in Eq. \eqref{fv}. In the case $\mu_A=m_1,\mu_B=m_2$ one gets \cite{bosons}:
%
\be
{}_{A,B}\lan 0(t)|a^\dag_{\G k,j} \, a_{\G k,j}|0(t)\ran_{A,B} \ = \ {}_{A,B}\lan 0(t)|b^\dag_{-\G k,j} \, b_{-\G k,j}|0(t)\ran_{A,B} \ = \ \sin^2\theta \, |V_\G k|^2 \, 2 \, \om_{\G k, j} \, (2 \pi)^3\, , 
\ee
%
where $|V_\G k| \ \equiv |\la_\G k|_{\mu_A=m_1,\mu_B=m_2}$, analogous to the function encountered in Eq.\eqref{Vk}. This structure is responsible for the $CPT$ symmetry breaking discussed below. 

A flavor state is defined as an excitation of the flavor vacuum:
%
\be \label{invflavstate}
|a_{\G k,\si}(t)\ran \ \equiv \  a^\dag_{\G k,\si}(t)|0(t)\ran_{A,B} \, .
\ee
%
These are eigenstates of the flavor charge
%
\bea \non
Q_{\si}(t) & = & i \, \intx \, \ph^\dag_\si(x) \, \stackrel{\leftrightarrow}{\pa_0} \, \ph_\si(x) \\[2mm]
& = &\int \!\! \frac{\dr^3 \G k}{2 \om_{\G k,\si} \, (2 \pi)^3} \, \lf(a^\dag_{\G k,\si}(t) \, a_{\G k,\si}(t)-b^\dag_{\G k,\si}(t) \, b_{\G k,\si}(t)\ri) \, , \label{fc}
\eea
%
at fixed time $t$
%
\be
Q_{\si}(t) \, |a_{\G k,\si}(t)\ran \ = \ |a_{\G k,\si}(t)\ran \, , \quad \si=A,B.
\ee
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discrete symmetries} \label{Sec4}
In this Section we study the behavior of flavor annihilation and creation operators under parity, charge conjugation and time reversal. We will see that $CPT$ symmetry is spontaneously broken on the flavor vacuum. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Parity}
The parity transformation of the flavor scalar fields is given by:
%
\be
P \, \ph_\si(x) \, P^{-1} \ = \ \eta_{\si,{}_{P}} \, \ph_\si(\tilde{x}) \, , 
\ee
%
where $P$ is the unitary parity operator and $\tilde{x}=(t,-\G x)$. As usual, $|\eta_{\si,{}_{P}}|^2=1$. By using the explicit expansion \eqref{expph}, we find:
 %
\bea 
P \, \ph_\si(x) \, P^{-1} & = & \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[P \, a_{\G k,\si}(t) \, P^{-1}\, e^{-i\om_{\G k,\si} t}\ + \ P \, b^{\dagger}_{-\G k,\si}(t) \, P^{-1} \, e^{i\om_{\G k,\si} t}\ri] \, e^{i\G k \cdot \G x} \non \\[2mm]
& = &  \eta_{\si,{}_{P}} \, \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[a_{\G k,\si}(t) \, e^{-i\om_{\G k,\si} t}\ + \ b^{\dagger}_{-\G k,\si}(t) \, \, e^{i\om_{\G k,\si} t}\ri] \, e^{-i\G k \cdot \G x} \, .
\eea 
%
Transformations of creation and annihilation operators follow:
%
\bea
P \, a_{\G k,\si}(t) \, P^{-1} & = & \eta_{\si,{}_{P}} \, a_{-\G k,\si}(t) \, \qquad P \, b_{\G k,\si}(t) \, P^{-1} \ = \ \eta^*_{\si,{}_{P}} \, b_{-\G k,\si}(t) \, , \\[2mm]
P \, a^\dag_{\G k,\si}(t) \, P^{-1} & = & \eta^*_{\si,{}_{P}} \, a^\dag_{-\G k,\si}(t) \, \qquad P \, b^\dag_{\G k,\si}(t) \, P^{-1} \ = \ \eta_{\si,{}_{P}} \, b^\dag_{-\G k,\si}(t) \, .
\eea
%
The flavor vacuum is invariant under parity transformation:
%
\be \label{pifv}
P \, |0(t)\ran_{A,B} \ = \ |0(t)\ran_{A,B} \, .
\ee
%
In fact, the explicit form of $P$ is the same as in the free case \cite{greiner}:
%
\bea
P & = & \exp\lf\{i \frac{\pi}{2} \, \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[ \lf(a^\dag_{-\G k,\si}(t) \, a_{\G k,\si}(t)+b^\dag_{-\G k,\si}(t) \, b_{\G k,\si}(t)\ri) \ri. \ri. \non \\[2mm]
&  &-\lf. \lf.  \eta_{\si,{}_{P}}  \lf(a^\dag_{\G k,\si}(t) \, a_{\G k,\si}(t)+b^\dag_{\G k,\si}(t) \, b_{\G k,\si}(t)\ri) \ri]\ri\} \, ,
\eea
from which Eq.\eqref{pifv} follows. A flavor state \eqref{invflavstate} transforms as
%
\be
P\, |a_{\G k,\si}(t)\ran \ = \ |a_{-\G k,\si}(t)\ran \, , 
\ee
%
and the flavor charge \eqref{fc} remains invariant, i.e. 
%
\be
\lf[P \, , \, Q_\si(t)\ri] \ = \ 0 \, .
\ee
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Charge conjugation}
The charge conjugation transformation of the flavor scalar fields is given by:
%
\be
C \, \ph_\si(x) \, C^{-1} \ = \ \eta_{\si,{}_{C}} \, \ph^\dag_\si(x) \, , 
\ee
%
where $C$ is the unitary charge conjugation operator. As usual, $|\eta_{\si,{}_{C}}|^2=1$. Once more, by using the explicit expansion \eqref{expph}, we find:
 %
\bea 
C \, \ph_\si(x) \, C^{-1} & = & \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[C \, a_{\G k,\si}(t) \, C^{-1}\, e^{-i \, k \, x}\ + \ C \, b^{\dagger}_{\G k,\si}(t) \, C^{-1} \, e^{i \, k \, x}\ri] \non \\[2mm]
& = &  \eta_{\si,{}_{C}} \, \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[a^\dag_{\G k,\si}(t) \, e^{i \, k \, x}\ + \ b_{\G k,\si}(t) \, \, e^{-i \, k \, x}\ri] \, .
\eea 
%
Transformations of creation and annihilation operators follow:
%
\bea
C \, a_{\G k,\si}(t) \, C^{-1} & = & \eta_{\si,{}_{C}} \, b_{\G k,\si}(t) \, \qquad C \, b_{\G k,\si}(t) \, C^{-1} \ = \ \eta^*_{\si,{}_{C}} \, a_{\G k,\si}(t) \, , \\[2mm]
C \, a^\dag_{\G k,\si}(t) \, C^{-1} & = & \eta^*_{\si,{}_{C}} \, b^\dag_{\G k,\si}(t) \, \qquad C \, b^\dag_{\G k,\si}(t) \, C^{-1} \ = \ \eta_{\si,{}_{C}} \, a^\dag_{\G k,\si}(t) \, .
\eea
%
The flavor vacuum is invariant under charge conjugation:
%
\be \label{ccfv}
C \, |0(t)\ran_{A,B} \ = \ |0(t)\ran_{A,B} \, .
\ee
%
In fact, the explicit form of $C$ is
%
\bea
C & = & \exp\lf\{i \frac{\pi}{2} \, \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[ \lf(b^\dag_{\G k,\si}(t) \, a_{\G k,\si}(t)+a^\dag_{\G k,\si}(t) \, b_{\G k,\si}(t)\ri) \ri. \ri. \non \\[2mm]
&  &-\lf. \lf.  \eta_{\si,{}_{C}}  \lf(a^\dag_{\G k,\si}(t) \, a_{\G k,\si}(t)+b^\dag_{\G k,\si}(t) \, b_{\G k,\si}(t)\ri) \ri]\ri\} \, ,
\eea
%a
from which Eq.\eqref{ccfv} follows. A flavor state \eqref{invflavstate} transforms as
%
\be
C\, |a_{\G k,\si}(t)\ran \ = \ |b_{\G k,\si}(t)\ran \, , 
\ee
%
while flavor charge \eqref{fc} reverses its sign
%
\bea 
C \, Q_{\si}(t) \, C^{-1} \ = \ \int \!\! 
\frac{\dr^3 \G k}{2 \om_{\G k,\si} 	 (2 \pi)^3} 
\, \lf(b^\dag_{\G k,\si}(t) \, 
b_{\G k,\si}(t)-a^\dag_{\G k,\si}(t) \, a_{\G k,\si}(t)\ri)\ = \ - Q_{\si}(t)\, , 
\eea
%
as expected.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Time reversal}
The time reversal transformation of the flavor scalar fields is given by:
%
\be
T \, \ph_\si(x) \, T^{-1} \ = \ \eta_{\si,{}_{T}} \, \ph_\si(-\tilde{x}) \, , 
\ee
%
where $T$ is the \emph{antiunitary} time reversal operator. As usual, $|\eta_{\si,{}_{T}}|^2=1$. By using the explicit expansion \eqref{expph}, we find:
 %
\bea 
T \, \ph_\si(x) \, T^{-1} & = & \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[T \, a_{\G k,\si}(t) \, T^{-1}\, e^{i\om_{\G k,\si} t}\ + \ T \, b^{\dagger}_{-\G k,\si}(t) \, T^{-1} \, e^{-i\om_{\G k,\si} t}\ri] \, e^{-i\G k \cdot \G x} \non \\[2mm]
& = &  \eta_{\si,{}_{T}} \, \int \frac{\dr^3 \G k}{2\om_{\G k,\si}(2 \pi)^3} \, \lf[a_{\G k,\si}(-t) \, e^{i\om_{\G k,\si} t}\ + \ b^{\dagger}_{-\G k,\si}(-t) \, \, e^{-i\om_{\G k,\si} t}\ri] \, e^{i\G k \cdot \G x} \, .
\eea 
%
Transformations of creation and annihilation operators follow:
%
\bea
T \, a_{\G k,\si}(t) \, T^{-1} & = & \eta_{\si,{}_{T}} \, a_{-\G k,\si}(-t) \, \qquad T \, b_{\G k,\si}(t) \, T^{-1} \ = \ \eta^*_{\si,{}_{T}} \, b_{-\G k,\si}(-t) \, , \\[2mm]
T \, a^\dag_{\G k,\si}(t) \, T^{-1} & = & \eta^*_{\si,{}_{T}} \, a^\dag_{-\G k,\si}(-t) \, \qquad T \, b^\dag_{\G k,\si}(t) \, T^{-1} \ = \ \eta_{\si,{}_{T}} \, b^\dag_{-\G k,\si}(-t) \, .
\eea
%
By using canonical commutation relations \eqref{flavccr} one gets:
%
\bea
&&\hspace{-11mm}
a_{-\G k,\si}(-t)   
 =  \!\!\sum_{\rho=A,B} \frac{1}{2 \om_{\G k,\rho}(2\pi)^3}\lf(\lf[a_{-\G k,\si}(-t)  ,  a^\dag_{ -\G k,\rho}(t)\ri]  a_{-\G k,\rho}(t) - \lf[a_{-\G k,\si}(-t) ,  b_{\G k,\rho}(t)\ri]  b^\dag_{\G k,\rho}(t)\ri)  , 
\eea
%%
and similar relations  for the other operators. If one now look at flavor vacuum transformation properties 
%
\be \label{tsb}
|0(-t)\ran^T_{A,B} \ = \ T|0(t)\ran_{A,B} \, , 
\ee
%
it is evident that time-reversal symmetry is spontaneously broken. This could also be deduced by looking at the transformation of the  flavor charge \eqref{fc}:
%
\bea 
T \, Q_{\si}(t) \, T^{-1} \ = \ \int \!\! \frac{\dr^3 \G k}{2 \om_{\G k,\si} 	 (2 \pi)^3} \, \lf(a^\dag_{\G k,\si}(-t) \, a_{\G k,\si}(-t)-b^\dag_{\G k,\si}(-t) \, b_{\G k,\si}(-t)\ri)\ = \ Q_{\si}(-t)\, , 
\eea
%
i.e. $[Q_\si(t),T] \neq 0$ in a non-trivial way. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{CP and CPT symmetry}
From the previous considerations it is evident that $CP$ is an exact symmetry in the flavor representation\footnote{This is not true for the three flavor case, where $CP$ symmetry can be \emph{explicitly} broken because of a complex phase in the mass matrix.}:
%
\be
C \, P \, |0(t)\ran_{A,B} \ = \  |0(t)\ran_{A,B} \, .
\ee
%
However, from Eq.\eqref{tsb}, it follows that $CPT$ symmetry is spontaneously broken on the flavor vacuum:
%
\be \label{cptsb}
|0(-t)\ran^\Theta_{A,B} \ = \ \Theta \, |0(t)\ran_{A,B} \, ,
\ee
%
where $\Theta\equiv C \, P\, T$. This is a consequence of the transformation law of creation and annihilation operators:
%
\bea
\Theta \, a_{\G k,\si}(t) \, \Theta^{-1} & = & \eta_{\si} \, b_{\G k,\si}(-t) \, \qquad \Theta \, b_{\G k,\si}(t) \, \Theta^{-1} \ = \ \eta^*_{\si} \, a_{\G k,\si}(-t) \, , \\[2mm]
\Theta \, a^\dag_{\G k,\si}(t) \, \Theta^{-1} & = & \eta^*_{\si} \, b^\dag_{\G k,\si}(-t) \, \qquad \Theta \, b^\dag_{\G k,\si}(t) \, \Theta^{-1} \ = \ \eta_{\si} \, a^\dag_{\G k,\si}(-t) \, .
\eea
%
where $\eta_{\si}\equiv \eta_{\si,{}_{C}} \eta_{\si,{}_{P}} \eta_{\si,{}_{T}}$. This implies the charge transformation:
%
\bea \non
\Theta \, Q_{\si}(t) \, \Theta^{-1} & = & \int \!\! \frac{\dr^3 \G k}{2 \om_{\G k,\si} 	 (2 \pi)^3} \, \lf(b^\dag_{\G k,\si}(-t) \, b_{\G k,\si}(-t)-a^\dag_{\G k,\si}(-t) \, a_{\G k,\si}(-t)\ri)\\[2mm]
& = & -Q_{\si}(-t)\, .
\eea
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}\label{conclusions}
In this paper, we have briefly reviewed the mechanism of dynamical generation of field mixing via flavor vacuum condensate, as introduced in Ref.~\cite{ccpaper}. Then, we have studied the properties of flavor creation and annihilation operators under discrete symmetry transformations, in an effective model describing a flavor scalar doublet with mixing. We have proved that these transformations are symmetries of the Lagrangian, but they do not annihilate the flavor vacuum. Therefore spontaneous breaking of $CPT$ symmetry occurs. This SSB can be driven by a mechanism analogous to the one described in Section~\ref{Sec1}.

A similar symmetry breaking occurs in literature, e.g. 
in Refs.~\cite{Colladay:1996iz,Kostelecky2004}, where explicit Lorentz and $CPT$ violating terms are added to the Standard Model Lagrangian. A link between our approach and these previous studies should be determined. In this connection, also Lorentz/Poincar\'e symmetry breaking should be explicitly analyzed~\cite{poi}.

Recently \cite{Lobanov}, an extension of the Standard Model were proposed, by noticing that flavor mixing naturally implies Poincar\'e symmetry breaking. Then it is evident that a careful study of the symmetry properties of the flavor vacuum may represent an important indication of new physics. In this direction, the analysis of vacuum structure in connection with dynamical mixing generation \cite{mavroman,DynMix,ccpaper,ccpaper3} reveals its important r\^{o}le.    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments} P.J.  was  supported  by the Czech  Science  Foundation Grant No. 17-33812L.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{References}
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\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document} 