The Higgs Seesaw Induced Neutrino Masses and Dark Matter

In this paper we propose a possible explanation of the active neutrino Majorana masses with the TeV scale new physics which also provide a dark matter candidate. We extend the Standard Model (SM) with a local U(1)' symmetry and introduce a seesaw relation for the vacuum expectation values (VEVs) of the exotic scalar singlets, which break the U(1)' spontaneously. The larger VEV is responsible for generating the Dirac mass term of the heavy neutrinos, while the smaller for the Majorana mass term. As a result active neutrino masses are generated via the modified inverse seesaw mechanism. The lightest of the new fermion singlets, which are introduced to cancel the U(1)' anomalies, can be a stable particle with ultra flavor symmetry and thus a plausible dark matter candidate. We explore the parameter space with constraints from the dark matter relic abundance and dark matter direct detection.


I. INTRODUCTION
With the discovery of the Higgs-like scalar at the CERN LHC, the Standard Model Higgs mechanism for spontaneous breaking of the SU(2) L ×U(1) Y gauge symmetry appears to be a correct description of the nature. In addition to explaining the spontaneous breaking of the electroweak symmetry, the Higgs boson is also responsible for the origin of fermion masses, via the Yukawa interactions. On the other hand, the minimal Higgs mechanism is not able to address the the fermion mass hierarchy problem, where the quark-lepton masses range from the top quark with mass of order electroweak scale, M t = 172 GeV, down to electron of mass, M e = 0.511 MeV, and the first order phase transition, relevant for baryon asymmetry of the Universe. More precise measurement of Higgs boson properties will help determine whether there are new degrees of freedom that participate in electroweak symmetry breaking or otherwise involve in new Higgs boson interactions.
Furthermore, the discovery of the neutrino oscillation has confirmed the theoretical expectation that neutrinos are massive and lepton flavors are mixed [1], which provided the first piece of evidence for physics beyond the Standard Model (SM). In order to accommodate the tiny neutrino masses, one can extend the SM by introducing several right-handed neutrinos, which are taken to be singlets under the SU(2) L × U(1) Y gauge group. In this case, the gauge invariance allows right-handed neutrinos to have Majorana mass M R , which is not subject to the electroweak symmetry breaking scale. Thus the effective mass matrix of three light Majorana neutrinos can be highly suppressed if M R is much larger than the electroweak scale, which is the so-called canonical seesaw mechanism [2]. Two other types of tree-level seesaw mechanisms have also been proposed [3,4]. Despite its simplicity and elegance, the canonical seesaw mechanism is impossible to be tested in current collider experiments, especially at the Large Hadron Collider, due to its inaccessibly high right-handed Majorana mass scale. Heavy Majorana neutrinos can also give large radiative corrections to the SM Higgs mass, which causes the seesaw hierarchy problem [5]. An alternative way to generate tiny Majorana neutrino masses at the TeV scale is the inverse seesaw mechanism [6,7], in which the neutrino mass m ν is proportional to a small effecitive Majorana mass term µ. But there is no dynamical explanation of the smallness of µ. The argument is that neutrinos become massless in the limit of vanishing µ and the global lepton number, U(1) L , is then restored, leading to a larger symmetry [8]. This argument, however, only works when we give the left-handed singlets (S L ) the same quantum(lepton) number as that of the right-handed heavy neutrinos (N R ). If the lepton number of S L is zero, the argument above does not hold up any more. Besides, the lepton number is only an accidental symmetry of the SM and is explicitly broken by anomalies.
Since neutrino is the only neutral matter field in the SM, it is reasonable to conjecture that neutrinos are correlated with the dark matter, which provides another evidence of the new physics beyond the SM from the precise cosmological observations, through certain dark symmetry. The nature of the dark matter and the way it interacts with ordinary matter are still mysteries. The discovery of the Higgs boson opens up new ways of probing the world of the dark matter. The neutrino flux from the annihilation of the dark matter at the center of the dark matter halo also provides a way of indirect detecting the dark matter.
In this paper, we propose a possible explanation of the smallness of neutrino masses and a possible candidate of the dark matter. The discovery of the Higgs-like boson makes the Higgs mechanism more promising as a possible way to understand the origin of the fermion masses. We study the possibility of generating a small Majorana mass term with the help of the seesaw mechanism in the Higgs sector. We extend the SM with a local U(1) ′ gauge symmetry, which is spontaneously broken by the vacuum expectation value (VEV) ϕ of an extra scalar singlet. Furthermore there is a seesaw mechanism in the scalar singlet sector: a second scalar singlet gets a tiny VEV Φ in a way similar to that of the Higgs triplet in the type-II seesaw model [3]. ϕ is responsible for the origin of the dark matter mass and the Dirac neutrino mass term, while Φ is responsible for the origin of a small Majorana neutrino mass term. The active neutrino mass matrix arises from the modified inverse seesaw mechanism. A crucial feature of our model is that all the mass terms originate from the spontaneous breaking of local gauge symmetries, and dark matter is correlated with the neutrino physics via the U(1) ′ gauge symmetry. We study constraints on the parameter space of this model from astrophysical observation and dark matter direct detections.
The paper is organized as follows: In section II we describe our model, including the full Lagrangian, Higgs VEVs and mass spectrum. In section III we study the neutrino masses and the effective lepton mixing matrix of the model. Section IV is devoted to the study of the dark matter phenomenology. We summarize in section V.

II. THE MODEL
We extend the SM with three generations of right-handed neutrinos N R and singlets S L as in the inverse seesaw mechanism, together with two extra scalar singlets, ϕ and Φ, as well as a spontaneously broken U(1) ′ gauge symmetry and a global U(1) D flavor symmetry. The quantum numbers of the fields are given in Table I, where ℓ L is left-handed lepton doublet, E R is the right-handed charged lepton, H is the SM Higgs doublet, and χ L and χ R are the fermion singlet pair carrying the same U(1) D quantum number. Three generations of gauge singlets χ L,R are needed to cancel anomalies [9][10][11][12][13][14][15] of the U(1) ′ gauge symmetry. The lightest generation of χ L,R is stable due to the global U(1) D flavor symmetry and thus plays the role of dark matter [30][31][32].  The Higgs potential of the model can be written as After imposing the conditions of the global minimum, one has Then the VEVs can be solved in terms of the parameters where v 2 is proportional to Λ and suppressed by m 2 2 . Thus v 2 can be a small value given a large m 2 2 or small Λ. In the basis (h 0 , φ 0 , Φ 0 ), the mass matrix of the CP-even Higgs can be written as The mass eigenstates of the CP-even Higgs are then denoted as h i including the SM-like Higgs h and two exotic Higgs, h 1 and h 2 . There is no mixing between the SM CP-odd Higgs A, which is the Goldstone boson eaten by the Z gauge boson, and those of the Higgs singlets, i.e. δ and ρ. The mass matrix of the CP-odd Higgs singlets in the basis of (δ, ρ) is The massless eigenstate of the eq. (7) is the Goldstone boson eaten by the Z ′ and the nonzero mass eigenstate of the CP-odd scalar is then denoted as A ′ , the mass squared of which can be written as m 2 Since the SM particles are not charged under U(1) ′ , there is no experimental constraint on the new symmetry. Besides, there is no tree-level mixing between Z and Z ′ . Thus the mass and coupling constant of Z ′ are not constrained by current experiments either.

III. NEUTRINO MASSES
Now we investigate how to realize the neutrino masses in our model. The Yukawa interactions of the lepton sector can be given by where the first and second terms are the charged lepton and neutrino Yukawa interactions separately, the third and fourth terms are the Yukawa coupling of heavy neutrinos to the scalar singlets, and the last term is the Yukawa coupling of the additional fermions. We assume that there is no N C R MN R type of mass term, which can be easily forbidden by an extra global U(1) symmetry, in which all the right-handed fermions, H and S L are singly charged, Φ doubly charged and all other particles neutral. The symmetry is explicitly broken by the last term of the Higgs potential in Eq. (1). We can write down the mass matrix of neutrinos in the basis (ν L , N C R , S L ) T : where v, v 1 , v 2 are given in Eq. (5). Given v 1 ∼ 1 T eV and v 2 ∼ 1 MeV , the inverse seesaw mechanism is naturally realized. The matrix M can be diagonalized by the unitary transformation U † MU * =M; or explicitly, whereM ν,N,S are 3 × 3 diagonal matrices. The nine mass eigenstates correspond to three observed light neutrinosν and six heavy Majorana neutrinosŜ andN, which pair up to form three pseudo-Dirac neutrinos.
Alternatively, the neutrino mass matrix can be block diagonalized and the effective Majorana mass matrix of the active neutrinos can be approximately written as The mass eigenvalues of the three pairs of heavy neutrinos are of the order M R , and the mixing between SU(2) L singlets and doublets is suppressed by M D /M R . In the basis where the flavor eignestates of the three charged leptons are identified with their mass eigenstates, the charged-current interactions between neutrinos and charged leptons turn out to be Obviously A describes the charged-current interactions of light Majorana neutrinos, while B and C are relevant to the charged currents of heavy neutrinos. The neutral current interactions between Majorana neutrinos and neutral gauge boson or Higgs can be also written down in a similar way.
The explicit expression of A can be obtained by integrating out heavy neutrinos and performing the normalization to the light neutrino wave functions. So the effective leptonmixing matrix can be written as where U is the standard PMNS matrix. Obviously the effective neutrino mixing matrix is not unitary. The deviation of A from a unitary matrix is proportional to |M D M −1 R | 2 . Constraints on the elements of the leptonic mixing matrix, combining data from neutrino oscillation experiments and weak decays was studied in Ref. [16] . So far neutrino mixing angles have all been measured to a good degree of accuracy, and a preliminary hint for a nontrivial value of δ has also been obtained from a global analysis of current neutrino oscillation data. But the constraint on the non-unitarity of the lepton mixing matrix still need to be improved and the future neutrino factory can measure this effect through the "zero-distance" effect and extra CP violations. The Daya Bay [17] reactor neutrino experiment has measured a nonzero value for the neutrino mixing angle θ 13 with a significance of 5.2 standard deviations. For this case, even though the neutrino mixing matrix U, which diagonalizes the active neutrino mass matrix, takes the well-known lepton mixing pattens, such as Tri-Bimaximal [18] , Bimaximal [19] and Democratic [20] pattens, where θ 13 is exactly zero, it is still possible to get relatively large θ 13 from the non-unitarity factors in eq. (13) [21]. One can also check the non-unitary effect from the lepton-flavor-violating SM Higgs decays, which, interesting and important but beyond the scope of this paper, will be shown somewhere else.

IV. DARK MATTER
Precise cosmological observations have confirmed the existence of the non-baryonic cold dark matter. The lightest generation of χ L,R , the only odd particles under the global U (1) symmetry, can be a stable dark matter candidate. In order to produce the dark matter relic abundance observed today Ω DM h 2 = 0.1187±0.017 [22], the thermally averaged annihilation rate σ A v should approximately be 3 × 10 −27 cm 3 s −1 /Ω DM h 2 . Interactions relevant to dark matter phenomenology can be written as II where θ is the mixing angle between the SM Higgs boson and the Higgs singlet. It's the 1-2 mixing angle of matrix given in Eq. (6). F is either D or G, the 21 and 31 entry in U. The from which it's easily seen that the active neutrinos mainly mix with S L , while the mixing with N C R is highly suppressed by the factor µM −1 R . The major contributions to the annihilation cross section come from two types of channels, where X represents the SM fields including h i but other than neutrinos. The relevant Feynman diagrams for dark matter annihilation are given in Fig. 1. Obviously the dark matter in our model is the hybrid of neutrino portal and Higgs portal.
To investigate the viability of this model of providing a good dark matter candidate, we fix those parameters irrelevant to the dark matter properties and vary the others. Without loss of generality we also simplify the calculation by taking diagonal Yukawa coupling matrices, which are relevant for the generation of neutrino mixing but irrelevant for the dark matter phenomenology. The typical input parameters are given in the table. II. The relics density and direct detection cross section are calculated with micrOMEGAs [26], which solves the Boltzmann equations numerically and utilizes CalcHEP [27] to calculate the relevant cross section. We show in the left panel of Fig.2 the contour plot of λ 3 as a function of (λ H , λ 1 ) with m h = 120, 126 GeV . The approximate range for λ 3 is roughly (0, 5, 1.5) for value chosen in Table. II. We also show in the right panel of Fig.2 the contour plot of the CP-even exotic Higgs mass M h 1 as a function of (λ H , λ 1 ) with the SM-like Higgs mass fixed at 126 GeV, which shows that the mass of the exotic CP-even Higgs is in the range of 300 − 700 GeV.  Table. II. Right panel: Contour plot of the mass of h 1 as a function of (λ H , λ 1 ) which gives the right SM-like Higgs mass.
parameters values or range parameters values or range [ 10,2000]   Dark matter is also constrained by direct detection experiments such as LUX [28] and XENON 100 [29]. The dark matter -quark interactions in the effective models naturally induce the dark matter-nucleus interactions. The effective Hamiltonian in our model can be written as where c θ = cos θ and s θ = sin θ. Parameterizing the nucleonic matrix element as N q m qq q = f N m N , where m N is the proton or neutron mass and f N are the nucleon form factors. We refer to [30][31][32] for explicit values of f p,n . The cross section for the DM scattering elastically from a nucleus is given by where µ = m χ m N /(m χ +m N ) is the reduced mass of the WIMP-nucleon system, with m N the target nucleus mass. Z and (A − Z) are the numbers of protons and neutrons in the nucleus. M Z ′ = 1000 GeV respectively. The red solid line is the LUX limit. One can see from (21) that the scattering cross section is sensitive to λ 3 , which determines the mixing angle, θ, between the SM-like Higgs and the Heavier scalar singlet. The direct detection cross section gets bigger when λ 3 increases.

V. CONCLUDING REMARKS
In this paper we extend the SM with a local and a global U(1) symmetry. The smallness of active neutrino Majorana masses is explained by the modified inverse seesaw mechanism.
Extra fermion singlets introduced to cancel anomalies of the model can play the role of dark matter. Constraints on the model parameter space from dark matter relic density as well as dark matter direct searches are studied. All the fermion masses arise from the spontaneous breaking of local gauge symmetries, which is a very appealing feature of the model in the era of Higgs physics.