Neutrinophilic two Higgs doublet model with dark matter under an alternative $U(1)_{B-L}$ gauge symmetry

Abstract

We propose a Dirac type active neutrino with rank two mass matrix and a Majorana fermion dark matter candidate with an alternative local $U(1)_{B-L}$ extension of neutrinophilic two Higgs doublet model. Our dark matter candidate can be stabilized due to charge assignment under the gauge symmetry without imposing extra discrete $Z_2$ symmetry and the relic density is obtained from an $Z'$ boson exchanging process. Taking into account collider constraints on the $Z'$ boson mass and coupling, we estimate the relic density.

1 Introduction

The origin of tiny neutrino masses is one of the unsolved issues in the standard model (SM) where the neutrino mass type can be either of Dirac or of Majorana type. The Majorana type neutrino mass can be realized in many scenarios such as the type-I seesaw mechanism in which heavy SM singlet right-handed neutrinos are introduced. On the other hand, the Dirac type neutrino mass can also be obtained as charged leptons by introducing right-handed neutrinos without Majorana mass term. In such a scenario, a neutrinophilic two Higgs doublet model is suggested to avoid a large hierarchy in the Yukawa couplings where only one Higgs doublet with small vacuum expectation value (VEV) has a Yukawa interaction among lepton doublets and right-handed neutrinos, giving the neutrino masses [1,2,3]. This kind of Higgs doublet model can be constructed by imposing a symmetry such as global U(1) symmetry [1, 3]. It is also interesting to consider the realization of a neutrinophilic two Higgs doublet model based on an exotic U(1) gauge symmetry such as $U(1)_{B-L}$. Then we consider an alternative $U(1)_{B-L}$ charge assignment for right-handed neutrinos [4,5,6,7], since the original $U(1)_{B-L}$ charge assignment is not suitable due to the universal $B-L$ charge for leptons including right-handed neutrinos.

In this paper, we construct a neutrinophilic two Higgs doublet model based on an alternative $U(1)_{B-L}$ gauge symmetry, which introduces three right-handed neutrinos $\nu _{R_1}$, $\nu _{R_2}$ and $\nu _{R_3}$ with $B-L$ charge $-4$, $-4$ and 5 to cancel gauge anomalies. We also assign $B-L$ charge $-3$ to one of the Higgs doublets, and the $B-L$ charged Higgs doublet only has Yukawa couplings among SM lepton doublets and right-handed neutrinos $\nu _{R_{1,2}}$. Thus a Dirac type neutrino mass matrix is obtained after electroweak symmetry breaking and the smallness of the neutrino mass can be explained by the small VEV of the $B-L$ charged Higgs doublet by choosing parameters in the scalar potential appropriately. Furthermore $\nu _{R_3}$ is stabilized by an accidental $Z_2$ symmetry due to the charge assignment and is a good dark matter (DM) candidate. Here we emphasize that three fermion contents inducing active neutrino mass and providing DM are required by anomaly cancellation condition, and our charge assignments for the fermions naturally guarantee the stability of DM, without inducing a further symmetry such as the discrete $Z_2$ symmetry. Then we discuss the phenomenology of the model such as the $Z'$ boson at collider experiments and DM physics. The constraints on the $Z'$ mass and $U(1)_{B-L}$ gauge coupling can be obtained by the data of a LEP experiment and the current LHC experiments. Taking into account the constraints, the DM relic density is estimated by calculating the annihilation process via $Z'$ exchange in the s-channel.

This paper is organized as follows. In Sect. 2, we present our model, and we formulate the neutral fermion sector, boson sector and lepton sector. In Sect. 3, we discuss the phenomenology of the model such as $Z'$ boson at collider experiments and dark matter physics. Finally we conclude and discuss results in Sect. 4.

Table 1 Field contents of bosons and fermions and their charge assignments under $SU(2)_L\times U(1)_Y\times U(1)_{B-L}$, where $a=1-3$ and $i=1,2$ are flavor indices

2 Model setup and particle contents

In this section, we introduce our model and discuss masses in the scalar sector and fermion sector. First of all, we introduce the gauged $U(1)_{B-L}$ symmetry, introducing non-trivial $B-L$ charge assignments of the right-handed neutrinos, $(-4,-4,5)$ for $(\nu _{R_1},\nu _{R_2}, X_{R})$ where we write the third right-handed neutrino as $X_R$. The first of the two right-handed neutrinos $(\nu _{R_1},\nu _{R_2})$ lead to Dirac type active neutrinos via a Dirac Yukawa term [3]. On the other hand the third right-handed neutrino $X_{R}$ becomes a Majorana fermion by itself after the $B-L$ symmetry breaking. Thus $X_{R}$ is a good DM candidate, which is stabilized by an accidental $Z_2$ symmetry due to the gauge symmetry.¹ To construct mass terms for the SM fermions appropriately, we adopt the neutrinophilic two Higgs doublet model, in which a new isospin doublet boson $\Phi$ with non-zero $B-L$ charge is introduced in order to induce the neutrino masses, while the other SM fermion masses are generated with the SM-like Higgs doublet H, which is the same as SM. In addition, we introduce isospin singlet scalars $\{ \varphi _{10}, \varphi _3 \}$ with non-zero $B-L$ charges denoted by subscripts; here $\varphi _{10}$ is required to give $X_{R}$ mass and $\varphi _{3}$ is necessary to avoid a massless Goldstone boson from the Higgs doublet $\Phi$. All the field contents and their assignments are summarized in Table 1. Under these framework, one finds the following renormalizable Lagrangian:

$$\begin{aligned} {}-{\mathcal {L}}_{L}= & {} (y_\ell )_{aa}{\bar{L}}_{L_{a}} e_{R_{a}} H + (y_{\nu })_{ai}{\bar{L}}_{L_{a}} {\tilde{\Phi }} \nu _{R_{i}} \nonumber \\&+f_X \bar{X}^c_{R} X_{R} \varphi _{10}^* + {\mathrm{c.c.}}, \end{aligned}$$

(2.1)

$$\begin{aligned} V= & {} -\mu _H^2 H^\dag H + \mu _\Phi ^2 \Phi ^\dag \Phi - \mu ^2_{10} \varphi _{10}^* \varphi _{10}^{}\nonumber \\&- \mu ^2_{3} \varphi _{3}^*\varphi _{3}^{} -( \mu \Phi ^\dagger H \varphi _3^* + h.c. ) \nonumber \\&+\lambda _1 (\Phi ^\dag \Phi )^2 +\lambda _2 (H^\dag H)^2 +\lambda _{\varphi _{10}^{}} (\varphi _{10}^* \varphi _{10}^{})^2 \nonumber \\&+\lambda _{\varphi _{3}} (\varphi _{3}^* \varphi _{3}^{})^2 \nonumber \\&+ \lambda _{3} (H^\dag H)(\Phi ^\dag \Phi ) +\lambda _{4} (H^\dag \Phi )(\Phi ^\dag H)\nonumber \\&+ \lambda _{H\varphi _{10}} (H^\dag H)(\varphi _{10}^*\varphi _{10}^{})\nonumber \\&+ \lambda _{\Phi \varphi _{10}} (\Phi ^\dag \Phi )(\varphi _{10}^*\varphi _{10}^{}) \nonumber \\&+ \lambda _{H\varphi _{3}} (H^\dag H)(\varphi _{3}^*\varphi _{3}^{}) + \lambda _{\Phi \varphi _{3}} (\Phi ^\dag \Phi )(\varphi _{3}^*\varphi _{3}^{}) \nonumber \\&+ \lambda _{\varphi _{3} \varphi _{10}} (\varphi _{3}^*\varphi _{3}^{})(\varphi _{10}^*\varphi _{10}^{}) \end{aligned}$$

(2.2)

where we assume the parameters in the potential are real, $\tilde{H} \equiv (i \sigma _2) H^*$ with $\sigma _2$ being the second Pauli matrix, a runs over 1–3, and i runs over 1–2.

2.1 Scalar sector

The scalar fields are parameterized as

$$\begin{aligned} H&=\left[ \begin{array}{c} w^+\\ \frac{v_H + h +i z}{\sqrt{2}} \end{array}\right] ,\quad \Phi =\left[ \begin{array}{c} \phi ^+\\ \frac{v_\phi + \phi _R + i \phi _I}{\sqrt{2}} \end{array}\right] ,\nonumber \\ \varphi _{Q}&= \frac{v_{Q} + \varphi _{QR} + iz'_Q}{\sqrt{2}}, \end{aligned}$$

(2.3)

where $Q = \{3, 10 \}$ indicates $B-L$ charges for singlet scalar fields, the lightest mass eigenstate after diagonalizing the matrix in the basis of $(w^\pm$ , $\phi ^\pm )$, which is massless, is absorbed by the SM $W^\pm$ boson, and two of the mass eigenstates in the CP-odd boson sector $(\phi _I, z, z'_{Q})$ are also absorbed by the SM Z boson and the $B-L$ gauge boson $Z'$, as Nambu–Goldstone (NG) bosons. The VEVs are obtained by applying the conditions $\partial V/ \partial v_Q =0$, $\partial V/ \partial v_H =0$ and $\partial V/ \partial v_\phi =0$ such that

$$\begin{aligned}&v_{10} \simeq \sqrt{\frac{\mu _{10}^2}{\lambda _{\varphi _{10}}}}, \quad v_3 \simeq \sqrt{\frac{2\mu _3^2 - \lambda _{H \varphi _{3}} v_{H}^2 - \lambda _{\varphi _3 \varphi _{10}} v_{10}^2 }{2\lambda _{\varphi _3}}}, \nonumber \\&v_H \simeq \sqrt{\frac{2\mu _H^2 - \lambda _{H \varphi _{3}} v_{3}^2 - \lambda _{H \varphi _{10}} v_{10}^2 }{2\lambda _H}} \nonumber \\&v_\phi \simeq \frac{\sqrt{2} \mu v_H v_3 }{2 \mu _\Phi ^2 + (\lambda _{3} + \lambda _{4})v_H^2 + \lambda _{\Phi \varphi _{3}} v_{3}^2+ \lambda _{\Phi \varphi _{10}} v_{10}^2 } \end{aligned}$$

(2.4)

where we assume the relation $v_{\phi }^2 \ll \{v_{H}^2, v_3^2, \mu _3^2 \} \ll \{v_{10}^2, \mu _{10}^2\}$. The small $v_\phi$ can be realized taking the trilinear coupling $\mu$ to be sufficiently small. Note that $z'_{10}$ is dominant component of NG boson which is absorbed by the $Z'$ boson since we consider the VEV of $\varphi _{10}$ is much larger than the others. After $\varphi _{10}$ developing a VEV at high energy scale, our scalar sector has the same structure as discussed in Ref. [3]. Then the CP-odd component of $\varphi _3$; $z'_{3}$, becomes a physical massless Goldstone boson (GB) due to a global symmetry in the scalar potential. However, the existence of this physical Goldstone boson does not cause serious problems in particle physics or cosmology since it does not couple to SM particles directly except for the Higgs boson whose couplings are well controlled by the parameters in the potential, and it decouples from the thermal bath in the early Universe. Here we discuss the condition that the GB decouples from the thermal bath at a sufficiently early Universe, following the discussion in Ref. [8]. Since scalar and gauge bosons which couple to the GB are heavy, their interactions with GB decouple at sufficiently high temperature. We thus focus on the interaction between GB and the SM fermions. The ratio between collision and expansion rates is roughly given by $R(T) \sim \lambda _{\varphi _{3} H}^2 m_\mathrm{f}^2 T^5 m_{\mathrm{pl}}/(m_{\varphi _{3R}}^4 m_\mathrm{h}^4 )$ [8] where we take Boltzmann constant as 1, $m_{\mathrm{pl}}$ is the Planck mass, and $m_\mathrm{f}$ denotes an SM fermion mass; the process GB GB $\leftrightarrow ff$ is induced by the Higgs–$\varphi _3$ mixing for $m_\mathrm{f} < T$. The decoupling occurs when $R(T) \sim 1$ and we obtain a decoupling temperature $T_\mathrm{d} \sim 0.42/\lambda _{\varphi _{3} H}^{2/5}$ GeV assuming $m_\mathrm{f} = m_\tau$. In such a case $\lambda _{\varphi _3 H} = 0.001$ gives $T_\mathrm{d} \sim 2.7$ GeV, which is consistent with the condition $T > m_\tau$. Thus if $\lambda _{\varphi _3 H} \lesssim 0.001$ the GB decouples from the thermal bath around O(1) GeV temperature and cosmology is not affected by the GB; note that SM fermions with $m_\mathrm{f} < m_\tau$ decouple earlier due to smaller couplings.

In our scenario, one finds that $v\equiv \sqrt{v_H^2+v_\phi ^2}\sim v_H$ where $v \simeq 246$ GeV since $v_\phi$ is expected to be tiny in order to generate the active neutrino masses. Thus the charged component $w^\pm$ in H is approximated as being the NG boson which is absorbed by the $W^\pm$ boson while the $\phi ^\pm$ from $\Phi$ is a physical charged Higgs boson. Similarly the CP-odd boson z is absorbed by the neutral SM gauge boson Z, while $\phi _I$ is a physical CP-odd Higgs. The masses of physical charged Higgs and CP-odd Higgs are approximately given by [3]

$$\begin{aligned} m_{\phi ^\pm }^2&\simeq \frac{v_2 (\sqrt{2} \mu v_3 - \lambda _4 v_1 v_2)}{2 v_1}, \end{aligned}$$

(2.5)

$$\begin{aligned} m_{\phi _I}^2&\simeq \frac{\mu v_2 v_3}{\sqrt{2} v_1}. \end{aligned}$$

(2.6)

The mass matrix for the CP-even scalars has $4 \times 4$ structure in a basis of $(h, \phi _R, \varphi _{3R}, \varphi _{10R})$. In our analysis, we omit details of the matrix assuming the SM Higgs is the lightest component among four physical CP-even scalar bosons. In addition, we assume mixing among SM Higgs and other CP-even scalars are small to avoid experimental constraints for simplicity. More details of the scalar sector can be found in Refs. [1, 3], and we focus on $Z'$ and DM physics in the analysis below.

2.2 Fermion sector

The masses for charged leptons are induced via $y_\ell$ after symmetry breaking, and active neutrino masses are also done via $y_\nu$ term where neutrinos are supposed to be Dirac type fermions. Their masses are symbolized by $m_{\ell _a}\equiv v_H y_{\ell _a}/\sqrt{2}$ and $m_{\nu _{ai}}\equiv v_\phi y_{\nu _{ai}}/\sqrt{2}$. Since the charged-lepton mass matrix is diagonal, the neutrino mixing matrix $V_{ab}$ has arisen from diagonalizing the neutrino mass matrix squared:

$$\begin{aligned} (M_\nu ^\mathrm{diag})^2 = (V^\dagger )_{aa'} \sum _{i=1-2}\left( m_{\nu _{a' i}}m_{\nu _{ib'}}^\dag \right) V_{b' b}, \end{aligned}$$

(2.7)

where V is the measured PMNS matrix by the neutrino oscillation data [9]. Notice here that one of the active neutrinos is massless due to the rank two matrix. Thus one can parametrize the neutrino mass matrix in terms of observables and arbitrary parameters in the following form:

$$\begin{aligned} m_{\nu _{a'i}} = V_{aa'} (M_\nu ^\mathrm{diag})_{a'a'} \mathcal{O}_{a' i} , \end{aligned}$$

(2.8)

where $\mathcal{O}_{a' i}$ is generally an arbitrary three by two matrix with complex values, satisfying $\mathcal{O}\mathcal{O}^\dag \ne 1_{3\times 3}$ and $\mathcal{O}^\dag \mathcal{O}=1_{2\times 2}$. However, since we have enough theoretical parameters, we simply reduce the parameterization of $\mathcal{O}$ for the normal hierarchy (NH) and the inverted hierarchy (IH) [10], which is in analogy of the case of the Majorana neutrino mass matrix:

$$\begin{aligned} O =\left[ \begin{array}{cc} 0 &{} 0\\ \cos z &{} -\sin z \\ \zeta \sin z &{} \zeta \cos z \\ \end{array}\right] , \quad O =\left[ \begin{array}{cc} \cos z &{} -\sin z \\ \zeta \sin z &{} \zeta \cos z \\ 0 &{} 0\\ \end{array}\right] , \end{aligned}$$

(2.9)

respectively, where $\zeta$ is complex number satisfying $|\zeta |=1$, and we parametrize z to be a real value with $z \in [0, 2\pi ]$ and $\zeta = e^{i \theta }$ with $\theta \in [0, 2\pi ]$. In our numerical analysis, we will use the global fit of the current neutrino oscillation data as the best fit values for NH and IH [9]:

$$\begin{aligned} \mathrm{NH}:\ s_{12}^2&=0.304,\quad s_{23}^2=0.452,\quad s_{13}^2=0.0218,\nonumber \\ \delta _{CP}&=\frac{306}{180}\pi ,\quad (m_{\nu _1},\ m_{\nu _2},\ m_{\nu _3})\approx (0, 8.66, 49.6)\ \mathrm{meV}, \end{aligned}$$

(2.10)

$$\begin{aligned} \mathrm{IH}:\ s_{12}^2&=0.304,\quad s_{23}^2=0.579,\quad s_{13}^2=0.0219,\nonumber \\ \delta _{CP}&=\frac{254}{180}\pi ,\quad (m_{\nu _1},\ m_{\nu _2},\ m_{\nu _3})\approx (49.5,50.2,0)\ \mathrm{meV}, \end{aligned}$$

(2.11)

where $s_{12,13,23}$ are the short-hand notations of $\sin \theta _{12,13,23}$ for the three mixing angles of V, while two Majorana phases are taken to be zero. We show some samples of the allowed regions for z and the mass scale of $m_{\nu }$ in Fig. 1, where the upper figures represent NH and the lower one IH, while the left figures shows the mass scale of $m_{\nu _{11}}$ in terms of z and the right ones the mass scale of $m_{\nu _{11}}$ and $m_{\nu _{22}}$. They suggest that the typical mass scale of $m_\nu$ is $10^{-12} \sim 10^{-11}$ GeV.² Note here that correlations between them seem to occur due to the manner of our parametrization. Since $(m_\nu )_{ai} = v_\phi y_{\nu _{ai}}/\sqrt{2}$, the order of the Yukawa coupling is $10^{-12\sim 11}/v_\phi$; when $v_\phi \sim \mathcal {O}(\mathrm{KeV})$ the order of the coupling is around $10^{-6}$, which is similar to the electron Yukawa coupling.

Fig. 1

The allowed regions to satisfy the neutrino oscillation data, where the upper figures represent NH and the lower one IH, while the left figures shows the mass scale of $m_{\nu _{11}}$ in terms of z and the right ones the mass scale of $m_{\nu _{11}}$ and $m_{\nu _{22}}$

The third right-handed neutrino obtains Majorana mass term from the term with $f_X$ after $\varphi _{10}$ developing the VEV is $\langle \varphi _{10} \rangle = v_{10}/\sqrt{2}$. The Majorana mass is simply given by

$$\begin{aligned} M_X = \frac{f_X}{\sqrt{2}} v_{10}. \end{aligned}$$

(2.12)

3 Phenomenology

In this section, we consider the phenomenology of our model focusing on the $Z'$ boson and the dark matter candidate.

3.1 $Z'$ boson

Here we consider constraints from collider experiments for the $Z'$ boson mass $m_{Z'}$ and the $U(1)_{B-L}$ gauge coupling $g_{BL}$. The gauge interactions of $Z'$ and fermions are given by

$$\begin{aligned} -\mathcal{L}&= 5g_{BL} \bar{X}\gamma ^\mu P_R X Z'_\mu + g_{BL} Q^f_{BL} \bar{f}_{\mathrm{SM}} \gamma ^\mu f_{\mathrm{SM}} Z'_\mu \nonumber \\&\quad - g_{BL} (\bar{\nu }_a \gamma ^\mu P_L\nu _a + 4 \bar{\nu }_i \gamma ^\mu P_R\nu _i) Z'_\mu , \end{aligned}$$

(3.1)

where $Q^f_{BL}$ is the charge of the $U(1)_{B-L}$ symmetry, and $f_{\mathrm{SM}}$ denotes all the electrically charged fermions in SM. Here $Z'$ mass is given by $m_{Z'} \simeq 10 g_{BL} v_{10}$ as we assume $v_{10} \gg \{v_{3}, v_{\phi } \}$.

Firstly, we discuss the constraint from the LEP experiment. Since $Z'$ couples to SM leptons, we obtain the following effective interactions considering $Z'$ to be sufficiently heavy:

$$\begin{aligned} L_{\mathrm{eff}} = \frac{1}{1+\delta _{e \ell '}} \frac{g_{BL}^2}{m_{Z'}^2} (\bar{e} \gamma ^\mu e)( \bar{\ell }' \gamma _\mu \ell ') \end{aligned}$$

(3.2)

where $\ell ' = e$, $\mu$ and $\tau$. In this case, we obtain constraints from the analysis for the process $e^+ e^- \rightarrow \ell '^+ \ell '^-$ with the data of measurement at LEP [11]:

$$\begin{aligned} \frac{m_{Z'}}{g_{BL}} \gtrsim 6.9\ \mathrm{TeV}. \end{aligned}$$

(3.3)

In the following analysis, we take into account the constraint.

We next discuss the $Z'$ production at LHC. In hadron collider experiments, the $Z'$ boson can be produced via the process $q \bar{q} \rightarrow Z'$ where q indicates SM quarks. Here we estimate the production cross section using CalcHEP [12] implementing the relevant interactions with the CTEQ6 parton distribution functions (PDFs) [13]. Then $Z'$ decays into $B-L$ charged particles where we only consider fermions assuming the masses of the scalar bosons are greater than $m_{Z'}/2$. The decay width is given by

$$\begin{aligned} \Gamma _{Z'}&= \frac{g_{BL}^2m_{Z'}}{12 \pi } \sum _{f} N_c^{f} C_\mathrm{f} |Q_{BL}^f|^2 \left( 1 + \frac{2 m_\mathrm{f}^2}{m_{Z'}^2} \right) \sqrt{1- \frac{4 m_\mathrm{f}^2}{m_{Z'}^2}} \nonumber \\&\simeq \frac{g_{BL}^2 m_{Z'}}{12 \pi } \left[ \frac{133}{6} + \frac{1}{3} \left( 1 + \frac{2 m_\mathrm{t}^2}{m_{Z'}^2} \right) \sqrt{1- \frac{4 m_\mathrm{t}^2}{m_{Z'}^2}}\right. \nonumber \\&\quad \left. + \frac{25}{2} \left( 1 + \frac{2 m_X^2}{m_{Z'}^2} \right) \sqrt{1- \frac{4 m_X^2}{m_{Z'}^2}} \right] , \end{aligned}$$

(3.4)

where f denotes any fermion in the model, $N_c^f$ is a color factor, and we used $m_{\mathrm{f}}/m_{Z'} \ll 1$ for fermions except for the top quark and X. The branching ratio for a mode $Z' \rightarrow f \bar{f}$ is given by

$$\begin{aligned} BR(Z' \rightarrow f \bar{f})\simeq & {} \frac{g_{BL}^2m_{Z'}}{12 \pi } N_c^{f} C_\mathrm{f} |Q_{BL}^f|^2 \left( 1 + \frac{2 m_\mathrm{f}^2}{m_{Z'}^2} \right) \nonumber \\&\times \sqrt{1- \frac{4 m_\mathrm{f}^2}{m_{Z'}^2}} \times \Gamma _{Z'}^{-1}. \end{aligned}$$

(3.5)

The branching ratios for charged-lepton modes are less than $\sim 0.05$ since $Z'$ dominantly decays into $\nu _{R_{1,2}}$ due to the charge assignment. In Fig. 2, we show the product of cross section and branching ratio, $\sigma (pp \rightarrow Z')\mathrm{BR}(Z' \rightarrow \ell ^+ \ell ^-)$ where $\ell = e$ and $\mu$, at $\sqrt{s} = 13$ TeV as a function of $m_{Z'}$ with $g_{BL} = \{0.1, 0.05, 0.01\}$, which is to be compared with the limit from the LHC data [14]. We thus find that $m_{Z'} \gtrsim 2.8$ TeV is required for $g_{BL} = 0.1$, while $m_{Z'} \simeq 500$ GeV is still allowed for smaller gauge coupling, $g_{BL} = 0.01$. Note that the constraint on the $Z'$ mass is weaker than that in the original $U(1)_{B-L}$ case [15] since our $Z'$ dominantly decays into light right-handed neutrinos $\nu _i$; it has larger charge than the other SM fermions. A greater parameter region will be tested by the data of the future LHC experiments with larger integrated luminosity.

Fig. 2

$\sigma (pp \rightarrow Z')BR(Z' \rightarrow \ell ^+ \ell ^-)$, $\ell = e$ and $\mu$, as a function of $m_{Z'}$ at $\sqrt{s} = 13$ TeV with $g_{BL}=0.1$, 0.05 and 0.01 as indicated in the plot. The red curve shows the upper limit from the ATLAS experiment [14]

3.2 Dark matter

In this subsection we discuss a dark matter candidate: $X_R$. Firstly, we assume that any contributions from the Higgs mediating interactions are negligibly small and DM annihilation processes are dominated by the gauge interaction with $Z'$; we thus can easily avoid the constraints from direct detection searches such as LUX [16], XENON1T [17], and PandaX-II [18, 19]. Here we discuss the condition for the Higgs portal interaction from direct detection constraints. The nucleon–DM interaction is induced by the Higgs portal interaction via mixing between the SM Higgs and $\varphi _{10}$. The relevant effective interaction is given by [3]

$$\begin{aligned} L_\mathrm{eff} = \sum _q \frac{f_X m_q \sin \theta \cos \theta }{2 \sqrt{2} v m_h^2} \bar{X} X \bar{q} q, \end{aligned}$$

(3.6)

where $\theta$ is the mixing angle for Higgs–$\varphi _{10}$ mixing, $m_q$ is a mass of quark q and we assumed the heavier scalar boson is much heavier than $m_h$. Then the X–N spin-independent scattering cross section is obtained:

$$\begin{aligned} \sigma _{XN} \simeq \frac{1}{2 \pi } \frac{\mu _{NX}^2 f_N^2 m_N^2 f_X^2 \sin ^2 \theta \cos ^2 \theta }{v^2 m_h^4} \end{aligned}$$

(3.7)

where $m_N$ is the nucleon mass, $\mu _{NX} = m_N m_X/(m_N+m_X)$ is the reduced mass of nucleon and DM, and $f_N$ is the effective coupling constant for the Higgs–nucleon interaction. For simplicity, we estimate the cross section applying $f_n \simeq 0.287$ for the neutron. Finally, we estimate the cross section as

$$\begin{aligned} \sigma _{Xn} \sim 4.2 \times 10^{-43} \left( \frac{100 \, \mathrm{GeV}}{m_X} \right) ^2 f_X^2 \sin ^2 \theta \cos ^2 \theta \ \mathrm{cm}^2. \end{aligned}$$

(3.8)

Therefore direct detection constraints can be satisfied with small mixing angle such as $\sin \theta \ll 0.1$ even if the coupling $f_X$ is $\mathcal {O}(1)$. We next consider direct detection via $Z'$ exchange. The relevant effective interaction between DM and the SM quarks is given by

$$\begin{aligned} \frac{g_{BL}^2}{m_{Z'}^2} \frac{5}{6} (\bar{X} \gamma ^\mu \gamma _5 X)(\bar{q} \gamma ^\mu q) \equiv \frac{(\bar{X} \gamma ^\mu \gamma _5 X)(\bar{q} \gamma ^\mu q)}{\Lambda _{Z'}^2}, \end{aligned}$$

(3.9)

where DM has only an axial vector current due to the Majorana property and we defined $\Lambda _{Z'} \equiv 6 m_{Z'}/(5 g_{BL})$. The operator in Eq. (3.9) induces the spin-dependent operator $\vec {s}^\bot _X \cdot \vec {q}$ and $\vec {s}_X \cdot (\vec {s}_N \times q)$ [20,21,22]; $\vec {q}$ is the transferred momentum, $s_{X(N)}$ is the spin operator of DM (nucleon) and $\bot$ indicates a direction perpendicular to the $\vec {q}$ direction. In Ref. [22], the lower limit of $\Lambda _{Z'}$ is given as $\sim 1$ TeV which is obtained by data from PandaX, LUX and XENON1T including spin-dependent direct detection results [18]. This constraint is much weaker than the constraint from collider search of $Z'$ as shown in Fig. 2. Thus, the constraint from direct detection is not stringent in our model.

Relic density: We have annihilation modes via the gauge interaction as $X \bar{X} \rightarrow Z' \rightarrow f \bar{f}$ to explain the relic density of DM: $\Omega h^2\approx 0.12$ [23], and their relevant Lagrangian in a basis of mass eigenstates is given in Eq. (3.1). Then the relic density of DM is estimated by [24]

$$\begin{aligned}&\Omega h^2 \approx \frac{1.07\times 10^9}{\sqrt{g_*(x_\mathrm{f})}M_{\mathrm{Pl}} J(x_\mathrm{f})[\mathrm{GeV}]}, \end{aligned}$$

(3.10)

where $g^*(x_\mathrm{f}\approx 25)$ is the number of degrees of freedom for relativistic particles at temperature $T_\mathrm{f} = M_X/x_\mathrm{f}$, $M_{\mathrm{Pl}}\approx 1.22\times 10^{19}$ GeV, and $J(x_\mathrm{f}) (\equiv \int _{x_\mathrm{f}}^\infty \mathrm{d}x \frac{\langle \sigma v_\mathrm{rel}\rangle }{x^2})$ is given by [25]

$$\begin{aligned} J(x_\mathrm{f})&=\int _{x_\mathrm{f}}^\infty \mathrm{d}x\left[ \frac{\int _{4M_X^2}^\infty \mathrm{d}s\sqrt{s-4 M_X^2} W(s) K_1\left( \frac{\sqrt{s}}{M_X} x\right) }{16 M_X^5 x [K_2(x)]^2}\right] ,\end{aligned}$$

(3.11)

$$\begin{aligned} W(s)&\approx \frac{4}{3\pi } (s-M_X^2) \left| \frac{5 g_{BL}^2 }{s-m_{Z'}^2+i m_{Z'} \Gamma _{Z'}}\right| ^2\nonumber \\&\quad \times \sum _f C_\mathrm{f}\sqrt{1-\frac{4 m_{\mathrm{f}}^2}{s}} (s+2 m^2_{\mathrm{f}})|Q_{BL}^{f}|^2, \end{aligned}$$

(3.12)

where we implicitly impose the kinematical constraint above, $C_\mathrm{f} = 1/2$ for neutrino pairs including two light right-handed neutrino $\nu _i$ in the second line of Eq. (3.1) otherwise $C_\mathrm{f} =1$, and the width of $Z'$ is given by Eq. (3.4).

In Fig. 3, we show the relic density in terms of $M_X$, where we fix the parameters $m_{Z'}=\{500, 1000 \}$ GeV and $g_{BL}=0.01$, which are allowed by the collider experiments. We find that the correct relic density can be obtained near the $Z'$ pole since we need resonant enhancement due to the small gauge coupling.

Fig. 3

The relic density of dark matter X as a function of $m_{X}$ where solid (dashed) lines correspond to $m_{Z'} = 500(1000)$ GeV and $g_{BL} = 0.01$ is applied as the allowed sampling point

4 Conclusion

In this paper, we have discussed a neutrinophilic two Higgs doublet model with alternative anomaly free $U(1)_{B-L}$ gauge symmetry under which the three right-handed neutrinos $\nu _{R_1}$, $\nu _{R_2}$ and $\nu _{R_3}$ have charges $Q_{B-L} = -4$, $-4$ and 5. The neutrinophilic structure is realized by assigning a non-zero $U(1)_{B-L}$ charge to one of the Higgs doublets due to the charge assignment for the right-handed neutrinos. Then the two right-handed neutrinos $\nu _{R_{1,2}}$ have a Yukawa coupling with the SM left-handed neutrinos and we obtain a $2 \times 3$ Dirac neutrino mass matrix predicting one massless neutrino. In addition, $X_R \equiv \nu _{R_3}$ can be a good dark matter candidate since it is stabilized by an accidental $Z_2$ symmetry in our model due to the charge assignment of $U(1)_{B-L}$. In the scalar sector, we have introduced two SM singlet scalar fields $\varphi _{10}$ and $\varphi _3$ with $U(1)_{B-L}$ charge 10 and 3, respectively, where the former one is introduced to break $U(1)_{B-L}$, giving the $Z'$ boson mass, and the latter one is introduced to avoid a massless Goldstone boson from Higgs doublet. Then the CP-odd component of $\varphi _3$ becomes a physical Goldstone boson which is harmless since it does not directly couple to SM particles except for the Higgs boson and the coupling to the Higgs boson can be controlled by the parameters in the scalar potential.

Then we have discussed the phenomenology of the model such as $Z'$ boson production at collider experiments and dark matter physics. Our $Z'$ can be produced at LHC and can decay into SM leptons. We thus have estimated the cross section and branching ratios for $Z' \rightarrow \ell ^+ \ell ^-$ and compared the resulting values of the product of the cross section and branching ratio with the current LHC constraint. We find that $Z'$ should be heavier than $\sim 2.8$ TeV for a gauge coupling $g_{BL} = 0.1$, while $m_{Z'} \simeq 500$ GeV is still allowed for a smaller gauge coupling $g_{BL} = 0.01$. We have found that the constraint on $Z'$ mass is weaker than that in original $U(1)_{B-L}$ case, since our $Z'$ dominantly decays into light right-handed neutrinos $\nu _i$; it has larger charge than the other SM fermions. We have also estimated the relic density of dark matter, which is determined by the thermally averaged cross section of the processes $XX \rightarrow Z' \rightarrow f f$ where f is for any of the fermions in the model. We have shown that the observed relic density can be obtained with $Z'$ mass and $g_{BL}$ allowed by the collider constraints. Our model can be further tested in the future by both collider and dark matter experiments.