Phenomenology in supersymmetric neutrinophilic Higgs model with sneutrino dark matter

We study a supersymmetric neutrinophilic Higgs model with large neutrino Yukawa couplings where neutrinos are Dirac particles and the lightest right-handed (RH) sneutrino is the lightest supersymmetric particle (LSP) as a dark matter candidate. Neutrinophilic Higgs bosons need to be rather heavy by the precise determination of the muon decay width and dark radiation constraints for large Yukawa couplings. From the Large Hadron Collider constraints, neutrinophilic Higgsino mass need to be heavier than several hundred GeV or close to the RH sneutrino LSP mass. The latter case is interesting because the muon anomalous magnetic dipole moment can be explained with a relatively large lightest neutrino mass, if RH sneutrino mixings are appropriately fine tuned in order to avoid stringent lepton flavor violation constraints. Dark matter is explained by asymmetric RH sneutrino dark matter in the favoured region by the muon anomalous magnetic dipole moment. In other regions, RH sneutrino could be an usual WIMP dark matter.


I. INTRODUCTION
A scalar field could be responsible for the breakdown of a large gauge symmetry and the generation of the masses of gauge bosons and fermions. In fact, a scalar boson discovered at the Large Hadron Collider (LHC) appears to be the Higgs boson in the standad model (SM) of particle physics [1,2]. After the spontaneous symmetry breaking, the mass of each fermion, namely quarks and charged leptons, is given by the product of each Yukawa coupling constant and the vacuum expectation value (VEV) of the Higgs field.
However, one might suspect a special mechanism for the generation of neutrino masses and a special reason for its smallness, because masses of neutrinos are very small compared with other SM fermions. One approach is the so-called seesaw mechanism with very heavy righthanded (RH) Majorana neutrinos, where the smallness of neutrino mass can be understood as a consequence of the high scale of RH neutrino mass [3].
Another approach is the neutrinophilic Higgs model [4][5][6]. In this model, neutrino has Dirac mass terms generated by another Higgs field whose VEV is much smaller than that of the SM Higgs, where the smallness of neutrino mass is a consequence of the smallness of the other Higgs VEV. In this case, the neutrino Yukawa couplings can be much larger than those with only the SM Higgs field if the Higgs VEV is small, because the neutrino mass is the product of the VEV of the neutrinophilic Higgs field and the Yukawa couplings.
The large neutrino Yukawa couplings in the neutrinophilic Higgs model shows many interesting features, such as the possibility of the RH sneutrino as thermal dark matter (DM) [7][8][9] or the low scale thermal leptogenesis [10,11]. In addition, the large Yukawa couplings have more implications in the flavour structures and astrophysical phenomenon.
In this paper, we examine various phenomenological aspects of the large Yukawa interactions in the supersymmetric extended neutrinophilic Higgs model. Those include the anomalous magnetic moments of muon, lepton flavour violation, experimental constraints on the couplings and the masses of new particles, and cosmological and astrophysical constraints including indirect detection signatures by asymmetric sneutrino DM through gamma ray and neutrinos.
In Section II, we consider the constraints on the Yukawa couplings from neutrino masses and mixings, the muon decay, collider searches, and lepton flavour violation. Taking these constraints into account, we study the possibility to explain the muon anomalous magnetic moment in this model. In Section III, we consider the cosmological constraints from dark radiation, and in Section IV we study the possibility of the lightest RH sneutrino as DM and the astrophysical constraints on the models. We conclude our study in Section V. We provide some formulas in the Appendicies.

II. PHENOMENOLOGICAL CONSTRAINTS AND IMPLICATIONS
In a supersymmetric model, the interaction is described by the term in superpotential where L α is the lepton doublet of the Standard Model, ν Ri is a gauge singlet RH neutrino superfield and H ν is a scalar doublet in addition to the standard two Higgs doublets in the MSSM and i, α = 1, 2, 3 denotes the generation index. In the so called neutrinophilic Higgs model with the neutrino Dirac mass given by a small VEV of neutrinophilic Higgs field, H 0 ν , neutrino Yukawa couplings can be as large as of the order of unity. We consider the Yukawa interaction of Dirac neutrino and neutinophilic Higgs as After the electroweak symmetry breaking, the neutrinophilic Higgs field develops the VEV H 0 ν = v ν / √ 2 and generates the neutrino mass. Since neutrinos are Dirac particles in this model, their mass matrix is simply proportional to 3 × 3 neutrino Yukawa coupling matrix.
The neutrino mass matrix, or equivarently Yukawa interaction, is given by with here and hereafter we assume the Yukawa couplings are real, for simplicity. Therefore, the left-and right-handed neutrinos compose the four component Dirac mass eigenstates.
In a supersymmetric theory, there exists the Yukawa interaction of scalar RH neutrino with the same Yukawa coupling of Eq. (2), given by Here, RH sneutrinosν Ri are defined as the superpartner of each ν Ri . If the lightest RH sneutrino is the lightest supersymmetric particle (LSP), it can be a good candidate for DM as shown in Refs [7,8]. Throughout this paper, we consider this case of RH sneutrino DM and study the phenomenological constraints and implications.

A. Neutrino mass and mixing
Without loss of the generality, we can regard that the ν Ri is already mass eigenstate.
The neutrino mass matrix is diagonalized with the Maki-Nakagawa-Sakata (MNS) matrix U MNS , which transfers LH neutrinos from mass eigenstates (ν L,i ) to flavor eigenstates (ν L,α ) The neutrino oscillation data gives two independent mass squared differences and three mixing angles [12], The neutrino Yukawa couplings can be expressed in terms of these neutrino oscillation parameters as Here, we neglect for simplicity any CP phase and take y ν to be real. In the second line, θ 23 ≃ π/4 and sin θ 13 ≪ 1 are used, while we use the full formula of MNS matrix in our numerical calculation.  The upper bound on the sum of neutrino masses is provided by cosmological arguments.
Since its value strongly depends on data set and a cosmological model used in the analysis, for reference, we here just quote one of less conservative values m ν < 0.23 eV, from Ref. [13].
In Table I, we list four benchmark points used in our analysis for a given lightest neutrino mass,m 1 = 0, 0.07 eV for normal hierarchy (m 1 < m 2 < m 3 ) and m 3 = 0, 0.05 eV for inverted hierarchy (m 3 < m 1 < m 2 ) of neutrino masses.
with R being a rotation matrix and the variables, σ 12 , σ 23 , σ 13 , are the corresponding mixing angle. The Yukawa interaction Eq. (5) becomes for the mass eigenstates. Thus, we defineỹ iα ≡ (S T y ν ) iα for Yukawa couplings of the RH sneutrino, lepton and Higgsino.
1. loop enhancement of µ → eν e ν µ First, let us estimate the decay width of the main decay mode. In the following, p 1 , p 2 , q 1 and q 2 are the momentum of incoming µ, outgoing e, ν µ andν e respectively. The amplitude of W boson mediated process is given by where g 2 is the SU(2) gauge coupling and M W is the W boson mass. That of RH neutrino and Higgs bosons loop is given by with the auxlinary function F 2 (x), which is defined in the Appendix. In this estimation of We obtain with the auxlinary function F 3 (x, y, z, w), which is given in the Appendix. Here m µ is the muon mass and v ≃ 246 GeV is the VEV of the SM Higgs field and we keep only the leading order loop corrections, namely the interferenece between tree and one loop.
Due to the lepton flavor violating neutrino Yukawa coupling, the flavor of the final state neutrino can be different from muon-type and anti-electron-type. However, this decay mode has only loop induced new contribution and is suppressed compared to the other contribution to the decay which has the interference term between tree-level and loop induced term. Thus, this mode is negligible.
This decay mode with RH neutrinos in the final state is induced by the tree level process mediated by the neutrinophilic charged Higgs boson [14]. A worth noting feature is that this is not a V −A interaction but a scalar interaction. The amplitude of the process µ → ν RiνRj e mediated by H + ν is given by Here, we normalise the effective coupling with G F , the Fermi constant measured in the experiment, and we introduce a new parameter g S LL defined by [15] The partial width is estimated as As we will see later, it turns out that this decay mode has to be highly suppressed due to the well consistency with the SM. Thus, in fact, this would not be significant for muon decay contribution.

Total
From Eqs. (17) and (21), the final total decay width of muon is given by By comparing Fermi constant G F measured from muon decay width and other SM quantities, the consistency of the SM can be tested [16,17]. If we express the Fermi coupling constant with a parameter which stands for a correction due to new physics by the new physics contribution ∆ is constrained to be [18] ∆ = 0 ± 0.0006.
The last term in Eq. (22) comes from the non-(V −A) interaction via H + ν mediation. The muon decay experiments can not measure helicity of produced neutrinos, but the inverse decay of muon, ν µ + e → µ + missing, well confirms the V − A form interaction and leave a small room for scalar interaction as [19] With Eq. (20), we find the constraint on the H + ν mass and Yukawa couplings. First, let us consider the constraints on charged Higgs boson from muon decay in the decoupling limit of Higgsino of the second term. Then the first term in Eq. (22) gives dominant contribution to ∆ and the third term to g S LL . The constraints on charged Higgs mass and Yukawa coupling is shown in Fig. 1, where for simplicity we take MH− ν = MH 0 ν . For Yukawa couplings of the order of unity, the mass of charged Higgs must be heavier than around 600 GeV 1 . The cosmological consideration of dark radiation imposes the further stringent lower bound on the masses of those extra Higgs bosons, as we will see later in Section III.
Next, we need to consider the decoupling limit of very heavy Higgs boson as found just above, the the dominant correction comes from the second term in Eq. (22) from the sneutrino and chargino loop, namely In Fig. 2, we show the contour plot of 2∆ for O(1) Yukawa couplings in the plane in the left window , and that in the plane of GeV in the right window. We can see that the charged HiggsinoH − ν need to be heavier than a few hundreds GeV for degenerate sneutrino case, MÑ j ≃ MÑ i , or two of RH sneutrinos need to be several times heavier than chargino with 100 GeV mass and only one RH neutrino can be light. exchange for pp collision. They subsequently decay to leptons (l i ) and the lightest sneutrino where ν i is the mass eigenstate of neutrino, and we assume thatÑ DM is the lightest RH sneutrino as dark matter. For a case where those particle are too heavy to be produced at the on-shell, the production cross section is kinematically very suppressed. For cases that RH sneutrinos are light but degenerate, only three-body decay is possible via virtual Higgsino (H ν ) and the energy of the produced lepton is much suppressed.
The corresponding diagrams are given in Fig. 3. The final decay products are multi leptons plus a large missing energy byÑ DM , with n being an integer, or mono photon plus a large missing energy byÑ DM ,

Constraints from LEP
As far as the LEP bound on chargino is concerned, if the mass of the Higgsinos are greater than the threshold energy scale for e + e − collision ( √ s/2 ≃ 104 GeV), it is natural to expect that the constraints are relaxed drastically. Thus, we take mH ν 103.5 GeV [22] as the kinematical lower bound to avoid the LEP searches for direct production of charginos.
For the direct production ofÑ i , the t-channelH ν exchange for e + e − collision is dominant as in FIG. 4. e + e − → γ + missing E.
In this case, the result of monophoton searches for the MSSM neutralino e + e − →χ 0 1χ 0 1 γ can be used to constrain the mass and couplings of the RH sneutrino to electrons [23]. In this analysis, using the effective operatorχeēχ/Λ, the cutoff scale Λ should be greater than about 330 GeV for the fermion dark matter mass m χ < 80 GeV.
The neutrinophilic Higginos (H ν ) production and its sequent decay into RH sneutrino.
H ν s are produced by the s-channel of Z 0 /γ, t-channel ofÑ , and s-channel of W ± exchanges. from the interaction term yÑ DMψH ν P L e + h.c.. We take mH ν = 110 GeV, and the cases with two Yukawa couplings (y = 0.65, 0.7) are presented (Blue and Magenta). E beam is taken as 100 GeV, which is the average value for the LEP search. We find that our model parameters (y and mH ν ) are constrained by the LEP search as did in Ref. [23], i.e. the region above the red dashed line is ruled out. decay into light leptons, namely e or µ. For the decay into τ lepton, the bound is relaxed and requires MH 350 GeV and MÑ 150 GeV [24].

D. lepton flavor violation (µ → eγ)
The off-diagonal components of Yukawa couplings (y ν ) iα in Eq. (2) induce lepton flavor violating (LFV) decay of leptons. The general effective operator can be written as with σ µν = i 2 [γ µ , γ ν ]. The decay rate of l j → l i γ is given by For the muon decay, µ → eγ, the branching ratio is given by The present bounds on the branching ratios of the LFV decays are [25,26] Br(µ → eγ) < 5.7 × 10 −13 (90%C.L.), For our model with Yukawa interactions in Eq. (2), we obtain with F (x) being an auxlinary function. The experimental limits (34) give strong bounds on Yukawa couplings as well as masses of mediated particle, MÑ k , MH− ν and M H ± ν , as we will show. In fact, for O(100) GeV masses of sneutrinos and the neutrinophilic chargino, we find the LFV decaying branching ratios are of O(10 −6 ), those are very large compared with current bounds (34). Since Yukawa couplings are fixed from the neutrino mass and are the order of unity, the charged Higgs must be heavier than around 10 TeV in the second term.
On the other hand, for the first term, we have a possibility that one of sneutrinos and charged Higgsinos are relatively light in the case thatỹ = S T y ν are suitably aligned by appropriate sneutrino mixings σ ij in such a way that the flavor-violating processes are suppressed enough.
Such mixing angles can be found by requiring some off-diagonal components ofỹ = S T y ν to be almost vanishing.
Thus, this discrepancy has been regarded as a hint and provided a motivation to investigate new physics beyond the standard model of particle physics. The resultant electric and magnetic dipole moment of l j lepton are given as The additional contribution to the induced magnetic moment of muon in the supersymmetric neutrinophilic Higggs model with large Yukawa couplings is given by where we assumed Yukawa cpuplings are real and the negligible charged Higgs boson contribution is omited. We might expect a large g − 2 of the muon for light sneutrinos and the lightH ν -like chargino because Yukawa coupling constants are O(1). However, as mentioned above, the experimental limits on the lepton flavor violation in Eq. (34) are very stringent and we need a special mixing of sneutrinos.

F. Compatibility in benchmarks
As mentioned previous subsections, LFV constraints are very stringent. Indeed, for cases of vanishing lightest neutrino mass m 1 as in benchmark point 1 and 3 in Table I, we could not find viable parameter sets. Thus, here we mention viable parameter sets based on the benchmark point 2 and 3 in the Table I. At first, for the benchmark 2, we find that LFV constraints are avoided with the sneutrino mixing angles (σ 12 , σ 23 , σ 13 ) ≃ (0.75, 0.68, 0.20), and the resultantỹ is given bỹ the lightest RH sneutrino (Ñ 2 ) and the electron is negligibly small. This small coupling automatically suppresses the LEP mono-photon constraint from Eq. (30).
In Fig. 6, we show the viable sneutrino mixing angles from the constrains on the LFV processes for the benchmark 2. In those plots, we fixed one mixing angle σ 12 = 0.75, the neutrinophilic Higgsino mass MH ν = 110 GeV, and the lightest sneutrino mass MÑ 2 = 60 GeV. Heavy sneutrino masses, (MÑ 1 , NÑ 3 ) are taken from around TeV to 10 TeV As the figures show, the constraint from µ → eγ is most serious, and very small regions are allowed.
Including the constrains from τ → µγ, we find that the value of σ 23 should be around 0.68, which corresponds to (S T y ν ) 2e ≈ 0. The constraints do not restrict the value of σ 13 much for the case with two heavy sneutrino masses. In the allowed parameter space, the sizable δa µ can be obtained as denoted by the blue colored region.
In Fig. 7, we show the viable parameter space for the benchmark 2 with contours of the contribution to the muon anomalous magnetic moment from Eq. (39) in the plane of the mass of neutrinophilic Higgsino and the RH sneutrino mass MÑ 2 , which is the lightest supersymmetric particle. Here, we use MÑ 1 = 7 TeV and MÑ 3 = 10 TeV for reference. The region of Red and Orange color respectively show 1σ and 2σ range of Eq. (36). The blue region corresponds mÑ 2 > mH ν . The yellow region, where the mass spliting between sneutrino and chargino is too large, is constrianted by the LHC results. In the blue region, MÑ > MH ν is realized. The yellow region, where the mass spliting between sneutrino and chargino is too large, is constrianted by the LHC results.
The scatterings between charged lepton and RH neutrinos are mediated by the neutrinophilic charged Higgs boson H ± ν . We obtain with s being the energy at the center of mass frame. Taking thermal average, we find σv ≃ |y ν y ν | 2 32πM 4

Decoupling condition
The decoupling condition of RH neutrino at the quark-hadron transition epoch is expressed as which is rewritten as Therefore the neutrinopholic Higgs must be heavier than around 3 TeV for order of unity Yukawa couplings. We note that the similar bound has already been obtained but H 0 ν and A 0 ν contributions were missing in Ref. [9,20]. Here we have re-estimated and corrected it.

IV. DARK MATTER
The lightest RH sneutrino is stable when R-parity is preserved and can be a good candidate for dark matter. The possibility in the neutrinophilic Higgs model was suggested in Ref. [7,8] by two of the present authors. In this section we generalise the previous results considering the benchmark points in the previous section and examine the cosmological and astrophysical phenomenon.

A. Relic density of dark matter
Due to the large Yukawa coupling in Eq. (2), the RH sneutrino interacts with fermions and Higgsinos efficiently so that they could be in the thermal equilibrium at high temperature.
In the rapidly expanding early Universe, those RH sneutrinos decouple from the thermal plasma and the comoving abundance is conserved after that. The relic density of WIMPs is determined by the annihilation cross section which determines the freeze-out temperature of DM. However, for complex fields, there might be the nonvanishing DM asymmetry. With a large DM asymmetry, the final relic density of DM may depend on the annihilation cross section and the DM asymmetry [29][30][31][32][33][34][35]. This is the case for light RH sneutrino DM in our scenario.
The annihilation cross section of RH sneutrino DM is dominantly determined by the annihilations into the leptons, that is given in partial wave expansion method by [7,8] σv There is another subdominant contribution from the induced annihilation into photons, where we used the approximation of M Hν = M H ′ ν = Ml for simplicity. In fact, it gives small subdominant contribution to determine the relics density of DM in our consideration with Since the RH sneutrinos were in the thermal equilibrium, the asymmetry could be generated from non-zero baryon asymmetry during the sphaleron process. The asymmetry of RH sneutrinos will depend on the specific model of baryogenesis, mass spectrum and the electroweak phase transition. In the simple case, the leptonic asymmetry is expected to be the order of baryon asymmetry, as 10 −10 [9]. In this paper, in order to see the asymmetry dependence, we treat it as a free parameter taking a value of a certain range.
For the WIMP with a nonvanishing asymmetry, the resulting relic density can be estimated by [36] Here with where x F (x F ) denotes the value of x at the "freeze out" time of (anti-)dark matter particle.
The asymmetry of dark matter is given by In the figure 9 we show the contour plot of the corresponding DM asymmetry to give the correct relic density of DM. For a given asymmetries C = (5 × 10 −11 , 10 −11 , 5 × 10 −12 , 3 ×

C. Indirect signal from sneutrino dark matter
For the asymmetric DM, their annihilation in the galaxy is negligible. However, they can scatter off the cosmic rays and produce secondary particles such as gamma-ray or neutrinos [42]. Since we are considering the large Yukawa couplings, the indirect signature might be very promising or even harmful. However, in our benchmark point 2 and 3, the Yukawa coupling between the lightest RH sneutrino (N 2 ) and the electron is quite negligible as a consequence of suppressing LFV and therefore the indirect signature is also very suppressed.
For symmetric DM case with the heavyH ± ν , the DM annihilation signal can be seen most likely as a gamma-ray line because annihilation into a fermion pair is helicity suppresed [7].

D. Decay of Cosmic Neutrino Background
As the muon can decay to the electron through the charged neutrinophilic Higgs(ino), and sneutrino loops, the heavier neutrino can decay to the lighter one through similar diagrams.
Assuming that the mass of lighter one is much smaller than that of the heavier one, the decay rate of neutrino is where Although the GIM mechanism is not applied, so there is no suppression by the mass of charged leptons, the suppression by small neutrino mass as 5th powers is enough to satisfy the present constraint on the life-time of the neutrinos, which is from the analysis of the cosmic infrared background [43].

V. CONCLUSION
We have studied an extended supersymmetric model where neutrinos are Dirac particle and those masses are given by large neutrino Yukawa couplings and a small VEV of the neutrinophilic Higgs field. Provided the lightest RH sneutrino is LSP as the dark matter candidate, we have examined various aspects of the model with Dirac Yukawa couplings of the order of unity.
By only considering the muon decay width, it turns out that the neutrinophilic Higgs bosons must be heavier than several hundreds GeV and some supersymmetric particles among RH sneutrinos and neutrinophilic Higgsino need to be heavier than several hundreds GeV. In fact, we have found that the neutrinophilic Higgsino and one of the RH sneutrino can be relatively light of the order of 100 GeV if the other two RH neutrinos are heavy enough. The current collider experiment, most importantly the LHC, constraints require a viable parameter space;H ± ν is heavy enough, or the mass difference ofH ± ν and the lightest RH sneutrino is smaller than about 50 GeV, according to Ref. [24]. The LEP constrains thẽ H ± ν −Ñ DM − e coupling to be less than about 0.6. For general mixing of RH sneutrinos, due to the large flavor mixings of neutrino sector, the lepton flavor violating processes induced through neutrino Yukawa interactions are also typically as large as 10 −6 in the decay branching ratio, with new particles of O(100) GeV mass. Only with appropriately tuned RH sneutrino mixings, we can avoid the LFVs.
With the chosen parameters, we found that the deviation of the muon g − 2 can be explained with a relatively large lightest neutrino mass around m 1 ≃ 0.05 eV and the lightest RH sneutrino and Higgsino mass, MÑ = 10−100 GeV and MH ν = 60−160 GeV respectively.
In other words, if the muon g−2 is explained in this model, then m 1 can not be so small. As a result of tuned RH sneutrino mixings, theH ± ν −Ñ DM −e coupling is almost vanishing, which means the LEP data does not significantly constrain this model and the international linear collider also would not be able to produce a mono-photon signal, while theH ± ν −Ñ DM − µ coupling is about unity. A muon collider can easily test this model if it will be indeed constructed, as the Fermilab plans [44].
In this muon g − 2 favored parameter region, the DM relic density is explained by RH sneutrino with the asymmetry of C ∼ 5 × 10 −12 . If we do not mind the deviation of the muon g − 2, RH sneutrino dark matter could be an usual WIMP with heavierH ± ν .