h→γγ in U(1 ) R − lepton number model with a right-handed neutrino
aa r X i v : . [ h e p - ph ] A ug Prepared for submission to JHEP
HRI-P-14-10-001 h → γ γ in U (1) R − lepton number model with aright-handed neutrino Sabyasachi Chakraborty, a, AseshKrishna Datta, b Sourov Roy a a Department of Theoretical Physics, Indian Association for the Cultivation of Science, 2A & b Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, INDIA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We perform a detailed study of the signal rate of the lightest Higgs boson in thediphoton channel ( µ γγ ), recently analyzed by both the ATLAS and CMS collaborations atthe Large Hadron Collider, in the framework of U (1) R − lepton number model with a righthanded neutrino superfield. The corresponding neutrino Yukawa coupling, ‘ f ’, plays a veryimportant role in the phenomenology of this model. A large value of f ∼ O (1) provides anadditional tree level contribution to the lightest Higgs boson mass along with a very light(mass ∼ a few hundred MeV) bino like neutralino and a small tree level mass of one of theactive neutrinos that is compatible with various experimental results. In the presence ofthis 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 µ γγ obtained in this scenario is compatible with the recent results from both the ATLAS andthe CMS collaborations at 1 σ level. A small value of ‘ f ’, on the other hand, is compatiblewith a sterile neutrino acting as a 7 keV dark matter that can explain the observation of amono-energetic X-ray photon line by the XMM-Newton X-ray observatory. We also studythe impact of µ γγ in this case. Keywords:
Supersymmetry Phenomenology Corresponding author. ontents U (1) R -lepton number model with a right handed neutrino 43 The scalar sector 6 m h µ γγ h → gg h → γγ f ∼ O (1) µ γγ f ∼ O (10 − ) 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, anelementary scalar boson of nature [1, 2], with mass around
GeV. Almost all the decaychannels 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,– 1 –his is the discovery mode of the Higgs boson and second, being a loop induced processit may potentially carry indirect hints of new physics. The results reported so far showsome deviations with respect to the Standard Model (SM) prediction. For example, theATLAS collaboration reported µ γγ = 1 . ± . [3], where µ γγ = σ ( pp → h → γγ ) σ ( pp → h → γγ ) SM . On theother hand, CMS collaboration reported a best-fit signal strength in their main analysis[4] where, µ γγ = 1 . +0 . − . . Moreover, a cut-based analysis by CMS produced a slightlydifferent value, which is quoted as µ γγ = 1 . +0 . − . . This enhancement or suppression inthe h → γγ channel with respect to the SM provide a natural testing ground for physicsBeyond the SM (BSM). Detailed studies have already been carried out for this particularchannel. For example, h → γγ is studied in a wide variety of supersymmetric (SUSY)models namely, the minimal supersymmetric standard model (MSSM) [5–25], its next-to-minimal version (NMSSM) [26–34], the constrained MSSM (CMSSM) [35–40] and also in(B-L)SSM [41–44], 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 inthe context of a supersymmetric scenario known as U (1) R − lepton number model, whichis augmented by a single right-handed neutrino superfield. It is rather well known thatsupersymmetry is one of the very popular frameworks that provides a suitable dark mattercandidate and can also explain the origin of neutrino masses and mixing. However, the non-observation of superpartners so far has already put stringent lower bounds on their massesin 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 opposedto 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 crosssection compared to MSSM thus relaxing the bound on squark masses. Detailed studieson R -symmetric models and Dirac gauginos can be found in the literature [48–98]. Flavorand CP violating constraints are also suppressed in these class of models [58]. To constructDirac gaugino masses, the gauge sector of the supersymmetric Standard Model has to beextended to incorporate chiral superfields in the adjoint representations of the SM gaugegroup. A singlet ˆ S , an SU (2) triplet ˆ T and an SU (3) octet ˆ O , help obtain the Dirac gauginomasses.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 R-charges are identified with lepton numbers such that the lepton number of SM fermionsand their superpartners are negative of the corresponding R-charges. Such an identificationleaves the lepton number assignments of the SM fermions unchanged from the usual oneswhile the same for the superpartners become non-standard. This has an interesting conse-quence for the sneutrinos which now do not carry any lepton number. Hence, although inthis model sneutrinos get non-zero vacuum expectation value ( vev ) in general, the latter donot get constrained from neutrino Majorana masses which require lepton number violationby two units. A sneutrino thus can play the role of a down type Higgs boson, a phenomenonwhich has crucial implications [74, 79, 84, 85, 88] for our purpose that we would discusslater in this work. The right handed neutrino, on the other hand, not only provides a small– 2 –ree level Dirac neutrino mass but also gives rise to an additional tree level contributionto the Higgs boson mass proportional to the neutrino Yukawa coupling [88]. When theR-symmetry is broken, a small ( < ∼ .A large Yukawa coupling f ∼ O (1) facilitates having the mass of the lightest Higgsboson around GeV without resorting to radiative contributions. Large values of f also result in a very light neutralino with mass around a few hundred MeV. Cosmologicalimplications 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 [100–109]. On the otherhand, in the regime of small Yukawa coupling f ∼ − , the Higgs boson mass is devoid ofany large tree level contribution. Therefore, to obtain the mass of the lightest Higgs bosonin the right ballpark, radiative corrections have to be incorporated, which are required to belarge 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 tothe diphoton final states arising from the decay of the lightest Higgs boson. We study thisscenario in conjunction with the recent results of µ γγ obtained from the latest results of LHCcollaborations. This particular case under consideration has some important implicationssince we can now afford rather light top squarks which potentially affect the resonantproduction rate of the lightest Higgs boson and its decay pattern. Furthermore, presenceof a very light neutralino opens up new decay modes of the Higgs bosons which in turnis subject to the constraints from Higgs invisible branching fractions. 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 2 we briefly discuss the main features ofthe model. The principal motivation and the artifacts of the U (1) R − lepton number modelare also discussed with reference to its scalar and the electroweak gaugino sector. In section3, we discuss the scalar sector of the model in detail. In Section 4 we address the neutralinoand the chargino sectors. The masses and the couplings in these sectors play importantroles in the computation of µ γγ . A thorough analysis of µ γγ requires the knowledge of bothproduction and decays of the Higgs boson. In Section 5 issues pertaining to the productionof Higgs boson in the present scenario is discussed in some detail. Analytical expressions ofHiggs boson decaying to two photons in our model are also given in the same section. Section6 is dedicated to the computation of the invisible decay width of the Higgs boson. Here wealso discuss the impact of the findings from the LHC pertaining to the Higgs sector on thescenario under discussion for two distinct cases: a) when the neutrino Yukawa coupling islarge, i.e., O (1) and b) when it is O (10 − ) . We also provide µ γγ and show its variation withrelevant parameters, along with the points representing the 7 keV sterile neutrino warmdark matter in this model. We conclude in Section 7 with some future outlooks. The Higgsboson couplings to neutralino and charginos in this model are relegated to the appendix. For a review on other models of keV sterile neutrino dark matter, see ref. [99]. – 3 – U (1) R -lepton number model with a right handed neutrino We consider a minimal extension of an R -symmetric model, first discussed in [84, 85], byextending the field content with a single right handed neutrino superfield [88]. Along withthe MSSM superfields, ˆ H u , ˆ H d , ˆ U ci , ˆ D ci , ˆ L i , ˆ E ci , two inert doublet superfields ˆ R u and ˆ R d with opposite hypercharge are considered in addition to the right handed neutrino superfield ˆ N c . These two doublets ˆ R u and ˆ R d carry non zero R-charges (The R-charge assignmentsare provided in table 1 and therefore, in order to avoid spontaneous R-breaking and theemergence of R-axions, the scalar components of ˆ R u and ˆ R d do not receive any nonzero vev and because of this they are coined as inert doublets. ˆ Q i ˆ U ci ˆ D ci ˆ L i ˆ E ci ˆ H u ˆ H d ˆ R u ˆ R d ˆ S ˆ T ˆ O ˆ N c U (1) R Table 1 . U (1) R charge assignments of the chiral superfields. R-symmetry prohibits the gauginos to have Majorana mass term and trilinear scalarinteractions ( A -terms) are also absent in a U (1) R invariant scenario. However, the gauginoscan acquire Dirac masses. In order to have Dirac gaugino masses one needs to include chiralsuperfields in the adjoint representations of the standard model gauge group. Namely asinglet ˆ S , an SU (2) L triplet ˆ T and an octet ˆ O under SU (3) c . These chiral superfieldsare essential to provide Dirac masses to the bino, wino and gluino respectively. We wouldlike to reiterate that the lepton numbers have been identified with the (negative) of R-charges such that the lepton number of the SM fermions are the usual ones whereas thesuperpartners of the SM fermions carry non-standard lepton numbers. With such leptonnumber 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 W = y uij ˆ H u ˆ Q i ˆ U cj + µ u ˆ H u ˆ R d + f i ˆ L i ˆ H u ˆ N c + λ S ˆ S ˆ H u ˆ R d + 2 λ T ˆ H u ˆ T ˆ R d − M R ˆ N c ˆ S + µ d ˆ R u ˆ H d + λ ′ S ˆ S ˆ R u ˆ H d + λ ijk ˆ L i ˆ L j ˆ E ck + λ ′ ijk ˆ L i ˆ Q j ˆ D ck + 2 λ ′ T ˆ R u ˆ T ˆ H d + y dij ˆ H d ˆ Q i ˆ D cj + y eij ˆ H d ˆ L i ˆ E cj + λ N ˆ N c ˆ H u ˆ H d . (2.1)For simplicity, in this work we have omitted the terms κ ˆ N c ˆ S ˆ S and η ˆ N c from the superpo-tential. As long as η ∼ M USY and κ ∼ we do not expect any significant change in theanalysis and the results presented in this paper.In order to have a realistic model one should also include supersymmetric breakingterms, which are the scalar and the gaugino mass terms. The Lagrangian containing theDirac gaugino masses can be written as [71, 73] L Diracgaugino = Z d θ W ′ α Λ [ √ κ W α ˆ S + 2 √ κ tr( W α ˆ T ) + 2 √ κ tr( W α ˆ O )] + h.c., (2.2)where W ′ α = λ α + θ α D ′ is a spurion superfield parametrizing D-type supersymmetry break-ing. This results in Dirac gaugino masses as D ′ acquires vev and are given by M Di = κ i < D ′ > Λ , (2.3)– 4 –here 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 generatelogarithmic divergence to the scalar masses whereas in ref. [54], it was shown that thepurely scalar loop, obtained from the adjoint superfields cancels this logarithmic divergencein the case of Dirac gauginos. Hence it is not unnatural to consider the Dirac gauginomasses to be rather large.The R-conserving but soft supersymmetry breaking terms in the scalar sector are gen-erated from a spurion superfield ˆ X , where ˆ X = x + θ F X such that R [ ˆ X ] = 2 and < x > = 0 , < F X > = 0 . The non-zero vev of F X generates the scalar soft terms and the correspondingpotential is given by V soft = m H u H † u H u + m R u R † u R u + m H d H † d H d + m R d R † d R d + m L i ˜ L † i ˜ L i + m R i ˜ l † Ri ˜ l Ri + M N ˜ N c † ˜ N c + m S S † S + 2 m T tr( T † T ) + 2 m O tr( O † O ) + ( BµH u H d + h . c . ) − ( bµ iL H u ˜ L i + h . c . ) + ( t S S + h . c . ) + 12 b S ( S + h . c . ) + b T (tr( T T ) + h . c . )+ B O (tr( OO ) + h . c . ) . (2.4)The presence of the bilinear terms bµ iL H u ˜ L i implies that all the three left handed sneutrinoscan acquire non-zero vev ’s. However, it is always possible to make a basis rotation in whichonly one of the left handed sneutrinos get a non-zero vev and one must keep in mind thatthe physics is independent of this basis choice.Such a rotation can be defined as ˆ L i = v i v a ˆ L a + X b e ib ˆ L b . (2.5)Note that the index ( i ) runs over three generations whereas a = 1( e ) and b = 2 , µ, τ ) .This basis rotation implies that the scalar component of the superfield ˆ L a acquires a nonzero vev (i.e. h ˜ ν i ≡ v a = 0) whereas the other two sneutrinos do not get any vev . Onecan 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) cannotget mass from this Yukawa couplings because of SU (2) L invariance but can be generatedfrom R-symmetric supersymmetry breaking operators [84]. Moreover, we also choose theneutrino Yukawa coupling in such a way that only ˆ L a couples to ˆ N c . In such a scenario theleft-handed sneutrino can play the role of a down type Higgs boson since its vev preserveslepton number and is not constrained by neutrino Majorana mass. Hence one has thefreedom to keep a very large µ d such that the superfields ˆ H d and ˆ R u get decoupled fromthe 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 chargedlepton and down type quark fields the superpotential now has the following form (integratingout ˆ H d and ˆ R u ) W = y uij ˆ H u ˆ Q i ˆ U cj + µ u ˆ H u ˆ R d + f ˆ L a ˆ H u ˆ N c + λ S ˆ S ˆ H u ˆ R d + 2 λ T ˆ H u ˆ T ˆ R d − M R ˆ N c ˆ S + W ′ , (2.6) For a detailed discussion we refer the reader to ref. [88]. – 5 –here W ′ = X b =2 , f lb ˆ L a ˆ L ′ b ˆ E ′ cb + X k =1 , , f dk ˆ L a ˆ Q ′ k ˆ D ′ ck + X k =1 , ,
12 ˜ λ k ˆ L ′ ˆ L ′ ˆ E ′ ck + X j,k =1 , , b =2 , ˜ λ ′ bjk ˆ L ′ b ˆ Q ′ j ˆ D ′ ck , (2.7)and includes all the trilinear R-parity violating terms in this model. In the subsequentdiscussion we shall confine ourselves to this choice of basis but get rid of the primes fromthe fields and make the replacement ˜ λ , ˜ λ ′ → λ , λ ′ .In this rotated basis the soft supersymmetry breaking terms look like V soft = m H u H † u H u + m R d R † d R d + m L a ˜ L † a ˜ L a + X b =2 , m L b ˜ L † b ˜ L b + M N ˜ N c † ˜ N c + m R i ˜ l † Ri ˜ l Ri + + m S S † S + 2 m T tr( T † T ) + 2 m O tr( O † O ) − ( bµ L H u ˜ L a + h . c . ) + ( t S S + h . c . )+ 12 b S ( S + h . c . ) + b T (tr( T T ) + h . c . ) + B O (tr( OO ) + h . c . ) . (2.8)With this short description of the theoretical framework let us now explore the scalar andthe fermionic sectors in some detail in order to prepare the ground for the study of thediphoton decay of the lightest Higgs boson. The scalar potential receives contributions from the F-term, the D-term, the soft SUSYbreaking terms and the terms coming from one-loop radiative corrections. Thus, schemat-ically, V = V F + V D + V soft + V one − loop . (3.1)The F-term contribution is given by V F = X i (cid:12)(cid:12)(cid:12)(cid:12) ∂W∂φ i (cid:12)(cid:12)(cid:12)(cid:12) , (3.2)where the superpotential W is given by eq. (2.6). The D -term contribution can be writtenas V D = 12 X a D a D a + 12 D Y D Y , (3.3)where D a = g ( H † u τ a H u + ˜ L † i τ a ˜ L i + T † λ a T ) + √ M D T a + M D T a † ) . (3.4)The τ a ’s and λ a ’s are the SU (2) generators in the fundamental and adjoint representationrespectively. The weak hypercharge contribution D Y is given by D Y = g ′ H + u H u − ˜ L + i ˜ L i ) + √ M D ( S + S † ) , (3.5)– 6 –here g and g ′ are SU (2) L and U (1) Y gauge couplings respectively. The expanded formsof V F and V D in terms of various scalar fields can be found in [88]. The soft SUSY breakingterm V soft is given in Eq. (2.8) whereas the dominant radiative corrections to the quarticpotential are of the form δλ u ( | H u | ) , δλ ν ( | ˜ ν a | ) and δλ | H u | | ˜ ν a | . The coefficients δλ u , δλ ν and δλ are given by δλ u = 3 y t π ln (cid:18) m ˜ t m ˜ t m t (cid:19) + 5 λ T π ln (cid:18) m T v (cid:19) + λ S π ln (cid:18) m S i v (cid:19) − π λ S λ T m T − m S (cid:18) m T (cid:26) ln (cid:18) m T v (cid:19) − (cid:27) − m S (cid:26) ln (cid:18) m S v (cid:19) − (cid:27)(cid:19) , (3.6) δλ ν = 3 y b π ln (cid:18) m ˜ b m ˜ b m b (cid:19) + 5 λ T π ln (cid:18) m T v (cid:19) + λ S π ln (cid:18) m S v (cid:19) − π λ S λ T m T − m S (cid:18) m T (cid:26) ln (cid:18) m T v (cid:19) − (cid:27) − m S (cid:26) ln (cid:18) m S v (cid:19) − (cid:27)(cid:19) , (3.7) δλ = 5 λ T π ln( m T v ) + 132 π λ S ln (cid:18) m S v (cid:19) + 132 π λ S λ T m T − m S (cid:18) m T (cid:26) ln (cid:18) m T v (cid:19) − (cid:27) − m S (cid:26) ln (cid:18) m S v (cid:19) − (cid:27)(cid:19) . (3.8)We shall see later that for large values of the couplings λ T and λ S or large stop massesthese one-loop radiative contributions to the Higgs quartic couplings could play importantroles in obtaining a CP-even lightest Higgs boson with a mass around 125 GeV. Let us assume that the neutral scalar fields H u , ˜ ν a ( a = 1( e )) , S and T acquire real vacuumexpectation values v u , v a , v S and v T , respectively. The scalar fields R d and ˜ N c carryingR-charge 2 are decoupled from these four scalar fields. We can split the fields in termsof their real and imaginary parts: H u = h R + ih I , ˜ ν a = ˜ ν aR + i ˜ ν aI , S = S R + iS I and T = T R + iT I . The resulting minimization equations can be found easily and with the helpof these minimization equations, the neutral CP-even scalar squared-mass matrix in thebasis ( h R , ˜ ν R , S R , T R ) can be written down in a straightforward way, where h correspondsto the lightest CP even mass eigenstate [88]. In the R-symmetry preserving scenario the– 7 –lements of this symmetric × matrix are found to be ( M S ) = ( g + g ′ )2 v sin β + ( f M R v S − bµ aL )(tan β ) − + 2 δλ u v sin β, ( M S ) = f v sin 2 β + bµ aL − ( g + g ′ − δλ )4 v sin 2 β − f M R v S , ( M S ) = 2 λ S v S v sin β + 2 µ u λ S v sin β + 2 λ S λ T vv T sin β + √ g ′ M D v sin β − f M R v cos β, ( M S ) = 2 λ T v T v sin β + 2 µ u λ T v sin β + 2 λ S λ T v S v sin β − √ gM D v sin β, ( M S ) = ( g + g ′ )2 v cos β + ( f M R v S − bµ aL ) tan β + 2 δλ ν v cos β, ( M S ) = −√ g ′ M D v cos β − f M R v sin β, ( M S ) = √ gM D v cos β, ( M S ) = − µ u λ S v sin βv S − λ S λ T v T v sin βv S − t S v S + g ′ M D v cos 2 β √ v S + f M R v sin 2 β v S , ( M S ) = λ S λ T v sin β, ( M S ) = − µ u λ T v v T sin β − λ S λ T v S v v T sin β − gM D √ v v T cos 2 β, (3.9)where tan β = v u /v a and v = v u + v a . The W ± - and the Z -boson masses can be writtenas m W = 12 g ( v + 4 v T ) ,m Z = 12 g v / cos θ W . (3.10)Note that the electroweak precision measurements of the ρ -parameter requires that thetriplet vev v T must be 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 singletscalar S . In such a simplified but viable scenario in which the singlet and the SU (2) L tripletscalars get decoupled from the theory, we are left with a × scalar mass matrix. In thiscase the angle α represents the mixing angle between h R and ˜ ν R and can be expressed interms of other parameters as follows tan 2 α = − f v sin 2 β + bµ aL − ( g + g ′ − δλ )4 v sin 2 β ( g + g ′ ) v cos 2 β + 2 bµ aL cot 2 β − v (cid:8) δλ u sin β − δλ ν cos β (cid:9) . (3.11) m h In addition, in such a situation (with v S , v T ≪ v ) it can be shown easily that the lightestCP-even Higgs boson mass is bounded from above at tree level [88], ( m h ) tree ≤ m z cos β + f v sin β. (3.12)The bound in Eq. (3.12) is saturated for v s < ∼ − GeV, i.e., when the singlet has alarge soft supersymmetry breaking mass and is integrated out. The f v term grows at– 8 – h =
125 GeV H m h L Tree H m h L Tree - bound - v S H GeV L m h H G e V L Figure 1 . The tree level mass of the lightest Higgs boson as a function of the singlet ( S ) vacuumexpectation value v S with f = 1.5, tan β =4 and other parameter choices are as described in thetext. The upper bound on the tree level mass of the Higgs boson from eq. 3.12 is also shown. f = v s = - GeV f = v s = - GeV f = v s = - GeV f = v s = - GeV
200 400 600 800 1000510152025303540 m t Ž H GeV L t a n Β Figure 2 . Mass-contours for the lightest Higgs boson with m h =
125 GeV in the m ˜ t – tan β planefor large values of f and λ T = 0.5. small tan β and thus the largest Higgs boson mass is obtained with low tan β and largevalues of f . We shall show in the next section that f ∼ can be accommodated in thisscenario 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 ∼ GeV where the peak in– 9 –he 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 orso and couplings λ T and λ S can be small ( ∼ − ) 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 verysmall v S ( < ∼ − GeV) the tree level Higgs boson mass is 150 GeV and is reduced to 125GeV for a v S ∼ . GeV. As v S increases further, ( M h ) Tree starts decreasing rapidly andthe Higgs boson mass becomes lighter than 100 GeV. In such a case one requires largerradiative corrections to the Higgs boson mass and this can be achieved with the help oflarge triplet/singlet couplings ( O (1) ) and/or large stop mass. For example, with a choiceof λ S = 0 . and λ T = 0.5, the one-loop radiative corrections to the Higgs boson massarising from these two couplings are sizable . In this case, in order to have a 125 GeV Higgsboson, the tree level contribution should be smaller and for a very small v S ( ∼ − GeV)and large f ( > ∼ tan β . The one loop corrections fromthe stop loop must also be small and this is realized for small m ˜ t and large tan β . Thisis illustrated in figure (2) where we plot mass-contours for the lightest Higgs boson witha mass of 125 GeV in the m ˜ t – tan β plane for different choices of f and v S . One can seefrom this figure the effect of a larger v S , which requires a larger stop loop contribution tohave a Higgs boson mass of 125 GeV. The fermionic sector of the scenario, involving the neutralinos and the charginos, has richnew features. In the context of the present study, when analyzed in conjunction with thescalar sector of the scenario, this sector plays a pivotal role by presenting the defining issuesfor the phenomenology of this scenario. Its influence ranges over physics of the Higgs bosonat current experiments and the physics of the neutrinos before finally reaching out to thedomain of astrophysics and cosmology by offering a possible warm dark matter candidatewhose actual presence may find support in the recent observations of a satellite-borne X-rayexperiment. Thus, it is of crucial importance to study the structure and the content of thissector in appropriate detail.A thorough discussion of µ γγ in the present scenario requires a study of the masses andthe mixing angles of the neutralinos and the charginos. One of the natural consequencesof such a U (1) R -lepton number model with a right-handed neutrino is that one of theleft-handed neutrinos (the electron-type one) and the right-handed neutrino become partsof the extended neutralino mass matrix. The electron-type neutrino of the SM can beidentified with the lightest neutralino eigenstate. We also address the issue of tree levelneutrino mass. Subsequently, we show that in certain region of the parameter space thelightest neutralino-like state can be very light (with a mass of order 100 MeV). This maycontribute to the invisible decay width of the lightest Higgs boson. We study the validity These choices of λ T and λ S are not completely independent. Rather they follow a relationship derivedfrom the requirement of small tree level mass of the active neutrino. This will be discussed in the nextsection. – 10 –f the parameter space when it is subject to the constraint from invisible decay width ofthe Higgs boson. 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 theelectroweak symmetry is broken. The part of the Lagrangian that corresponds to the neutralfermion mass matrix is given by L = ( ψ ) T M Dχ ( ψ − ) where ψ = (˜ b , ˜ w , ˜ R d , N c ) , withR-charges +1 and ψ − = ( ˜ S, ˜ T , ˜ H u , ν e ) with R-charges -1. The neutral fermion massmatrix M Dχ is given by M Dχ = M D g ′ v u √ − g ′ v a √ M D − gv u √ gv a √ λ S v u λ T v u µ u + λ S v S + λ T v T M R − f v a − f v u . (4.1)The above matrix can be diagonalized by a biunitary transformation involving two unitarymatrices V N and U N and results in four Dirac mass eigenstates e χ i ≡ e ψ i e ψ − i ! , with i = 1 , , , and e ψ i = V Nij ψ j , e ψ − i = U Nij ψ − j . The lightest mass eigenstate e χ isidentified with the light Dirac neutrino. The other two active neutrinos remain masslessin this case. Generically the Dirac neutrino mass can be in the range of a few eV to tensof MeV. However, one can also accommodate a mass of 0.1 eV or smaller for the Diracneutrino by assuming certain relationships involving different parameters [88], which are λ T = λ S tan θ W (4.2)and M R = √ f M D tan βg tan θ W . (4.3)With these choices the Dirac mass of the neutrino can be written as m Dν e = v f g sin β √ γM D M D λ T ( M D − M D ) . (4.4)where γ = µ u + λ S v S + λ T v T . It is straightforward to check from eq. (4.4) that by suitablechoices of the parameters f , λ T and ǫ ≡ ( M D − M D ) , one can have a Dirac neutrino massin the right ballpark of < ∼ f ∼ O (1) is possible for a small λ T ( ∼ − ) and nearly degenerate Dirac gauginos ( ǫ < ∼ − ) assuming µ u , M D , M D inthe few hundred GeV range. R-symmetry is not an exact symmetry and is broken by a small gravitino mass. One cantherefore consider the gravitino mass as the order parameter of R-breaking. The breaking– 11 –f R-symmetry has to be communicated to the visible sector and in this work we con-sider anomaly mediation of supersymmetry breaking playing the role of the messenger ofR-breaking. This is known as anomaly mediated R-breaking (AMRB) [79]. A non-zerogravitino mass generates Majorana gaugino masses and trilinear scalar couplings. We shallconsider the R-breaking effects to be small thus limiting the gravitino mass ( m / ) around10 GeV.The R-breaking Lagrangian contains the following terms L = M e b e b + M e w e w + M e g e g + X b =2 , A lb ˜ L a ˜ L b ˜ E cb + X k =1 , , A dk ˜ L a ˜ Q k ˜ D ck + X k =1 , , A λ k ˜ L ˜ L ˜ E ck + X j,k =1 , , b =2 , A λ ′ bjk ˜ L b ˜ Q j ˜ D ck + A ν H u ˜ L a ˜ N c + H u ˜ QA u ˜ U c (4.5)where M , M and M 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 writtenin the basis ψ = (˜ b , ˜ S, ˜ w , ˜ T , ˜ R d , ˜ H u , N c , ν e ) T as L mass˜ χ = 12 ( ψ ) T M Mχ ψ + h.c. (4.6)where the symmetric ( × ) neutralino mass matrix M Mχ is given by M Mχ = M M D g ′ v u √ − g ′ v a √ M D λ S v u M R
00 0 M M D − gv u √ gv a √ M D λ T v u λ S v u λ T v u µ u + λ S v S + λ T v T g ′ v u √ − gv u √ µ u + λ S v S + λ T v T − f v a M R − f v a − f v u − g ′ v a √ gv a √ − f v u . (4.7)The above mass matrix can be diagonalized by a unitary transformation given by N ⋆ M Mχ N † = ( M χ ) diag . (4.8)The two-component mass eigenstates are defined by χ i = N ij ψ j , i, j = 1 , ..., (4.9)and one can arrange them in Majorana spinors defined by ˜ χ i = χ i ¯ χ i ! , i = 1 , ... . (4.10)– 12 –imilar to the Dirac case, the lightest eigenvalue ( m ˜ χ ) of this neutralino mass matrixcorresponds to the Majorana neutrino mass. Using the expression of M R in eq. (4.3) andthe relation between λ S and λ T in eq. (4.2), the active neutrino mass is given by [88], ( m ν ) Tree = − v (cid:2) gλ T v ( M D − M D ) sin β (cid:3) [ M α + M δ ] (4.11)where α and δ are defined as α = 2 M D M D γ tan βg tan θ w + √ v λ S tan β ( M D sin β + M D cos β ) ,δ = √ M D v λ T tan β (4.12)and the quantity γ has been defined earlier in section 4.1. This shows that to have anappropriate 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 anappropriately small value of λ T . Such a choice, in turn, would imply an almost negligibleradiative contribution to the lightest Higgs boson mass. Interestingly, the Yukawa couplingdoes not appear in the expression for ( m ν ) Tree in eq. (4.11). This is precisely because of therelation between M R and f in eq. (4.3). However, ‘ f ’ has some interesting effects on thenext-to-lightest eigenstates of the mass matrix. The following situations are phenomeno-logically important: • A large value of f ∼ O (1) generates a very light bino-like neutralino ( ˜ χ ) with massaround 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 massparameter M . A very light neutralino has profound consequences in both cosmologyas well as in collider physics [100–109]. In the context of the present model one caneasily 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 takeinto account the constraints coming from the invisible decay branching ratio of thelightest Higgs boson. In our scenario h → ˜ χ ˜ χ (where ˜ χ is the light active neutrino)could effectively contribute to the invisible final state. This is because, although ˜ χ would undergo an R-parity violating decay, for example, ˜ χ → e + e − ν , the resultingfour body final state presumably has to be dealt with as an invisible mode for thelightest Higgs boson. Such constraints are discussed in detail later in this paper. Notethat Γ( h → ˜ χ ˜ χ ) is negligibly small because of suppressed h - ˜ χ - ˜ χ coupling for abino-dominated, ˜ χ .A 10 GeV gravitino NLSP could also decay to a final state comprising of the lightestneutralino accompanied by a photon. In order to avoid the strong constraint on sucha decay process coming from big-bang nucleosynthesis (BBN) one must consider anupper bound on the reheating temperature of the universe T R < ∼ GeV [104, 111].In addition, such a light state is subjected to various collider bounds [100] and boundscoming from rare meson decays such as the decays of pseudo-scalar and vector mesonsinto light neutralino should also be investigated [105] in this context. The spectra of– 13 –ow lying mass eigenstates for the large f case will be shown later for a few benchmarkpoints. • For small f ∼ O (10 − ) , ˜ χ is a sterile neutrino state, which is a plausible warm darkmatter candidate with appropriate relic density. Its mass can be approximated fromthe × neutralino mass matrix as follows: M RN ≈ M f tan βg ′ . (4.13)For a wide range of parameters, the active-sterile mixing angle, denoted as θ , canbe estimated as θ = ( m ν ) T ree M RN . (4.14)Furthermore, the sterile neutrino can be identified with a warm dark matter candidateonly if the following requirements are fulfilled. These are: (i) it should be heavier than0.4 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 constraintcoming from different X-ray experiments [113].Under the circumstances, the lightest neutralino-like state is the next-to-next-to-lightest eigenstate ( ˜ χ ) of the neutralino mass matrix. Its composition is mainlycontrolled by the parameter µ u , chosen to be rather close to the electroweak scale( M D , M D > µ u ). The masses of the lighter neutralino states for this case (small f )will be presented later. We shall now discuss the chargino sector in some detail as it plays a crucial role in thedecay h → γγ . The relevant Lagrangian after R-breaking in the AMRB scenario obtainsthe following form: L ch = M e w + e w − + M D e T + u e w − + √ λ T v u e T + u e R − d + gv u e H + u e w − − µ u e H + u e R − d + λ T v T e H + u e R − d − λ S v S e H + u e R − d + gv a e w + e − L + M D e T + u e w − + m e e cR e − L + h.c. (4.15)The chargino mass matrix, in the basis ( e w + , e T + u , e H + u , e cR ) and ( e w − , e T − d , e R − d , e − L ) , is writtenas M c = M M D gv a M D √ v u λ T gv u − µ u − λ S v S + λ T v T
00 0 0 m e . (4.16)This matrix can be diagonalized by a biunitary transformation, U M c V T = M ± D . Thechargino 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 χ − i = U ij ψ − j ,χ + i = V ij ψ + j , (4.17)– 14 –here ψ + i = e w + e T + u e H + u e cR , ψ − i = e w − e T − d e R − d e − L . (4.18)The four-component Dirac spinors can be written in terms of these two-component spinorsas e χ + i = χ + i χ − i ! , ( i = 1 , ..., . (4.19)It is to be noted that e χ ci ≡ ( e χ + i ) c = e χ − i is a negatively charged chargino. Hence, the lightestchargino ( e χ − ) corresponds to the electron and the structure of the chargino mass matrixensures (see eq. 4.16) that the lightest mass eigenvalue remains unaltered from the inputmass parameter for the electron , i.e., m e = 0 . MeV.Let us now analyze the composition of different chargino states and how they affect thedecay width Γ( h → γγ ) in this model. Due to constraints from the electroweak precisionmeasurements one must consider a heavy Dirac wino mass [84]. Furthermore, a smalltree level Majorana neutrino mass demands a mass-degeneracy of the electroweak Diracgauginos as is obvious from eq. (4.11). In addition, we assume an order one λ T which weuse throughout this work for numerical purposes. With these, we observe the followingfeatures of the next-to-lightest physical chargino state which could potentially contributeto µ γγ : • In the limit when M D >> µ u , the next-to-lightest chargino, χ − (which is actuallythe lightest chargino-like state in the MSSM sense), comprises mainly of e R − d with avery little admixture of e T − d while χ +3 is dominated by e H + u with a small admixture of e w + . • For M D ≪ µ u , χ − is predominantly e w − while χ +3 is composed mainly of e T + u . • Finally, for M D ≈ µ u , χ − is dominantly e w − and χ +3 is mostly made up of e T + u .Apart from the electron, the mass of the chargino states are controlled mainly bythe parameters M D and µ . We have varied the input parameters in such a way that thelightest chargino-like state is always heavier than 104 GeV [110]. The chargino mass spectracorresponding to different benchmark points will be presented later. µ γγ The resonant production of the Higgs boson at the LHC, with the dominant contributioncoming from gluon fusion, is related to its decay to gluons by ˆ σ ( gg → h ) = π Γ( h → – 15 – g ) / m h . Thus, µ γγ can be expressed entirely in terms of various decay widths of theHiggs boson as follows [23, 24]: µ γγ = σ ( pp → h → γγ ) σ ( pp → h → γγ ) SM , = Γ( h → gg )Γ( h → gg ) SM , Γ SMTOT Γ TOT . Γ( h → γγ )Γ( h → γγ ) SM . = k gg .k − .k γγ , (5.1)where we use k gg ≡ ˆ σ ( gg → h )ˆ σ ( gg → h ) SM = Γ( h → gg )Γ( h → gg ) SM and k TOT = Γ TOT Γ SMTOT , Γ TOT being the total decaywidth of the Higgs boson in the present scenario. The decay of h → γγ is mediated mainlyby the top quark and the W ± -loops in the SM and in addition, by top squark, chargedHiggs and chargino loops in our scenario. In the subsequent discussion we investigate thesewidths in some detail.As discussed before, note that in this model we have integrated out the down typeHiggs ( b H d ) superfield and the sneutrino e ν a ( a = 1( e ) ) plays the role of the down typeHiggs boson acquiring a large non-zero vev . The sneutrino ( e ν a ) couples to charged leptons(second and third generation) and down type quarks via R -parity violating couplings whichare identified with the standard Yukawa couplings. Thus, the couplings of the Higgs bosonto charged leptons and quarks remain the same as in the MSSM. This is apparent from thefirst term given in eq. (2.7). h → gg The partial width of the Higgs boson decaying to a pair of gluons via loops involving quarksand squarks is given by Γ( h → gg ) = G F α s m h √ π (cid:12)(cid:12)(cid:12) X Q g hQ A hQ ( τ q ) + X e Q g h e Q A h e Q ( τ e Q ) (cid:12)(cid:12)(cid:12) , (5.2)where τ i = m h / m i , G F is the Fermi constant, α s is the strong coupling constant and A hQ ( τ ) = 32 [ τ + ( τ − f ( τ )] /τ ,A h e Q ( τ ) = −
34 [ τ − f ( τ )] /τ , (5.3)with f ( τ ) given by f ( τ ) = arcsin √ τ τ ≤ , − " log 1 + √ − τ − − √ − τ − − iπ τ > . (5.4)– 16 –he couplings are given by g hQ ( u ) = cos α sin β ,g hQ ( d ) = − sin α cos β ,g h e Q = m f m e Q g hQ ∓ m Z m e Q ( I f − e f sin θ W ) sin( α + β ) , (5.5)where the angle α is defined in eq. (3.11) and tan β = v u /v a . The couplings of the Higgsboson 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 dueto the absence of the µ -term and the A -terms .As far as the production of the Higgs boson is concerned, we shall show later that arather light top squark with mass around − GeV enhances the value of k gg comparedto the SM. The SM and the MSSM results for the decay h → gg can be found in [114–116]. h → γγ In the SM, the primary contribution to the decay h → γγ comes from the W boson loop andthe top quark loop with the former playing the dominant role. In supersymmetric models,the charged Higgs ( H ± ) , top squark ( e t ) and the chargino ( e χ ± ) provide extra contributionsin addition to the W boson and the top quark loop. The authors of ref. [24] have noted thatthe relative strengths of the loop contributions involving the vector bosons, the fermionsand the scalars with mass around 100 GeV follow a rough ratio of . . . 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 tothe charged Higgs state is negligible. On the other hand, charginos in loop could enhancethe h → γγ decay width, in particular, when they are light and/or diagrams involving theminterfere constructively with the W -mediated loop diagram.The Higgs to diphoton decay rate can be written down as [114] Γ( h → γγ ) = G F α m h √ π (cid:12)(cid:12)(cid:12) X f N c Q f g hf A h / + g hW + W − A h + g hH + H − A h + X e c g h e χ + i e χ − j A h / + X e f N c e e f g h e f e f A h (cid:12)(cid:12)(cid:12) , (5.6)where A h = − [2 τ + 3 τ + 3(2 τ − f ( τ )] /τ ,A h / = 2[ τ + ( τ − f ( τ )] /τ ,A h = − [ τ − f ( τ )] /τ , (5.7) Actually, tiny ‘ A ’-terms are generated because of the breaking of R -symmetry but we can neglect themin the present context. – 17 –ith the loop functions already defined in eq. (5.4). The relevant couplings are given by, g huu = cos α sin β ,g hdd = − sin α cos β ,g hW W = sin( β − α ) ,g hH + H − = m W m H ± [sin( β − α ) + cos 2 β sin( β + α )2 cos θ W ] ,g h e f e f = m f m e f g hff ∓ m Z m e f [ I f − e f sin θ w ] sin( α + β ) ,g h e χ + i e χ − j = 2 m W m e c k ( ξ ij sin α − η ij cos α ) . (5.8)Here ξ ij = − √ V i U j and η ij = − √ (cid:16) √ λ T g U i V j + U i V j (cid:17) . The masses which appear inthe denominator of the couplings given above, represent physical masses propagating in theloop. For example, m e c k are the physical chargino masses, m e f are the physical masses of thesfermions and so on. We present the complete set of Higgs-chargino-chargino interactionvertices in Appendix A .As noted earlier, the largest contribution in the Higgs decay rate to two photons comesfrom the W boson loop. Similar to the MSSM, the hW W coupling gets modified by thefactor sin( β − α ) . Hence, in order to have a significant contribution from the W boson loopin our model, the angles α and β need to be aligned in such a way that one obtains a largevalue of sin( β − α ) , which can be achieved in the decoupling regime, i.e., the coupling tothe lightest Higgs boson becomes SM like.In fig. 3, we illustrate the variations of the couplings g hW + W − and g h e χ +3 e χ − , whichmight play important roles in the decay h → γγ . We choose M D = 1 . TeV, µ u = 200 GeV, m / = 10 GeV, m e t = 500 GeV, v S = 10 − GeV, v T = 10 − GeV and retain anear degeneracy between the Dirac gaugino masses with ǫ ≡ ( M D − M D ) = 10 − GeV,with f = 0 . and Bµ L = − (400) ( GeV ) . From the left panel of fig. 3 we observe thatthe hW W coupling is almost SM like as we are essentially working in the decoupling limit.This implies that the W -loop contribution in the h → γγ process remains almost unchangedwith varying tan β . On the other hand, as µ u << M D , , the next-to-lightest chargino stateis dominantly controlled by the µ u parameter. For this case, the coupling g h e χ +3 e χ − is plottedas a function of tan β in the right panel of fig. 3. One can clearly see that g h e χ +3 e χ − is alreadymuch suppressed compared to g hW + W − , for the entire range of tan β . From the expressionfor g h e χ +3 e χ − in eq. (5.8) it is straightforward to verify that this coupling remains very muchsuppressed for all the different cases mentioned in section 4.3. The Higgs boson couplingsto heavier charginos are also highly suppressed as can be seen from fig. 4. Thus, thecontribution of charginos in Γ( h → γγ ) would, in any case, be insignificant. Referring backto eq. (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 fig. 5 we illustrate their variations For the MSSM case see refs. [116, 117]. – 18 –
10 15 20 25 30 35 400.9999980.9999980.9999990.9999991.000000 tan Β g h W + W - - ´ - - ´ - - ´ - - ´ - tan Β g h Χ Ž + Χ Ž - Figure 3 . Couplings of the lightest Higgs boson to a pair of W -bosons (left) and to a pair of lightcharginos ( e χ ± ) (right).
10 15 20 25 30 35 402. ´ - ´ - ´ - ´ - ´ - ´ - ´ - tan Β g h Χ Ž , + Χ Ž , - g h ΧŽ + ΧŽ - g h ΧŽ + ΧŽ - Figure 4 . Couplings of the Higgs boson to heavier charginos. The thick black line represents thecoupling to the heaviest chargino ( e χ ± ) whereas the blue dashed one represents the same to thechargino immediately lighter to it ( e χ ± ) . ( k gg in red and k γγ 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 productionvia loops involving the top quark and the top squark. The couplings involved there carrya factor cos α/ sin β , which varies only marginally with respect to tan β . Similarly k γγ alsoremains insensitive with tan β . The reason being, Γ( h → γγ ) receives major contributionfrom the W boson induced loop where the involved coupling goes as sin( β − α ) . As shown– 19 –ividly in the left panel of fig. 3 that hW W coupling remains almost unchanged with tan β .As a result, k γγ shares the same feature as k gg as far as variation with respect to tan β isconcerned.
400 600 800 1000 1200 14000.951.001.051.101.151.20 m t Ž H GeV L k k ΓΓ tan Β= k ΓΓ tan Β= k ΓΓ tan Β= k gg tan Β= k gg tan Β= k gg tan Β= Figure 5 . Variations of k gg (in red) and k γγ (in blue) as functions of M e t for tan β = 4 , 10, 35. 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 somewhatlarger for higher tan β . However, it is found that k gg > while k γγ < , all through. Wehave also checked that the illustrated variations of k gg and k γγ are following their respectivegross 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 theHiggs boson and the requirement of having no tachyonic scalar states. In section 6, whilediscussing the quantitative impact of the recent LHC results on such a scenario, we presentresults of detailed scan of the parameter space by including all these constraints.All the previous plots consider a large values of ‘ f ’ ( f ∼ O (1) ) for which one obtainsa large tree level correction to the Higgs boson mass as well as an appropriate mass for theactive 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. In the presence of much lighter charginos and neutralinos (as discussed in sections 4.2 and4.3), an SM-like Higgs boson with mass around 125 GeV could undergo decays to a pair ofthese states. We study these things in detail in this section.It has been noted in section 4.2, that the smallest eigenvalue ( m e χ ) of the neutralinomass matrix corresponds to the neutrino mass. The next-to-lightest neutralino ( e χ ) turnsout to be a bino-like neutralino (the sterile neutrino) for large (small) values of ‘ f ’. More-– 20 –ver, the mass of the next-to-next-to-lightest neutralino state ( e χ ) is mostly controlled by µ u . Since we have chosen µ u to be very close to the electroweak scale, the Higgs bosondecay to a pair of e χ is not possible. The presence of light neutralino states may enhancethe invisible decay width of the Higgs boson considerably. Amongst them, the most domi-nant contribution comes from h - e χ - e χ coupling. However, a detailed study reveals that thecontribution of this coupling is not substantial and hence the corresponding decay width israther small. It is clear from fig. 6 that the h - e χ - e χ coupling grows for small tan β . This tan Β g h Χ Ž Χ Ž Figure 6 . Variation of the h − e χ − e χ coupling as a function of tan β . is essentially because for smaller values of tan β , the sneutrino component of the lightestHiggs boson mass eigenstate is large, which results in a slightly larger value of this coupling.This fact also shows up for the invisible decay widths of the Higgs boson, which we willdiscuss later.On the other hand, the lightest chargino eigenstate ( e χ ± ) corresponds to the electron.The mass of the next-to-lightest chargino ( e χ ± ) is again controlled by µ u if µ u < M D . Thus,decay of the Higgs boson to a pair of e χ ± is not possible. The most general expressions forthe partial widths of the Higgs boson decaying to a pair of neutralinos ( Γ( h → e χ i e χ j )) or apair of charginos ( Γ( h → e χ + i e χ − j )) can be found in the Appendix A and B.– 21 – .4 The total decay width of the Higgs boson In this section we collect the partial decay widths of the lightest Higgs boson that domi-nantly contribute to its total decay width. The latter is thus given by Γ TOT = Γ( h → bb ) + Γ( h → τ τ ) + Γ( h → gg ) + Γ( h → W W ∗ ) + Γ( h → ZZ ∗ )+ Γ( h → γγ ) + Γ( h → e χ i e χ j ) + Γ( h → e χ + i e χ − j ) . (5.9)For completeness, we present here the analytical expressions for all the decay rates whichgo into our analysis but were not presented earlier. These are as follows: Γ( h → bb ) = 3 G F m b m h π √ (cid:16) sin α cos β (cid:17) h − m b m h i / , Γ( h → τ τ ) = G F m τ m h π √ (cid:16) sin α cos β (cid:17) h − m τ m h i / , Γ( h → W W ∗ ) = 3 G F m W m h π sin ( α − β ) R (cid:16) m W m h (cid:17) , Γ( h → ZZ ∗ ) = 3 G F m Z m h π h −
109 sin θ W + 4027 sin θ W i R (cid:16) m Z m h (cid:17) . (5.10)The function R ( x ) is defined as [116, 118, 119] R ( x ) = 3 (1 − x + 20 x ) p (4 x −
1) arccos (cid:16) x − x / (cid:17) − (cid:16) − x x (cid:17) (2 − x + 47 x ) −
32 (1 − x + 4 x ) log x. (5.11)The Higgs boson decay rates to charginos and neutralinos are shown in Appendix A and Brespectively. The recent CMS analysis constrains the total decay width of the Higgs bosonto be less than 14 MeV or so [120]. In the subsequent sections we present the numericalresults of our analysis pertaining to the diphoton signal strength µ γγ and subject this toimportant experimental findings. In this section, we discuss the impact of the findings from the LHC pertaining to the Higgssector on the scenario under discussion. As pointed out earlier, two broad scenarios basedon the magnitude of ‘ f ’ worth special attention: the scenario with large ‘ f ’ ( ∼ O (1) ) andthe one for which ‘ f ’ is rather small. f ∼ O (1) A large neutrino Yukawa coupling ( f ∼ O (1) ) already enhances the tree level Higgs bosonmass. Thus, such a scenario banks less on large radiative contributions from the top squark We neglect the rare decay modes like H → Zγ , γ ∗ γ , µ + µ − , e + e − etc. – 22 –oops 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 ( e χ , thenext-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 Majoranamass term of the U (1) gaugino (the bino), i.e., M , and hence related to the gravitino mass m / . Since we assume m / ∼ GeV, the next-to-lightest neutralino acquires a mass ofthe order of a few hundred MeV. The presence of such a light bino like neutralino impliesan additional contribution to the total decay width of the Higgs boson. We also looked atthe diphoton signal strength µ γγ and compared it with the latest ATLAS and CMS results. tan Β B r H h - > i n v i s i b l e L % Λ T = Λ T = Λ T = Λ T = Λ T = Figure 7 . The lightest Higgs boson invisible branching ratio as a function of tan β for differentvalues of λ T . The horizontal line corresponds to the upper limit on the invisible branching ratiofrom model independent analysis [121]. To take into account the constraints coming from the invisible decay branching ratioof the lightest Higgs boson we have fixed M D = M D = 1 . TeV, µ u = 200 GeV, (i.e., M , M << µ u ), m / = 10 GeV, m e t = 500 GeV, v S = 10 − GeV and v T = 10 − GeV with f = 0 . , Bµ L = − (200) ( GeV ) . As discussed earlier, the partial decay width of the Higgsboson decaying to a neutrino and a neutralino ( h → e χ e χ ) could essentially contribute tothe invisible final state. This can be understood from the fact that although e χ wouldundergo R -parity violating decays e χ → qqν , e + e − ν , ννν , qq ′ e − , where q , q ′ are the SMlight quark states from the first two generations, these decay modes involve very smallcouplings and as a result, the decay length happens to be much larger than the colliderdimension. Therefore, the LSP neutralino contributes to missing energy (MET) signals– 23 –122]. Note that Γ( h → e χ e χ ) is negligibly small because of suppressed h - e χ - e χ couplingfor a bino-dominated, e χ .We observe from fig. 7 that this partial decay width is comparatively larger for smallervalues of tan β and λ T . However, it is clear that the presence of a bino-like neutralino stateis not yet constrained from the invisible decay mode of the Higgs boson in our scenario withall the curves staying well below the experimentally derived [121] upper bound of ∼ for the invisible branching fraction of the Higgs boson. µ γγ It is now important to analyse the signal strength corresponding to the h → γγ channel. Infig. 8 we fix λ T = 0 . , and f = 0.8, with all other parameters held at the values mentionedin section 5.2. The red dashed lines represent the contours of m h = 124 GeV and 126.2 GeV
124 GeV 126.2 GeV Μ ΓΓ = m t Ž t a n Β Figure 8 . Contours of various fixed values of m h (124 GeV and 126.2 GeV), µ γγ and k T OT in the m e t – tan β plane. λ T and f are fixed at 0.45 and 0.8, respectively. Other parameters are set at thevalues as mentioned in the text. respectively and enclose the experimentally allowed range of m h . The black thick lines arethe contours of fixed µ γγ with values 1.15, 1.1, 1.05 and 1.03 respectively. Figure 8 showsthat there is an available region of parameter space consistent with the latest experimentalfindings involving m h and µ γγ . Relatively low values of the top squark mass results in anincrease of the cross section for the resonant Higgs boson production through gluon fusionand thus enhances µ γγ . On the other hand µ γγ is almost insensitive to tan β for tan β ≥ .This is because hbb coupling (which controls the total decay width of the Higgs boson in asignificant way) becomes independent of tan β for larger values of this parameter.– 24 –
24 GeV 126.2 GeV Μ ΓΓ = m t Ž t a n Β Figure 9 . Same as in fig. 8 but with λ T = 0 . and f = 1 . Figure 9 addresses the same issue but with f = 1 and λ T = 0 . . Since a larger valueof λ T already provides a significant contribution to the Higgs boson mass via radiativecorrection, only light top squarks are compatible with the measured range of m h . Moreover,a larger value of ‘ f ’ implies a larger tan β to have the Higgs boson mass in the correct range.It is pertinent to mention that these plots use spectra of particles which are consistent withthe lower bound on the lightest chargino mass ( > GeV, from the LEP experiments)and are also free from tachyonic scalar states.
In this subsection we briefly discuss how other final states arising from the lightest Higgsboson are expected to be affected in our scenario relative to the γγ final state and wherethey stand vis-a-vis the experimental results. Such a study of relative strengths over theparameter space of our scenario would be indicative of how well the same is compatible withthe experimental observations in the Higgs sector, in a global sense. The recent results fromthe ATLAS and the CMS collaborations on different decay modes of the lightest Higgs bosonare presented in table 2. In fig. 10, we present the µ -values reported by the ATLAS andthe CMS collaborations for different final states in the so-called signature (ratio) space, inreference to µ γγ .In each plot, the blue circle (green square) represents the experimentally reported cen-tral values for a given pair of observables from ATLAS and CMS collaborations, respectively.The solid grey lines show the range of µ values as observed by the CMS experiment whilethe dashed ones delineate the same as obtained by the ATLAS experiment. In order togenerate fig. 10 we vary tan β within the range < tan β < . We have also varied the– 25 –hannel µ (CMS) µ (ATLAS) h → γγ . + 0 . − . [4] . + 0 . − . [3] h ZZ ∗ −−−→ l . + 0 . − . [123] . + 0 . − . [3] h W W ∗ −−−−→ l ν . + 0 . − . [124] . +0 . − . [125] h → bb . + 0 . − . [126] . + 0 . − . [127] h → τ τ . + 0 . − . [128] . + 0 . − . [129] Table 2 . Signal strengths ( µ ) in different decay final states of the SM-like Higgs boson as reportedby the CMS and the ATLAS collaborations (with the corresponding references). ææàà - - Μ ΓΓ Μ bb ææàà Μ ΓΓ Μ ΤΤ ææàà Μ ΓΓ Μ WW ææàà Μ ΓΓ Μ ZZ Figure 10 . Bands representing mutual variation of relative signal strengths in various possiblefinal states arising from the decay of the lightest Higgs boson as obtained by scanning the parameterspace of the scenario under consideration. The ranges of different parameters used in the scan areas follows: < tan β < ,
350 GeV < m e t < . , . < f < and . < λ T < . . Thesolid grey lines give 1- σ ranges from the MVA based analysis (main analysis) performed by theCMS collaboration (blue circles represent the respective central values) whereas the dashed greylines represent the corresponding results from the ATLAS collaboration (green squares representthe respective central values). mass of the top squark within the range
350 GeV < m e t < . with . < f < and . < λ T < . . All other parameters are kept fixed at the previously mentioned valuesin section 5.2. While scanning, care has been taken to reject spectra with tachyonic scalarstates and to conform with the lower bound on the lightest chargino mass of GeV asobtained from the LEP experiment. Also, the scan required m h to be within the range of . − . GeV as reported by the LHC experiments. The spread in the upper two plots– 26 –n fig. 10 are due to the variation of f which affects µ bb and µ γγ whereas µ W W and µ ZZ remain unaffected. The values of µ γγ is very much consistent with the recent ATLAS andCMS findings. Finally, in order to have an idea of the mass-spectra of the light neutralinoand the chargino states, we provide a few benchmark points in table 3, for the large ‘ f ’scenario. Parameters BP- BP- BP- M D M D µ u
200 GeV 200 GeV 200 GeV m /
20 GeV 20 GeV 10 GeV tan β
25 35 40 m e t
500 GeV 400 GeV 400 GeV f λ T v S − GeV − GeV − GeV v T − GeV − GeV − GeV Bµ L − (400) (GeV) − (400) (GeV) − (400) (GeV) Observables BP- BP- BP- m h ( m ν ) Tree m e χ
168 MeV 169 MeV 84 MeV m e χ m e χ m e χ m e χ m e χ × GeV 1.11 × GeV . × GeV m e χ × GeV 1.11 × GeV . × GeV m e χ +3 m e χ +2 m e χ +1 µ γγ Table 3 . Benchmark sets of input parameters in the large Yukawa coupling ( f ) scenario and theresulting mass-values for some relevant excitations. The Higgs signal strength in the diphoton finalstate ( µ γγ ) is also indicated. f ∼ O (10 − ) In the limit when the Yukawa coupling is small ( f ∼ − ), the next-to-lightest neutralinostate 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 activeneutrino is given by eq. (4.11) whereas the sterile neutrino mass and the mixing angle– 27 –etween the active and the sterile neutrino are given by eqs. (4.13) and (4.14). We havementioned in the previous section that an X-ray line at around 3.5 keV was observed in theX-ray spectra of the Andromeda galaxy and in the same from various other galaxy clustersincluding the Perseus cluster. The observed flux and the best fit energy peak are shownin [130, 131]. The origin of this line is disputed since atomic transitions in the thermalplasma may also be responsible for this energy line. Nevertheless, a possible explanationcan be provided by taking into account a 7 keV dark matter, in this case a sterile neutrino[130, 131]. As discussed earlier, the observed flux and the peak of the energy can be trans-lated to an active-sterile mixing in the range . × − < sin θ < × − . To satisfysuch small active sterile mixing, the tree level neutrino mass turns out to be very small( O (10 − ) eV). Therefore, in order to explain the neutrino mass and mixing, one needs toinvoke radiative corrections. For a detailed discussion, we refer the reader to [88]. It is alsoimportant to study the signal strength of h → γγ in the light of this 7 keV sterile neutrinowith appropriate active-sterile mixing. m h =
124 GeV M NR = Θ = ´ - ´ - f t a n Β Figure 11 . Contours of fixed values of m h , µ γγ , M RN and sin θ in the f − tan β parameterspace. The respective values of the contour lines are as shown in the figure. The shaded regionin grey corresponds to the experimentally allowed band of the lightest Higgs boson mass. Otherparameters are fixed at values mentioned in the text. In fig. 11 we present the contours of m h , µ γγ , M RN and sin θ in the f – tan β plane.The contour of the sterile neutrino mass of 7 keV is shown with the thick black line. Thered dashed lines represent the contours of active-sterile mixing fixed at . × − and × − . We have fixed M D at 1 TeV, maintaining a degeneracy ǫ = ( M D − M D ) = 10 − GeV. µ u is fixed at 500 GeV. The other fixed parameters are m / = 10 GeV , m e t = 400 GeV , λ T = 0 . , v S = − .
01 GeV , v T = 0 .
01 GeV and Bµ L = − (400) (GeV) . The not so– 28 –eavy top squark, as justified in section 6.1.2, enhances µ γγ considerably and we showthe contours of µ γγ at 1.1 and 1.114 respectively with blue dashed lines. Finally, thegrey shaded region is the parameter space consistent with the observed Higgs boson mass . < m h < . . Figure 11 clearly shows that for this choice of parameters µ γγ > ∼ . is completely consistent with a 7 keV sterile neutrino dark matter and theexperimentally allowed range of Higgs boson mass. We have seen that charginos do notprovide much enhancement to µ γγ due to its very suppressed couplings under the presentset-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 viasuppression 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 radiativecorrections from λ S and λ T reduces the necessity of having heavy top squarks. In thescatter plot of fig. 12 we show the possible range of variation of µ γγ with varying m e t . To
400 600 800 1000 1200 14000.981.001.021.041.061.081.101.12 m t Ž H GeV L Μ ΓΓ Figure 12 . Scatter plot showing possible range of variation of µ γγ with varying m e t . The bluepoints are consistent with .
01 keV < M RN < .
11 keV . All points satisfy 124.0 GeV < m h < generate this plot we have chosen relevant parameters over the following ranges: 20 GeV , < tan β < , 300 GeV < m e t < . , − < f < × − , . < λ T < and − . 01 GeV < v S < − . Other parameters are retained at theirpreviously mentioned values (used to obtain fig. 11), maintaining the degeneracy betweenthe Dirac gaugino masses as already mentioned. Again, all these points are consistent with . < m h < . and free from any tachyonic scalar states. The effects of thelight top squarks results in some enhancement in µ γγ . The blue points are consistent witha keV sterile neutrino with mass ranging between . 01 keV < M RN < . 11 keV and is known– 29 – æàà Μ ΓΓ Μ bb ææàà Μ ΓΓ Μ ΤΤ ææàà Μ ΓΓ Μ WW ææàà Μ ΓΓ Μ zz Figure 13 . Same as in figure 10 except for a small input value of f . to be a fit warm dark matter candidate having the right relic density. Finally, it is againvery relevant to check the relative signal strengths for different decay modes of the lightestHiggs boson in such a scenario with small ‘ f ’; similar to what we have done in section 6.1.3for the large ‘ f ’ scenario. Figure 13 shows scattered points consistent with the CMS or/andthe ATLAS results at 1 σ level. However, note that the scatter plot in the µ γγ – µ W W planeis consistent only with the results from the ATLAS experiments at the 1 σ level whereasthe the scatter plot in the µ γγ – µ bb plane is consistent only with the results from the CMSexperiments at the 1 σ level. In the near future, a more precise measurement together withan improved analysis is likely to become more decisive on this issue. Finally, for the sakeof completeness, in table 4 we provide three more benchmark sets comprising of the inputparameters of the small Yukawa coupling scenario (with ( f ∼ − ) ), the correspondingmass-values of the relevant excitations and the Higgs signal strengths in the diphoton finalstate ( µ γγ ). In this paper we study the h → γγ channel in the U (1) R lepton number model with a righthanded neutrino. We show that the recent results from ATLAS and CMS on µ γγ is verymuch consistent with our outcomes for both the cases, i.e., f ∼ O (1) and f ∼ O (10 − ) .We also show for large neutrino Yukawa coupling, f the light bino-like neutralino state isnot yet constrained from the invisible branching fraction of the Higgs boson.– 30 –arameters BP- BP- BP- M D µ u 300 GeV 600 GeV 600 GeV m / tan β 35 25 15 m e t 500 GeV 500 GeV 500 GeV f × − × − × − λ T v S - − GeV - − GeV - − GeV v T − GeV − GeV − GeVObservables BP- BP- BP- m h 125 GeV 124.257 GeV 124.448 GeV m RN m e χ m e χ m e χ m e χ m e χ m e χ m e χ +3 m e χ +2 m e χ +1 sin θ . × − . × − . × − µ γγ Table 4 . Same as in table 3 but for small Yukawa coupling with f ∼ O (10 − ) . In all three caseswe have chosen ǫ = 10 − GeV. Neutrino mass at the tree level is very small ( O (10 − ) eV) and notshown in the table (See text for more details). So far we have seen that the model under consideration have already demonstratedits ability to attract constraints from recent experiments in diverse areas ranging from theneutrino to astro-particle physics and finally from the LHC experiments pertaining to theHiggs sector and other BSM searches. It will be really interesting to see if the model canprovide any novel signatures as far as the collider experiments are concerned. A The Higgs-chargino-chargino coupling In this appendix we work out the Higgs-chargino-chargino coupling in the scenario underdiscussion and present the analytical expression for the width of the lightest Higgs bosondecaying into a pair of charginos. The relevant Lagrangian in the two-component notation– 31 –ontaining the Higgs-chargino-chargino interaction is given by L h e χ + e χ − = g (cid:18) v a + S i √ h i (cid:19) e w + e − L + √ λ T e T + u (cid:18) v u + S i √ h i (cid:19) e R − d + g (cid:18) v u + S i √ h i (cid:19) e H + u e w − − λ S (cid:18) v S + S i √ h i (cid:19) e H + u e R − d + λ T (cid:18) v T + S i √ h i (cid:19) e H + u e R − d + g (cid:18) v T + S i √ h i (cid:19) ˜ T + u ˜ w − − g (cid:18) v T + S i √ h i (cid:19) ˜ w + ˜ T − d + h.c., (A.1)where the matrix S connects the mass and gauge eigenstates of the CP even scalar masssquared matrix, written in the basis ( h R , ˜ ν R , S R , T R ). To be more precise the physicalCP-even scalar states are related to the gauge eigenstates in the following manner: h h h h = S S S S S S S S S S S S S S S S h R ˜ ν R S R T R . (A.2)In our notation the lightest physical state ( h ) of the CP even scalar mass matrix corre-sponds to the physical Higgs boson, h . Moreover, the charginos ˜ χ ± i are four componentDirac fermions which arise due to the mixing between the charged gauginos and higgsinosas well as the charged lepton of first generation. In order to evaluate find out the Higgs-chargino-chargino coupling and to evaluate the Higgs boson partial decay width to a pair ofcharginos, it is pertinent to write down the interaction Lagrangian in the four-componentnotation. We now define the 4-component spinors as f W = e w + ¯ e w − ! , e H = e H + u ¯ e R − d ! , e T = e T + u ¯ e T − d ! , L (4) e = e cR ¯ e − L ! . (A.3)Using the transformation relations, e w + e − L = ¯ L (4) e P L f W e T + u e R − d = e HP L e T e H + u e w − = f W P L e H e H + u e R − d = e HP L e H, (A.4)the Lagrangian in eq. (A.1) can be expressed in the four component notation as L (4) h e χ + e χ − = g S √ hL (4) e P L f W + √ λ T S √ h e H P L e T + g S √ h f W P L e H − λ S S √ h e HP L e H + λ T S √ h e HP L e H + g S √ h f W P L e T − g S √ e T P L f W + h.c. (A.5)– 32 –he chargino masses can have any sign. By demanding that the four component Lagrangiancontains only positive masses for the charginos, we define the chargino states in the followingmanner [132, 133] e χ + i = ( ǫ i P L + P R ) χ + i ¯ χ − i ! , i = 1 , ..., (A.6)where ǫ i carries the sign of the chargino masses, which can be ± . When ǫ = − , P R − P L = γ , which essentially implies a γ rotation to the four component spinors to absorb the sign.Hence, the transformation relations involving only P L changes, which modifies the Feynmanrules. The two-component mass eigenstates ( χ ± i ) of the charginos are related to the gaugeeigenstates in a manner shown in eq. (4.17).Using the following set of relations P L f W = P L V ∗ i ǫ i e χ i P L e T = P L V ∗ i ǫ i e χ i P L e H = P L V ∗ i ǫ i e χ i P R f W = P R U i e χ i P R e H = P R U i e χ i P R e T = P R U i e χ i P R L (4) e = P R U i e χ i , (A.7)we rewrite eq. (A.5) in the mass eigenstate basis as L (4) mh ˜ χ + i ˜ χ − j = gh ˜ χ i (cid:0) ζ ∗ ij P L + ζ ji P R (cid:1) ˜ χ j , (A.8)where ζ ij = (cid:20) S √ U i V j + √ λ T g S √ U i V j + S √ U i V j − λ S g S √ U i V j + λ T g S √ U i V j + S √ U i V j − S √ U i V j (cid:21) ǫ i . (A.9)The coupling is obtained from Eq. (A.8) as g (cid:2) ζ ∗ ij (1 − γ ) + ζ ji (1 + γ ) (cid:3) . (A.10)It is now straightforward to compute the lightest Higgs boson decay width to a pair ofcharginos, which we find as Γ h → e χ + i e χ − j = g πm h h(cid:8) m h − ( m e χ + i + m e χ − j ) (cid:9) − m e χ + i m e χ − j i / h ( ζ ij + ζ ji )( m h − m e χ + i − m e χ − j ) − ζ ij ζ ji m e χ + i m e χ − j i . (A.11)– 33 – χ + i ˜ χ − j h Figure 14 . The Higgs-chargino-chargino vertex. Finally, if we assume the singlet and the triplet vev ’s to be very small, this would imply thatthe 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 ˜ ν R ≃ v a + 1 √ H cos α − h sin α ) h R ≃ v u + 1 √ H sin α + h cos α ) , (A.12)where we have chosen S = cos α , S = − sin α , and S ∼ S ∼ . With this simplifica-tion we can write ζ ij = (cid:20) − sin α √ U i V j + cos α √ (cid:18) √ λ T g U i V j + U i V j (cid:19)(cid:21) ǫ i = ξ ij sin α − η ij cos α, (A.13)where ξ ij = − U i V j √ ǫ i η ij = 1 √ (cid:18) √ λ T g U i V j + U i V j (cid:19) ǫ i . (A.14)– 34 – The Higgs-neutralino-neutralino coupling In a similar manner the interaction of the Higgs boson with neutralinos can be constructedfrom the following (two-component) Lagrangian L h e χ e χ = g ′ √ (cid:18) v u + S i √ h i (cid:19) e b e H u − g ′ √ (cid:18) v a + S i √ h i (cid:19) e bν e + λ S (cid:18) v u + S i √ h i (cid:19) e S e R d − g √ (cid:18) v u + S i √ h i (cid:19) e w e H u + g √ (cid:18) v a + S i √ h i (cid:19) e wν e + λ T (cid:18) v u + S i √ h i (cid:19) e T e R d + (cid:20) λ S (cid:18) v s + S i √ h i (cid:19) + λ T (cid:18) v T + S i √ h i (cid:19)(cid:21) e R d e H u − f (cid:18) v a + S i √ h i (cid:19) e H u N c − f (cid:18) v u + S i √ h i (cid:19) N c ν e + h.c. (B.1)We stick to the notation for the lightest CP even physical scalar state being denoted by h and identified with the lightest Higgs boson h . We again define the 4-component spinorsas [134] ˜ B = ˜ b ¯ e b T ! , ˜ S = ˜ S ¯ e S T ! , ˜ R d = ˜ R d ¯ e R Td ! , ˜ H u = ˜ H u ¯ e H Tu ! , ˜ T = ˜ T ¯ e T T ! , ˜ W = ˜ W ¯ f W T ! , ν e = ν e ¯ ν Te ! , N c = N c ¯ N cT ! . (B.2)In terms of these spinors the 4-component Lagrangian takes the following form L (4) h e χ e χ = g ′ √ S √ h ¯ e BP L e H u − g ′ √ S √ h ¯ e BP L ν e + λ S S √ h ¯ e SP L e R d − g √ S √ h ¯ f W P L ˜ H u + g √ S √ h ¯ f W P L ν e + λ T S √ h ¯ e T P L e R d + λ S S √ h ¯ e R d P L e H u + λ T S √ h ¯ e R d P L e H u − f S √ h ¯ e H u P L N c − f S √ h ¯ N c P L ν e + h.c. (B.3)Eq. (B.3) represents the interactions in the gauge eigenstate basis. Neutralinos are physicalMajorana spinors, arising due to the mixing of the neutral gauginos, higgsinos as well as theactive (first generation) and sterile neutrino states. The four component neutralino stateis defined as e χ i = ( ǫ i P L + P R ) χ i ¯ χ i ! , i = 1 , ..., (B.4)where χ i are two component neutralino mass eigenstates and they are related to the gaugeeigenstates as χ i = N ij ψ j , i, j = 1 , ..., (B.5)– 35 –here ψ = (cid:0)e b, e S, f W , e T , e R d , e H u , N c , ν e (cid:1) T . As presented in Appendix A, in a similar fashionwe use the following transformation relations to write down the interaction Lagrangiangiven in Eq. (B.3) in the mass eigenstate basis P L e B = N ∗ i P L ǫ i e χ i , P R e B = N i P R e χ i P L e S = N ∗ i P L ǫ i e χ i , P R e S = N i P R e χ i P L f W = N ∗ i P L ǫ i e χ i , P R f W = N i P R e χ i P L e T = N ∗ i P L ǫ i e χ i , P R e T = N i P R e χ i P L e R d = N ∗ i P L ǫ i e χ i , P R e R d = N i P R e χ i P L e H u = N ∗ i P L ǫ i e χ i , P R e H u = N i P R e χ i P L N c = N ∗ i P L ǫ i e χ i , P R N c = N i P R e χ i P L ν e = N ∗ i P L ǫ i e χ i , P R ν e = N i P R e χ i . (B.6)It is now straightforward to write down the Higgs-neutralino-neutralino interaction in the4-component notation as L (4) mh ˜ χ ˜ χ = g ¯˜ χ i h (cid:0) ζ ′∗ ij P L + ζ ′ ji P R (cid:1) ˜ χ j , (B.7)where ζ ′ ij = S h g ′ g N i N j λ S g N i N j √ − N i N j λ T g N i N j √ − fg N i N j √ i ǫ i + S h N i N j − g ′ g N i N j − fg N i N j √ i ǫ i + S h λ S g N i N j √ i ǫ i + S h λ T g N i N j √ i ǫ i + ( i ↔ j ) . (B.8) ˜ χ i ˜ χ j h Figure 15 . The Higgs-neutralino-neutralino vertex. Finally, the partial decay width Γ( h → e χ i e χ j ) is given as Γ h → e χ i e χ j = g πm h (1 + δ ij ) h { m h − ( m e χ i + m e χ j ) } − m e χ i m e χ j i / × h (cid:0) ζ ′ ij + ζ ′ ji (cid:1) (cid:16) m h − m e χ i − m e χ j (cid:17) − ζ ′ ij ζ ′ ji m e χ i m e χ j i . (B.9)– 36 –gain in the limit where the singlet and triplet vev ’s are very small, we can safelyignore the contributions from S and S . Furthermore, replacing S by cos α and S by- sin α , we can write ζ ′ ij = η ′ ij cos α + ξ ′ ij sin α, (B.10)where, η ′ ij = (cid:20) g ′ g N i N j λ S g N i N j √ − N i N j λ T g N i N j √ − fg N i N j √ (cid:21) ǫ i + ( i ↔ j ) ,ξ ′ ij = (cid:20) g ′ g N i N j fg N i N j √ − N i N j (cid:21) ǫ i + ( i ↔ j ) . (B.11) Acknowledgments SC would like to thank the Council of Scientific and Industrial Research, Government ofIndia for the financial support received as a Senior Research Fellow. AD acknowledges thehospitality of the Department of Theoretical Physics, IACS during the course of this work.SR would like to thank the hospitality of the University of Helsinki and Helsinki Instituteof Physics during the final stages of this work. It is also a pleasure to thank Dilip KumarGhosh, Katri Huitu and Oleg Lebedev for helpful discussions. References [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B (2012) 1 [arXiv:1207.7214 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125 GeVwith the CMS experiment at the LHC,” Phys. Lett. B (2012) 30 [arXiv:1207.7235[hep-ex]].[3] G. Aad et al. [ ATLAS Collaboration], “Measurement of Higgs boson production in thediphoton decay channel in pp collisions at center-of-mass energies of 7 and 8 TeV with theATLAS detector,” arXiv:1408.7084 [hep-ex].[4] V. Khachatryan et al. [CMS Collaboration], “Observation of the diphoton decay of the Higgsboson and measurement of its properties,” arXiv:1407.0558 [hep-ex].[5] A. Arbey, M. Battaglia, A. Djouadi and F. Mahmoudi, “An update on the constraints on thephenomenological MSSM from the new LHC Higgs results,” Phys. Lett. B (2013) 153[arXiv:1211.4004 [hep-ph]].[6] P. Bechtle, S. Heinemeyer, O. Stal, T. Stefaniak, G. Weiglein and L. Zeune, “MSSMInterpretations of the LHC Discovery: Light or Heavy Higgs?,” Eur. Phys. J. C (2013)2354 [arXiv:1211.1955 [hep-ph]].[7] K. Schmidt-Hoberg, F. Staub and M. W. Winkler, “Enhanced diphoton rates at Fermi andthe LHC,” JHEP (2013) 124 [arXiv:1211.2835 [hep-ph]].[8] M. Drees, “A Supersymmetric Explanation of the Excess of Higgs–Like Events at the LHCand at LEP,” Phys. Rev. D (2012) 115018 [arXiv:1210.6507 [hep-ph]]. – 37 – 9] A. Arbey, M. Battaglia, A. Djouadi and F. Mahmoudi, “The Higgs sector of thephenomenological MSSM in the light of the Higgs boson discovery,” JHEP (2012) 107[arXiv:1207.1348 [hep-ph]].[10] K. Schmidt-Hoberg and F. Staub, “Enhanced h → γγ rate in MSSM singlet extensions,”JHEP (2012) 195 [arXiv:1208.1683 [hep-ph]].[11] M. Carena, I. Low and C. E. M. Wagner, “Implications of a Modified Higgs to DiphotonDecay Width,” JHEP (2012) 060 [arXiv:1206.1082 [hep-ph]].[12] M. Carena, S. Gori, N. R. Shah and C. E. M. Wagner, “A 125 GeV SM-like Higgs in theMSSM and the γγ rate,” JHEP (2012) 014 [arXiv:1112.3336 [hep-ph]].[13] L. J. Hall, D. Pinner and J. T. Ruderman, “A Natural SUSY Higgs Near 126 GeV,” JHEP (2012) 131 [arXiv:1112.2703 [hep-ph]].[14] S. Heinemeyer, O. Stal and G. Weiglein, “Interpreting the LHC Higgs Search Results in theMSSM,” Phys. Lett. B (2012) 201 [arXiv:1112.3026 [hep-ph]].[15] A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, “Implications of a 125GeV Higgs for supersymmetric models,” Phys. Lett. B (2012) 162 [arXiv:1112.3028[hep-ph]].[16] P. Draper, P. Meade, M. Reece and D. Shih, “Implications of a 125 GeV Higgs for the MSSMand Low-Scale SUSY Breaking,” Phys. Rev. D (2012) 095007 [arXiv:1112.3068 [hep-ph]].[17] N. Chen and H. -J. He, “LHC Signatures of Two-Higgs-Doublets with Fourth Family,” JHEP (2012) 062 [arXiv:1202.3072 [hep-ph]].[18] X. G. He, B. Ren and J. Tandean, “Hints of Standard Model Higgs Boson at the LHC andLight Dark Matter Searches,” Phys. Rev. D (2012) 093019 [arXiv:1112.6364 [hep-ph]].[19] A. Djouadi, O. Lebedev, Y. Mambrini and J. Quevillon, “Implications of LHC searches forHiggs–portal dark matter,” Phys. Lett. B (2012) 65 [arXiv:1112.3299 [hep-ph]].[20] K. Cheung and T. -C. Yuan, “Could the excess seen at 124-126 GeV be due to theRandall-Sundrum Radion?,” Phys. Rev. Lett. (2012) 141602 [arXiv:1112.4146 [hep-ph]].[21] B. Batell, S. Gori and L. -T. Wang, “Exploring the Higgs Portal with 10/fb at the LHC,”JHEP (2012) 172 [arXiv:1112.5180 [hep-ph]].[22] N. D. Christensen, T. Han and S. Su, “MSSM Higgs Bosons at The LHC,” Phys. Rev. D (2012) 115018 [arXiv:1203.3207 [hep-ph]].[23] A. Belyaev, S. Khalil, S. Moretti and M. C. Thomas, “Light sfermion interplay in the 125GeV MSSM Higgs production and decay at the LHC,” JHEP (2014) 076[arXiv:1312.1935 [hep-ph]].[24] M. Hemeda, S. Khalil and S. Moretti, “Light chargino effects onto h → γγ in the MSSM,”Phys. Rev. D (2014) 1, 011701 [arXiv:1312.2504 [hep-ph]].[25] A. Chakraborty, B. Das, J. L. Diaz-Cruz, D. K. Ghosh, S. Moretti and P. Poulose, “The 125GeV Higgs signal at the LHC in the CP Violating MSSM,” Phys. Rev. D (2014) 055005[arXiv:1301.2745 [hep-ph]].[26] S. F. King, M. Mühlleitner, R. Nevzorov and K. Walz, “Natural NMSSM Higgs Bosons,”Nucl. Phys. B (2013) 323 [arXiv:1211.5074 [hep-ph]].[27] J. F. Gunion, Y. Jiang and S. Kraml, “Could two NMSSM Higgs bosons be present near 125GeV?,” Phys. Rev. D (2012) 071702 [arXiv:1207.1545 [hep-ph]]. – 38 – 28] G. Belanger, U. Ellwanger, J. F. Gunion, Y. Jiang, S. Kraml and J. H. Schwarz, “HiggsBosons at 98 and 125 GeV at LEP and the LHC,” JHEP (2013) 069 [arXiv:1210.1976[hep-ph]].[29] J. F. Gunion, Y. Jiang and S. Kraml, “The Constrained NMSSM and Higgs near 125 GeV,”Phys. Lett. B (2012) 454 [arXiv:1201.0982 [hep-ph]].[30] U. Ellwanger and C. Hugonie, “Higgs bosons near 125 GeV in the NMSSM with constraintsat the GUT scale,” Adv. High Energy Phys. (2012) 625389 [arXiv:1203.5048 [hep-ph]].[31] U. Ellwanger, “A Higgs boson near 125 GeV with enhanced di-photon signal in the NMSSM,”JHEP (2012) 044 [arXiv:1112.3548 [hep-ph]].[32] J. -J. Cao, Z. -X. Heng, J. M. Yang, Y. -M. Zhang and J. -Y. Zhu, “A SM-like Higgs near 125GeV in low energy SUSY: a comparative study for MSSM and NMSSM,” JHEP (2012)086 [arXiv:1202.5821 [hep-ph]].[33] Z. Kang, J. Li and T. Li, “On Naturalness of the MSSM and NMSSM,” JHEP (2012)024 [arXiv:1201.5305 [hep-ph]].[34] K. Huitu and H. Waltari, “Higgs sector in NMSSM with right-handed neutrinos andspontaneous R-parity violation,” arXiv:1405.5330 [hep-ph].[35] S. Chatrchyan et al. [CMS Collaboration], “Search for microscopic black holes in pp collisionsat √ s = 7 TeV,” JHEP (2012) 061 [arXiv:1202.6396 [hep-ex]].[36] H. Baer, V. Barger and A. Mustafayev, “Implications of a 125 GeV Higgs scalar for LHCSUSY and neutralino dark matter searches,” Phys. Rev. D (2012) 075010[arXiv:1112.3017 [hep-ph]].[37] L. Aparicio, D. G. Cerdeno and L. E. Ibanez, “A 119-125 GeV Higgs from a string derivedslice of the CMSSM,” JHEP (2012) 126 [arXiv:1202.0822 [hep-ph]].[38] J. Ellis and K. A. Olive, “Revisiting the Higgs Mass and Dark Matter in the CMSSM,” Eur.Phys. J. C (2012) 2005 [arXiv:1202.3262 [hep-ph]].[39] H. Baer, V. Barger and A. Mustafayev, “Neutralino dark matter in mSUGRA/CMSSM witha 125 GeV light Higgs scalar,” JHEP (2012) 091 [arXiv:1202.4038 [hep-ph]].[40] J. Cao, Z. Heng, D. Li and J. M. Yang, “Current experimental constraints on the lightestHiggs boson mass in the constrained MSSM,” Phys. Lett. B (2012) 665 [arXiv:1112.4391[hep-ph]].[41] A. Elsayed, S. Khalil and S. Moretti, “Higgs Mass Corrections in the SUSY B-L Model withInverse Seesaw,” Phys. Lett. B (2012) 208 [arXiv:1106.2130 [hep-ph]].[42] L. Basso and F. Staub, “Enhancing h → γγ with staus in SUSY models with extended gaugesector,” Phys. Rev. D (2013) 015011 [arXiv:1210.7946 [hep-ph]].[43] S. Khalil and S. Moretti, “Heavy neutrinos, Z’ and Higgs bosons at the LHC: new particlesfrom an old symmetry,” J. Mod. Phys. (2013) 7 [arXiv:1207.1590 [hep-ph]].[44] S. Khalil and S. Moretti, “A simple symmetry as a guide toward new physics beyond theStandard Model,” Front. Phys. (2013) 10 [arXiv:1301.0144 [physics.pop-ph]].[45] M. Frank, D. K. Ghosh, K. Huitu, S. K. Rai, I. Saha and H. Waltari, “Left-rightsupersymmetry after the Higgs discovery,” arXiv:1408.2423 [hep-ph].[46] M. Frank and S. Mondal, “Light Neutralino Dark Matter in U (1) ′ models,” Phys. Rev. D (2014) 075013 [arXiv:1408.2223 [hep-ph]]. – 39 – 47] T. Basak and S. Mohanty, “130 GeV gamma ray line and enhanced Higgs di-photon rate fromTriplet-Singlet extended MSSM,” JHEP (2013) 020 [arXiv:1304.6856 [hep-ph]].[48] P. Fayet, “Supersymmetry and Weak, Electromagnetic and Strong Interactions,” Phys. Lett.B (1976) 159.[49] J. Polchinski and L. Susskind, “Breaking Of Supersymmetry At Intermediate-Energy,” Phys.Rev. D , 3661 (1982).[50] L. J. Hall, “Alternative Low-energy Supersymmetry,” Mod. Phys. Lett. A (1990) 467.[51] L. J. Hall and L. Randall, “U(1)-R symmetric supersymmetry,” Nucl. Phys. B (1991)289.[52] I. Jack and D. R. T. Jones, “Nonstandard soft supersymmetry breaking,” Phys. Lett. B (1999) 101 [hep-ph/9903365].[53] A. E. Nelson, N. Rius, V. Sanz and M. Unsal, “The Minimal supersymmetric model withouta mu term,” JHEP (2002) 039 [hep-ph/0206102].[54] P. J. Fox, A. E. Nelson and N. Weiner, “Dirac gaugino masses and supersoft supersymmetrybreaking,” JHEP (2002) 035 [hep-ph/0206096].[55] Z. Chacko, P. J. Fox and H. Murayama, “Localized supersoft supersymmetry breaking,”Nucl. Phys. B (2005) 53 [hep-ph/0406142].[56] I. Antoniadis, K. Benakli, A. Delgado, M. Quiros and M. Tuckmantel, “Splitting extendedsupersymmetry,” Phys. Lett. B , 302 (2006) [arXiv:hep-ph/0507192]; “Split extendedsupersymmetry from intersecting branes,” Nucl. Phys. B , 156 (2006)[arXiv:hep-th/0601003].[57] I. Antoniadis, K. Benakli, A. Delgado and M. Quiros, “A new gauge mediation theory,” Adv.Stud. Theor. Phys. , 645 (2008) [arXiv:hep-ph/0610265].[58] G. D. Kribs, E. Poppitz and N. Weiner, “Flavor in supersymmetry with an extendedR-symmetry,” Phys. Rev. D (2008) 055010 [arXiv:0712.2039 [hep-ph]].[59] S. Y. Choi, M. Drees, A. Freitas and P. M. Zerwas, “Testing the Majorana Nature of Gluinosand Neutralinos,” Phys. Rev. D (2008) 095007 [arXiv:0808.2410 [hep-ph]].[60] S. D. L. Amigo, A. E. Blechman, P. J. Fox and E. Poppitz, “R-symmetric gauge mediation,”JHEP , 018 (2009) [arXiv:0809.1112 [hep-ph]]; A. E. Blechman, “R-symmetric GaugeMediation and the MRSSM,” Mod. Phys. Lett. A (2009) 633 [arXiv:0903.2822 [hep-ph]].[61] K. Benakli and M. D. Goodsell, “Dirac Gauginos in General Gauge Mediation,” Nucl. Phys.B (2009) 185 [arXiv:0811.4409 [hep-ph]].[62] G. Belanger, K. Benakli, M. Goodsell, C. Moura and A. Pukhov, “Dark Matter with Diracand Majorana Gaugino Masses,” JCAP (2009) 027 [arXiv:0905.1043 [hep-ph]].[63] K. Benakli and M. D. Goodsell, “Dirac Gauginos and Kinetic Mixing,” Nucl. Phys. B (2010) 315 [arXiv:0909.0017 [hep-ph]].[64] A. Kumar, D. Tucker-Smith and N. Weiner, “Neutrino Mass, Sneutrino Dark Matter andSignals of Lepton Flavor Violation in the MRSSM,” JHEP (2010) 111 [arXiv:0910.2475[hep-ph]].[65] B. A. Dobrescu and P. J. Fox, “Uplifted supersymmetric Higgs region,” Eur. Phys. J. C (2010) 263 [arXiv:1001.3147 [hep-ph]]. – 40 – 66] K. Benakli and M. D. Goodsell, “Dirac Gauginos, Gauge Mediation and Unification,” Nucl.Phys. B (2010) 1 [arXiv:1003.4957 [hep-ph]].[67] S. Y. Choi, D. Choudhury, A. Freitas, J. Kalinowski, J. M. Kim and P. M. Zerwas, “DiracNeutralinos and Electroweak Scalar Bosons of N=1/N=2 Hybrid Supersymmetry atColliders,” JHEP (2010) 025 [arXiv:1005.0818 [hep-ph]].[68] L. M. Carpenter, “Dirac Gauginos, Negative Supertraces and Gauge Mediation,” JHEP (2012) 102 [arXiv:1007.0017 [hep-th]].[69] G. D. Kribs, T. Okui and T. S. Roy, “Viable Gravity-Mediated Supersymmetry Breaking,”Phys. Rev. D (2010) 115010 [arXiv:1008.1798 [hep-ph]].[70] S. Abel and M. Goodsell, “Easy Dirac Gauginos,” JHEP (2011) 064 [arXiv:1102.0014[hep-th]].[71] K. Benakli, M. D. Goodsell and A. -K. Maier, “Generating mu and Bmu in models withDirac Gauginos,” Nucl. Phys. B (2011) 445 [arXiv:1104.2695 [hep-ph]].[72] J. Kalinowski, “Phenomenology of R-symmetric supersymmetry,” Acta Phys. Polon. B (2011) 2425.[73] K. Benakli, “Dirac Gauginos: A User Manual,” Fortsch. Phys. (2011) 1079[arXiv:1106.1649 [hep-ph]].[74] C. Frugiuele and T. Gregoire, “Making the Sneutrino a Higgs with a U (1) R Lepton Number,”Phys. Rev. D (2012) 015016 [arXiv:1107.4634 [hep-ph]].[75] C. Brust, A. Katz, S. Lawrence and R. Sundrum, “SUSY, the Third Generation and theLHC,” JHEP (2012) 103 [arXiv:1110.6670 [hep-ph]].[76] K. Rehermann and C. M. Wells, “Weak Scale Leptogenesis, R-symmetry, and a DisplacedHiggs,” arXiv:1111.0008 [hep-ph].[77] R. Davies and M. McCullough, “Small neutrino masses due to R-symmetry breaking for asmall cosmological constant,” Phys. Rev. D (2012) 025014 [arXiv:1111.2361 [hep-ph]].[78] H. Itoyama and N. Maru, “D-term Dynamical Supersymmetry Breaking Generating SplitN=2 Gaugino Masses of Mixed Majorana-Dirac Type,” Int. J. Mod. Phys. A (2012)1250159 [arXiv:1109.2276 [hep-ph]].H. Itoyama and N. Maru, “D-term Triggered Dynamical Supersymmetry Breaking,” Phys.Rev. D (2013) 025012 [arXiv:1301.7548 [hep-ph], arXiv:1301.7548 [hep-ph]].H. Itoyama and N. Maru, “126 GeV Higgs Boson Associated with D-term TriggeredDynamical Supersymmetry Breaking,” arXiv:1312.4157 [hep-ph].[79] E. Bertuzzo and C. Frugiuele, “Fitting Neutrino Physics with a U (1) R Lepton Number,”JHEP (2012) 100 [arXiv:1203.5340 [hep-ph]].[80] R. Davies, “Dirac gauginos and unification in F-theory,” JHEP (2012) 010[arXiv:1205.1942 [hep-th]].[81] R. Argurio, M. Bertolini, L. Di Pietro, F. Porri and D. Redigolo, “Holographic Correlatorsfor General Gauge Mediation,” JHEP (2012) 086 [arXiv:1205.4709 [hep-th]].[82] R. Fok, G. D. Kribs, A. Martin and Y. Tsai, “Electroweak Baryogenesis in R-symmetricSupersymmetry,” Phys. Rev. D (2013) 5, 055018 [arXiv:1208.2784 [hep-ph]].[83] R. Argurio, M. Bertolini, L. Di Pietro, F. Porri and D. Redigolo, “Exploring HolographicGeneral Gauge Mediation,” JHEP (2012) 179 [arXiv:1208.3615 [hep-th]]. – 41 – 84] C. Frugiuele, T. Gregoire, P. Kumar and E. Ponton, “’L=R’ - U (1) R as the Origin ofLeptonic ’RPV’,” JHEP (2013) 156 [arXiv:1210.0541 [hep-ph]].[85] C. Frugiuele, T. Gregoire, P. Kumar and E. Ponton, “’L=R’ – U (1) R Lepton Number at theLHC,” JHEP (2013) 012 [arXiv:1210.5257 [hep-ph]].[86] K. Benakli, M. D. Goodsell and F. Staub, “Dirac Gauginos and the 125 GeV Higgs,” JHEP (2013) 073 [arXiv:1211.0552 [hep-ph]].[87] F. Riva, C. Biggio and A. Pomarol, “Is the 125 GeV Higgs the superpartner of a neutrino?,”JHEP (2013) 081 [arXiv:1211.4526 [hep-ph]].[88] S. Chakraborty and S. Roy, “Higgs boson mass, neutrino masses and mixing and keV darkmatter in an U (1) R − lepton number model,” JHEP (2014) 101 [arXiv:1309.6538[hep-ph]].[89] C. Csaki, J. Goodman, R. Pavesi and Y. Shirman, “The m D − b M Problem of DiracGauginos and its Solutions,” arXiv:1310.4504 [hep-ph].[90] E. Dudas, M. Goodsell, L. Heurtier and P. Tziveloglou, “Flavour models with Dirac and fakegluinos,” Nucl. Phys. B (2014) 632 [arXiv:1312.2011 [hep-ph]].[91] H. Beauchesne and T. Gregoire, “Electroweak precision measurements in supersymmetricmodels with a U(1) R lepton number,” JHEP (2014) 051 [arXiv:1402.5403 [hep-ph]].[92] E. Bertuzzo, C. Frugiuele, T. Gregoire and E. Ponton, “Dirac gauginos, R symmetry and the125 GeV Higgs,” arXiv:1402.5432 [hep-ph].[93] K. Benakli, M. Goodsell, F. Staub and W. Porod, “The Constrained Minimal Dirac GauginoSupersymmetric Standard Model,” Phys. Rev. D , 045017 (2014) [arXiv:1403.5122[hep-ph]].[94] S. Chakraborty, D. K. Ghosh and S. Roy, “7 keV Sterile neutrino dark matter in U (1) R − lepton number model,” JHEP (2014) 146 [arXiv:1405.6967 [hep-ph]].[95] M. D. Goodsell and P. Tziveloglou, “Dirac Gauginos in Low Scale Supersymmetry Breaking,”arXiv:1407.5076 [hep-ph].[96] S. Ipek, D. McKeen and A. E. Nelson, “CP Violation in Pseudo-Dirac Fermion Oscillations,”arXiv:1407.8193 [hep-ph].[97] D. Busbridge, “Constrained Dirac gluino mediation,” arXiv:1408.4605 [hep-ph].[98] P. Dieçner, J. Kalinowski, W. Kotlarski and D. StÃűckinger, “Higgs boson mass andelectroweak observables in the MRSSM,” arXiv:1410.4791 [hep-ph].[99] A. Merle, “keV Neutrino Model Building,” Int. J. Mod. Phys. D (2013) 1330020[arXiv:1302.2625 [hep-ph]].[100] H. K. Dreiner, S. Heinemeyer, O. Kittel, U. Langenfeld, A. M. Weber and G. Weiglein,“Mass Bounds on a Very Light Neutralino,” Eur. Phys. J. C (2009) 547 [arXiv:0901.3485[hep-ph]].[101] J. F. Gunion, D. Hooper and B. McElrath, “Light neutralino dark matter in the NMSSM,”Phys. Rev. D (2006) 015011 [hep-ph/0509024].[102] H. K. Dreiner, J.S. Kim and O. Lebedev, “First LHC constraints on neutralinos,” Phys.Lett. B 715 (2012) 199 [arXiv:1206.3096] [inSPIRE]. – 42 – (2013) 132 [arXiv:1307.4119].[104] H. K. Dreiner, M. Hanussek, J. S. Kim and S. Sarkar, “Gravitino cosmology with a verylight neutralino,” Phys. Rev. D (2012) 065027 [arXiv:1111.5715 [hep-ph]].[105] H. K. Dreiner, S. Grab, D. Koschade, M. Kramer, B. O’Leary and U. Langenfeld, “Raremeson decays into very light neutralinos,” Phys. Rev. D , 035018 (2009) [arXiv:0905.2051[hep-ph]].[106] D. Choudhury, H. K. Dreiner, P. Richardson and S. Sarkar, “A Supersymmetric solution tothe KARMEN time anomaly,” Phys. Rev. D (2000) 095009 [hep-ph/9911365].[107] R. Adhikari and B. Mukhopadhyaya, “Can we identify a light neutralino in B Factories?,”Phys. Lett. B (1995) 228 [hep-ph/9411208].[108] R. Adhikari and B. Mukhopadhyaya, “Light neutralinos in B decays,” Phys. Rev. D (1995) 3125 [hep-ph/9411347].[109] R. Adhikari and B. Mukhopadhyaya, “Some signals for a light neutralino,” hep-ph/9508256.[110] PARTICLE DATA GROUP collaboration, J.Beringer et al., Review of particle physics , Phys. Rev. D 86 (2012) 010001 [inSPIRE].[111] M. Kawasaki and T. Moroi, “Gravitino production in te inflatinary universe and the effectson big ban nucleosynthesis,” Prog. Theor. Phys. (1995) 879 [hep-ph/9403364][inSPIRE].[112] A. Boyarsky, O. Ruchayskiy and D. Iakubovskyi, “A Lower bound on the mass of DarkMatter particles,” JCAP (2009) 005 [arXiv:0808.3902 [hep-ph]].[113] A. Boyarsky, O. Ruchayskiy and M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. (2009) 191[arXiv:0901.0011 [hep-ph]].[114] A. Djouadi, “The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in thestandard model,” Phys. Rept. (2008) 1 [hep-ph/0503172].[115] A. Djouadi, “The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in theminimal supersymmetric model,” Phys. Rept. (2008) 1 [hep-ph/0503173].[116] M. Spira, “QCD effects in Higgs physics,” Fortsch. Phys. (1998) 203 [hep-ph/9705337].[117] M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, “Higgs boson production at theLHC,” Nucl. Phys. B (1995) 17 [hep-ph/9504378].[118] W. -Y. Keung and W. J. Marciano, “Higgs Scalar Decays: H —> W+- X,” Phys. Rev. D (1984) 248.[119] T. G. Rizzo, “Decays of Heavy Higgs Bosons,” Phys. Rev. D (1980) 722.[120] V. Khachatryan et al. [CMS Collaboration], “Constraints on the Higgs boson width fromoff-shell production and decay to Z-boson pairs,” Phys. Lett. B (2014) 64[arXiv:1405.3455 [hep-ex]].[121] K. Cheung, J. S. Lee and P. Y. Tseng, “Higgcision Updates 2014,” arXiv:1407.8236 [hep-ph].[122] W. Porod, M. Hirsch, J. Romao and J. W. F. Valle, Phys. Rev. D (2001) 115004doi:10.1103/PhysRevD.63.115004 [hep-ph/0011248].[123] S. Chatrchyan et al. [CMS Collaboration], “Measurement of the properties of a Higgs bosonin the four-lepton final state,” Phys. Rev. D , 092007 (2014) [arXiv:1312.5353 [hep-ex]]. – 43 – et al. [CMS Collaboration], “Measurement of Higgs boson production andproperties in the WW decay channel with leptonic final states,” JHEP , 096 (2014)[arXiv:1312.1129 [hep-ex]].[125] [ATLAS Collaboration], “Combined coupling measurements of the Higgs-like boson with theATLAS detector using up to 25 fb − of proton-proton collision data,”ATLAS-CONF-2013-034.[126] S. Chatrchyan et al. [CMS Collaboration], “Search for the standard model Higgs bosonproduced in association with a W or a Z boson and decaying to bottom quarks,” Phys. Rev.D , 012003 (2014) [arXiv:1310.3687 [hep-ex]].[127] The ATLAS collaboration, “Search for the bb decay of the Standard Model Higgs boson inassociated W/ZH production with the ATLAS detector,” ATLAS-CONF-2013-079.[128] S. Chatrchyan et al. [CMS Collaboration], “Evidence for the 125 GeV Higgs boson decayingto a pair of τ leptons,” JHEP , 104 (2014) [arXiv:1401.5041 [hep-ex]].[129] The ATLAS collaboration, “Evidence for Higgs Boson Decays to the τ + τ − Final State withthe ATLAS Detector,” ATLAS-CONF-2013-108.[130] E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein and S. W. Randall,arXiv:1402.2301 [astro-ph.CO].[131] A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi and J. Franse, arXiv:1402.4119 [astro-ph.CO].[132] J. F. Gunion and H. E. Haber, “Higgs Bosons in Supersymmetric Models. 1.,” Nucl. Phys. B (1986) 1 [Erratum-ibid. B (1993) 567].[133] J. F. Gunion and H. E. Haber, “Higgs Bosons in Supersymmetric Models. 2. Implications forPhenomenology,” Nucl. Phys. B (1986) 449.[134] M. Drees, R. Godbole and P. Roy, “Theory and phenomenology of sparticles: An account offour-dimensional N=1 supersymmetry in high energy physics,” Hackensack, USA: WorldScientific (2004) 555 p(1986) 449.[134] M. Drees, R. Godbole and P. Roy, “Theory and phenomenology of sparticles: An account offour-dimensional N=1 supersymmetry in high energy physics,” Hackensack, USA: WorldScientific (2004) 555 p