Predictive scotogenic model with flavor dependent symmetry
Abstract
In this paper, we propose a viable approach to realise two texturezeros in the scotogenic model with flavor dependent \(U(1)_{B2L_\alpha L_\beta }\) gauge symmetry. These models are extended by two righthanded singlets \(N_{Ri}\) and two inert scalar doublets \(\eta _{i}\), which are odd under the dark \(Z_2\) symmetry. Among all the six constructed textures, texture \(A_1\) and \(A_2\) are the only two allowed by current experimental limits. Then choosing texture \(A_1\) derived from \(U(1)_{B2L_eL_\tau }\), we perform a detail analysis on the corresponding phenomenology such as predictions of neutrino mixing parameters, lepton flavor violation, dark matter and collider signatures. One distinct nature of such model is that the structure of Yukawa coupling \({\bar{L}}{\tilde{\eta }}N_R\) is fixed by neutrino oscillation data, and can be further tested by measuring the branching ratios of charged scalars \(\eta _{1,2}^\pm \).
1 Introduction
It is well known that the standard model (SM) needs extensions to accommodate two missing spices: the tiny but nozero neutrino masses and the cosmological dark matter (DM) candidates. One way of incorporating above two issues in a unified framework is the scotogenic model [1, 2, 3], where neutrinos are radiatively generated and the DM field serves as intermediate messengers propagating inside the loop diagram. With all new particles around TeV scale, the scotogenic model leads to testable phenomenologies [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Therefore, viable models are extensively studies in recent years [32].
On the other hand, the understanding of the leptonic flavor structure is still one of the major open questions in particle physics. The consensus is that the leptonic mass texture is tightly restricted under the present experimental data. An attractive approach is to consider two texturezeros in neutrino mass matrix (\(M_{\nu }\)) so that the number of parameters in the Lagrangian is reduced [33]. The phenomenological analysis of two texturezeros models have been studied in Refs. [34, 35]. Among fifteen logically patterns, seven of them are compatible to the lowenergy experimental data.
On the theoretical side, the simplest way of realizing texturezeros is to impose the discrete \(Z_{N}\) flavor symmetry [36]. However, it might be more appealing to adopt gauge symmetries instead of discrete ones, because the latter may be treated as the residual of U(1) gauge symmetry. It is noted that one can not set any restriction on lepton mass matrix by means of fields with flavor universal charges. Thus the flavor dependent U(1) gauge symmetry is the reasonable choice. Along this thought of idea, specific models are considered in the context of seesaw mechanisms. In Ref. [37], the two texturezeros are realized based on the anomalyfree \(U(1)_{X}\) gauge symmetry with \(X\equiv B\sum x_{\alpha }L_{\alpha }(\alpha =e,\mu ,\tau )\) being the linear combination of baryon number B and the lepton numbers \(L_{\alpha }\) per family. In Ref. [38], more solutions are found in the typeI and/or III seesaw framework.
It is then natural to ask, if predictive texturezeros in \(M_{\nu }\) can be realized in the scotogenic scenario and several attempts have been made in this direction. For example, one texturezero is recently considered in Ref. [39]. Texture \(B_1\)\(B_4\) have been discussed in a modelindependent way in Ref. [40]. Texture C is obtained by introducing \(U(1)_{L_{\mu }L_{\tau }}\) gauge symmetry [41, 42, 43, 44]. Texture \(B_2\) is realised with \(U(1)_{L_e+L_\mu L_\tau }\) gauge symmetry in Ref. [45]. If the quark flavor is also flavor dependent, e.g., \(U(1)_{xB_3xL_eL_\mu +L_\tau }\), then one can further interpret the \(R_{K}\) anomaly with texture \(A_1\) [46]. Other viable two texturezeros are systematically realised in Ref. [47] by considering the \(U(1)_{B2L_\alpha L_\beta }\) gauge symmetry with three righthanded singlets. In this paper, we provide another viable approach. Under the same flavor dependent \(U(1)_{B2L_\alpha L_\beta }\) gauge symmetry, we introduce only two righthanded singlets but two inert scalars, leading to different texturezeros compared with Ref. [47]. In aspect of predicted phenomenology, the texture \(B_1\) considered in Ref. [47] is marginally allowed by current Planck result for \(\sum m_i<0.12\) eV [48], we thus consider texture \(A_1\) with latest neutrino oscillation data [49] as the benchmark model. In this case, the gauge symmetry is \(U(1)_{B2L_eL_\tau }\) in our approach.
2 The model setup
2.1 Classic scotogenic model
Particle content and corresponding charge assignments
Group  Lepton fields  Scalar fields  

\(L_\alpha \)  \(\ell _{\alpha R}\)  \(L_\beta \)  \(\ell _{\beta R}\)  \(L_\gamma \)  \(\ell _{\gamma R}\)  \(N_{R1}\)  \(N_{R2}\)  \(\Phi \)  \(\eta _{1}\)  \(\eta _{2}\)  \(S_1\)  \(S_2\)  
\(SU(2)_L\)  2  1  2  1  2  1  0  0  2  2  2  1  1 
\(U(1)_{Y}\)  \(\frac{1}{2}\)  \(1\)  \(\frac{1}{2}\)  \(1\)  \(\frac{1}{2}\)  \(1\)  1  1  \(\frac{1}{2}\)  \(\frac{1}{2}\)  \(\frac{1}{2}\)  0  0 
\(Z_2\)  \(+\)  \(+\)  \(+\)  \(+\)  \(+\)  \(+\)  −  −  \(+\)  −  −  \(+\)  \(+\) 
\(U(1)_{B2L_\alpha L_\beta }\)  \(2\)  \(2\)  \(1\)  \(1\)  0  0  \(1\)  \(2\)  0  \(1\)  0  2  3 
2.2 Two texturezeros in scotogenic model
Two texturezeros and corresponding \(U(1)_{B2L_\alpha L_\beta }\) symmetry. Here, \(\times \) denotes a nonzero matrix element
Texture of \(M_{\nu }\)  Group  Texture of \(M_{\nu }\)  Group  Status 

\(A_1: \left( \begin{array}{l@{\quad }l@{\quad }l} 0&{}0&{}\times \\ 0&{}\times &{}\times \\ \times &{}\times &{}\times \end{array}\right) \)  \(U(1)_{B2L_{e}L_{\tau }}\)  \(A_2: \left( \begin{array}{l@{\quad }l@{\quad }l} 0&{}\times &{}0\\ \times &{}\times &{}\times \\ 0&{}\times &{}\times \end{array}\right) \)  \(U(1)_{B2L_{e}L_{\mu }}\)  Allowed 
\(B_3: \left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}0&{}\times \\ 0&{}0&{}\times \\ \times &{}\times &{}\times \end{array}\right) \)  \(U(1)_{B2L_{\mu }L_{\tau }}\)  \(B_4: \left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}\times &{}0\\ \times &{}\times &{}\times \\ 0&{}\times &{}0 \end{array}\right) \)  \(U(1)_{B2L_{\tau }L_{\mu }}\)  Marginally allowed 
\(D_1: \left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}\times &{}\times \\ \times &{}0&{}0\\ \times &{}0&{}\times \end{array}\right) \)  \(U(1)_{B2L_{\mu }L_{e}}\)  \(D_2: \left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}\times &{}\times \\ \times &{}\times &{}0\\ \times &{}0&{}0 \end{array}\right) \)  \(U(1)_{B2L_{\tau }L_{e}}\)  Excluded 
Same as 2, but for one texturezero
Texture of \(M_{\nu }\)  Group  Texture of \(M_{\nu }\)  Group  Status texture of \(M_{\nu }\)  Group 

\(\left( \begin{array}{l@{\quad }l@{\quad }l} 0&{}\times &{}\times \\ \times &{}\times &{}\times \\ \times &{}\times &{}\times \end{array}\right) \)  \(U(1)_{B2L_{e}L_{\mu ,\tau }}\)  \(\left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}\times &{}\times \\ \times &{}0&{}\times \\ \times &{}\times &{}\times \end{array}\right) \)  \(U(1)_{B2L_{\mu }L_{e,\tau }}\)  \(\left( \begin{array}{l@{\quad }l@{\quad }l} \times &{}\times &{}\times \\ \times &{}\times &{}\times \\ \times &{}\times &{}0 \end{array}\right) \)  \(U(1)_{B2L_{\tau }L_{e,\mu }}\) 
Note if \(\lambda _1\ne 0\), hence mixing between \(\eta _1\) and \(\eta _2\) exists, then new contributions as \(hM_Nf^T+fM_Nh^T\) to neutrino masses are possible. In this way, the resulting neutrino mass matrix will only have one texturezero [50], thus less constrained and less predictive than the two texturezero. The obtained one texturezeros are presented in Table 3, which are all allowed by neutrino oscillation data [50]. Since two different gauge symmetry lead to same one texturezero, it is then not possible to distinguish them only by precise neutrino oscillation measurements.
3 Phenomenology
3.1 Neutrino mixing
In Fig. 2, we show the scanning results of texture \(A_1\). It is worth to note that the best fit value of neutrino oscillation parameters by global analysis [49] is only marginally consistent with predictions of texture \(A_1\), which is clearly seen in Fig. 2a. From Fig. 2b, we obtain that \(m_1\sim 0.007\) eV, \(m_2\sim 0.01\) eV, and \(m_3\approx \sqrt{\Delta m^2}\sim 0.05\) eV. The resulting sum of neutrino mass is then \(\sum m_i\sim 0.07\) eV, thus it satisfies the bound from cosmology, i.e., \(\sum m_i<0.12\) eV [48, 54]. The Dirac phase should fall in the range \(\delta \in [0.75\pi , 1.77\pi ]\), meanwhile Fig. 2c, d indicate that \(\rho \approx \frac{\delta }{2}\) and \(\sigma \approx \frac{\delta }{2}\frac{\pi }{2}\).
3.2 Lepton flavor violation
Predictions for \(\Delta a_\mu \). In this figures, we have fix \(M_{N_1}=200\) GeV
3.3 Dark matter
In this work, \(N_1\) is the viable DM candidate. In the original scotogenic model [1], the possible annihilation channels are \(N_1 N_1 \rightarrow \ell ^+\ell ^,{\bar{\nu }}\nu \) via the Yukawa interaction \({\bar{L}}_\ell {\tilde{\eta }} N_{1}\) [61]. However, such annihilation channels are tightly constrained by nonobservation of LFV [51]. Thanks to relative loose constraints from \(\tau \) decays, the scanning results of Ref. [51] suggested that \(N_1\) should have a large coupling to \(L_\tau \). Thus, the dominant annihilation channel is \(\tau ^+\tau ^\) and \({\bar{\nu }}_\tau \nu _\tau \) with \(M_{N_1}\lesssim 3\) TeV.
Predicted relic density as a function of \(m_\eta \), where we have fix \(M_{N_1}=200\) GeV. The green line corresponds to the observed relic density \(\Omega h^2=0.120\pm 0.001\) [48]
Combined results for the Yukawaportal DM. Left pattern: in the \(h_{\mu 1}\)\(M_\eta \) plane; right pattern: in the \(M_{N_1}\)\(M_\eta \) plane. The green lines satisfy the condition for correct relic density. And the blue regions are excluded by LHC direct search, which will be discussed in Sect. 3.4
Branching ratios of scalar singlet \(H_1\) for \(\sin \alpha =0.1\) (left) and \(\sin \alpha =0.02\) (right). In this figures, we have also fix \(M_{N_1}=200\) GeV and \(v_S=10\) TeV. Note in left pattern, BR\((H_1\rightarrow N_1 N_1)\) is less than 0.01, thus is not shown in the plot
In Fig. 7, we show the combined results from LFV, \(\Delta a_\mu \), relic density and LHC search. In left pattern of Fig. 7, it indicates that for \(M_{N_1}=200\) GeV, the only exclusion region is from LHC search. Hence, either \(M_\eta \lesssim 300\) GeV with \(h_{\mu 1}\lesssim 0.9\) or \(M_\eta \gtrsim 500\) GeV with \(h_{\mu 1}\gtrsim 1.3\) is required. In right pattern of Fig. 7, two benchmark value \(h_{\mu 1}=1.0,1.5\) are chosen to illustrate. For \(h_{\mu 1}=1.0\), we have \(250~\text {GeV}\lesssim M_{N_1}\lesssim M_\eta \sim 400\) GeV. Therefore, the only viable region is \(M_{N_1}\sim M_{\eta }\lesssim 400\) GeV for \(h_{\mu 1}\lesssim 1\). Meanwhile for \(h_{\mu 1}=1.5\), \(M_{N_1}\gtrsim 120\) GeV with \(M_\eta \gtrsim 520\) GeV is able to escape LHC limit.
3.4 Collider signature
Decay branching ratio of \(U(1)_{B2L_eL_\tau }\) gauge boson \(Z'\), where we have show the lepton flavor individually
\(q{\bar{q}}\)  \(e^+e^\)  \(\mu ^+\mu ^\)  \(\tau ^+\tau ^\)  \(\nu \nu \)  NN  \(H_1H_1\) 

0.154  0.308  0  0.077  0.192  0.192  0.077 
Left pattern: predicted cross section ratios in \(U(1)_{B2L_eL_\tau }\) and corresponding limit from LHC. Right pattern: allowed parameter space in the \(g'\)\(M_{Z'}\) plane
Branching ratios of charge scalar \(\eta _{1,2}^\pm \)
Final state  \(e^\pm N_1\)  \(\mu ^\pm N_1\)  \(\tau ^\pm N_1\)  \(e^\pm N_2\)  \(\mu ^\pm N_2\)  \(\tau ^\pm N_2\) 

\(\eta ^\pm _1\)  0.000  0.465  0.099  0.000  0.178  0.258 
\(\eta ^\pm _2\)  0.043  0.000  0.611  0.113  0.000  0.233 
4 Conclusion
The scotogenic model is an elegant pathway to explain the origin of neutrino mass and dark matter. Meanwhile, texturezeros in neutrino mass matrix provide a promising way to under stand the leptonic flavor structure. Therefore, it is appealing to connect the scotogenic model with texturezeros. In this paper, we propose a viable approach to realise two texturezeros in the scotogenic model with flavor dependent \(U(1)_{B2L_\alpha L_\beta }\) gauge symmetry. These models are extended by two righthanded singlets \(N_{Ri}\) and two inert scalar doublets \(\eta _{i}\), which are odd under the dark \(Z_2\) symmetry. Six kinds of texturezeros are realised in our approach, i.e., texture \(A_1\), \(A_2\), \(B_3\), \(B_4\), \(D_1\) and \(D_2\). Among all the six texturezeros, we find that texture \(A_1\) and \(A_2\) are allowed by current experimental limits, while texture \(B_3\) and \(B_4\) are marginally allowed. Besides, texture \(D_1\) and \(D_2\) are already excluded by neutrino oscillation data.

The texture \(A_1\) predicts vanishing neutrinoless double beta decay rate. And only normal neutrino mass hierarchy is allowed. It predicts \(m_1\sim 0.007\) eV, \(m_2\sim 0.01\) eV, and \(m_3\approx \sqrt{\Delta m^2}\sim 0.05\) eV, then \(\sum m_i\sim 0.07\) eV. There are also strong correlation between the Dirac and Majorana phases, i.e., \(\rho \approx \frac{\delta }{2}\) and \(\sigma \approx \frac{\delta }{2}\frac{\pi }{2}\).
 The ratios of corresponding Yukawa couplings are also predicted by neutrino oscillation data, e.g.,$$\begin{aligned} \frac{h_{\tau 2}}{h_{\mu 1}}:\frac{f_{\tau 1}}{h_{\mu 1}} :\frac{f_{e2}}{h_{\mu 1}}\simeq 0.745:0.933:0.401. \end{aligned}$$

Due to specific Yukawa structure, the LFV process \(\mu \rightarrow e\gamma \) is missing at oneloop level. Meanwhile, large cancellations are possible for \(\tau \rightarrow \mu \gamma \) and \(\tau \rightarrow e\gamma \) with degenerate righthanded singlets. More stringent constraint comes from muon anomalous magnetic moment \(\Delta a_\mu \). Although \({\mathcal {O}}(1)\) Yukawa couplings are easily to avoid such limit.

Satisfying all constraints, correct relic density of dark matter \(N_1\) is achieved for \(M_{N_1}\lesssim M_{\eta }<500\) GeV with \(h_{\mu 1}\lesssim 1\) or \(M_\eta >500\) GeV with \(h_{\mu 1}>1\).As for direct detection, we have shown that the predicted spinindependent DMnucleon cross section \(\sigma ^\text {SI}\) with \(\sin \alpha =0.1\) satisfies the current XENON1T limit, but is within future reach of LZ.

The Majoron J contributes to invisible decay of SM Higgs. The additional scalar singlet \(H_1\) can be probe in the channel \(gg\rightarrow H_1\rightarrow W^+W^, ZZ\) at LHC. Decays of charged scalars \(\eta _{1,2}^\pm \) lead to Open image in new window signature. Note that the corresponding branching ratios are also correlated with neutrino oscillation parameters.
 The neutral gauge boson \(Z'\) is promising via the dielectron signature \(pp\rightarrow Z' \rightarrow e^+e^\). Its \(B2L_eL_\tau \) nature can be confirmed by$$\begin{aligned}&\text {BR}(Z'\rightarrow b{\bar{b}}):\text {BR}(Z'\rightarrow e^+e^):\text {BR}(Z'\rightarrow \mu ^+\mu ^)\\&\quad :\text {BR}(Z'\rightarrow \tau ^+\tau ^)=\frac{1}{3}:4:0:1, \end{aligned}$$
Notes
Acknowledgements
The work of Weijian Wang is supported by National Natural Science Foundation of China under Grant Numbers 11505062, Special Fund of Theoretical Physics under Grant Numbers 11447117 and Fundamental Research Funds for the Central Universities under Grant Numbers 2014ZD42. The work of ZhiLong Han is supported by National Natural Science Foundation of China under Grant Nos. 11805081 and 11605075, Natural Science Foundation of Shandong Province under Grant Nos. ZR2019QA021, ZR2018MA047, ZR2017JL006 and ZR2014AM016.
References
 1.E. Ma, Phys. Rev. D 73, 077301 (2006). arXiv:hepph/0601225 ADSCrossRefGoogle Scholar
 2.L.M. Krauss, S. Nasri, M. Trodden, Phys. Rev. D 67, 085002 (2003). arXiv:hepph/0210389 ADSCrossRefGoogle Scholar
 3.M. Aoki, S. Kanemura, O. Seto, Phys. Rev. Lett. 102, 051805 (2009). arXiv:0807.0361 [hepph]ADSCrossRefGoogle Scholar
 4.E. Ma, Mod. Phys. Lett. A 21, 1777 (2006). arXiv:hepph/0605180 ADSCrossRefGoogle Scholar
 5.T. Hambye, K. Kannike, E. Ma, M. Raidal, Phys. Rev. D 75, 095003 (2007). arXiv:hepph/0609228 ADSCrossRefGoogle Scholar
 6.D. Aristizabal Sierra, J. Kubo, D. Restrepo, D. Suematsu, O. Zapata, Phys. Rev. D 79, 013011 (2009). arXiv:0808.3340 [hepph]
 7.D. Suematsu, T. Toma, T. Yoshida, Phys. Rev. D 79, 093004 (2009). arXiv:0903.0287 [hepph]ADSCrossRefGoogle Scholar
 8.D. Schmidt, T. Schwetz, T. Toma, Phys. Rev. D 85, 073009 (2012). arXiv:1201.0906 [hepph]ADSCrossRefGoogle Scholar
 9.R. Bouchand, A. Merle, JHEP 1207, 084 (2012). arXiv:1205.0008 [hepph]ADSCrossRefGoogle Scholar
 10.A. Merle, M. Platscher, JHEP 1511, 148 (2015). arXiv:1507.06314 [hepph]ADSCrossRefGoogle Scholar
 11.E. Ma, A. Natale, A. Rashed, Int. J. Mod. Phys. A 27, 1250134 (2012). arXiv:1206.1570 [hepph]ADSCrossRefGoogle Scholar
 12.M. Klasen, C.E. Yaguna, J.D. RuizAlvarez, D. Restrepo, O. Zapata, JCAP 1304, 044 (2013). arXiv:1302.5298 [hepph]ADSCrossRefGoogle Scholar
 13.S.Y. Ho, J. Tandean, Phys. Rev. D 87, 095015 (2013). arXiv:1303.5700 [hepph]ADSCrossRefGoogle Scholar
 14.A. Ahriche, S. Nasri, JCAP 1307, 035 (2013). arXiv:1304.2055 [hepph]ADSCrossRefGoogle Scholar
 15.M. Chekkal, A. Ahriche, A.B. Hammou, S. Nasri, Phys. Rev. D 95(9), 095025 (2017). arXiv:1702.04399 [hepph]
 16.K.P. Modak, JHEP 1503, 064 (2015). arXiv:1404.3676 [hepph]ADSCrossRefGoogle Scholar
 17.E. Molinaro, C.E. Yaguna, O. Zapata, JCAP 1407, 015 (2014). arXiv:1405.1259 [hepph]ADSCrossRefGoogle Scholar
 18.G. Faisel, S.Y. Ho, J. Tandean, Phys. Lett. B 738, 380 (2014). arXiv:1408.5887 [hepph]ADSCrossRefGoogle Scholar
 19.A. Merle, M. Platscher, Phys. Rev. D 92(9), 095002 (2015). arXiv:1502.03098 [hepph]ADSCrossRefGoogle Scholar
 20.A. Ahriche, K.L. McDonald, S. Nasri, JHEP 1602, 038 (2016). arXiv:1508.02607 [hepph]ADSCrossRefGoogle Scholar
 21.A. Ahriche, K.L. McDonald, S. Nasri, JHEP 1606, 182 (2016). arXiv:1604.05569 [hepph]ADSCrossRefGoogle Scholar
 22.A. Ahriche, A. Manning, K.L. McDonald, S. Nasri, Phys. Rev. D 94(5), 053005 (2016). arXiv:1604.05995 [hepph]
 23.M. Lindner, M. Platscher, C.E. Yaguna, A. Merle, Phys. Rev. D 94(11), 115027 (2016). arXiv:1608.00577 [hepph]ADSCrossRefGoogle Scholar
 24.A.G. Hessler, A. Ibarra, E. Molinaro, S. Vogl, JHEP 1701, 100 (2017). arXiv:1611.09540 [hepph]ADSCrossRefGoogle Scholar
 25.D. Borah, A. Gupta, Phys. Rev. D 96(11), 115012 (2017). arXiv:1706.05034 [hepph]ADSCrossRefGoogle Scholar
 26.A. Abada, T. Toma, JHEP 1804, 030 (2018). arXiv:1802.00007 [hepph]CrossRefGoogle Scholar
 27.T. Hugle, M. Platscher, K. Schmitz, Phys. Rev. D 98(2), 023020 (2018). arXiv:1804.09660 [hepph]ADSCrossRefGoogle Scholar
 28.S. Baumholzer, V. Brdar, P. Schwaller, JHEP 1808, 067 (2018). arXiv:1806.06864 [hepph]ADSCrossRefGoogle Scholar
 29.D. Borah, P.S.B. Dev, A. Kumar, arXiv:1810.03645 [hepph]
 30.L. Bian, X. Liu, arXiv:1811.03279 [hepph]
 31.E. Ma, I. Picek, B. Radovčič, Phys. Lett. B 726, 744 (2013). arXiv:1308.5313 [hepph]ADSCrossRefGoogle Scholar
 32.Y. Cai, J. HerreroGarcía, M.A. Schmidt, A. Vicente, R.R. Volkas, Front. Phys. 5, 63 (2017). arXiv:1706.08524 [hepph]CrossRefGoogle Scholar
 33.P.H. Frampton, S.L. Glashow, D. Marfatia, Phys. Lett. B 536, 79 (2002). arXiv:hepph/0201008]ADSCrossRefGoogle Scholar
 34.H. Fritzsch, Z.Z. Xing, JHEP 1109, 083 (2011). arXiv:1108.4534 [hepph]
 35.J. Alcaide, J. Salvado, A. Santamaria, JHEP 1807, 164 (2018). arXiv:1806.06785 [hepph]ADSCrossRefGoogle Scholar
 36.W. Grimus, A.S. Joshipura, L. Lavoura, M. Tanimoto, Eur. Phys. J. C 36, 227 (2004). arXiv:hepph/0405016 ADSCrossRefGoogle Scholar
 37.T. Araki, J. Heeck, J. Kubo, JHEP 1207, 083 (2012). arXiv:1203.4951 [hepph]ADSCrossRefGoogle Scholar
 38.L.M. Cebola, D. EmmanuelCosta, R. Gonzalez Felipe, Phys. Rev. D 88(11), 116008 (2013). arXiv:1309.1709 [hepph]
 39.T. Kitabayashi, Phys. Rev. D 98(8), 083011 (2018). arXiv:1808.01060 [hepph]ADSCrossRefGoogle Scholar
 40.T. Kitabayashi, S. Ohkawa, M. Yasuè, Int. J. Mod. Phys. A 32(32), 1750186 (2017). arXiv:1703.09417 [hepph]ADSCrossRefGoogle Scholar
 41.S. Baek, H. Okada, K. Yagyu, JHEP 1504, 049 (2015). arXiv:1501.01530 [hepph]ADSCrossRefGoogle Scholar
 42.S. Baek, Phys. Lett. B 756, 1 (2016). arXiv:1510.02168 [hepph]ADSCrossRefGoogle Scholar
 43.S. Lee, T. Nomura, H. Okada, Nucl. Phys. B 931, 179 (2018). arXiv:1702.03733 [hepph]ADSCrossRefGoogle Scholar
 44.K. Asai, K. Hamaguchi, N. Nagata, S.Y. Tseng, K. Tsumura, arXiv:1811.07571 [hepph]
 45.T. Nomura, H. Okada, Phys. Dark Univ. 21, 90 (2018). arXiv:1712.00941 [hepph]CrossRefGoogle Scholar
 46.P. Ko, T. Nomura, H. Okada, Phys. Lett. B 772, 547 (2017). arXiv:1701.05788 [hepph]ADSMathSciNetCrossRefGoogle Scholar
 47.T. Nomura, H. Okada, arXiv:1806.09957 [hepph]
 48.N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astroph.CO]
 49.I. Esteban, M.C. GonzalezGarcia, A. HernandezCabezudo, M. Maltoni, T. Schwetz, arXiv:1811.05487 [hepph]
 50.E.I. Lashin, N. Chamoun, Phys. Rev. D 85, 113011 (2012). arXiv:1108.4010 [hepph]ADSCrossRefGoogle Scholar
 51.A. Vicente, C.E. Yaguna, JHEP 1502, 144 (2015). arXiv:1412.2545 [hepph]ADSCrossRefGoogle Scholar
 52.S.M. Bilenky, C. Giunti, W. Grimus, B. Kayser, S.T. Petcov, Phys. Lett. B 465, 193 (1999). arXiv:hepph/9907234]ADSCrossRefGoogle Scholar
 53.F. Vissani, JHEP 9906, 022 (1999). arXiv:hepph/9906525]ADSCrossRefGoogle Scholar
 54.S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, M. Lattanzi, Phys. Rev. D 96(12), 123503 (2017). arXiv:1701.08172 [astroph.CO]ADSCrossRefGoogle Scholar
 55.T. Toma, A. Vicente, JHEP 1401, 160 (2014). arXiv:1312.2840 [hepph]ADSCrossRefGoogle Scholar
 56.R. Ding, Z.L. Han, Y. Liao, H.J. Liu, J.Y. Liu, Phys. Rev. D 89(11), 115024 (2014). arXiv:1403.2040 [hepph]ADSCrossRefGoogle Scholar
 57.B. Aubert et al., BaBar Collaboration, Phys. Rev. Lett. 104, 021802 (2010). arXiv:0908.2381 [hepex]
 58.K. Hayasaka [Belle and BelleII Collaborations], J. Phys. Conf. Ser. 408, 012069 (2013)Google Scholar
 59.M. Lindner, M. Platscher, F.S. Queiroz, Phys. Rept. 731, 1 (2018). arXiv:1610.06587 [hepph]ADSCrossRefGoogle Scholar
 60.T. Blum, A. Denig, I. Logashenko, E. de Rafael, B.L. Roberts, T. Teubner, G. Venanzoni, arXiv:1311.2198 [hepph]
 61.J. Kubo, E. Ma, D. Suematsu, Phys. Lett. B 642, 18 (2006). arXiv: hepph/0604114 ADSCrossRefGoogle Scholar
 62.T. Li, W. Chao, Nucl. Phys. B 843, 396 (2011). arXiv:1004.0296 [hepph]ADSCrossRefGoogle Scholar
 63.G. Bertone, D. Hooper, J. Silk, Phys. Rept. 405, 279 (2005). arXiv:hepph/0404175]ADSCrossRefGoogle Scholar
 64.N. Okada, O. Seto, Phys. Rev. D 82, 023507 (2010). arXiv:1002.2525 [hepph]ADSCrossRefGoogle Scholar
 65.S. Kanemura, O. Seto, T. Shimomura, Phys. Rev. D 84, 016004 (2011). arXiv:1101.5713 [hepph]ADSCrossRefGoogle Scholar
 66.N. Okada, Y. Orikasa, Phys. Rev. D 85, 115006 (2012). arXiv:1202.1405 [hepph]ADSCrossRefGoogle Scholar
 67.W. Wang, Z.L. Han, Phys. Rev. D 92, 095001 (2015). arXiv:1508.00706 [hepph]ADSCrossRefGoogle Scholar
 68.N. Okada, S. Okada, Phys. Rev. D 93(7), 075003 (2016). arXiv:1601.07526 [hepph]ADSCrossRefGoogle Scholar
 69.N. Okada, S. Okada, D. Raut, Phys. Lett. B 780, 422 (2018). arXiv:1712.05290 [hepph]ADSCrossRefGoogle Scholar
 70.Z.L. Han, W. Wang, R. Ding, Eur. Phys. J. C 78(3), 216 (2018). arXiv:1712.05722 [hepph]ADSCrossRefGoogle Scholar
 71.S. Okada, Adv. High Energy Phys. 2018, 5340935 (2018). arXiv:1803.06793 [hepph]CrossRefGoogle Scholar
 72.Z.L. Han, W. Wang, Eur. Phys. J. C 78(10), 839 (2018). arXiv:1805.02025 [hepph]ADSCrossRefGoogle Scholar
 73.N. Okada, S. Okada, D. Raut, arXiv:1811.11927 [hepph]
 74.D. Borah, D. Nanda, N. Narendra, N. Sahu, arXiv:1810.12920 [hepph]
 75.Y.G. Kim, K.Y. Lee, C.B. Park, S. Shin, Phys. Rev. D 93(7), 075023 (2016). arXiv:1601.05089 [hepph]ADSCrossRefGoogle Scholar
 76.R. Ding, Z.L. Han, L. Huang, Y. Liao, Chin. Phys. C 42(10), 103101 (2018). arXiv:1802.05248 [hepph]ADSCrossRefGoogle Scholar
 77.J.R. Ellis, A. Ferstl, K.A. Olive, Phys. Lett. B 481, 304 (2000). arXiv:hepph/0001005 ADSCrossRefGoogle Scholar
 78.E. Aprile et al. [XENON Collaboration], Phys. Rev. Lett. 121(11), 111302 (2018). arXiv:1805.12562 [astroph.CO]
 79.D.S. Akerib et al. [LUXZEPLIN Collaboration], arXiv:1802.06039 [astroph.IM]
 80.G. Aad et al., ATLAS Collaboration, Phys. Lett. B 716, 1 (2012). arXiv:1207.7214 [hepex]
 81.S. Chatrchyan et al., CMS Collaboration, Phys. Lett. B 716, 30 (2012). arXiv:1207.7235 [hepex]
 82.W. Wang, Z.L. Han, Phys. Rev. D 94(5), 053015 (2016). arXiv:1605.00239 [hepph]ADSMathSciNetCrossRefGoogle Scholar
 83.D. de Florian et al. [LHC Higgs Cross Section Working Group], arXiv:1610.07922 [hepph]
 84.V. Khachatryan et al., CMS Collaboration, JHEP 1702, 135 (2017). arXiv:1610.09218 [hepex]
 85.C. Bonilla, J.W.F. Valle, J.C. Romão, Phys. Rev. D 91(11), 113015 (2015). arXiv:1502.01649 [hepph]ADSCrossRefGoogle Scholar
 86.T. Robens, T. Stefaniak, Eur. Phys. J. C 76(5), 268 (2016). arXiv:1601.07880 [hepph]ADSCrossRefGoogle Scholar
 87.A. Ilnicka, T. Robens, T. Stefaniak, Mod. Phys. Lett. A 33(10–11), 1830007 (2018). arXiv:1803.03594 [hepph]ADSCrossRefGoogle Scholar
 88.M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C 78(1), 24 (2018). arXiv:1710.01123 [hepex]
 89.M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. D 98(5), 052008 (2018). arXiv:1808.02380 [hepex]
 90.A.M. Sirunyan et al., CMS Collaboration, Phys. Lett. B 788, 7 (2019). arXiv:1806.00408 [hepex]
 91.M. Aaboud et al., ATLAS Collaboration, JHEP 1811, 040 (2018). arXiv:1807.04873 [hepex]
 92.L. Basso, A. Belyaev, S. Moretti, C.H. ShepherdThemistocleous, Phys. Rev. D 80, 055030 (2009). arXiv:0812.4313 [hepph]ADSCrossRefGoogle Scholar
 93.E.J. Chun, A. Das, J. Kim, J. Kim, arXiv:1811.04320 [hepph]
 94.G. Cacciapaglia, C. Csaki, G. Marandella, A. Strumia, Phys. Rev. D 74, 033011 (2006). arXiv:hepph/0604111 ADSCrossRefGoogle Scholar
 95.M. Aaboud et al., ATLAS Collaboration, JHEP 1710, 182 (2017). arXiv:1707.02424 [hepex]
 96.G. Aad et al. [ATLAS Collaboration], arXiv:1903.06248 [hepex]
 97.A.M. Sirunyan et al., CMS Collaboration, JHEP 1806, 120 (2018). arXiv:1803.06292 [hepex]
 98.J. Alwall et al., JHEP 1407, 079 (2014). arXiv:1405.0301 [hepph]ADSCrossRefGoogle Scholar
 99.M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C 78(12), 995 (2018). arXiv:1803.02762 [hepex]
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}