New model for radiatively generated Dirac neutrino masses and lepton flavor violating decays of the Higgs boson

TL;DR

This paper introduces a new two-loop mechanism for generating Dirac neutrino masses that conserves lepton number and involves dark matter, predicting potentially observable lepton flavor violating Higgs decays.

Contribution

The paper presents a novel model for neutrino mass generation that allows large lepton flavor violating Higgs decays without conflicting with existing constraints.

Findings

Lepton flavor violating Higgs decays can be significantly larger than charged lepton decays.

The model remains consistent with current neutrino, lepton flavor violation, and dark matter constraints.

Future collider observations of Higgs decays could support this model.

Abstract

We propose a new mechanism to explain neutrino masses with lepton number conservation, in which the Dirac neutrino masses are generated at the two-loop level involving a dark matter candidate. In this model, branching ratios of lepton flavor violating decays of the Higgs boson can be much larger than those of lepton flavor violating decays of charged leptons. If lepton flavor violating decays of the Higgs boson are observed at future collider experiments without detecting lepton flavor violating decays of charged leptons, most of the models previously proposed for tiny neutrino masses are excluded while our model can still survive. We show that the model can be viable under constraints from current data for neutrino experiments, searches for lepton flavor violating decays of charged leptons and dark matter experiments.

Tables

Table 1. Table 1: The list of new fields in our model.

	${\nu }_{iR}$	${\psi }_{aR}$	$\Phi$	$s_1^{+}$	$\eta$	$s_2^{+}$
Spin $J$	$1/2$	$1/2$	$0$	$0$	$0$	$0$
$\mathrm{SU}{\left(2\right)}_L$	1	1	2	1	2	1
$U{\left(1\right)}_Y$	0	0	$3/2$	1	$1/2$	1
$Z_2^{\prime }$	$-$	+	(+)	$-$	+	+
$L$	1	0	$-2$	$-2$	$-1$	$-1$
$Z_2$	+	$-$	+	+	$-$	$-$

Table 2. Table 2: Current experimental constrains on branching ratios of LFV processes.

Process	Upper limit
$\mu \to e\gamma$	$4.2\times {10}^{-13}\text{<cite><a>TheMEG:2016wtm </a></cite>}$
$\tau \to e\gamma$	$3.3\times {10}^{-8}\text{<cite><a>Aubert:2009ag </a></cite>}$
$\tau \to \mu \gamma$	$4.4\times {10}^{-8}\text{<cite><a>Aubert:2009ag </a></cite>}$

Table 3. Table 3: Numerical results for the LFV branching ratios in the benchmark scenario for the normal ordering case.

Process	Numerical result
$\mu \to e\gamma$	$3.36\times {10}^{-16}$
$\tau \to e\gamma$	$1.25\times {10}^{-14}$
$\tau \to \mu \gamma$	$1.18\times {10}^{-9}$

Table 4. Table 4: Numerical results for the LFV branching ratios in the benchmark scenario for the inverted ordering case of neutrino masses.

Process	Numerical result
$\mu \to e\gamma$	$3.34\times {10}^{-16}$
$\tau \to e\gamma$	$1.25\times {10}^{-14}$
$\tau \to \mu \gamma$	$1.21\times {10}^{-9}$

Equations

$${\mathcal{L}}_{\mathrm{Yukawa}}=$$

$$V=$$

$$+\left({\sigma }_1{\Phi }^{†}\varphi s_1^{+}+h.c.\right)+\left({\sigma }_2{\Phi }^{†}\eta s_2^{+}+h.c.\right)+\left({\sigma }_3{\varphi }^{†}{\eta }^c{s}_2^{+}+h.c.\right)$$

$$+{\lambda }_{\varphi }|\varphi {|}^4+{\lambda }_{\Phi }|\Phi {|}^4+{\lambda }_1|s_1^{+}{|}^4+{\lambda }_{\eta }|\eta {|}^4+{\lambda }_2|s_2^{+}{|}^4$$

$$+{\lambda }_{\varphi \Phi }|\varphi {|}^2|\Phi {|}^2+{\lambda }_{\varphi \Phi }^{\prime }|{\varphi }^{†}\Phi {|}^2+{\lambda }_{\varphi \eta }|\varphi {|}^2|\eta {|}^2+{\lambda }_{\varphi \eta }^{\prime }|{\varphi }^{†}\eta {|}^2+{\lambda }_{\Phi \eta }|\Phi {|}^2|\eta {|}^2+{\lambda }_{\Phi \eta }^{\prime }|{\Phi }^{†}\eta {|}^2$$

$$+\sum _{k=1}^2\left\{{\lambda }_{\varphi k}|\varphi {|}^2|s_k{|}^2+{\lambda }_{\Phi k}|\Phi {|}^2|s_k{|}^2+{\lambda }_{\eta k}|\eta {|}^2|s_k{|}^2\right\}+{\lambda }_{12}|s_1^{+}{|}^2|s_2^{+}{|}^2$$

$$+\left({\xi }_1{\eta }^{†}\Phi {\eta }^{†}{\varphi }^c+h.c.\right)+\left({\xi }_2{\Phi }^{†}{\varphi }^c{\left(s_2^{+}\right)}^2+h.c.\right).$$

$$m_{{\Phi }^{++}}^2$$

$$m_{{\eta }^0}^2$$

$$\left(\begin{array}{c}{\pi }_1^{+}\\ {\pi }_2^{+}\end{array}\right)=U_{\theta }\left(\begin{array}{c}{\Phi }^{+}\\ s_1^{+}\end{array}\right),U_{\theta }=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right),$$

$$\tan 2\theta =\frac{-2{\left(M_{\Phi s_1}^2\right)}_{12}}{{\left(M_{\Phi s_1}^2\right)}_{22}-{\left(M_{\Phi s_1}^2\right)}_{11}},M_{\Phi s_1}^2=\left(\begin{array}{cc}{\mu }_2^2+\frac{1}{2}\left({\lambda }_{\varphi \Phi }+{\lambda }_{\varphi \Phi }^{\prime }\right)v^2& \frac{1}{\sqrt{2}}{\sigma }_{1}v\\ \frac{1}{\sqrt{2}}{\sigma }_{1}v& {\mu }_3^2+\frac{1}{2}{\lambda }_{\varphi 1}{v}^2\end{array}\right).$$

$$m_{{\pi }_1}^2$$

$$m_{{\pi }_2}^2$$

$$\left(\begin{array}{c}{\omega }_1^{+}\\ {\omega }_2^{+}\end{array}\right)=U_{\chi }\left(\begin{array}{c}{\eta }^{+}\\ s_2^{+}\end{array}\right),U_{\chi }=\left(\begin{array}{cc}\cos \chi & \sin \chi \\ -\sin \chi & \cos \chi \end{array}\right),$$

$$\tan 2\chi =\frac{-2{\left(M_{\eta s_2}^2\right)}_{12}}{{\left(M_{\eta s_2}^2\right)}_{22}-{\left(M_{\eta s_2}^2\right)}_{11}},M_{\eta s_2}^2=\left(\begin{array}{cc}{\mu }_4^2+\frac{1}{2}{\lambda }_{\varphi \eta }{v}^2& -\frac{1}{\sqrt{2}}{\sigma }_{3}v\\ -\frac{1}{\sqrt{2}}{\sigma }_{3}v& {\mu }_5^2+\frac{1}{2}{\lambda }_{\varphi 2}{v}^2\end{array}\right).$$

$$m_{{\omega }_1}^2$$

$$m_{{\omega }_2}^2$$

$${\left(m_{\nu }\right)}_{\ell i}=$$

$$m_{\nu }=U_{\text{MNS}}\mathrm{diag}(m_1,m_2,m_3),$$

$$U_{\text{MNS}}$$

$$\frac{\text{Br}\left(\ell \to {\ell }^{\prime }\gamma \right)}{\text{Br}\left(\ell \to {\ell }^{\prime }{\nu }_{\ell }\overline{{\nu }_{{\ell }^{\prime }}}\right)}=\frac{3}{16\pi }\frac{\alpha }{G_F^2{m}_{\ell }^4}\left(|A_R^{s_1}+A_R^{\omega }{|}^2+|A_L^{\omega }{|}^2\right),$$

$$\displaystyle\Gamma\left(h\to\ell\ell^{\prime}\right)=\Gamma\left(h\to\overline{\ell}\ell^{\prime}\right)+\Gamma(h\to\overline{\ell}{}^{\prime}\ell)=\hskip 1.0pt\frac{m_{h}}{8\pi}\left(\frac{1}{16\pi^{2}}\right)^{2}\biggl{(}\bigl{|}B_{R}^{s_{1}}+B_{R}^{\omega}\bigr{|}^{2}+\bigl{|}B_{L}^{\omega}\bigr{|}^{2}\biggr{)},$$

$$\displaystyle\frac{\text{Br}\bigl{(}\ell_{m}\to\overline{\ell}_{n}\,\ell_{p}\,\ell_{q}\bigr{)}}{\text{Br}\bigl{(}\ell_{m}\rightarrow\ell_{p}\,\nu_{\ell_{m}}\overline{\nu}_{\ell_{p}}\bigr{)}}$$

$$+{\left|{\left(C_{LLRR}^{\omega }\right)}_{mnpq}\right|}^2+{\left|{\left(C_{LLRR}^{\omega }\right)}_{mnqp}\right|}^2-\text{Re}\left[{\left(C_{LLRR}^{\omega }\right)}_{mnpq}{\left(C_{LLRR}^{\omega }\right)}_{mnqp}^{\ast }\right]$$

$$+{\left|{\left(C_{RRLL}^{\omega }\right)}_{mnpq}\right|}^2+{\left|{\left(C_{RRLL}^{\omega }\right)}_{mnqp}\right|}^2-\text{Re}\left[{\left(C_{RRLL}^{\omega }\right)}_{mnpq}{\left(C_{RRLL}^{\omega }\right)}_{mnqp}^{\ast }\right]$$

$$\displaystyle{}+\left|\left(C_{LRRL}^{\omega}\right)_{mnpq}\right|^{2}+\left|\left(C_{RLLR}^{\omega}\right)_{mnpq}\right|^{2}+\left|\left(C_{LRLR}^{\omega}\right)_{mnpq}\right|^{2}+\left|\left(C_{RLRL}^{\omega}\right)_{mnpq}\right|^{2}\Biggr{\}},$$

$$\displaystyle\left<\sigma v_{\mathrm{rel}}\right>\simeq\frac{1}{8\pi}\bigl{(}Y_{2}^{\dagger}Y_{2}\bigr{)}_{aa}^{2}\frac{M_{\psi_{a}}^{2}\bigl{(}M_{\psi_{a}}^{4}+m_{\omega_{2}}^{4}\bigr{)}}{\bigl{(}M_{\psi_{a}}^{2}+m_{\omega_{2}}^{2}\bigr{)}^{4}}\,\frac{1}{\,x\,},$$

$${\Omega }_{{\psi }_a}{h}^2=1.04\times 10^9\times 2\times x_f^2\frac{\sqrt{g_{\ast }}}{g_{\ast s}}\left\{\frac{{\text{GeV}}^{-1}}{m_{\text{Pl}}⟨\sigma v_{\mathrm{rel}}⟩{|}_{x=1}}\right\},$$

$$x_f=$$

$$\displaystyle-\frac{3}{2}\ln\biggl{\{}\ln\bigl{[}0.038\times 2(g_{\psi}/\sqrt{g_{*}})m_{\text{Pl}}M_{\psi_{a}}\left<\sigma v_{\mathrm{rel}}\right>|_{x=1}\bigr{]}\biggr{\}},$$

$${\sigma }_2=50\text{GeV},\sin 2\theta \simeq -3.38\times 10^{-2},\sin 2\chi \simeq 4.99\times 10^{-7},$$

$${\lambda }_{\varphi 1}=1.0,{\lambda }_{\varphi 2}=-1.0,{\lambda }_{\varphi \eta }=-1.0,{\lambda }_{\varphi \Phi }={\lambda }_{\varphi \Phi }^{\prime }=1.0,$$

$$M_{{\psi }_1}=97.6\text{GeV},M_{{\psi }_2}=150\text{GeV},M_{{\psi }_3}=200\text{GeV},$$

$$m_{{\pi }_1}=520\text{GeV},m_{{\pi }_2}=510\text{GeV},$$

$$\displaystyle m_{\omega_{1}}\bigl{(}m_{\eta^{+}}\bigr{)}=1000\,{\text{GeV}},\hskip 10.0ptm_{\omega_{2}}\bigl{(}m_{s_{2}}\bigr{)}=550\,{\text{GeV}},\hskip 10.0ptm_{\eta^{0}}=1000\,{\text{GeV}},$$

$$Y_1=\left(\begin{array}{ccc}10^{-4}& 10^{-4}& 0.10\\ 0.80& 0.80& 10^{-4}\\ 1.5& 1.0& 10^{-4}\end{array}\right),$$

$$Y_2=\left(\begin{array}{ccc}10^{-4}& 10^{-4}& 0.10\\ -1.5& -0.40& 10^{-4}\\ 1.5& 1.0& 10^{-4}\end{array}\right),Y_{\eta }\simeq \left(\begin{array}{ccc}6.60\times 10^{-4}& -4.56\times 10^{-4}& 4.22\times 10^{-3}\\ 1.98\times 10^{-4}& -2.65\times 10^{-4}& 1.85\times 10^{-2}\\ -2.88\times 10^{-4}& 2.98\times 10^{-4}& 2.19\times 10^{-2}\end{array}\right).$$

$${\sin }^2{\theta }_{12}=0.307,{\sin }^2{\theta }_{13}=2.12\times 10^{-2},{\sin }^2{\theta }_{23}=0.417,$$

$$\Delta m_{21}^2=7.53\times 10^{-5}{\text{eV}}^2,$$

$$\Delta m_{32}^2=2.51\times 10^{-3}{\text{eV}}^2,$$

Full text

††preprint: OU-HET 993††preprint: UT-HET 130

I Introduction

Although the Standard Model (SM) is consistent with the current data of collider experiments, there are still mysterious phenomena which cannot be explained in the SM, such as the origin of neutrino masses, the nature of dark matter and the baryon asymmetry of the Universe. To explain these phenomena by extending the SM is one of the central interests of today’s high energy physics. Various models and mechanisms have also been proposed.

For the origin of neutrino masses, many new models have been studied along with the idea of the seesaw mechanism, which explains Majorana-type tiny neutrino masses by introducing new heavy particles, such as right-handed neutrinos ref:seesaw ; Schechter:1980gr , an additional isospin triplet scalar field Schechter:1980gr ; ref:HTM and isospin triplet fermions Foot:1988aq . There is also an alternative scenario where tiny neutrino masses are generated by quantum effects. The first model along this line was proposed by Zee Zee:1980ai , in which one-loop effects due to an additional Higgs doublet field and a charged singlet scalar field yield Majorana-type tiny neutrino masses. There have been many variation models Zee:1985id ; Babu:1988ki ; Cheng:1980qt ; ref:GNR ; ref:KNT ; ref:AKS ; ref:Ma , some of which introduce an unbroken discrete symmetry in order not only to forbid tree-level generation of neutrino masses but also to guarantee the stability of extra particles in the loop so that the lightest one can be identified as a dark matter candidate ref:GNR ; ref:KNT ; ref:AKS ; ref:Ma . In Ref. ref:AKS , an extended scalar sector for inducing neutrino masses at the three loop level with a dark matter candidate is also used to cause the strongly first order electroweak phase transition, which is required for successful electroweak baryogenesis Kuzmin:1985mm . In addition, models which generate Dirac-type tiny neutrino masses by quantum effects have also been proposed in Refs. ref:1loopDirac ; Gu:2007ug ; Kanemura:2017haa . In Ref. Kanemura:2017haa , introducing right-handed neutrinos with an odd quantum number under a new discrete symmetry, Dirac-type tiny neutrino masses are generated at the two-loop level. This model also has a dark matter candidate and can realize the strongly first order phase transition.

In Ref. Kanemura:2015cca , a class of models in which Majorana-type tiny neutrino masses are generated by quantum effects has been comprehensively studied by using flavor structures of induced neutrino mass matrices. Classification of models to generate Dirac-type neutrino masses has also been performed in Ref. Kanemura:2016ixx .

Several years ago, anomaly for a lepton flavor violating (LFV) decay process of the Higgs boson $h\to\mu\tau$ at the LHC was reported by ATLAS Aad:2015gha and CMS Khachatryan:2015kon ; CMS:2016qvi , although it disappeared soon later Sirunyan:2017xzt . Motivated by this anomaly, the authors of Ref. Aoki:2016wyl examined in a systematic way what kind of models for neutrino masses can predict a significant amount of signals for $h\to\mu\tau$. It was shown that most of the proposed models radiatively generating Majorana-type neutrino masses and Dirac-type neutrino masses, as well as minimal models of Type-I, II and III seesaw mechanisms are excluded if the signal of LFV decays of the Higgs boson is observed at future collider experiments without detecting LFV process for charged leptons. They also found that only a few models, in which Dirac-type neutrino masses are generated radiatively, may not be excluded even in this case.

In this paper, we concretely build one of such models, where additional scalar fields as well as right-handed fermions are introduced with even or odd charge under new discrete symmetries, so that Dirac-type tiny neutrino masses are generated at the two-loop level and a dark matter candidate is also contained. The branching ratio for LFV decays of the Higgs boson is not too small in spite of the stringent constraints from LFV processes for charged lepton decays. We will show that the model can be viable under the constraints from current data for neutrino experiments, searches for flavor violating decays of charged leptons and dark matter experiments.

This paper is organized as follows. In Sec. II, we define our model and introduce new fields and symmetries. In Sec. III, we give the formula of neutrino mass matrix which is generated at two-loop level. In Sec. IV, we consider the LFV processes $\ell\to\ell^{\prime}\gamma,h\to\ell\ell^{\prime}~{}\text{and}~{}\ell_{m}\to\overline{\ell}_{n}\ell_{p}\ell_{q}$. In Sec. V, we show formulae of the thermally averaged cross sections of annihilation processes of the dark matter and the relic abundance. In Sec. VI, we present two benchmark scenarios and give numerical results of various phenomena in Secs. III, IV and V. The first scenario is for the normal ordering mass hierarchy of neutrinos, and the second one is for the inverted ordering one. Conclusions are shown in Sec. VII. Some formulae are presented in Appendices.

II Model

In our model, fields listed in Table 1 are added to the SM ones. We impose the conservation of the lepton number $L$ to our model. Gauge singlet right-handed fermions $\nu_{iR}$ $(i=1,2,3)$ have $L=1$, which compose three Dirac neutrinos with left-handed neutrinos $\nu_{\ell L}$ $(\ell=e,\mu,\tau)$ of the SM lepton doublet fields $L_{\ell}=(\nu_{\ell L},\,\ell_{L})^{T}$. On the other hand, lepton numbers of the other gauge singlet fermions $\psi_{aR}$ $(a=1,2,3)$ are zero. They have Majorana mass terms, $\frac{1}{2}M_{\psi_{a}}\overline{\psi_{aR}^{c}}\psi_{aR}$, without breaking the lepton number conservation, while Majorana mass terms of $\nu_{iR}$ are forbidden. If neutrinos have Yukawa interactions $(Y_{\nu})_{\ell i}\overline{L_{\ell}}\,\phi^{c}\nu_{iR}$ with the SM Higgs doublet field $\phi=(\phi^{+},\phi^{0})^{T}$, masses of Dirac neutrinos can be generated with the vacuum expectation value $\langle\phi^{0}\rangle$. However, required values of Yukawa coupling constants $(Y_{\nu})_{\ell i}$ for tiny neutrino masses seem to be unnaturally small. Thus, we impose a softly broken discrete symmetry $(Z_{2}^{\prime})$ to our model in order to forbid tree-level Yukawa interaction of neutrinos, where $\nu_{iR}$ are odd under $Z_{2}^{\prime}$ while fields in the SM are even. Assignments of $Z_{2}^{\prime}$ quantum number to the new fields are shown in Table 1. Although neutrino masses in the lagrangian are forbidden by $Z_{2}^{\prime}$, they can be generated at the loop level via the soft breaking effect in the scalar sector. Throughout this paper, we take the basis where $\ell$, $\nu_{iR}$, and $\psi_{aR}$ are mass eigenstates.

Four new scalar fields ($\Phi$, $s_{1}^{+}$, $\eta$, and $s_{2}^{+}$) are involved in our model in addition to the Higgs doublet field $\phi$ of the SM. Both of $s_{1}^{+}$ with $L=-2$ and $s_{2}^{+}$ with $L=-1$ are ${\text{SU}}(2)_{L}$-singlet fields with $Y=1$. On the other hand, $\Phi=(\Phi^{++},\Phi^{+})^{T}$ with $Y=3/2$ and $\eta=(\eta^{+},\eta^{0})^{T}$ with $Y=1/2$ are ${\text{SU}}(2)_{L}$-doublet fields. The doublet field $\Phi$ has $L=-2$, and the even parity under $Z_{2}^{\prime}$ is assigned to $\Phi$.111 Actually, the $Z_{2}^{\prime}$ parity of $\Phi$ is irrelevant to our study in this article so that the odd-parity is also acceptable for $\Phi$. Although $\eta$ belongs to the same representation as $\phi$ under the SM gauge symmetry, $\eta$ has $L=-1$ in contrast with $L=0$ for $\phi$. We restrict ourselves to the case where $\eta^{0}$, the neutral component of $\eta$, does not have a vacuum expectation value in order to keep the lepton number conservation. The other new scalar fields do not also have vacuum expectation values because they are electrically charged.

Apart from $Z_{2}^{\prime}$, an accidental unbroken discrete symmetry $(Z_{2})$ appears in our model due to the lepton number conservation, Majorana mass terms of $\psi_{aR}$ and some of new Yukawa interactions,222 These Majorana mass terms, $Y_{1}$ and $Y_{2}$ terms in Eq. (1) explicitly break ${\text{U}}(1)_{L+2J}$ into its $Z_{2}$ subgroup.

where the parity is given by $(-1)^{L+2J}$. Three fields ($\psi_{aR}$, $\eta$, and $s_{2}^{+}$) are odd under $Z_{2}$. The lightest $Z_{2}$-odd particle is stable. If $\psi_{aR}$ or $\eta^{0}$ is the lightest one, it can be a dark matter candidate.

In our model, there are three new Yukawa interactions as

[TABLE]

The scalar potential is given by

[TABLE]

Notice that $\sigma_{1}$ is the soft breaking parameter for $Z_{2}^{\prime}$.333 If $\Phi$ is taken to be odd under $Z_{2}^{\prime}$, the soft breaking parameter is $\sigma_{2}$. Therefore, a product $\sigma_{1}\sigma_{2}$ breaks $Z_{2}^{\prime}$ independently of the $Z_{2}^{\prime}$-parity of $\Phi$.

There are five complex coupling constants ($\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$, $\xi_{1}$, and $\xi_{2}$), and two CP-violating phases remain as physical parameters after redefinitions of phases of fields.444 If we take $\Phi$ as a $Z_{2}^{\prime}$-odd field, terms of $\xi_{1}$ and $\xi_{2}$ are replaced with $\xi_{3}\Phi^{\dagger}\eta^{c}\hskip 1.0pts_{1}^{+}\hskip 1.0pts_{2}^{+}$. Then, only one CP-violating phase is physical.

In this article, coupling constants in the scalar potential are taken to be real, just for simplicity.

The SM Higgs doublet field $\phi$ does not mix with the other scalar fields in our model. Thus, identically to the SM, the field can be expressed as $\phi=(G^{+},(v+h+iG^{0})/\sqrt{2}\,)^{T}$, where $v$ ($=\sqrt{-\mu_{1}^{2}/\lambda_{\phi}}=246\,{\text{GeV}}$) is the vacuum expectation value. The real component $h$ corresponds to the SM Higgs boson, whose mass is given by $m_{h}=\sqrt{2\lambda_{\phi}}\,v$. Nambu-Goldston bosons ($G^{+}$ and $G^{0}$) are absorbed by the longitudinally polarized weak gauge bosons by the electroweak symmetry breaking.

Fields $\Phi^{++}$ and $\eta^{0}$ are mass eigenstates. Their squared masses are given by

[TABLE]

Mass eigenstates $\pi_{1}^{+}$ and $\pi_{2}^{+}$, which are singly-charged and have $L=-2$, are obtained by linear combinations of $\Phi^{+}$ and $s_{1}^{+}$ as

[TABLE]

where the mixing angle $\theta$ is defined as

[TABLE]

Squared masses of $\pi_{1}^{+}$ and $\pi_{2}^{+}$ are given by

[TABLE]

Mass eigenstates $\omega_{1}^{+}$ and $\omega_{2}^{+}$, which are $Z_{2}$-odd with $L=-1$, are constructed by linear combinations of $\eta^{+}$ and $s_{2}^{+}$ as follows:

[TABLE]

where the mixing angle $\chi$ is defined as

[TABLE]

Squared masses of $\omega_{1}^{+}$ and $\omega_{2}^{+}$ are given by

[TABLE]

III Neutrino Mass

Mass terms $(m_{D})_{\ell i}\,\overline{\nu_{\ell L}}\,\nu_{iR}$ of Dirac neutrinos are generated in our model via two-loop diagrams in Fig. 1.

The Dirac neutrino mass matrix $(m_{D})_{\ell i}$ is calculated as

[TABLE]

where the coupling constant $\sigma_{1}$ in Fig. 1 is replaced by using $2\sigma_{1}\left<\phi^{0}\right>=\left(m^{2}_{\pi_{1}}-m^{2}_{\pi_{2}}\right)\sin(2\theta)$. The explicit formula for the loop function $I_{\ell^{\prime}ak}$ is given in Appendix A. Notice that $\sigma_{2}\sin(2\theta)$ softly breaks $Z_{2}^{\prime}$ that forbids $\overline{L_{\ell}}\,\phi^{c}\nu_{iR}$.

Since we take the basis where $\nu_{iR}$ are mass eigenstates, the neutrino mass matrix $\left(m_{\nu}\right)_{\ell i}$ is diagonalized as

[TABLE]

where $m_{i}$ $(i=1,2,3)$ denote masses of Dirac neutrinos. The mixing matrix $U_{\text{MNS}}$ is the Maki-Nakagawa-Sakata matrix Maki:1962mu , which can be parameterized as

[TABLE]

where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$, and $\delta$ is a CP-violating phase in the lepton sector.

IV Lepton Flavor Violation

Matrices $Y_{1}$, $Y_{2}$, and $Y_{\eta}$ are not diagonal and cause LFV processes. Radiative decays of charged leptons, $\ell\to\ell^{\prime}\gamma$, can be caused via the one-loop diagrams in Fig. 2. Ignoreing $m_{\ell^{\prime}}$, branching ratios of these decays are expressed as

[TABLE]

where $G_{F}=1.17\times 10^{-5}\,{\text{GeV}}^{-2}$ is the Fermi constant, $\text{Br}\left(\tau\to e\nu_{\tau}\overline{\nu_{e}}\right)\simeq 0.178$, $\text{Br}\left(\tau\to\mu\nu_{\tau}\overline{\nu_{\mu}}\right)\simeq 0.174$ and $\text{Br}\left(\mu\to e\nu_{\mu}\overline{\nu_{e}}\right)\simeq 1$ Tanabashi:2018oca . Formulae of $A_{R}^{s_{1}}$, $A_{R}^{\omega}$ and $A_{L}^{\omega}$ are presented in Appendix B. $A_{R}^{s_{1}}$ corresponds to the contribution from $s_{1}^{+}$ to $\ell\to\ell^{\prime}_{R}\gamma$. Contributions of $s_{2}^{+}$ and $\eta^{+}$ to $\ell\to\ell^{\prime}_{R}\gamma$ are given by $A_{R}^{\omega}$, while $A_{L}^{\omega}$ is for their contributions to $\ell\to\ell^{\prime}_{L}\gamma$.

Scalar fields that contribute to $\ell\to\ell^{\prime}\gamma$ affect also $h\to\ell\ell^{\prime}$ ($\ell\neq\ell^{\prime}$) via diagrams in Fig. 3. Decay widths for $h\to\ell\ell^{\prime}$ ($\ell\neq\ell^{\prime}$) are given by

[TABLE]

where we take $m_{\ell^{\prime}}=0$. Formulae of $B_{R}^{s_{1}}$, $B_{R}^{\omega}$ and $B_{L}^{\omega}$ are shown in Appendix C. The contribution from $s_{1}^{+}$ is given by $B_{R}^{s_{1}}$, while those from $s_{2}^{+}$ and $\eta^{+}$ are involved in both of $B_{R}^{\omega}$ and $B_{L}^{\omega}$. The subscript $X$ $(=L,R)$ in these $B_{X}$’s indicates the chirality of the lighter charged lepton $\ell^{\prime}_{X}$ in the final state.

New scalar bosons in our model contribute also to $\ell_{m}\to\overline{\ell}_{n}\,\ell_{p}\,\ell_{q}$ ($m=2,3$ and $n,p,q=1,2$) with new Yukawa interactions, where $\ell_{1},\ell_{2}$ and $\ell_{3}$ corresponds to $e,\mu$ and $\tau$, respectively. Contributions from penguin diagrams can be ignored because of the constraint from $\ell\to\ell^{\prime}\gamma$. However, if some coupling constants of new Yukawa interactions are $O(1)$, box diagrams in Fig. 4 should be considered. Branching ratios for $\ell_{m}\to\overline{\ell}_{n}\,\ell_{p}\,\ell_{q}$ via the box diagrams are given by

[TABLE]

where $S=1$ ($2$) for $p=q$ ($p\neq q$). The variable $\left(C_{RRRR}^{s_{1}}\right)_{mnpq}$ ( $\left(C_{RRRR}^{s_{2}}\right)_{mnpq}$ ) corresponds to the contribution from $s_{1}^{+}$ ($s_{2}^{+}$) in the first diagram (the second and the third diagrams) in Fig. 4: the structure of chiralities is $\ell_{mR}\to\overline{\ell_{nR}}\,\ell_{pR}\,\ell_{qR}$ because charged leptons that have Yukawa interactions with $s_{1}^{+}$ and $s_{2}^{+}$ are only right-handed ones. The contribution from $\eta^{+}$ to $\ell_{mL}\to\overline{\ell_{nL}}\,\ell_{pL}\,\ell_{qL}$ via the second and the third diagrams in Fig. 4 is given by $\left(C_{LLLL}^{\eta}\right)_{mnpq}$. The other $\left(C^{\omega}\right)_{mnpq}$’s arise due to the mixing between $s_{2}^{+}$ and $\eta^{+}$ in the second and the third diagrams in Fig. 4. See Appendix D for formulae of $\left(C\right)_{mnpq}$’s. Current constraints on the branching ratios for LFV processes ($\ell\to\ell^{\prime}\gamma$, $h\to\ell\ell^{\prime}$, and $\ell_{m}\to\overline{\ell}_{n}\,\ell_{p}\,\ell_{q}$) are summarized in Table 2.

V Dark Matter

In our model, dark matter candidates are the lightest of fermions $\psi_{a}$ and a scalar $\eta^{0}$, which are neutral $Z_{2}$-odd particles. Notice that $\eta^{0}$ from a doublet field is a complex scalar with the lepton number $L=-1$. In other words, there is no mass spritting between CP-even and odd parts of $\eta^{0}$. According to Ref. Escudero:2016gzx , the scenario where the dark matter is such a complex scalar is stringently constrained from direct search experiments because it interacts with nuclei at tree level. Therefore, we consider the case where the dark matter is the lightest one of gauge singlet Majorana fermions $\psi_{a}$.

The dark matter candidate $\psi_{a}$ can be annihilated via tree-level diagrams shown in Fig. 5. The thermal averages $\left<\sigma v_{\mathrm{rel}}\right>$, where $\sigma$ is annihilation cross section of $\psi_{a}$ and $v_{\mathrm{rel}}$ denotes the relative velocity of the initial particles, is given by a sum of two processes as $\left<\sigma v_{\mathrm{rel}}\right>=\left<\sigma_{\ell}v_{\mathrm{rel}}\right>+\left<\sigma_{\nu}v_{\mathrm{rel}}\right>$. Thermal avalages $\left<\sigma_{\ell}v_{\mathrm{rel}}\right>$ and $\left<\sigma_{\nu}v_{\mathrm{rel}}\right>$ correspond to the effects of left and right diagrams in Fig. 5, respectively. Formulae of $\left<\sigma_{\ell}v_{\mathrm{rel}}\right>$ and $\left<\sigma_{\nu}v_{\mathrm{rel}}\right>$ are shown in Appendix E. Notice that the $s$-wave annihilation is only involved in $\left<\sigma_{\ell}v_{\mathrm{rel}}\right>$ with a mixing $\chi$.

For the case where the elements of $Y_{\eta}$ and the mixing angle $\chi$ are negligible (we take such a benchmark scenario in the next section), the dominant contribution to $\left<\sigma v_{\mathrm{rel}}\right>$ comes from the mediation of $s_{2}^{+}$ ($\simeq\omega_{2}^{+}$) in the left diagram in Fig. 5. Then, $\left<\sigma v_{\mathrm{rel}}\right>$ is approximately calculated as

[TABLE]

where $x=m_{\psi_{a}}/T$ at the temperature $T$. In Appendix E, $\left<\sigma v_{\mathrm{rel}}\right>$ for more general case is presented. The relic abundance of $\psi_{a}$ with the $p$-wave annihilation is calculated as

[TABLE]

where $m_{\text{Pl}}=1.2\times 10^{19}\,{\text{GeV}}$ stands for the Planck mass, and $g_{*}=106.75$ ($g_{*S}=106.75$) is the effective degree of freedom for energy (entropy) density in the era of the freeze out of the dark matter Kolb:1990vq , and $x_{f}$ is defined by

[TABLE]

where $g_{\psi}=2$ is the degree of freedom of $\psi_{a}$.

VI Benchmark Scenarios and Numerical Evaluation

We here consider the possibility that $h\to\mu\tau$ is enhanced in comparison with LFV decays of charged leptons. First, we take the following benchmark scenario for $m_{1}<m_{3}$ (the normal ordering case of neutrino masses):

[TABLE]

The small value of the mixing angle $\chi$ implies $\omega_{1}^{+}\simeq\eta^{+}$ and $\omega_{2}^{+}\simeq s_{2}^{+}$. Since $\pi_{k}^{+}$ and $\omega_{k}^{+}$ have Yukawa interactions only with leptons, their masses are constrained by the slepton searches in the context of supersymmetric models at the LHC, which give about $500\,{\text{GeV}}$ as the lower bound Aaboud:2018jiw .

The generated neutrino mass matrix results in the following values, which are consistent with the current constraint from neutrino oscillation experiments Tanabashi:2018oca :

[TABLE]

where $\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$. The values of $m_{1}$ is also consistent with $\sum_{i}m_{i}<0.26\,{\text{eV}}$ that is given by cosmological observations Loureiro:2018pdz , although $m_{1}$ is not constrained by the oscillation data.

In Table 3, we show branching ratios for the LFV processes $\ell\to\ell^{\prime}\gamma$, $\ell_{m}\to\overline{\ell}_{n}\ell_{p}\ell_{q}$ and $h\to\ell\ell^{\prime}$ in our benchmark scenario given in Eq. (VI). They satisfy the constraints from the current data in Table 2. Since the elements of $Y_{\eta}$ are rather small as seen in Eq. (VI), the contribution from $\eta^{+}$ to $\ell\to\ell^{\prime}\gamma$ ($A_{L}^{\omega}$ in Eq. (16)) is negligible. Then, values of $\text{BR}(\ell\to\ell^{\prime}\gamma)$ in our benchmark scenario are suppressed due to the cancellation of $A_{R}^{s1}$ and $A_{R}^{\omega}$, which are contributions from $s_{1}^{+}$ and $s_{2}^{+}$, respectively. This is an interesting utilization of scalar bosons ($s_{1}^{+}$ and $s_{2}^{+}$) that are originally introduced for generating neutrino masses. On the other hand, the contribution from $\eta^{+}$ to $h\to\ell\ell^{\prime}$ ($B_{L}^{\omega}$ in Eq. (17)) is also negligible due to small values of components of $Y_{\eta}$. Even though contributions from $s_{1}^{+}$ and $s_{2}^{+}$ to $\ell\to\ell^{\prime}\gamma$ are destructive with each other, their contributions to $h\to\ell\ell^{\prime}$ ($B_{R}^{s1}$ and $B_{R}^{\omega}$ in Eq. (17)) are not necessarily cancelled with each other because of the sign flip by using coupling constants in the scalar sector, $\Lambda^{\pi}_{22}$ and $\Lambda^{\omega}_{22}$ in Appendix C. 555Notice that other $\Lambda^{\pi}$’s and $\Lambda^{\omega}$’s do not contribute to the cancellation dominantly because $\theta$ and $\chi$ are small in the benchmark scenario. In our benchmark scenario, $\text{BR}(h\to\mu\tau)$ is indeed much larger than $\text{BR}(\tau\to\mu\gamma)$. This hierarchy is what expected in Ref. Aoki:2016wyl , and our calculation explicitly shows that the expectation is correct.

In Fig. 6, we show plots of the branching ratio for $\tau\to\mu\gamma$ versus that for $h\to\mu\tau$. In the left one, we change only the values of $\left(Y_{1}\right)_{\mu 2}$ between $-1.5$ and $1.5$. In the right one, we assume that the form of the matrices $Y_{1}$ and $Y_{2}$ is

[TABLE]

where $k=1,2$, and then we vary eight unfixed parameters between $-1.5$ and $1.5$. The orange points are predictions with same sign $\lambda$’s, $\lambda_{\phi 1}=\lambda_{\phi 2}=\lambda_{\phi\eta}=\lambda_{\phi\Phi}=\lambda_{\phi\Phi}^{\prime}=1.0$. The blue points are ones with opposite sign $\lambda$’s, $\lambda_{\phi 1}=\lambda_{\phi\Phi}=\lambda_{\phi\Phi}^{\prime}=1.0$ and $\lambda_{\phi 2}=\lambda_{\phi\eta}=-1.0$, as in the benchmark scenario. In both of the plots, values of fixed parameters are taken to be the same with those of the benchmark scenario. Two branching ratios are equal on the solid line in the figures. The upper dashed line is the current upper limit for $\text{BR}(\tau\to\mu\gamma)$, $4.4\times 10^{-8}$, and the lower one is the expected upper limit, $1.0\times 10^{-9}$, from the Belle-II experiment Kou:2018nap with the integrated luminosity $50~{}\mathrm{ab^{-1}}$. In the case with same sign $\lambda$’s, the correlation between branching ratios is almost linear, and $\text{BR}(\tau\to\mu\gamma)$ is larger than $\text{BR}(h\to\mu\tau)$ in most of the orange points. In the case with opposite sign $\lambda$’s, on the other hand, $\text{BR}(h\to\mu\tau)$ are significantly larger than $\text{BR}(\tau\to\mu\gamma)$ in some of the blue points. This is just what we anticipated. The red point represents the result in the benchmark scenario.

In Fig. 7, we show the plot for $\text{BR}(\tau\to\overline{\mu}\mu\mu)$ versus $\text{BR}(h\to\mu\tau)$ under the same assumptions as in the right one of Fig. 6. The upper dashed line is the current upper limit for $\text{BR}(\tau\to\overline{\mu}\mu\mu)$, $2.1\times 10^{-8}$, and the lower one is the expected upper limit, $3.3\times 10^{-10}$, from the Belle-II experiment Kou:2018nap with the integrated luminosity $50~{}\mathrm{ab^{-1}}$. We cannot find any correlation between the branching ratios, because these processes are given by different kind of Feynman diagrams.

Although $\text{BR}(h\to\mu\tau)=1.04\times 10^{-7}$ is about $10^{2}$ times larger than our prediction on $\text{BR}(\tau\to\mu\gamma)$, the value is rather below the expected sensitivities, $O(10^{-4})$ at HL-LHC Calibbi:2017uvl ; Banerjee:2016foh and $O(10^{-5})$ at ILC250 Chakraborty:2016gff . We can in principle enhance $\text{BR}(h\to\mu\tau)$ further by taking larger values666 It is difficult to take lighter masses of $s_{1}^{+}$ and $s_{2}^{+}$ because of constraint from the slepton searches. of $Y_{1}$, $Y_{2}$, and $\lambda$’s, although we should worry about unitality bounds. On the other hand, the values for $\text{BR}(\tau\to\mu\gamma)$ and $\text{BR}(\tau\to\bar{\mu}\mu\mu)$ in our benchmark scenario are close to the sensitivity in Belle II experiment Kou:2018nap , and then the scenario might be tested. In the case that $\tau\to\mu\gamma$ or $\tau\to\overline{\mu}\mu\mu$ is observed, we can distinguish our benchmark scenario from other models for tiny neutrino masses by the searches for the signal of $h\to\mu\tau$.

The dark matter in the benchmark scenario is the lightest $Z_{2}$-odd Majorana fermion $\psi_{1}$. The density of the thermal relic abundance $\Omega_{\psi_{1}}h^{2}$ can be evaluated with Eqs. (19)-(21), which are valid for the case where $Y_{\eta}$ and $\chi$ are negligible. The Planck experiment shows that $\Omega_{\mathrm{DM}}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx . In Fig. 8, we show $\Omega_{\psi_{1}}h^{2}$ as a function of the dark matter mass $M_{\psi_{1}}$, where $Y_{2}$ and $m_{\omega}$ are fixed to the values of the benchmark scenario. The blue curve is the mass dependence in our model, and the horizontal line shows the observed value. It is clear that the appropriate value of $\Omega_{\psi_{1}}h^{2}$ is obtained for $M_{\psi_{1}}=97.6\,{\text{GeV}}$ in the benchmark scenario.

There is no tree-level contribution to the dark matter scattering off nuclei, because $\psi_{a}$ are gauge singlet fermions. The scattering occurs at one-loop level via three penguin diagrams with $\omega_{1},\ \omega_{2}$ and $\eta^{0}$ in the loop. In our benchmark scenario, the elements of the matrix $Y_{\eta}$ are typically smaller than those of $Y_{2}$, so that we consider only the contribution from the diagram with $\omega_{2}$ in the loop. In Ref. Herrero-Garcia:2018koq , the authors studied in detail the gauge singlet Majorana dark matter which is coupled to a dark scalar and charged leptons. They also considered the scenario where the dark matter has no interaction with electrons, which is similar to our benchmark scenario. They gave the constraint from the direct searches with the combined data from XENON1T Aprile:2017iyp , PandaX Cui:2017nnn and LUX Akerib:2016vxi . From their results, we can estimate that the upper limits on $(Y_{2})_{\mu 1}$ and $(Y_{2})_{\tau 1}$ in our benchmark scenario are both about $3.2$. Since $\left(Y_{2}\right)_{\mu 1}$ and $\left(Y_{2}\right)_{\tau 1}$ in Eq. (VI) are below this upper limit, the dark matter in our benchmark scenario satisfies the constraint from the current direct detection experiments.

Next, we consider the benchmark scenario for the inverted ordering case ($m_{3}<m_{1}$). The difference from the normal ordering case appears on $Y_{\eta}$, and we here take

[TABLE]

All the other parameters are taken to be the same with those in Eq. (VI).

The neutrino mass matrix generated at two loop gives the following values, which are consistent with the current constraint from neutrino oscillation experiments Tanabashi:2018oca ,

[TABLE]

The value of $m_{3}$ satisfies the condition from the Planck ovservation, $\sum_{i}m_{i}<0.26\ \mathrm{eV}$  Loureiro:2018pdz .

Branching ratios for the LFV processes in this scenario are listed in Table 4.

Most of all branching ratios are the same as those in the scenario in Eq.(VI), because the elements of $Y_{\eta}$ are typically smaller than those of $Y_{1}$ and $Y_{2}$ in the both scenarios. $\text{BR}(h\to\mu\tau)$ is about $10^{2}$ times larger than our prediction on $\text{BR}(\tau\to\mu\gamma)$.

The dark matter in this scenario is again the lightest $Z_{2}$-odd Majorana fermion $\psi_{1}$. The density of the thermal relic abundance depends on only $(Y_{2}^{\dagger}Y_{2})_{11},\ M_{\psi_{1}}$ and $m_{\omega_{2}}$ in the case, where $Y_{\eta}$ and $\chi$ are negligibly small. Values of these parameters are the same with those of Eq.(VI). Therefore, $M_{\psi_{1}}=97.6\ {\text{GeV}}$ can still explain the observed relic density $\Omega_{\mathrm{DM}}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx , just like in the benchmark scenario for the case of $m_{1}<m_{3}$. The constraint from the direct detection experiments is also the same with the previous scenario.

VII Conclusions

We have proposed a new mechanism to explain neutrino masses with lepton number conservation, in which the Dirac neutrino masses are generated at the two-loop level involving a dark matter candidate. In this model branching ratios of lepton flavor violating decays of the Higgs boson can be much larger than those of lepton flavor violating decays of charged leptons. We have found the benchmark scenarios for normal ordered masses of neutrinos and inverted ones, where the neutrino mass matrix, the relic density of dark matter and the branching ratios for LFV processes can satisfy the constraints from current experimental data. We have showed that $\text{BR}(h\to\mu\tau)$ is about $10^{2}$ lager than $\text{BR}(\tau\to\mu\gamma)$ in our benchmark scenarios. If the lepton flavor violating decays of the Higgs boson are observed at the future collider experiments without detecting lepton flavor violating decays of charged leptons, most of the previously proposed models are excluded, while our model can still survive.

In this paper, we did not discuss collider signature of new scalars and fermions. Collider phenomenology for $Z_{2}$-even/odd charged singlet scalars in different models can be found in the literature Kanemura:2000bq ; Aoki:2010tf / Aoki:2010tf ; Ahriche:2014xra ; Aoki:2010aq , while that for $\Phi$ $(Y=3/2)$ has been discussed in Ref. Rentala:2011mr ; Aoki:2011yk in the different context. We will discuss these issues elsewhere in the future future_work .

Acknowledgements.

The work of K. E. was supported in part by the Sasakawa Scientific Research Grant from The Japan Science Society. The work of S. K. was supported in part by Grant-in-Aid for Scientific Research on Innovative Areas, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), No. 16H06492, No. 18H04587, and Grant H2020-MSCA-RISE-2014 no. 645722 (Non Minimal Higgs). The work of K. S. was supported in part by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 18J12866 (JSPS Research Fellow). The work of H. S. was supported in part by MEXT KAKENHI Grant No. 18H05543 (Innovative Areas) and JSPS KAKENHI Grant No. 18K03625 (Scientific Research (C)).

Appendix A The loop function in the neutrino mass matrix

The neutrino mass matrix formula given by Eq. (13) in Sec. III contains the loop function $I_{\ell^{\prime}ak}$ $(k=1,2,\ a=1,2,3~{}\text{and}~{}\ell^{\prime}=e,~{}\mu,~{}\tau)$. We here show the explicit formula for $I_{\ell^{\prime}ak}$;

[TABLE]

where the function $f_{ak}$ is defined as follows

[TABLE]

with

[TABLE]

Appendix B Some formulae for $\ell\to\ell^{\prime}\gamma$

In Sec. IV, blanching ratios for $\ell\to\ell^{\prime}\gamma$ are given by Eq. (16), which depend on $A_{R}^{s_{1}},\ A_{R}^{\omega}$ and $A_{L}^{\omega}$. We here present their explicit formulae. They are given by

[TABLE]

where $F_{2}(x)$ and $G(x)$ are defined as

[TABLE]

Terms that proportional to $M_{\psi_{a}}/m_{\ell}$ in formulae of $A_{R}^{\omega}$ and $A_{L}^{\omega}$ appear due to the mixing between $s_{2}^{+}$ and $\eta^{+}$.

Appendix C Some formulae for $h\to\ell\ell^{\prime}$

In Sec. IV, blanching ratios for $h\to\ell\ell^{\prime}$ are given by Eq. (17), which depend on $B_{R}^{s_{1}},\ B_{R}^{\omega}$ and $B_{L}^{\omega}$. We here give their explicit formulae. They are defined as

[TABLE]

Coefficients $\Lambda^{\pi}_{kk^{\prime}}$ and $\Lambda^{\omega}_{kk^{\prime}}$ are defined in order to satisfy

[TABLE]

and given by

[TABLE]

Appendix D Some formulae for $\ell_{m}\to\overline{\ell}_{n}\ell_{p}\ell_{q}$

In Sec. IV, blanching ratios for $\ell_{m}\to\overline{\ell}_{n}\ell_{p}\ell_{q}$ are given by Eq. (18), which depend on $(C_{RRRR}^{s_{1}})_{mnpq},\ (C_{RRRR}^{s_{2}})_{mnpq},\ (C^{\eta})_{mnpq}$ and $(C^{\omega})_{mnpq}{}^{\prime}\mathrm{s}$. We here give their explicit formulae. They are given by

[TABLE]

[TABLE]

where $\Delta$ and $\Sigma$ are defined as

[TABLE]

and the symbol $\int$ denotes the integration with respect to $x,y,z$ and $\omega$ as follows;

[TABLE]

By exchanging $p$ and $q$ for $\left(C_{RLLR}^{\omega}\right)_{mnpq}$ and $\left(C_{LRRL}^{\omega}\right)_{mnpq}$, we obtain

[TABLE]

Appendix E Annihilation of dark matter $\psi_{a}$

In Sec. V, we have shown only the approximate formula for the thermal averaged cross section for annihilation of the dark matter $\psi_{a}$, $\left<\sigma v_{\mathrm{rel}}\right>$. In this appendix, we show the complete formula at tree level. First, the contribution from annihilation to a pair of charged leptons, $\left<\sigma_{\ell}v_{\mathrm{rel}}\right>$, which is shown by the left of Fig. 5, is given by

[TABLE]

Second, the contribution from annihilation to a pair of neutrinos, $\left<\sigma_{\nu}v_{\mathrm{rel}}\right>$, which is represented by the right of Fig. 5, is given by

[TABLE]

The complete formula for $\left<\sigma v_{\mathrm{rel}}\right>$ is given at tree level by the sum of Eq. (69) and (70).