A Supersymmetric Explanation of the Excess of Higgs--Like Events at the LHC and at LEP
OOctober 2012
A Supersymmetric Explanation of the Excess of Higgs–LikeEvents at the LHC and at LEP
Manuel Drees
Bethe Center for Theoretical Physics and Physikalisches Institut d. Universit¨at Bonn,Nussallee 12, 53115 Bonn, Germany
Abstract
The LHC collaborations have recently announced evidence for the production of a “Higgs–like” boson with mass near 125 GeV. The properties of the new particle are consistent (withinstill quite large uncertainties) with those of the Higgs boson predicted in the Standard Model(SM). This discovery comes nearly ten years after a combined analysis of the four LEP exper-iments showed a mild excess of Higgs–like events with a mass near 98 GeV. I show that bothgroups of events can be explained simultaneously in the minimal supersymmetric extension ofthe SM, in terms of the production and decay of the two neutral CP–even Higgs bosons predictedby this model, and explore the phenomenological consequences of this explanation. a r X i v : . [ h e p - ph ] N ov Introduction
Recently the LHC collaborations ATLAS and CMS announced the discovery of a “Higgs–like” bosonwith mass near 125 GeV [1]. This new boson has been detected at the LHC chiefly in the γγ andfour lepton final states. In addition, there is evidence, at the ∼ σ level, for decays into b ¯ b pairs fromthe Tevatron experiments CDF and D0 [2].Within the still quite large experimental (and theoretical [3]) uncertainties, the properties of thenew boson are consistent with those of the single physical Higgs boson in the Standard Model (SM).However, as well known, the scalar sector of the SM is technically unnatural, since it suffers fromquadratic divergencies. These divergencies are canceled in supersymmetric extensions of the SM [4].Even the simplest such theory, the Minimal Supersymmetric extension of the SM (MSSM), con-tains two Higgs doublets, the second doublet being required both for the cancellations of anomalies(from the higgsinos), and in order to give masses to all quarks [5]. As a result, the MSSM containsthree neutral physical Higgs states. In the absence of CP violation, these can be classified as twoCP–even states h, H (with m h < m H ) and one CP–odd state A . Most interpretations of the newboson discovered by the LHC experiments (many of which were published after the first experimentalevidence was announced in December 2011) within the MSSM focus on the possibility that it is thelighter CP–even state h [6, 7, 8]. However, achieving m h (cid:39)
125 GeV is only possible if stop squarksare very heavy. By most definitions, this requires a somewhat uncomfortable amount of finetuning ∗ In this scenario the heavier neutral Higgs bosons
H, A as well as the charged Higgs bosons H ± can beessentially arbitrarily heavy. In fact, in simple (constrained) scenarios of supersymmetry breaking,the large lower bounds on (first generation) squark masses typically require these states to be quiteheavy; within such constrained scenarios the new state therefore has to be interpreted as h .The possibility that instead the heavier state H has been discovered has also been entertained[10, 7, 11]. In this case the lighter CP–even state would obviously have to be lighter than 125GeV, and would need to satisfy limits from Higgs searches both at the LHC [1] and at LEP [12].As pointed out in refs.[7, 11] this is not difficult to achieve, if H is SM–like in agreement withexperimental observations of the new state at 125 GeV. In particular, sufficiently large branchingratios for H into four leptons, and into two photons, require the couplings of H to two massive gaugebosons to not differ very much from the corresponding SM values. In the context of the MSSM thisautomatically implies that h has suppressed couplings to W and Z , making it difficult to detect.Here I wish to point out that in this scenario h might be put to good use, by explaining an excessof Higgs–like events observed some ten years ago by the four LEP collaborations [12]. † Actually, thecombined LEP data showed two regions of reconstructed Higgs mass where some excess occurred.One was right at the kinematic limit, near 115 GeV. This excess was observed mostly by the ALEPHcollaboration [14]; in the combined data, its statistical significance reached only 1.7 standard devia-tions. It is compatible with an SM–like Higgs with this mass. However, this interpretation is at oddswith the interpretation of the new particle discovered at the LHC as an SM–like Higgs boson.The combination of the data from all four LEP experiments also revealed [12] a somewhat moresignificant excess near 98 GeV, with significance of about 2.3 standard deviations. This excess is not compatible with an SM Higgs at that mass; rather, it’s compatible with an about ten times smaller ∗ See however ref.[9] for an example spectrum with a 125 GeV Higgs boson and heavy stops nevertheless requiringlittle “electroweak–scale finetuning”. † Very recently it has been pointed out that both the LEP excess and the LHC discovery can be explained in theNMSSM [13], where the spectrum contains three CP–even and two CP–odd neutral Higgs bosons. hZZ (and hence also the hW W ) coupling [15]. Thisimplies that the other MSSM Higgs bosons have to be relatively light: if they were heavy, h wouldbecome SM–like. A detailed analysis found an upper bound on m H of about 140 GeV [15]. The newparticle at 125 GeV therefore falls right in the middle of the allowed range for m H in this scenario,where the lower bound (of about 114 GeV) comes from LEP Higgs searches. ‡ The LHC discovery obviously greatly constrains the allowed parameter space of this scenario,where now the masses of both
CP–odd Higgs bosons are fixed within a few GeV theoretical andexperimental uncertainty. However, as well known the MSSM Higgs sector is subject to large radia-tive corrections. This introduces several new parameters, the most important ones being the onesappearing in the stop mass matrix. Here I present a detailed analysis of this scenario. This not onlyupdates ref.[15] by including the constraint m H (cid:39)
125 GeV; I also carefully compute the relevantdecay widths, and resulting branching ratios and signal strengths, of the neutral Higgs bosons, wherethe latter are normalized to the signal strength of the SM Higgs boson. I also include constraints fromnull searches for neutral MSSM Higgs bosons decaying into tau pairs performed by CMS [16], and forcharged Higgs bosons produced in top quark decays performed by ATLAS [17]. The former are moreimportant, considerably limiting the allowed parameter space of this scenario. I nevertheless findthat m A masses roughly between 95 and 150 GeV are allowed in this scenario. The charged Higgsboson mass can reach up to about 170 GeV. The signals for H production in both the di–photon andfour lepton channels can be considerably enhanced, but the ratio of these two signals cannot exceedits SM prediction by more than about 35%.This analysis has been performed in the framework of the general MSSM, where all relevantparameters are fixed directly at the weak (or TeV) scale. In order to limit the size of the parameterspace, I will not specify soft breaking parameters for the first two generations of sfermions, whichhave almost no impact on the masses and couplings of Higgs bosons. This approach also permits meto ignore all constraints from flavor physics, which depend very strongly on the flavor structure ofthe soft breaking terms.The rest of this paper is organized as follows. In Sec. 2 I describe details of the analysis. InSec. 3 I explore the parameter space that is compatible with this explanation; in particular, I giveallowed ranges for physical quantities of interest, and explore correlations between them. Finally,Sec. 4 contains a brief summary and conclusions. This analysis is performed in the framework of the general MSSM, where all relevant parameters arefixed at the weak scale, and no high–scale constraints on the spectrum of superpartners are imposed.Obviously at least the leading radiative corrections [18] to the masses and mixing angle of theMSSM Higgs bosons have to be included in any quantitative analysis. This is most easily done usingthe effective potential (or, equivalently, Feynman diagrammatic calculations with vanishing externalmomentum). Recall that the entire Higgs spectrum should be relatively light in this scenario; this ‡ Ref.[10] also considered the MSSM with m H (cid:39)
125 GeV, including scenarios with m h ≤
110 GeV and suppressed
ZZh couplings, in the context of an analysis of scenarios with a light neutralino as Dark Matter candidate, basedon very early, preliminary LHC results. No bounds on the strengths of the H signals were imposed, and h was notrequired to explain the LEP excess near 98 GeV. m h and m H . Note that the SUSY QCD corrections tothe bottom mass are also included in the calculation of the corresponding Yukawa couplings, whichaffect both the partial widths of the neutral Higgs bosons into b ¯ b pairs and the b loop contributionto the partial widths of the decays into gluon and photon pairs.The running top mass, m t ( m t ) is fixed to 165 GeV (in the DR scheme). This corresponds toa pole mass near 173 GeV, the current central value [26]. I also fix m b ( m b ) = 4 .
25 GeV. As finalsimplification, I have taken the soft breaking parameters in the stop and sbottom mass matrices tobe the same. This is always true for the masses of the superpartners of the left–handed squarks, dueto SU (2) invariance, but the masses of the SU (2) singlet squarks as well as the two A − parameterscould in principle be different. However, we will see that the CMS di–tau search requires the ratioof vacuum expectation values tan β to be relatively small, below 13; as a result, sbottom loops arealways subdominant, and thus need not be treated as carefully as stop loops.The most convincing signals for the new state at 125 GeV have been found in the di–photonchannel. This is obviously only accessible through loop diagrams. In addition to the diagramsinvolving W bosons or third generation fermions, diagrams involving charged Higgs bosons as wellas all third generation sfermions are included. It had been noticed [27, 11] that loops involving stausleptons could significantly change the di–photon widths of neutral CP–even MSSM Higgs bosons.I therefore allow the ˜ τ L,R soft breaking masses as well as the trilinear soft breaking parameter A τ to vary independently from the parameters of the stop sector. However, it turns out that in thegiven scenario, stau loops always make very small contributions. Similarly, the partial widths of theHiggs bosons into gluons are computed including loops of third generation quarks as well as squarks(squark loops are absent in case of the CP–odd Higgs boson). The relevant expressions are takenfrom [28].Altogether we are thus left with ten free parameters: tan β, m A , µ, m ˜ t L , m ˜ t R , A t , m ˜ τ L , m ˜ τ R , A τ , m ˜ g .This ten–dimensional parameter space has been scanned randomly, subject to the following con-3traints not involving Higgs bosons: | µ | , m ˜ t R , m ˜ t L , m ˜ g , m ˜ τ L , m ˜ τ R ≤ | µ | , m ˜ t , m ˜ b , m ˜ τ ≥
100 GeV; (1b) | m ˜ t − m ˜ b | ≤
50 GeV or max( m ˜ t , m ˜ b ) >
300 GeV; (1c) m ˜ g ≥
600 GeV; (1d) | A t | , | µ | ≤ . (cid:0) m ˜ t R + m ˜ t L (cid:1) ; (1e) | A τ | , | µ | ≤ . m ˜ τ R + m ˜ τ L ) ; (1f) δρ ˜ t ˜ b ≤ · − . (1g)The first of these constraints is a (quite conservative) naturalness criterion. Conditions (1b) ensurethat higgsino–like charginos (with mass ∼ | µ | ) as well as the lighter physical stop (˜ t ), sbottom (˜ b )and stau (˜ τ ) states escaped detection at LEP [29]. Condition (1c) ensures that only one of the twolighter squark states can be below 300 GeV, unless they are close in mass. In the latter case theycould both be close in mass to the lightest neutralino, in which case ˜ t and ˜ b pair production wouldlead to events with a small amount of visible energy, which are difficult to detect. Condition (1d)is a rather conservative interpretation of gluino search limits in the general MSSM. Note that loopsinvolving gluinos affect the Higgs masses and mixing angle only at two–loop order, but modify the hb ¯ b and Hb ¯ b couplings already at one–loop. The upper bounds (1e,f) on the parameters determiningmixing in the stop and stau sectors have been imposed to avoid situations where ˜ t or ˜ τ fields havenon–vanishing VEVs in the absolute minimum of the scalar potential [30]. § Finally, (1g) requires thecontribution of stop–sbottom loops to the electroweak ρ parameter [32] to be sufficiently small.In order to be able to describe the (mild) excess of Higgs–like events at LEP, and the propertiesof the new boson discovered at the LHC, the Higgs sector has to simultaneously satisfy the followingconstraints: 95 GeV ≤ m h ≤
101 GeV ; (2a)123 GeV ≤ m H ≤
128 GeV ; (2b)0 . ≤ sin ( α − β ) ≤ .
144 ; (2c)0 . ≤ R V VH ≤ . V = W, Z ) ; (2d)0 . ≤ R γγH . (2e)The first of these constraints places m h in the range where an excess of events had been observed atLEP [12]. Similarly, (2b) ensures that m H agrees with the value reported by the LHC experiments[1]. In both cases, the range is a crude estimate of theoretical and experimental uncertainties. Notethat the peak at the LHC is somewhat narrower than at at LEP, since the latter has been observedchiefly in multi–hadron final states.The third constraint [15] ensures that the Zh production cross section at LEP is roughly tentimes smaller than the corresponding cross section in the SM, for given mass of the Higgs boson;as noted above, the excess at LEP is compatible with Higgs production only if the ZZh coupling issuppressed. § Note that ref.[27], where the influence of light staus on the two–photon widths of MSSM Higgs bosons was firstexplored, only imposes the much weaker constraint on | µ | that follows from the requirement that the zero temperaturetunnel rate into the false vacuum [31] is sufficiently small. W W ∗ , four lepton and di–photonchannels come out roughly correct. They are described by the quantities R XXH ≡ Γ( H → gg )Γ( H SM → gg ) · Γ( H → XX )Γ( H SM → XX ) · Γ( H SM , tot )Γ( H tot ) . (3)They describe the strength of the H signal in the XX channel normalized to the strength of thecorresponding signal for the SM Higgs boson H SM . Here I have assumed that Higgs production atthe LHC is dominated by gluon fusion; this is true both in the SM and in the relevant parameterrange of the MSSM. The strength of the four lepton signal observed at the LHC, which in the Higgsinterpretation of the signal is due to the decay of the Higgs boson into a real and a virtual Z boson,agrees quite well with the SM prediction; I therefore allow this signal to be at most a factor of twostronger or weaker than in the SM. In contrast, the di–photon signal appears somewhat strongerthan in the SM; I therefore only impose a lower bound on the strength of the signal in this channel.Since the γγ invariant mass peak has a finite width of roughly 1 GeV, essentially given by theexperimental resolution, R γγH includes the contribution from gg → A → γγ whenever | m H − m A | < A → γγ decays, which in turn is due to the absence of an AW + W − coupling.Note also that in the MSSM, R W WH = R ZZH , so that no independent constraint can be imposed onthe strength in the di–lepton channel.Finally, null results of additional searches for Higgs bosons have to be imposed. In particular, forcharged Higgs bosons with mass (well) below m t − m b , ATLAS searches for t → H + b decays, with H + → τ + ν τ , exclude [17] both small and large values of tan β , leaving an allowed strip centered attan β (cid:39) (cid:112) m t ( m t ) /m b ( m t ) (cid:39) H + tb coupling is minimal. At least in the present context,the CMS search for neutral MSSM Higgs bosons in the di–tau channel [16] is even more constraining.Here I have taken both analyses at face value. Since the ATLAS charged Higgs search is basicallyindependent of the details of the neutral Higgs spectrum, it should indeed apply to the presentscenario. CMS states its bounds on MSSM parameter space in the context of the “maximal mixing”scenario, which maximizes m h for given average stop mass. In this scenario the CP–odd state istypically quite closely degenerate with either h or H , especially for large tan β where this searchis most sensitive. Such a degeneracy obviously increases the yield of tau pairs of a given invariantmass. In the present context the mass splittings between all three neutral Higgs bosons are oftensizable; this should lead to somewhat smaller signals in the di–tau channel than in the “maximalmixing” scenario. Incorporating the CMS constraints in the ( m A , tan β ) plane without modificationtherefore probably overstates their impact somewhat. However, this is not easy to quantify withouta full simulation including experimental resolutions.This concludes the description of the analysis. Let us now turn to the results. This Section contains a discussion of the results of the scan of parameter space, subject to theconstraints discussed in the previous Section. Of course, the first and quite nontrivial result is thatallowed sets of parameter sets can indeed be found, i.e. the (phenomenological) MSSM can indeedexplain at the same time the (mild) excess of Higgs–like events at LEP and the detection of aHiggs–like particle by the LHC experiments. 5n order to further test this scenario, one has to know what it implies for the relevant observables.To that end, I will first describe upper and/or lower bounds on quantities of interest that were foundin the scan, before discussing correlations between pairs of these quantities.
Let us first look at observables in the Higgs sector. Note first of all that the upper and lower limitson both the h and H mass can be saturated, i.e. the scenario doesn’t allow to further shrink eitherof these mass regions beyond the limits imposed as constraints in eqs.(2a,b).However, not surprisingly there are nontrivial bounds on the masses of the CP–odd and chargedHiggs bosons. Some bounds already follow [15] from the constraint (2c) on the Zhh coupling: if m A or m H + becomes very large, h automatically becomes SM–like; in this “decoupling scenario”the upper bound on the Zhh coupling is therefore badly violated. At the same time the constraint m H >
123 GeV imposes a non–trivial lower bound on the mass of the charged Higgs. Altogether Ifind 120 GeV ≤ m H + ≤
170 GeV . (4)The upper bound can be saturated, implying that t → H + b decays can be closed kinematically. Thisis in (mild) conflict with a statement of [11], probably due to the large range of parameters I exploredhere. Saturating this upper bound requires very large µ , a large hierarchy between the ˜ t L and ˜ t R masses, a top mixing parameter | A t | saturating its upper bound (with A t < , µ > β (cid:39) A reads96 GeV ≤ m A ≤
152 GeV . (5)The upper bound on m A is saturated for the same choice of parameters as the upper bound on m H + .The lower bound on m A together with the constraint (2a) on m h implies that limits from searchesfor hA production at LEP are always satisfied.The LHC searches for non–SM Higgs bosons discussed at the end of Sec. 2 considerably restrictthe allowed values of tan β , leading to 5 . ≤ tan β ≤ . . (6)The lower bound is largely determined by the ATLAS search for charged Higgs bosons, while theupper limit is chiefly due to the CMS search for neutral Higgs bosons in the di–tau channel. Theallowed range of tan β thus looks quite narrow. However, closing it entirely may not be easy. Asnoted above, the H + tb coupling reaches its minimum near tan β = (cid:112) m t /m b , which falls in the range(6). The signal strength in the di–tau channel scales essentially like tan β , so reducing the upperbound on tan β by a factor of about 2.5 requires an increase of the sensitivity of the search by afactor of six. Recall also that my interpretation of the CMS bound might be overly strict, i.e. thetrue bound might be somewhat weaker.As noted earlier, the constraint (2c) implies that the HW W and
HZZ couplings have close toSM strength. However, this doesn’t imply that the gg → H → ZZ ∗ → (cid:96) signal also has close toSM strength. On the one hand, loops of new strongly interacting sparticles, in particular stops, canchange the H production cross section significantly. On the other hand, the couplings of H to SMfermions, in particular to b quarks and τ leptons, can still differ considerably from their SM values,6hereby modifying the H decay branching ratios. As a result, both the upper and the lower limits on R ZZH in (2d) can be saturated, if ˜ t is not too heavy. The lower bound on R γγH can also be saturated,and the upper bound is R γγH ≤ . . (7)A significant enhancement of the H → γγ signal, which is hinted at by present data, is thus possiblein this scenario. However, this enhancement is mostly due to the increase of the H production crosssection and/or decrease of its total width; both these effects also increase R ZZH . In fact, when consid-ering the ratio of signal strengths ∗ in the γγ and 4 (cid:96) (or, more generally, V V ∗ ) channels normalizedto their respective SM values, only a moderate deviation from unity is possible in this scenario:0 . ≤ R γγH R ZZH ≤ . . (8)I do not find any scenarios where the branching ratio for H → γγ decays is affected significantly by˜ τ loops; this is probably due to the vacuum stability constraint [30] | µ | ≤ m ˜ τ L + m ˜ τ R ) /
2, which hasnot been imposed in refs.[27] and [11]. The contribution of charged Higgs loops to this branchingratio is also always very small, although the charged Higgs boson is quite light in this scenario, asshown in (4).The di–tau channel is currently poorly constrained by the data. In fact, the strength of the signalin this channel can deviate quite significantly from its SM value in the present scenario:0 . ≤ R ττH ≤ . . (9)Note that the size of the τ Yukawa coupling exceeds its SM value for tan β >
1. However, the sizeof the Hτ + τ − coupling also depends on the mixing angle α between the neutral CP–even Higgsbosons, and even vanishes if cos α = 0. This limit cannot be realized in the present scenario, but asubstantial suppression of the di–tau signal strength is possible. On the other hand, the maximalenhancement of this signal occurs when stop and sbottom loops simultaneously enhance the gg → H production cross section and suppress the Hb ¯ b coupling, while the Hτ + τ − coupling is enhancedsince | cos α | > | cos β | . Moreover, if | m H − m A | ≤ R ττA has been added to R ττH , since the H and A di–tau signals would be difficult to distinguish experimentally in this case.However, the A → τ + τ − signal is quite small in the relevant region of parameter space, which hastan β (cid:39)
6; here the tan β enhancement of the Ab ¯ b coupling cannot yet compensate for the cot β suppression of the At ¯ t coupling, leading to an A production cross section from gluon fusion which issignificantly smaller than the corresponding SM value. † Nevertheless the upper end of the range (9)is probably already disfavored by present data, given the absence of a clear signal in this channel.Note, however, that requiring R ττH < h at the LHC in this scenario. The γγ signal is very weak for this state, R γγh ≤ . ∗ Such ratios have very recently also been discussed in [8], which however assumes that the LHC signals are due tothe production of h , not H . † In general there is also a significant contribution to the inclusive A production cross section from the tree–levelprocess gg → b ¯ bA ; however, the presence of two additional b − jets should allow to discriminate this process from SMHiggs production, so this contribution should not simply be added to R ττH even if m A = m H . hW + W − coupling implied by the constraint (2c). On the other hand, the gg → h production cross section need not be suppressed relative to its SM value, so that0 . ≤ R ττh ≤ . . (10)Even the upper end of this range might be difficult to probe at the LHC, since h is quite close inmass to the Z boson which yields a much stronger signal in the τ + τ − channel. For this reason, thelower end will probably remain unobservable even for LHC upgrades.Before concluding this Subsection, let me briefly mention some constraints on quantities relatedto the stop sector; these are the only MSSM parameters not directly related to the Higgs sector forwhich some non–trivial constraints can be derived in the present context. For example, requiringthat the heavy CP–even Higgs boson H explains the LHC signals makes it quite difficult to findacceptable scenarios with small values of the Higgs(ino) mass parameter µ : only for tan β > ∼ | µ | <
400 GeV survive; for tan β = 10, some solutions with µ > | µ | = 100 GeV. Similarly, | A t | has to exceed 400 GeV, and the sum | A t | + | µ | > t , ˜ b and ˜ τ masses can all saturate the lower bounds of 100GeV; moreover, no meaningful upper bounds on these masses can be derived. On the other hand,the ˜ t mass must exceed 600 GeV in this scenario, and the sum of ˜ t and ˜ t masses must exceed 900GeV. For comparison: demanding m h >
123 GeV, as required if the recent LHC discovery is to beinterpreted in terms of the production and decay of h , leads to the lower bound m ˜ t + m ˜ t > Let us now analyze correlations between observables that result from the constraints (2) as well asthe upper limits on MSSM Higgs searches at the LHC, beginning with correlations between physicalmasses. The most obvious such correlations are shown in Figs. 1a,b, which show the correlationbetween the mass of the CP–odd Higgs boson and the mass of the charged Higgs boson and lighterstop eigenstate, respectively. These, and all following, scatter plots are based on scans over param-eter space containing several million sets of parameters (not all of which are plotted), with specialemphasis on those regions of parameter space where an observable reaches an extremum.As shown in Fig. 1a, the masses of the charged and CP–odd Higgs bosons are strongly correlated.This is not surprising, given the tree–level relation m H + = (cid:112) m A + M W , which is indicated by thesolid red line. We see that the radiative corrections to this relation are usually negative, but rathermodest in size in the allowed region of parameter space. This is a consequence of the upper bounds(1e,f) on the parameters determining stop mixing, which also largely determine the size of trilinearcouplings between Higgs bosons and stop and sbottom squarks; the upper bound (6) on tan β alsoplays a role in limiting the size of the corrections to this relation.The right frame in Fig. 1 shows that an upper bound on the ˜ t mass results in the present scenarioif m A > ∼
110 GeV; this results from the upper bound on m H . Notice that for relatively light ˜ t , theheavier CP–even Higgs boson can be significantly lighter than the CP–odd Higgs boson; in contrast,at the tree level one has m H > m A . The magnitude of these negative corrections to m H is limitedby the upper bound on | A t | and | µ | given in (1e).Figure 2 shows correlations between a mass and a (ratio of) signal strength(s). Recall that onlythe gluon fusion contribution to various Higgs signals is included here; this should be the dominant8
00 120 140 160 m A [GeV] m H + [ G e V ] a)
90 100 110 120 130 140 150 m A [GeV] m t ~ [ G e V ] b) Figure 1: Allowed region in the ( m A , m H ± ) (a) and the ( m A , m ˜ t ) plane (b), after the constraints (2)as well as the various sparticle and Higgs search limits discussed in the text have been imposed. Thesolid (red) line in (a) shows the tree–level relation between m H + and m A .channel in general, but, depending on the cuts, there might be significant contributions also from W W and ZZ fusion. Associate production with a b ¯ b pair, which can become quite important atlarge tan β [28], is not expected to be very important, given the upper bound (6).Frame (a) shows correlations between the mass of the CP–odd Higgs boson and the h signalstrength in the τ + τ − channel. The h → τ + τ − signal strength shows a first peak at m A = 100 GeV (cid:39) m h ; due to this near–degeneracy, the signal from A → τ + τ − has been added, which increases R ττh by up to one unit. This signal reaches its (local) maximum for the largest allowed value of tan β .Here the cross sections for producing an h or A boson are slightly larger than the corresponding crosssection for producing an SM Higgs boson with equal mass; the enhancement of the bottom Yukawacoupling over–compensates the suppression of the couplings of these two lighter Higgs bosons to topquarks. Both stop squarks need to be fairly heavy in this region of parameter space, in order toobtain a sufficiently large value of m H .Recall that the A and h contributions to this channel are added only for | m A − m h | ≤ m A the maximal strength of this signal therefore decreases, before reachingits absolute maximum near m A = 130 GeV. At the absolute maximum of R ττh , the lower bound on m ˜ t is saturated, and sbottom loops suppress the hb ¯ b coupling. It is still significantly larger thanthe corresponding coupling in the SM, but the enhancement of the hτ + τ − coupling is even larger,leading to an enhanced branching ratio for h → τ + τ − . Moreover, the h → gg width, and hencethe h production cross section, is dominated by ˜ t and b loops, which have the same sign, while thesubleading t loop contribution has opposite sign. As a result, the gluonic decay width of h exceedsits SM value by about a factor of two. In combination, this enhances the h → τ + τ − signal by up toa factor of 3 . m h is quite close to M Z , it is not clear whetherthis enhancement is sufficient to make h detectable at the LHC in this channel. Note also that valuesof R ττh well below 1 are possible for nearly all values of m A . This is chiefly due to the suppression of the h → gg width, which in turn is caused by strong cancellations between the t and b loop contributionsin this region of parameter space. There is a nontrivial, although phenomenologically probably not9 m A [GeV] R h ττ a) -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 µ [GeV] R H γγ b)
100 1000 m t [GeV] R H ττ
200 2000 c)
100 1000 m t [GeV] R H γγ / R HVV
200 500 d) Figure 2: Allowed region in the ( m A , R ττh ) (a), the ( µ, R γγH ) (b), the ( m ˜ t , R ττH ) (c) and the( m ˜ t , R γγH /R V VH ) plane (d), after the constraints (2) as well as the various sparticle and Higgs searchlimits discussed in the text have been imposed.very interesting, lower bound on this quantity, as shown in (10), since the b loop contribution has asizable imaginary part, which cannot be canceled by loops involving much heavier t or ˜ t particles.Fig. 2b shows that obtaining | µ | < . β is near the upper end of the allowed range (6). Moreover, a large mass splitting between the˜ t L and ˜ t R masses is required. The masses of the CP–even Higgs bosons h and H are then near thelower and upper ends of their allowed ranges, respectively. In this region of parameter space boththe ˜ t and the real part of the b loop contributions to H → gg as well as H → γγ have the samesign as the t loop contributions. This enhances the partial width for H → gg by up to a factor of1 .
6, but suppresses the partial width for H → γγ , which is dominated by W loops, by up to 20%.In addition, the partial widths for H → b ¯ b and H → τ + τ − are enhanced, further suppressing thebranching ratio for H → γγ . This overcompensates the increase of the H production cross section.On the other hand, R γγH can exceed unity for | µ | > µ = 2 TeV. Here both ˜ t and ˜ b are quite light, while m ˜ t (cid:39) β (cid:39)
6. The light ˜ t H → γγ by about 5%, but suppresses the partial width for H → gg by about 30%. This suppression of the total cross section for H production is over–compensated bythe greatly reduced partial widths for H → b ¯ b and, to a lesser extent, H → τ + τ − decays; the light˜ b significantly reduces the Hb ¯ b coupling vial SUSY QCD loop corrections in this case. The signalsin the V V ∗ channels ( V = W ± or Z ) are therefore also enhanced by almost a factor of two.Note finally that Fig. 2b shows more solutions with µ > µ <
0; moreover, the LEPlower bound on | µ | can only be saturated for positive µ . The reason for this asymmetry is that onlypositive values of the gluino mass parameter were considered. The relative sign (more generally,relative phase) between µ and the gluino mass parameter has physical meaning, just as the relativesigns (or phases) between µ and the soft breaking A − parameters are significant. Since only these relative signs are physical, the gluino mass parameter can be chosen to be positive without lack ofgenerality, so long as both signs for µ and the A − parameters are considered, as is done in the currentanalysis.Fig. 2c shows that the H → τ + τ − signal strength can be enhanced by more than a factor of threeonly if ˜ t is very light, m ˜ t ≤
200 GeV. In this region of parameter space light ˜ t loops increase the H production cross section by about a factor of three over its SM value. Since sin ( β − α ) saturatesits upper bound, the partial width for H → W W ∗ is reduced by about 15%, while the partial widthsinto the τ + τ − and b ¯ b final states are increased by factors of 4 and 2 .
4, respectively; the signals in the
V V channels are therefore slightly smaller than in the SM, in spite of the increased production crosssection. Note also that light ˜ b loops again play an important role in suppressing the Hb ¯ b coupling.If all squarks are heavier than a few hundred GeV, the H → τ + τ − signal can still be enhancedby up to a factor of three, essentially by enhancing the Hτ + τ − couplings since | cos α | > cos β ; since˜ b is heavy, the Hb ¯ b coupling will then be enhanced by a similar amount.Of perhaps greater interest, given current trends in the data, is that the H → τ + τ − signal canalso be considerably weaker than in the SM. This signal is weakest for the smallest allowed valueof tan β , and requires ˜ t and ˜ b to be relatively light; the former reduces the H production ratevia gluon fusion, whereas the latter partly compensates for the reduction of the Hb ¯ b coupling thatoriginates from the very small values of | cos α | that can be realized in this region of parameter space.The lower bound on the strength in the τ + τ − channel is then essentially set by the upper bound (2d)on the strength of the signal in the V V channels, which imposes an upper bound on the branchingratios for these channels. For larger squark masses the ratio of the Hτ + τ − and Hb ¯ b couplings isessentially fixed, independent of the parameters of the Higgs sector, leading to a slightly strongerlower bound on the H → τ + τ − signal strength. However, even this increased lower bound is stillbelow conceivable near–future sensitivities in this channel.Finally, Fig. 2d shows that the double ratio R γγH /R V VH can differ by more than 10% from unityonly if m ˜ t <
300 GeV. Note that the production cross section, i.e. the partial width for H → gg , aswell as the total width of H cancel out in this double ratio, which is simply given by the ratio of thecorresponding partial widths, Γ( H → γγ ) / Γ( H → W + W − ), normalized to the same ratio of partialwidths of the SM Higgs boson. Since the HW + W − coupling, which is proportional to cos( α − β ), isonly slightly reduced from its SM value, the biggest contribution to radiative H → γγ decays alwayscomes from W loops in this scenario. For m ˜ t >
300 GeV the only other significant contributioncomes from top loops, which always interfere destructively with the W loops here, just as in the SM.Due to this destructive interference the reduction of the HW + W − (and HZZ ) coupling implied bythe constraint (2c) reduces the H → γγ partial width slightly more than the H → V V ∗ ( V = W, Z )partial widths. However, this reduction of the double ratio by ∼
5% will likely remain unobservable11t the LHC. R H γγ R HVV a) R H ττ R h ττ b) R H γγ / R HVV R h ττ c) R H γγ / R HVV R H ττ d) Figure 3: Allowed region in the ( R γγH , R V VH ) (a), the ( R ττH , R ττh ) (b), the ( R , R ττh ) (c) and the ( R , R ττH )plane (d), after the constraints (2) as well as the various sparticle and Higgs search limits discussed inthe text have been imposed; here R = R γγH /R V VH . The solid (red) line in frame a) shows R γγH = R V VH .For smaller ˜ t mass the double ratio may differ by up to ∼
30% from unity. This is due to the effectof ˜ t and, to a lesser extent, ˜ b loops on Γ( H → γγ ); recall that equal soft breaking parameters havebeen used in the ˜ t and ˜ b sectors here. Depending on the sign of the H ˜ t ˜ t † coupling, this contributioncan interfere constructively or destructively with the dominant W loop contribution; this explainsthe bifurcation of the results for m ˜ t <
400 GeV. In either case the effect is maximized for large masssplitting between the ˜ t mass eigenstates and large | µ | , with maximal suppression (enhancement) ofthe double ratio requiring positive (negative) µ . A light ˜ τ can suppress the double ratio by anadditional 3% or so.The correlation between the two signal rates defining the double ratio is explored in the firstframe of Fig. 3, which shows correlations between various (ratios of) signal strengths. The red linecorresponds to R γγH = R V VH , leading to a unit value for the double ratio. We see that both signalstrengths can saturate their lower bounds defined in (2d,e), and that R V VH can also saturate its upper12ound. Evidently relaxing the bounds on R V VH would also lead to an increased allowed range for R γγH , beyond the range shown in (7). On the other hand, neither R γγH nor R V VH is strongly correlatedwith the ratio between these two quantities: the allowed range of the ratio of signal strengths movesonly slightly towards smaller values as R V VH increases, such that R γγH < ∼ . R V VH when R V VH saturatesits upper bound of 2; this is to be compared with the absolute upper bound of 1 . h and H signals in the di–tau channel are positively correlated. The reasonis that increasing tan β increases the basic τ Yukawa coupling in the Lagrangian, which therefore alsotends to increase the couplings of both h and H to τ leptons. However, for small value of R ττh thiscorrelation is not particularly strong: the H → τ + τ − signal can then be both significantly strongerand significantly weaker than in the SM. On the other hand, the h → τ + τ − signal strength can onlyexceed that of the SM significantly if the H → τ + τ − signal is also enhanced. Current data disfavoran enhanced signal in the di–tau channel for the new boson near 125 GeV; in the present context asignificant upper bound on this signal would make it even more difficult to detect h at the LHC.The correlation between the h → τ + τ − signal strength and the double ratio R γγH /R V VH is exploredin Fig. 3c. We see that the former can only be enhanced significantly beyond its SM value if thelatter is somewhat below unity. Recall from the discussion of Fig. 2a that maximizing R ττh requiressmall m ˜ t . In the case at hand this enhances the partial width for H → gg , but reduces the partialwidth for H → γγ , leading to a reduction of the ratio of signal strengths in the γγ and V V ∗ channelsrelative to their SM value. Again the current data favor this double ratio to be enhanced; Fig. 3cshows that this would reduce the upper bound on the h → τ + τ − signal strength in this scenario.Finally, Fig. 3d shows the correlation between the double ratio of H → γγ and H → V V ∗ signalstrengths and the H → τ + τ − signal. These quantities are clearly anti–correlated in the presentscheme. The H → τ + τ − signal is maximized in a similar region of parameter space as the h → τ + τ − signal; we just saw that this leads to a suppression of the double ratio.Conversely, the double ratio reaches its maximum when both the H → γγ and H → V V ∗ signalsare suppressed by destructive interference of top and stop loop contributions to H → gg ; stop loopcontributions then maximally enhance Γ( H → γγ ). The suppression of the H production crosssection also reduces the strength of the signal in the di–tau channel. One can also find configurationswith slightly less enhanced double ratio where both the H → γγ and the H → V V ∗ signals areenhanced over their SM values. This can be achieved if | cos α | < cos β , which suppresses the H → b ¯ b and H → τ + τ − partial widths, and thus the total decay width of H . This mechanism also leads toa suppression of the H → τ + τ − signal. Finally, the branch with R γγH /R V VH (cid:39) . , R ττH (cid:39) . µ , large and positive A t , and (as usual) large splitting between m ˜ t L and m ˜ t R . The light ˜ t loops then again suppress H → gg decays and enhance H → γγ decays, whereas | cos α | < cos β reduces the total width of H ; the former effect is dominant, i.e. the H → γγ and H → V V ∗ signals are both suppressed relative to their SM values. In this case the di–tau signalis further suppressed because the light ˜ b enhances, rather than suppresses, the ratio of Hb ¯ b and Hτ + τ − coupling. In this paper I have shown that one can explain both the recent discovery of a “Higgs–like particle”by the LHC experiments, and the 2 . σ excess of Higgs–like events found by the LEP collaborations13ome ten years ago, in the “phenomenological” MSSM, where all (relevant) weak–scale soft breakingparameters are treated as independent free parameters. In this interpretation the masses of thetwo CP–even Higgs bosons h and H are essentially fixed by the data, to ∼
98 and ∼
125 GeV,respectively. Radiative corrections to the Higgs sector are crucial for the viability of this scheme. Asa result, the masses of the remaining Higgs bosons, the CP–odd state A and the charged state H ± ,can still vary considerably. Nevertheless stringent upper bounds on the masses of these states canbe derived, which would be straightforward to test at an e + e − collider operating at √ s ≥
350 GeV.Much of the allowed parameter space can probably also be probed by searches for these states at theLHC; in particular, t → H + b decays are open over almost the entire parameter space. However, thisis not sufficient to guarantee that such decays, or other A or H ± production processes, can actuallybe detected at the LHC.The upper bound on m H ± implies that loops involving the charged Higgs boson and the t quarkwill give significant positive contributions to the partial width for radiative b → sγ decays [33]; ifthese were the only new contributions the predicted partial width would exceed the measured value[29], which is quite close to the SM prediction. However, it is well known that even within theMSSM with minimal flavor violation, chargino–stop loops can cancel the charged Higgs loops [34], soa portion of the parameter space (with rather light ˜ t ) is most likely allowed even in this constrainedscenario. Moreover, as argued in the Introduction, the general MSSM contains many additionalparameters that can be tuned to satisfy flavor constraints. In particular, a small amount of ˜ b − ˜ s mixing would lead to large gluino loop contributions to b → sγ [35] of either sign.The light state h is also very difficult to detect at the LHC. It has greatly reduced couplings to Z and W bosons, and hence also a greatly reduced branching ratio into γγ final states. Part of theallowed parameter space could perhaps be probed through h → τ + τ − decays, but the small value of m h implies that Z → τ + τ − decays will be a formidable background.Although the couplings of H to W and Z bosons are quite SM–like in this scenario, both theproduction cross section and the decay branching ratios of H can still differ significantly from thoseof the SM Higgs. On the one hand, the couplings of H to third generation fermions can be quitedifferent from those of the SM Higgs; here light ˜ b loop contributions to the Hb ¯ b coupling can play asignificant role. A good measurement of, or upper bound on, the strength of the di–tau signal wouldtherefore narrow down the allowed parameter space of this scenario. A reliable observation of the H → b ¯ b signal and/or t ¯ tH production would be similarly useful, but are experimentally (even) morechallenging. Moreover, light ˜ t loops can modify the partial widths for H → gg and, to a somewhatlesser extent, for H → γγ decays significantly. In particular, in this scenario one can simultaneouslyreduce the di–tau signal and enhance the di–photon signal, in agreement with the (statistically notvery compelling) trend of current data.However, the di–photon signal can be enhanced relative to the V V ∗ ( V = W ± , Z ) signals only if˜ t is rather light. On–going and future searches for light stop and sbottom squarks therefore have thepotential to further constrain the parameter space of this model. Unfortunately the interpretation ofsuch searches also depends on the chargino and neutralino sectors of the MSSM, which have not beenspecified here, since they hardly affect the Higgs sector. Note, however, that the LHC experimentsshould eventually be able to probe ˜ t masses well above 200 GeV even in the experimentally mostdifficult case where ˜ t is nearly degenerate with a stable neutralino [36]. If all squarks are heavy, H can be so SM–like that it probably cannot be distinguished from the Higgs boson of the SM. Inparticular, the squared couplings to W and Z are just 5 to 15% smaller than in the SM; this followsfrom the normalization of the excess observed at LEP.14t should be admitted that this scenario is theoretically not especially appealing. In particular,the LEP excess cannot be explained in a constrained version of the MSSM [37]. Moreover, the lowerbound on the sum of the stop masses, and hence the required finetuning associated with radiativecorrections from the top and stop sector, is similar to that in the more common MSSM interpretationof the LHC discovery in terms of production and decay of the light CP–even state h . It is neverthelessamusing to note that the MSSM can simultaneously explain two sets of observations of excesses ofHiggs–like events. Acknowledgments
This work was supported in part by the BMBF–Theorieverbund, and in part by the DFG TransregioTR33 “The Dark Universe”.
References [1] ATLAS Collab., G. Aad et al., Phys. Lett.
B716 (2012) 1, arXiv:1207.7214 [hep-ex]; CMSCollab., S. Chatrchyan et al., Phys. Lett.
B716 (2012) 30, arXiv:1207:7235 [hep-ex].[2] I.A. Oksuzian for the CDF and D0 Collaborations, arXiv:1209.1586 [hep-ex].[3] J. Baglio, A. Djouadi and R.M. Godbole, Phys. Lett.
B716 (2012) 203, arXiv:1207.1451 [hep-ph].[4] E. Witten, Nucl. Phys.
B188 , 513 (1981); N. Sakai, Z. Phys.
C11 , 153 (1981); S. Dimopoulosand H. Georgi, Nucl. Phys.
B193 , 150 (1981); R.K. Kaul and P. Majumdar, Nucl. Phys.
B199 ,36 (1982).[5] M. Drees, R.M. Godbole and P. Roy,
Theory and phenomenology of sparticles: An account offour-dimensional N=1 supersymmetry in high energy physics , World Scientific (2004).[6] L.J. Hall, D. Pinner and J.T. Ruderman, JHEP (2012) 131, arXiv:1112.2703 [hep-ph]; J.L.Feng, K.T. Matchev and D. Sanford, Phys. Rev.
D85 (2012) 075007, arXiv:1112.3021 [hep-ph];A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, Phys. Lett.
B708 (2012)162, arXiv:1112.3028 [hep-ph]; P. Draper, P. Meade, M. Reece and D. Shih, Phys. Rev.
D85 (2012) 095007, arXiv:1112.3068 [hep-ph]; M. Carena, S. Gori, N.R. Shah and C.E.M. Wagner,JHEP (2012) 014, arXiv:1112.3336 [hep-ph]; S. Akula, B. Altunkaynak, D. Feldman, P.Nath, and G. Peim, Phys. Rev.
D85 (2012) 075001, arXiv:1112.3645 [hep-ph]; J.-J. Cao, Z.-X.Heng, J.M. Yang, Y.-M. Zhang and J.-Y. Zhu, JHEP (2012) 086, arXiv:1202.5821 [hep-ph]; O. Buchm¨uller et al., arXiv:1207.7315 [hep-ph]; J. Cao, Z. Heng, J.M. Yang and J. Zhu,arXiv:1207.3698 [hep-ph].[7] S. Heinemeyer, O. Stal and G. Weiglein, Phys. Lett.
B710 (2012) 201, arXiv:1112.3026 [hep-ph];N.D. Christensen, T. Han and S. Su, Phys. Rev.
D85 (2012) 115018, arXiv:1203.3207 [hep-ph];A. Arbey, M. Battaglia, A. Djouadi and F. Mahmoudi, arXiv:1207.1348 [hep-ph]; R. Benbrik,M. Gomez Bock, S. Heinemeyer, O. Stal, G. Weiglein and L. Zeune, arXiv:1207.1096 [hep-ph].[8] S.S. AbdusSalam and D. Choudhury, arXiv:1210.3322 [hep-ph].159] H. Baer, V. Barger, P. Huang, A. Mustafayev and X. Tata, arXiv:1207.3343 [hep-ph].[10] A. Bottino, N. Fornengo and S. Scopel, Phys. Rev.
D85 (2012) 095013, arXiv:1112.5666 [hep-ph].[11] K. Hagiwara, J.S. Lee and J. Nakamura, arXiv:1207.0802 [hep-ph].[12] The ALEPH, DELPHI, L3 and OPAL Collab.s, Phys. Lett.
B565 , 61 (2003), hep–ex/0306033.[13] G. B´elanger, U. Ellwanger, J.F. Gunion, Y. Jiang, S. Kraml and J.H. Schwarz, arXiv:1210.1976[hep-ph].[14] ALEPH Collab., A. Heister et al., Phys. Lett.
B526 , 191 (2002), hep–ex/0201014.[15] M. Drees, Phys. Rev.
D71 (2005) 115006, hep-ph/0502075.[16] CMS Collab., S. Chatrchyan et al., Phys. Lett.
B713 (2012) 68, arXiv:1202.4083 [hep-ex].[17] ATLAS Collab., G. Aad et al., JHEP (2012) 039, arXiv:1204.2760 [hep-ex].[18] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. , 1 (1991), and Phys. Lett. B262 , 54 (1991); R. Barbieri, M. Frigeni and F. Caravaglio, Phys. Lett.
B258 , 167 (1991); H.E.Haber and R. Hempfling, Phys. Rev. Lett. , 1815 (1991); J. Ellis, G. Ridolfi and F. Zwirner,Phys. Lett. B257 , 83 (1991), and
B262 , 477 (1991).[19] M. Drees and M.M. Nojiri, Phys. Rev.
D45 , 2482 (1992).[20] S.Y. Choi, M. Drees and J.S. Lee, Phys. Lett.
B481 , 57 (2000), hep–ph/0002287.[21] H.E. Haber, R. Hempfling and A.H. Hoang, Z. Phys.
C75 , 539 (1997), hep–ph/9609331.[22] M. Carena, J.R. Espinosa, M. Quiros and C.E.M. Wagner, Phys. Lett.
B355 , 209 (1995),hep–ph/9504316; M. Carena, M. Quiros and C.E.M. Wagner, Nucl. Phys.
B461 , 407 (1996),hep–ph/9508343.[23] H. Eberl, K. Hidaka, S. Kraml, W. Majerotto and Y. Yamada, Phys. Rev.
D62 , 055006 (2000),hep–ph/9912463.[24] S. Heinemeyer, Int. J. Mod. Phys.
A21 (2006) 2659, hep–ph/0407244.[25] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Eur. Phys. J.
C28 , 133(2003), hep–ph/0212020.[26] CDF and D0 Collab.s, T. Aaltonen et al., arXiv:1207.1069 [hep-ex]; CMS Collab., S. Chatrchyanet al., arXiv:1209.2319 [hep-ex].[27] M. Carena, S. Gori, N.R. Shah, C.E.M. Wagner and L.-T. Wang, JHEP (2012) 175,arXiv:1205.5842 [hep-ph].[28] A. Djouadi, Phys. Rept. (2008) 1, hep-ph/0503172, and Phys. Rept. (2008) 1, hep-ph/0503173. 1629] J. Beringer et al. (Particle Data Group), Phys. Rev.
D86 , 010001 (2012).[30] J.M. Fr`ere, D.R.T. Jones and S. Raby, Nucl. Phys.
B222 , 11 (1983); M. Claudson, L. Hall andI. Hinchliffe, Nucl. Phys.
B228 , 501 (1983).[31] J. Hisano and S. Sugiyama, Phys. Lett.
B696 (2011) 92, arXiv:1011.0260 [hep-ph].[32] R. Barbieri and L. Maiani, Nucl. Phys.
B224 , 32 (1983); C.S. Lim, T. Inami and N. Sakai,Phys. Rev.
D29 , 1488 (1984); M. Drees and K. Hagiwara, Phys. Rev.
D42 , 1709 (1990).[33] M. Ciuchini, G. Degrassi, P. Gambino and G.F. Giudice, Nucl. Phys.
B527 (1998) 21, hep-ph/9710335; F. Borzumati and C. Greub, Phys. Rev.
D58 (1998) 074004, hep-ph/9802391, andPhys. Rev.
D59 (1999) 057501, hep-ph/9809438.[34] S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, Nucl. Phys.
B353 (1991) 591; R. Barbieriand G.F. Giudice, Phys. Lett.
B309 (1993) 86, hep-ph/9303270.[35] K.-i. Okumura and L. Roszkowski, Phys. Rev. Lett. (2004) 161801, hep-ph/0208101; J.Foster, K.-i. Okumura and L. Roszkowski, JHEP (2008) 109, arXiv:0808.2298 [hep-ph]; S.Bornhauser, M. Drees, S. Grab and J.S. Kim, Phys. Rev.
D83 (2011) 035008, arXiv:1011.5508[hep-ph]; M. Drees, M. Hanussek and J.S. Kim, Phys. Rev.
D86 (2012) 035024, arXiv:1201.5714[hep-ph].[37] M. Asano, S. Matsumoto, M. Senami and H. Sugiyama, Phys. Rev.