Two Loop Radiative Seesaw Model with Inert Triplet Scalar Field

We propose a radiative seesaw model with an inert triplet scalar field in which Majorana neutrino masses are generated at the two loop level. There are fermionic or bosonic dark matter candidates in the model. We find that each candidate can satisfy the WMAP data when its mass is taken to be around the half of the mass of the standard model like Higgs boson. We also discuss phenomenology of the inert triplet scalar bosons, especially focusing on the doubly-charged scalar bosons at Large Hadron Collider in parameter regions constrained by the electroweak precision data and WMAP data. We study how we can distinguish our model from the minimal Higgs triplet model.


I. INTRODUCTION
Recent several experiments require the serious modifications of the standard model (SM) in spite of the great success. For an example, the SM Higgs boson search in the diphoton mode h → γγ at Large Hadron Collider (LHC) is shown that its signal strength is 1.65±0.24 at ATLAS [1,2] and 1.6±0.4 at CMS [3]. For another example, the existence of non-baryonic dark matters (DMs), which cannot be included in SM, dominates about 23% from the CMB observation by WMAP [4]. The fact is strongly supported by the cosmological observations such as the rotation curves of the galaxy [5] and the gravitational lensing [6] in our universe. In recent years, direct detection experiments of DM; XENON100 [7], CRESSTII [8], CoGeNT [9] and DAMA [10], show the scattering events with nuclei. XENON100 has not shown a result of DM signal but shown an upper bound with the minimal bound around 100 GeV. On the other hand, CoGeNT, DAMA and CRESSTII have reported the observations which can be interpreted as DM signals that favor a light DM with several GeV mass and rather large cross section. As far as we consider these experiments, the mass scale of DM should be O(1-100) GeV.
In order to explain the excess in the h → γγ channel, a modified diphoton event has been discussed in a lot of paper. If a model contains charged new particles which couple to the SM-like Higgs boson such as charged Higgs bosons, the decay rate of h → γγ can be enhanced due to loop effects of these charged particles. However, in a model with only one pair of singly-charged scalar bosons such as two Higgs doublet models, it is difficult to predict around 60% enhancement of the decay rate unless the mass of the charged scalar bosons is taken to be smaller than about 100 GeV 1 . The minimal Higgs triplet model (HTM) motivated from the type II seesaw mechanism [12], introducing the isospin triplet scalar field ∆, can easily explain the diphoton anomaly. Moreover, if ∆ can be inert scalar, then its neutral component can be a promising DM candidate that can strongly correlate with neutrinos. This is like a radiative seesaw models . Hence the theory can be realized at TeV scale, so be well-tested at current experiments like LHC.
In this paper, we propose a two-loop induced neutrino model with gauged B − L symme- 1 If the decay rate of the Higgs to bb mode is sufficiently suppressed compared to the SM value, the branching ratio of the Higgs to diphoton mode can be enhanced without changing its decay rate. In that case, the branching fraction of the Higgs to W W * and ZZ * modes are also enhanced [11]. try that is an extension of the HTM. In the bosonic sector, three scalar fields are introduced in addition to the SM particles; ∆ and η are an SU(2) L triplet and doublet fields, respectively which do not have vacuum expectation values (VEVs), and χ is SU(2) L singlet which acquires a VEV after the spontaneous B − L symmetry breaking 2 . In the fermionic sector, we introduce an exotic vector-like lepton with SU(2) L doublet [44], and three right-handed neutrino with SU(2) L singlet, both of which can contribute to the radiative neutrino mass.
Due to the abundant extra fields, we have several DM candidates. Also we can discuss the testability of the Higgs sector, especially, the doubly-charged scalar boson, in which we could have a discrimination to the HTM, since it couples to the exotic lepton. As a notice, such a complicated model can be realized within non-Abelian symmetries. So we briefly show how to realize our model in the appendix. This paper is organized as follows. In Sec. II, we show our model building including the Higgs potential, stationary condition, neutrino mass, and lepton flavor violation (LFV).
In Sec. III, we analyze DM phenomenologies. In Sec. IV, we analyze Higgs phenomenology including electroweak precision observables, signatures of the doubly-charged scalar boson at LHC. We summarize and conclude in Sec. V. In appendices, some result of detail calculations are write down; gauge boson two point functions, decay rate of the doubly-charged scalar boson, and the assignments for each particles in non-Abelian symmetries.  We propose a two-loop radiative seesaw model with U(1) B−L gauge symmetry which is an extended model of the minimal HTM motivated from the type II seesaw mechanism [12].

II. THE RADIATIVE SEESAW MODEL
The particle contents are shown in Table II. We add three right-handed neutrinos N c , vectorlike SU(2) L doublet leptons L ′ and L ′c , an SU(2) L triplet scalar ∆, an SU(2) L doublet scalar η and B − L charged scalar χ to the SM, where η and ∆ do not have VEV. The Z 2 parity is also imposed so as to forbid the undesired terms. As a result, the neutrino mass is obtained not through the one-loop level (just like Ma-Model [13]) but through the two-loop level and the stability of DM candidates can be assured. Where we define L ′ ≡ (N 4 , E 4 ) [44].
The renormalizable Lagrangians for Yuakawa sector and Higgs potential are given by where α, β, γ, δ, a, b are the flavor indices. In the scalar potential, the couplings λ 1 , λ 2 , λ 6 , and λ 9 have to be positive to stabilize the potential. Notice here that one straightforwardly probes the term ΦN c L, which derives neutrino mass at tree level, cannot be forbidden by any Abelian symmetries. If we introduce non-Abelian discrete symmetries [45,46], one finds some successful groups such as T 7 [47][48][49][50] and ∆(27) [51][52][53] to forbid the term, remaining that η and ∆ are one generation 3 . It implies that such an extension does not affect on analyses of dark matter and Higgs phenomenology focused on the inert scalar bosons. So we stay to analyze our model with flavor independent way hereafter for simplicity.
The scalar fields Φ, χ, η and ∆ can be parameterized as where v is VEV of the doublet Higgs field Φ satisfying v 2 = 1/( In this model, the Z 2 -even fields Φ and χ cannot be mixed with the Z 2 -odd fields ∆ and η, so that the mass matrices for the component scalar fields from Φ and χ and those from ∆ and η can be separately considered. By inserting the tadpole conditions; m 2 1 = −λ 1 v 2 − λ 7 v ′2 /2 and m 2 3 = −λ 6 v ′2 − λ 7 v 2 /2 (which is exactly the same as the results in Ref. [28]), the mass matrix for the CP-even scalar bosons in the basis of (φ 0 , χ 0 ) is calculated as The mass matrices of the Z 2 -odd scalar bosons are respectively obtained in the basis of (∆ ± W , η ± W ), (Re∆ 0 , Reη 0 ) and (Im∆ 0 , Imη 0 ) by where we define as follows: The mass eigenstates for the Z 2 -even Higgs bosons and those for the Z 2 -odd scalar bosons can be defined by introducing the mixing angles α, β, γ and δ as where h can be regarded as the SM-like Higgs boson. These mixing angles are expressed as The mass eigenvalues are calculated by using the mixing angles given in Eq. (II.13) and the mass matrices given in Eqs. (II.5) and (II.6) as (II.14) The doubly-charged triplet scalar bosons do not mix with others, and those masses are given by We note that there is a characteristic relations for the mass spectrum among the tripletlike scalar bosons ∆ ±± , ∆ ± , ∆ 0 R and ∆ 0 I in the limit of µ → 0 as Therefore, two of four mass parameters for the triplet-like scalar bosons are determined by using above two equations. The same relations also appear in HTM when the lepton number violating coupling constant is taken to be zero [54].

B. Neutrino mass matrix
The active neutrino mass matrix (depicted in Fig.1) through two loop contribution is given by where As can be seen in the above equation, we can reproduce observed neutrino masses ∼ O(0.1) eV and the mixing data because of many parameters.

C. Lepton Flavor Violation
We investigate the LFV process ℓ α → ℓ β γ (ℓ α , ℓ β = e, µ, τ ) as shown in Fig. 2. The experimental upper bounds of the branching ratios are B (µ → eγ) ≤ 2.4 × 10 −12 [55], [56]. The branching ratios of the processes ℓ α → ℓ β γ are calculated as G F is the Fermi constant and the loop function F 2 (x) is given by The µ → eγ process gives the most stringent constraint. The mixing angle sin β directly does not contribute to the neutrino mass, however, sin 2β should not be zero to retain µv = 0 in Eq. (II.13).

III. DARK MATTERS
We discuss DM candidates in this section. We have six DM candidates in general; that is, , the lightest one of N c , and N 4 . However N 4 cannot be the candidate because N 4N4 annihilation via Z boson gives too large cross section to obtain the observed relic density Ωh 2 ≃ 0.11 [4] as well as too large scattering cross section in the direct detection search [7] 4 . Moreover, we restrict ourselves that only the real part of η R(I) and ∆ R(I) are considered as a DM candidates, since the DM property of the imaginary part is more or less the same as the real one. Hence we analyze three DM candidates; η R , ∆ R , the lightest one of N c , below. Hereafter we symbolize the DM mass as m DM .

A. Fermionic Dark Matter
We discuss a fermionic DM candidate N c , assuming the following mass hierarchy M 1 < M 2 < M 3 for the right-handed neutrinos N c i . Notice here that the DM mass range be less than M = 100 GeV to avoid the too short lifetime of DM.
WMAP: At first, we analyze the DM relic density from WMAP. We have only the schannel process via the Higgs bosons h/H as shown in the upper panel of Fig. 3. The effective cross section to f f /W + W − /2Z 0 is given as where V = W ± , Z 0 and we neglected the light quarks contributions and put only bottom , is the relative velocity of incoming DM. Here we neglect the contribution of H, assuming the mass is enough heavy.
The SM-like Higgs mass is fixed to m h = 125 GeV. Notice here that the channel of V V . Moreover, in m DM ≤ m h /2, the channel h → 2DM is also added and given by which is known as an invisible decay and recently reported by the LHC experiment that the branching ratio B inv is excluded to the region 0.4 ≤ B inv [60].
Direct Detections: Let us move on to the discussion of direct detections. Our DM interacts with quarks via Higgs exchange. Thus it is possible to explore DM in direct detection experiments like XENON100 [7]. The Spin Independent (SI) elastic cross section σ SI with nucleon N is given by is the DM-nucleon reduced mass and the heavy Higgs contribution is neglected. The parameters f N q which imply the contribution of each quark to nucleon mass are calculated by the lattice simulation [61,62]  Ref. [63].
Considering all the above constraints, we find that the allowed region is sharp at around m h /2.

B. Bosonic Dark Matters
We discuss bosonic DM candidates η R and ∆ R [64]. Notice here that the DM mass range be less than M W = 80 GeV to satisfy the constraint of the antiproton no excess reported by PAMELA [65] as well as WMAP 5 .
WMAP: At first, we analyze the DMs relic density from WMAP. We have two annihilation modes; t and u channel of 2η R /∆ R → N 4 (E 4 ) → 2ν(ℓl) and s-channel of 2η R /∆ R → h/H → ff 6 . However since the t and u channel does not have s wave contribution, the dominant cross section is given only through the s channel [31] as shown in the lower diagram of Fig. 3.
The effective cross section to f f is given as When our DMs are less than m h /2, the invisible decay width is give by [31] Direct Detections: In the direct detection, the spin independent elastic cross section σ SI with nucleon N is given by (III.10) where µ DM and f N q has been defined in the fermionic part. We here comment on the scenario based on the neutral component of the inert triplet field being the DM candidate, which has been discussed in Ref. [63]. Such a scenario is severely constrained by the direct detection experiments because of the Z boson exchanging contribution if the CP-even scalar boson and the CP-odd scalar boson from the triplet field are degenerate in mass. According to Ref. [63], the mass of the DM candidate has to be around 2.8 TeV in the model with Y=1 inert triplet field in order to satisfy both the WMAP data and the direct search data. However, this result cannot be applied to our model, because we can take a mass splitting between ∆ 0 R and ∆ 0 I due to the mixing between η and ∆. Thus, we can avoid the Z boson exchanging contribution to the direct search experiments.
Considering all the above constraints, we find that the allowed region is the same as the fermionic case; that is, around m h /2. As a result, we assume to analyze that DM mass (especially ∆ 0 ) is m h /2 in the next section.

A. Electroweak precision observables
We discuss the constraints of the Higgs parameters from the electroweak precision observables; i.e., the Peskin-Takeuchi S, T and U parameters [66]. If the mixing angle α is taken to be zero, the new physics contributions to the S and T parameters can be separated from those from the SM, so that we here assume α = 0 for simplicity. Then the new physics contributions to S and T are calculated as where Σ 11 T , Σ 3Q T and Σ 33 T are obtained by calculating the 1PI diagrams of the gauge boson two point functions at the one-loop level whose analytic expressions are given in Appendix A. The

B. Signature of ∆ ±± at LHC
We discuss how our model can be tested at collider experiments. We focus on the signature from the doubly-charged scalar bosons ∆ ±± , because appearance of ∆ ±± is one of the striking properties of the model. In order to focus on the phenomenology of the triplet-like scalar bosons, we assume that the mixing between ∆ and η are taken to be quite small (β ≃ γ ≃ δ ≃ 0), and the masses of η ± and η 0 are much heavier than those of the triplet-like scalar bosons.
There are an indirect way and a direct way to identify existence of ∆ ±± . in the case with m ∆ ±± to be smaller than about 200 GeV [68] without contradiction with the constraints from the vacuum stability [69] and the perturbative unitarity [69,70]. The prediction of the deviation in the decay rate of h → γγ in our model is almost the same as that in the HTM as long as the contributions from η ± are neglected.
Discovery of ∆ ±± can be a direct evidence of our model. We consider the case where a lighter neutral scalar boson from the triplet field (∆ R or ∆ I ) is assumed to be DM. In that case, the neutral scalar boson should be the lightest of all the triplet-like scalar bosons; i.e., m ∆ 0 ≤ m ∆ + ≤ m ∆ ++ to guarantee the stability of DM, and its mass is around the half of the Higgs boson mass m h to satisfy WMAP data and direct detection experiments. In addition, the mass differences among the triplet-like scalar bosons are restricted from the S and T parameter as shown in Fig. 4; e.g., ∆m is constrained to be less than 15 GeV in the case with m ∆ 0 = 63 GeV. This implies that the upper limit for m ∆ ++ is about 90.5 GeV by using the mass relation given in Eq. (II.16).
A search for doubly-charged Higgs bosons has been done at LEP [71], Tevatron [72] and LHC [73]. All these searches have been done under the assumption where doubly-charged Higgs bosons decay into the same sign dilepton. The most stringent lower bound for the mass of doubly-charged Higgs bosons is about 400 GeV given at LHC [73]. In our model, ∆ ±± cannot decay into the same sign dilepton associated without any other particles, because they are the Z 2 odd particles. In that case, the mass bound given at LHC cannot be applied to that in our model.
The decay branching fractions for (1) and (2) are determined by the magnitude of y ∆ , ∆m 7 The magnitudes of the branching fractions of ∆ ±± in the same sign dilepton modes: E ± 4 e ± , E ± 4 µ ± and E ± 4 τ ± depend on the value of y i ∆ . In the following discussion for the collider phenomenology, we do not specify the flavor of ℓ ± . In Fig. 5, the branching fraction of ∆ ±± is shown as a function of M. We take m ∆ ++ = 90.5 GeV and m ∆ + = 78 GeV which correspond to the case with m ∆ 0 = 63 GeV and ∆m = 15 GeV. The Yukawa coupling y ∆ is taken to be 0.1 and 0.01. It is seen that the main decay mode is changed from ∆ ++ → E + 4 ℓ + to ∆ ++ → ∆ + W + * when M is getting larger values. For example, 50% of B(∆ ++ → E + 4 ℓ + ) can be obtained in the case of M ≃ 89 GeV (84 GeV) and y ∆ = 0.1 (0.01).
In the following, we discuss the case where ∆ ±± mainly decay into the same sign dilepton ). In that case, the exotic lepton mass M should be between m ∆ ++ and m ∆ + , otherwise there is no possible decay channel for E ± 4 . As an benchmark scenario, we take the following mass spectrum and the coupling constant which is allowed from the electroweak precision data and also the LFV data: In this parameter set, the branching fraction of ∆ ±± → E ± 4 ℓ ± is about 99%, and that of ∆ ± → W ± ∆ 0 and E ± 4 → ∆ ± ν are 100%, where ∆ 0 is ∆ R or ∆ I . Thus, the decay process of ∆ ++ is expected as follows: so that the final state contains the same sign dilepton and the missing energy.
We then discuss the signal of ∆ ±± at LHC in the parameter set given in Eq. (IV.4). At LHC, ∆ ±± can be mainly produced via the Drell-Yan processes: qq → γ * /Z * → ∆ ++ ∆ −− and qq ′ → W ± * → ∆ ±± ∆ ∓ . Thus, the signal events are expected to be 8 The bosons H ± in HTM can decay into W ± Z in this case as long as the mass difference between H ± and H ± is not too large. Therefore, the same events as expressed in Eqs. (IV.6) and (IV.7) can appear in the following way:  also be The invariant mass distributions in the system of the same sign dilepton M ℓ + ℓ + can be useful to discriminate between our model and HTM. In Fig. 6, the distribution for M ℓ + ℓ + in the ℓ + ℓ + ℓ − ℓ − E T / system is shown in which the number of event is assumed to be 10 4 and We then conclude that the four lepton events with the missing energy ℓ + ℓ + ℓ − ℓ − E T / from our model and HTM with the diboson decay and the cascade decay of H ±± may be able to be discriminated by using the M ℓ + ℓ + distribution and the number of event.

V. CONCLUSIONS
We have constructed a two-loop radiative seesaw model that provides neutrino masses in a TeV scale theory. We have also studied DM properties, in which our model has fermionic (N c ) or bosonic DM (η 0 , ∆ 0 ) candidate with the same mass scale, which is at around m h /2, from the constraint of WMAP and the direct detection search in XENON100. We have also m ∆ R , m ∆ I ) + (c γ c δ + 2s γ s δ ) 2 B 5 (p 2 , m η R , m η I ) + (−2c γ s δ + s γ c δ ) 2 B 5 (p 2 , m ∆ R , m η I ) + (−2s γ c δ + c γ s δ ) 2 B 5 (p 2 , m η R , m ∆ I ) , (A.4) where c θ = cos θ and s θ = sin θ. In the above equations, B 3 (p 2 , m 1 , m 2 ) and B 5 (p 2 , m 1 , m 2 ) functions [75] are respectively expressed in terms of the Passarino-Veltman functions [76] by The functions Σ 11 T , Σ 3Q T and Σ 33 T are given in terms of above the gauge boson two point functions by where λ(x, y) and G(x, y) are the phase space functions which are given by λ(x, y) = 1 + x 2 + y 2 − 2x − 2y − 2xy, (B.6) G(x, y) = 1 12y 2 (−1 + x) 3 − 9 −1 + x 2 y + 6 (−1 + x) y 2 + 6 (1 + x − y) y −λ(x, y) tan −1 −1 + x − y −λ(x, y) + tan −1 −1 + x + y −λ(x, y)  Here we show an example how to realize our model for the lepton sector by using non-Abelian discrete symmetry. The minimal extension is to introduce T 7 flavor symmetry [47][48][49][50]. Each of the field assignment is given in Table III, where the other assignments are same. The extended Lagrangian to Eq. (II.1) is modified as 9 where ω ≡ e 2iπ/3 . Here notice that all the terms except χN c N c are diagonal. It suggests that the observed neutrino mass and lepton mixing can be obtained only through the χN c N c term. After the B − L spontaneously breaking; χ = v (i) 2 , the right- 9 The charged-lepton sector has to be improved in order to forbid the universal Yukawa coupling of the SM-like Higgs boson, since it has been ruled out by the current Higgs boson search data at LHC. Straightforward ways to solve it is to change the flavor symmetry group and/or introduce some additional Higgs fields.
handed neutrino mass matrix is given by As a result, we can easily find the observed neutrino mass difference and their mixings by controlling each of the VEV. In the Higgs potential, the modifications are as follows: