Pati-Salam unification with a spontaneous CP violation

TL;DR

This paper explores a Pati-Salam unification model where spontaneous CP violation explains the origin of CP-violating phases in quark and lepton mixing matrices, linking low-energy physics to high-energy unification.

Contribution

It presents a novel Pati-Salam framework where CP violation arises spontaneously, providing a potential explanation for the phases in the CKM and PMNS matrices.

Findings

CP phases can be generated spontaneously in the model

The framework connects low-energy CP violation to high-energy physics

Phenomenological implications for neutrino and quark sectors

Abstract

Recent neutrino oscillation experiments suggest that the PMNS matrix in the lepton sector has a $CP$ violating phase as the CKM matrix in the quark sector. However, origin of these phases in both matrices are not clarified by now. Although complex Yukawa couplings could induce these phases, they remain as free parameters of the model even in that case. If the $CP$ symmetry is considered to be spontaneously broken, they are expected to be determined by some physics at a much lower energy scale than the Planck scale. We study such a possibility in a framework of Pati-Salam type unification. We also discuss other phenomenological issues in it.

Equations

$$f_{L_i}(4,2,0),h_{R_i}(4,1,1/2),k_{R_i}(4,1,-1/2),$$

$$\Sigma_{R_{\alpha}}\equiv\sum_{a=1}^{3}\frac{\tau^{a}}{2}\Sigma^{a}_{R_{\alpha}}=\frac{1}{2}\left(\begin{array}[]{cc}\Sigma^{0}_{R_{\alpha}}&\sqrt{2}\Sigma^{+}_{R_{\alpha}}\\ \sqrt{2}\Sigma^{-}_{R_{\alpha}}&-\Sigma^{0}_{R_{\alpha}}\\ \end{array}\right).$$

$$\Phi (4,1,1/2),\Psi (4,1,1/2),\varphi (1,2,-1/2),\eta (1,2,-1/2),$$

$$\sigma (1,1,0),S(1,1,0),s(1,1,0).$$

$$f_{L_i},h_{R_i},k_{R_i}\Rightarrow (0,1),F_L,\Rightarrow (0,7),F_R\Rightarrow (2,1),{\Sigma }_{R_{\alpha }}\Rightarrow (1,1),S\Rightarrow (0,6),$$

$$\sigma \Rightarrow (2,2),\eta \Rightarrow (-1,1),\Phi \Rightarrow (0,4),\Psi ,\varphi \Rightarrow (0,0),s\Rightarrow (0,1).$$

$$-{\mathcal{L}}_y=y_{ij}^h{\bar{f}}_{L_i}\varphi h_{R_j}+y_{ij}^k{\bar{f}}_{L_i}\tilde{\varphi }{k}_{R_j}+y_{i}S{\bar{F}}_L{k}_{R_i}+x{\sigma }^{\ast }{\bar{F}}_L{F}_R+{\gamma }_{{\Sigma }_{\alpha }}{\sigma }^{\ast }{\bar{\Sigma }}_{R_{\alpha }}^c{\Sigma }_{R_{\alpha }}+h.c.,$$

$$V$$

$$SU(4)\times SU(2)\times U(1{)}_X\stackrel{⟨\Phi ⟩,⟨\Psi ⟩}{⟶}SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y.$$

$$Y=\frac{2}{\sqrt{6}}{T}_{15}+X,$$

$$f_{L_i}=(q_{L_i},{\ell }_{L_i}),h_{R_i}=(u_{R_i},N_{R_i}),k_{R_i}=(d_{R_i},e_{R_i}),$$

$$-{\mathcal{L}}_y$$

$$y_{ij}^h=y_{ij}^u=y_{ij}^{\nu },y_{ij}^k=y_{ij}^d=y_{ij}^e,y_i=y_i^D=y_i^E,{\tilde{y}}_i={\tilde{y}}_i^D={\tilde{y}}_i^E,x=x_D=x_E,$$

$$-{\mathcal{L}}_y^{\prime }$$

$$⟨\sigma ⟩=w{e}^{i\chi },⟨S⟩=u{e}^{i\rho },⟨s⟩=v{e}^{i\psi }.$$

$$V$$

$$m_{\varphi }^2={\tilde{m}}_{\varphi }^2+{\kappa }_{\sigma \varphi }{w}^2+{\kappa }_{S\varphi }{u}^2+{\kappa }_{s\varphi }{v}^2+2{\zeta }_{\varphi }{\Lambda }^2,m_{\eta }^2={\tilde{m}}_{\eta }^2+{\kappa }_{\sigma \eta }{w}^2+{\kappa }_{S\eta }{u}^2+{\kappa }_{s\eta }{v}^2+2{\zeta }_{\eta }{\Lambda }^2.$$

$$10^9\mathrm{GeV}<w<10^{12}\mathrm{GeV}.$$

$$(\bar{d}_{Li},\bar{D}_{L})\left(\begin{array}[]{cc}m^{d}_{ij}&0\\ {\cal F}^{d}_{j}&\mu_{D}\\ \end{array}\right)\left(\begin{array}[]{c}d_{R_{j}}\\ D_{R}\\ \end{array}\right).$$

$$\left(\begin{array}[]{cc}A&B\\ C&D\\ \end{array}\right)\left(\begin{array}[]{cc}m^{d}m^{d\dagger}&m^{d}{\cal F}^{d\dagger}\\ {\cal F}^{d}m^{d\dagger}&\mu_{D}\mu_{D}^{\dagger}+{\cal F}^{d}{\cal F}^{d\dagger}\\ \end{array}\right)\left(\begin{array}[]{cc}A^{\dagger}&C^{\dagger}\\ B^{\dagger}&D^{\dagger}\\ \end{array}\right)=\left(\begin{array}[]{cc}m^{2}&0\\ 0&M^{2}\\ \end{array}\right),$$

$$m^d{m}^{d†}=A^{†}{m}^{2}A+C^{†}{M}^{2}C,{\mathcal{F}}^d{m}^{d†}=B^{†}{m}^{2}A+D^{†}{M}^{2}C,$$

$${\mu }_D{\mu }_D^{†}+{\mathcal{F}}^d{\mathcal{F}}^{d†}=B^{†}{m}^{2}B+D^{†}{M}^{2}D.$$

$$B\simeq -\frac{A{m}^d{\mathcal{F}}^{d†}}{{\mu }_D{\mu }_D^{†}+{\mathcal{F}}^d{\mathcal{F}}^{d†}},C\simeq \frac{{\mathcal{F}}^d{m}^{d†}}{{\mu }_D{\mu }_D^{†}+{\mathcal{F}}^d{\mathcal{F}}^{d†}},D\simeq 1,$$

$$A^{-1}{m}^{2}A=m^d{m}^{d†}-\frac{1}{{\mu }_D{\mu }_D^{†}+{\mathcal{F}}^d{\mathcal{F}}^{d†}}(m^d{\mathcal{F}}^{d†})({\mathcal{F}}^d{m}^{d†}).$$

$$(\bar{e}_{Li},\bar{E}_{L})\left(\begin{array}[]{cc}m^{e}_{ij}&0\\ {\cal F}^{e}_{j}&\mu_{E}\\ \end{array}\right)\left(\begin{array}[]{c}e_{R_{j}}\\ E_{R}\\ \end{array}\right),$$

$${\tilde{A}}^{-1}{\tilde{m}}^2\tilde{A}=m^e{m}^{e†}-\frac{1}{{\mu }_E{\mu }_E^{†}+{\mathcal{F}}^e{\mathcal{F}}^{e†}}(m^e{\mathcal{F}}^{e†})({\mathcal{F}}^e{m}^{e†}),$$

$$(\bar{\nu}_{L}^{c},\bar{N}_{R}^{c})\left(\begin{array}[]{cc}0&y^{\nu}\langle\phi\rangle\\ y^{\nu T}\langle\phi\rangle&y^{N}ue^{i\rho}+\tilde{y}^{N}ue^{-i\rho}\\ \end{array}\right)\left(\begin{array}[]{c}\nu_{L}\\ N_{R}\\ \end{array}\right).$$

$${\mathcal{M}}_{ij}^{(a)}=\sum _{k=1}^3{y}_{ik}^{\nu }{y}_{jk}^{\nu }\frac{⟨\varphi {⟩}^2}{y_k^{N}u{e}^{i\rho }+{\tilde{y}}_k^{N}u{e}^{-i\rho }}.$$

$${\mathcal{M}}_{ij}^{(b)}$$

$$h_{e1}=0,h_{\mu 1}=h_{\tau 1}\equiv h_1;h_{e2}=h_{\mu 2}=-h_{\tau 2}\equiv h_2,$$

$$h_1^2\simeq 9.3\times 10^{-3}\left(\frac{10^{-2}}{{\lambda }_5}\right)\left(\frac{M_{{\Sigma }_1}}{10^{10}\mathrm{GeV}}\right)\left[1-{\left(\frac{y_3^{\nu }}{4.4\times 10^{-3}}\right)}^2\left(\frac{10^{10}\mathrm{GeV}}{M_{N_3}}\right)\right],$$

$$h_2^2\simeq 9.3\times 10^{-4}\left(\frac{10^{-2}}{{\lambda }_5}\right)\left(\frac{M_{{\Sigma }_2}}{10^{10}\mathrm{GeV}}\right),y_1^{\nu 2}<9.2\times 10^{-7}\left(\frac{M_{N_1}}{10^9\mathrm{GeV}}\right),$$

Full text

KANAZAWA-19-03

June, 2019

Pati-Salam unification with a spontaneous $CP$ violation

Daijiro Suematsu111e-mail:  suematsu@hep.s.kanazawa-u.ac.jp

Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan

**Abstract

**Recent neutrino oscillation experiments suggest that the PMNS matrix in the lepton sector has a $CP$ violating phase as the CKM matrix in the quark sector. However, origin of these phases in both matrices are not clarified by now. Although complex Yukawa couplings could induce these phases, they remain as free parameters of the model even in that case. If the $CP$ symmetry is considered to be spontaneously broken, they are expected to be determined by some physics at a much lower energy scale than the Planck scale. We study such a possibility in a framework of Pati-Salam type unification. We also discuss other phenomenological issues in it.

1 Introduction

A $CP$ violation in a quark sector has been confirmed to be explained by a CKM phase through experiments of the $B$ meson system. However, its origin is not known still now. Although the CKM phase can be derived from complex Yukawa couplings of quarks [1], the $CP$ symmetry is considered to be explicitly broken in such a case and then the CKM phase remains as a free parameter of the model. Even if its origin could be explained in some physics at the Planck scale, it seems to be difficult to confirm it through experiments. As another problem related to the $CP$ violation, we have strong $CP$ problem [2]. The experimental bound of the electric dipole moment of a neutron suggests that $\bar{\theta}~{}{{}^{<}_{\sim}}~{}10^{-10}$ should be satisfied [3], where $\bar{\theta}$ is defined as $\bar{\theta}\equiv\theta_{\rm QCD}+{\rm arg}(\det{\cal M}_{u}{\cal M}_{d})$ for up and down type quark mass matrices ${\cal M}_{u}$ and ${\cal M}_{d}$. Since a QCD parameter $\theta_{\rm QCD}$ and the second term caused from the quark masses are irrelevant each other, the required smallness of $\bar{\theta}$ seems to be unnatural, which is called the strong $CP$ problem in the standard model (SM).

One of the solutions for this problem is known to be presented by the Peccei-Qiunn (PQ) mechanism [4]. Since its validity could be examined through the existence of a light pseudo scalar called axion [5, 6, 7], axion search is now performed in various experiments [8]. As another solution for the strong $CP$ problem, the Nelson-Barr (NB) model is known [9]. In this scenario, the $CP$ symmetry is assumed to be an exact symmetry and then $\theta_{\rm QCD}=0$ is satisfied. If quark mass matrices take a special form based on some symmetry to satisfy ${\rm arg}(\det{\cal M}_{u}{\cal M}_{d})=0$, $\bar{\theta}=0$ could be realized at least at a tree-level even after the spontaneous $CP$ violation. On the other hand, this spontaneous $CP$ violation could give a $CP$ phase in the CKM matrix. In this point, the scenario is interesting since it could explain an origin of the $CP$ violation at a much lower energy scale than the Planck scale. Moreover, if a $CP$ breaking sector couples also with leptons, a $CP$ phase in the PMNS matrix [10], whose existence is suggested through the long baseline neutrino oscillation experiments such as NOvA and T2K, might be explained by the same source as the quark sector.

A minimal simple example of the NB type model has been proposed by Bento, Branco and Parada (BBP) [11]. In this model, extra heavy vector-like down type quarks are introduced, and a $Z_{2}$ symmetry is imposed on the model which controls the down type quark mass matrix so as not to bring about a contribution to $\bar{\theta}$ through ${\rm arg}(\det{\cal M}_{u}{\cal M}_{d})$ after the spontaneous $CP$ violation. If we impose a global $U(1)$ symmetry instead of the $Z_{2}$ symmetry and assign its charge to these extra heavy quarks, it is easy to find that the required form of the mass matrix could be realized in the same way. In such a case, interestingly enough, the model has the similar structure to an invisible axion model by KSVZ [6], which solves the strong $CP$ problem through the PQ mechanism. If the introduced global $U(1)$ works as the PQ symmetry, the contribution to $\bar{\theta}$ through radiative corrections to ${\rm arg}(\det{\cal M}_{u}{\cal M}_{d})$ could be erased out. In that case, one of the problems in the NB model which are pointed out in [12] could disappear. In this paper, we study this scenario in a Pati-Salam type unified model, in which the $CP$ phases in the CKM matrix and the PMNS matrix could be related.

The remaining part of the paper is organized as follows. In section 2, we introduce our model and discuss a possible origin of $CP$ phases in both the CKM and PMNS matrices. The generation of small neutrino masses is also addressed. We additionally examine a possible spontaneous $CP$ violation in the model. In section 3, we discuss several phenomenological issues in the model. Section 4 is devoted to the summary of the paper.

2 Origin of $CP$ violation

2.1 A Pati-Salam type unified model

We consider a unification model of quarks and leptons via Pati and Salam [13]. The gauge symmetry is taken to be $SU(4)\times SU(2)\times U(1)_{X}$ in which the forth color is identified with a lepton. Fermion contents and their representations under this gauge group are assumed to be

[TABLE]

where $i$ is the generation index $(i=1,2,3)$. As easily found, these contain all ordinary quarks and leptons. We also introduce additional vector-like colored fermions $F_{L,R}~{}(4,1,-1/2)$, and $n$ triplet fermions $\Sigma_{R_{\alpha}}~{}(1,3,0)$ where $\alpha=1-n$ and they are defined as

[TABLE]

On the other hand, scalar contents and their representations are taken to be

[TABLE]

In addition to this structure, we impose a global $U(1)\times Z_{8}$ symmetry. Its charge is assigned to these fields as follows,

[TABLE]

We also assume that $CP$ is an exact symmetry of the model. Although $\Sigma_{R_{\alpha}}$ and $\eta$ might be considered needless in the model for the explanation of features shown through several experiments which cannot be explained in the SM framework,aaaIf the axion is identified with the dark matter, they might be needless. However, we would like to consider much wider possibilities because of reasons which are addressed later. we start our discussion in these field contents.

If we adopt these field contents, Yukawa couplings invariant under the imposed symmetry are written as

[TABLE]

where $\tilde{\phi}=i\tau_{2}\phi^{\ast}$. On the other hand, scalar potential is expressed as

[TABLE]

where $V_{b}$ contains potential terms which are invariant under the symmetry mentioned above but it violates the $S$ number conservation. Since $CP$ is assumed to be exact, all coupling constants are real. If $\Phi$ and $\Psi$ get vacuum expectation values (VEVs) such as $\langle\Phi\rangle=\langle\Psi\rangle=(0,0,0,\Lambda)^{T}$ for example,bbb$\langle\Phi\rangle=\langle\Psi\rangle$ is assumed just for simplicity. the gauge symmetry is broken to the one of the SM

[TABLE]

The weak hypercharge $U(1)_{Y}$ whose charge is normalized as $Q_{EM}=\frac{\tau_{3}}{2}+Y$ is obtained as a linear combination of a diagonal generator $T_{15}$ of $SU(4)$ and a charge $X$ of $U(1)_{X}$ as

[TABLE]

where $T_{15}=\frac{1}{2\sqrt{6}}{\rm diag}(1,1,1,-3)$. We note that the imposed global $U(1)$ symmetry remains unbroken but $Z_{8}$ is broken to $Z_{4}$ at this stage. All fermions remain massless since they have no Yukawa couplings only with $\Phi$ and $\Psi$.

After this symmetry breaking, each fermion is decomposed to the contents of the SM such as

[TABLE]

where $q_{L_{i}}$ and $\ell_{L_{i}}$ are $SU(2)_{L}$ doublet quarks and leptons, and $u_{R_{i}}$, $d_{R_{i}}$ and $e_{R_{i}}$ are singlet quarks and charged leptons, respectively. The vector-like fermions $F_{L,R}$ are decomposed as $(D_{L,R},E_{L,R})$. If we use these decomposed fermions, Yukawa couplings in eq. (5) are expressed ascccWe note that a non-renormalizable operator such as $S^{\ast}(\Psi\bar{F}_{L})(\Phi^{\dagger}k_{R_{i}})$ which is invariant under the imposed symmetry induces the Yukawa terms $S^{\ast}\bar{D}_{L}d_{R_{i}}$ and $S^{\ast}\bar{E}_{L}e_{R_{i}}$.

[TABLE]

where the Yukawa coupling constants are expected to satisfy the conditions

[TABLE]

at a unification scale $\Lambda$. After the spontaneous breaking of $SU(4)$ via $\langle\Phi\rangle$ and $\langle\Psi\rangle$, new Yukawa couplings are expected to be induced effectively as invariant ones under the remaining symmetry,dddIt should be noted that the Yukawa term $S^{\ast}\bar{N}^{c}_{R_{i}}N_{R_{i}}$ can be induced by a non-renormalizable operator $S^{\ast}(\Phi^{\dagger}h_{R_{i}})(\Psi h_{R_{i}})$ invariant under the imposed symmetry, for example.

[TABLE]

where we list up the terms up to dimension five. The couplings $y^{N}_{i}$ and $\tilde{y}^{N}_{i}$ are assumed to be diagonal. We also note that there is a nonrenormalizable dimension five operator $\tilde{\lambda}_{5}\frac{\sigma}{\Lambda}(\phi^{\dagger}\eta)^{2}$ as an invariant one. It plays a crucial role in the small neutrino mass generation as seen later.

In this effective model, we consider symmetry breaking due to VEVs of the singlet scalars $\sigma$, $S$ and $s$ such aseeeThe global symmetry $U(1)$ is broken to $Z_{2}$ by these VEVs. The $Z_{2}$ guarantees the stability of DM as discussed later.

[TABLE]

They could also break the $CP$ symmetry spontaneously. Although we will discuss whether this spontaneous $CP$ violation could be realistic or not in the present model later, we assume it for a while. Here we note that for $\bar{D}_{L}d_{R_{i}}$, $\bar{E}_{L}e_{R_{i}}$ and $\bar{N}^{c}N_{R_{i}}$ in eqs. (10) and (12) there are contributions from the dimension four and five operators. We can expect that the formers give the dominant contribution as long as $v~{}{{}^{<}_{\sim}}~{}u$ is satisfied at least. We suppose such a situation and take account of these contributions only in the following study.

After this symmetry breaking, the potential for the remaining scalars $\phi$ and $\eta$ can be written as

[TABLE]

where $\lambda_{5}$ is defined as $\lambda_{5}=\tilde{\lambda}_{5}\frac{w}{\Lambda}$ and it is real.fffThe $CP$ phase $\chi$ can be removed by the field redefinition of $\eta$. It changes $h_{i\alpha}$ in eq. (12) to $h_{i\alpha}e^{-i\frac{\chi}{2}}$. The scalar masses are shifted through the symmetry breaking effect as

[TABLE]

Since $m_{\phi}$ and $m_{\eta}$ are supposed to take much smaller values than $\Lambda$, serious fine tunings are required. However, we do not treat this hierarchy problem in the present study and just assume that both $m_{\phi}$ and $m_{\eta}$ are of $O(1)$ TeV. The coupling constants $\lambda_{i}$ are also related to the ones at high energy regions through threshold corrections at each symmetry breaking scales [14].

An interesting feature of the present model is that the spontaneous $CP$ violation through eq. (13) could derive both $CP$ phases in the CKM matrix and the PMNS matrix keeping $\bar{\theta}=0$. In the next part, we discuss how the $CP$ phases in both CKM and PMNS matrices are induced.

2.2 A $CP$ phase in the CKM matrix

The $CP$ symmetry is assumed to be exact in the model and then all the coupling constants in the Lagrangian are real. Thus, we cannot expect any origin of $CP$ violation in the up type quark sector, which has no extended structure compared with the SM. Since the up sector mass matrix $m^{u}_{ij}=y^{u}_{ij}\langle\phi\rangle$ is real, they can be diagonalized by orthogonal transformations $u_{L}^{\prime}=O^{L}u_{L}$ and $u_{R}^{\prime}=O^{R}u_{R}$. In the present effective model, on the other hand, we find that the down type quark sector has the same structure as the BBP model [11]. The BBP model is an extension of the SM by extra colored vector-like down type heavy quarks $(D_{L},D_{R})$ and a singlet complex scalar $S$. We can apply their discussion to the present model to show how the $CP$ phase could be induced in the CKM matrix. Although the $Z_{2}$ symmetry is imposed to control the mass matrix in their model, the global $U(1)$ symmetry in eq. (4) could play the same role as it in the present model. Moreover, since this $U(1)$ is chiral and has a color anomaly, it can play a role as the PQ symmetry which has a domain wall number one as in the KSVZ model [6]. As a result, a Nambu-Goldstone boson produced as a result of its spontaneous breaking through the VEV $\langle\sigma\rangle$ could work as an axion to solve the strong $CP$ problem without inducing the domain wall problem [15]. On the other hand, since the axion phenomenology constrains a breaking scale of this symmetry, we have to fix the scale $w$ to be [16]

[TABLE]

The Yukawa couplings of the down type quarks shown in eq. (10) derive a $4\times 4$ mass matrix ${\cal M}_{d}$ as

[TABLE]

where $m_{ij}^{d}=y_{ij}^{d}\langle\tilde{\phi}\rangle$, ${\cal F}_{j}^{d}=(y_{j}^{D}ue^{i\rho}+\tilde{y}_{j}^{D}ue^{-i\rho})$ and $\mu_{D}=x_{D}we^{i\chi}$. Due to the PQ mechanism, $\bar{\theta}=\theta_{QCD}+{\rm arg}(\det{\cal M}_{u}{\cal M}_{d})=0$ is satisfied via the axion even after we take account of radiative corrections including the phases caused by the spontaneous $CP$ violation. Next we see that this phase can generate the CKM phase following the BBP model.

We consider the diagonalization of a matrix ${\cal M}_{d}{\cal M}_{d}^{\dagger}$ by a unitary matrix such as

[TABLE]

where a $3\times 3$ matrix $m^{2}$ is diagonal in which the generation indices are abbreviated. Eq. (18) requires

[TABLE]

If $\mu_{D}\mu_{D}^{\dagger}+{\cal F}^{d}{\cal F}^{d\dagger}$ is much larger than each components of ${\cal F}^{d}m^{d\dagger}$, which means $u,w\gg\langle\tilde{\phi}\rangle$, we find that $B,C$ and $D$ can be approximated as

[TABLE]

which guarantee the approximate unitarity of the matrix $A$. In such a case, it is also easy to find that

[TABLE]

The right-hand side is an effective mass matrix of the ordinary down type quarks which is derived through the mixing with the extra heavy quarks. Since the second term can have complex phases in off-diagonal components as long as $y^{D}_{i}\not=\tilde{y}^{D}_{i}$ is satisfied, the matrix $A$ could be complex. Moreover, if $\mu_{D}\mu_{D}^{\dagger}<{\cal F}^{d}{\cal F}^{d\dagger}$ is satisfied, the complex phase in $A$ could have a substantial magnitude since the second term is comparable with the first term. Since the CKM matrix is determined as $V_{CKM}={O^{L}}^{T}A$, the $CP$ phase of $V_{CKM}$ is caused through the one of $A$. Here, we have to note whether such phases could be physical or not is dependent on the flavor structure of Yukawa coupling $y^{d}$, $y^{D}$ and $\tilde{y}^{D}$. It should also be noted that the matrix $A$ needs to take an almost diagonal form as long as there is no correlation between $A$ and $O^{L}$ since $V_{CKM}$ has a nearly diagonal form. It may be instructive to show how the physical phase could be induced through this mechanism using a concrete example. We give such an example in Appendix.

2.3 Neutrino masses and the PMNS matrix

In the lepton sector, we can treat the charged lepton sector in the same way as the down type quark sector. In fact, the Yukawa couplings in eq. (10) induce the charged lepton mass matrix as follows,

[TABLE]

where $m_{ij}^{e}=y_{ij}^{e}\langle\tilde{\phi}\rangle$, ${\cal F}_{j}^{e}=(y_{j}^{E}ue^{i\rho}+\tilde{y}_{j}^{E}ue^{-i\rho})$ and $\mu_{E}=x_{E}we^{i\chi}$. Since the mass matrix takes the same form as the one of the down type quarks (18), the diagonalization matrix $\tilde{A}$ for the above charged lepton mass matrix could be complex and it should satisfy the relation

[TABLE]

where $\tilde{m}^{2}$ corresponds to the diagonalized mass matrix $m^{2}$ in eq. (18). As long as $\mu_{E}\mu_{E}^{\dagger}<{\cal F}^{e}{\cal F}^{e\dagger}$ is satisfied, non-negligible $CP$ phases could be expected in $\tilde{A}$ in the same way as the down type quark sector.

On the other hand, small neutrino masses are expected to be produced not only by the type I seesaw [17] but also by the scotogenic type III seesaw [18] in this model. In fact, the lepton sector of the model has the structure in which the scotogenic type III seesaw mechanism could work as found from the terms contained in eqs. (10) and (12). Diagrams which contribute to the neutrino mass generation are shown in Fig.1.

(a) Neutrino masses due to the type I seesaw

The singlet fermions $N_{R_{i}}$ get Majorana mass via the VEV $\langle S\rangle$. On the other hand, they have Yukawa couplings with the doublet leptons and the ordinary Higgs doublet scalar $\phi$. Thus, the ordinary type I seesaw makes neutrinos $\nu_{L_{i}}$ massive through the diagram shown in the left of Fig. 1. Neutrino mass matrix caused by this can be written as

[TABLE]

Since $u\gg\langle\phi\rangle$ is supposed in this model, the contribution to neutrino masses from this diagram is estimated as

[TABLE]

The neutrino Yukawa couplings $y_{ik}^{\nu}$ satisfy the same relation as the Yukawa couplings of the up type quarks as found in eq. (11). Since $\tilde{A}$ is expected to take an almost diagonal form as $A$, the PMNS matrix is consider to have the similar form as the CKM matrix. This means that other contributions to the neutrino masses are indispensable for the explanation of large flavor mixing required by the neutrino oscillation data. This is one of the reasons why we consider the extended structure with $\eta$ and $\Sigma_{R_{\alpha}}$. These fields could give additional contributions to the neutrino masses in the following way.

(b) Neutrino masses due to the scotogenic type III seesaw

As found in eq. (12), $\Sigma_{R_{\alpha}}$ has Yukawa couplings with $\nu_{L_{i}}$. However, since $\phi$ has no coupling with these and $\eta$ is assumed to have no VEV, neutrino masses via $\Sigma_{R_{\alpha}}$ are not generated at a tree level but generated at a one-loop level. The coupling $\frac{\lambda_{5}}{2}(\eta^{\dagger}\phi)^{2}+{\rm h.c.}$ brings about a small mass difference between the real and imaginary components of $\eta^{0}$. As its result, the one-loop diagram shown in the right of Fig. 1 gives a contribution to the neutrino masses. It can be estimated as

[TABLE]

where $M_{\Sigma_{\alpha}}=\gamma_{\Sigma_{\alpha}}w$ and $M_{\eta}^{2}=m_{\eta}^{2}+(\lambda_{3}+\lambda_{4})\langle\phi\rangle^{2}$. The second similarity is satisfied for $M_{\eta}=O(1)$ TeV since $w$ is much larger than a TeV scale as discussed in the previous part. Although neutrino mass eigenvalues are determined through ${\cal M}_{ij}^{\nu}={\cal M}_{ij}^{(a)}+{\cal M}_{ij}^{(b)}$, ${\cal M}_{ij}^{(a)}$ should be sufficiently small compared with ${\cal M}_{ij}^{(b)}$ for large flavor mixings. If we consider that this matrix is diagonalized by a unitary matrix $U$ as $U^{T}{\cal M}^{\nu}U={\cal M}^{\rm diag}$, the PMNS matrix is obtained as $V_{PMNS}=\tilde{A}^{\dagger}U$ which could have a Dirac phase and two Majorana phases. An example of $V_{PMNS}$ obtained through this framework in a simple model is given in Appendix.

Next, we address the constraint on the relevant parameters caused by the neutrino oscillation data. Since ${\cal M}^{(a)}$ should be a subdominant contribution to the neutrino masses, we have to extend the model at least with two triplet fermions $(n_{\Sigma}=2)$ for the explanation of the neutrino oscillation data. In order to estimate the required magnitude of the neutrino Yukawa couplings in such a case, we suppose, for simplicity and definiteness, the tri-bimaximal flavor structure for $h_{i\alpha}$ as [19]

[TABLE]

and also diagonal $y_{ij}^{\nu}$ such as $y_{ij}^{\nu}=y_{i}^{\nu}\delta_{ij}$ with $y_{1}^{\nu}\ll y_{2}^{\nu}\ll y_{3}^{\nu}$.gggThis assumption is adopted due to the relation (11) to the up type quarks which is caused by the $SU(4)$ symmetry. We also assume $y^{N}_{1,2}=0$ and $\tilde{y}^{N}_{3}=0$, for simplicity. Under this assumption, if the normal hierarchy for the neutrino masses is assumed, squared mass differences required by the neutrino oscillation data suggest [20]

[TABLE]

where $M_{N_{1,2}}=\tilde{y}^{N}_{1,2}u$ and $M_{N_{3}}=y^{N}_{3}u$, and $M_{\eta}=1$ TeV is also assumed.

Finally, it may be useful to present a remark on the extension by the vector-like fermions. Although these fermions are introduced to the down sector in the above discussion, the CKM phase could be derived in the same way even if we introduce them to the up sector. However, the situation could be largely changed for the $CP$ phases in the PMNS matrix and the small neutrino mass generation. The present choice seems to be crucial for the present scenario. It could also play an important role when we consider an embedding of the model into a fundamental model at the Planck scale region.hhhThe model might be embedded into an effective model derived by a suitable compactification of $E_{8}\times E_{8}^{\prime}$ superstring [21].

2.4 Spontaneous $CP$ violation

In the previous part, we just assume that eq. (13) is realized as a potential minimum. Here, we discuss in what situation the spontaneous $CP$ violation could occur in a realistic way in the present model. The condition required for the spontaneous $CP$ violation has been studied in detail in [22]. If we follow their results, the VEVs of $\sigma$ is found not to break the $CP$ symmetry spontaneously, and then $\chi=0$. The reason is that the spurions for it cannot be introduced since the imposed global $U(1)$ symmetry is assumed to be exact except for the color anomaly effect. On the other hand, we can introduce the spurions for $S$ which has no global $U(1)$ charge. In fact, if we introduce the terms such as $S^{4}$ and $S^{2}$ which break a $U(1)$ symmetry corresponding to the $S$ number, a nonzero $\rho$ could appear as a potential minimum.iiiWe do not consider such terms for $s$ in the present study.

Relevant potential is found from eq. (6) to be

[TABLE]

where $\bar{m}_{a}^{2}=\tilde{m}_{a}^{2}+\zeta_{a}\Lambda^{2}$ $(a=S,\sigma)$ and $\tilde{m}_{a}^{2}>0$ and $\zeta_{a}<0$ are assumed since we suppose that the potential minimum is fixed as a result of the $SU(4)$ breaking. As examples, we consider two cases for $V_{b}$ in (29) such asjjjWe note that terms proportional to $S^{2}$ are induced through the $SU(4)$ breaking from an operator $\Phi^{\dagger}\Psi S^{2}$ wihich is invariant under the imposed symmetry.

[TABLE]

Here, we confine our study to the situation where the VEVs $u$ and $w$ are determined by a part of $V_{CP}$ except for $V_{b}$. It could be realized for $\kappa_{S}\gg\alpha$ and $|\bar{m}_{S}^{2}|\gg|\mu^{2}|$ in the case (i) and also for $\kappa_{S}\gg\alpha$ and $|\beta|\ll 1$ in the case (ii). The potential minimum could be found for sufficiently small $|\kappa_{S\sigma}|$ in both cases

[TABLE]

and also the $CP$ phase is determined as

[TABLE]

in each case. These examples show that the spontaneous $CP$ violation could occur through the scalar $S$ as long as suitable values of the parameters are chosen. In fact, for example, if $\mu^{2}=-4\alpha u^{2}$ is satisfied for $\alpha\ll 1$ and $|\mu^{2}|\ll u^{2}$, the maximum $CP$ phase $\rho\simeq\frac{\pi}{2}$ could be realized in the case (i). We should note that these conditions on $\alpha$ and $\mu^{2}$ is consistent with the requirement for which $u$ and $w$ are determined as eq. (31). In the case (ii), the maximum $CP$ phase is obtained for $\beta w^{2}\simeq 4\alpha u^{2}$ which is consistent with the determination of $u$ and $w$ as found from eq. (31). As a result of this symmetry breaking, the mass of $S$ is fixed as $m_{S_{R}}=\sqrt{4\kappa_{S}}u$.

On the other hand, in order for this breaking to cause large $CP$ phases in both the CKM and PMNS matrices, the conditions $\mu_{D}\mu_{D}^{\dagger}<{\cal F}^{d}{\cal F}^{d\dagger}$ and $\mu_{E}\mu_{E}^{\dagger}<{\cal F}^{e}{\cal F}^{e\dagger}$ should be satisfied as discussed before. They are supposed to require

[TABLE]

as long as the relevant Yukawa couplings has a similar magnitude. This condition can be easily satisfied for suitable parameters as found from eq. (31). Although the tuning of parameters is necessary, the present scenario is found to work as long as the scalar potential takes a suitable form. We can expect that the required $CP$ violation is induced in both the quark and lepton sectors based on the same origin.

3 Phenomenology

In the previous part, we addressed that the $CP$ problem in the SM could be solved in this model. In this section, we order several discussions and comments on other phenomenological issues.

3.1 Inflation

The model has candidates for the inflaton such as $\sigma$ and $S$. They can have non-minimal couplings with Ricci scalar $R$ [23]

[TABLE]

Although the real and imaginary components of $\sigma$ have the same coupling $\xi_{\sigma}$, only the real part of $S$ could have a nonzero coupling $\frac{1}{2}\xi S_{R}^{2}R$ in the case $\xi_{S_{1}}=\xi_{S_{2}}$ where $S\equiv\frac{1}{\sqrt{2}}(S_{R}+iS_{I})$ and $\xi\equiv\xi_{S_{1}}+\xi_{S_{2}}$. If we suppose that a coupling $\xi$ takes a sufficiently large value in such a case, inflation via $S_{R}$ is expected to occur in the same way as the Higgs inflation [24]. A nice feature in this scenario is that the dangerous unitarity violation caused by a higher order mixing between $S_{R}$ and $S_{I}$ [25] is not induced at $\frac{M_{\rm pl}}{\xi}$ but could be suppressed at least up to an inflation scale $\frac{M_{\rm pl}}{\sqrt{\xi}}$ [26].

The potential of the inflaton can be expressed in the Einstein frame as

[TABLE]

Since the canonically normalized inflaton $\chi$ is defined as

[TABLE]

$\chi$ and $S_{R}$ are related each other as $S_{R}\propto\exp\frac{\chi}{\sqrt{6}M_{\rm pl}}$ at a large field region $S_{R}^{2}\gg\frac{M_{\rm pl}^{2}}{\xi}$. In that region, the potential of $\chi$ becomes constant $V_{E}=\frac{\kappa_{S}M_{\rm pl}^{4}}{4\xi^{2}}$ as long as $S_{R}\gg S_{I}$ is satisfied. The slow roll parameters for $\chi$ can be expressed as

[TABLE]

by using the e-foldings number $N_{e}$. If we take $N_{e}=60$, we obtain the spectral index $n_{s}=0.97$ and the tensor-to-scalar ratio $r=3.3\times 10^{-3}$. On the other hand, since the amplitude of scalar perturbation is given as $A_{S}=\frac{V_{E}}{24\pi^{2}M_{\rm pl}^{4}\epsilon}$ and the CMB observation constrains it as $A_{S}=2.4\times 10^{-9}$ at $k_{\ast}=0.002~{}{\rm Mpc}^{-1}$ [27], $\kappa_{S}$ has to satisfy $\kappa_{S}=4.7\times 10^{-10}\xi^{2}$ for $N_{e}=60$. Using this constraint, the inflaton mass is found to be determined as

[TABLE]

The inflaton mass should be fixed in a consistent way with eqs. (16) and (33). We also note that the assumed vacuum with the spontaneous $CP$ violation could be consistently realized for suitable parameters in this inflation framework.

The reheating after the end of inflation is expected to be caused by the inflaton decay to the singlet neutrino pairs $N_{i}N_{i}$ through the couplings in eq. (12). In the case $y^{N}_{3}>\tilde{y}^{N}_{1,2}$ which is assumed in this study, a dominant process is $S_{R}\rightarrow N_{3}N_{3}$. Since singlet fermions $N_{i}$ interact with other fields only through the neutrino Yukawa couplings except for the couplings with $S$ and $S^{\ast}$, instantaneous reheating is expected to occur for the case $M_{S_{R}}>2M_{N_{3}}$ and $H\simeq\Gamma_{S}~{}{{}^{>}_{\sim}}~{}\Gamma_{N_{3}}$, where $\Gamma_{S}$ and $\Gamma_{N_{3}}$ are the decay width of $S_{R}\rightarrow N_{3}N_{3}$ and $N_{3}\rightarrow\bar{\ell}_{i}\phi^{\dagger}$, respectively.kkkHere, we do not consider a possibility for non-thermal leptogenesis which could be expected to occur for the case $\Gamma_{N_{i}}>\Gamma_{S}$ [28]. If we take account of these conditions which may be expressed as $\sqrt{\kappa_{S}}>y_{3}^{N}$ and $2\sqrt{\kappa_{S}}y^{N}_{3}~{}{{}^{>}_{\sim}}~{}y_{3}^{\nu 2}$, the reheating temperature $T_{R}$ could be bounded aslll The restoration of the PQ symmetry could occur in the reheating process depending on the parameters. However, since the domain wall number is one in this model, no domain wall problem is induced even if the PQ symmetry is restored.

[TABLE]

Although this shows that $T_{R}>M_{N_{1}}(\equiv\tilde{y}_{1}^{N}u)$ could be satisfied for suitable parameters, $N_{1}$ is not expected to be thermalized as a relativistic particle since the Yukawa coupling $y_{1}^{\nu}$ of $N_{1}$ is supposed to be very small. Fortunately, it could be expected to reach the thermal equilibrium through the scattering process $N_{3}N_{3}\rightarrow N_{1}N_{1}$ mediated by the scalar $S_{R}$ before it becomes non-relativistic $(T<M_{N_{1}})$. This allows the present model to generate the lepton number asymmetry sufficiently through the out-of-equilibrium decay of $N_{1}$ although the Yukawa coupling $y_{1}^{\nu}$ of $N_{1}$ is very small. We discuss this possibility in the next part.

3.2 Leptogenesis

The model has two possible decay processes $N_{1}\rightarrow\ell_{i}\phi^{\dagger}$ and $\Sigma_{\alpha}\rightarrow\ell_{i}\eta^{\dagger}$ which could contribute to the generation of the lepton number asymmetry since these processes violate the lepton number. However, $\Sigma_{\alpha}$ has the $SU(2)$ gauge interaction so that its out-of-equilibrium decay is impossible at least before the electroweak symmetry breaking. On the other hand, as addressed above, $N_{1}$ could reach the equilibrium abundance through the scattering mediated by $S_{R}$ even if its neutrino Yukawa coupling $y_{1}^{\nu}$ is very small. In that case, its decay could generate the lepton number asymmetry through the out-of-equilibrium decay at $T<M_{N_{1}}$. As long as the couplings $y_{i}^{N}$ or $\tilde{y}^{N}_{i}$ between the inflaton and $N_{i}$ have sufficient magnitude such as $y_{3}^{N}>\tilde{y}_{2}^{N}~{}{{}^{>}_{\sim}}~{}\tilde{y}_{1}^{N}~{}{{}^{>}_{\sim}}~{}10^{-3}$, the equilibrium number density of $N_{1}$ can be easily realized at $T>M_{N_{1}}$ as shown below.mmmWe should recall that $y^{N}_{1,2}=0$ and $\tilde{y}^{N}_{3}=0$ are assumed. On the other hand, since the mass of $N_{i}$ is generated through the Yukawa coupling $(y_{i}^{N}S+\tilde{y}^{N}_{i}S^{\ast})\bar{N}_{i}^{c}N_{i}$, $N_{i}$ cannot be light if we take account of the values of $y_{i}^{N}$ and $\tilde{y}^{N}_{i}$ mentioned above. In fact, under the constraints (16) and (33), the mass of $N_{1}$ has to be $M_{1}>10^{7}$ GeV at least.

In order to check whether this scenario works, we present a typical solution of the Boltzmann equations for $Y_{N_{1}}$ and $Y_{N_{3}}$ as functions of $z(\equiv\frac{M_{N_{1}}}{T})$ in the left panel of Fig. 2. Here, $Y_{N_{i}}$ is defined as $Y_{N_{i}}=\frac{n_{N_{i}}}{s}$ with the $N_{i}$ number density $n_{N_{i}}$ and the entropy density $s$. In this calculation, as an example, we assume $u=2\times 10^{12}~{}{\rm GeV}$ and $\xi=500$ and then the inflaton mass is fixed as $m_{S_{R}}=4.3\times 10^{10}$ GeV. Taking account of the constraints in eq. (28), we fix other relevant parameters at the following values,nnnThese values of $y^{\nu}_{i}$ require some overall suppression effect compared with the Yukawa couplings of the up type quarks in eq. (11). We just assume it in this setting.

[TABLE]

Since $m_{S_{R}}>2M_{N_{3}}$ is satisfied, $N_{3}$ is allowed to be produced through the inflaton decay $S_{R}\rightarrow N_{3}N_{3}$. The following $N_{3}$ decay $N_{3}\rightarrow\ell_{i}\phi^{\dagger}$ caused by the coupling $y_{3}^{\nu}$ is considered to be a substantial process for the thermalization. Thus, the initial value of $Y_{N_{3}}$ is fixed as the one produced through this inflaton decay assuming the instantaneous reheating. The figure shows that $Y_{N_{1}}$ reaches the equilibrium value $Y_{N_{1}}^{\rm eq}$ around $z\simeq 1$ for the assumed value of $\tilde{y}_{1}^{N}$ and leaves its equilibrium value at $z~{}{{}^{>}_{\sim}}~{}1$ where the out-of-equilibrium decay could generate the lepton number asymmetry.oooFor a smaller value of $\tilde{y}_{1}^{N}$ or a larger value of $y^{\nu}_{3}$, $Y_{N_{1}}$ cannot reach an equilibrium value $Y_{N_{1}}^{\rm eq}$ for $z<1$ although $Y_{N_{1}}$ keeps a constant value in the same way as the one in the left panel of Fig. 2. A larger value of $\tilde{y}_{1}^{N}$ realizes $Y_{N_{1}}=Y_{N_{1}}^{\rm eq}$ at an earlier stage.

The generated lepton number asymmetry through the $N_{1}$ decay is converted to the baryon number asymmetry through the sphaleron process as in the usual leptogenesis [29, 30]. In the present model, the $CP$ asymmetry $\varepsilon$ for the decay $N_{1}\rightarrow\ell\phi^{\dagger}$ [31] is dominantly caused by an interference between a tree diagram and a one-loop diagram mediated by $N_{3}$ which are shown in Fig. 3. Under the assumption given in eq. (27), it can be estimated as

[TABLE]

where $F(x)$ is defined as

[TABLE]

In the following analysis, we assume $\sin(-2\rho)=1$ which makes $\varepsilon$ maximal.

If $N_{1}$ is in the thermal equilibrium at $z<1$, the out-of-equilibrium decay of $N_{1}$ could start at $z\sim 1$ and the lepton number asymmetry is effectively generated at $z>1$. By introducing an efficiency factor for the washout of the generated lepton number asymmetry as $\kappa$, the lepton number asymmetry $Y_{L}$ which is defined as $Y_{L}\equiv\frac{n_{L}}{s}$ by using a net lepton number density $n_{L}$ is roughly estimated as $Y_{L}=\varepsilon\kappa Y_{N_{1}}^{\rm eq}|_{z=1}$. It suggests that $\varepsilon~{}{{}^{>}_{\sim}}~{}8\times 10^{-8}\kappa^{-1}$ is necessary to realize a value $Y_{L}~{}{{}^{>}_{\sim}}~{}2.5\times 10^{-10}$ at a sphaleron decoupling temperature in order to produce the sufficient baryon number asymmetry in the Universe for $Y_{N_{1}}^{\rm eq}|_{z=1}\simeq 3.1\times 10^{-3}$. Since $y_{1}^{\nu}$ is supposed to be very small in this model, $N_{1}$ is considered to start its substantial decay at a later stage such as $z\gg 1$ where the washout caused by $N_{3}$ and $\Sigma_{\alpha}$ could be largely Boltzmann suppressed as long as $\frac{M_{N_{3}}}{M_{N_{1}}},\frac{M_{\Sigma_{\alpha}}}{M_{N_{1}}}>1$ are satisfied. Thus, in such a case, the almost all lepton number asymmetry generated there could be kept and the sufficient lepton number asymmetry is expected to be generated through the out-of-equilibrium decay of $N_{1}$.

In the right panel of Fig. 2, we present the evolution of the lepton number asymmetry $Y_{L}$ generated through the out-of-equilibrium decay of $N_{1}$ using the same parameters given in eq. (40) which can prepare an initial value $Y_{N_{1}}(1)\simeq Y_{N_{1}}^{\rm eq}(1)$ as shown in the left panel. In this analysis of Boltzmann equations, we fully take account of the washout processes and use the neutrino Yukawa couplings $h_{1,2}$ which are fixed by taking account of the condition (28) with $M_{\eta}=10^{3}~{}{\rm GeV}$, $M_{\Sigma_{1,2}}=3M_{N_{1}}$ and $|\lambda_{5}|=10^{-1.5}$. Since the small neutrino Yukawa coupling $y_{1}^{\nu}$ makes the $N_{1}$ decay be delayed to the temperature where the washout processes could be frozen out due to the Boltzmann suppression. This feature can be found in the behavior of $Y_{N}$ and $Y_{L}$ in the right panel. As its result, almost all the lepton number asymmetry generated through the out-of-equilibrium $N_{1}$ decay could be converted to the baryon number asymmetry in the Universe as discussed above. The model is found to present a successful leptogenesis framework. Results of the analysis for several parameter settings are also listed in Table 1.

Here, we order a few remarks related to these results. First, since a smaller $|\lambda_{5}|$ makes $h_{1,2}$ larger through the neutrino mass condition (28) for the fixed $M_{\Sigma_{1,2}}$, the washout processes mediated by $\Sigma_{1,2}$ are considered to suppress the generation of the lepton number asymmetry at an early stage where it is not frozen out. Second, the $N_{1}$ mass seems to be bounded as $M_{1}>10^{9}$ GeV in the present model in order to produce the required baryon number asymmetry. This bound is similar to the one given in [32]. Third, for the present parameter settings, $w~{}{{}^{>}_{\sim}}~{}10^{10}~{}{\rm GeV}$ seems to be required to avoid the washout of the generated lepton number asymmetry, which is consistent with the requirement from the PQ symmetry breaking scale. Finally, the coexistence of the couplings $y^{N}_{i}$ and $\tilde{y}^{N}_{i}$ such as $y^{N}_{i}\not=\tilde{y}^{N}_{i}$ in eq. (12) is crucial for the leptogenesis. We should recall that the same feature is required in the explanation of the CKM phase through the mass matrix (17).

3.3 Dark matter

The model has three dark matter (DM) candidates, that is, the axion, the neutral component of $\Sigma_{\alpha}$ and the lightest neutral component of $\eta$. The axion could explain the required DM abundance as long as $w\simeq 10^{12}$ GeV is satisfied [16]. The latter two have odd parity of the remnant $Z_{2}$ of the global $U(1)$ symmetry, which makes them stable and then DM candidates. However, $\Sigma^{0}_{\alpha}$ is supposed to have a large mass so that it cannot be DM in the present model.pppThe DM study in the cases where $\Sigma$ has a mass of $O(1)$ TeV can be found in [18]. On the other hand, $\eta$ is assumed to have a mass of $O(1)$ TeV as discussed in the neutrino mass generation. In that case, the lightest neutral component of $\eta$ can be DM. Moreover, even if the VEV $w$ is not large enough to guarantee the sufficient axion density for the explanation of the DM energy density, the thermal relics of $\eta^{0}$ could explain it as long as the quartic couplings $\lambda_{3,4}$ in eq. (14) take suitable values [33, 34]. As a result, the breaking scale $w$ of the PQ symmetry could be free from the explanation of the DM energy density in this model.

3.4 Quark and lepton mass hierarchy

Yukawa coupling constants for quarks and leptons are related each other by eq. (11) at an $SU(4)$ breaking scale $\Lambda$. On the other hand, their weak scale values which determine mass eigenvalues of the quarks and the leptons are fixed through the renormalization group equations taking them as the initial values. It can bring about a difference of a factor three due to the color effect between quarks and leptons. The mass difference between the down type quarks and the charged leptons seems to be partially explained by this effect but it is not satisfactory. Even if corrections caused by the mixing with heavy fermions in these sectors are taken into account, this situation is not improved and then some new ingredients are needed to be introduced for it.

On the other hand, in the up type quarks and the neutrinos, several additional parameters related to the neutrino mass generation could give a different feature in these sectors. Especially, since neutrino masses are determined by the type III seesaw contribution, the relation among the Yukawa couplings of quarks and leptons at the high energy scale is not directly affect their mass matrices. These features could make the large difference found in the CKM and PMNS matrices be consistently realized in the present unification scheme. Since details depend on the model parameters and this issue is beyond the scope of present study, we will not discuss it further here and leave it to future study. Finally, it may be useful to note the fact that the present unification scheme could make the leptogenesis work well. A requirement that the third generation Yukawa coupling of the up quark sector should be much larger than others brings about the relation $y_{1,2}^{\nu}\ll y_{3}^{\nu}$ in the neutrino sector, which plays a crucial role in the present leptogenesis scenario as shown in the above study

4 Summary

We proposed a model which gives the origin of the $CP$ violation at an intermediate scale. In this model, the $CP$ symmetry is supposed to be spontaneously broken but it does not cause the strong $CP$ problem and $\bar{\theta}=0$ is kept even if the radiative corrections are taken into account. We showed that such a model could be realized in a Pati-Salam type unification model, in which $CP$ phases in both the CKM and PMNS matrices are derived from the same source. Neutrino masses are generated in a hybrid way by the tree level type I seesaw and the one-loop type III seesaw. The required baryon number asymmetry can be produced through the leptogenesis. The out-of-equilibrium decay of $N_{1}$ occurs at a later stage where the washout effects are almost frozen out. As a result, the generated lepton number asymmetry could be effectively converted to the baryon number asymmetry. This feature comes from the present unification based on a fact that the top Yukawa coupling is much larger than others. The model has two DM candidates and the dominant DM is fixed depending on the intermediate symmetry breaking scale. Since the axion needs not to be DM, the PQ symmetry breaking scale can be free from the condition for the DM energy density realization. We also note a possibility such that the model might be derived as the low energy effective model of the $E_{8}\times E_{8}^{\prime}$ superstring. It will be discussed elsewhere.

Appendix

In this Appendix, we present a simple example which could bring about a phase in the CKM matrix. We assume the relevant couplings $y^{d}$, $y^{D}$ and $\tilde{y}^{D}$ to be written asqqqA similar Yukawa coupling matrix for the down type quarks has been considered in a different context [35]. There is no background to explain its hierarchical structure in the present model.

[TABLE]

by using real constants $a_{i}$, $b_{i}$, $c$ and $x,~{}y$. As long as $\epsilon$ satisfies $\epsilon\ll 1$, the down type quark mass matrix $m^{d}(\equiv y^{d}\langle\tilde{\phi}\rangle)$ has hierarchical mass eigenvalues. Here, we introduce $X_{ij}$ and $Y_{ij}$ whose definition is given as

[TABLE]

where $p_{i}$ is fixed as $p_{1}=x,~{}p_{2}=y$ and $p_{3}=-1$. If we define $R_{ij}$ and $\theta_{ij}$ by using these quantities as

[TABLE]

the component of eq. (23) is found to be expressed as

[TABLE]

where $\mu_{D}^{2}\ll{\cal F}^{d}{\cal F}^{d\dagger}$ is assumed. $\epsilon_{ij}$ is defined as

[TABLE]

By solving eq. (46), we find that $A$ is approximately written as

[TABLE]

where the constants $\lambda$, $\alpha$ and $\theta$ are defined as

[TABLE]

This expression shows that $A$ could have a non-trivial phase which gives the origin of the CKM phase as long as $a_{2}b_{3}-a_{3}b_{2}\not=0$ and $x\not=y$ are satisfied. If the diagonalization matrix $O^{L}$ for the mass matrix of the up type quarks takes an almost diagonal form, an interesting matrix could be obtained as the CKM matrix such as $V_{CKM}\simeq A$. In this case, the mass eigenvalues for the down type quarks are obtained as

[TABLE]

A diagonalization matrix $\tilde{A}$ for the charged lepton mass matrix takes the same form as $A$ as a result of the Pati-Salam $SU(4)$ symmetry in the model. However, since the Yukawa couplings which induce the neutrino mass matrix could be irrelevant to the ones in the up type quarks as discussed in the text, the large mixing in the PMNS matrix could be obtained if large flavor mixings are realized in the neutrino mass matrix. If we use the assumption in eq. (27), the PMNS matrix in this example is found to be written as

[TABLE]

where $\beta=\frac{X_{23}}{|\alpha|^{2}X_{33}}e^{i\theta}$ and the Majorana phases are not taken into account.

Acknowledgements

This work is partially supported by MEXT Grant-in-Aid for Scientific Research on Innovative Areas (Grant No. 26104009) and a Grant-in-Aid for Scientific Research (C) from Japan Society for Promotion of Science (Grant No. 18K03644).