-even higgs boson mass bound

PHYSICAL REVIEW D 73, 115004 (2006)

Four-body decay of the lighter top squark constrainedby the lighter CP-even Higgs boson mass bound

Siba Prasad Das*Department of Physics, Jadavpur University, Kolkata- 700 032, India

(Received 31 December 2005; published 13 June 2006)

*Electronic

1550-7998=20

We reinvestigated the parameter space allowing a large branching ratio (BR) of the 4-body decay of thelighter top squark (~t1) accessible at Tevatron Run-II by imposing the lighter CP-even Higgs boson mass(mh0 ) bound from LEP. Important constraints were obtained in mSUGRA as well as in the unconstrainedsupersymmetric models. Our results show that the prospect of searching the lighter top squark via the 4-body decay mode, in particular, the ‘� n� jets� E6 T signal, is not promising in mSUGRA due to theabove bound on mh0 . The existing bounds on m~t1 from Tevatron Run-I and LEP assuming 100% BR of theloop decay of ~t1 are, therefore, valid to a good approximation. We also find that large BRs of the above 4-body decay are allowed in the unconstrained model over significant regions of parameter spaces and thepossibility that this decay mode is the main discovery channel at Tevatron Run-II is open. We have brieflyreviewed the theoretical uncertainties in the calculation of mh0 and their consequences for the constraintsobtained by us. We have commented upon, with illustrative examples, how the above parameter space isaffected if future experiments push the Higgs boson mass bound upward.

DOI: 10.1103/PhysRevD.73.115004 PACS numbers: 11.30.Pb, 12.60.Jv, 14.80.Ly, 12.60.Fr

1

I. INTRODUCTION

The mass of the lighter top squark (~t1), superpartner ofthe top quark in the standard model (SM) may turn out tobe below the top-quark mass. This happens in a wideregion of the supersymmetry (SUSY) parameter spaceboth in the minimal supergravity (mSUGRA) model andin the minimal supersymmetric standard model (MSSM). Itcould even be the next-to-lightest supersymmetric particle(NLSP), the lightest neutralino (~�0

1) being the lightestsupersymmetric particle (LSP) by assumption. Being thelightest among the strongly interacting sparticles with arelatively large cross section compared to other sparticles,the lighter top squark may show up in direct searches atTevatron Run-II. In our convention, ~t1 � ~tL cos�~t �~tR sin�~t and ~t2 � �~tL sin�~t �~tR cos�~t, where �~t is themixing angle and ~t2 is the heavier top squark. The top-quark mass (mt) appears in the off-diagonal elements of thetop-squark mass matrix. So, the physical top-squarkmasses (m~t1 ,m~t2 ) and the mixing angle depend on the inputvalue of the top-quark mass.

In the lighter top-squark-NLSP (~t1-NLSP) scenario (i.e.,m~�0

1<m~t1< the mass of any other sparticle), ~t1 has few

allowed decay modes in the R-parity conserving (RPC)SUSY model. They are the flavor changing neutral current(FCNC) induced loop decay, ~t1 ! c~�0

1 [1], the 4-bodydecay, ~t1 ! b~�0

1f �f0, where f �f0 � u; d; c; s; ‘; �‘ [2], thetree level two body decay, ~t1 ! t~�0

1 and the three bodydecay, ~t1 ! bW ~�0

1. If m~t1 * mt �m~�01, the production

cross section of ~t1 is rather low for the Tevatron experi-ments. The ~t1 ! bW ~�0

1 mode may be suppressed if thecoupling W ~�0

1 ~��1 is small or m~t1 <MW �m~�01�mb. This

address: [email protected]

06=73(11)=115004(11) 115004

may happen if ~�01 is bino like as in the mSUGRA model.

However, in this report we have considered only thoseparameter spaces where the last two decay modes are eitherkinematically or dynamically suppressed.

So far most of the ~t1-NLSP searches at Tevatron Run-Iand at the Large Electron Positron (LEP) collider havebeen performed assuming 100% BR of the loop decay,1

leading to the signal: jets�missing energy (E6 T). The moststringent constraint comes from the Tevatron Run-I experi-ments [5]. However, the above assumption becomes unre-alistic, if ~t1 has other decay modes with comparablewidths. This possibility exists in a large region of theSUSY parameter spaces in both the models mentionedabove [2,6]. The 4-body decay and the loop decay inRPC models may have decay widths of the same order ofmagnitude in a wide region of the parameter space [2].

An interesting feature of all supersymmetric models isthe prediction of the existence of at least one light Higgsboson [7,8]. So far the LEP and Tevatron Run-I experi-ments found no significant evidence in favor of a lightHiggs boson. The direct searches from the four LEP ex-periments concluded that the lighter Higgs boson must beheavier than 114.4 GeV at 95% C.L. [9]. Generally thisbound depends on the value of sin2�� , where� and�are the ratio of the vacuum expectation values (VEVs) ofthe two Higgs doublets and the mixing angle in theCP-even Higgs boson mass matrix, respectively. How-ever, the Higgs boson mass bound is 114.4 GeV whensin2�� ’ 1. We have found that this factor is indeed’ 1 for the parameter space interesting for us. This Higgs

Recently the D0 Collaboration [3] and the ALEPHCollaboration [4] looked for the ~t1 ! b~�0

1f �f0 channel withspecial assumptions about its BR.

-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.73.115004

SIBA PRASAD DAS PHYSICAL REVIEW D 73, 115004 (2006)

boson mass bound is directly applicable in SUSY when theCP-odd Higgs boson (MA) mass is very large compared toZ-boson mass (MZ), i.e., MA � MZ, commonly known asthe decoupling limit.

In SUSY the mass of the light Higgs boson at tree levelis given by

m2h0 �

1

2�M2

Z �M2A �

��M2

Z �M2A�

2 � 4M2ZM

2Acos22�

q:

(1)

Equation (1) shows that mh0 & MZ at the tree level. Butradiative corrections involving top quark and squarks (~t) inthe loop indicate that m2

h0 grows like GFNcCm4t where GF

is the Fermi coupling constant, Nc is the color factor and Cis a model dependent loop factor. It is clear that therelatively large experimental error in top mass may leadto a sizable uncertainty in the Higgs boson mass prediction.The loop corrections from bottom quark and squarks (~b)are also sizable for large values of tan� and � [10], where� is the Higgs mass parameter in the superpotential. Weshall discuss the Higgs boson mass as a function of theSUSY model parameters in Sec. II.

Theoretically the mass of the lighter CP-even Higgsboson is bounded from above: mh0 & 135 GeV. Thisbound has been obtained by including radiative correctionsup to two-loop level [8,11–15]. So,mh0 is expected to lie inthe range 114:4 GeV & mh0 & 135 GeV. Because of theyet unknown higher-order corrections the theoretical erroron the lighter CP-even Higgs boson mass consists ofseveral pieces [16–20]. They are the momentum-independent two-loop corrections, the momentum-dependent two-loop corrections, the higher loop correc-tions from t=~t sector etc. Taking into account of all shortsof uncertainties the intrinsic error has been estimated to be�mh0 3 GeV.2 However the magnitude of this uncer-tainty crucially depends on the SUSY parameter space.In view of the above theoretical errors the viability of aparameter space will be judged in this work by using thefollowing two constraints on the calculated Higgs mass:(i) mh0 * 111:4 GeV (the weaker bound) (ii) mh0 *

114:4 GeV (the experimental bound). The first bound ob-viously leads to more conservative results as will be shownin Sec. II.

We now make a few comments on the ~t1 decay widthsconsidered in this paper. In the approximation of neglect-ing the charm quark mass, ~cR does not mix with topsquarks. The mixing of ~cL with the top-squarks masseigenstates result in a redefined lighter top squark given

2Inclusion of the full two-loop corrections as well as theleading higher loop corrections is expected to yield a reduceduncertainty �mh0 & 0:5 GeV [16,18].

115004

approximately by ~t1 � ~t1 � �~cL [1,21], where

� ��L cos�~t � �R sin�~t

m2~t1�m2

~cL

: (2)

The parameters �L;R [1,21] are given by

�L ��

4s2W

log�M2

GUT

M2W

�V�tbVcbm

2b

2M2Wcos2�

� �m2~cL�m2

~bR�m2

Hd� A2

b�; (3)

�R ��

4s2W

log�M2

GUT

M2W

�V�tbVcbm

2b

2M2Wcos2�

mtAb; (4)

where m~cL , m~bR, mHd

, and Ab are the left-handed scharm,right-handed sbottom, down-type Higgs scalar, and trilin-ear sbottom mass parameters, respectively. MGUT is thegrand unification theory (GUT) scale in the SUSY model.

Assuming proper electroweak symmetry breaking(EWSB), the Higgs scalar mass squared parameter [inEq. (3)] can be written as

m2Hd� M2

Asin2�� cos2�M2W ��

2: (5)

The loop decay width can then be expressed as [1]:

��~t1 ! c�01� �

�4m~t1

�1�

m2�0

1

m2~t1

�2jfL1j

2j�j2; (6)

where fL1 is given by

fL1 ��2p �

2

3�cWN11 � sWN12� �

�1

2�

2

3s2W

�

��sWN11 � cWN12

cWsW

�: (7)

Here N11 and N12 denote the elements of the neutralinomixing matrix projecting the LSP on to the photino (j � 1)and zino (j � 2) states. After renormalization one obtainsthe residual logarithmic terms in Eqs. (3) and (4) whichlead to a large contribution in the amplitude. Addingvarious diagrams contributing to the loop, a divergentterm is obtained in general. In principle this divergencemust be cancelled by adding appropriate counterterms. Thedominant effects of renormalization are expected to comefrom the logarithmic divergences caused by soft breakingterms at MGUT in supergravity (SUGRA) models. Thus inpractice a large logarithmic factor ln�M2

GUT=m2W� 65 is

included in the decay amplitude.3

The 4-body decay mode ~t1 ! b�01f �f0, proceeds through

various diagrams [2]. An approximate analytical expres-sion for the 4-body decay amplitude can be found in [2].

3For SUSY-breaking mechanisms other than gravity media-tion, such as gauge mediation, the SUSY-breaking scale can bemuch lower than the GUT scale, which lead to a somewhatsmaller value of � in Eq. (6).

-2

FOUR-BODY DECAY OF THE LIGHTER TOP SQUARK . . . PHYSICAL REVIEW D 73, 115004 (2006)

However, the decay rate has been been calculated in [2]without any approximation by taking into account all dia-grams and the interferences among them. We thank theauthors of [2] for providing their FORTRAN code. The pack-ages SDECAY [22], and CALCHEP V2.1 [23] are now availablefor calculating the decay width in the context of MSSM.We have cross checked the FORTRAN code with CALCHEP

V2.1.In general, in any diagram the virtuality of the ex-

changed particle in the propagator is important for gettingan appreciable contribution. This gives an idea of theparameter space where the 4-body width is significant. Ithas been checked that for the ~t1 mass range and the modelparameters used in this analysis the chargino and/or theslepton mediated diagrams are the dominant ones and the4-body width is indeed comparable to the loop decaywidth.

The 4-body and the loop decay of the ~t1-NLSP are of thesame order in perturbation theory, i.e. O��3�. So, they maycompete with each other. Our prime objective is to figureout the SUSY parameter space where the 4-body BR maycompete with or overwhelm the loop decay BR. In thisperspective a few more points are noteworthy.

(i) A
s discussed above the large logarithmic factor inthe width [see Eqs. (3) and (4)] presupposes theexistence of a grand unification scale. Moreover, itis also assumed that MSSM is the correct theory allthe way up toMGUT. If this is not the case this factorcan be rather small. This would enhance the 4-bodyBR compared to the typical values presented in thispaper.
(ii) I
f the ~t1 is purely right handed, the amplitudeinvolves only �R in Eq. (2). This will suppressthe amplitude for small values of the trilinear cou-pling Ab [in Eq. (4)]. Additional suppression maycome if the (common) SUSY-breaking scalar massof the first two generations is large [Eq. (2)].
(iii) A
ny scenario with tan�~t ’ �L=�R immediatelyleads to a negligible � [Eq. (2)].
(iv) F
rom Eqs. (3) and (4) we note that the mixingbetween ~t1 and ~cL is proportional to tan2�. It is,therefore, relatively suppressed at small tan�which is favorable for a large 4-body BR [2,6].
4Other public codes, for example, FeynHiggs [27] andCPsuperH [28] are also available for calculating the propertiesof the Higgs sector in various SUSY models.

It has already been pointed out that the ~t1 searches in the4-body leptonic decay channel is especially promising for3 & tan� & 5 at the upgraded Tevatron [6]. However, it isalso well known that the radiatively corrected mh0 in-creases with tan�. So, what happens to the parameterspaces favorable for ~t1 searches in the 4-body decay chan-nel if the Higgs boson mass bound is invoked? We areaddressing this issue in this paper.

The input value of top-quark mass in this paper isimportant for two reasons. Firstly, it changes the parameterspace where the ~t1-NLSP condition is satisfied. Con-sequently the 4-body decay BR will be affected as wellfor a particular choice of SUSY model parameters.

115004

Secondly, as discussed above the, radiative corrections tothe Higgs boson mass is very sensitive to mt.

Before delving into the numerical discussions let usbriefly review the current status of the top mass measure-ment. From the published Tevatron Run-I and preliminaryTevatron Run-II data, the present world average for the top-quark mass is mt � 172:7� 2:9 GeV [24]. Since there isconsiderable uncertainty in mt we shall consider in thispaper two representative values: (i) mt � 172:7 GeVwhich is the central value of the Run-I and Run-II averageand (ii) mt � 178:0 GeV which is on the higher side of thecurrently allowed range (approximately 2 away from thepresent central value). The second choice predicts rela-tively large mh0 for fixed values of other SUSY parametersand hence, leads to weaker constraints on the parameterspace allowed by the bound on mh0 .

The plan of the paper is as follows. In Sec. II, we beginwith a discussion of the numerical tools used for calculat-ing the sparticle masses and the lighter CP-even Higgsboson mass. The Secs. II A and II B are devoted to thephenomenological discussions of the lighter top-squark 4-body decay modes, respectively, in mSUGRA and MSSMmodels consistent with the lighter CP-even Higgs bosonmass limit from LEP experiments. Consequences of thetheoretical uncertainties in the mh0 prediction are alsoanalyzed. We shall conclude in Sec. III.

II. NUMERICAL ANALYSIS:

We begin with representative parameter spaces in whichthe ~t1-NLSP condition is realized. We then draw the con-tours of its total 4-body decay BR in this parameter space.Finally we impose the lighter CP-even Higgs boson massconstraint on this parameter space. The sparticle spectrumin mSUGRA has been calculated using ISAJET V7.48 [25].The sparticle spectrum in MSSM has been calculated usingour own codes. The Higgs boson mass has been calculatedusing SUSPECT V2.33 [26] in both the models. This codeuses the full one-loop and dominant two-loop contributionsto the self energy corrections in the SUSY Higgs bosonmass matrix in the dimensional reduction (DR) scheme.4

The supersymmetric radiative corrections to the particle(top, bottom, tau) and sparticles masses (squarks and gau-ginos) have also been incorporated. We have set the fol-lowing input values in SUSPECT v2.33 at the weak scale(Mweak): 1

�em� 1

127:934 , �s � 0:1172, mt�pole� � 172:7 or178.0 GeV, m��pole� � 1:777 GeV, mb�mb� � 4:25 GeV,sin2

�W� 0:2221.

The MA and � are treated as inputs in MSSM in thispaper. All masses and mass parameters in this paper are inGeV.

-3

5The top-quark mass is very crucial for calculating mHuand

�R in Eqs. (8) and (9).6One can easily find m~�0

1and m~��1

from the parameter set usedin the figures.


The magnitude of � can be constrained from the radia-tive electroweak symmetry breaking (REWSB) conditionswhich produce the observed Z-boson mass. Minimizing theHiggs potential with radiative corrections one obtains

1

2M2Z �

m2Hd�m2

Hutan2�

tan2�� 1��2 ��R (8)

and

�B �1

2�m2

Hd�m2

Hu� 2�2� sin2�; (9)

where mHu, mHd

are the soft Higgs mass parameters atMweak and �R is the radiative correction which is given in[25,26]. This REWSB criterion restrict the j�Bj and j�j atMweak and leaves the sign�� as a free parameter inmSUGRA. One can trade the parameter B for tan� (ratioof Higgs VEVs). Equations (8) and (9) are also valid inMSSM but for arbitrary soft masses.

The soft SUSY-breaking terms cannot be arbitrary atMweak in mSUGRA. The soft terms are renormalizationgroup equation (RGE) driven from high scale inputs likethe common scalar mass, see Sec. II A. The trilinear cou-plings atMweak can also be computed from a common inputvalue at MGUT using the RGE. The soft terms are treated asindependent inputs in MSSM which we shall discuss in theSec. II B.

However, in both the models one can show that a deepcharge-color-breaking (CCB) minima [29–31] appears inthe scalar potential at Mweak unless

jAtj2 & 3�m2~tL�m2

~tR�m2

Hu��2�;

jAbj2 & 3�m2

~bL�m2

~bR�m2

Hd��2�;

jA�j2 & 3�m2

~�L�m2

~�R�m2

Hd��2�;

(10)

where At, Ab, and A� are the trilinear soft parameters of thetop squark, bottom-squark and tau-slepton sector,respectively.

In our analysis we have considered the ranges of modelparameters such that the CCB minima [29–31] of thepotential do not arise. Moreover all sparticle masses arerequired to satisfy the recent experimental bounds. We nowmake a few comments on the obtained ~t1 mass limits.

The published mass limits on ~t1 not only depend on theproduction cross sections, they also depend on �M �m~t1 �mLSP, where LSP can be either ~�0

1 or ~�. In additionthese limits depend on the BRs of various decay channelskinematically allowed. Moreover, the limits from LEP alsodepend on �~t.

Most of the mass limits from Tevatron [32–35] and LEP[36,37] have been obtained by assuming 100% BR of the~t1 ! c~�0

1 or the ~t1 ! b~��1 decay. However, the presence ofother competent decay mode, for example the 4-bodydecay mode emphasized in this paper, substantially

115004

changes these limits. The CDF limit [33] with the aboveassumption has been included in the analysis of Sec. II A.

Recently the D0 Collaboration [3] and the ALEPHCollaboration [4] put limits on m~t1 assuming completedominance of the channel ~t1 ! b~�0

1‘�‘ and coexistenceof the loop decay and the 4-body decay, respectively. FromFig. 4b of [4] the absolute lower limits ofm~t1 is found to be 63 at 95% C.L. This limit is obtained for �M � 5,BR�~t1 ! c~�0

1� � 22% and BR�~t1 ! b~�01‘�‘� � 55% and

�~t � 56�. The m~t1 limit becomes stronger when BR�~t1 !c~�0

1� is increased from 22%. It may be noted that theassumed BRs in the above works are not realistic. Forexample it was shown in [6] that the BR of the leptonic(‘ � e and �) 4-body decay mode is unlikely to exceed20% since competing hadronic 4-body decay modes areinevitably present. In a different approach [38] modelindependent limits on the product of branching ratiosBR�~t1 ! be��e ~�0

1� � BR�~t�1 ! �bq �q0 ~�01�, where q and q0

correspond quarks of all flavors kinematically allowed,were obtained as a function of m~t1 using Tevatron Run-Idata.

The LEP-II Collaboration puts the limit m~b1* 96 [39]

assuming complete dominance of ~b1 ! b~�01 decay. The

following constraints on sparticle masses have also beenincluded in the analysis of this paper: m~�1

* 86, m~� *

43:7, m ~�R* 94:9, m~eR * 99:4 [40], and m~��1

* 103 [41].

A. The mSUGRA model:

The input parameters of mSUGRA at MGUT are thecommon scalar mass (m0), the common gaugino mass(m1=2), the trilinear scalar coupling term (A0), tan� andsign of � (the Higgsino mass parameter) [42]. The magni-tude of � is fixed by the REWSB condition [29,43] anddepends on the input top-quark mass.5

Since the NLSP nature of the ~t1 is very crucial for ourwork, we have demarcated different regions in the A0 �tan� parameter space where ~t1 becomes the NLSP. Adetailed discussion is given in Sec. II of [6] (see Fig. 1and 26 of [6]), where it was assumed that mt � 175:0.Similar information is given in Fig. 1 and 2 in this paperfor mt � 172:7 and mt � 178:0 respectively. Several con-tours of the combined BR of all 4-body decay modes arealso shown. In order to identify the parameter space al-lowed by the mh0 constraints, we present the contours ofmh0 ’ 111:4 (the weaker bound) in both the figures andmh0 ’ 114:4 (the experimental bound) in the right panel ofFig. 2 only.

The choice of other mSUGRA input parameters used aregiven in the figure captions. These choices are guided by

-4

mχ 01

51Br 0.2Br 0.9Br 0.5Br 0.3Br 0.1

111.4mt = 172.7

A0 (GeV)ta

n β-490-500-510-520-530-540-550-560-570

18

16

14

12

10

8

6

4

2

Br 0.2Br 0.9Br 0.5Br 0.3Br 0.1

111.4

m t =172.7

A0 (GeV)-540-560-580-600-620-640

18

16

14

12

10

8

6

4

2

tan β

FIG. 1. The ~t1 becomes the NLSP in the whole marked region in the A0 � tan� plane in the mSUGRA model for m0 � 200�140�,m1=2 � 145�180�, �> 0 and mt � 172:7 in the left (right) panel. The bounds on sparticle masses and other SUSY constraints, see thetext, are also imposed. The CP-even Higgs boson mass contour for 111.4 is shown and the region above the contour is allowed (see thetext for a brief review of the theoretical uncertainties in the calculated mh0 ). The whole marked region is ruled out by the experimentalbound mh0 * 114:4. Several contours of total 4-body decay BR lighter top squark are shown by different point styles, see the legends.The ‘‘�’’ shows the regions where BR�~t1 ! b~�0

1‘�‘� ’ 20% for ‘ � e and �. The regions marked with heavy dotted square box areexcluded by the CDF limit (m~t1 * 102 GeV for m~�0

1 51 GeV) obtained by assuming 100% BR of the loop decay.


the following facts. If m0 � m1=2, the gauginos are natu-rally light. This disfavors the ~t1-NLSP scenario. Form1=2 � m0, on the other hand, one may require largejA0j for this scenario. However, such high values of jA0jmay violate of the CCB conditions, see Eq. (10). So, in ouranalysis we focused on m0 m1=2 regions for mSUGRA.However, the ~t1-NLSP is quite common in larger regions inthe parameter space of the unconstrained MSSM since thesoft breaking parameters are arbitrary. This model will beanalyzed in the next section.

For each tan� there is a range of jA0j (jA0jmin < jA0j<jA0jmax) at MGUT which corresponds to the ~t1-NLSP (seeFig. 1 and 2). Otherwise ~�1 or ~�0

2 or ~��1 happens to be theNLSP and ~t1 decays into other 2-body or 3-body decaychannels.

For jA0j> jA0jmax, ~t1 becomes the LSP which is for-bidden. On the other hand if jA0j< jA0jmin for the givenset of mSUGRA parameters, ~t1 becomes heavier than thelighter chargino and the ~t1-NLSP condition is violated. We,therefore, concentrate on jA0j jA0jmin so that the char-gino virtuality is small [2,6] and consequently the 4-body

mχ01

51Br 0.2Br 0.9Br 0.5Br 0.3Br 0.1

111.4

mt = 178

A 0 (GeV)

tan β

-530-540-550-560-570-580-590-600-610-620

18

16

14

12

10

8

6

4

2

FIG. 2. The ~t1 becomes the NLSP in the whole marked region in tm1=2 � 145�180�, �> 0 and mt � 178:0 in the left (right) panel. Thtext, are also imposed. The CP-even Higgs boson mass contours for 1regions above the contours are allowed from the respective bounds.mh0 * 114:4�117:4� (see the text). The total 4-body decay BR’s contthe legends. The ‘‘�’’ shows the regions where BR�~t1 ! b~�0

1‘�‘� ’square box can be excluded by the CDF limit (see Fig. 1 caption).

115004

BR is appreciable. The off-diagonal terms in the top-squark mass matrix have the form (At �

�tan� ). So as long

as At at Mweak and � have opposite signs the NLSPscenario is favored.

Formt � 172:7, the ~t1-NLSP is realized in a large regionof the parameter space (see Fig. 1). As expected the totalBR of the 4-body decays is appreciable for relatively lowvalue of tan� [2,6]. The region above the mh0 ’ 111:4contour cannot be excluded due to the theoretical uncer-tainties mentioned in the introduction. It should be notedthat even with the weaker bound on mh0 the total 4-bodydecay BR is & 5% for m0 � 200, m1=2 � 145 (see the leftpanel of Fig. 1). For m0 � 140, m1=2 � 180, on the otherhand, the total 4-body decay BR can be as large as 40%(see the right panel of Fig. 1). However, the whole parame-ter space in Fig. 1 is disfavored if the experimental bound isimposed.

A similar analysis for mt � 178:0 is shown in Fig. 2. Inthe left panel of Fig. 2 with m0 � 200, m1=2 � 145 wehave some allowed region if the weaker bound on mh0 isimposed. Still the total 4-body decay BR can be at most

Br 0.2Br 0.9Br 0.5Br 0.3Br 0.1

114.4

111.4

mt = 178

A 0 (GeV)-580-600-620-640-660-680-700

18

16

14

12

10

8

6

4

2

tan β

he A0 � tan� plane in the mSUGRA model for m0 � 200�140�,e bounds on sparticle masses and other SUSY constraints, see the11.4 (111.4 and 114.4) are plotted in the left (right) panel and theThe whole marked region in the left (right) are ruled out from

ours of lighter top squark are shown by different point styles, see20% for ‘ � e and �. The regions marked with heavy dotted

-5

FIG. 3. The m~t1 (mh0 ) as a function of A0 for for m0 � 140, m1=2 � 180, �> 0, and tan� � 5; 10; 15 in the left (right) panel inmSUGRA for mt � 178:0. The ~t1-NLSP criterion is relaxed in this particular figure.


20%. In the right panel of Fig. 2 with m0 � 140, m1=2 �

180 we have some allowed region even if the experimentalbound is imposed. However, the total 4-body decay BR canbe at most 20%. Of course with the weaker bound larger 4-body decay BRs ( & 60%) are allowed. It should also benoted that for a fixed m0, m1=2 cannot be increased arbi-trarily because that would lead to other instabilities of thescalar potential [29–31,44].

We have shown the variation ofm~t1�mh0� as a function ofA0 in the left (right) panel of Fig. 3 formt � 178:0 for threevalues of tan�. Other model parameters are given in thefigure caption. The ~t1-NLSP criterion is not imposed in thisparticular figure. However, the range of m~t1 satisfying the~t1-NLSP condition can be easily found out by comparingwith the right panel of Fig. 2. Then one can read off thelighter chargino and neutralino masses from the left panelof Fig. 3. A ~t1 having mass in the range shown in Fig. 3 maybe copiously produced at Tevatron Run-II. From the rightpanel it is clear that tan� � 5 is marginally allowed only ifthe weaker bound on mh0 is required. On the other hand iftan� * 9 thenmh0 is consistent even with the experimentalbound, but the total 4-body decay BR is negligible as canbe seen from the right panel of Fig. 2.

The scatter plots in Fig. 4 demarcate the ~t1-NLSP regionconsistent with the bound on mh0 in the m0 �m1=2 pa-rameter space for positive �, mt � 178:0, tan� � 5 (left

m˜01

51Br 0.5Br 0.3Br 0.1

111.4

mt = 178

m 0 (GeV)

m1/

2(G

eV)

30025020015010050

190

180

170

160

150

140

FIG. 4. The ~t1 becomes the NLSP in the whole marked region in�> 0, mt � 178:0 and tan� � 5�9� in the left (right) panel. The bouare also imposed. The CP-even Higgs boson mass contour for 111.4* 111:4 in the whole marked region in the right panel. The regions aThe whole marked regions in the left (right) panel are ruled out fromthe lighter top squark are shown by different point styles, see the legexcluded by the CDF limit (see Fig. 1 caption).

115004

panel) and tan� � 9 (right panel). In the left panel of Fig. 4with A0 � �630 we have some region allowed by theweaker bound on mh0 . The total 4-body decay BR can beas large as 50%. In the right panel of Fig. 4 for tan� � 9we have some allowed region even after imposing theexperimental bound on mh0 but the total 4-body decayBR can be at most 10%. With the weaker bound on mh0

the whole ~t1-NLSP region is allowed. In the regions withmh0 ’ 111:4 the total 4-body BR is <5%.

We have repeated the analysis for mt � 172:7 using thesame parameters as in Fig. 4. In this case the entire~t1-NLSP region in the m0 �m1=2 parameter space is dis-favored for tan� � 5 even for the weaker bound on mh0 *

111:4. For tan� � 9 on the other hand almost the entireparameter space is allowed by the weaker bound on mh0 .However, the total 4-body BRs can be at most 10% fortan� � 9.

In [6], we have studied the ~t1 search prospects atTevatron Run-II using the one lepton with 2 or more jetsaccompanied by a large amount of missing energy signalassuming mt � 175. This signal topology arises from thepair production of the lighter top squark followed by theleptonic 4-body decay of one lighter top squark (~t1 !b~�0

1‘�‘) while the other decays hadronically (~t1 !b~�0

1q �q0). We have shown that, BR�~t1 ! b~�01‘�‘� 20%,

where ‘ � e and � (see Figs. 3 and 4 of [6]), is adequate

m˜01

51Br 0.1

mh0 ≥ 111.4

114.4mt = 178

m0 (GeV)

m1/

2(G

eV)

300250200150100

190

180

170

160

150

140

130

the m0 �m1=2 plane in the mSUGRA model for A0 � �630,nds on sparticle masses and other SUSY constraints, see the text,(114.4) is plotted in the left (right) panel. The mh0 happens to bebove the contour are allowed from the respective bound on mh0 .mh0 * 114:4 (117.4). The contours of total 4-body decay BR ofends. The regions marked with heavy dotted square box can be

-6


for detection at Tevatron Run-II. From the present analysiswe find that such large leptonic BRs cannot be realized atleast in the mSUGRA model for mt � 172:7 and not evenwith the rather conservative choice mt � 178:0 irrespec-tive of the mh0 bound invoked. Our analysis also show thatthe ~t1 mass limits from LEP and Tevatron Run-I assuming100% BR of the loop decay of ~t1 are approximately valid inthe context of mSUGRA if mt is close to its current centralvalue (172.7) irrespective of the uncertainties in the pre-dicted mh0 . However, in view of the uncertainties inmt andin the calculated mh0 one cannot conclusively exclude thepossibility of a relatively large total BR of all 4-body decaymodes (see, e.g., the right panel of Fig. 2).

We have shown some points in the parameter spaceexcluded by the CDF limits on m~t1 in Fig. 1 (left panel),Fig. 2 (left panel) and Fig. 4 using the heavy dottedsquares. For each of these points m~�0

1’ 51, m~t1 & 102

and 4-body BR & 10%. It is heartening to note that eventhe modest statistics from Tevatron Run-I can already putconstraints on A0 which is rather difficult to constrain fromother measurements. However, the CDF excluded pointsare already ruled even by the weaker boundmh0 * 111:4 inthe context of mSUGRA. One notable exception is foundin the right panel of Fig. 4. Here we find that a small regionof parameter space allowed by the weaker bound on mh0 isruled out by the CDF constraint.

B. The MSSM model:

In the MSSM with a universal gaugino mass, the SU(2)gaugino mass parameter (M2), � and tan� completelydescribe the neutralino and chargino masses and mixings.Depending on the relative magnitudes of M2 and �, themodel is categorized in three different scenarios. They arethe Higgsino (M2 � �), the gaugino (M2 � �) and themixed (M2 �) scenarios. The lighter chargino (~��1 ) andthe two lighter neutralinos (~�0

1 and ~�02) all have approxi-

mately the same mass ( �) in the Higgsino scenario.Thus it is difficult to accommodate the ~t1-NLSP withoutfine adjustments of the parameters. The ~t1 happens to bethe NLSP in large regions of the parameter space in thegaugino and the mixed scenarios. In this paper we studiedthe mixed scenario. However, the main results and con-clusions are qualitatively valid in the gaugino scenario.

To calculate m~t1 and mh0 in MSSM, we use the parame-ter Xr (instead of A0) as input, where

Xr ��At �� cot��m~uLm~uRp ; (11)

m~uL and m~uR are the soft SUSY-breaking mass parametersfor the first generation left and right-squark, respectively, atthe weak scale.

In our analysis the first generation left and right-squarksoft masses are assumed to be the same, i.e., m~uL � m~uR �

m~dR� m~q. The charged slepton sector is also treated simi-

115004

larly, i.e., m~eL � m~eR � m~l, but the universality of squarkand slepton masses is not assumed. These mass patternscan be realized in some variations of the mSUGRA model.

For the third generation squarks the above equality maynot hold even in mSUGRA due to, for example, RGEdriven by the Yukawa terms. However, the SU�2�L invari-ance require m~bL

� m~tL and m~�L � m~‘L, neglecting the

D-term contributions. In this paper the right and left-squarks soft masses of the third generation are assumedto be equal to m~q. The physical masses are calculated afterincorporating the corresponding fermion masses, theD-terms etc. in the mass matrix.

For a given �, tan�, m~q, and Xr, the trilinear softbreaking mass parameters (At) of the top-squark sectorcan be calculated from Eq. (11). The same parameter forbottom-squark (Ab) and tau-slepton (A�) sectors are treatedas free input parameters. Using these inputs the thirdgeneration squark and slepton masses and mixing anglescan be calculated.

We have used theCP-odd Higgs boson mass (MA) and�as the free parameters in SUSPECT V2.33 to calculate thelighter CP-even Higgs boson mass.

The off-diagonal terms in the top-squark mass matrixhave the form mtXrm~q. So, the masses and the mixingangle of the top-squark sector are controlled by theseparameters. For negative Xr, the m~t1 and m~t2 are exactlythe same as that for positive Xr. The mixing angle dependon the relative magnitude of the diagonal and off-diagonalterms in the top-squark mass matrix.

The m~t1 (mh0 ) as a function of Xr has been shown formt � 172:7 in the left (right) panel of Fig. 5 for four valuesof tan� (see the figure caption for other input MSSMparameters). m~t1 is insensitive to mt and tan�.

The lighter CP-even Higgs boson mass has a periodicdependence on Xr. For a given Xr, if tan� increases theHiggs boson mass also increases. If tan� � 5�15� andmt � 172:7, mh0 ’ 114:4 for Xr ’ 2:0�1:5� and mh0 ’111:4 for Xr ’ 1:75�1:30�. It is clear from Fig. 5 thatboth the ~t1-NLSP condition and the experimental boundon mh0 may be satisfied for large positive values of Xr. Inour subsequent analyses we shall only consider such valuesof Xr.

Because of the large number of free parameters inMSSM we varied two important parameters at a timekeeping the rest fixed.

(i) X

-7

r � tan�—We have demarcated the ~t1-NLSP re-gions in Xr � tan� plane in the left (right) panel ofFig. 6 for mt � 172:7�178:0�. Several contours forfixed values of the total BR of 4-body decays arealso shown. Comparing with Fig. 5 we find that themass of the ~t1-NLSP is kinematically accessible atTevatron Run-II. For each tan� there is a range ofXr�jXrjmin < jXrj< jXrjmax� satisfying the ~t1-NLSPcondition (recall the A0 ranges in Figs. 1 and 2).From each BR contour it follows that when Xr is

FIG. 6.MSSM(right)shows t(114.4,

FIG. 5. The m~t1 �mh0 � as a function of Xr in the MSSM in the left (right) panel for mt � 172:7. The MSSM model parameters are:M2 � 300, � � 300, MA � 500, Ab � 200, A� � 300, m~q � 400, and m~‘ � 250. The choices of tan� are shown in the legends. The~t1-NLSP criterion is relaxed in this particular figure.


increasing smaller values of tan� are required. Forlarge Xr, m~t1 decreases and the virtuality of theexchanged chargino and/or slepton becomes largeand tends to suppress the 4-body decay width. Inorder to keep the BR fixed tan�must be correspond-ingly reduced.We have also plotted several contours for mh0 ’114:4 and a few other Higgs boson masses.Comparing the left panel of Fig. 6 with the rightpanel of Fig. 5 we find that for mh0 ’ 111:4 the~t1-NLSP condition is violated.As has already been noted, for mt � 178:0 the con-straints imposed on the parameter space by thebound on mh0 are less severe (see the right panelof Fig. 6). The total 4-body BR can be as large as90% for tan� ’ 9 for both choices of mt in Fig. 6.Even at tan� ’ 24 the total 4-body BR is small butnot negligible ( 10%). In this case it will beinteresting to search for hadronic 4-body decays of~t1 at the Large Hadron Collider (LHC) or theInternational Linear Collider (ILC). Even if tan� ’3, mh0 ’ 114:4 is allowed. Consequently almost theentire parameter space in the right panel of Fig. 6 isallowed. Such low values of tan� naturally guaran-tees large total BR of the 4-body decay modes.Moreover we find that the BR of leptonic 4-bodydecays can be as large as 20%, required for an

The ~t1 is the NLSP in the whole marked region consistent with omodel for M2 � 300, � � 300, MA � 500, Ab � 200, A� � 300,

panel. The total 4-body decay BRs of the lighter top squark are shohe regions where BR�~t1 ! b~�0

1‘�‘� ’ 20% for ‘ � e and�. The Hig118.0, 120.0, and 122.0) are also plotted in the left (right) panel.

115004-8

observable signal at Tevatron Run-II [6].It is expected that the upcoming collider searcheswill either discover the Higgs boson or push its massbound upward. Moreover, the theoretical uncertain-ties on the Higgs boson mass is likely to be signifi-cantly reduced (�mh0 & 0:5) [16,18]. We wouldnow like to comment on the impact of such possibleimprovements in our understanding of mh0 on ~t1searches. For illustration we have plotted contoursof mh0 ’ 116:0, 118.0 and 119.0 (118.0, 120.0 and122.0) for mt � 172:7�178:0� in the left (right)panel of Fig. 6. The total 4-body BR * 30% canbe completely (nearly) ruled out by an improvedHiggs boson mass bound, mh0 * 119:0�122:0� ifmt � 172:7�178:0� for our choice of other SUSYparameters.

(ii) X
r �MA—We have shown the ~t1-NLSP regions, afew 4-body BR and mh0 contours in the Xr �MAplane for mt � 172:7 and tan� � 5�9� in the left(right) panel of Fig. 7. The other MSSM modelparameters used are mentioned in the figure caption.The lighter top-squark mass is almost independentof MA if the other MSSM parameters are fixed. So,the ~t1-NLSP region remains the same when MA isvaried for fixed Xr. The input value ofMA can affectthe CCB condition (see Eqns. (5), (8), and (10))through the soft breaking Higgs boson masses.
ther sparticle mass bounds in the Xr � tan� plane in them~q � 400, m~‘ � 250, and mt � 172:7�178:0� in the leftwn by the contours with different point styles. The ‘‘�’’gs boson mass contours for 114.4, 116.0, 118.0, and 119.0

Br 0.2

Br 0.9Br 0.5Br 0.3Br 0.1

tanβ = 5

115

115.7

111.4

114.4

mt = 172.7

XrM

A(G

eV)

2.42.32.22.12

1000

900

800

700

600

500

400

300

200

Br 0.2

Br 0.9Br 0.5Br 0.3Br 0.1

tanβ = 9

mt = 172.7

119

118.4

117.8117.4117116

Xr

MA

(GeV

)

2.52.42.32.22.12

1000

900

800

700

600

500

400

300

200

FIG. 7. The ~t1 is the NLSP in the whole marked region in the Xr �MA plane in the MSSM model for M2 � 300, � � 300, m~q �400, m~‘ � 250, Ab � 200, A� � 300, mt � 172:7 and tan� � 5�9� in the left (right) panel. The total 4-body decay BR of lighter topsquark are shown by different point styles. The ‘‘�’’ shows the regions where BR�~t1 ! b~�0

1‘�‘� ’ 20% for ‘ � e and �. The Higgsboson mass contours for 111.4, 114.4, 115.0, and 115.7 (116.0, 117.0, 117.4, 117.8, 118.4, and 119.0) are also plotted in the left (right)panel.


This disallows some regions of the parameter spacefor large Xr. However, the CP-even Higgs bosonmass depends on MA. On the other hand MA has anontrivial impact on the loop decay width, see,Eqns. (3) and (5) or Ref. [1]. For a fixed Xr if MAincreases the loop decay width also increases.However, the 4-body decay width is insensitive toMA. So, for large MA the total 4-body BR decreasesfor a given Xr. This is very transparent in Fig. 7 forrelatively large values of Xr corresponding to largevirtuality of the exchanged particles in the 4-bodydecay amplitude.We find that the total 4-body BR (leptonic BR) canbe as large as 90% (20%) in significant regions ofthe parameter space for tan� � 5. If the experimen-tal bound on the Higgs boson mass is invoked thenthe total 4-body BR (leptonic BR) ’ 90% (20%) isallowed if MA * 300�400� and Xr & 2:17�2:08� fortan� � 5 (see the left panel of Fig. 7). We haveshown mh0 ’ 111:4 contour in the left panel ofFig. 7. It is clear that almost entire parameter spaceis allowed by the weaker bound on the Higgs bosonmass. It has also been checked that the whole~t1-NLSP region is disfavored for mh0 * 117:4.For tan� � 9, in the entire ~t1-NLSP region in Fig. 7corresponds to mh0 * 114:4. However, the total 4-body BR (leptonic BR) ’ 90% (20%) occurs only inrestricted regions, i.e., MA & 500 and Xr & 2:025as can be seen from the right panel of Fig. 7.Since the lower bound on Higgs boson mass mayimprove in future, we have shown several mh0 con-tours, e.g., 115.0 and 115.7 (116.0, 117.0, 117.4,117.8, 118.4, and 119.0) for tan� � 5�9� in the left(right) panel of Fig. 7 to illustrate the consequencesfor the BR under study.

III. CONCLUSIONS

The lighter top squark turns out to be the NLSP in asignificant region of the supersymmetric parameter spaceboth in constrained and unconstrained models like

115004

mSUGRA and MSSM, respectively. The ~t1-NLSP hastwo competing decay modes for the top-squark masseswithin the reach of the Tevatron Run-II. They are theloop induced ~t1 ! c~�0

1 and the 4-body decay, ~t1 !b~�0

1f �f0. These two decay modes may have comparabledecay widths over large regions of parameter spaces inboth the models [2,6]. However, for low tan� the loopdecay width is suppressed which in turn enhances the 4-body decay BRs. On the other hand the current boundmh0 * 114:4 from LEP disfavors low values of tan�. Itis therefore important to identify the parameter spacesallowing large total BRs of the 4-body decays after impos-ing the mh0 constraint.

The theoretical uncertainties in the calculated mh0 usingany currently available code is 3 due to the yet unknownhigher-order corrections. In view of this we have assumedthat realistically the bound on the calculated mh0 can berelaxed to mh0 * 111:4.

The masses of ~t1 and mh0 crucially depend on the inputvalue of mt. We have considered the current world averagecentral value of mt which is mt � 172:7 as well as mt �178:0, which is approximately 2 away from the centralvalue and yields somewhat weaker constraints.

We have demarcated the ~t1-NLSP regions in themSUGRA model for mt � 172:7 and 178.0 in the A0 �tan� (Figs. 1 and 2) and m0 �m1=2 planes (Fig. 4). Wefound that for a given tan�, a limited range of A0 withA0 jA0jmin allows large 4-body BRs. For demonstrationwe have presented in the A0 � tan� plane a few 4-body BRcontours for two representative choices ofm0 andm1=2 (seeSec. II A and the figure captions). For mt � 172:7 allparameter spaces with the total BR of the 4-body decays>10 (40)% are disfavored even if the weaker Higgs bosonmass bound is invoked [see the left (right) panel of Fig. 1].The experimental bound on mh0 rules out any parameterspace with numerically significant BR of the above decays.For mt � 178:0 the experimental bound on mh0 leaves noroom for total BR >10 (20)% (see the left (right) panel ofFig. 2). The weaker bound, however, allows BRs as large as30–50% in a small region of the parameter space (the rightpanel of the above figure).

-9


Complementary information in them0 �m1=2 plane, forA0 � �630, sign��> 0 and mt � 178:0 are presented inFig. 4. For tan� � 5 the weaker constraint on mh0 allowsBRs as large as 60% (see the left panel) in a small parame-ter space, while the experimental bound on mh0 practicallyrules out the possibility of a sizable BR of the decaychannels of interest. For tan� � 9, the total 4-body BRcan be at most 10% due to the experimental bound onHiggs boson mass.

We have also excluded small regions in A0 � tan� andm0 �m1=2 plane from the CDF limits on the mass of ~t1 inthe jets� E6 T channel. In most cases, however, these con-straints are superseeded even by the weaker bound on mh0 .

In conclusion we note that in mSUGRA the uncertaintiesinmh0 andmt leave open the possibility that the total BR ofthe 4-body decay of the ~t1-NLSP is sizable although theparameter space corresponding to this scenario is severelysqueezed by the bound on mh0 . Perhaps the main futureinterest in this decay channel in the context of mSUGRAwould be to observe it as a rare decay mode at the LHC orthe ILC. Moreover, the existing bounds on m~t1 obtained byassuming 100% BR of the loop decay are approximatelyvalid in most of the parameter space.

In MSSM we have used the parameter Xr [see Eq. (11)]as an input. The interplay of the ~t1-NLSP condition and theHiggs boson mass constraints can be easily followed byvarying Xr (see Fig. 5). One of the important conclusions isthat sizable total BR of the 4-body decays of ~t1 can occurfor the stronger constraints mt � 172:7, mh0 * 114:4 andintermediate values of tan� (10–15) (see Fig. 6). Forexample, this BR can be as large as 90% in the Xr �tan� plane even for tan� 9, mh0 * 114:4 and mt �172:7 and 178.0. For lower values of tan� the leptonic 4-body BR ’ 20%, which can lead to an observable signal at

115004

Tevatron Run-II [6], is also consistent with the aboveconstraints. The total BR can be restricted to the level of10–20% only if the bound onmh0 is significantly improvedand mt is determined with better precision. Com-plementary information leading to similar conclusionsare presented also in the Xr �MA plane (see Fig. 7).

In conclusion we emphasize that the bounds onm~t1 fromTevatron obtained by assuming the complete dominance ofthe loop decay of ~t1 requires revision if we want thesebounds to be model independent. This is true even thoughthe strong constraint on mh0 from LEP squeezes the pa-rameter space which favors the competing 4-body decaysof ~t1. Based on our earlier analyses we conclude that the 4-body mode could still be the main discovery channel for ~t1at both Tevatron Run-II and LHC. As future strategies forthe discovering ~t1 are planned, attention should be paid tothe lighter CP-even Higgs boson search. If its mass boundis significantly improved, the interest in the 4-body decaywould be to look for it as a rare decay. It is expected that theHiggs boson may show up at an early stage of LHC theexperiments and, at least, some preliminary information onits mass will be available. This information will finallydecide the viability of ~t1 search in the 4-body decaychannel.

ACKNOWLEDGMENTS

The author thanks Amitava Datta and AmitavaRaychaudhuri for suggestions, comments and carefulreading of the manuscript. The author is thankful toM. Guchait, M. Maity, and S. Poddar for discussions. Apart of the work was supported by the Project No. (SP/S2/K-10/2001) of the Department of Science and Technology(DST), India.

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