# Phenomenology in supersymmetric neutrinophilic Higgs model with sneutrino dark matter

###### Abstract

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.

## 1 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 standard model (SM) of particle physics Aad:2012tfa (); Chatrchyan:2012ufa (). 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 right-handed (RH) Majorana neutrinos, where the smallness of neutrino mass can be understood as a consequence of the high scale of RH neutrino mass Seesaw ().

Another approach is the neutrinophilic Higgs model Ma (); Wang (); Nandi (). 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) Choi:2012ap (); Choi:2013fva (); Mitropoulos:2013fla () or the low scale thermal leptogenesis HabaSeto (); HSY (). 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 2, 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 3, we consider the cosmological constraints from dark radiation, and in Section 4 we study the possibility of the lightest RH sneutrino as DM and the astrophysical constraints on the models. We conclude our study in Section 5. We provide some formulas in the Appendices.

## 2 Phenomenological constraints and implications

In a supersymmetric model, the interaction is described by the term in superpotential

(1) |

where is the lepton doublet of the Standard Model, is a gauge singlet RH neutrino superfield and is a scalar doublet in addition to the standard two Higgs doublets in the MSSM and 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, , neutrino Yukawa couplings can be as large as of the order of unity.

We consider the Yukawa interaction of Dirac neutrino and neutrinophilic Higgs as

(2) |

After the electroweak symmetry breaking, the neutrinophilic Higgs field develops the VEV and generates the neutrino mass. Since neutrinos are Dirac particles in this model, their mass matrix is simply proportional to neutrino Yukawa coupling matrix. The neutrino mass matrix, or equivalently Yukawa interaction, is given by

(3) |

with

(4) |

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

(5) |

Here, RH sneutrinos are defined as the superpartner of each . If the lightest RH sneutrino is the lightest supersymmetric particle (LSP), it can be a good candidate for DM as shown in Refs Choi:2012ap (); Choi:2013fva (). Throughout this paper, we consider this case of RH sneutrino DM and study the phenomenological constraints and implications.

### 2.1 Neutrino mass and mixing

Without loss of the generality, we can regard that the is already mass eigenstate. The neutrino mass matrix is diagonalized with the Maki-Nakagawa-Sakata (MNS) matrix , which transfers LH neutrinos from mass eigenstates () to flavor eigenstates () ,

(6) |

The neutrino oscillation data gives two independent mass squared differences and three mixing angles PDG (),

(7) |

The neutrino Yukawa couplings can be expressed in terms of these neutrino oscillation parameters as

(8) | |||||

Here, we neglect for simplicity any CP phase and take to be real. In the second line, and 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

(9) |

from Ref. Ade:2013zuv ().

In Table 1, we list four benchmark points used in our analysis for a given lightest neutrino mass, for normal hierarchy () and for inverted hierarchy () of neutrino masses. To estimate Yukawa coupling constants, we have to fix one extra free parameter . Example value sets of in Tab. 1 are obtained under the assumption that the largest coupling is unity. The actual value can be somewhat larger or smaller by changing .

Point | 1: normal | 2: normal | 3: inverted | 4: inverted |

[eV] | ||||

[eV] | 0.06 | 0.23 | 0.19 | 0.10 |

[eV] | 0.05 | 0.08 | 0.08 | 0.05 |

### 2.2 Muon decay width

Muon decays into electron, electron-type anti-neutrino and muon-type neutrino, in the Standard Model. However, in the neutrinophilic Higgs model, the Yukawa interaction Eq. (1) gives additional contribution to the SM prediction via one loop induced vertices and changes the muon decay width. Those come from box diagrams with a loop ( and ) or ( and ). Moreover, muon has additional decay mode into electron, RH neutrino and anti-RH neutrino, through the mediation. This new contribution can be severely constrained by the precise measurement of the decay rate and inverse decay rate of muon.

Now we are going to estimate those additional contributions to muon decay width. For this, we define mass eigenstates from flavour states with an unitary matrix as defined by

(10) | |||||

(11) |

with being a rotation matrix and the variables, , are the corresponding mixing angle. The Yukawa interaction Eq. (5) becomes

(12) |

for the mass eigenstates. Thus, we define for Yukawa couplings of the RH sneutrino, lepton and Higgsino.

#### 2.2.1 loop enhancement of

First, let us estimate the decay width of the main decay mode. In the following, and are the momentum of incoming , outgoing , and respectively. The amplitude of boson mediated process is given by

(13) |

where is the gauge coupling and is the boson mass. That of RH neutrino and Higgs bosons loop shown in the left window of Fig. 1 is given by

(14) | |||||

with the auxiliary function , which is defined in the Appendix. In this estimation of , we neglect terms with being the mass scale of new particles, those are much smaller than the leading corrections of . That of RH sneutrino and Higgsino loop shown in the right window of Fig. 1 is given by

(15) | |||||

We obtain

(16) | |||||

(17) |

with the auxiliary function , which is given in the Appendix. Here is the muon mass and is the VEV of the SM Higgs field and we keep only the leading order loop corrections, namely the interference between tree and one loop.

#### 2.2.2 with

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.

#### 2.2.3

This decay mode with RH neutrinos in the final state is induced by the tree level process mediated by the neutrinophilic charged Higgs boson 0809.5221 (). A worth noting feature is that this is not a interaction but a scalar interaction. The amplitude of the process mediated by shown in the Fig. 2 is given by

(18) | |||||

(19) |

Here, we normalise the effective coupling with , the Fermi constant measured in the experiment, and we introduce a new parameter defined by Fet ()

(20) |

The partial width is estimated as

(21) |

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.

#### 2.2.4 Total

By comparing Fermi constant measured from muon decay width and other SM quantities, the consistency of the SM can be tested Herczeg:1997bu (); Marciano:1999ih (). If we express the Fermi coupling constant with a parameter which stands for a correction due to new physics by

(23) |

the new physics contribution is constrained to be Erler:2004cx ()

(24) |

The last term in Eq. (22) comes from the non-() interaction via mediation. The muon decay experiments can not measure helicity of produced neutrinos, but the inverse decay of muon, , well confirms the form interaction and leave a small room for scalar interaction as charm ()

(25) |

With Eq. (20), we find the constraint on the 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 .
The constraints on charged Higgs mass and Yukawa coupling
is shown in Fig. 4, where for simplicity we take .
For Yukawa couplings of the order of unity,
the mass of charged Higgs must be heavier than around ^{2}^{2}2Neutrinophilic Higgs
bosons with mass O(100) GeV is possible
for Davidson:2009ha (); Haba:2011nb (). .
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 3.

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

(26) |

In Fig. 4, we show the contour plot of for Yukawa couplings in the plane of with in the left window , and that in the plane of with GeV in the right window. We can see that the charged Higgsino need to be heavier than a few hundreds GeV for degenerate sneutrino case, , or two of RH sneutrinos need to be several times heavier than chargino with GeV mass and only one RH neutrino can be light.

### 2.3 Collider constraints

In our model, the neutrinophilic Higgsinos ) are doublet and can be light enough to be produced at the on-shell from or collisions, and subsequently decay to leptons () and lightest sneutrino (), which similar to the production of Wino/Zino and subsequent decay to the lightest neutralino in the minimal supersymmetric standard model (MSSM). However, the collider constraints on our model with the RH sneutrino LSP is slightly different from the current searches on SUSY based on the neutralino LSP.

The production channels of the neutrinophilic Higgsinos are -channel boson exchange for , collisions, -channel exchange for collision, and -channel exchange for collision. They subsequently decay to leptons () and the lightest sneutrino (),

(27) |

where is the mass eigenstate of neutrino, and we assume that 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 () and the energy of the produced lepton is much suppressed.

The corresponding diagrams are given in Fig. 6. The final decay products are multi leptons plus a large missing energy by ,

(28) |

with being an integer, or mono photon plus a large missing energy by ,

(29) |

#### 2.3.1 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 collision (), it is natural to expect that the constraints are relaxed drastically. Thus, we take LEPSUSY () as the kinematical lower bound to avoid the LEP searches for direct production of charginos.

For the direct production of , the -channel exchange for collision is dominant as in FIG. 6.

(30) |

In this case, the result of monophoton searches for the MSSM neutralino can be used to constrain the mass and couplings of the RH sneutrino to electrons Fox:2011fx (). In this analysis, using the effective operator , the cutoff scale should be greater than about GeV for the fermion dark matter mass GeV as shown in figure 7.

#### 2.3.2 LHC bound

The current 8 TeV LHC gives lower bound on the charged Higgsino mass of , leaving the degenerate region with unconstrained from the chargino decay into light leptons, namely or . For the decay into lepton, the bound is relaxed and requires and Guo:2013asa ().

### 2.4 lepton flavor violation ()

The off-diagonal components of Yukawa couplings in Eq. (2) induce lepton flavor violating (LFV) decay of leptons. The general effective operator can be written as

(31) |

with . The decay rate of is given by

(32) |

For the muon decay, , the branching ratio is given by

(33) |

The present bounds on the branching ratios of the LFV decays are Adam:2013mnn (); Aubert:2009ag ()

(34) |

For our model with Yukawa interactions in Eq. (2), from the diagrams given in figure 8, we obtain

(35) |

with being an auxiliary function. The experimental limits (34) give strong bounds on Yukawa couplings as well as masses of mediated particle, and , as we will show. In fact, for GeV masses of sneutrinos and the neutrinophilic chargino, we find the LFV decaying branching ratios are of , 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 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 that are suitably aligned by appropriate sneutrino mixings in such a way that the flavor-violating processes are suppressed enough. Such mixing angles can be found by requiring some off-diagonal components of to be almost vanishing.

### 2.5 Anomalous magnetic dipole moment of muon

The muon anomalous magnetic dipole moment has discrepancy between experimental data and the SM prediction as Bennett:2006fi (); Hagiwara:2011af ()

(36) |

Thus, this discrepancy has been regarded as a hint and provided a motivation to investigate new physics beyond the standard model of particle physics.

and in Eq. (31) contribute to the magnetic and electric dipole moment, respectively. The resultant electric and magnetic dipole moment of lepton are given as

(37) | |||||

(38) |

The additional contribution to the induced magnetic moment of muon in the supersymmetric neutrinophilic Higggs model with large Yukawa couplings is given by

(39) |

where we assumed Yukawa couplings are real and the negligible charged Higgs boson contribution is omitted. We might expect a large of the muon for light sneutrinos and the light -like chargino because Yukawa coupling constants are . 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.

### 2.6 Compatibility in benchmarks

As mentioned previous subsections, LFV constraints are very stringent. Indeed, for cases of vanishing lightest neutrino mass as in benchmark point 1 and 3 in Table 1, 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 1.

At first, for the benchmark 2, we find that LFV constraints are avoided with the sneutrino mixing angles , and the resultant is given by

(40) |

Then, the mass of the lightest RH sneutrino can be GeV without inducing a large LFV, while the mass of the other two and need to be TeV. Notice that with the definition of by Eq. (10), we take the mass ordering of RH neutrinos as . With this choice of sneutrino mixing angles, the muon decay width constraint discussed Sec. 2.2 is avoided. Here, we can see that the coupling between the lightest RH sneutrino () and the electron is negligibly small. This small coupling automatically suppresses the LEP mono-photon constraint from Eq. (30).

In Fig. 9, 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 , the neutrinophilic Higgsino mass GeV, and the lightest sneutrino mass GeV. Heavy sneutrino masses, are taken from around TeV to TeV As the figures show, the constraint from is most serious, and very small regions are allowed. Including the constrains from , we find that the value of should be around , which corresponds to . The constraints do not restrict the value of much for the case with two heavy sneutrino masses. In the allowed parameter space, the sizable can be obtained as denoted by the blue colored region.

In Fig. 10, 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 , which is the lightest supersymmetric particle. Here, we use TeV and TeV for reference. The region of Red and Orange color respectively show and range of Eq. (36). The blue region corresponds . The yellow region, where the mass splitting between sneutrino and chargino is too large, is constrained by the LHC results.

For the benchmark 3, if the sneutrino mixing angles are , the resultant is given by

(41) |

Then, the mass of the lightest RH sneutrino can be GeV without inducing a large LFV, while the mass of the other two and need to be TeV. In Fig. 11, we show the viable region with TeV and TeV. Again, we take the mass ordering of RH sneutrinos as .

## 3 Cosmological constraints

### 3.1 dark radiation

RH component of neutrinos could contribute to the additional relativistic degrees of freedom in the early Universe, which is constrained by big bang nucleosysthesis and cosmic microwave background radiation observation as . In our model, RH neutrinos could be in thermal equilibrium due to the scatterings with charged leptons (left-handed neutrinos) through neutrinophilic charged (neutral) Higgs bosons via the Yukawa interaction in Eq. (2).

In order to suppress enough, RH neutrinos should have decoupled from the thermal bath much before the quark-hadron phase transition which takes place at the cosmic temperature MeV.

#### 3.1.1

The scatterings between charged lepton and RH neutrinos are mediated by the neutrinophilic charged Higgs boson . We obtain

(42) |

with being the energy at the center of mass frame. Taking thermal average, we find

(43) |

#### 3.1.2

Similarly, the thermal scattering cross section between left-handed and right-handed components of neutrinos via and is estimated as

(44) |

#### 3.1.3 Decoupling condition

The decoupling condition of RH neutrino at the quark-hadron transition epoch is expressed as

(45) |

which is rewritten as

(46) |

Therefore the neutrinophilic Higgs must be heavier than around for order of unity Yukawa couplings. We note that the similar bound has already been obtained but and contributions were missing in Ref. Davidson:2009ha (); Mitropoulos:2013fla (). Here we have re-estimated and corrected it.

## 4 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. Choi:2012ap (); Choi:2013fva () 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.

### 4.1 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 non-vanishing DM asymmetry. With a large DM asymmetry, the final relic density of DM may depend on the annihilation cross section and the DM asymmetry Barr:1990ca (); Barr:1991qn (); Kaplan:1991ah (); Thomas:1995ze (); Hooper:2004dc (); Kitano:2004sv (); Kaplan:2009ag (). 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 Choi:2012ap (); Choi:2013fva ()

(47) |

There is another subdominant contribution from the induced annihilation into photons,

(48) |

where we used the approximation of for simplicity and the soft term . 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