Connecting Neutrino Masses and Dark Matter by High-dimensional Lepton Number Violation Operator

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 O_7=\bar l_R^c \gamma^\mu L_L(D_mu \Phi) \Phi \Phi. Note that it is necessary to impose an Z_2 symmetry in order that 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.


I. INTRODUCTION
The presence of the tiny neutrino masses and mixings between different neutrino flavors have been established by many neutrino oscillation experiments [1][2][3][4][5][6][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][11][12][13]). We would like to push this connection further in the present paper.
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][31][32][33][34][35][36][37][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 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 models [13,[39][40][41], which realize the dimension-9 effective operator O 9 = l c R l R [(D µ Φ)Φ] 2 with the DM embedded in the loop.
In this study, we focus on a specific dimension-7 LNV operator O 7 =l c R γ µ L L (D µ Φ)ΦΦ [32,34,37] and construct a UV complete model with an unbroken Z 2 symmetry to accomplish the above general arguments. In this model, Majorana neutrino masses arise radiatively at two-loop level, and neutrinoless double beta (0νββ) decays are dominated by a new "long- range" contribution 1 , as the results of the existence of O 7 , while DM can also be embedded naturally as the lightest Z 2 -odd neutral state. This paper is organized as follows. In Sec. II, we first describe the particle content and write down the relevant part of the Lagrangian for the model. We then calculate the two-  In order to produce O 7 at one-loop level, we introduce two scalars: s : (1, 2) and χ : (2, 1/2), and three vector-like fermions D (L,R)i : (2, 1/2) with i = 1, 2 and 3 to the SM under SU(2) L × U(1) Y . A Z 2 symmetry is also imposed, in which only the new particles carry odd 1 The definitions of "short" and "long" range contributions to 0νββ follow Refs. [42,43].
charges. The relevant new parts of the Lagrangian are given by where Φ = (Φ + , Φ 0 ) T is the SM Higgs doublet and σ 2 is the Pauli matrix for the SU(2) L gauge group. After the EW spontaneous symmetry breaking, Φ acquires a vacuum expectation value v ≡ √ 2 Φ 0 ≃ 246 GeV, while µ 2 χ > 0 is necessary for preserving the Z 2 symmetry. Notice that the lepton number is explicitly broken only when κ, λ 5 , and at least one of ξ il ζ il ′ are non-zero simultaneously. For convenience, we define M 2 The trilinear coupling constant κ in Eq. (1) makes s ± mix with χ ± , which can be formulated as As for the new fermions, we have the tree-level relation M D ± i = M D 0 i for each D i doublet. 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: where λ L ≡ 1 2 (λ 3 + λ 4 + λ 5 ), and the other independent coupling constants from quarter terms: λ Φs , λ χs , λ χ , and λ s , which are less relevant in our discussion.

B. Two-Loop Majorana Neutrino Masses
As seen in Fig. 1b where m l (l = e, µ, τ ) are charged lepton masses, and I ij are defined by Subsequently, one can diagonalize M ν by where m 1,2,3 are three neutrino mass eigenvalues, which can have the normal ordering, m 1 < m 2 ≪ m 3 , or inverted ordering, m 3 ≪ m 1 < m 2 , and V is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [47,48]. Without loss of generality, V can be written as the standard 2 There is a similar realization of O 7 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 O 7 does not give the dominant contribution to Majorana neutrino masses in Ref. [37]. 3 Similar 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.
From Eq. (8), we can get two important features for this mass generation mechanism.
Firstly, the overall size of M ν is proportional to the mass difference of the neutral scalars, A , and the combined factor of the charged states, s 2θ (I i1 − I i2 ), in which the former is generated by λ 5 and the latter corresponds to the size of κ. Turning off one of them will make all neutrinos massless. Secondly, the neutrino masses are positively correlated to the coupling matrix elements ξ il and ζ il , as well as the sizes of m l . As the existence of the charged lepton mass hierarchy, m e ≪ m µ < m τ , if both matrices of (ξ il ) and (ζ il ) are in uniform textures, the magnitude of (M ν ) ee should be much smaller than those of other M ν entries. We make a great advantage of this general expectation by taking the following limit which is shown in Refs. [49,50] to rule out the inverted ordering of neutrino mass spectrum at more than 2σ confident level. Thus, the normal ordering is predicted for the present model.
Note that in the limit of Eq. (13), Ref. [49] even shows that the lightest neutrino mass m 1 can only be located within the range 0.001eV m 1 0.01eV. Moreover, the smallness of (M ν ) ee is also required by the 0νββ decay, which will be clear in the next subsection.
We now search for possible coupling matrix forms to realize the above four CP conserving neutrino textures. For simplicity, hereafter we will take M D1 = M D2 = M D3 = M D , and also set ξ proportional to the identity matrix with the diagonal matrix element to be ξ d . Taking a symmetric form of ζ, the mass matrix element should be proportional to ξ d ζ l ′ l (m l + m l ′ ).
We remark that by an appropriate phase absorption to the fermion fields, one can always have a positive ξ d without loss of generality. Comparing with Eqs. (14)- (17), the forms of ζ ll ′ in the four neutrino matrix textures can be obtained as 0.12 0.89 0.14 0.052 0.14 0.068 where the cross means that the value of ζ ee is still arbitrary at this stage, which will be constrained by 0νββ decays. The overall scale of ζ ll ′ can be determined by Eq. (8) when all new particle masses, s θ and ξ d are known. We will also leave the discussion about the correlation between ξ and ζ from the LFV constraints to Sec. III.
is well studied in Refs. [51,52,[54][55][56], and the new contribution is much larger than that from Fig. 2a by orders of magnitude of 10 8 . For our model or those with O 7 as the main LNV source, the 0νββ decays are dominated by the diagram in Fig. 2b, which is not suppressed by the nearly-vanishing (M ν ) ee . We can write down a general formula for the half lifetime T 0νββ 1/2 of the 0νββ decay with the contributions from Figs. 2a and 2b, given by [57] [T 0νββ with where C mm , C mη , and C ηη include the phase space integrations and nuclear matrix elements defined in Ref. [57], and η is the coupling of the interaction (4G F / √ 2)(ū L γ µ d L )(l R γ µ ν c L ), which is originated from the one-loop generation for O 7 (in Fig. 2c). By using the numerical results therein and also in Ref. [58], we find that the contribution proportional to C ηη is much larger than those to C mm and C mη .
As given in Eq. (20) that η is proportional to ξ d ζ ee , the upper bound on |ξ d ζ ee | can be obtained by comparing with the current experimental sensitivities on T 0νββ 1/2 [59][60][61][62][63][64][65] in Table I Table I also shows the maximum value of (M ν ) ee for each nucleus. would also change the leading modes accordingly.

A. Electroweak Precision Tests
As discussed previously, in order to have the two-loop neutrino masses in our model, the non-zero coupling constants λ 5 and κ are both required. The former splits the masses between H 0 and A 0 , and the latter mixes the charged states χ ± and s ± which carry different 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] ∆T = 1 with the function F defined by The value of F x,y becomes zero when M x → M y , and it increases with the mass splitting among the new scalars. Note that ∆T has little to do with D i since there is neither mixing between D i and the SM leptons nor tree-level mass splitting among D i , while the deviation for the S parameter can also be ignored [66]. The formulae of Eq. (21) is a general result

B. Dark Matter
In this model, the lightest of the extra neutral particles: H 0 , A 0 , and D 0 1,2,3 could be a DM candidate, whose stability is guaranteed by the imposed Z 2 symmetry. In the following, we will focus on the case that DM is constituted solely by H 0 with a small charged scalar mixing s θ , in which our DM would be very similar to that in the well-studied inert doublet model [68][69][70]. Furthermore, we concentrate on the low DM mass region with 50 GeV M H 80 GeV [69,70], in which a large H 0 -A 0 mass splitting can be allowed for the generation of the right two-loop neutrino masses. In addition, the mass of S ± 2 should be higher than 90 GeV in order to escape the LEP bounds [67], so that the co-annihilation channels, such as H 0 -A 0 and H 0 -S ± 2 , would be strongly suppressed and thus ignored. We use the package micrOMEGAs [71] to accurately calculate the relic abundance Ω H in the above parameter space, including all possible annihilations and co-annihilations. When M H approaches the half of the SM Higgs mass M h /2 ≃ 62.5 GeV [72,73], the Higgs resonance in the s-channel would become prominent, which is characterized by the coupling λ L controlling the trilinear vertex (λ L v)hH 0 H 0 . However, in other regions, the DM annihilation cross section is dominated by the W W ( * ) mode. Therefore, the correct DM relic abundance is achieved mainly by the balance of the W W ( * ) and Higgs resonance channels.  [8,9,49]. Note that when DM is heavier than 73 GeV, the W W ( * ) channel would give a too large annihilation cross section to accommodate the DM relic abundance [69,70], which is omitted in Fig. 4.
The DM H 0 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 spinindependent 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 provided by the LUX experiment [74], with the minimum cross section of 7.6 × 10 −46 cm 2 for a DM mass of 33 GeV. It is shown in Fig. 4 that the LUX experiment has already probed some parameter space required by the DM relic abundance. Especially, the low DM mass region with M H 52 GeV is actually ruled out, as indicated by the shaded area in the plot.
However, most parameter spaces are still allowed by LUX.

C. Lepton Flavor Violation
The current experimental constraints on LFV processes, such as the radiative decays l → l ′ γ [75,76], µ − e conversions [77][78][79][80], and three-lepton decays l → l 1 l 2l3 [81,82], are all dominated by one-loop diagrams with Z 2 odd particles inside. We take µ → eγ as an illustration because the current experimental upper bound Br(µ + → e + γ) < 5.7 × 10 −13 [75] usually constrains a model in the most stringent way. In our model, we have where the loop integrals K x and K ′ x are defined as with z = M 2 D /M 2 x . As is expected, the decay branching ratio is proportional to the coupling constant combination l ζ lµ ζ le Br(µ → eγ) = 10 −13 (3 × 10 −15 ) for M S 2 = 90 (120) GeV, which might be measured by the next-generation experiments in the future.

D. 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: It is clear that the matrix ζ corresponds to the neutrino texture T A . We also plot this benchmark point by a black dot in Figs. 5a, 3b, and 4, where the experimental results from LFV processes, oblique parameters, and DM searches are well satisfied, respectively.

IV. 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 the Z 2 symmetry to generate the dimension-7 operator O 7 =l c R γ µ L L (D µ Φ)ΦΦ. We have shown that the Majorana neutrino mass matrix structure and the leading 0νββ decay contribution are closely related to O 7 . Especially, the neutrino mass matrix is predicted to be of the normal ordering due to the hierarchy in the charged lepton masses, and the 0νββ decay rate can be large enough to be tested in the next-generation experiments. If we impose an additional CP symmetry in the lepton sector, we can even determine the form of the neutrino mass matrix completely. We have also focused on a specific parameter region with a small mixing between charged scalars, and considered the constraints from the electroweak precision tests, dark matter searches, and LFV processes.