CP symmetry violation in the scalar sector of 331 models

TL;DR

This paper investigates CP violation in the scalar sector of a 331 model with SU(3) symmetry, focusing on spontaneous CP violation with a single CP phase and analyzing scalar mass states.

Contribution

It introduces a specific 331 model with a discrete symmetry that enables spontaneous CP violation with one independent phase, expanding understanding of CP violation mechanisms.

Findings

Identified conditions for spontaneous CP violation in the model

Derived scalar mass eigenstates and their rotations

Provided a framework for analyzing CP violation in extended gauge models

Abstract

In order to understand some frameworks for CP Violation scenarios in the scalar sector, a 331 model was considered which its main property is the incorporation of a local group symmetry SU(3) in the electroweak sector. In particular, a 331 model with a particular choice of free parameter. CP Violation scenarios were obtained by introducing a discrete symmetry in the scalar triplets, which exhibit a spontaneous CP Violation frame with just one independent CP phase associated. Mass state rotations were obtained.

Tables

Table 1. Table 1: Scalar Spectrum for 331 model with β = 1 3 𝛽 1 3 \beta=\frac{1}{\sqrt{3}}

scalar triplet	$Q_{\Phi }$	$Y_{\Phi }$	$X_{\Phi }$
$\chi =\left(\begin{array}{c}{\chi }^{\pm }\\ \frac{1}{\sqrt{2}}\left({\xi }_{{\chi }_2}\pm i{\zeta }_{{\chi }_2}\right)\\ \frac{1}{\sqrt{2}}\left({\xi }_{{\chi }_3}\pm i{\zeta }_{{\chi }_3}\right)\end{array}\right)$	$\left(\begin{array}{c}\pm 1\\ 0\\ 0\end{array}\right)$	$\left(\begin{array}{c}\pm \frac{1}{2}\\ \pm \frac{1}{2}\\ 0\end{array}\right)$	$\frac{1}{3}$
$\rho =\left(\begin{array}{c}{\rho }^{\pm }\\ \frac{1}{\sqrt{2}}\left({\xi }_{{\rho }_2}\pm i{\zeta }_{{\rho }_2}\right)\\ \frac{1}{\sqrt{2}}\left({\xi }_{{\rho }_3}\pm i{\zeta }_{{\rho }_3}\right)\end{array}\right)$	$\left(\begin{array}{c}\pm 1\\ 0\\ 0\end{array}\right)$	$\left(\begin{array}{c}\pm \frac{1}{2}\\ \pm \frac{1}{2}\\ 0\end{array}\right)$	$\frac{1}{3}$
$\eta =\left(\begin{array}{c}\frac{1}{\sqrt{2}}\left({\xi }_{\eta }\pm i{\zeta }_{\eta }\right)\\ {\eta }_2^{\mp }\\ {\eta }_3^{\mp }\end{array}\right)$	$\left(\begin{array}{c}0\\ \mp 1\\ \mp 1\end{array}\right)$	$\left(\begin{array}{c}\mp \frac{1}{2}\\ \mp \frac{1}{2}\\ \mp 1\end{array}\right)$	$-\frac{2}{3}$

Table 2. Table 2: Physical Scalar Spectrum

Scalar Boson	Squared Mass
$G_{1,2}^{\pm }$	0
$G_{1,2,3,4}^0$	0
$H_1^{\pm }$	$\frac{1}{2}{l}_7{u}^2$
$H_2^{\pm },H_4^0,H_5^0$	$\frac{f}{\sqrt{2}}u\left(\frac{{\nu }_{ew}^2}{\left|v\right|\left|w\right|}\right)$
$H_1^0$	$\frac{1}{2}\left(l_8+l_{10}\right)u^2$
$H_2^0$	$\frac{1}{2}\left(l_8-l_{10}\right)u^2$
$H_3^0$	$2l_1{u}^2$
$h^0$	$\frac{1}{2}\left(\frac{l_c{\left|w\right|}^4-2l_b{\left|w\right|}^2{\left|v\right|}^2-l_a{\left|v\right|}^4}{{\nu }_{ew}^2}\right)$

Table 3. Table 3: Physical Vectorial Fields Spectrum

Vector Boson	Squared Mass
$A_{\mu }$	0
$W_{\mu }^{\pm }$	$\frac{g_L^2}{4}{\nu }_{ew}^2$
$V_{\mu }^{\pm }$	$\frac{g_L^2}{4}\left(u^2+{\left|w\right|}^2\right)$
$V_{\mu }^0,{\overline{V^0}}_{\mu }$	$\frac{g_L^2}{4}\left(u^2+{\left|\nu \right|}^2\right)$
$Z_{\mu }$	$\frac{g_L^2}{4C_W^2}{v}_{ew}^2$
$Z_{\mu }^{\prime }$	$\frac{g_X^2}{3T_W^2}{u}^2$

Equations

$$\hat{Q}=\alpha G_3+\beta G_8+X,$$

$$Q_L^{(a)}={\left(\begin{array}{c}{u}^{(a)}\\ d^{(a)}\\ J^{(a)}\end{array}\right)}_L$$

$$Q_L^{(3)}={\left(\begin{array}{c}b\\ -t\\ J^{(3)}\end{array}\right)}_L$$

$${\ell }_L^{(i)}={\left(\begin{array}{c}{e}^{(i)}\\ -{\nu }^{(i)}\\ N^{(i)}\end{array}\right)}_L$$

$$u_R^{(i)},J_R^{(3)}$$

$$d_R^{(i)},J_R^{(a)}$$

$$e_R^{(i)}$$

$${\nu }_R^{(i)},N_R^{(i)}$$

$$D_{\mu }={\partial }_{\mu }-ig{W}_{\mu }^{\alpha }{G}_{\alpha }-i{g}_{X}X{B}_{\mu },$$

$$W_{\mu }=\frac{1}{2}\left(\begin{array}{ccc}{W}^3+\frac{1}{\sqrt{3}}{W}^8& \sqrt{2}{W}_{\mu }^{+}& \sqrt{2}{V}_{\mu }^{+}\\ \sqrt{2}{W}_{\mu }^{-}& -W^3+\frac{1}{\sqrt{3}}{W}^8& \sqrt{2}{V}_{\mu }^0\\ \sqrt{2}{V}_{\mu }^{-}& {\bar{V}}_{\mu }^0& -\frac{2}{\sqrt{3}}{W}^8\end{array}\right).$$

$$SU(3{)}_L\otimes U(1{)}_X\underset{1.T.}{\overset{⟨\chi ⟩}{\to }}SU(2{)}_L\otimes U(1{)}_Y\underset{2.T.}{\overset{⟨\rho ,\eta ⟩}{\to }}U(1{)}_Q.$$

$$⟨\chi ⟩=\left(\begin{array}{c}0\\ 0\\ v_{\chi }\end{array}\right),⟨\rho ⟩=\left(\begin{array}{c}0\\ v_{{\rho }_1}\\ v_{{\rho }_2}\end{array}\right),⟨\eta ⟩=\left(\begin{array}{c}{v}_{\eta }\\ 0\\ 0\end{array}\right).$$

$$V_H=$$

$${\mathcal{L}}_Y^{{}_Q}$$

$${\mathcal{L}}_Y^{\ell }=$$

$$⟨\chi ⟩=\left(\begin{array}{c}0\\ 0\\ |v_{\chi }|e^{i{\alpha }_{\chi }}\end{array}\right),⟨\rho ⟩=\left(\begin{array}{c}0\\ |v_{{\rho }_1}|e^{i{\alpha }_{{\rho }_1}}\\ |v_{{\rho }_2}|e^{i{\alpha }_{{\rho }_2}}\end{array}\right),⟨\eta ⟩=\left(\begin{array}{c}|v_{\eta }|e^{i{\alpha }_{\eta }}\\ 0\\ 0\end{array}\right).$$

$$\mathrm{SU}(3{)}_{\mathrm{L}}:{\varphi }_a\to {\varphi }_a^{\prime }=e^{i{\vec{\Omega }}_a\cdot \vec{T}}{\varphi }_a,$$

$${\vec{\Omega }}_{\chi }={\vec{\Omega }}_{\rho },{\vec{\Omega }}_{\eta }=-2{\vec{\Omega }}_{\rho }.$$

$$e^{i\mathbb{M}}=\sum _{n=0}^{N-1}{\mathrm{f}}_n(\mathbb{M}){\mathbb{M}}^n,$$

$${\mathrm{f}}_n(\mathbb{M})=\sum _{j=1}^N{P}_{jn}{e}^{i{m}_j},$$

$$\prod _{i\ne j}\frac{\mathbb{M}-m_i}{m_j-m_i}=\sum _{n=0}^{N-1}{P}_{jn}{\mathbb{M}}^n.$$

$$e^{i{\vec{\Omega }}_a\cdot \vec{T}}=$$

$$\mathcal{F}=\frac{e^{\frac{2}{\sqrt{3}}i{\alpha }_a\sin (\varphi +2\pi k/3)}}{1-2\cos (2\varphi +4\pi k/3)},$$

$$\varphi =-\frac{1}{3}\arcsin \left(\frac{3\sqrt{3}}{2}\text{det}(\mathbb{M})\right).$$

$$e^{i\vec{\Omega }\cdot \vec{T}}\equiv \mathbb{T}=\left(\begin{array}{ccc}{T}_1& m_1+n_1& m_2+n_2\\ m_1-n_1& T_2& m_3+n_3\\ m_2-n_2& m_3-n_3& T_3\end{array}\right),$$

$$\mathbb{T}=\left(\begin{array}{ccc}{\omega }^{\prime }/v_{\eta }& 0& 0\\ 0& v^{\prime }/v_{{\rho }_1}& 0\\ 0& -(v_{{\rho }_2}{u}^{\prime })/(v_{{\rho }_1}{v}_{\chi })& u^{\prime }/v_{\chi }\end{array}\right),$$

$$⟨\chi ⟩=\left(\begin{array}{c}0\\ 0\\ |u^{\prime }|e^{i{\alpha }_{\chi }}\end{array}\right),⟨\rho ⟩=\left(\begin{array}{c}0\\ |{\upsilon }^{\prime }|e^{i{\alpha }_{\rho }}\\ 0\end{array}\right),⟨\eta ⟩=\left(\begin{array}{c}|{\omega }^{\prime }|e^{i{\alpha }_{\eta }}\\ 0\\ 0\end{array}\right).$$

$$\mathrm{U}(1{)}_{\mathrm{X}}:{\varphi }_a\to {\varphi }_a^{\prime }=e^{i{\theta }_a{X}_a}{\varphi }_a,$$

$$V_H=...$$

$${\theta }_{\rho }={\theta }_{\eta }={\theta }_{\chi }=\theta .$$

$$⟨\chi ⟩=\left(\begin{array}{c}0\\ 0\\ |u^{\prime }|\end{array}\right),⟨\rho ⟩=\left(\begin{array}{c}0\\ |{\upsilon }^{\prime }|e^{i{\delta }_{\rho }}\\ 0\end{array}\right),⟨\eta ⟩=\left(\begin{array}{c}|{\omega }^{\prime }|e^{i{\delta }_{\eta }}\\ 0\\ 0\end{array}\right),$$

$${\delta }_{\rho }={\alpha }_{\rho }-{\alpha }_{\chi },{\delta }_{\eta }={\alpha }_{\eta }+2{\alpha }_{\chi },$$

$$⟨\chi ⟩=\left(\begin{array}{c}0\\ 0\\ u\end{array}\right),⟨\rho ⟩=\left(\begin{array}{c}0\\ v_1+i{v}_2\\ 0\end{array}\right),⟨\eta ⟩=\left(\begin{array}{c}{w}_1+i{w}_2\\ 0\\ 0\end{array}\right).$$

Full text

Keywords:

CP Violation, 331 model, Complex Scalar Triplets, $Z_{2}$ Symmetry.

I Introduction

The models based on the gauge symmetry group $SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}$, called 331 models, extend the electroweak sector of the Standard Model (SM) with many interesting consequences. For example, they can explain the existence of three fermion families by considering the cancellation of chiral anomalies and the asymptotic freedom in QCD pisano1992 ; *Frampton_1992; buras2013B . In addition, since the model is not universal of flavor families (the third family transform with a different representation), there arises an approach to understand the hierarchy of masses in the quark sector ochoa2005 ; buras2013B . In scenarios associated with $B$ physics, we can use specific versions of 331 models to explore new physics signals, as for example, in flavor changing neutral current (FCNC) contributions at tree level induced by the new $Z^{\prime}$ boson emerged from this model martinez2008 ; montano2013B ; cogollo2012 ; promberger2007flavor .

Another interesting aspect that can be study in the framework of 331 models, are their charge-parity symmetry (CP) properties. For example, one important scenario where CP violation plays a fundamental role is in the mechanism of leptogenesis WEINBERG_1979 ; *YOSHIMURA:1979aa; *Barr:1979ye; *NANOPOULOS:1979aa; *FRY_1980; *KOLB:1980aa; fukugita_1986 ; *Luty:1992un which provide us an attractive explanation for the generation of the observed Baryon Asymmetry of the Universe (BAU). In the framework of the Electroweak Baryogenesis (EWB), the three necessary conditions proposed by Sakharov sakharov1967violation are fulfilled in the extension of the SM with right-handed neutrinos. In fact, the Majorana masses violate the lepton number through the sphaleron processes Manton:1983nd ; *Kuzmin:1985mm; fukugita_1986 ; *Luty:1992un; barbieri_2003 ; *Giudice:2004aa; *nardi_2008, while CP violation occurs between lepton doublets and the right-handed neutrinos in the Yukawa couplings, and out-of-equilibrium neutrino decay arise fukugita_1986 ; *Luty:1992un; Covi:1996aa ; barbieri_2003 ; *Giudice:2004aa; *nardi_2008. In the SM, because of the large Higgs mass, it is not possible to obtain an out-of-equilibrium stage at the electroweak phase transition, while the CKM phase is not large enough kajantie_phase_1995 , then the resultant CP violation is too small SMCPV_93 ; SMCPV_quimbay_94 . Thus, feasible models of EWB need a modification of the scalar potential in order to introduce new sources of CP violation. In the 331 models, the leptonic number violation can be obtained through the contribution of new singlet neutrinos with heavy Majorana masses ($m_{N}\ll\upsilon_{EW}$) concha_neutrino_2003 ; nardi_2008 .

In order to incorporate CP violation, the introduction of complex couplings in the Higgs potential or complex vacuum expectation values (VEV’s) in the scalar fields can be proposed to generate explicit or spontaneous CP violation. In multi-Higgs models, as the popular two-Higgs-doublet model (2HDM), CP violation in the scalar sector can be controlled through discrete symmetries. Usually, various CP-violation terms appears in the Higgs potential. These terms can be forbidden by requiring discrete symmetries, as for example $Z_{2}$, obtaining a CP-conservative model. However, by soft-breaking the discrete symmetries accepting quadratic non-invariant terms, it is possible to obtain CP-violation interactions.

In this work, we address the CP properties of the scalar sector in the $\beta=1/\sqrt{3}$ 331 model; specifically, we do a systematic study for spontaneous CP breaking. In contrast to the 2HDM, we control the CP-violation terms through global continuous symmetries. In this scenario, we can obtain spontaneous CP-violation by requiring: 1.) complex VEVs of the scalar sector and 2.) the breaking of the global symmetry into a discrete symmetry.

This article is organized as follows. First, the next section is devoted to review the general properties of the CP conservative 331 models. Later, in section (III); an extension of the 331 model with CP violation in the Higgs sector, including minimal conditions for its realization is proposed. In particular, the model explores both explicitly and spontaneous CP violation. Finally, rotations into mass eigenstates are obtained in section (IV), where CP-even and -odd states are mixed.

II 331 model

A 331 model is a group extension of the SM group to the $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ gauge symmetries. One consequence of this extension, is that the 331 models have an extended Higgs sector, and a two-stage Spontaneous Symmetry Breaking (SSB) is required, where the first transition occurs at a higher scale than the electroweak breaking buras2013B . Also, they introduce new particles in the fermionic sector, which may produce new physics. In addition, the extension of the SM gauge group from $SU(2)_{L}$ to $SU(3)_{L}$ implies the existence of five new gauge bosons.

The $SU(3)_{L}$ group must satisfy the Lie algebra, thus the generators are proportional to the Gell Mann matrices; $G_{\alpha}=\frac{1}{2}\lambda_{\alpha}$ with $\alpha=1,...,8$. The electric charge operator is defined as the following combination of the diagonal generators:

[TABLE]

where we introduced the new quantum number ($X$), proportional to the identity matrix. By requiring the Gell-Mann Nijishima relation; $\hat{Q}=G_{3}+\hat{Y}$, where $G_{3}$ is an extension of the third Pauli matrix $\sigma_{3}$, we may choose $\alpha=1$ and $\hat{Y}=\beta G_{8}+X$ for the hipercharge, obtaining $\beta$ as a free parameter of the model. The phenomenology of 331 models depends on the value assigned to the $\beta$ parameter diaz2004 ; diaz2005 ; ochoa2005 . In particular, in order to avoid exotic charges (non-SM electric charges) this parameter must take the values $\beta=\pm\frac{1}{\sqrt{3}}$.

Finally, to be consistent with the SM phenomenology, the new particles must acquire masses at a very high energy. This can be achieved in a natural way by introducing a two-stage SSB scheme at different energy scales, where the new particles become massive in the first stage at a high energy scale above the electroweak scale.

II.1 Particles Spectrum

In 331 models, the left-handed leptons form triplets or antitriplets, and the right-handed leptons have singlet representations, similar to the SM, while in the quark sector, two left-handed generations comes in triplets while the other family has an anti-triplet representation in order to cancel the chiral anomalies. Thus, in the quark sector, the model is naturally nonuniversal of families, which provide a hint to understand the hierarchy mass problem exhibited by the heaviest family and the other ones. The above structure is summarize as follows:

[TABLE]

[TABLE]

In the fermionic spectrum shown above, we indicate the corresponding ($SU(3)_{C},SU(3)_{L},U(1)_{X}$) representations, where $J^{(i)}$ are new quarks, and $N^{(i)}$ are new neutral leptons. The spectrum also includes right-handed neutrinos which give us a better framework to address the phenomenology of neutrinos. The superscript convention is: $a=1,2$ for the first two families and $i=1,2,3$ run over all the three families.

In the bosonic sector, the covariant derivative over triplets is :

[TABLE]

where we must introduce 9 vector fields: eight fields $W^{a}_{\mu}$ associated with each $SU(3)$ generator and one field $B_{\mu}$, associated to the $U(1)$ group. We obtain the following representation for the gauge fields:

[TABLE]

where $W^{3}_{\mu}\,,\,W^{8}_{\mu}\,,\,B_{\mu}$ are three neutral vector bosons, which after rotations into mass eigenstates become the photon, the neutral weak boson $Z$ and a new neutral weak boson $Z^{\prime}$. In addition, there arise other two neutral gauge bosons $V_{\mu}^{0}$ and $\bar{V}_{\mu}^{0}$. For the charged sector, we obtain the weak bosons $W^{\pm}$ and new charged weak bosons $V^{\pm}$.

On the other hand, an appropriate scalar sector must be introduced to provide the SSB mechanism that gives masses to the vector gauge bosons. In addition, the Yukawa Lagrangian which couple scalar fields with fermions, must provide masses to all fermions. In our case, the 331 group exhibit a SSB scheme that must contain at least two transitions: one that lead us from 331 to the SM gauge symmetry (321), and another one from 321 to the chromodynamics and electrodynamics symmetries (31). In order to provide masses to all fermions, we must to introduce two scalar triplets in the second transition. So, the model exhibits the following breaking scheme through their scalar Vacuum Expectation Values (VEV):

[TABLE]

In the first transition (1.T.): we have five broken generators due to the VEV $\langle\chi\rangle$, where the five new gauge bosons acquire masses. In the 2.T. due to $\langle\rho,\eta\rangle$, three other generators are broken, which lead us to three massive gauge bosons and one massless gauge boson (the photon). The representation of the scalar fields are summarized in Table 1.

In order to obtain the above SSB pattern, the most general VEV structure of each triplet is:

[TABLE]

Since each field $\rho$ and $\eta$ does not fit VEVs separately in the first and second components simultaneously, it is necessary to take both scalar fields in the second transition in order to ensure that all fermions become massive after the SSB.

As we can see in Table 1, the scalar triplets that breaks the symmetry in the second transition, $\rho$ and $\eta$, have different quantum number on $X$, defining an unique scalar basis for CP studies, in contrast to models as in 2HDM which exhibits two identical scalar multiplets, and any combination between them define one possible basis that respect the gauge symmetry. Thus, we do not need to propose a general CP transformation, as occurs with 2HDM ecker1987GCP ; branco2011 .

II.2 Higgs and Yukawa Lagrangian

The most general hermitian, renormalizable and gauge invariant Higgs potential, considering $\beta=\frac{1}{\sqrt{3}}$, is:

[TABLE]

while the most general Yukawa Lagrangian that couple left- and right-handed quarks to Higgs fields has the following structure:

[TABLE]

We can note in the above Lagrangian, that the new quarks $J$ obtain masses from the coupling to the scalar triplet $\chi$ in the first transition. Also, there exists mixing terms with the light sector through the triplets $\rho$ and $\eta$ at the scale of the second transition. Although the gauge invariance does not forbid those terms, we can restrict these mixing terms through extra global symmetries.

Analogous to the quark sector, for the leptonic part in the Lagrangian we have the following terms:

[TABLE]

where couplings with right-handed neutrinos and Majorana terms are introduced.

III CP Violation

CP violation in the scalar sector can be introduced in two forms: explicit or spontaneous. The first form occurs when some terms in the scalar Lagrangian are non-invariant under the CP transformation. In the second one, the Lagrangian of the model conserves CP, but the vacuum of the scalar sector is not CP-invariant, so that the theory breaks CP together with the SSB haber2012group . In the above section, if we consider that all the Lagrangian parameters and VEVs are real, then the model does not break CP neither explicitly nor spontaneously. Now, we are going to consider the situation when couplings and VEVs are complex. In the literature, some particular CP-violation scenarios in these models have already been considered before promberger2007flavor ; doff2006spontaneous ; montero1999soft . Here, we undertake a systematic study of the conditions for CP violation in the model with $\beta=1/\sqrt{3}$. For that, we first note that only seven of the nineteen couplings of the Higgs potential in equation (5) can be complex. They are: one quadratic coupling ($\mu_{4}^{2}$), one cubic coupling ($f$), and five quartic couplings ($l_{10,11,12,13,14}$). Thus, we may introduce up to 7 phases in the potential. In addition, the SSB mechanism allows four VEVs as shown in equation (4), which may introduce 4 more complex phases as:

[TABLE]

However, because of the rephasing invariance of the potential, most of the above phases can be reabsorbed, and not all of them will lead us to interactions with CP violation. Furthermore, because of the gauge symmetry, we may induce appropriate transformations to eliminate two VEVs, as shown below.

III.0.1 $SU(3)_{L}$ Rotation

The Higgs potential must be invariant under transformations of the $SU(3)_{L}$ group. First, we define the general transformation over the scalar fields as:

[TABLE]

where $\vec{T}$ are the group generators ot the $SU(3)_{L}$ group and $\vec{\Omega}_{a}$ are the transformation parameters for each field $a=\chi,\rho$ and $\eta$. Applying the transformation on the Higgs potential and by invoking the $SU(3)_{L}$ invariance ($V^{\prime}_{H}=V_{H}$), we obtain the following conditions between the parameters:

[TABLE]

In order to study how the scalar triplets transform, we use the spectral theorem for general $SU(N)$ groups to expand the exponential as weigert1997baker :

[TABLE]

with $\mathbb{M}$ an hermitian matrix, while the coefficients are:

[TABLE]

where $m_{j}$ are the eigenvalues of the matrix $\mathbb{M}$, and $P_{jn}$ are coefficients obtained from:

[TABLE]

In our case, for $SU(3)$, we have that $\mathbb{M}=\vec{\Omega}_{a}\cdot\vec{T}$, and $N=3$. Upon identifying $\vec{\Omega}_{a}=\alpha_{a}\hat{n}$ and $\hat{\mathbb{M}}=\hat{n}\cdot\vec{T}$ the following expansion can be obtained curtright2015elementary :

[TABLE]

where:

[TABLE]

[TABLE]

Making the explicit calculations, as shown in the Appendix A, the following transformation matrix is obtained:

[TABLE]

where the parameters $m_{\{1,2,3\}}$, $n_{\{1,2,3\}}$ and $T_{\{1,2,3\}}$ are defined in Equation (93) from the appendix A.

Through the above matrix representation of a general SU(3) transformation, we can rotate the scalar triplets in (8) in order to reduce: 1.) the number of complex phases of the VEV’s from 4 to 3, and 2.) the number of VEVs in $\langle\rho\rangle$ from 2 to 1. In particular, we choose to rotate this triplet so that one VEV remains in the second component in order to ensure that all fermions acquire masses. As shown in the Appendix A, the matrix that lead us to the reduction described above is:

[TABLE]

obtaining a new basis with the following complex VEV structure:

[TABLE]

Using this basis, three complex phases remains in the vacuum state. However, further reduction can be obtained by applying an $U(1)_{X}$ phase rotation, as shown below.

III.0.2 $U(1)_{X}$ Rotation

The transformation associated to $U(1)_{X}$ group, which operates over the scalar fields $\phi$ can be defined as:

[TABLE]

which corresponds to a phase rotation according to the value $X_{a}$ of each triplet. After a general $U(1)_{X}$ phase transformation, the terms that obtain a global phase in the scalar potential are:

[TABLE]

From the invariance of the potential, we find that all the triplets must transform in the same amount:

[TABLE]

In particular, we can do a $U(1)_{X}$ phase rotation in order to eliminate up to one complex phase in (17). For example, with a rotation $\theta=-3\alpha_{\chi}$, we can eliminate the phase associated to the scalar field $\chi$ to avoid CP violation in the first transition, obtaining the following VEV structures:

[TABLE]

where the new phases are defined as:

[TABLE]

or, in cartesian form:

[TABLE]

In summary, we obtain only two possible phases as candidates to produce spontaneous CP violation. However, as we will see below, from the stationary conditions of the Higgs potential, these phases are not mutually independent.

III.1 Higgs Potential with $U(1)$ Global Symmetry

The full Higgs potential in Equation (5) can be separated into two parts: the first one with real couplings, and the other one with complex coupling constants, $V_{H}=V_{\mathbb{R}}+V_{\mathbb{C}}$, where:

[TABLE]

We can restrict the complex sector if we demand additional global symmetries. Let us assume, for example, invariance of the Higgs potential under the following $U(1)$ transformations:

[TABLE]

where $\chi$ remains invariant, while the other two scalar fields, $\rho$ and $\eta$, change by an opposite phase. Invariance under this transformation leaves us with only the f-term in the complex part of the potential:

[TABLE]

By using this simplified potential, we will evaluate below the CP-properties of the scalar sector.

III.1.1 Explicit CP invariance

In order to evaluate if the potential in (25) violates CP or not explicitly, let us assume real VEVs. In this case, the only complex phase come from the cubic coupling constant $f=|f|e^{i\alpha_{f}}$ in the potential. However, explicit CP violation arises only if rephasing transformations of the fields that transform the complex parameters into real do not exist Branco1999 ; ginzburg2005 . In our case, if we transform the three scalar triplets as

[TABLE]

then, the potential in Eq. (25) become:

[TABLE]

In particular, any phase transformation that satisfies the relation:

[TABLE]

will transform the above potential into a real one, i.e., there exists a real basis. Thus, the model with the global symmetry (24) is explicitly CP invariant in the scalar sector.

III.1.2 Spontaneous CP invariance

Knowing that the potential is explicitly CP invariant, we will evaluate if the model with complex vacuum given by (20) is also CP-invariant. After replacing the scalar triplets into the potential, evaluated at the vacuum state, and including the complex phase of the f-term, $|f|e^{i\alpha_{f}}$, the complex part results with additional phases as:

[TABLE]

obtaining terms with the global phase: $\delta=\delta_{\eta}+\delta_{\rho}-\alpha_{f}$. However, the vacuum must accomplish the minimum conditions with respect to the phases, i.e:

[TABLE]

from where we obtain the constraint:

[TABLE]

So, the phases from the VEVs are not independent of the phase from the coupling constant $f$. Spontaneous CP violation arises if rephasing transformations of the fields that transform all the complex VEVs into real, and simultaneously preserve the real basis do not exist. In our case, as we saw above, a real basis is obtained through transformations of the form (26) - (28) with the phase relation given by (30). Under these transformations, the VEVs in (20) become:

[TABLE]

so that, after considering the constraint (33), the phases of the VEVs are:

[TABLE]

Thus, real VEVs in the real basis can be obtained if all the above phases can be cancelled simultaneously. Indeed, if $\theta^{\prime}_{\rho}=0$ and $\theta^{\prime}_{\eta}=0$ (i.e. we rotate as $\theta_{\rho(\eta)}=-\delta_{\rho,(\eta)}$), we obtain automatically that also $\theta^{\prime}_{\chi}=0$.

In conclusion, the model with the global symmetry defined by (24) is both explicitly and spontaneously CP invariant in the scalar sector.

III.2 Breaking of Symmetry $U(1)$ $\rightarrow$ $Z_{2}$

Let us now consider a less restricted scenario by breaking the global $U(1)$ symmetry into a discrete $Z_{2}$ symmetry. For example, if we limit the symmetry (24) only for $\Theta=\pi$, we obtain:

[TABLE]

so that the complex part of the Higgs potential preserves, in addition to the f-terms, the $l_{10}$-terms:

[TABLE]

Below, we analyze the CP properties of this potential, as we did in the previous section.

III.2.1 Explicit CP invariance

By assuming real VEVs ($\{\langle\chi\rangle,\,\langle\rho\rangle,\,\langle\eta\rangle\}\in\mathbb{R}$), we propose a general basis rotation as in equations (26 - 28), obtaining the following scalar potential:

[TABLE]

where we include the complex phases of the $f$ and $l_{10}$ couplings. A real basis is found if we choose rotations that satisfy:

[TABLE]

obtaining a real potential. Thus, the model with the $\left.Z_{2}\right|_{\pi}$ symmetry is explicitly CP invariant in the scalar sector.

III.2.2 Spontaneous CP Violation

By considering general complex VEVs as in (20) into the scalar potential (37), we find that the $l_{10}$-terms do not contribute, obtaining the same result as in equation (31). Thus, the phases of the VEVs are constrained by the same minimum condition found in (33), so that the relations in (39) become:

[TABLE]

On the other hand, the rotations (26 - 28) lead us to new transformed VEVs with the following phases:

[TABLE]

which must accomplish both relations from equation (40) in order to obtain a real basis. Real VEVs are obtained if there exists some rotation that cancel all the above phases. For example, if we choose $\theta_{\chi}=0$, we obtain from (40) the solutions: $\theta_{\rho}=(1/2)\alpha_{10}$ and $\theta_{\eta}=-(1/2)\alpha_{10}-\delta_{\rho}-\delta_{\eta}$, which convert the phases in (41) into:

[TABLE]

obtaining, at best, the constraint $\theta^{\prime}_{\eta}=-\theta^{\prime}_{\rho}$. Since we are assuming from the beginning a general complex potential with arbitrary phases $\alpha_{10}$ and $\delta_{\rho}$ different from zero, then we obtain VEVs with one irreducible complex phase 111There is a very particular case for a model with $\alpha_{10}=-2\,\delta_{\rho}$, which cancel all the phases, and the model become CP conservative. Thus, the model with the symmetry $\left.Z_{2}\right|_{\pi}$ is spontaneously CP violating.

IV Mass states in CPV 331 model

Now that we have found a 331 model with a $Z_{2}$ symmetry which admits spontaneous CP violation in the scalar sector, we are going to analyze the mass eigenstates to obtain the physical spectrum.

IV.1 Physical Spectrum - Scalar Sector

Table (1) define the scalar spectrum in the electroweak basis. We must rotate this basis in order to obtain the physical states for the scalar fields and identify the Goldstone bosons and the Higgs massive fields. For that, it is more convenient to take the complex VEVs of the scalar fields in its cartesian form, as in Eq. (Eq. 22), so that the minimum conditions of the Higgs potential become:

[TABLE]

with $V_{H}$ the Higgs potential with $Z_{2}$ symmetry from (37). First, after minimization, we find one condition that relates the VEVs of the $\rho$ and $\eta$ triplets:

[TABLE]

Second, we obtain the following tadpoles equations:

[TABLE]

After replacing the above tadpoles equations into the Higgs potential, and through their second derivatives, we may obtain the mass matrices. For the charged fields, the mixing mass elements are:

[TABLE]

in the basis $\Phi_{C_{i}}=\{\chi^{\pm}\,,\,\eta^{\pm}_{3}\,,\,\rho^{\pm}\,,\,\eta^{\pm}_{2}\}$, which can be separated into two independent $2\times 2$ blocks in the $\{\chi^{\pm}\,,\,\eta^{\pm}_{3}\}$ and $\{\rho^{\pm}\,,\,\eta^{\pm}_{2}\}$ basis, as shown in Eqs. (104)-(106) in Appendix B.1. The diagonalization of these matrices lead us to the following mass eigenstates:

[TABLE]

where the mixing angles and the complex phase are defined as:

[TABLE]

with $|w|,\,|u|,\,|\nu|$ the norm of the VEV’s and $v_{ew}\approx 246$ GeV the electroweak breaking scale. The fields $G_{1,2}^{\pm}$ are the corresponding massless Goldstone bosons associated to the massive $W^{\pm}$ and $V^{\pm}$ gauge bosons. The other charged Higgs bosons remains as physical particles with the following squared masses:

[TABLE]

where the approximations can be considered if we assume that the first transition occurs at much larger scale than the second one, i.e., if $u\gg v_{1},v_{2},w_{1},w_{2}$.

For the neutral components, we have:

[TABLE]

which lead us to two independent mass matrices in the basis $\{\xi_{\chi_{2}},\zeta_{\chi_{2}},\xi_{\rho_{3}},\zeta_{\rho_{3}}\}$ and $\{\zeta_{\chi_{3}},\zeta_{\rho_{2}},\zeta_{\eta},\xi_{\rho_{2}},\xi_{\eta},\xi_{\chi_{3}}\}$ written in Appendices B.2 and B.3, where we can see that mixing terms between CP-even ($\xi_{i}$) and CP-odd ($\zeta_{i}$) fields emerge, which will induce CP violation. For the $4\times 4$ mass matrix, we make a block diagonalization by organizing the matrix in $2\times 2$ submatrices, as shown in equations (107) - (110), which exhibits a hierarchy structure that allow us apply a recursive expansion to obtain the set of eigenvalues and eigenstates, as shown in App. B.2. This procedure results in the following mass eigenstates:

[TABLE]

[TABLE]

where the mixing angle is:

[TABLE]

The neutral Higgs bosons have the following squared masses:

[TABLE]

For the other neutral matrix, we also define a block structure, as shown in Eq. (114), where we identify three scales: very light ($\sim v_{ew}^{3}/u$), light ($\sim v_{ew}^{2}$) and heavy ($\sim v_{ew}u$ and $u^{2}$). This hierarchy structure allow us to obtain the diagonalization of the $6\times 6$ matrix, as shown in Appendix B.3, obtaining the following mass eigenstates:

[TABLE]

with

[TABLE]

and $\theta$ the same mixing angle obtained in Eq. (49). In the above spectrum, we find two Goldstone bosons ($G_{3,4}^{0}$) and four Higgs bosons with the following masses

[TABLE]

where:

[TABLE]

We observe that one of the Higgs bosons ($h^{0}$) has mass at the electroweak scale, which we will identify with the observe one at LHC.

In summary, Table 2 shows the physical scalar spectrum of the model.

IV.2 Physical Spectrum - Vector Sector

The interactions with the scalar fields are incorporated through the kinetic Higgs Lagrangian.

[TABLE]

where $D_{\mu}$ is the covariant derivative, defined in the equation (2). The mass matrix can be obtained through:

[TABLE]

where $\Phi_{i}$ are the scalar fields in weak basis and $V_{i}$ are the vector fields. The charged bosons and the complex neutral bosons ($V^{0}$), are already in mass states with masses:

[TABLE]

Since $v_{ew}^{2}=|v|^{2}+|w|^{2}\ll u^{2}$, we obtain the hierarchy $M_{W^{\pm}}\ll M_{V}$. On the other hand, the other neutral gauge bosons exhibit mixing mass terms. This matrix is singular, ensuring one massless boson (the photon). In 331 models, the electroweak gauge bosons rotate into mass eigenstates as:

[TABLE]

where the Weinberg angle is defined as

[TABLE]

and $f[T_{W}]=\sqrt{1-T_{W}^{2}/3}$. However, in the above rotation, there is still a small mixing in the basis $(Z,Z^{\prime})$, which diagonalize through a rotation with angle $\varphi$ given approximately by:

[TABLE]

so that the mass eigenstates rotate as:

[TABLE]

where the rotation matrix is defined as:

[TABLE]

$Z^{1}_{\mu}$ can be identified as the phenomenological neutral weak boson and $Z^{2}_{\mu}$ is a new heavy physical boson. As we can see, the angle $\varphi$ that mix the components to obtain $Z^{1}_{\mu}$ and $Z^{2}_{\mu}$ is suppressed as $s_{\varphi}\propto v_{ew}/u^{2}$. In the limit without mixing, $Z^{1}_{\mu}\rightarrow Z$ and $Z^{2}_{\mu}\rightarrow Z^{\prime}$.

As in the SM, we obtain that $C_{W}=\frac{M^{2}_{W}}{M^{2}_{Z}}$, which allow us to identify the angle $\theta_{W}$ with the Weinberg angle. Table 3 shows the physical vector bosons and their masses.

IV.3 Physical Spectrum - Yukawa Sector

Due to the $Z_{2}$ symmetry which was chosen in the Higgs potential for the model, some terms in the Yukawa Lagrangian must be removed, spoiling the mass acquisition of the fermions. In particular, introducing this symmetry, all the light fermions (those terms that couple with $\rho$ and $\eta$) become massless.

Thus, it is necessary to include additional $Z_{2}$ symmetries in the quarks sector in order to guarantee mass for all the fermionic fields. We choose the following symmetries:

Right-handed Sector

:

[TABLE]

Left-handed Sector

:

[TABLE]

Taking into account the above symmetries, it is possible to set the Yukawa Lagrangian. After that and computing in the vacuum states of scalar fields, we obtain the following mass matrix for the up sector:

[TABLE]

and the matrix for the down sector:

[TABLE]

In general, those coupling parameters ($\Gamma$) are complex. Then we can obtain the physical states with a bi-unitary transformation.

After diagonalizing, the fields are obtained in mass eigenstates, however, this sector exhibits many phenomenological aspects as the problem of Flavor Changing Neutral Currents (FCNC’s) that must be controlled in this model, the hierarchy of quarks masses and the violation effects through the CKM matrix. These aspects are outside of the central purpose of this work but they can be considered for a later research.

V Conclusions

We study the most general scalar potential for the 331 model with a parameter $\beta=1/\sqrt{3}$. Using the Gauge symmetries of the model we rotate the scalar fields in order to eliminate non-physical phases to remain exclusively with the physical phases222In the appendix A, it is illustrated in detail; how to use the Gauge symmetries of $SU(3)\otimes U(1)$ to rotate the complex phases and stay with only the physical phases. which allow generating a CP violation mechanism in the scalar sector.

We find that taking the model with three Higgs triplets only two complex physical phases are required. We consider a particular phenomenological scenario in which these phases are fixed in the scalar doublets associated with the weak transition ($2^{\text{nd}}$.T.), where the CP violation would exhibit at low energies on an electroweak scale. Although it is possible to consider the scenario where one of these phases is on the electroweak scale while the other one is in the TeV breaking scale, this scenario will be considered in a later work.

For our case, we find the mass eigenstates spectrum of the even and odd fields, in both the neutral and the charged sector of the model. From the conditions of the minimum potential, it is possible to eliminate one of the complex phases and rewrite the mass eigenstates and rotations according to only one of the complex phases. This phase only appears in the rotation matrix of the charged Higgs sector. The neutral sector is free of this complex phase. The presence of CP violation is also reflected in the mixture of even scalar fields with the odd fields. However, from LHC data in the Higgs decay in two photons, there is a very small gap in which the Higgs boson has a mixture from the odd sector Brod:2013cka ; Inoue:2014nva ; Keus:2015hva , therefore the mixture of the odd’s with the even’s it is very restricted by LHC data.

We also consider additional global symmetries where it is shown that complex phases can be reabsorbed into the parameters of the scalar potential and therefore in those particular scenarios there is no CP violation, leaving only the global symmetry scenario $Z_{2}$ which allows a framework with spontaneous CP violation.

Appendix A Matrix Transformation for SU(3) and rotations

The spectral theorem for smooth functions of a Hermitian matrix with eigenvalues $m_{j}$ allow the expansion:

[TABLE]

with

[TABLE]

and expanded as in Equation (13), from where weigert1997baker ; curtright2015elementary :

[TABLE]

for example, for our purpose, the exponential function ($f(\mathbb{M}):=e^{i\mathbb{M}}$), can be written as:

[TABLE]

On the other hand, the hermitian matrix $\mathbb{M}$ can be written as the product between the rotation angle ($\Omega$) with the group generators as $\mathbb{M}=\vec{\Omega}\cdot\vec{T}$. By writing ($\Omega=\alpha\hat{n}_{\alpha}$), we can rewrite the matrix as ($\mathbb{M}=\alpha(\hat{n}_{\alpha}\cdot\vec{T})=\alpha\hat{\mathbb{M}}$). For the $SU(3)$ group, we have that $\hat{n}_{\alpha}=\{n_{1},...,n_{8}\}$, so that the powers of the matrix $\hat{\mathbb{M}}$ are:

[TABLE]

changing the space of parameters from 8 real parameters ($n_{1,2,...,8}$) to 3 complex parameters ($b,c,e$) and 2 real parameters ($a,d$). From the spectral expansion in Equation (82) for $SU(3)$, the exponential can be written as:

[TABLE]

where curtright2015elementary

[TABLE]

and

[TABLE]

Using equations (83, 84 and 85), we obtain the transformation in matrix form ( $e^{i\mathbb{M}}=\mathbb{T}$) as:

[TABLE]

where

[TABLE]

The matrix (90) can be written as in equation (15),

[TABLE]

if we define:

[TABLE]

We can use this matrix representation $\mathbb{T}$, to make a $SU(3)$ rotation that eliminates one VEV from ($\rho$). On the other hand, if we apply an element of $SU(3)$ over the VEV of the scalar field $\eta$, the unbroken element must remain invariant, i.e.,

[TABLE]

Using explicit matrix representation we have:

[TABLE]

and considering the mentioned invariance, we obtain the following relations;

[TABLE]

obtaining the following form:

[TABLE]

The scalar field ($\rho$) has two VEVs in general different from zero. However, we can rotate the triplet to obtain only one VEV in the second component. So, we demand that:

[TABLE]

from where we obtain the following relations:

[TABLE]

which lead us to:

[TABLE]

Finally, the VEV of the scalar field $\chi$ transforms as:

[TABLE]

obtaining:

[TABLE]

so that the transformation matrix become:

.

[TABLE]

Appendix B Mass Matrices

B.1 Mass Matrix for the Charged Sector

Applying the second derivative as defined as in Eq. (44), we obtain the following mass matrix for the charged sector:

[TABLE]

where;

[TABLE]

Both matrices are singular, so we obtain at least two Goldstone bosons. By direct diagonalization of a $2\times 2$ matrix, we find the eigenvectors and eigenvalues written in Eqs. (45)-(51).

B.2 Mass Matrix for Neutral Sector - $\mathbb{M}_{a}$

Applying the second derivative from Eq. (52) restricted to the basis $\Phi_{N_{i}}=(\xi_{\chi_{2}}\,,\,\zeta_{\chi_{2}}\,,\,\xi_{\rho_{3}}\,,\,\zeta_{\rho_{3}}$), we find the following blocks with hierarchy structure:

[TABLE]

where

[TABLE]

As we can see, matrix $C$ is proportional to the energy scale of the first transition ($C\sim u^{2}$), the matrix $B$ has intermediate energy scale ($B\sim u$) and finally, the matrix $A$ is at the electroweak scale ($A\sim\nu_{ew}$), so that:

[TABLE]

Therefore, the matrix (110) can be block diagonalized by a unitary rotation of the form Grimus_2000 :

[TABLE]

where $1$ is the identity submatrix and $F$ is a submatrix that satisfy ($F<<1$). Keeping only up to linear terms on $F$, the matrix rotation has the form:

[TABLE]

From the cancellation of the nondiagonal block, we obtain the solution

[TABLE]

Therefore, from the diagonal blocks, we obtain at dominant order that:

[TABLE]

By replacing each submatrix into the above results, we find that $\mathbb{D}_{a1}=0$, obtaining two Goldstone bosons, while $\mathbb{D}_{a2}$ is already diagonal, obtaining the two massive Higgs bosons written in (53) and masses in (56) and (57).

B.3 Neutral sector mass Matrix - $\mathbb{M}_{b}$

The mass matrix in the basis $\Phi_{N_{i}}=(\zeta_{\chi_{3}}\,,\,\zeta_{\rho_{2}}\,,\,\zeta_{\eta}\,,\,\xi_{\rho_{2}}\,,\,\xi_{\eta}\,,\,\xi_{\chi_{3}}$) appears as an independent matrix separated from the above matrix, $\mathbb{M}_{a}$, obtaining the following structure:

[TABLE]

where:

[TABLE]

By assuming that the VEVs of the triplets $\rho$ and $\eta$, and the cubic coupling constant $f$ are of the order of the electroweak scale (i.e. $f\sim v_{1,2}\sim w_{1,2}\sim v_{ew}$) while $\chi$ breaks the symmetry at a larger scale ($u\gg v_{ew}$), then the above blocks contains four energy scales: $M_{b_{11}}$ is the lightest scale at $v_{ew}^{3}/u$, $D$ is the intermediate scale at $v_{ew}^{2}$, $E$ and $\mathbb{H}$ contains heavy terms at $v_{ew}u$, and $M_{b_{66}}$ is the heaviest scale at $u^{2}$. We may further reduce the matrix by putting the $v_{ew}u$ and $u^{2}$ scales into a single block, so that the mass matrix in (114) become:

[TABLE]

where:

[TABLE]

and $M_{b_{11}}\ll B\ll\mathbb{C}$. On the other hand, the exact $6\times 6$ matrix is singular with multiplicity two, containing two massless Goldstone bosons. One of this Goldstone bosons come from the submatrix $\mathbb{C}$, so that the other massless Goldstone must arise from the other blocks. Thus, we propose a block diagonalization of the form:

[TABLE]

where $U$ is an unitary rotation, which, as in (111), can be approximately written through a small transformation $F_{n\times m}$ of dimension $n\times m$ as:

[TABLE]

so that at dominant order, we obtain:

[TABLE]

Eq. (125) represents a constraint over the components of the transformation $F_{1\times 5}$, while (126) tell us that, at dominant order, the sub-matrix $\mathbb{C}$ in (122) decouple from the other components, and can be block diagonalized separately. Taking into account that $\mathbb{H}\sim E\ll M_{b66}$, we can use the same recursive method as before, where a rotation matrix $R$ block diagonalize the matrix as:

[TABLE]

with

[TABLE]

Thus, we obtain:

[TABLE]

The component $d_{2}$ gives directly the neutral Higgs boson written in Eq. (59) with mass given by Eq. (62), while $d_{1}$ is a new singular $4\times 4$ matrix that can be diagonalized by analytic methods, obtaining the rotation matrix, mass eigenvectors and mass eigenvalues written in Eqs. (60) - (64). Finally, the Goldstone boson that arise in the $11$ component in (123) rotates through the transformation $F_{1\times 5}$ that must accomplish the constraint from (125). As a first approximation, if we neglect this transformation, we obtain the Goldstone boson written in (58)