Implication of a Higgs boson at 125 GeV within the stochastic superspace framework

Manimala Chakraborti,1,* Utpal Chattopadhyay,1,† and Rohini M. Godbole2,‡

1Department of Theoretical Physics, Indian Association for the Cultivation of Science,2A & B Raja S. C. Mullick Road, Jadavpur, Kolkata 700 032, India

2Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India(Received 19 November 2012; published 19 February 2013)

We revisit the issue of considering stochasticity of Grassmannian coordinates in N ¼ 1 superspace,

which was analyzed previously by Kobakhidze et al. In this stochastic supersymmetry (SUSY) frame-

work, the soft SUSY breaking terms of the minimal supersymmetric Standard Model (MSSM) such as the

bilinear Higgs mixing, trilinear coupling, as well as the gaugino mass parameters are all proportional to a

single mass parameter , a measure of supersymmetry breaking arising out of stochasticity. While a

nonvanishing trilinear coupling at the high scale is a natural outcome of the framework, a favorable

signature for obtaining the lighter Higgs boson mass mh at 125 GeV, the model produces tachyonic

sleptons or staus turning to be too light. The previous analyses took , the scale at which input parameters

are given, to be larger than the gauge coupling unification scale MG in order to generate acceptable scalar

masses radiatively at the electroweak scale. Still, this was inadequate for obtaining mh at 125 GeV. We

find that Higgs at 125 GeV is highly achievable, provided we are ready to accommodate a nonvanishing

scalar mass soft SUSY breaking term similar to what is done in minimal anomaly mediated SUSY

breaking (AMSB) in contrast to a pure AMSB setup. Thus, the model can easily accommodate Higgs data,

LHC limits of squark masses, WMAP data for dark matter relic density, flavor physics constraints, and

XENON100 data. In contrast to the previous analyses, we consider ¼ MG, thus avoiding any

ambiguities of a post-grand unified theory physics. The idea of stochastic superspace can easily be

generalized to various scenarios beyond the MSSM.

DOI: 10.1103/PhysRevD.87.035022 PACS numbers: 12.60.Jv, 04.65.+e, 95.30.Cq, 95.35.+d

I. INTRODUCTION

Low energy supersymmetry (SUSY) [1–4] has beenone of the most promising candidates for a theory offundamental particles and interactions going beyond theStandard Model (SM); the so-called BSM physics. Theminimal extension of the SM including SUSY, namely,the minimal supersymmetric Standard Model (MSSM),extends the particle spectrum of the SM by one additionalHiggs doublet and the supersymmetric partners of all theSM particles—the sparticles. The supersymmetric exten-sion of the SM provides a particularly elegant solution tothe problem of stabilizing the electroweak (EW) symmetrybreaking scale against large radiative correction and keepsthe Higgs ‘‘naturally’’ light. In fact, a very robust upperlimit on the mass of the lightest Higgs boson is perhaps oneof the important predictions of this theory. Further, thisupper limit is linked in an essential way to the values ofsome of the SUSY breaking parameters in the theory. Inaddition, in R-parity conserving SUSY, the lightest super-symmetric particle (LSP) emerges as the natural candidatefor the dark matter (DM), the existence of which has beenproved beyond any doubt in astrophysical experiments.The search for evidence of the realization of this symmetry

in nature (in the context of high energy collider experi-ments, precision measurements at the high intensity Bfactories, and in the DM detection experiments) has there-fore received enormous attention of particle physicists,perhaps only next to the Higgs boson. The recent observa-tion of a boson with mass around 125 GeV at the LargeHadron Collider (LHC) [5] and the rather strong lowerlimits on the masses of sparticles that possess strong inter-actions that the LHC searches have yielded [6] necessitatescareful studies of the MSSM in the context of all the recentlow energy data. In these studies, it is also very importantto seek suitable guiding principles which could possiblyreduce the associated large number of SUSY breakingparameters of MSSM. Thus, looking for modes of specificSUSY breaking mechanisms that involve only a few inputquantities given at a relevant scale can be useful. Here, thesoft SUSY breaking parameters at the electroweak scaleare found via renormalization group (RG) analyses. Apartfrom the simplicity of having a few parameters as input,such schemes create challenging balancing acts. On theone hand, various soft breaking masses and couplingsbecome correlated with one another in such schemes. Onthe other hand, overall one has to accommodate a largenumber of very stringent low energy constraints in a com-prehensive model consisting of only a few parameters. Asimple and well-motivated example of a SUSY model isthe minimal supergravity (mSUGRA) [7]. Here, SUSY isbroken spontaneously in a hidden sector and the breaking

*tpmc@iacs.res.in†tpuc@iacs.res.in‡rohini@cts.iisc.ernet.in

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is communicated to the observable sector where MSSMresides via Planck mass suppressed supergravity interac-tions. The model involves soft SUSY breaking parameterslike (i) universal gaugino mass parameter m1

2, (ii) the uni-

versal scalar mass parameterm0, (iii) the universal trilinearcoupling A0, and (iv) the universal bilinear coupling B0, allgiven at the gauge coupling unification scale. In addition,one has the superpotential related Higgsino mixingparameter 0 with its associated sign parameter. The tworadiative electroweak symmetry breaking (REWSB) con-ditions may then be used so as to replace B0 and 0 withthe Z-boson massMZ and tan, the ratio of Higgs vacuumexpectation values. Similar to mSUGRA, one has otherSUSY breaking scenarios like the gauge mediated SUSYbreaking and models with anomaly mediated SUSY break-ing (AMSB), etc. [2,3]. Apart from direct collider physicsdata, one has to satisfy constraints from flavor changingneutral current (FCNC), as well as flavor conserving phe-nomena like the anomalous magnetic moment of muon,constraints like electric dipole moments associated withCP violations, or to check whether there is a proper amountof dark matter content in R-parity conserving scenariosor proper neutrino masses in R-parity violating scenarios[1–3]. A single model is yet to be found that can adequatelyexplain various stringent experimental results, and at thesame time possesses a sufficient degree of predictiveness.However, it is always important to continue the quest of asimple model and check the degree of agreement with lowenergy constraints.

In this work, we pursue a predictive theory of SUSYbreaking by considering a field theory on a superspacewhere the Grassmannian coordinates are essentially fluc-tuating/stochastic [8–10]. We note that in a given SUSYbreaking scenario, our limitation of knowing the actualmechanism of breaking SUSY is manifested in the softparameters. Here, in a stochastic superspace framework weassume that a manifestation of an unknown but fundamen-tal mechanism of SUSY breaking may effectively lead tostochasticity in the Grassmannian parameters of the super-space. With a suitably chosen probability distribution, thiscauses a given Kahler potential and a superpotential to leadto soft breaking terms that carry signatures of the stochas-ticity. As we will see, the SUSY breaking is parametrizedby which is nothing but 1=h i, where the symbol hirefers to averaging over the Grassmannian coordinates.The other scale that is involved is . Values of varioussoft parameters at this scale are the input parameters of thescheme. The values of the same at the electroweak scaleare then obtained from these input values by using therenormalization group evolution. Considering the super-potential of MSSM, the soft terms obtained are readilyrecognized as the ones supplied by the externally addedsoft SUSY breaking terms of constrained MSSM(CMSSM) [3], except that the model is unable to producea scalar mass soft term [8]. Reference [8] used and as

free parameters while analyzing the low energy signatureswithin MSSM. was chosen betweenMG toMP, the scaleof gauge coupling unification and the Planck mass scale,respectively. The model as given in Ref. [8] is called thestochastic supersymmetric model (SSM) and it is charac-terized by universal gaugino mass parameter m1

2, universal

trilinear soft SUSY breaking parameter A0, and universalbilinear soft SUSY breaking parameter B0, all beingrelated to , the parameter related to SUSY breaking. Wenote that with the bilinear soft SUSY parameter beinggiven, tan, becomes a derived quantity.However, as already mentioned, in spite of the fact that

the SSM generates soft SUSY breaking terms, it producesno scalar mass soft term. Scalar masses start from zero atthe scale and renormalization group evolution is used togenerate scalar masses at the electroweak scaleMZ. Scalarmasses at MZ severely constrain the model because typi-cally scalars are very light. In particular, quite often slep-tons turn to be the lightest supersymmetric particles oreven become tachyonic over a large part of the parameterspace. This is partially ameliorated when one takes thehigh scale to be larger than the gauge coupling unifica-tion scale MG 2 1016 GeV. However, in spite of ob-taining valid parameter space that would provide us with alightest neutralino as a possible dark matter candidate inR-parity preserving framework, we must note that the lowvalues that one obtains for the masses of the first twogeneration of squarks are hardly something of an advan-tage in view of the constraints coming from FCNC as wellas those from the LHC data [6]. On the other hand, SSMhas a natural advantage of being associated with a non-vanishing trilinear coupling that is favorable to produce arelatively light spectra for a given value of Higgs bosonmass mh. The recent announcement from the CMS andATLAS Collaborations of the LHC experiment about thediscovery of a Higgs-like boson at 125 GeV [5] thusmakes this model potentially attractive. However, as ex-plored in Ref. [10], SSM as such is unable to accommodatesuch a large mh in spite of having a built-in feature ofhaving a nonvanishing A0. It could at most reach 116 GeVformh [9,10] and the constraint due to BrðBs ! þÞ asused in Ref. [10] was much less stringent in comparison tothe same of present day [11]. Furthermore, it is alsoimportant to investigate the effect of the direct detectionrate of dark matter as constrained by the recentXENON100 data [12].In this analysis, we would like to give all the input

parameters at the grand unification scale MG, the scale atwhich the Standard Model gauge group, namely, SUð3Þ SUð2Þ Uð1Þ, comes into existence. Any evolution aboveMG would obviously demand choosing a suitable gaugegroup; a question that is not going to be addressed inthis work. In this way we would like to avoid unknownissues arising out of a post-grand unified theory (GUT)[13] physics. However, we would rather try to meet the

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phenomenological demand of confronting the issue ofsleptons becoming tachyonic or avoiding scalar massesto become light in general in a minimal modification byconsidering an externally given scalar mass soft parameterm0 as a manifestation of an additional origin of SUSYbreaking. It would be useful to have the first two gener-ations of scalar masses adequately heavy so as to overcomethe FCNC related constraints and LHC data [6] on squarkmasses. Additionally, this will also be consistent withhaving the lighter Higgs boson mass (mh) to be in thevicinity of 125 GeV. We will henceforth denote the modelas Mod-SSM.

We may note that traditionally minimal versions ofmodels of SUSY breaking have been extended for phe-nomenological reasons. It is also true that extending aminimal model often lowers predictiveness and may evencause partial dilution of the main motivations associatedwith the building of the model. For example, consideringnonuniversal gaugino or scalar mass scenarios may bemore suitable than CMSSM or mSUGRA so as to obtaina relatively lighter spectra in the context feasibility ofexploring via LHC. Another example may be given inthe context of the minimal AMSB model. As we know, apure AMSB scenario [14] is associated with form invari-ance of the renormalization group equations (RGE) ofscalar masses and absence of flavor violation. However,it produces tachyonic sleptons. In the minimal AMSBmodel [3,15], one introduces an additional common massparameter m0 for all the scalars of the theory. This ameli-orates the tachyonic slepton problem but it is true that wesacrifice the much cherished feature of form invariance andaccept some degree of flavor violations at the end. Wewould like to explore a nonminimal scenario of stochasticsupersymmetry model, namely, Mod-SSM in this spirit.We particularly keep in mind that the stochastic supersym-metry formalism may be used not only within the MSSMframework but it may be extended to superpotentialsbeyond that of the MSSM [9]. The fact that the modelwith a minimal modification can easily accommodate therecent Higgs boson mass range by its generic feature ofhaving a nonvanishing trilinear coupling parameter makesit further attractive. We believe that our approach of con-sidering an additional SUSY breaking scalar mass term isjustified for phenomenological reasons.

Thus, in Mod-SSM we consider the input of soft termparametersm1

2, B0, and A0 (all are either proportional to jj

or ) and a universal scalar mass soft parameter m0, allbeing given at a suitable scale which we simply choose asthe gauge coupling unification scale MG considering theLEP data on gauge couplings could be a hint of theexistence of grand unification. As in Ref. [8], we wouldalso restrict to be real, either positive or negative.

We clearly like to emphasize that the SSM frameworkproduces no scalar mass soft term at a scale . With setto the gauge coupling unification scale MG, SSM is

plagued with tachyonic sleptons. With >MG, one canavoid tachyonic scalar, but there is no scope of obtainingthe currently accepted Higgs boson mass. In Mod-SSM, weconsider ¼ MG and add the extra scalar mass term for anadditional SUSY breaking effect. With the trilinear cou-pling parameter A0 and the bilinear coupling parameter B0

becoming correlated with m12, Mod-SSM parameter space

is essentially a subset of the one for the CMSSM. In Mod-SSM, the SSM inspired nonvanishing A0 parameter issuitable for producing an appropriately large loop correc-tion to the lighter Higgs boson mass while keeping theoverall sparticle spectra relatively at a lower range.CMSSM, on the other hand, does not pinpoint with anyfundamental physical distinction/motivation for such azone of parameter space that is capable of producing thecorrect Higgs mass with similarly smaller sparticle massscale. Even if we consider the stochastic superspace modelin its original form as only a toy idea, we believe that itmay be worthwhile to explore it with minimal modifica-tions in regard to the current Higgs boson mass, as well asother phenomenological constraints.

II. STOCHASTIC GRASSMANNIANCOORDINATES AND SUSY BREAKING

As seen in Ref. [8], we consider an N ¼ 1 superspacewhere the Grassmannian coordinates and are taken tobe stochastic in nature. One starts with identifying theterms involving superfields in the superpotential andthe kinetic energy terms that could be used to constructthe SUSY invariant Lagrangian density for a given model.Each term is then multiplied with a probability distributionfunction P ð; Þ and integrated over the Grassmanniancoordinates appropriately. P ð; Þ can be expanded intoterms involving and which obviously has a finitenumber of terms because of the Grassmannian nature of and . One then imposes the normalization conditionRd2d2 P ð; Þ ¼ 1 and vanishing of Lorentz nonscalar

moments like hi, h i, h i, h2 i, and h 2i. The stochas-ticity parameter is defined as hi ¼ 1=. Here, is acomplex parameter with mass dimension unity. As com-puted in Ref. [8], and as worked out in this analysisexplicitly in the Appendix, the above leads to the followingHermitian probability distribution:

P ð; Þjj2 ¼ ~P ð; Þ¼ 1þ ðÞ þ ð Þ þ jj2ðÞð Þ:

(1)

For the simple case of a Wess-Zumino type of scenario [3]where the kinetic term is obtained from y and thesuperpotential is given as W ¼ 1

2m2 þ 13h

3, where

is a chiral superfield, one finds that the effect of stochas-ticity as described above leads to the following SUSYbreaking term:

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Lsoft ¼ 1

2m2 þ 2

3h3 þ H:c: (2)

Applying the stochasticity idea to the superpotential ofMSSM, along with considering the effect on the gaugekinetic energy function, the above formalism leads to thefollowing tree level soft SUSY breaking parameters to begiven at the high scale :

(a) universal gaugino mass parameter m12¼ 1

2 jj,(b) universal trilinear soft parameter A0 ¼ 2,(c) universal bilinear Higgs soft parameter B0 ¼ .

Recall that there is no scalar mass soft SUSY breakingterm in SSM. For convenience, we take to be a realpositive number with an additional input signðÞ. If wecount the universal gaugino mass parameter m1

2as the

independent parameter, we have

A0 ¼ signðÞ:4m12; B0 ¼ signðÞ:2m1

2: (3)

As has already been discussed before, we introduce anonvanishing scalar mass parameter m0 and fix atMG.

1 Thus with the above extension, the input quantitiesfor the stochastic SUSY model are

m12; m0; signðÞ; and signðÞ:

We note that the model quite naturally is associated withnonvanishing trilinear soft breaking terms. As we will see,this is quite interesting in view of the recent LHC an-nouncement for the Higgs mass range centering around125 GeV [5]. In this analysis, we will discuss only the caseof < 0 because the other sign of does not produce aspectra compatible with the dark matter relic densityconstraint.

The requirement of the REWSB then results in thefollowing relations at the electroweak scale:

2 ¼ 1

2M2

Z þm2

HDm2

HUtan 2

tan 2 1þ 1 2tan

2

tan 2 1;

(4)

and

sin 2 ¼ 2B=ðm2HD

þm2HU

þ 22 þ 1 þ2Þ; (5)

where i denote the one-loop corrections [16,17]. Here, Brefers to the value of bilinear Higgs coupling at the elec-troweak scale which has to be consistent with its givenvalue B0 at MG. B0 is determined via m1

2apart from a sign

of the stochasticity parameter as mentioned before.Consequently, tan is a derived quantity in the model.B0 at the scale MG and B at the electroweak scale are

connected via the following RGE written here at the one-loop level:

dB

dt¼

3~2 ~m2 þ 3

5~1 ~m1

þ ð3YtAt þ 3YbAb þ YAÞ;

(6)

where t ¼ lnðM2G=Q

2Þ with Q being the renormalization

scale. ~i ¼ i=ð4Þ for i ¼ 1, 2, 3 refer to scaled gaugecoupling constants (with 1 ¼ 5

3Y) and ~mi for i ¼ 1, 2, 3

are the running gaugino masses. Yi are the squared Yukawacouplings, e.g., Yt y2t =ð4Þ2 where yt is the top Yukawacoupling. In this analysis, the value of tan is determinedvia Eqs. (4)–(6) along with B0 ¼ signðÞ2m1

2at the scale

MG. We use SuSpect [18] for solving the RGEs andobtaining the spectra. The code takes tan as an inputquantity. Hence, we implement a self-consistent method ofsolution that starts from a guess value of tan resultinginto a BðMGÞ that in general would not agree with the inputof B0. Use of a Newton-Raphson root finding schemeensures a fast convergence toward the correct value oftan when BðMGÞ matches with the input of B0. Herewe stress that we do not encounter any parameter pointwith multiple values of tan in our analysis.2

III. RESULTS

The fact that the model has tan as a derived quantitynecessitates studying the behavior of the evolution of thebilinear Higgs parameter B. Figure 1 shows the evolutionof a few relevant couplings for a specimen input of

100

104

108

1012

1016

Scale Q (GeV)

-3000

-2000

-1000

0

1000

2000

Cou

plin

gs (

GeV

)

B

At

Ab

Ατ

µξ < 0m

1/2=600 GeV, m

0=2 TeV

m1

m2

FIG. 1 (color online). Evolution of a few relevant couplings fora specimen input of m1=2 ¼ 600 GeV, m0 ¼ 2 TeV, and > 0.

With < 0, one has B0 ¼ 2m1=2 and A0 ¼ 4m1=2. For a

valid parameter point within the model with > 0, we requireB ¼ BðMZÞ> 0 a necessity in order to have a positive sin 2from Eq. (5).

1We note that a vanishing scalar mass parameter at a post-GUTscale with RG evolution corresponding to an appropriate gaugegroup would indeed generate nonvanishing scalar mass terms atthe unification scale MG [13].

2See Ref. [19] for such a general possibility in REWSB whereB0 is given as an input.

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m1=2 ¼ 600 GeV, m0 ¼ 2 TeV, and > 0 in Mod-SSM.

For < 0, both B0 as well as A0 are negative, namely,B0 ¼ 2m1=2 and A0 ¼ 4m1=2. For a valid parameter

point within the model with > 0, we require BðMZÞ>0a necessity in order to have a positive sin 2 from Eq. (5).3

We note that the denominator in the right-hand side ofEq. (5) is the square of pseudoscalar Higgs mass whichneeds to be positive. The fact that B is originally negativeat MG and has to change to a positive value at MZ puts astrong constraint on the parameter space of the model.Numerically, this results in tan assuming large values.In regard to the evolution of A parameters, we defer ourdiscussion until Fig. 3.

Figure 2 shows a scatter plot of parameter points in thetanm1

2plane that satisfy the REWSB constraints of

Eqs. (4) and (5). As m0 is varied up to 7 TeV and m12up to

2 TeV, tan is seen to have a range of 32 to 48. The spreadof tan for a given m1

2arises from variation of m0. For

smaller values of m12, there is a larger dependence of m0 on

tan. Hence there is a larger spread of tan when m0 isvaried. For largerm1

2, the valid solution of tan has a lesser

dependence on m0. Hence, for larger m12values the spread

in values of tan decreases. The white region embeddedwithin the blue-green area corresponds to parameter points

with no valid solution satisfying REWSB. Typically, thetwo REWSB conditions are nonlinear in nature with re-spect to m1

2and m0. The code unsuccessfully tries with a

large number of iterations to find consistent 2 and m2A

solutions for parameter points within the white region (forall m0Þ. Thus, a lack of a valid tan for any value of m0

results in the above white region. It is worth mentioningthat the range of valid tan is much larger for the case of > 0 where B stays positive throughout the range fromMG to MZ. This is unlike the case of < 0 under discus-sion, where B is negative at MG and necessarily has tobecome positive at the electroweak scale, thus addingstringency to tan in its range. However, as already men-tioned, we will not discuss the case of > 0 further be-cause of the resulting overabundance of dark matter for theentire parameter space for this sign of . Wewill study nowthe effect of low energy constraints particularly in thecontext of the recent discovery of the Higgs-like boson[5]. Figure 3 shows the result in them1

2m0 plane for <0.

A sufficiently nonvanishing At (see, for example, Fig. 1)helps in producing a large loop correction to the lighterCP-even Higgs boson h. This can be understood by

500 1000 1500 2000m

1/2 (GeV)

0

1000

2000

3000

4000

5000

6000

7000

m0 (

GeV

)

Bs --> µ+µ-

mh = 122 GeV

mh = 128 GeV

III

III

b --> s+γ

ξ < 0

FIG. 3 (color online). Constraints shown in the m12m0 plane

for > 0 and < 0 for Mod-SSM. A0 and B0 in the modelsatisfy A0 ¼ signðÞ:4m1

2and B0 ¼ signðÞ:2m1

2. tan becomes

a derived quantity that varies between 32 and 48. The Higgsboson limits are shown as two solid blue lines. Brðb ! sÞ limitis shown as a maroon dot-dashed line. The lower part corre-sponds to discarded region via Eq. (9) where the branching ratiogoes below the lower limit of the constraint. BrðBs ! þÞlimit is shown as a brown solid line of which the lower regionexceeds the upper limit of Eq. (10). The top green region (I)corresponds to discarded zone via REWSB. The bottom blue-green region (III) refers to the zone where stau becomes LSP ortachyonic. The gray region (II) has discontinuous patches ofvalid parameter zones, the details of which are mentioned in thetext. Red points/areas falling in region II satisfy the WMAP-7data only for the upper limit of Eq. (12). Typically, the red pointsbordering region III and some part of the extreme left red points(for very small m1

2) satisfy both the upper and the lower limits

of Eq. (12).

200 400 600 800 1000 1200 1400 1600 1800 2000m

1/2 (GeV)

5

10

15

20

25

30

35

40

45

50

55

60ta

nβξ < 0

FIG. 2 (color online). Scatter plot of parameter points in them1

2 tan plane whenm0 andm1

2are scanned up to 7 and 2 TeV,

respectively. Here we only consider the validity of REWSBconstraints of Eqs. (4) and (5). The values of tan that satisfythe REWSB constraint vary from 32 to 48. The white regioninside the shaded (blue-green) region corresponds to invalidparameter points where no consistent solution satisfyingREWSB could be found even after trying with a large numberof iterations. The spread of tan for a given m1

2arises from

variation of m0. This spread decreases as m12becomes larger

because of decreased sensitivity onm0 while satisfying REWSB.

3tan which is the ratio of two vacuum expectation values ispositive. Hence, sin 2 ¼ 2 tan

ð1þtan 2Þ is also positive.

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looking at the expression for the dominant part of loopcorrection to the Higgs boson mass coming from the top-stop sector [20–22]

m2h ¼

3 mt4

22v2sin 2

log

M2S

mt2þ X2

t

2M2S

1 X2

t

6M2S

: (7)

Here, MS ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffim~t1m~t2

p, Xt ¼ At cot, v ¼ 246 GeV,

and mt is the running top-quark mass that also takes intoaccount QCD and electroweak corrections. The loop cor-

rection is maximized if Xt ¼ffiffiffi6

pMS. Clearly, a nonvanish-

ing A0 can be useful to increasem2h so thatmh reaches the

LHC specified zone without a need to push up the averagesparticle mass scale by a large amount. As particularlymentioned in Ref. [8], the SSM is additionally attractivein this context since it naturally possesses a nonvanishingand large jA0j. We further note that in this model, thelighter Higgs boson h has couplings similar to those inthe Standard Model because the CP-odd Higgs boson mass(mA) is in the decoupling zone [23]. The ATLAS andCMS results for the possible Higgs boson masses are126:0 0:4ðstatÞ 0:4ðsystÞ GeV and 125:30:4ðstatÞ0:5ðsystÞGeV, respectively [5]. In regard to MSSM lightHiggs boson mass, we note that there is about a 3 GeVuncertainty arising out of uncertainties in the top-quarkmass, renormalization scheme, as well as scale dependenceand uncertainties in higher order loop corrections up tothree loop [24–28]. Hence, in this analysis we consider thefollowing limits for mh:

122 GeV<mh < 128 GeV: (8)

We will outline the other relevant limits used in thisanalysis. A SUSY model parameter space finds a strongconstraint from Brðb ! sÞ. In the SM, the principal con-tribution that almost saturates the experimental valuecomes from the loop comprised of top quark and W boson[29]. In the MSSM, principal contributions arise fromloops containing top quark and charged Higgs bosons,and the same containing top squarks and charginos [30].The chargino loop contributions are proportional to At,and this may cause cancellations or enhancements betweenthe principal terms of the MSSM contribution dependingon the sign of At. Similar to mSUGRA, both the SSM andMod-SSM with < 0 also typically have At < 0. With> 0, this primarily means cancellation between thechargino and the charged Higgs contributions. This leadsto a valid region for Brðb ! sÞ for a larger sparticle massscale compared to the case of < 0. Brðb ! sÞ con-straint thus favors the positive sign of by allowing largerareas of parameter space. We consider the experimentalvalue Brðb ! sÞ ¼ ð355 24 9Þ 106 [31]. This re-sults in the following 3 level zone as used in this analysis:

2:78 104 < Brðb ! sÞ< 4:32 104: (9)

The above constraint is displayed as a maroon dot-dashedline. The left region of this line would be a discarded zone.

Next, the fact that the stochastic model with < 0 selectsappreciably large values for tan necessitates checking theBs ! þ limit. This is required because Bs ! þincreases with tan as tan 6 and decreases with increasein mA, the mass of pseudoscalar Higgs boson as m4

A [32].We use the recent experimental limit from LHCb [11]:BrðBs!þÞexp¼ð3:2þ1:4

1:2ðstat:Þþ0:50:3ðsyst:ÞÞ109. This

is contrasted with the SM evaluation BrðBs!þÞSM¼ð3:230:27Þ109 [33]. As in Ref. [34], combining theerrors of the LHCb data along with that of the SM result,one finds the following:

0:67 109 < BrðBs ! þÞ< 6:22 109: (10)

The upper limit of BrðBs ! þÞ is shown as a brownsolid line going across Fig. 3. Parameter values in theregion below this curve lead to values of BrðBs!þÞhigher than the above limit.We also compute BrðB ! Þ in this analysis. The

SUSY contribution to BrðB ! Þ is typically effectivefor large tan and small charged Higgs boson mass scenar-ios [35]. The experimental data from BABAR [36] readsBrðBþ ! þÞ ¼ ð1:83þ0:53

0:49ðstat:Þ 0:24ðsyst:ÞÞ 104.

The recent result from Belle [37] for B ! that usedthe hadronic taggingmethod is given byBrðB ! Þ ¼ð0:72þ0:27

0:25ðstat:Þ 0:11ðsyst:ÞÞ 104. The same branch-

ing ratio from Belle extracted by using a semileptonictagging method is ð1:54þ0:38

0:37ðstat:Þþ0:290:31ðsyst:ÞÞ 104

[38]. We use Ref. [39] for the result of averaging all of therecent Belle and BABAR data which is BrðB ! Þexp ¼ð1:16 0:22Þ 104. The SM result strongly depends onthe CKM element jVubj and the B-meson decay constant.We use BrðB ! ÞSM ¼ ð0:97 0:22Þ 104 [39].Using the above theoretical and experimental errors appro-priately, we obtain the following:

RðB!Þ ¼BrðB ! ÞSUSYBrðB ! ÞSM ¼ 1:21 0:30: (11)

This translates into 0:31<RðB!Þ < 2:10 at 3. Here,

BrðB ! ÞSUSY denotes the branching ratio in a SUSYframework, of course including the SM contribution. Ingeneral, we find that the model parameter space of Mod-SSM is not constrained by B ! since charged Higgsbosons are sufficiently heavy.We have not, however, included the constraint from

muon g 2 in this analysis, considering the tension arisingout of large deviation from the SM value, uncertainty inhadronic contribution evaluations, and accommodatingSUSY models in view of the LHC sparticle mass lowerlimits [40].We now explore the cosmological constraint for neutra-

lino dark matter relic density [41]. At 3, the WMAP-7data [42] are considered as shown below

0:094<~01h2 < 0:128: (12)

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The conclusions in regard to the relic density constraint isadditionally found to be sensitive on the top-quark mass inthis model. We divide the dark matter analysis into twoparts depending on (a) the top-quark pole mass set at173.3 GeV, and (b) using a spread of top-quark pole masswithin its range mt ¼ 173:3 2:8 GeV following the re-sult of the recent analysis performed in Ref. [43]. In thiscontext, we note that the experimental value as measuredby the CDF and D0 Collaborations of Tevatron is m

expt ¼

173:2 0:9 GeV [44].4

A. Analysis with mt ¼ 173:3 GeV:Underabundant LSP

The lightest neutralino, the LSP of the model, is typi-cally highly bino dominated except in a few regions wherethe Higgsino mixing parameter turns out to be small.The parameter points within all of the white region in Fig. 3have bino-dominated LSP. At this point we note that theimplementation of the REWSB conditions as manifest inEqs. (4) and (5) has to be done by keeping in mind(i) positivity of sin 2, (ii) positivity of 2, and (iii) therequirement of B0, A0 being related to m1

2as given by

Eq. (3), as well as requiring that the lighter charginomass lower limit is respected. All these requirementslead the top green shaded region (labeled as I) to be adiscarded zone. On the other hand, the gray shadedregion (II) has discontinuous zones of valid parameterpoints shown in red. The red points have considerablysmall values of , thus giving the LSP a large degree ofHiggsino mixing. Apart from the red points, there are nosolutions in the gray areas of region II. The need to satisfythe point (i) as explained above along with the requirementto satisfy the condition (iii) (implemented via a Newton-Raphson method of finding the correct tan) stringentlynegates the existence of solution zones within region II. Asa result, either there are solutions with appreciably smallor no solution at all within this region. This on the otherhand leads to a large amount of ~0

1 ~1 coannihilation.

Such degrees of coannihilations indeed cause the LSP to beonly a subdominant component of DM. Thus, in this part ofthe analysis, we consider the possibility of an underabun-dant dark matter candidate and ignore the lower limit of theWMAP-7 data. Typically we see that the relic density fallsbelow the lower limit of Eq. (12) in region II by an order ofmagnitude. Such underabundant LSP scenarios have beendiscussed in several works [45].

The lower shaded region III is disallowed as the mass ofthe stau (~1) turns negative or it is the LSP. Typically thered strip near region III refers to the LSP-stau coannihila-tion5 zone where the relic density can be consistent withboth the upper and lower limits of Eq. (12). Quite naturally,

the coannihilation may be stronger, and this would addi-tionally produce some underabundant DM points for thiszone. Finally, the leftmost red region (with very small m1

2)

satisfying WMAP-7 data is discarded by all other con-straints. We note that only a small region satisfying theHiggs mass bound is discarded via Brðb ! sÞ. On theother hand, the Higgs mass bound line of 122 GeV super-sedes the constraint imposed by the recent data on inclu-sive search for SUSY by the ATLAS experiment [6].The most potent constraint to eliminate a large region ofparameter space with m0 up to 1.5 TeV or so is due toBrðBs ! þÞ data Eq. (10). The constraint is effectivesimply because of the large values of tan involved in themodel. The discarded part of parameter space via the aboveconstraint includes a large zone that satisfies the darkmatter limit via LSP-stau coannihilation.We will now describe the spin-independent direct detec-

tion scattering cross section for scattering of LSP withproton. The scalar cross section depends on t-channelHiggs exchange diagrams and s-channel squark diagrams.Unless the squark masses are close to that of the LSP, theHiggs exchange diagrams dominate [47]. We note that forthe cases of parameter points with ~h

2 < ðCDMh2Þmin ,

where ðCDMh2Þmin refers to the lower limit of Eq. (12),

one must appropriately include the fraction of local DMdensity contributed by the specific candidate of DM underdiscussion while evaluating the event rate.This translates into multiplying SI

p~01

for such under-

abundant scenarios by = 0. Here, is the actual DM

density contributed by the specific DM candidate contrib-uting to 0 where the latter is the local dark matter density.We thus use ¼ 0 where ¼ ~h

2=ðCDMh2Þmin .

On the other hand, is simply 1 for abundant or over-abundant dark matter cases. Thus, we conveniently define ¼ min f1;~h

2=ðCDMh2Þmin g [48]. Figure 4 shows the

rescaled cross section as computed via micrOMEGAsversion 2.4 [49]. We wish to emphasize that while someregion of parameter space where the LSP typically has alarge degree of Higgsino component is eliminated viaXENON100 data, as announced in the summer of 2012[12], a large section of parameter space remains to beexplored via future direct detection of DM experiments.This consists of both types of coannihilation zones;namely, the chargino as well as the stau coannihilationzones. We must also keep in mind the issue of theoreticaluncertainties, particularly the hadronic uncertainties inevaluating SI

p~01

. The strangeness content of nucleon finds

a large reduction in the evaluation of relevant couplingsvia lattice calculations [50]. This is not incorporated inour computation while using micrOMEGAs to calculatethe cross section. Thus, the above itself will cause areduction of SI

p~01

by almost an order of magnitude.

There is also an appreciable amount of uncertainty ofthe local dark matter density [51]. All these points need

4For bottom quark mass we have usedmbMSðmbÞ ¼ 4:19 GeV.

5See, for example, works in Ref. [46] for various annihilationprocesses in relation to SUSY parameter space in general.

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to be kept in mind while evaluating the implications ofFig. 4 for SUSY models.

B. Analysis with mt ¼ 173:3 2:8 GeV:LSP of right abundance

It is to be noted that in the gray area (shown as region II)of Fig. 3, the existence of a valid solution depends verycritically on the parameters of the model. It was found thateither (i) we obtain a very small (barely satisfying thelighter chargino mass lower limit) in the gray area thatwould only provide us with extreme coannihilation be-tween lighter chargino and LSP leading to underabundanceof DM, or (ii) we find no valid solution at all. The sensi-tivity arises from the stringency of satisfying REWSB onthe parameter space for this region. In other words, for agiven m1

2, a small change in assumed m0 for a valid

parameter point would produce a small change in tanin the white region producing most probably another validparameter point. However, the change may not be allowedvia REWSB, particularly Eq. (5) if the parameter point isconsidered in the gray region. Equation (5) means sin 2needs to be a positive quantity less than unity. This typi-cally becomes a severe constraint even if the condition ofsatisfying the lighter chargino mass lower limit is met.With the top-quark mass having a strong influence onREWSB, it may be useful to investigate whether varyingmt may extract newer valid points within the gray regionthat would have a suitable so as to satisfy a well-tempered [52] LSP situation. This will then have the rightabundance of dark matter satisfying both the lower andupper limits of DM in Eq. (12).

The present experimental data on top-quark mass readm

expt ¼ 173:2 0:9 GeV [44]. Recently, Ref. [43] pre-

dicted the pole mass of top quark to be mpolet ¼ 173:3

2:8 GeV. The analysis used the next-to-next-to-leading

order (NNLO) in the QCD prediction of the inclusivepp ! ttþ X cross section and the Tevatron and LHCdata of the the same cross section. The comparisonbetween the experimental and theoretical results helpedextract the top-quark mass in the modified minimal

subtraction (MS) scheme. This was then used to compute

the pole mass mpolet . We now extend our analysis by inves-

tigating the effect of varying top-quark pole mass within

the above range (mpolet ¼ 173:3 2:8 GeV) on the solu-

tion space of the model. Indeed, we will see that even witha variation of 0.9 GeV, the range of experimental errorwould be enough to have a substantial effect on theconclusions.Figure 5 shows the scattered points that satisfy the

WMAP-7 given range of relic density for the aforesaid

variation ofmt mpolet . Here,m1

2andm0 are varied up to 2

and 7 TeV, respectively. We see from this figure that thecentral value of mtð¼ 173:3Þ GeV as considered in Fig. 3is indeed the one which has the least amount of possibilityto satisfy the WMAP-7 data. The occurrence of pointswhich satisfies WMAP-7 data is particularly rare aroundthis value of mt, near the lower part of the limit of relicdensity. We already found that the gray region (region II) ofFig. 3 is a sensitive zone because of REWSB, where canbe quite small. For such small values of , one can onlyexpect a large degree of ~0

1 ~1 coannihilation which

results into very small relic density. The latter goes belowthe lower limit of Eq. (12). Hence no red point exists nearthe bottom blue line of Fig. 5 for this value of mt. On thecontrary, a value of mt less than 1 GeV from the central

0 200 400 600 800 1000mχ1

0 (GeV)

10-12

10-11

10-10

10-9

10-8

10-7

ζσSI

(p-χ

) (pb)

XENON100

FIG. 4 (color online). Scaled spin-independent ~01 p scatter-

ing cross section vs LSP mass. The scaling factor is given as ¼ ~h

2=ðCDMh2Þmin , where ðCDMh

2Þmin refers to the

lower limit of Eq. (12).

170 171 172 173 174 175 176mt (GeV)

0.095

0.1

0.105

0.11

0.115

0.12

0.125

0.13

Ωχh2

ξ<0

(m1/2 < 2 TeV, m0 < 7 TeV)

FIG. 5 (color online). Relic density satisfied points shown inred that fall within the lower and upper limits of WMAP-7 datafor the neutralino relic density, when mt is varied by 2.8 GeVoneither side of 173.3 GeV. Herem1

2andm0 are scanned up to 2 and

7 TeV, respectively, for > 0 and < 0. The two blue lines arethe WMAP-7 limits of Eq. (12). The value of mt ¼ 173:3 GeVas used in Fig. 3 visibly falls in a disfavored zone in the contextof obtaining the correct relic density, particularly toward thelower limit of Eq. (12).

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value, which is only within the experimental error of mexpt ,

would cause to have a LSP with correct abundance for DM.The most favored zone formt, however, would be from 171to 172 GeV for satisfying the relic density limits. In fact, areduced degree of sensitivity for satisfying REWSB is thereason for obtaining a well-tempered LSP while consider-ing a top-quark mass little away from the central value.

Figure 6 shows the effect of scanning the top-quark masson the m1

2m0 plane. Region I shown in green is a dis-

allowed area of parameter space via REWSB similar toFig. 3, except that it is now an invalid region for all valuesof mt within its limit. Thus, this region is smaller inextension than the corresponding region of Fig. 3. Theregion II is disallowed because of stau turning tachyonicor the least massive. The constraints from Brðb ! sÞ andBrðBs ! þÞ are as shown. The lines denote theboundary of purely discarded zones irrespective of varia-tion of mt within its range. The blue line for mh ¼124 GeV means that mh < 124 GeV for all the regionleft of the line irrespective of the value of mt within therange. The ATLAS specified limit [6] for squarks also fallswell within this left zone. The neutralino relic densitysatisfied areas are below region I and above region II.The red points satisfy both the limits of the WMAP-7data, and thus correspond to having the right degree ofabundance of DM. On the other hand, we have also shownblue-green points that only satisfy the upper limit of theWMAP-7 data. In this part of the analysis, we consider the

LSP to have the correct abundance so as to be a uniquecandidate for DM.Finally, we show the effect of varying mt on the

spin-independent LSP-proton scattering cross section inFig. 7. Only the WMAP-7 satisfied points are shown (inred). Considering an order of magnitude of uncertainty(reduction), primarily because of the issue of strangenesscontent of nucleon as well as astrophysical uncertainties asmentioned before, we believe that the recent XENON100data still can accommodate Mod-SSM even while consid-ering the LSP as a unique candidate of DM.Table I shows three benchmark points of the model. The

top-quark massmt is as shown for each of the cases. PointsA and B correspond to values of mt which are entirelywithin the experimental error. The points A and C corre-spond to the upper region of Fig. 6, and these two pointscorrespond to the hyperbolic branch (HB)/focus point (FP)zone [53,54]. These points are associated with a largedegree of ~0

1 ~1 coannihilation. The degree of agree-

ment between the desired value of B0 and the one obtainedvia the Newton-Raphson iteration may be seen from thefifth and sixth rows. We allowed a maximum deviation of1% in the iterative procedure while scanning the parameterspace. A majority of the points are consistent within 0.1%in this regard. The charginos and the neutralinos are rela-tively lighter for points A and C in comparison to those inpoint B. The scalar particles, on the other hand, are rela-tively lighter for point B. Undoubtedly, like many SUSYmodels analyzed or reanalyzed after the Higgs boson dis-covery, the spectra is on the heavier side. The fact that A0 isnonvanishing and adequately large helps in reducing theaverage sparticle mass to a great extent compared to van-ishing A0 scenarios satisfying the current lighter Higgsboson limit. The mh for point C, however, goes belowthe assumed limit of Eq. (8). However, we still believe

500 1000 1500 2000m

1/2 (GeV)

0

1000

2000

3000

4000

5000

6000

7000m

0 (G

eV)

I

II

mtop

scanned

(ξ<0)

mh=124 GeV

b-->sγ Bs-->µ+µ−

A

B

C

FIG. 6 (color online). Scattered points in the m12m0 plane

for > 0 and < 0 when top-quark mass is varied by 2.8 GeVon either side of 173.3 GeV. Region I is the same as that in Fig. 3except that it is now a discarded zone for all values of mt withinits limit. The region II is discarded via stau turning the LSPor turning itself tachyonic for all mt. The constraints fromBrðb ! sÞ and BrðBs ! þÞ are as shown. The lines showthe boundary of purely discarded zones irrespective of variation ofmt within its range. There is no region toward the left of the blueline labeled as mh ¼ 124 GeV for which mh may become largerthan 124 GeV irrespective of values of mt. Three benchmarkpoints A, B, and C are shown corresponding to Table I.

0 200 400 600 800 1000mχ1

0 (GeV)

10-12

10-11

10-10

10-9

10-8

10-7

σ SI(p

- χ) (p

b)

(mtop

scanned)

XENON100

FIG. 7 (color online). Spin-independent LSP-proton scatteringcross section vs LSP mass when mt is scanned along with m1

2and

m0 for > 0 and < 0. Only WMAP-7 satisfied points (forboth the lower and the upper limit) are shown along with theXENON100 exclusion limit.

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that it is within an acceptable zone considering the variousuncertainties to computemh as mentioned before. Points Aand C have larger spin-independent scattering cross sectionSI

p than the XENON100 limit. But, we believe this is

within an acceptable limit considering the existing uncer-tainties arising out of strangeness content of nucleon, aswell as those from astrophysical origins particularly fromlocal DM density. Finally, it is also possible to satisfy allthe limits in full subject to a 5%–10% heavier spectraand/or considering a multicomponent DM scenario.

IV. CONCLUSION

In the SSM [8], one assumes that a manifestation of anunknown but a fundamental mechanism of SUSY breakingmay effectively lead to stochasticity in the Grassmannianparameters of the superspace. With a suitable probabilitydistribution decided out of physical requirements, stochas-ticity in Grassmannian coordinates for a given Kahlerpotential and a superpotential may lead to well-knownsoft breaking terms. When applied to the superpotentialof the MSSM, the model leads to soft breaking terms like

the bilinear Higgs coupling term, a trilinear soft term, aswell as a gaugino mass term, all related to a parameter ,the scale of SUSY breaking. The other scale considered inthe model of Ref. [8] at which the input quantities are givenis , where the latter can assume a value between thegauge coupling unification scale MG and the Planck massscaleMP. There is an absence of a scalar mass soft term at in the original model that leads to stau turning to be theLSP or even turning itself tachyonic if is chosen as MG.This is only partially ameliorated when is above MG.However, the model because of its nonvanishing trilinearparameter is potentially accommodative to have a largerlighter Higgs boson mass via large stop scalar mixing. Asrecently shown in Ref. [10], the model in spite of its nicefeature of bringing out the desired soft breaking terms ofMSSM is not able to producemh above 116 GeV, and it hasa LSP which is only a subdominant component of DM.In this work, a minimal modification (referred to

as Mod-SSM) is made by allowing a nonvanishing singlescalar mass parameterm0 as an explicit soft breaking term.Phenomenologically, the above addition is similar to whatwas considered in minimal AMSB while confronting the

TABLE I. Spectra of three specimen parameter points A, B, and C as shown in Fig. 6. Resultswith marginal deviation from the assumed limits are shown in italics (red). Muon g 2 is notimposed as a constraint in this analysis. B0 ¼ BðMGÞ has two entries. The first one is the desiredvalue while the second one is the value obtained with a suitable tan as found from the Newton-Raphson root finding scheme. See text for further details.

Parameter A B C

mt 173.10 173.87 171.58

m1=2 838.78 1239.16 579.69

m0 6123.75 1817.69 4200.55

ðA0 ¼ 4m1=2Þ 3355:13 4956:64 2318:75ðB0 ¼ 2m1=2Þ 1677:56 2478:32 1159:37B0 (as output) 1683:56 2478:32 1160:27tan (as output) 45.86 40.92 45.11

sgnðÞ 1 1 1

403.86 2508.85 310.43

m~g 2145.53 2727.64 1525.80

m~uL 6247.84 2994.87 4292.70

m~t1 , m~t2 3758.76, 4376.60 1333.10, 2078.60 2587.56, 3026.20

m~b1, m~b2

4397.10, 4886.58 2054.58, 2339.25 3037.22, 3381.67

m~eL , ~e 6119.53, 6119.05 1983.72, 1982.21 4197.61, 4196.89

m~1 , m ~4750.43, 5482.91 549.62, 1536.41 3281.31, 3770.25

m~1, m~

2406.25, 741.03 1038.00, 2491.42 304.84, 518.70

m~01, m~0

2356.62, 417.02 548.95, 1038.00 241.80, 316.70

m~03, m~0

4424.40, 741.06 2489.56, 2491.14 322.35, 518.80

mA, mH 2573.69, 2573.69 1967.83, 1967.50 1846.04, 1846.04

mh 124.42 126.55 121.57

~1h2 0.1105 0.1002 0.1106

BFðb ! sÞ 3:23 104 2:96 104 3:15 104

BFðBs ! þÞ 2:98 109 5:23 109 2:89 109

RðB!Þ 0.98 0.98 0.97

a 5:78 1011 1:65 1010 1:22 1010

SIp 3:05 108 1:04 1011 2:64 108

CHAKRABORTI, CHATTOPADHYAY, AND GODBOLE PHYSICAL REVIEW D 87, 035022 (2013)

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issue of sleptons turning tachyonic in a pure AMSB frame-work. The modified model successfully accommodates thelighter Higgs boson mass near 125 GeV. Additionally, itcan accommodate the stringent constraints from the darkmatter relic density, BrðBs ! þÞ, Brðb ! sÞ, andXENON100 data on direct detection of dark matter.A variation of top-quark mass within its allowed range isincluded in the analysis, and this shows the LSP to be asuitable candidate for dark matter satisfying both the limitsof WMAP-7 data. Finally, we remind that the idea ofstochastic superspace can easily be generalized to variousscenarios beyond the MSSM.

ACKNOWLEDGMENTS

M.C. would like to thank the Council of Scientific andIndustrial Research, Government of India for support.U. C. and R.M.G. are thankful to the CERN THPH divi-sion (where the work was initiated) for its hospitality.R.M.G. wishes to acknowledge the Department ofScience and Technology of India for financial supportunder Grant No. SR/S2/JCB-64/2007. We would like tothank B. Mukhopadhyaya, S. Kraml, P. Majumdar, K. Ray,S. Roy, and S. SenGupta for valuable discussions.

APPENDIX

Using Ref. [8], we consider the following Hermitianprobability distribution:

P ð; Þ ¼ Aþ þ _ _ þ Bþ _

_C

þ _

_V þ _

_ þ _ _

þ _ _D: (A1)

Here, A, B, C, D, and V are complex numbers. , , ,

and are Grassmann numbers.In order to arrive at the results of Ref. [8], we use the

following [3]: d2 ¼ 14d

d, d2 ¼ 14d

_d _,

d4 ¼ d2d2 ,Rd2ðÞ ¼ 1,

Rd2 ð Þ ¼ 1,

Rd2 ¼R

d2 ¼ 0, andRd2 ¼ R

d2 _ ¼ 0.Normalization: First, P ð; Þ should satisfy the normal-

ization conditionRd2d2 P ð; Þ ¼ 1. All the terms

except the one with the coefficient D vanishes inRd2d2 P ð; Þ. Thus, D ¼ 1.Vanishing fermionic moments: Next, we require

vanishing of moments of fermionic type because of therequirement of the Lorentz invariance. The fact that hi¼Rd2d2 P ð; Þ¼0 means ¼0. Similarly, h _i ¼ 0

means _ ¼ 0, and h _i ¼ 0 leads to V ¼ 0. Finally,

h2 _i ¼ 0 gives _ ¼ 0, and h 2i ¼ 0 gives ¼ 0.

Bosonic moments: We now compute the bosonicmoments.

hi ¼Z

d2d2 P ð; Þ

¼Z

d2d2 ðÞð ÞC ¼ C:

Similarly, h i ¼ B. Calling B ¼ 1=, one has B ¼ C ¼1=. Furthermore, h i ¼ A. The fact that h i ¼hih i leads to A ¼ 1=jj2.Thus we find the following Hermitian probability

measure for the stochastic Grassmann variables:

P ð; Þjj2 ¼ ~P ð; Þ¼ 1þ ðÞ þ ð Þ þ jj2ðÞð Þ:

(A2)

We consider the Wess-Zumino model with a singlechiral superfield . has the following expansion [3]:

¼ ðxÞ i @ðxÞ 1

42 2@@

ðxÞ

þ ffiffiffi2

pc ðxÞ þ iffiffiffi

2p 2@c ðxÞ þ 2FðxÞ: (A3)

Correspondingly, for y we have

y¼ðxÞþ i @ðxÞ1

42 2@@

ðxÞ

þ ffiffiffi2

p c ðxÞ iffiffiffi

2p 2@ c ðxÞþ 2FðxÞ: (A4)

The kinetic term of the Lagrangian L is ½yD. Onefinds

y ¼ jj2 þ ffiffiffi2

pc þ ffiffiffi

2p

c þ 2Fþ 2F

þ 2 c c þ iffiffiffi2

p2 c ½@ þ ffiffiffi

2p

2 c F

2i ½@þ iffiffiffi2

p2 c ½@

þ ffiffiffi2

p2cF þ 2 2

FFþ 1

2@

½@

1

2½@@þ ic½@ c

: (A5)

Here, X½@Y ¼ 12 ðX@Y Y@XÞ. Upon vanishing the

appropriate surface terms, the D term in particular reads

½yD ¼FFþ@@þ i

2ðc@ c @c c Þ:

(A6)

Next, we consider the superpotential W given byW ¼ 1

2m2 þ 13h

3. One has

2 ¼ 2 þ 2ffiffiffi2

pcþ 2ð2F c c Þ; and

3 ¼ 3 þ 3ffiffiffi2

pc2 þ 32ðF2 c cÞ:

(A7)

Thus,

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W ¼1

2m2 þ 1

3h3

þ ffiffiffi

2p

c ðmþ h2Þ

þ 2mF 1

2mc c þ hF2 hc c

: (A8)

The potential energy density term will be as follows:

½WþH:c:F¼mF1

2mc c þhF2hc c

þH:c:

(A9)

Kinetic and potential terms averaged over and andemergence of soft SUSY breaking terms:

Averaging over the Grassmannian coordinates, we

compute L ¼ hLi ¼ Rd2d2 ~P ð; ÞL. Here L is the

usual super-Lagrangian density: L ¼ yþWð2Þð Þ þWyð2ÞðÞ. Then, using Eqs. (A2) and (A5) hLkinetici,namely, the kinetic part of L is found as

hLkinetici ¼ ½yD þ jj2jj2 þ F þ Fzfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflSUSY

:

(A10)

Similarly, the potential energy density averaged over and is given by

hW þ H:c:i ¼Z

d2d2 eP ð; ÞðWð2Þð Þ þWyð2ÞðÞÞ:(A11)

Using Eqs. (A2) and (A8) we find

hW þ H:c:i ¼1

2m2 þ 1

3h3

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflSUSY

þmF 1

2mc c þ hF2 hc c

þ H:c: (A12)

The total Lagrangian is then

L ¼ hLi ¼ hLkinetici þ hW þ H:c:i: (A13)

Using Eqs. (A9), (A10), and (A12) we break L intoSUSY invariant and SUSY breaking parts as follows:

L ¼ hLiSUSY þ hLiSUSY

; (A14)

where

hLiSUSY

¼ ½yD þ ½W þ H:c:F; (A15)

and

hLiSUSY

¼ jj2jj2 þ F þ F

þ1

2m2 þ 1

3h3

þ H:c:

: (A16)

The equations of motion of auxiliary fields are then

F ¼ ðþm þ h2Þ; and

F ¼ ð þmþ h2Þ:(A17)

Substituting F and F in L; one finds

L ¼ LOn-shell-SUSY þ Lsoft; (A18)

where LOn-shell-SUSY is the usual on shell SUSY invariantLagrangian for the interacting Wess-Zumino model and isgiven by

LOn-shell-SUSY

¼ @@þ i

2ðc@ c @c c Þ

m2jj2 h2ðjj2Þ2

mhjj2þ 1

2mc c þ hc c

þ H:c:

; (A19)

and Lsoft is given by

Lsoft ¼

1

2m2 þ 2

3h3

þ H:c:

: (A20)

We remind that a negative sign in the left-hand side ofEq. (A20) appears simply because of considering a positivesign before hW þ H:c:i in Eq. (A13) while writing thetotal Lagrangian. We note that it is only the superpotentialterm in the original theory that leads to soft breaking termsin the resulting Lagrangian (m ! m and h ! 2hgoing from W to Lsoft). The presence of a vector fieldwill not lead to any soft SUSY breaking term. This caneasily be seen by considering a vector superfield in Wess-Zumino gauge.MSSM: In the MSSM, as mentioned in Ref. [8] the

superpotential term HuHd will lead to ~Hu~Hd, and

terms like yupQUcHu will lead to 2yup ~Q ~Uc~Hu as soft

SUSY breaking terms. Here, fields with tildes denote thescalar component of the corresponding chiral superfields.

One further obtains a gaugino mass term 2 i

ðiÞðiÞ. Thus,one finds a universal gaugino mass m1=2 ¼ 1

2 jj, a bilinearHiggs soft parameter B ¼ , and a universal trilinear

soft parameter A0 ¼ 2. With no resulting scalar massterm, one has the universal scalar mass parameter m0 ¼ 0.These are the input quantities to be given at a scale. Lowenergy spectra are then found via RG evolutions.Reference [8] considered and as the input quantitiesand considered MG <<MP.

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[1] For reviews on supersymmetry, see, e.g., H. P. Nilles,Phys. Rep. 110, 1 (1984); J. D. Lykken, arXiv:hep-th/9612114; J. Wess and J. Bagger, Supersymmetry andSupergravity (Princeton University, Princeton, NJ, 1991),2nd ed.

[2] D. J. H. Chung, L. L. Everett, G. L. Kane, S. F. King, J. D.Lykken, and L. T. Wang, Phys. Rep. 407, 1 (2005); H. E.Haber and G. Kane, Phys. Rep. 117, 75 (1985); S. P.Martin, arXiv:hep-ph/9709356.

[3] M. Drees, P. Roy, and R.M. Godbole, Theory andPhenomenology of Sparticles (World Scientific,Singapore, 2005).

[4] H. Baer and X. Tata, Weak Scale Supersymmetry: FromSuperfields to Scattering Events (Cambridge UniversityPress, Cambridge, England, 2006), p. 537.

[5] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716, 1(2012); S. Chatrchyan et al. (CMS Collaboration), Phys.Lett. B 716, 30 (2012).

[6] G. Aad et al. (ATLAS Collaboration), Phys. Rev. D 86,092002 (2012); , Phys. Lett. B 710, 67 (2012); S.Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett.107, 221804 (2011).

[7] A. H. Chamseddine, R. Arnowitt, and P. Nath, Phys. Rev.Lett. 49, 970 (1982); R. Barbieri, S. Ferrara, and C.A.Savoy, Phys. Lett. 119B, 343 (1982); L. J. Hall, J. Lykken,and S. Weinberg, Phys. Rev. D 27, 2359 (1983); P. Nath,R. Arnowitt, and A.H. Chamseddine, Nucl. Phys. B227,121 (1983); N. Ohta, Prog. Theor. Phys. 70, 542 (1983); P.Nath, R. Arnowitt, and A.H. Chamseddine, AppliedN ¼ 1 Supergravity (World Scientific, Singapore, 1984).

[8] A. Kobakhidze, N. Pesor, and R. R. Volkas, Phys. Rev. D79, 075022 (2009).

[9] A. Kobakhidze, N. Pesor, and R. R. Volkas, Phys. Rev. D81, 095019 (2010).

[10] A. Kobakhidze, N. Pesor, R. R. Volkas, and M. J. White,Phys. Rev. D 85, 075023 (2012).

[11] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 110,021801 (2013).

[12] E. Aprile et al. (XENON100 Collaboration), Phys. Rev.Lett. 109, 181301 (2012).

[13] R. L. Arnowitt and P. Nath, Phys. Rev. D 56, 2833 (1997).[14] G. F. Giudice, M.A. Luty, H. Murayama, and R. Rattazzi,

J. High Energy Phys. 12 (1998) 027; L. Randall and R.Sundrum, Nucl. Phys. B557, 79 (1999); J. Bagger, T.Moroi, and E. Poppitz, J. High Energy Phys. 04 (2000)009.

[15] T. Gherghetta, G. F. Giudice, and J. D. Wells, Nucl. Phys.B559, 27 (1999); J. L. Feng, T. Moroi, L. Randall, M.Strassler, and S. Su, Phys. Rev. Lett. 83, 1731 (1999); J. L.Feng and T. Moroi, Phys. Rev. D 61, 095004 (2000); U.Chattopadhyay, D. K. Ghosh, and S. Roy, Phys. Rev. D 62,115001 (2000).

[16] R. Arnowitt and P. Nath, Phys. Rev. D 46, 3981 (1992).[17] G. Gamberini, G. Ridolfi, and F. Zwirner, Nucl. Phys.

B331, 331 (1990); V.D. Barger, M. S. Berger, and P.Ohmann, Phys. Rev. D 49, 4908 (1994); S. P. Martin,Phys. Rev. D 66, 096001 (2002).

[18] A. Djouadi, J.-L. Kneur, and G. Moultaka, Comput. Phys.Commun. 176, 426 (2007).

[19] M. Drees and M.M. Nojiri, Nucl. Phys. B369, 54 (1992).[20] A. Djouadi, Phys. Rep. 459, 1 (2008).

[21] H. E. Haber, R. Hempfling, and A.H. Hoang, Z. Phys. C75, 539 (1997).

[22] H. E. Haber and R. Hempfling, Phys. Rev. Lett. 66, 1815(1991); Y. Okada, M. Yamaguchi, and T. Yanagida, Prog.Theor. Phys. 85, 1 (1991); Phys. Lett. B 262, 54 (1991); J.Ellis, G. Ridolf, and F. Zwirner, Phys. Lett. B 257, 83(1991); 262, 477 (1991).

[23] H. E. Haber and Y. Nir, Phys. Lett. B 306, 327 (1993);H. E. Haber, arXiv:hep-ph/9505240; A. Dobado, M. J.Herrero, and S. Penaranda, Eur. Phys. J. C 17, 487(2000); J. F. Gunion and H. E. Haber, Phys. Rev. D 67,075019 (2003); A. Djouadi and R.M. Godbole,arXiv:0901.2030.

[24] A. Arbey, M. Battaglia, A. Djouadi, and F. Mahmoudi,J. High Energy Phys. 09 (2012) 107.

[25] S. Heinemeyer, O. Stal, and G. Weiglein, Phys. Lett. B710, 201 (2012).

[26] B. C. Allanach, A. Djouadi, J. L. Kneur, W. Porod, andP. Slavich, J. High Energy Phys. 09 (2004) 044.

[27] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich, andG. Weiglein, Eur. Phys. J. C 28, 133 (2003).

[28] R. V. Harlander, P. Kant, L. Mihaila, and M. Steinhauser,Phys. Rev. Lett. 100, 191602 (2008); 101, 039901 (2008);S. P. Martin, Phys. Rev. D 75, 055005 (2007).

[29] S. Bertolini, F. Borzumati, and A. Masiero, Phys. Rev.Lett. 59, 180 (1987); N.G. Deshpande, P. Lo, J. Trampetic,G. Eilam, and P. Singer, Phys. Rev. Lett. 59, 183 (1987);B. Grinstein and M.B. Wise, Phys. Lett. B 201, 274(1988); B. Grinstein, R. P. Springer, and M.B. Wise,Phys. Lett. B 202, 138 (1988); W.-S. Hou and R. S.Willey, Phys. Lett. B 202, 591 (1988); B. Grinstein,R. P. Springer, and M. B. Wise, Nucl. Phys. B339, 269(1990).

[30] S. Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi,Nucl. Phys. B353, 591 (1991); R. Barbieri and G. F.Giudice, Phys. Lett. B 309, 86 (1993); R. Garisto andJ. N. Ng, Phys. Lett. B 315, 372 (1993); P. Nath and R. L.Arnowitt, Phys. Lett. B 336, 395 (1994); M. Ciuchini, G.Degrassi, P. Gambino, and G. F. Giudice, Nucl. Phys.B534, 3 (1998).

[31] D. Asner et al. (Heavy Flavor Averaging GroupCollaboration), arXiv:1010.1589.

[32] S. R. Choudhury and N. Gaur, Phys. Lett. B 451, 86(1999); K. S. Babu and C. Kolda, Phys. Rev. Lett. 84,228 (2000); A. Dedes, H. K. Dreiner, and U. Nierste, Phys.Rev. Lett. 87, 251804 (2001); P. H. Chankowski and L.Slawianowska, Phys. Rev. D 63, 054012 (2001); R.Arnowitt, B. Dutta, T. Kamon, and M. Tanaka, Phys.Lett. B 538, 121 (2002); J. K. Mizukoshi, X. Tata, andY. Wang, Phys. Rev. D 66, 115003 (2002); S. Baek, P. Ko,and W.Y. Song, J. High Energy Phys. 03 (2003) 054; G. L.Kane, C. Kolda, and J. E. Lennon, arXiv:hep-ph/0310042;T. Ibrahim and P. Nath, Phys. Rev. D 67, 016005 (2003);J. R. Ellis, K.A. Olive, and V.C. Spanos, Phys. Lett. B624, 47 (2005); S. Akula, D. Feldman, P. Nath, and G.Peim, Phys. Rev. D 84, 115011 (2011); A. J. Buras, J.Girrbach, D. Guadagnoli, and G. Isidori, arXiv:1208.0934.

[33] A. J. Buras, J. Girrbach, D. Guadagnoli, and G. Isidori,Eur. Phys. J. C 72, 2172 (2012).

[34] L. Roszkowski, S. Trojanowski, K. Turzynski, and K.Jedamzik, arXiv:1212.5587.

IMPLICATION OF A HIGGS BOSON AT 125 GeV . . . PHYSICAL REVIEW D 87, 035022 (2013)

035022-13

[35] G. Isidori and P. Paradisi, Phys. Lett. B 639, 499(2006); B. Bhattacherjee, A. Dighe, D. Ghosh, and S.Raychaudhuri, Phys. Rev. D 83, 094026 (2011).

[36] J. P. Lees et al. (BABAR Collaboration), arXiv:1207.0698.[37] I. Adachi et al. (Belle Collaboration), arXiv:1208.4678.[38] K. Hara et al. (Belle Collaboration), Phys. Rev. D 82,

071101 (2010).[39] W. Altmannshofer, M. Carena, N. Shah, and F. Yu,

arXiv:1211.1976.[40] S. Bodenstein, C. A. Dominguez, and K. Schilcher, Phys.

Rev. D 85, 014029 (2012); G.-C. Cho, K. Hagiwara, Y.Matsumoto, and D. Nomura, J. High Energy Phys. 11(2011) 068; D. Ghosh, M. Guchait, S. Raychaudhuri, andD. Sengupta, Phys. Rev. D 86, 055007 (2012); S. Akula,P. Nath, and G. Peim, Phys. Lett. B 717, 188 (2012).

[41] G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279(2005); G. Jungman, M. Kamionkowski, and K. Griest,Phys. Rep. 267, 195 (1996).

[42] E. Komatsu et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 192, 18 (2011).

[43] S. Alekhin, A. Djouadi, and S. Moch, Phys. Lett. B 716,214 (2012).

[44] Tevatron Electroweak Working Group, CDFCollaboration, and D0 Collaboration, arXiv:1107.5255.

[45] F. Boudjema and G.D. La Rochelle, Phys. Rev. D 86,115007 (2012); S. Caron, J. Laamanen, I. Niessen, andA. Strubig, J. High Energy Phys. 06 (2012) 008; U.Chattopadhyay and D. P. Roy, Phys. Rev. D 68, 033010(2003).

[46] S. Mohanty, S. Rao, and D. P. Roy, J. High Energy Phys.11 (2012) 175; M.A. Ajaib, T. Li, and Q. Shafi, Phys. Rev.D 85, 055021 (2012); D. Feldman, Z. Liu, P. Nath, andB.D. Nelson, Phys. Rev. D 80, 075001 (2009); U.Chattopadhyay, D. Das, D. K. Ghosh, and M. Maity,Phys. Rev. D 82, 075013 (2010); D. Feldman, Z. Liu,and P. Nath, Phys. Rev. D 80, 015007 (2009); U.Chattopadhyay, D. Das, and D. P. Roy, Phys. Rev. D 79,095013 (2009); U. Chattopadhyay and D. Das, Phys.Rev. D 79, 035007 (2009); R.M. Godbole, M. Guchait,and D. P. Roy, Phys. Rev. D 79, 095015 (2009);

U. Chattopadhyay, D. Das, P. Konar, and D. P. Roy,Phys. Rev. D 75, 073014 (2007); U. Chattopadhyay, D.Das, A. Datta, and S. Poddar, Phys. Rev. D 76, 055008(2007); G. Belanger, F. Boudjema, A. Cottrant, R.M.Godbole, and A. Semenov, Phys. Lett. B 519, 93 (2001).

[47] M. Drees and M.M. Nojiri, Phys. Rev. D 48, 3483(1993).

[48] N. Fornengo, S. Scopel, and A. Bottino, Phys. Rev. D 83,015001 (2011); A. Bottino, F. Donato, N. Fornengo, and S.Scopel, Phys. Rev. D 81, 107302 (2010); A. Bottino, V. deAlfaro, N. Fornengo, S. Mignola, and S. Scopel, Astropart.Phys. 2, 77 (1994); T. K. Gaisser, G. Steigman, and S.Tilav, Phys. Rev. D 34, 2206 (1986).

[49] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov,Comput. Phys. Commun. 180, 747 (2009).

[50] M. Perelstein and B. Shakya, arXiv:1208.0833; C.Beskidt, W. de Boer, D. I. Kazakov, and F. Ratnikov,J. High Energy Phys. 05 (2012) 094; J. Giedt, A.W.Thomas, and R.D. Young, Phys. Rev. Lett. 103, 201802(2009); H. Ohki, H. Fukaya, S. Hashimoto, T. Kaneko, H.Matsufuru, J. Noaki, T. Onogi, E. Shintani, and N.Yamada, Phys. Rev. D 78, 054502 (2008); J. R. Ellis,K. A. Olive, and C. Savage, Phys. Rev. D 77, 065026(2008).

[51] See, for example, the second reference of Ref. [50].[52] N. Arkani-Hamed, A. Delgado, and G. F. Giudice, Nucl.

Phys. B741, 108 (2006).[53] K. L. Chan, U. Chattopadhyay, and P. Nath, Phys. Rev. D

58, 096004 (1998); U. Chattopadhyay, A. Corsetti, and P.Nath, Phys. Rev. D 68, 035005 (2003); S. Akula, M. Liu,P. Nath, and G. Peim, Phys. Lett. B 709, 192 (2012).

[54] J. L. Feng, K. T. Matchev, and T. Moroi, Phys. Rev. D 61,075005 (2000); Phys. Rev. Lett. 84, 2322 (2000); J. L.Feng, K. T. Matchev, and F. Wilczek, Phys. Lett. B 482,388 (2000); J. L. Feng and F. Wilczek, Phys. Lett. B 631,170 (2005); U. Chattopadhyay, T. Ibrahim, and D. P. Roy,Phys. Rev. D 64, 013004 (2001); U. Chattopadhyay, A.Datta, A. Datta, A. Datta, and D. P. Roy, Phys. Lett. B 493,127 (2000); S. P. Das, A. Datta, M. Guchait, M. Maity, andS. Mukherjee, Eur. Phys. J. C 54, 645 (2008).

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