Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator

Journal of High Energy Physics, Aug 2015

We propose a new model with the Majorana neutrino masses generated at two-loop level, in which the lepton number violation (LNV) processes, such as neutrino-less double beta decays, are mainly induced by the dimension-7 LNV effective operator \( {\mathcal{O}}_7={\overline{l}}_R^c{\gamma}^{\mu }{L}_L\left({D}_{\mu}\varPhi \right)\varPhi \varPhi \). Note that it is necessary to impose an Z 2 symmetry in order that \( {\mathcal{O}}_7 \) dominates over the conventional dimension-5 Weinberg operator, which naturally results in a stable Z 2-odd neutral particle to be the cold dark matter candidate. More interestingly, due to the non-trivial dependence of the charged lepton masses, the model predicts the neutrino mass matrix to be in the form of the normal hierarchy. We also focus on a specific parameter region of great phenomenological interests, such as electroweak precision tests, dark matter direct searches along with its relic abundance, and lepton flavor violation processes.

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Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator

JHE Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator F. del Aguila 0 2 7 8 9 A. Aparici 0 2 7 8 9 S. Bhattacharya 0 2 7 8 9 A. Santamaria 0 2 7 8 9 J. Wudka 0 2 7 8 9 Effective 0 2 7 8 9 0 Hsinchu , 300 Taiwan 1 Department of Physics, Tsinghua University 2 Chao-Qiang Geng 3 Collaborative Innovation Center of Quantum Matter 4 Department of Physics, National Tsing Hua University 5 Physics Division, National Center for Theoretical Sciences 6 Chongqing University of Posts & Telecommunications 7 Open Access , c The Authors 8 Beijing 100084 , China 9 Chongqing , 400065 , China We propose a new model with the Majorana neutrino masses generated at two-loop level, in which the lepton number violation (LNV) processes, such as neutrinoless double beta decays, are mainly induced by the dimension-7 LNV effective operator O7 = ¯lc γμLL(DμΦ)ΦΦ. Note that it is necessary to impose an Z2 symmetry in order R that O7 dominates over the conventional dimension-5 Weinberg operator, which naturally results in a stable Z2-odd neutral particle to be the cold dark matter candidate. More interestingly, due to the non-trivial dependence of the charged lepton masses, the model predicts the neutrino mass matrix to be in the form of the normal hierarchy. We also focus on a specific parameter region of great phenomenological interests, such as electroweak precision tests, dark matter direct searches along with its relic abundance, and lepton flavor violation processes. Neutrino Physics; Global Symmetries; Effective field theories 1 Introduction 2 Two-loop Majorana neutrino masses Neutrinoless double beta decay Phenomenological constraints Electroweak precision tests Lepton flavor violation Numerical results Introduction The presence of the tiny neutrino masses and mixings between different neutrino flavors have been established by many neutrino oscillation experiments [1–7], while more and more evidences are accumulated for the existence of dark matter (DM) over the last several decades, with the most precise measurement of its relic abundance by PLANCK [8, 9]. Both phenomena cannot be explained within the Standard Model (SM), thus providing us with two windows towards new physics beyond it. An interesting idea is to connect neutrinos and DM in a unified framework as many existing attempts in the literature (see e.g. refs. [10–13]). We would like to push this connection further in the present paper. In order to understand the mass hierarchy problem in the neutrino sector, there are many models in the literature to naturally generate small Majorana neutrino masses such as the traditional Seesaw [14–26] and radiative mass generation mechanisms [10–13, 27–29]. Most of them can be summarized as a specific realization of the conventional dimension-5 Weinberg operator. However, the generation of Majorana neutrino masses only requires the lepton number violation (LNV) by two units, and there exist many other equally legitimate LNV effective operators [30–38], which are composed of the SM fields but with higher scaling dimensions. From the effective field theory perspective, it is generically believed that these high-dimensional effective operators are subdominated by the Weinberg operator due to the suppression from the corresponding high powers of the large cutoff. In order for these operators to show up as the leading contributions, one usually needs to impose an additional symmetry on the model to break the usual scaling arguments. Furthermore, if this symmetry is kept unbroken, then the lightest symmetry-protected neutral particle would provide a perfect DM candidate. In this way, the symmetry connects l− Di− l− one-loop realization of O7, and (c) the two-loop neutrino mass generation. Majorana neutrino masses and DM physics by the high-dimensional effective operators. Such a connection has been already exemplified by some recent three-loop neutrino mass with the DM embedded in the loop. [32, 34, 37] and construct a UV complete model with an unbroken Z2 symmetry to accomplish the above general arguments. In this model, Majorana neutrino masses arise by a new “long-range” contribution,1 as the results of the existence of O7, while DM can also be embedded naturally as the lightest Z2-odd neutral state. This paper is organized as follows. In section 2, we first describe the particle content and write down the relevant part of the Lagrangian for the model. We then calculate the the emphasis on their relations to the high-dimensional effective operator O7. In section 3, the constraints on the model are addressed from the electroweak (EW) precision tests, dark matter searches, and lepton flavor violating (LFV) processes. Finally, we give the conclusions in section 4. Figure 1a shows the neutrino mass generation induced by the one-loop diagram with O7. In SU(2)L × U(1)Y . A Z2 symmetry is also imposed, in which only the new particles carry odd charges. The relevant new parts of the Lagrangian are given by pectation value v ≡ eigenstates with their masses, given by MS21 = MS22 = M H2 = Mχ2 + (λ4 + λ5)v2 , MA2 = Mχ2 + (λ4 − λ5)v2 . The mass splittings between the charge and neutral components of the inert fermion doublets can only be induced by loop corrections with values around a few hundred MeV [44]. In this paper, we will characterize the model by using the physical quantities: which are less relevant in our discussion. Two-loop Majorana neutrino masses As seen in figure 1b, the effective operator O7 can be induced by the one-loop diagram, whereas the Weinberg operator cannot.2 Consequently, Majorana neutrino masses appear 2There is a similar realization of O7 in ref. [37], in which a triplet replaced the singlet s of our model. A fundamental distinction of their paper from the present one is that O7 does not give the dominant contribution to Majorana neutrino masses in ref. [37]. X (ml ζil ξil′ + ml′ ζil′ ξil)[Ii1 − Ii2] , (2.8) Iij ≡ 2(1 − 3x) x(1 − x) Z 1−x3 Z 1−x2−x3 6y1(2 − x) (1 − x)2 log(mi2j ) + −2y1(2 − x) M W2 x(1 − x) mi2j ≡ y1[x1M H2 + x2MA2 + x3M W2 ] + y2x(1 − x)MS2j +(1 − y1 − y2)x(1 − x)M D2i , x = x1 + x2 . where m1,2,3 are three neutrino mass eigenvalues, which can have the normal ordering, Nakagawa-Sakata mixing matrix [47, 48]. Without loss of generality, V can be written as the standard parametrization by appropriate rephasing in LL’s and lR’s, given by [49] V =  −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ From eq. (2.8), we can get two important features for this mass generation mechanism. will make all neutrinos massless. Secondly, the neutrino masses are positively correlated 3Similar topology with one W ± exchange in a two-loop neutrino mass model can also be found in refs. [45, 46], in which a different high-dimensional effective operator is realized without DM. which is shown in refs. [49, 50] to rule out the inverted ordering of neutrino mass spectrum at Note that in the limit of eq. (2.13), ref. [49] even shows that the lightest neutrino mass m1 can only be located within the range 0.001eV . m1 . 0.01eV. Moreover, the smallness of fitting for the neutrino oscillation data [49], the corresponding mass matrices for Textures A, B, C, and D (TA, TB, TC and TD) are given by TA : Mν =  0.12 1.9 2.7  (10−2) eV , −0.24 2.7  (10−2) eV , −1.1 −2.3 −2.1  (10−2) eV , −0.055 −2.1 −3.1 −0.90 −0.24  −1.1 −0.055  0.12 0.052  TA : ζ ∝  0.12 0.89 0.14  ,  0.052 0.14 0.068 −1. −1. −0.0031  We now search for possible coupling matrix forms to realize the above four CP conserv −0.013 0.14 −0.84 −0.013  −0.081 0.062  −0.086 1.1  −2.6 −2.2 −2.2  (10−2) eV , −2.9 −0.0031 −0.11 −0.088  −1.1 −0.11  , −1.2 −0.12  , −0.12 −0.081 insertion on the propagator, (b) the contribution involving O7, and (c) the one-loop construction which realizes O7. For (b) and (c), the corresponding upside-down diagrams also need be considered. Neutrinoless double beta decay In the previous section, we have built a two-loop neutrino mass model in which the LNV operator O7 provides the leading contribution. The next step is to study the LNV effect in this model induced by this high-dimensional operator. The most sensitive smoking gun for For the Majorana neutrino masses, there always exists the traditional long-range decay process by the exchange of neutrinos with a pair of left-handed electrons emitted as shown in figure 2a. Note that there is a chirality flipping on the internal neutrino propagator, which leads to the proportionality of the amplitude to the neutrino mass matrix element does not give the main part of this process, and one should have a prior consideration on the effects of other new diagrams. For example, a class of neutrino models [51–53] that can well studied in refs. [51, 52, 54–56], and the new contribution is much larger than that from figure 2a by orders of magnitude of 108. For our model or those with O7 as the main 1 s2θξζ ee (M H2 − MA2 )(I1′ − I2′) , Ij′ = Z 1−x3 Z 1−x2−x3 x1M H2 +x2MA2 +x3MS2j +(1−x1 −x2 −x3)M D2 , GERDA-1(76Ge) [59] KamLAND-Zen(136Xe) [60] NEMO-3(150Nd) [62] CUORICINO(130Te) [63] NEMO-3(82Se) [64, 65] NEMO-3(100Mo) [65] 4.4 × 10−9 8.3 × 10−9 2.9 × 10−7 2.3 × 10−8 1.5 × 10−8 3.5 × 10−8 3.6 × 10−4 2.8 × 10−4 1.5 × 10−3 4.2 × 10−4 1.5 × 10−3 5.6 × 10−4 which is originated from the one-loop generation for O7 (in figure 2c). By using the numerical results therein and also in ref. [58], we find that the contribution proportional Finally, we end this section by mentioning that the role of the Z2 symmetry is to make the dimension-7 operator O7 become the dominant contributions to the LNV processes and singlet (s) scalars are the same as those in the Zee’s model [27], the Majorana neutrino masses would be mainly generated by the corresponding one-loop diagrams related to the conventional Weinberg operator if the Z2 symmetry is absent. However, with the Z2 symmetry, O7 is singled out at 1-loop level, while other LNV effective operators, especially the dimension-5 Weinberg operator, are much suppressed since they would be only induced by higher-loop diagrams. In this way, the Z2 symmetry breaks the conventional effective would also change the leading modes accordingly. Phenomenological constraints Electroweak precision tests As discussed previously, in order to have the two-loop neutrino masses in our model, the  =  = EW gauge quantum numbers. Both effects could change the values of the EW oblique S and T parameters. In particular, the T parameter should yield a stronger constraint on this model. The deviation of T from the SM is given by [13] with the function F defined by Fx,y = − Mx2 − My2 log The value of Fx,y becomes zero when Mx → My, and it increases with the mass splitting between Di and the SM leptons nor tree-level mass splitting among Di, while the deviation for the S parameter can also be ignored [66]. The formulae of eq. (3.1) is a general result for the models with the mixings between the inert doublet and singlet scalars. The global MS2 can not be too small, since there exists a lower bound on MS2 located within 70 to 90 GeV [67] from the LEP experiments. In this model, the lightest of the extra neutral particles: H0, A0, and D10,2,3 could be a DM candidate, whose stability is guaranteed by the imposed Z2 symmetry. In the following, we will focus on the case that DM is constituted solely by H0 with a small studied inert doublet model [68–70]. Furthermore, we concentrate on the low DM mass region with 50 GeV 6 MH 6 80 GeV [69, 70], in which a large H0-A0 mass splitting can be allowed for the generation of the right two-loop neutrino masses. In addition, the mass of S2± should be higher than 90 GeV in order to escape the LEP bounds [67], so that the co-annihilation channels, such as H0-A0 and H0-S2±, would be strongly suppressed and We use the package micrOMEGAs [71] to accurately calculate the relic abundance Higgs resonance in the s-channel would become prominent, which is characterized by the cross section to accommodate the DM relic abundance [69, 70], which is omitted in figure 4. The DM H0 in this low mass region could be constrained by the DM direct detection experiments. Since we need a relatively large Higgs-mediation annihilation channel to generate DM relic abundance, the Higgs exchange channel can also give rise to sizeable spin-independent signals, with the corresponding DM-nucleon cross section as follows [69]: Currently, the most stringent bound on the spin-independent DM-nucleon cross section is for a DM mass of 33 GeV. It is shown in figure 4 that the LUX experiment has already probed some parameter space required by the DM relic abundance. Especially, the low DM mass region with MH 6 52 GeV is actually ruled out, as indicated by the shaded area in the plot. However, most parameter spaces are still allowed by LUX. Lepton flavor violation The current experimental constraints on LFV processes, such as the radiative decays l → usually constrains a model in the most stringent way. In our model, we have 1 KH′ + KA dance. The gray shaded area is excluded by the LUX experiment, and the black dot represents the where the loop integrals Kx and Kx′ are defined as Kx = 2z3 + 3z2 − 6z + 1 − 6z2 log z 6(1 − z)4Mx2 Kx′ = − z3 − 6z2 + 3z + 2 + 6z log z 6(1 − z)4Mx2 constant combination next-generation experiments in the future. Numerical results Based on the above constraints from the LFV processes, EW precision tests, direct searches of DM with the required relic abundance, we find a benchmark point from the allowed parameter space, given by: MH = 70 GeV , MA = 95 GeV , MS1 = 310 GeV , MS2 = 90 GeV , MD1 = MD2 = MD3 = MD = 200GeV , red, green, and yellow curves correspond to the neutrino textures TA, TB, TC , and TD, respectively, while the black dot is the benchmark point.       = = 0.005 0.0022  0.0022 0.0061 0.0029 benchmark point by a black dot in figures 5a, 3b, and 4, where the experimental results from LFV processes, oblique parameters, and DM searches are well satisfied, respectively. the self-consistence of the perturbation theory. Conclusions We have tried to make the connection between neutrino physics and dark matter searches. In particular, we have emphasized that every effective operator, which violates the lepton number by two units, can give an equally good mechanism to generate Majorana neutrino masses. The problem lies in the fact that the new high-dimensional operators might be buried by the overwhelming effects from the conventional Weinberg operator which possess the smallest scaling dimension. One way to break this effective field theory ordering is to impose some symmetry which would protect the lightest neutral symmetry-protected states to be the DM particle. We have explicitly realized this connection by constructing a UV complete model with contribution are closely related to O7. Especially, the neutrino mass matrix is predicted to be of the normal ordering due to the hierarchy in the charged lepton masses, and the we impose an additional CP symmetry in the lepton sector, we can even determine the form of the neutrino mass matrix completely. 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Chao-Qiang Geng, Da Huang, Lu-Hsing Tsai, Qing Wang. Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator, Journal of High Energy Physics, 2015, 141, DOI: 10.1007/JHEP08(2015)141