$h \rightarrow \gamma \gamma$ in $U(1)_{R}-$ lepton number model with a right-handed neutrino

We perform a detailed study of the signal rate of the lightest Higgs boson in the diphoton channel ($\mu_{\gamma \gamma}$), recently analyzed by both the ATLAS and CMS collaborations at the Large Hadron Collider, in the framework of $U(1)_R-$ lepton number model with a right handed neutrino superfield. The corresponding neutrino Yukawa coupling, `$f$', plays a very important role in the phenomenology of this model. A large value of $f\sim\mathcal O(1)$ provides an additional tree level contribution to the lightest Higgs boson mass along with a very light (mass $\sim$ a few hundred MeV) bino like neutralino and a small tree level mass of one of the active neutrinos that is compatible with various experimental results. In the presence of this light neutralino, the invisible decay width of the Higgs boson can become important. We studied this scenario in conjunction with the recent LHC results. The signal rate $\mu_{\gamma\gamma}$ obtained in this scenario is compatible with the recent results from both the ATLAS and the CMS collaborations at 1$\sigma$ level. A small value of `$f$', on the other hand, is compatible with a sterile neutrino acting as a 7 keV dark matter that can explain the observation of a mono-energetic X-ray photon line by the XMM-Newton X-ray observatory. We also study the impact of $\mu_{\gamma\gamma}$ in this case.


Introduction
Recently two CERN based Large Hadron Collider (LHC) experiments, ATLAS and CMS, have confirmed the existence of a neutral boson, widely accepted to be the Higgs boson, an elementary scalar boson of nature [1,2], with mass around 125 GeV. Almost all the decay channels have been probed with reasonable precision. Out of these, results in the h → γγ channel have attracted a lot of attention in recent times. The reason is two-fold: first, this is the discovery mode of the Higgs boson and second, being a loop induced process it may potentially carry indirect hints of new physics. The results reported so far show some deviations with respect to the Standard Model (SM) prediction. For example, the ATLAS collaboration reported µ γγ = 1.17 ± 0.27 [3], where µ γγ = σ(pp→h→γγ) σ(pp→h→γγ) SM . On the other hand, CMS collaboration reported a best-fit signal strength in their main analysis [4] where, µ γγ = 1.14 +0. 26 −0. 23 . Moreover, a cut-based analysis by CMS produced a slightly different value, which is quoted as µ γγ = 1.29 +0. 29 −0. 26 . This enhancement or suppression in the h → γγ channel with respect to the SM provide a natural testing ground for physics Beyond the SM (BSM). Detailed studies have already been carried out for this particular channel. For example, h → γγ is studied in a wide variety of supersymmetric (SUSY) models namely, the minimal supersymmetric standard model (MSSM) , its next-tominimal version (NMSSM) [26][27][28][29][30][31][32][33][34], the constrained MSSM (CMSSM) [35][36][37][38][39][40] and also in (B-L)SSM [41][42][43][44], left-right supersymmetric models [45], and in U (1) ′ extension of MSSM [46]. In [47], a triplet-singlet extension of MSSM has been studied and µ γγ is computed.
Motivated by these results we would like to investigate the Higgs to diphoton mode in the context of a supersymmetric scenario known as U (1) R − lepton number model, which is augmented by a single right-handed neutrino superfield. It is rather well known that supersymmetry is one of the very popular frameworks that provides a suitable dark matter candidate and can also explain the origin of neutrino masses and mixing. However, the nonobservation of superpartners so far has already put stringent lower bounds on their masses in different SUSY models, subject to certain assumptions. In the light of these constraints, R-symmetric models which generically contain Dirac gauginos in their spectra (as opposed to Majorana gauginos in usual SUSY scenarios) are very well motivated. In particular, the presence of Dirac gluino in this class of models reduces the squark production cross section compared to MSSM thus relaxing the bound on squark masses. Detailed studies on R-symmetric models and Dirac gauginos can be found in the literature . Flavor and CP violating constraints are also suppressed in these class of models [58]. To construct Dirac gaugino masses, the gauge sector of the supersymmetric Standard Model has to be extended to incorporate chiral superfields in the adjoint representations of the SM gauge group. A singletŜ, an SU (2) tripletT and an SU (3) octetÔ, help obtain the Dirac gaugino masses.
In this paper we consider the minimal extension of a specific U (1) R symmetric model [84,85] by introducing a right handed neutrino superfield [88]. In such a scenario the Rcharges are identified with lepton numbers such that the lepton number of SM fermions and their superpartners are negative of the corresponding R-charges. Such an identification leaves the lepton number assignments of the SM fermions unchanged from the usual ones while the same for the superpartners become non-standard. This has an interesting consequence for the sneutrinos which now do not carry any lepton number. Hence, although in this model sneutrinos get non-zero vacuum expectation value (vev) in general, the latter do not get constrained from neutrino Majorana masses which require lepton number violation by two units. A sneutrino thus can play the role of a down type Higgs boson, a phenomenon which has crucial implications [74,79,84,85,88] for our purpose that we would discuss later in this work. The right handed neutrino, on the other hand, not only provides a small tree level Dirac neutrino mass but also gives rise to an additional tree level contribution to the Higgs boson mass proportional to the neutrino Yukawa coupling [88]. When the R-symmetry is broken, a small ( < ∼ 0.05 eV) Majorana mass for one of the active neutrinos is generated at the tree level while the right handed sterile neutrino can have keV Majorana mass and can be accommodated as a warm dark matter candidate 1 .
A large Yukawa coupling f ∼ O(1) facilitates having the mass of the lightest Higgs boson around 125 GeV without resorting to radiative contributions. Large values of f also result in a very light neutralino with mass around a few hundred MeV. Cosmological implications of having such a light neutralino is briefly discussed in ref. [88] for this model. Some general studies regarding light neutralinos can be found in [100][101][102][103][104][105][106][107][108][109]. On the other hand, in the regime of small Yukawa coupling f ∼ 10 −4 , the Higgs boson mass is devoid of any large tree level contribution. Therefore, to obtain the mass of the lightest Higgs boson in the right ballpark, radiative corrections have to be incorporated, which are required to be large enough. This can be achieved either by having large singlet and triplet couplings [62], λ S , λ T ∼ O(1), or by having a large top squark mass.
In this work, we study the implications of such a scenario with particular reference to the diphoton final states arising from the decay of the lightest Higgs boson. As we shall see later in this work, the scenario under consideration would have significant bearing on µ γγ . This is because one can now afford rather light top squarks which potentially affect the resonant production rate of the lightest Higgs boson and its decay pattern. Furthermore, presence of a very light neutralino opens up new decay modes of the Higgs bosons which in turn may suppress its diphoton branching fraction. Also, in general, presence of new particle states and their involved couplings would affect the proceedings.
The plan of the work is as follows. In Section II we briefly discuss the main features of the model. The principal motivation and the artifacts of the U (1) R − lepton number model are also discussed with reference to its scalar and the electroweak gaugino sector. In Section III we address the neutralino and the chargino sectors in detail. The masses and the couplings in these sectors play important roles in the computation of µ γγ . A thorough analysis of µ γγ requires the knowledge of both production and decays of the Higgs boson. In Section IV issues pertaining to the production of Higgs boson in the present scenario is discussed in some detail. Analytical expressions of Higgs boson decaying to two photons in our model are given in Section V. Section VI is dedicated to the computation of the total and invisible decay width of the Higgs boson. In Section VII, we compute µ γγ and show its variation with relevant parameters, along with the points representing the 7 keV sterile neutrino warm dark matter in this model. We conclude in Section VIII with some future outlooks. The Higgs boson couplings to neutralino and charginos in this model are relegated to the appendix.
We consider a minimal extension of an R-symmetric model, first discussed in [84,85], by extending the field content with a single right handed neutrino superfield [88]. Along with the MSSM superfields,Ĥ u ,Ĥ d ,Û c i ,D c i ,L i ,Ê c i , two inert doublet superfieldsR u andR d with opposite hypercharge are considered in addition to the right handed neutrino superfield N c . These two doubletsR u andR d carry non zero R-charges (The R-charge assignments are provided in table 1 and therefore, in order to avoid spontaneous R-breaking and the emergence of R-axions, the scalar components ofR u andR d do not receive any nonzero vev and because of this they are coined as inert doublets. R-symmetry prohibits the gauginos to have Majorana mass term and trilinear scalar interactions (A-terms) are also absent in a U (1) R invariant scenario. However, the gauginos can acquire Dirac masses. In order to have Dirac gaugino masses one needs to include chiral superfields in the adjoint representations of the standard model gauge group. Namely a singletŜ, an SU (2) L tripletT and an octetÔ under SU (3) c . These chiral superfields are essential to provide Dirac masses to the bino, wino and gluino respectively. We would like to reiterate that the lepton numbers have been identified with the (negative) of Rcharges such that the lepton number of the SM fermions are the usual ones whereas the superpartners of the SM fermions carry non-standard lepton numbers. With such lepton number assignments this R-symmetric model is also lepton number conserving [84,85,88].
The generic superpotential carrying an R-charge of two units can be written as For simplicity, in this work we have omitted the terms κN cŜŜ and ηN c from the superpotential. As long as η ∼ M 2 SU SY and κ ∼ 1 we do not expect any significant change in the analysis and the results presented in this paper.
In order to have a realistic model one should also include supersymmetric breaking terms, which are the scalar and the gaugino mass terms. The Lagrangian containing the Dirac gaugino masses can be written as [71,73] where W ′ α = λ α + θ α D ′ is a spurion superfield parametrizing D-type supersymmetry breaking. This results in Dirac gaugino masses as D ′ acquires vev and are given by where Λ denotes the scale of SUSY breaking mediation and κ i are order one coefficients. It is worthwhile to note that these Dirac gaugino mass terms have been dubbed as 'supersoft' terms. This is because we know that the Majorana gaugino mass terms generate logarithmic divergence to the scalar masses whereas in ref. [54], it was shown that the purely scalar loop, obtained from the adjoint superfields cancels this logarithmic divergence in the case of Dirac gauginos. Hence it is not unnatural to consider the Dirac gaugino masses to be rather large.
The R-conserving but soft supersymmetry breaking terms in the scalar sector are generated from a spurion superfieldX, whereX = x+θ 2 F X such that R[X] = 2 and < x >= 0, < F X > = 0. The non-zero vev of F X generates the scalar soft terms and the corresponding potential is given by The presence of the bilinear terms bµ i L H uLi implies that all the three left handed sneutrinos can acquire non-zero vev's. However, it is always possible to make a basis rotation in which only one of the left handed sneutrinos get a non-zero vev and one must keep in mind that the physics is independent of this basis choice.
Such a rotation can be defined aŝ Note that the index (i) runs over three generations whereas a = 1(e) and b = 2, 3(µ, τ ). This basis rotation implies that the scalar component of the superfieldL a acquires a non zero vev (i.e. ν ≡ v a = 0) whereas the other two sneutrinos do not get any vev. One can further go to a basis where the charged lepton Yukawa couplings are diagonal. It is, however, important to note that the charged lepton of flavor a (i.e. the electron) cannot get mass from this Yukawa couplings because of SU (2) L invariance but can be generated from R-symmetric supersymmetry breaking operators [84]. Moreover, we also choose the neutrino Yukawa coupling in such a way that onlyL a couples to 2N c . In such a scenario the left-handed sneutrino can play the role of a down type Higgs boson since its vev preserves lepton number and is not constrained by neutrino Majorana mass. Hence one has the freedom to keep a very large µ d such that the superfieldsĤ d andR u get decoupled from the theory. This is what we shall consider in the rest of our discussion. With a single sneutrino acquiring a vev and in the mass eigenstate basis of the charged lepton and down type quark fields the superpotential now has the following form (integrating outĤ d andR u ) where and includes all the trilinear R-parity violating terms in this model. In the subsequent discussion we shall confine ourselves to this choice of basis but get rid of the primes from the fields and make the replacementλ,λ ′ → λ,λ ′ . In this rotated basis the soft supersymmetry breaking terms look like With this short description of the theoretical framework let us now explore the scalar and the fermionic sectors in some detail in order to prepare the ground for the study of the diphoton decay of the lightest Higgs boson.

The scalar sector
The scalar potential receives contributions from the F-term, the D-term, the soft SUSY breaking terms and the terms coming from one-loop radiative corrections. Thus, schematically, The F-term contribution is given by where the superpotential W is given by eq. (2.6). The D-term contribution can be written as where The τ a 's and λ a 's are the SU (2) generators in the fundamental and adjoint representation respectively. The weak hypercharge contribution D Y is given by where g and g ′ are SU (2) L and U (1) Y gauge couplings respectively. The expanded forms of V F and V D in terms of various scalar fields can be found in [88]. The soft SUSY breaking term V soft is given in Eq. (2.8) whereas the dominant radiative corrections to the quartic potential are of the form 1 2 δλ u (|H u | 2 ) 2 , 1 2 δλ ν (|ν a | 2 ) 2 and 1 2 δλ 3 |H 0 u | 2 |ν a | 2 . The coefficients δλ u , δλ ν and δλ 3 are given by We shall see later that for large values of the couplings λ T and λ S or large stop masses these one-loop radiative contributions to the Higgs quartic couplings could play important roles in obtaining a CP-even lightest Higgs boson with a mass around 125 GeV.

CP-even neutral scalar sector
Let us assume that the neutral scalar fields H 0 u ,ν a (a = 1(e)), S and T acquire real vacuum expectation values v u , v a , v S and v T , respectively. The scalar fields R d andÑ c carrying R-charge 2 are decoupled from these four scalar fields. We can split the fields in terms of their real and imaginary parts: H 0 u = h R + ih I ,ν a =ν a R + iν a I , S = S R + iS I and T = T R + iT I . The resulting minimization equations can be found easily and with the help of these minimization equations, the neutral CP-even scalar squared-mass matrix in the basis (h R ,ν R , S R , T R ) can be written down in a straightforward way, where h 4 corresponds to the lightest CP even mass eigenstate [88]. In the R-symmetry preserving scenario the elements of this symmetric 4 × 4 matrix are found to be The W ± -and the Z-boson masses can be written as Note that the electroweak precision measurements of the ρ-parameter requires that the triplet vev v T must be small ( < ∼ 3 GeV) [110]. In addition, our requirement of a doublet-like lightest CP-even Higgs boson, in turn, demands a small vev v S of the singlet S as well. This is because a small value of v S reduces the mixing between the doublets and the singlet scalar S. In such a simplified but viable scenario in which the singlet and the SU (2) L triplet scalars get decoupled from the theory, we are left with a 2 × 2 scalar mass matrix. In this case the angle α represents the mixing angle between h R andν R and can be expressed in terms of other parameters as follows

Tree level mass bound on m h
In addition, in such a situation (with v S , v T ≪ v) it can be shown easily that the lightest CP-even Higgs boson mass is bounded from above at tree level [88], The bound in Eq. (3.12) is saturated for v s < ∼ 10 −3 GeV, i.e., when the singlet has a large soft supersymmetry breaking mass and is integrated out. The f 2 v 2 term grows at small tan β and thus the largest Higgs boson mass is obtained with low tan β and large values of f . We shall show in the next section that f ∼ 1 can be accommodated in this scenario without spoiling the smallness of the neutrino mass at tree level. Therefore, for f ∼ O(1), the tree level Higgs boson mass can be as large as ∼ 125 GeV where the peak in the diphoton invariant mass has been observed and no radiative corrections are required. This means that in this scenario one can still afford a stop mass as small as 350 GeV or so and couplings λ T and λ S can be small (∼ 10 −4 ) as well. This is illustrated in figure (1) where, the lightest Higgs boson mass is shown as a function of v S for f = 1.5, tan β = 4 and for a set of other parameter choices discussed later. One can see that for a very small v S ( < ∼ 10 −3 GeV) the tree level Higgs boson mass is 150 GeV and is reduced to 125 GeV for a v S ∼ 0.2 GeV. As v S increases further, (M h ) Tree starts decreasing rapidly and the Higgs boson mass becomes lighter than 100 GeV. In such a case one requires larger radiative corrections to the Higgs boson mass and this can be achieved with the help of large triplet/singlet couplings (O(1)) and/or large stop mass. For example, with a choice of λ S = 0.91 and λ T = 0.5, the one-loop radiative corrections to the Higgs boson mass arising from these two couplings are sizable 3 . In this case, in order to have a 125 GeV Higgs boson, the tree level contribution should be smaller and for a very small v S (∼ 10 −4 GeV) and large f ( > ∼ 1), this can be achieved with a larger tan β. The one loop corrections from the stop loop must also be small and this is realized for small mt and large tan β. This is illustrated in figure (2)

The fermionic sector
The fermionic sector of the scenario, involving the neutralinos and the charginos, has rich new features. In the context of the present study, when analyzed in conjunction with the scalar sector of the scenario, this sector plays a pivotal role by presenting the defining issues for the phenomenology of this scenario. Its influence ranges over physics of the Higgs boson at current experiments and the physics of the neutrinos before finally reaching out to the domain of astrophysics and cosmology by offering a possible warm dark matter candidate whose actual presence may find support in the recent observations of a satellite-borne X-ray experiment. Thus, it is of crucial importance to study the structure and the content of this sector in appropriate detail. A thorough discussion of µ γγ in the present scenario requires a study of the masses and the mixing angles of the neutralinos and the charginos. One of the natural consequences of such a U (1) R -lepton number model with a right-handed neutrino is that one of the lefthanded neutrinos (the electron-type one) and the right-handed neutrino become parts of the extended neutralino mass matrix. The electron-type neutrino of the SM can be identified with the lightest neutralino eigenstate. We also address the issue of tree level neutrino mass. Subsequently, we show that in certain region of the parameter space the lightest neutralino-like state can be very light (with a mass of order 100 MeV). This may contribute to the total as well as to the invisible decay width of the lightest Higgs boson thus drawing constraints on the parameter space of the scenario from the latest LHC results.

The neutralino sector: R-conserving case
In the neutral fermion sector we have mixing among the Dirac gauginos, the higgsinos, the active neutrino of flavor 'a' (i.e., ν e ) and the single right-handed neutrino N c once the electroweak symmetry is broken. The part of the Lagrangian that corresponds to the neutral fermion mass matrix is given by The neutral fermion mass matrix M D χ is given by The above matrix can be diagonalized by a biunitary transformation involving two unitary matrices V N and U N and results in four Dirac mass eigenstates χ 0+ is identified with the light Dirac neutrino. The other two active neutrinos remain massless in this case. Generically the Dirac neutrino mass can be in the range of a few eV to tens of MeV. However, one can also accommodate a mass of 0.1 eV or smaller for the Dirac neutrino by assuming certain relationships involving different parameters [88], which are and With these choices the Dirac mass of the neutrino can be written as , one can have a Dirac neutrino mass in the right ballpark of < ∼ 0.1 eV. Note that a choice of large f ∼ O(1) is possible for a small λ T (∼ 10 −6 ) and nearly degenerate Dirac gauginos (ǫ < ∼ 10 −1 ) assuming µ u , M D 2 , M D 1 in the few hundred GeV range.

The neutralino sector: R-breaking case
R-symmetry is not an exact symmetry and is broken by a small gravitino mass. One can therefore consider the gravitino mass as the order parameter of R-breaking. The breaking of R-symmetry has to be communicated to the visible sector and in this work we consider anomaly mediation of supersymmetry breaking playing the role of the messenger of R-breaking. This is known as anomaly mediated R-breaking (AMRB) [79]. A non-zero gravitino mass generates Majorana gaugino masses and trilinear scalar couplings. We shall consider the R-breaking effects to be small thus limiting the gravitino mass (m 3/2 ) around 10 GeV.
The R-breaking Lagrangian contains the following terms where M 1 , M 2 and M 3 are the Majorana mass parameters corresponding to U (1), SU (2) and SU (3) gauginos, respectively and A's are the scalar trilinear couplings. The (Majorana) neutralino mass matrix containing R-breaking effects can be written in the basis where the symmetric (8 × 8) neutralino mass matrix M M χ is given by (4.7) The above mass matrix can be diagonalized by a unitary transformation given by The two-component mass eigenstates are defined by and one can arrange them in Majorana spinors defined bỹ Similar to the Dirac case, the lightest eigenvalue (mχ0 8 ) of this neutralino mass matrix corresponds to the Majorana neutrino mass. Using the expression of M R in eq. (4.3) and the relation between λ S and λ T in eq. (4.2), the active neutrino mass is given by [88], where α and δ are defined as and the quantity γ has been defined earlier in section 4.1. This shows that to have an appropriate neutrino mass we require the Dirac gaugino masses to be highly degenerate.
The requirement on the degree of degeneracy can be somewhat relaxed if one chooses an appropriately small value of λ T . Such a choice, in turn, would imply an almost negligible radiative contribution to the lightest Higgs boson mass. Interestingly, the Yukawa coupling does not appear in the expression for (m ν ) Tree in eq. (4.11). This is precisely because of the relation between M R and f in eq. (4.3). However, 'f ' has some interesting effects on the next-to-lightest eigenstates of the mass matrix. The following situations are phenomenologically important: • A large value of f ∼ O(1) generates a very light bino-like neutralino (χ 0 7 ) with mass around a few hundred MeV. In this case, this is the lightest supersymmetric particle (LSP) and its mass is mainly controlled by the R-breaking Majorana gaugino mass parameter M 1 . A very light neutralino has profound consequences in both cosmology as well as in collider physics [100][101][102][103][104][105][106][107][108][109]. In the context of the present model one can easily satisfy the stringent constraint coming from the invisible decay width of the Z boson because the light neutralino is predominantly a bino. One should also take into account the constraints coming from the invisible decay branching ratio of the lightest Higgs boson. In our scenario h →χ 0 7χ 0 8 (whereχ 0 8 is the light active neutrino) could effectively contribute to the invisible final state. This is because, althoughχ 0 7 would undergo an R-parity violating decay, for example,χ 0 7 → e + e − ν, the resulting four body final state presumably has to be dealt with as an invisible mode for the lightest Higgs boson. Such constraints are discussed in detail later in this paper. Note that Γ(h →χ 0 7χ 0 7 ) is negligibly small because of suppressed h-χ 0 7 -χ 0 7 coupling for a bino-dominated,χ 0 7 . A 10 GeV gravitino NLSP could also decay to a final state comprising of the lightest neutralino accompanied by a photon. In order to avoid the strong constraint on such a decay process coming from big-bang nucleosynthesis (BBN) one must consider an upper bound on the reheating temperature of the universe T R < ∼ 10 6 GeV [104,111].
In addition, such a light state is subjected to various collider bounds [100] and bounds coming from rare meson decays such as the decays of pseudo-scalar and vector mesons into light neutralino should also be investigated [105] in this context. The spectra of low lying mass eigenstates for the large f case will be shown later for a few benchmark points.
• For small f ∼ O(10 −4 ),χ 0 7 is a sterile neutrino state, which is a plausible warm dark matter candidate with appropriate relic density. Its mass can be approximated from the 8 × 8 neutralino mass matrix as follows: For a wide range of parameters, the active-sterile mixing angle, denoted as θ 14 , can be estimated as (4.14) Furthermore, the sterile neutrino can be identified with a warm dark matter candidate only if the following requirements are fulfilled. These are: (i) it should be heavier than 0.4 keV, which is the bound obtained from a model independent analysis [112] and (ii) the active-sterile mixing needs to be small enough to satisfy the stringent constraint coming from different X-ray experiments [113].
Under the circumstances, the lightest neutralino-like state is the next-to-next-tolightest eigenstate (χ 0 6 ) of the neutralino mass matrix. Its composition is mainly controlled by the parameter µ u , chosen to be rather close to the electroweak scale The masses of the lighter neutralino states for this case (small f ) will be presented later.

The chargino sector
We shall now discuss the chargino sector in some detail as it plays a crucial role in the decay h → γγ. The relevant Lagrangian after R-breaking in the AMRB scenario obtains the following form: The chargino mass matrix, in the basis ( w + , , is written as This matrix can be diagonalized by a biunitary transformation, U M c V T = M ± D . The chargino mass eigenstates are related to the gauge eigenstates by these two matrices U and V . The chargino mass eigenstates (two-component) are written in a compact form as where The four-component Dirac spinors can be written in terms of these two-component spinors as It is to be noted that χ c i ≡ ( χ + i ) c = χ − i is a negatively charged chargino. Hence, the lightest chargino ( χ − 4 ) corresponds to the electron and the structure of the chargino mass matrix ensures (see eq. 4.16) that the lightest mass eigenvalue remains unaltered from the input mass parameter for the electron , i.e., m e = 0.5 MeV.
Let us now analyze the composition of different chargino states and how they affect the decay width Γ(h → γγ) in this model. Due to constraints from the electroweak precision measurements one must consider a heavy Dirac wino mass [84]. Furthermore, a small tree level Majorana neutrino mass demands a mass-degeneracy of the electroweak Dirac gauginos as is obvious from eq. (4.11). In addition, we assume an order one λ T which we use throughout this work for numerical purposes. With these, we observe the following features of the next-to-lightest physical chargino state which could potentially contribute to µ γγ : • In the limit when M D 2 >> µ u , the next-to-lightest chargino, χ − 3 (which is actually the lightest chargino-like state in the MSSM sense), comprises mainly of R − d with a very little admixture of T − d while χ + 3 is dominated by H + u with a small admixture of w + .
• Finally, for M D 2 ≈ µ u , χ − 3 is dominantly w − and χ + 3 is mostly made up of T + u .
Apart from the electron, the mass of the chargino states are controlled mainly by the parameters M D 2 and µ. We have varied the input parameters in such a way that the lightest chargino-like state is always heavier than 104 GeV [110]. The chargino mass spectra corresponding to different benchmark points will be presented later. gg). Thus, µ γγ can be expressed entirely in terms of various decay widths of the Higgs boson as follows [23,24]: , Γ T OT being the total decay width of the Higgs boson in the present scenario. The decay of h → γγ is mediated mainly by the top quark and the W ± -loops in the SM and in addition, by top squark and chargino loops in our scenario. In the subsequent discussion we investigate these widths in some detail.
Note that in our model, we have integrated out the down type Higgs (H d ) superfield and the sneutrinoν a (a = 1(e)) plays the role of the down type Higgs boson acquiring a large non-zero vev. The sneutrino (ν a ) couples to charged leptons (second and third generation) and down type quarks via R-parity violating couplings which are identified with the standard Yukawa couplings. Thus, the couplings of the Higgs boson to charged leptons and quarks remain the same as in the MSSM. This is apparent from the first term given in eq. (2.7).

The decay h → gg
The partial width of the Higgs boson decaying to a pair of gluons via loops involving quarks and squarks is given by , G F is the Fermi constant, α s is the strong coupling constant and with f (τ ) given by The couplings are given by where the angle α is defined in eq. (3.11) and tan β = v u /v a .
The couplings of the Higgs boson with the left-and the right-handed squarks are exactly the same as in the MSSM. However, one can neglect the mixing between the left-and the right-handed squarks due to the absence of the µ-term and the A-terms 4 .
As has been argued in the beginning of this section, the couplings g h Q and g h Q , mentioned in Eq. (5.5) remain the same as those found in the MSSM. As far as the production of the Higgs boson is concerned, we shall show later that a rather light top squark with mass around 200 − 300 GeV enhances the value of k gg compared to the SM. The SM and the MSSM results for the decay h → gg can be found in [114][115][116].

The decay h → γγ
In the SM, the primary contribution to the decay h → γγ comes from the W boson loop and the top quark loop with the former playing the dominant role. In supersymmetric models, the charged Higgs (H ± ), top squark (t) and the chargino (χ ± ) provide extra contributions in addition to the W boson and the top quark loop. The authors of ref. [24] have noted that the relative strengths of the loop contributions involving the vector bosons, the fermions and the scalars with mass around 100 GeV follow a rough ratio of 8 : 1.5 : 0.4. Nonetheless, a light charged Higgs boson (H ± ) could contribute substantially if one considers a large hH + H − coupling. However, since the triplet vev is small, the contribution of the triplet to the charged Higgs state is negligible. On the other hand, charginos in loop could enhance the h → γγ decay width, in particular, when they are light and/or diagrams involving them interfere constructively with the W -mediated loop diagram.
The Higgs to diphoton decay rate can be written down as [115] with the loop functions already defined in Eq. (5.4). The relevant couplings are given by g hūu = cos α sin β , The masses which appear in the denominator of the couplings given above, represent physical masses propagating in the loop. For example, mc k are the physical chargino masses, mf are the physical masses of the sfermions and so on. We present the complete set of Higgs-chargino-chargino interaction vertices in Appendix-I 5 .
As noted earlier, the largest contribution in the Higgs decay rate to two photons comes from the W boson loop. Similar to the MSSM, the hW W coupling gets modified by the factor sin(β − α). Hence, in order to have a significant contribution from the W boson loop in our model, the angles α and β need to be aligned in such a way that one obtains a large value of sin(β − α).
In figure 3 we illustrate the variations of the couplings g hW + W − and g hχ +  figure 3 it is clear that the hW W coupling increases with increasing tan β and essentially saturates at a value of tan β ≈ 30. However, one should note that the absolute increase in the coupling is not that big (∼ 5%, between tan β = 5 and tan β = 30, leading to ∼ 10% increase in the loop contribution). On the other hand, as µ u << M D 1,2 , the next-to-lightest chargino state is dominantly controlled by the µ u parameter. For this case, the coupling g hχ + 3χ − 3 is plotted as a function of tan β in the right panel of figure 3. One can clearly see that g hχ + 3χ − 3 is already much suppressed compared to g hW + W − , for the entire range of tan β. From the expression for g hχ + 3χ − 3 in eq. (5.8) it is straightforward to verify that this coupling remains very much suppressed for all the different cases mentioned in section 4.3. The Higgs boson couplings to heavier charginos are also highly suppressed as can be seen from figure 4. Thus, the contribution of charginos in Γ(h → γγ) would, in any case, be insignificant. Referring back to equation 5.1, we are now in a position to have some quantitative estimates of the quantities k gg and k γγ which control the signal strength µ γγ . In figure 5 we illustrate their variations (k gg in red and k γγ   Figure 4. Couplings of the Higgs boson to heavier charginos. The thick black line represents the coupling to the heaviest chargino (χ ± 1 ) whereas the blue dashed one represents the same to the chargino immediately lighter to it (χ ± 2 ) .
in blue) as functions of the mass of the top squark for various values of tan β. We observe that k gg is not at all sensitive to tan β (all three curves in red for three tan β values are found to be overlapping). This is since we considered gg → h production via loops involving the top quark and the top squark. The couplings involved there carry a factor cos α sin β , which varies only marginally with respect to tan β. On the other hand, k γγ changes significantly with tan β because Γ(h → γγ) receives major contribution from the W -boson induced loop for which the involved coupling goes as the factor sin(β − α). This factor is sharply varying with tan β and is responsible for the prominent variations of k γγ with tan β. As can be seen from the figure, k γγ is large for high tan β and the vice versa. It is observed that for for light top squarks, k gg gets enhanced by a considerable amount. However, in that very region , k γγ is rather small for small tan β, and it becomes somewhat larger for higher tan β. However, it is found that k gg > 1 while k γγ < 1, all through. We have also checked that the illustrated variations of k gg and k γγ are following their respective gross trends in the MSSM closely in the limit of zero left-right mixing in the scalar sector. Note that for this plot we have not incorporated the constraints from the mass of the Higgs boson and the requirement of having no tachyonic scalar states. In section 6, while discussing the quantitative impact of the recent LHC results on such a scenario, we present results of detailed scan of the parameter space by including all these constraints. It is also evident from this figure that a low value of tan β would always lead to µ γγ < 1 while its intermediate values could render the latter smaller or larger than 1 including values close to 1.
All the previous plots consider a large values of 'f ' (f ∼ O(1)) for which one obtains a large tree level correction to the Higgs boson mass as well as an appropriate mass for the active neutrino at the tree level. We adopt such a scenario with relatively large values of 'f ' in our study of the Higgs boson decay rates which we present in the next subsection.

Higgs boson decaying to charginos and neutralinos
In the presence of much lighter charginos and neutralinos (as discussed in sections 4.2 and 4.3), an SM-like Higgs boson with mass around 125 GeV could undergo decays to a pair of these states. It is important to consider such possibilities as these contribute to the total decay width of the Higgs boson (Γ T OT ) appearing in the expression of µ γγ in equation 5.1.
It has been noted in section 4.2, that the smallest eigenvalue (mχ0 8 ) of the neutralino mass matrix corresponds to the neutrino mass. The next-to-lightest neutralino (χ 0 7 ) turns out to be a bino like neutralino (the sterile neutrino) for large (small) values of 'f '. Moreover, the mass of the next-to-next-to-lightest neutralino state (χ 0 6 ) is mostly controlled by µ u . Since we have chosen µ u to be very close to the electroweak scale, the Higgs boson decay to a pair ofχ 0 6 is not possible. The presence of light neutralino states may enhance the total decay width of the Higgs boson considerably. It is found that h →χ 0 7χ 0 8 dominates over all the other possible decay modes. This is because a bino-like neutralino (χ 0 7 ) has got the involved coupling enhanced. We show the variation of the coupling g hχ 0 7χ 0 8 as a function of tan β in figure 6. This phenomenon has a major implication in the light of the recent studies of the invisible branching ratio of the Higgs boson as well as its total decay width. A quantitative estimate of this will be presented later. It is clear from figure 6 that the h-χ 0 7 -χ 0 8 coupling gets larger at small values of tan β. This, in turn, implies a gradual enhancement in the total decay width of the Higgs boson with decreasing tan β.
On the other hand, the lightest chargino eigenstate (χ ± 4 ) corresponds to the electron. The mass of the next-to-lightest chargino (χ ± 3 ) is again controlled by µ u if µ u < M D 2 . Thus, decay of the Higgs boson to a pair ofχ ± 3 is not possible. The most general expressions for the partial widths of the Higgs boson decaying to a pair of neutralinos (Γ(h →χ 0 iχ 0 j )) or a pair of charginos (Γ(h →χ + iχ − j )) can be found in the appendix. In the presence of these additional decay modes of the lightest Higgs boson (mainly h →χ 0 7χ 0 8 ), it is expected that Γ T OT would increase thus lowering the signal rate µ γγ .

The total decay width of the Higgs boson
In this section we collect the partial decay widths of the lightest Higgs boson that dominantly contribute to its total decay width. The latter is thus given by 6 For completeness, we present here the analytical expressions for all the decay rates which go into our analysis but were not presented earlier. These are as follows: The function R(x) is defined as [116,118,119] In the subsequent sections we present the numerical results of our analysis pertaining to the total decay width of the lightest (SM-like) Higgs boson as well as the diphoton signal strength µ γγ and subject them to important experimental findings to obtain nontrivial constraints on the scenario under consideration.

Impact of the LHC results
In this section, we discuss the impact of the findings from the LHC pertaining to the Higgs sector on the scenario under discussion. As pointed out earlier, two broad scenarios based on the magnitude of 'f ' worth special attention: the scenario with large 'f ' (∼ O(1)) and the one for which 'f ' is rather small.

The case of large neutrino Yukawa coupling, f ∼ O(1)
A large neutrino Yukawa coupling (f ∼ O(1)) already enhances the tree level Higgs boson mass. Thus, such a scenario banks less on large radiative contributions from the top squark loops to uplift the same. Further, an appropriately small tree level Majorana neutrino mass (the lightest neutralino) can be obtained along with a light bino-like neutralino (χ 0 7 , the next-to-lightest neutralino) once R-symmetry is broken explicitly, via anomaly mediation. The mass of this neutralino is essentially controlled by the R-symmetry breaking Majorana mass term of the U (1) gaugino (the bino), i.e., M 1 , and hence related to the gravitino mass m 3/2 . Since we assume m 3/2 ∼ 10 GeV, the next-to-lightest neutralino acquires a mass of the order of a few hundred MeV. The presence of such a light bino like neutralino implies a substantial enhancement in the total decay width of the Higgs boson, which, however, is now constrained by the LHC experiments [120]. An enhancement satisfying such a constraint would result in weakening of the diphoton signal strength, µ γγ . We also note that the partial width for the Higgs boson decaying to a neutrino and a bino-like neutralino, Γ(h →χ 0 7χ 0 8 ), is large at small tan β and dominates over all the other channels. This enhances the total decay width of the lightest Higgs boson at small tan β. This is because the next-to-lightest neutralino is bino like, which makes the product (N 71 .N 88 ) of the neutralino mixing matrix elements appearing for this particular decay width to be large. Furthermore, this decay width also involves sin α, which controls the sneutrino component in the lightest CP even Higgs boson state. At small tan β, sin α reaches its maximum value (see eq. 3.11) thus enhancing this partial decay width (see figure 6).
However, as mentioned earlier this decay mode will presumably contribute to the invisible decay width of the Higgs boson. We have taken into account the current constraint on this invisible branching ratio (< 20%) from model independent Higgs precision analysis [121]. In figure 8 we show the constraints on tan β obtained from this invisible branching ratio for different choices of λ T . One can see from figure 8 that smaller tan β values are allowed for larger λ T from the consideration of Higgs boson invisible branching ratio. However, tan β < 5 is ruled out for any values of λ T in the range (0.1 -0.9). A comparison with figure 7 reveals that the invisible branching ratio of the Higgs boson restricts tan β in a more stringent way than does the total decay width. The grey shaded region is disallowed from the constraint on invisible branching ratio of the Higgs boson. Figure 9 shows that there is an available region of parameter space consistent with the latest experimental findings involving m h and µ γγ . One can find that in this scenario somewhat larger values of tan β ( > ∼ 20) are preferred and this facilitates 8 having µ γγ in the vicinity of 1. At the same time the constraint from invisible Higgs branching ratio does not allow k T OT to be greater than ∼ 1.2. This in turn sets a lower bound on µ γγ roughly around 0.85. Note that since larger k T OT corresponds to smaller µ γγ , in this case the allowed region of parameter space easily accommodates a value of µ γγ within the -1σ ranges of the central values quoted in the main analysis of the CMS collaboration and the analysis of the ATLAS collaboration. Furthermore, figure 9 reveals that a conservative situation characterized by both µ γγ and k T OT not so different from their SM values, i.e., 1 is realized for moderate values of the top squark mass in the range 1 TeV -1.2 TeV.  It is pertinent to mention that these plots use spectra of particles which are consistent with the lower bound on the lightest chargino mass (> 104 GeV, from the LEP experiments) and are also free from tachyonic scalar states. 8 It is pertinent to note that the upper bound on tan β is obtained from the contribution of the leptonic Yukawa coupling, fτ ≡ λ133, to the ratio Rτ = Γ(τ → eνeντ )/ Γ(τ → µνµντ ). The resulting constraint is fτ < 0.07( mτ R 100 GeV ) [84]. Choosing mτ R to be around 1 TeV corresponds to tan β < ∼ 70.

Relative signal strengths in different final states
In this subsection we briefly discuss how other final states arising from the lightest Higgs boson are expected to be affected in our scenario relative to the γγ final state and where they stand vis-a-vis the experimental results. Such a study of relative strengths over the parameter space of our scenario would be indicative of how well the same is compatible with the experimental observations in the Higgs sector, in a global sense. The recent results from the ATLAS and the CMS collaborations on different decay modes of the lightest Higgs boson are presented in table 2. In figure 11, we present the µ-values reported by the ATLAS and the CMS collaborations for different final states in the so-called signature (ratio) space, in reference to µ γγ .  In each plot, the blue circle (green square) represents the experimentally reported central values for a given pair of observables from CMS and ATLAS collaborations, respectively. The solid grey lines show the range of µ values as observed by the CMS experiment while the dashed ones delineate the same as obtained by the ATLAS experiment. We have already seen from figures 9 and 10 that the total decay width is rather large compared to the SM value for small tan β ∼ 5. Hence, to be conservative, we do not let tan β to be that low and thus, vary it within the range 10 < tan β < 40. We have also varied the mass of the top squark within the range 350 GeV < mt < 1.5 TeV with 0.1 < f < 1 and 0.1 < λ T < 0.55. All other parameters are kept fixed at the previously mentioned values. While scanning, care has been taken to reject spectra with tachyonic scalar states and to conform with the lower bound on the lightest chargino mass of 104 GeV as obtained from the LEP experiment. Also, the scan required m h to be within the range of 124.0 − 126.2 GeV as reported by the LHC experiments. As can be noticed in figure 11, this particular scenario with somewhat large values of 'f ' generally predicts low values of µ γγ . This can perhaps be attributed to the presence of an MeV neutralino in the spectrum for such values of 'f ' that increases the total decay width of the lightest Higgs boson. Finally, in order to have an idea of the mass-spectra of the light neutralino and the chargino states, we provide a few benchmark points in table 3, for the large 'f ' scenario. 7 The case of small Yukawa coupling, f ∼ O(10 −4 ) In the limit when the Yukawa coupling is small (f ∼ 10 −4 ), the next-to-lightest neutralino state becomes the sterile neutrino with negligible active-sterile mixing. The lightest neu- tralino state is again the active neutrino. The tree level Majorana mass of the active neutrino is given by eq. (4.11) whereas the sterile neutrino mass and the mixing angle between the active and the sterile neutrino are given by eqs. (4.13) and (4.14). In the recent past, an X-ray line at around 3.5 keV was observed in the X-ray spectra of the Andromeda galaxy and in the same from various other galaxy clusters including the Perseus cluster. The observed flux and the best fit energy peak are found to be at [129,130] Φ γ = 4 ± 0.8 × 10 −6 photons cm −2 sec −1 , E γ = 3.57 ± 0.02 keV.
It is understood that atomic transitions in the thermal plasma cannot account for this energy line. Hence, a possible explanation can be provided by taking into account a 7 keV dark matter, in this case a sterile neutrino [129,130]. The observed flux and the peak of the energy can be translated to an active-sterile mixing in the range 2.2 × 10 −11 < sin 2 2θ 14 < 2 × 10 −10 . It is also important to study the signal strength of h → γγ in the light of this 7 keV sterile neutrino with appropriate active-sterile mixing. The not so heavy top squark, as justified in section 6.1.2, enhances µ γγ considerably and we show the contours of µ γγ at 1.17, 1.14 and 1.11 respectively with blue dashed lines. Finally, the grey shaded region is the parameter space consistent with the observed Higgs boson mass 124.0GeV < m h < 126.2 GeV. Figure  12 clearly shows that for this choice of parameters µ γγ > ∼ 1.1 is completely consistent with a 7 keV sterile neutrino dark matter and the experimentally allowed range of Higgs boson mass. We have seen that charginos do not provide much enhancement to µ γγ due to its very suppressed couplings under the present set-up. Furthermore, avoiding possible appearance of tachyonic scalar states restricts the vev of the singlet from becoming large. Therefore, expecting an enhancement in µ γγ via suppression of the hbb coupling because of the singlet admixture seems unrealistic Thus, the only enhancement in µ γγ can come from light top squarks. In addition, large radiative corrections from λ S and λ T reduces the necessity of having heavy top squarks. In the scatter plot of figure 13 we show the possible range of variation of µ γγ with varying mt. To generate this plot we have chosen relevant parameters over the following ranges: 1 GeV < m 3/2 < 20 GeV, 5 < tan β < 35, 300 GeV < mt < 1.5 TeV, 10 −5 < f < 3 × 10 −4 , 0.1 < λ T < 1 and −0.01 GeV < v S < −1 GeV. Other parameters are retained at their previously mentioned values, maintaining the degeneracy between the Dirac gaugino masses as already mentioned. Again, all these points are consistent with 124.0 GeV < m h < 126.2 GeV and free from any tachyonic scalar states. The main difference between the small and the large 'f ' scenarios is the absence of a bino like next-to-lightest neutralino in the former case. This decreases the total decay width of the Higgs boson, potentially resulting in some enhancement in µ γγ . The blue points are consistent with a keV sterile neutrino with mass ranging between 7.01 keV < M R N < 7.11 keV and is known to be a fit warm dark matter candidate having the right relic density. Finally, it is again very relevant to check the relative signal strengths for different decay modes of the lightest Higgs boson in such a scenario with small 'f '; similar to what we have done in section 6.1.3 for the large 'f ' scenario. Figure 14 shows scattered points consistent with the CMS or/and the ATLAS results at 1σ level. However, note that the scatter plot in the µ γγ -µ W W plane is consistent only with the results from the ATLAS experiments at the 1σ level whereas the the scatter plot in the µ γγ -µ bb plane is consistent only with the results from the CMS experiments at the 1σ level. In the near future, a more precise measurement together with an improved analysis is likely to become more decisive on this issue. Finally, for the sake of completeness, in table 4 we provide three more benchmark sets comprising of the input parameters of the small Yukawa coupling scenario (with (f ∼ 10 −4 )), the corresponding mass-values of the relevant excitations and the Higgs signal strengths in the diphoton final state (µ γγ ).

Conclusion
In this work a detailed analysis of the h → γγ channel in a non-minimal U (1) R −lepton number scenario has been performed. Experimental results reported for other final states arising from the decay of the Higgs boson are also put in context. We introduce one right handed neutrino superfield which leads to a multitude of interesting phenomenological consequences. In a previous work [88], it was shown that the Yukawa coupling 'f ', which couples the right handed neutrino superfield with the Higgs boson and the lepton superfield, plays a very important role. f ∼ O(1) contributes heavily to the tree level Higgs boson mass. f ∼ O(10 −4 ), yields a keV dark matter in the form of a sterile neutrino with correct relic density. In this case, large triplet and singlet couplings, λ T and λ S respectively, help achieve m h in the range narrowed down by the LHC experiments without requiring large masses for the top squarks. Compatibility of the scenario with the results reported by the LHC collaborations pertaining to h → γγ channel is demonstrated by studying the parton level production of the Higgs boson and its subsequent decays. It is observed that for large values of 'f ', a smaller top squark mass along with a moderately large tan β (∼ 10) provides an enhancement in both production cross section and decay width of the Higgs boson when compared to their values predicted by the SM. Contribution to Γ(h → γγ) from charginos in the loop for is found to be insignificant due to their very weak coupling with the Higgs boson. In the present scenario, hW W coupling is modified by the factor sin(β − α). It is demonstrated that Γ(h → γγ) may receive a significant contribution from the W boson loop when 'f ' and bµ L are large. Moreover, a heavy charged Higgs boson does not provide any enhancement to the h → γγ rate either. We have seen that the large 'f ' case is accompanied by a very light bino like neutralino, which enhances the total decay width of the Higgs boson. Therefore, a relatively small value of µ γγ ∼ 0.9 is observed. Recent studies on the invisible decay width of the Higgs boson from ATLAS and CMS help constrain the parameter space significantly. As for example, for λ T = 0.5, the lower bound on tan β can be as large as 18. Similar but a less stringent constraint is also obtained by investigating the total decay width of the lightest Higgs boson. Overall, the signal strength µ γγ matches very well with the main analysis performed by the CMS collaboration as well as the observation made by the ATLAS collaboration.
Subsequently, we have also studied the case of small values of 'f '. The scenario is  characterized by the presence of a sterile neutrino with mass in keV's which is a potent warm dark matter candidate. Further, the scenario is contrasted against the large 'f ' scenario by the absence of a light bino-like neutralino in its spectrum. We then present the variation of µ γγ with the model parameters varied simultaneously over appropriate ranges.
In the absence of light bino-like neutralino, to which the Higgs boson could have otherwise decayed to, the total decay width of the latter remains to be smaller compared to the large 'f ' case. This in turn ensures µ γγ attaining values all the way up to 1.2. Such values of µ γγ are also compatible with the results of the main analysis performed by the CMS collaboration and also conforms with the observations made by the ATLAS collaboration at the 1σ level. We have also discussed and illustrated the possibility of having a 7 keV sterile neutrino with appropriate active-sterile mixing, that can be a fit warm dark matter candidate.

A The Higgs-chargino-chargino coupling
In this appendix we work out the Higgs-chargino-chargino coupling in the scenario under discussion and present the analytical expression for the width of the lightest Higgs boson decaying into a pair of charginos. The relevant Lagrangian in the two-component notation containing the Higgs-chargino-chargino interaction is given by where the matrix S connects the mass and gauge eigenstates of the CP even scalar mass squared matrix, written in the basis (h R ,ν R , S R , T R ). To be more precise the physical CP-even scalar states are related to the gauge eigenstates in the following manner:  In our notation the lightest physical state (h 4 ) of the CP even scalar mass matrix corresponds to the physical Higgs boson, h. Moreover, the charginosχ ± i are four component Dirac fermions which arise due to the mixing between the charged gauginos and higgsinos as well as the charged lepton of first generation. In order to evaluate find out the Higgschargino-chargino coupling and to evaluate the Higgs boson partial decay width to a pair of charginos, it is pertinent to write down the interaction Lagrangian in the four-component notation. We now define the 4-component spinors as Using the transformation relations, the Lagrangian in eq. (A.1) can be expressed in the four component notation as The chargino masses can have any sign. By demanding that the four component Lagrangian contains only positive masses for the charginos, we define the chargino states in the following manner [131,132] χ + i = (ǫ i P L + P R ) where ǫ i carries the sign of the chargino masses, which can be ±1. When ǫ = −1, P R −P L = γ 5 , which essentially implies a γ 5 rotation to the four component spinors to absorb the sign. Hence, the transformation relations involving only P L changes, which modifies the Feynman rules. The two-component mass eigenstates (χ ± i ) of the charginos are related to the gauge eigenstates in a manner shown in eq. (4.17).
Using the following set of relations we rewrite eq. (A.5) in the mass eigenstate basis as The coupling is obtained from Eq. (A.8) as It is now straightforward to compute the lightest Higgs boson decay width to a pair of charginos, which we find as Finally, if we assume the singlet and the triplet vev's to be very small, this would imply that the singlet and triplet mixing in the light CP-even Higgs boson states become negligible. Under such an assumption, the CP even states can be written as where we have chosen S 41 = cos α, S 42 = − sin α, and S 43 ∼ S 44 ∼ 0. With this simplification we can write (A.14)

B The Higgs-neutralino-neutralino coupling
In a similar manner the interaction of the Higgs boson with neutralinos can be constructed from the following (two-component) Lagrangian In terms of these spinors the 4-component Lagrangian takes the following form It is now straightforward to write down the Higgs-neutralino-neutralino interaction in the 4-component notation as Finally, the partial decay width Γ(h → χ 0 i χ 0 j ) is given as Again in the limit where the singlet and triplet vev's are very small, we can safely ignore the contributions from S 43 and S 44 . Furthermore, replacing S 41 by cos α and S 42 by -sin α, we can write where,