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PHYSICAL REVIEW D 1 DECEMBER 1996VOLUME 54, NUMBER 11

0556-28

Constraints on the minimal supergravity model from nonstandard vacua

Howard Baer, Michal Brhlik, and Diego Castan˜oDepartment of Physics, Florida State University, Tallahassee, Florida 32306

~Received 29 July 1996!

We evaluate regions of parameter space in the minimal supergravity model where ‘‘unbounded from below’~UFB! or charge or color-breaking minima~CCB! occur. Our analysis includes the most important terms fromthe one-loop effective potential. We note a peculiar discontinuity of results depending on how renormalizationgroup improvement is performed: One case leads to a UFB potential throughout the model parameter spacwhile the other typically agrees quite well with similar calculations performed using only the tree levelpotential. We compare our results with constraints from cosmology and naturalness and find a preferred regioof parameter space which impliesmg &725 GeV,mq &650 GeV,mW1

&225 GeV, andml R&220 GeV. We

discuss the consequences of our results for supersymmetry searches at various colliding beam facilitie@S0556-2821~96!03423-6#

PACS number~s!: 11.30.Pb, 11.15.Ex, 14.80.Ly

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I. INTRODUCTION

The minimal supersymmetric standard model~MSSM! isone of the leading candidate models@1# for physics beyondthe standard model~SM!. In this theory, one begins with theSM particles~but with two Higgs doublets to ultimately ensure anomaly cancellation!; supersymmetrization then leadto partner particles for each SM particle which differ by sp12 . Supersymmetry breaking is implemented by adding eplicit soft supersymmetry-breaking terms. This proceduleads to a particle physics model with*100 free parameterswhich ought to be valid at the weak scale.

To reduce the number of free parameters, one needtheory of how the soft SUSY-breaking terms arise, i.e., hosupersymmetry is broken. In the minimal supergrav~SUGRA! model @2#, supersymmetry is spontaneously brken via a hidden sector field vacuum expectation va~VEV!, and the SUSY breaking is communicated to the vible sector via gravitational interactions. For a flat Ka¨hlermetric, this leads to a common scalar massm0 and commontrilinear and bilinear termsA0 and B0 at some high scaleMGUT2MPl , where the former choice is usually taken dueapparent gauge coupling unification at;231016 GeV; inaddition, we assume gaugino mass unification atMGUT. Thehigh scale mass terms and couplings are then linked to wescale values via renormalization group evolution. Eletroweak symmetry breaking@3#, which is hidden at highscales, is then induced by the large top-quark Yukawa cpling, which drives one of the Higgs field masses to a netive value. Minimization of the scalar potential allows oneeffectively replaceB by tanb and express the magnitude~butnot the sign! of the Higgsino massm in terms ofMZ . Theresulting parameter space of this model is thus usually givby the set„m0 ,m1/2,A0 ,tanb,sgn~m!….

Not all values of the above~411!-dimensional parameterspace of the minimal SUGRA mode are allowed. For istance, the top and bottom Yukawa couplings are driveninfinity somewhere betweenMZ andMGUT for tanb&1.5 and*50 ~depending on the value ofmt! @4#. For other parameterchoices, the lightest chargino or lightest slepton~or topsquark! can be the lightest SUSY particle, which would vio

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late limits on, for instance, heavy exotic nuclei or atoms. Fyet other parameter choices, electroweak symmetry breakleads to the wrong value ofMZ , and so these parametechoices are ruled out. In addition, there are cosmologicbounds from the relic density of neutralinos produced in thbig bang @5#. Requiring the universe to be older than;103109 years leads to only a subset of the parameter spabeing allowed, although this bound could be evaded bylowing for a small amount ofR-parity violation. Finally,certain regions of parameter space are rejected by negasearches for sparticles at colliding beam experiments, suchthose at the CERNe1e2 collider LEP and the FermilabTevatron@6#.

An additional constraint on minimal SUGRA parametercan be obtained by requiring that the global minimum of thscalar potential be indeed the minimum that leads to apppriate electroweak symmetry breaking. In the SM, thereonly a single direction in the field space of the scalar potetial, and so appropriate electroweak symmetry breaking cbe assured. For the MSSM, the plethora of new scalar fiewhich are introduced leads to many possible directionsfield space where minima could develop which are deepthan the standard minimum. Thus parameter choices whlead to deeper minima should be excluded as well, since thwould lead to a universe with a nonstandard vacuum.

Constraints along the preceding lines were developedRef. @7# in the early 1980s, using the renormalization-groupimproved tree-level effective potential. It was noted by Gamberini et al. @8# that the renormalization-group-improvedtree-level potential was subject to large variations due to tuncertainty in the correct scale choiceQ at which it wasevaluated. Inclusion of one-loop corrections served to amliorate this condition. Recently, Casas, Lleyda, and Mun˜oz@9# have made a systematic survey of all possible dangerodirections in scalar field space that can potentially leadminima deeper than the standard one. These have beenegorized as field directions that are either unbounded frobelow ~UFB! ~at the tree level! or that lead to charge- orcolor-breaking~CCB! minima. For simplicity, their analysisuses the tree-level scalar potential, but evaluated at an omized mass scale where one-loop corrections ought to

6944 © 1996 The American Physical Society

l

54 6945CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

only a small effect. Working within the minimal SUGRAmodel, they considered models withB05A02m0 orB052m0 and showed that significant regions of paramespace could be excluded via this method.

In the present work, one of our main goals is to delinethe parameter space regions where nonstandard poteminima develop in such a manner as to facilitate compasons with other constraints, including recently calculatedsults on the neutralino relic density@10# and parameter spacregions favored by fine-tuning considerations@11#. In addi-tion, expectations for supersymmetry at LEP 2@12#, theTevatron MI and TeV33 upgrades@13#, the CERN LargeHadron Collider ~LHC! @14#, and Next Linear Collider~NLC! @15# have been calculated within the minimaSUGRA framework. We also compare the nonstandvacuum constraints with the various collider expectatioand draw some conclusions. For instance, combiningnonstandard vacuum constraints with the most favoredrameter space regions from fine-tuning and cosmology sgests that the Fermilab TeV33 upgrade stands a high chato discover SUSY via theW1Z2→3l signal.

In the present work, we also adopt a somewhat differcalculational scheme from that employed in Ref.@9#. For allfield directions considered, we implement a renormalizatgroup ~RG! improvement to calculate the one-loop effectivpotential. We find that the inclusion of the one-loop corretion has important consequences. The one-loop correcalmost always represents a significant contribution totree-level potential. Nevertheless, our overall results agvery well with those of@9# for a ‘‘proper’’ choice in RGimprovement scheme. For reference, we shall call this‘‘ a case,’’ and it represents our main results. However,other choices of RG improvement, we find that the one-locorrection can be so dominant as to lead to unbounded fbelow ~UFB! potentials everywhere in parametespace: We refer to this as the ‘‘v case.’’ Because this is amultiscale problem, it is not entirely clear how to procewith RG improvement, and it is this ambiguity that leadsthe two cases above. The validity of our results hingestwo main assumptions. The first concerns the adequaccutting the expansion at one-loop. It is beyond the presanalysis to ascertain the significance of the two-loop conbution in any of the cases considered. However, becausthe dependence of our results on the details of RG improment, we believe that the two-loop contribution may be iportant. Second, we have only included the contributionthe top-quark–top-squark sector in our calculations ofone-loop correction. It remains to be determined if the incsion of the other fields will significantly affect the results.

We note briefly that the work of@16,17# and @18# ad-vances the idea that we may indeed exist in a false vacuand that the tunneling rate from our present vacuum tUFB or CCB vacuum might be small relative to the agethe universe. In this case, the following derived constraiwould not be meaningful. Such a philosophy must, howevbe reconciled@19# with the fact that we live in a world inwhich the cosmological constant either vanishes or istremely small. This is empirical, albeit indirect, evidence fsome mechanism which seeks to enforce the principle‘‘the cosmological constant of the true vacuum is zero.’’ Itdifficult to conceive of circumstances where we could te

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ably entertain both the idea that we are living in a falsevacuum and the idea that the smallness of the cosmologicaconstant has a natural solution, since this would require aprinciple which would set the cosmological constant to zeroin a false, broken vacuum while simultaneously leaving thetrue, broken vacuum with a large negative cosmological con-stant.

The organization of the paper is as follows. In Sec. II wereview the MSSM scalar potential and give a brief summaryof the UFB and CCB directions delineated in Ref.@9#. InSec. III we present our calculational procedure, and in Sec.IV present results of our scans over SUGRA parameterspace. In Sec. V we give a brief summary of our results.Detailed formulas for the effective potential in various UFBand CCB directions are included in Appendix A, while somecomputationally useful formulas for evaluating the effectivepotential in the limit of large VEV’s are presented in Appen-dix B.

II. DANGEROUS DIRECTIONS IN FIELD SPACE

The scalar potential of the MSSM can be written as

V5VF1VD1Vsoft, ~2.1!

where

VF5(a

U ]W

]faU25U(

iui yuiQi1mFdU2

1U(i

~ydidiQi1yeieiLi !1mFuU2

1(i

uyuiFuQi u21(i

uydiFdQi u2

1(i

uyeiFdLi u21(i

uyuiuiFu1ydidiFdu2

1(i

uyeieiFdu2, ~2.2!

VD5 12(

aga2S (

afa†TafaD 25g82

2 F(i

~ 16 uQi u22

23 uui u2

1 13 udi u22

12 uLi u21uei u2!1 1

2 uFuu2212 uFdu2G2

1g22

8 F(i

~Qi†tWQi1Li

†tWLi !1Fu†tWFu1Fd

†tWFdG2

1g32

8 F(i

~Qi†lWQi2ui

†lW * ui2di†lW * di !G2, ~2.3!

where tW5(t1 ,t2 ,t3) are the SU~2! Pauli matrices andlW5(l1 ,...,l8) are the Gell-Mann SU~3! matrices:

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6946 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

Vsoft5(a

mfa

2 ufau21~BmFuFd1c.c.!

1(i

~AuiyuiuiFuQi1Adi

ydidiFdQi

1AeiyeieiFdLi1c.c.!, ~2.4!

and the superpotentialW is

W5(i

~yuiuiFuQi1ydidiFdQi1yeieiFdLi !1mFuFd .

~2.5!

In the above,fa runs over the scalar components of thchiral superfields anda andi are gauge group and generatioindices, respectively.Qi (Li) are the scalar partners of thquark ~lepton! SU~2!L doublets, andui , di , and ei are thescalar partners of the SU~2!L singlets.Fu andFd are the twoHiggs doublets. When all the above summations are pformed, one is left with a very lengthy expression for tscalar potential. In the following, usually only a small number of scalar fields develop VEV’s, and so only a subsetthe many terms of the scalar potential are relevant.

For the usual breaking of electroweak symmetry in tMSSM, onlyFu andFd develop VEV’s, so that the relevanpart of the above potential is just

V05m12uFdu21m2

2uFuu21m32~FuFd1H.c.!

1g82

8~Fu

†Fu2Fd†Fd!

21g22

8~Fu

†tWFu1Fd†tWFd!

2,

~2.6!

where the masses appearing above are defined as

m125mFd

2 1m2, ~2.7!

ene

er-he-of

het

m225mFu

2 1m2, ~2.8!

m325Bm. ~2.9!

The one-loop contribution to the scalar potential is given b

DV1~Q!51

64p2 StrHM4S lnM2

Q2 23

2D J5

1

64p2 (i

~21!2si~2si11!mi4S ln mi

2

Q223

2D ,~2.10!

whereM2 is the field-dependent squared mass matrix of thmodel andmi is the mass of thei th particle of spinsi . In theone-loop correction, we shall always include only the contrbution from the top-quark–top-squark sector, this being geerally the most significant.

Minimization, at a scaleQ usually taken to beMZ , yieldstwo conditions on the parameters:

1

2mZ25

m122m2

2tan2b

tan2b21, ~2.11!

wheremZ25g2v25(g821g 2

2)v2/2, v25v u21v d

2, and

Bm5 12 ~m1

21m22!sin2b, ~2.12!

with tanb5vu/vd , and where the barred masses are the onloop analogues of~2.7! and ~2.8!—see, e.g., Ref.@20#. Atthis point, the minimum of the tree potential is

Vmin521

4g2@~m1

22m22!1~m1

21m22!cos2b#2. ~2.13!

The top quark and top squark masses are given by

mt5ytvu , ~2.14!

mt 1,2

2 5mt21 1

2 ~mQ3

2 1mu3

2 !1 14mZ

2cos2b6A@ 12 ~mQ3

2 2mu3

2 !1 112 ~8mW

2 25mZ2!cos2b#21mt

2~At1m cotb!2, ~2.15!

’s

wheremW2 5g 2

2v2/2. We will compare the value of the po-tential at the MSSM minimum with the value at the minimum for other field configurations and use this to rejeMSSM scenarios with false vacua.

The dangerous directions in field space have been caterized in Ref.@9# as various UFB and CCB directions. For thUFB directions, the trilinear scalar terms are unimportant. Tfind the deepest directions in field space, one searchesdirections where theD terms of Eq.~2.3! will be small orvanishing. The various UFB directions are characterizedfollows.

UFB-1. Here, only the fieldsFu andFd obtain VEV’s,with ^Fu&5^Fd& in order to cancelD terms in Eq.~2.6!.

UFB-2. In addition to the VEV’sFu& and ^Fd&, one has

-ct

go-eofor

as

a VEV for the third generation slepton field in then direc-tion: ^L3&n . The VEV’s are related as in Eq.~A3!.

UFB-3a. In this case, the relevant VEV’s are^Fu&,^L3& e

25^e3&2, and ^L2&n . This direction reputedly leads to

the most stringent bounds on parameter space. The VEVare related as in Eq.~A10!.

UFB-3b. This case is similar to UFB-3a, but instead of thefirst two slepton fields,Q3& d

25^d3&2 develop VEV’s. The

VEV’s are related as in Eq.~A16!.The various CCB directions each involve a particular tri-

linear coupling. For each trilinear coupling, there are tworelevant directions: CCB~a! ~equivalent to Casaset al., CCB-1!, and CCB~b! ~which combines the CCB-2 and CCB-3cases of Casaset al.!. The CCB~a! direction is not relevant

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54 6947CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

for the top trilinear term. Summing over the various trilineaterms and CCB directions can yield at least 17 possiblerections ~some other possible directions lead to essentiathe same constraints!. For illustration, we investigated thefollowing cases, which include cases with the largest asmallest Yukawa couplings.

CCB~a!-UP. The relevant VEV’s areFu&, ^Q1&u , ^u1&,^Q3& d

25^d3&2, and ^L3&n , with ^Fd&50. The VEV’s are re-

lated as in Eq.~A22!.CCB~b!-UP. The relevant VEV’s areFu&, ^Fd&, ^Q1&u ,

^u1&, and^L3&n . The VEV’s are related as in Eq.~A28!.CCB~b!-TOP. The relevant VEV’s areFu&, ^Fd&, ^Q3&u ,

^u3&, and^L3&n . The VEV’s are related as in Eq.~A41!.CCB~a!-ELECTRON. The relevant VEV’s areFd&,

^L1&e , ^e1&, and^Q3& u25^u3&

2, with ^Fu&50. The VEV’s arerelated as in Eq.~A57!.

CCB~b!-ELECTRON. Last, the relevant VEV’s areFd&,^Fu&, ^L1&e , and^e1&. The VEV’s are related as in Eq.~A63!.

III. CALCULATIONAL DETAILS

The standard procedure for studying the MSSM has beas summarized above, to fix the parametersB0 and m0 inorder to achieve symmetry breaking as dictated by Eq~2.11! and~2.12! with v5174 GeV. Furthermore, the choicein minimization scale being in theMZ range is dictated bythe desired vacuum expectation value and is validated byuse of the one-loop correction. Unfortunately, this methodoes not lend itself to the present task, sincea priori theminimum of the potential in a given configuration is unknown. The task is complicated since we must be ableprobe the potential for significantly different field values.

In order to validate the use of the one-loop effective ptential, one must ensure that not only the couplings be pturbative, but that the logarithms be small as well. This is tprocess of RG improvement@21#. In problems with only onemass scale, RG improvement is straightforward. The logrithm appearing in the one-loop correction can be masmall, indeed to vanish, for any choice in field value by aappropriate choice in renormalization scaleQ. This proce-dure yields theQ-independent one-loop RG-improved potential. For the cases in which we are interested, there are seral mass scales. Since in general no scale existssimultaneously makes all the logarithms vanish, we settlethe scale at which the logarithms are, simultaneously, opmally small. In this way, we construct the one-loop effectivpotential. Note that in such cases the one-loop correctdoes not vanish and indeed may represent a significant ctribution to the tree-level part. In our subsequent results,always include the one-loop correction in our evaluationthe effective potential. Figure 1 demonstrates the signcance of the one-loop correction for a representative cawith A050, m05100 GeV,m1/25200 GeV, tanb~MZ!52,andm,0. In this example, we have employed thea scheme~see below! for RG improvement. From Fig. 1~b! we see thatthe difference between the value of the tree potential atminimum and the one-loop effective potential in the UFB3~a! direction is almost a factor of 4. There is also an effein the standard MSSM direction as seen in Fig. 1~a!. Thisparticular point in parameter space is ruled out sinVminUFB23~a!,Vmin

MSSM. We implement the RG improvemen

ri-ly

d

n,

s.

hed

-to

-r-e

a-en

-ev-hatorti-eonon-effi-se

he-t

e

procedure as follows:~1! At each RG scaleQ find the fieldvaluef that minimizes the function

f ~f,Q!5(i

@ ln$mi2~f,Q!/Q2%2x#2 ~3.1!

and ~2! store this value and the correspondingV1(f,Q).This results in the function„f,VRGI~f!… whose minimum canthen be calculated.

Given the standard numerical procedure involved in RGstudies, in which Runge-Kutta routines are used to integrateoverQ, the above procedure of finding the optimalf at eachscaleQ is the most efficient, since we construct the RG-improved potential simultaneously as we evolve the RG pa-rameters of the MSSM. Also, we note that includingmi

4

coefficients in f (f,Q) ~as they appear inDV1! leads topathological results. Namely, at all scalesQ, thef -minimizing field value tends to zero. This is a pathology ofthe method we are employing. Had we instead fixedf andfound thef -minimizing value ofQ, this problem would notbe present. Because of the ambiguities in RG improvementin multiscale problems, we examined several prescriptionsfor constructing the effective potential. Equation~3.1! withx53/2 led to thea case. We also triedx50. We used thefunctional form f5ln2$m2/Q2%, and tried the top quark andtop squark masses as possible choices form. All of thesechoices led to thev case. In this case, the potentials alongthe UFB-3~b! and CCB~a!-UP directions were unbounded

FIG. 1. Plots of the one-loop correction and tree and one-loopeffective potentials along the MSSM and UFB-3~a! vacuum direc-tions in the A050, m05100 GeV, m1/25200 GeV, m,0, andtanb52 case. Renormalization group improvement was imple-mented using thea prescription.

6948 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

from below everywhere in parameter space. Although thresult is interesting, we believe that two-loop leading logrithms may remedy this curious situation. Therefore we dcount these results for the moment. Along all other directioconsidered, thev results were similar to thea results as oneexpects. Figure 2 displays results in thev case similar to Fig.1. Comparing Figs. 1 and 2, the one-loop effective potentiaare essentially identical for the MSSM direction. And athough the UFB-3~a! potentials are clearly different, thevpotential in this direction remains well behaved. Figuredisplays the logarithm of the VEV versusx in two vacuumdirections. In the UFB-3~a! direction, there is some minorchange in the VEV, but not very significant asx varies fromthe a to v value. In contrast, the VEV suffers a drasticdiscontinuous jump at aroundx51 in the CCB~a!-UP direc-tion. This same discontinuity occurs in the UFB-3~b! case.

From Fig. 3, we see that alteringx ~which effectivelychanges ourQ choice! changes our results dramatically inthe CCB~a!-UP and UFB-3~b! directions. These two direc-tions are special in that their one-loop contributions adominant in the large VEV domain. This comes in particulafrom contributions to the top squark masses by they b,t

2 d2

terms @see Eqs.~A19!, ~A20!, ~A25!, and ~A26!#; bear inmind that the quadratic~orG6! contributions inx are alwayssmall or zero. For these cases, in the large VEV domain,one-loop correctionDV1 is obviously unstable againstQ.For the case depicted in Fig. 3, we have verified thV1~Q,Fu51016GeV! has a large, negative slope and changsign atQ'431015 GeV, leading to a potential which is veryunstable against variations in scale choice. We assumethis instability is due to a need to include higher terms in th

FIG. 2. Plots similar to Fig. 1, but with renormalization grouimprovement implemented using thev prescription.

isa-is-ns

lsl-

3

,

rer

the

ates

thate

effective potential for these cases.It is interesting to note that the standard procedure for

computing the MSSM minimum gives results that differ,sometimes greatly, from the above RG-improved procedure.We find that the one-loop correction to the MSSM potentialdoes a poor job of stabilizing the potential againstQ nearMZ . This was alluded to in@22#. The result is surprisinggiven the conventional belief that the one-loop correctionstabilizes the potential in the electroweak range. A more de-tailed study of this issue is in progress. In Fig. 4, we dem-

p FIG. 3. Plots of the logarithm of the VEV versusx, a parameterappearing in the renormalization group improvement function, forthe vacuum directions UFB-3~b! and CCB~a!-UP.

FIG. 4. Evolution of the VEV by using the renormalizationgroupg functions of the Higgs fields~solid line! and by tracking theminimum of the one-loop potential~dashed line! as a function ofthe renormalization scaleQ.

o

nl

lvw

tg

h

54 6949CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

onstrate the problem by displaying theQ evolution of theVEV using two different methods in the caseA050,m05m1/25100 GeV, tanb~MZ!52, andm,0. The solid linerepresents the evolution as dictated by the RGg functions ofthe Higgs fields~see@20#!. The dashed line represents thtracking of the minimum of the one-loop effective potentiaHere we have included the contributions fromall particles inDV1.

For all cases, we have explored the dangerous directidelineated in Ref.@9#; these directions were obtained usinonly the tree-level potential. A better procedure would ivolve optimization of the full one-loop effective potentiaThis may in principle lead to even more precipitous diretions. However, since this procedure leads to very unwieexpressions, we have opted to explore the tree-level-deridirections in field space, although in these directionsmake comparisons using the one-loop-corrected scalartential.

We emphasize the importance of using the one-loop crection to the scalar potential, since its inclusion can alterdepth of the minimum significantly as was evident in Fi1~b!. Furthermore, the startling results of thev case were aconsequence of using the one-loop correction to computepotential. Had the one-loop correction been ignored,choices of RG improvement we tried would have led to rsults similar to@9#. Figure 5~a! shows the potential along theCCB~a!-UP direction @in the case A050, m05100,m1/25200, tanb~MZ!52, andm,0# using only the tree-levelpotential. This point would have been excluded by tCCB~a!-UP constraint sinceVmin

CCB~a!-UP,VminMSSM. However,

including the one-loop correction in this case leads to t

FIG. 5. Plots of the potential in the CCB~a!-UP case. In~a! thetree potential is displayed. In~b! the one-loop potential is displayedfor botha andv prescriptions.

el.

nsg-.c-dyedepo-

or-he.

thealle-

e

he

potential displayed in Fig. 5~b!. It appears unbounded frombelow as last seen at field values nearing the Planck scale inthe v case, thus also ruling out this point. However, in thesame figure, thea-case one-loop effective potential is notunbounded from below and indeed does not rule out thispoint ~in this direction!.

IV. RESULTS

Using the procedures outlined in Sec. III and AppendixesA and B, we explored regions of minimal SUGRA parameterspace for minima deeper than the standard MSSM one. Ourinitial scans took place in them0 vsm1/2 plane, to facilitatecomparison with recent results on fine-tuning, cosmology,and present and future collider searches. We fix tanb(MZ) tobe 2 or 10, and takeA050 andmt5170 GeV. Our searchwas performed in the ranges 0<m0, m1/2<500 GeV and thegrid was scanned with 25 GeV resolution. Figures 6~a!–6~d!display the regions where nonstandard global minima werediscovered. Of all the directions scanned for these plots, non-standard global vacua were found only in the UFB-3a direc-tion. In Fig. 6, we have encoded information about the mag-nitude of the VEV in the plotting symbol. Usingh5log10$v/vMSSM%, the squares represent 2,h,3, thecrosses 3,h,4, and the3’s 4,h,5. The most dangerousregions are those populated by squares: For these pointsthe ‘‘distance’’ between the standard and nonstandardminima is smallest, which would admit the largest rate fortunneling between them. Performing this scan using the ex-act prescription of Ref.@9# leads to nearly identical excludedregions.

We see from Fig. 6 that for all four frames, the region oflow m0 becomes excluded. As noted in Ref.@9#, this rulesout the so-called ‘‘no-scale’’ models which requirem050. Inaddition, in string models where supersymmetry is broken inthe dilaton sector, one is led to GUT or string scale softterms related bym1/252A05)m0 @23,24#. For this precisechoice of soft-term boundary conditions, much of the param-eter space is excluded by nonstandard minima. We furthernote that the excluded region rules out much of the SUGRAparameter space associated with light sleptons. In particular,

FIG. 6. Exclusion plots for them0 vs m1/2 plane based on theUFB-3~a! constraint. The squares represent 2,h,3, the crosses3,h,4, and the3’s 4,h,5.

e

e

r

,

e

ed

l

r

6950 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

taken literally, our results exclude regions where such decas Z2→ l L l and Z2→ n n take place.

Although Fig. 6 is plotted forA050, a similar excludedregion results for other choices of theA0 parameter. This isshown in Fig. 7, where we plot regions excluded in them0 vsA0 plane, for the same values of tanb andm, but for m1/2fixed at 200 GeV. The vacuum constraints exist for allA0values, but are smallest forA0;300 GeV.

In Fig. 8, we display a combined plot of Fig. 6 with superposed dark matter@10# and fine-tuning@11# contours. Theregions to the right of the solid line contours are cosmolocally excluded because they predict a relic densityVh2.1;this corresponds to a lifetime of the universe of less th103109 years. Cosmological models which take into accouCOBE data, nucleosynthesis, and large-scale structuremation prefer an inflationary cosmology, with a matter cotent of the universe comprising 60% cold dark matter~e.g.,neutralinos!, 30% hot dark matter~e.g., neutrinos!, and 10%baryonic matter. In this case, the preferred relic densityneutralinos should be 0.15,Vh2,0.4, i.e., the region be-tween the dot-dashed contours. In addition, Fig. 8 contatwo naturalness contours with varying degrees

FIG. 7. Exclusion plots for them0 vs A0 plane based on theUFB-3~a! constraint and withm1/25200 GeV. The symbols are thesame as in Fig. 6.

FIG. 8. Same as Fig. 6 but with superposed dark matter~dot-dashed and solid lines! and naturalness~dashed line! contours.

ays

-

gi-

anntfor-n-

of

insof

acceptability: g255 and 10@11#. The more encompassingcontour is a conservative estimate of a reasonable ‘‘tolerancelimit’’ for weak-scale supersymmetry. We see that the con-straint from false vacua overlaps considerably with the pre-ferred regions from cosmology and fine-tuning, leaving onlya small preferred region of parameter space aroundm0;100–200 andm1/2;100–250 in each frame. We notethat the resulting preferred region of parameter space re-quiresmg , mq , mW1

, andml R&650, 600, 220, and 175

GeV, respectively, for tanb52, andmg , mq , mW1, and

ml R&725, 650, 225, and 220 GeV for tanb510. The only

exception to these bounds is if the neutralino is poised nearthe peak of ans-channel pole in its annihilation cross sec-tion. These regions correspond to the narrow horizontal cor-ridors in the relic density contours.

Recently, the reach of the Fermilab Main Injector andTeV33 has been calculated in the same parameter spacframes@13#. By comparing the results of Ref.@13# with thepreferred parameter space discussed above, we see that thTeV33 option covers most of the preferred region from Fig.8~a! via the clean trilepton signal fromW1Z2→3l , the ex-ception being the region withm1/2*180 GeV, where thespoiler decay modeZ2→Z1h turns on. For Fig. 8~b!, TeV33will cover theentirepreferred region via clean trileptons. Forthe large tanb510 cases of Figs. 8~c! and 8~d!, the TeV33upgrade can see most, but not all, of the preferred parametespace regions. The reach of the Tevatron Main Injector issignificantly less than TeV33 for these preferred regions ofparameter space. The CERN LHC collider can of courseprobe all the preferred regions of parameter space. In factevent rates will be enormous for various multilepton1 mul-tijet 1 E” channels, which should facilitate precision mea-surements of parameters@14#. In particular, sleptons havemass less than 250 GeV in these regions and so ought to bvisible at the LHC. Finally, we note that both the lightchargino and right selectron have mass less than 250 GeV inthe preferred regions, so that both of these sparticles wouldbe accessible to Next Linear Collider~NLC! experimentsoperating atAs5500 GeV@15#.

V. CONCLUSION

The minimal SUGRA model provides a well-motivatedand phenomenologically viable picture of how weak-scalesupersymmetry might occur. The~411!-dimensional param-eter space can be constrained in numerous ways as discussin the Introduction. To constrain the model further, we havepursued the idea that parameter values that lead to globaminima in nonstandard directions, such as those with chargeor color breaking, should be excluded from consideration.There are cosmological issues pertaining to tunneling that wehave knowingly ignored. Nevertheless, with this limitationnoted, we searched for the preferred regions of parametespace. We analyzed the potentials carefully employing theone-loop correction including the contributions of the topquark and top squark in the calculation. In generating thepotential we found that the one-loop correction can signifi-cantly alter results based only on the tree approximation.

Because of the various scales present in the MSSM,renormalization group improvement has ambiguities associ-

54 6951CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

ated with it. We tried several procedures, but were ultimatled to two distinct results that we refer to as thea andvcases. In thev case, to our surprise, the entire paramespace of the model suffers from global minima along nostandard directions@namely, the UFB-3~b! and CCB~a!-UPdirections#. Since the two-loop correction may be significangiven the large value of the top quark mass, the results inv case, while intriguing, must not be taken too seriously.

In thea case, we are still left with a very restricted regioof parameter space after imposing in addition dark maand naturalness constraints. Most of this region shouldaccessible to the Fermilab TeV33 collider upgrade viaclean trilepton channel. This parameter space region shbe entirely explorable at the LHC and should yield a riharvest of multilepton signals for supersymmetry whiought to allow for precision determination of underlying prameters. In addition, both charginos and sleptons oughbe accessible to NLC experiments operating at jAs5500 GeV, so that the underlying assumptions of tminimal SUGRA model can be well tested.

ACKNOWLEDGMENTS

We thank Xerxes Tata and Greg Anderson for discsions. This research was supported in part by the U.S.partment of Energy under Grant No. DE-FG-05-87ER403

APPENDIX A

In this appendix, the cases from@9# that are considered inour investigation are reviewed. Also, formulas for the tquarks or top squarks are displayed in each case and fobottom quarks or bottom squarks in the last case.

UFB-1. In the UFB-1 case, the only fields acquiring nozero VEV’s are Fu&5^Fd&5x. The resulting tree-level po-tential in this direction is

V5~m121m2

222um32u!x2, ~A1!

wherem125mFd

2 1m2,m225mFu

2 1m2, andm325Bm. The top

quark mass isMt5ytx, and the top squark mass matrix etries are

MLL2 5mQ3

2 1yt2x2,

MRR2 5mu3

2 1yt2x2,

MLR2 5yt~m2At!x. ~A2!

UFB-2. In the UFB-2 case, an additional slepton fieldincluded to help control theD terms. The shifted fields aretherefore,

^Fu&5x,

^Fd&5gx,

^L3&n5gLx. ~A3!

The n subscript represents the SU~2! direction that has ac-quired the VEV. The scalar potential in this case is

ely

tern-

t,the

ntterbetheouldchcha-t tousthe

us-De-19.

opr the

n-

n-

is,

V5g2m12x21m2

2x222gum32ux21gL

2mL32 x2

1 14 g

2@12g22gL2#2x4. ~A4!

Minimization with respect tog andg L2 gives

g5um3

2um122mL3

2 , ~A5!

gL2512g22

2mL32

g2x2. ~A6!

If g L2,0, theng L

250, and we recover the UFB-1 direction. Inthis case, the top quark mass isMt5ytx, and the top squarkmass matrix entries are

MLL2 5mQ3

2 1yt2x21~ 1

12g822 14g2

2!@12g22gL2#x2,

~A7!

MRR2 5mu3

2 1yt2x22 1

3g82@12g22gL2#x2, ~A8!

MLR2 5yt~mg2At!x. ~A9!

UFB-3~a!. In the UFB-3~a! case, the shifted fields are

^Fu&5x,

^Fd&50,

^L3&e25^e3&

25 l 25UmxytU,

^L2&n5gLx. ~A10!

The VEV’s of the (e3 ,e3) sleptons are fixed by imposingFdF-term cancellation in the potential. The scalar potential forthis case is

V5mFu

2 x21~mL32 1me3

2 !l 21gL2mL2

2 x2

1 14 g

2@x21 l 22gL2x2#2. ~A11!

Minimization with respect tog L2 gives

gL2511U m

ytxU2 2mL2

2

g2x2. ~A12!

If g L2,0, then g L

250. In this case the top quark mass isMt5ytx, and the top squark mass matrix entries are

MLL2 5mQ3

2 1yt2x21~ 1

12g822 14g2

2!@x21 l 22gL2x2#,

~A13!

MRR2 5mu3

2 1yt2x22 1

3g82@x21 l 22gL2x2#, ~A14!

MLR2 52Atytx. ~A15!

UFB-3~b!. In the UFB-3~b! case, the shifted fields are

f

6952 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

^Fu&5x,

^Fd&50,

^Q3&d25^d3&

25d25Umxyb U,^L3&n5gLx. ~A16!

The VEV’s of the (d3 ,d3) squarks are fixed by imposingFdF-term cancellation in the potential. The scalar potential apears as

V5mFu

2 x21~mQ3

2 1md3

2 !d21gL2mL3

2 x2

1 14 g

2@x21d22gL2x2#2. ~A17!

Minimization with respect tog L2 gives

gL2511U m

ybxU2 2mL3

2

g2x2. ~A18!

If g L2,0, then g L

250. In this case the top quark massMt5ytx, and the top squark mass matrix entries are

MLL2 5mQ3

2 1yt2x21yb

2d21 112g82@x21d22gL

2x2#

2 14g2

2@x22d22gL2x2#, ~A19!

MRR2 5mu3

2 1yt2x21yt

2d22 13g82@x21d22gL

2x2#, ~A20!

MLR2 52Atytx. ~A21!

CCB~a!-UP. In the CCB~a! case for the up trilinear, theshifted fields are

^Fu&5x,

^Fd&50,

^Q1&u5ax,

^u1&5bx,

^Q3&d25^d3&

25d25Umxyb U,^L3&n5gLx. ~A22!

SU~3! D flatness impliesb25a2. Also, U~1! and SU~2! Dflatness imply 12a22g L

250. Consequently, the scalar potential appears as

V5yu2~21a2!a2x422T1a

2x21~M2a21mFu

2 1ml2!x2,

~A23!

where T15uAuyuxu, M25mQ3

2 1mu3

2 2ml2, and ml

25mL32 .

Minimization with respect toa2 yields

a25T12M2/22yu

2x2

yu2x2

. ~A24!

p-

is

-

If a2,0, thena250 andg L251. If g L

2,0, then one should try^L3&e5^e3&5gLx and ^L3&n50. D flatness now implies12a21g L

250. This also changesml25mL3

2 1me3

2 . In this

case the top quark mass isMt5ytx, and the top squark massmatrix entries are

MLL2 5mQ3

2 1yt2x21yb

2d21~ 112g821 1

4g22!d2, ~A25!

MRR2 5mu3

2 1yt2x21yt

2d22 13g82d2, ~A26!

MLR2 52Atytx. ~A27!

CCB~b!-UP. In the CCB~b! case for the up trilinear, theshifted fields are

^Fu&5x,

^Fd&5gx,

^Q1&u5ax,

^u1&5bx,

^L3&n5gLx. ~A28!

Unlike the previous CCB case~g50!, this case has threeterms whose phases~ci5coswi! are undetermined:

2uAuyuu1FuQ1uc112umyuu1Fd†Q1uc212umBFuFduc3 .

~A29!

Reference@9# shows that this ambiguity can be resolved intotwo distinct possibilities. If sgn(Au)52sgn(B), the threeterms can be made simultaneously negative. Isgn(Au)5sgn(B), then the term of smallest magnitude istaken positive and the other two can be taken negative.

SU~3! D flatness impliesb25a2. Also, U~1! and SU~2! Dflatness imply 12a22g22g L

250. Consequently, the scalarpotential appears as

V5yu2~21a2!a2x412~c1T1a

21c2T2a2ugu1c3T3ugu!x2

1@M2a21~m122ml

2!g21m221ml

2#x2, ~A30!

where

T15uAuyuxu, ~A31!

T25uyumxu, ~A32!

T35umBu, ~A33!

M25mQ3

2 1mu3

2 2ml2, ~A34!

m125mFd

2 1m2, ~A35!

m225mFu

2 1m2, ~A36!

and whereml25mL3

2 and ci5coswi . Minimization with re-

spect toa2 andugu ~note that the potential is a function ofugu!yields

54 6953CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

ugu5c2T2a

21c3T3ml22m1

2 , ~A37!

a252yu2x21M2/21c1T11c2T2ugu

yu2x2

. ~A38!

It must be confirmed that botha2.0 andugu.0. Otherwise,these are set to zero. It must also be checked thatg L

2.0.Otherwise, one should tryL3&e5^e3&5gLx and^L3&n50.Dflatness now implies 12a22g21g L

250. This also changesml25mL3

2 1me3

2 . In this case the top quark mass isMt5ytx,

and the top squark mass matrix entries are

MLL2 5mQ3

2 1yt2x2, ~A39!

MRR2 5mu3

2 1yt2x2, ~A40!

MLR2 5yt~mg2At!x. ~A41!

CCB~b!-TOP. In the CCB~b! case for the top trilinear@there is no~a! case#, the shifted fields are

^Fu&5x,

^Fd&5gx,

^Q3&u5ax,

^u3&5bx,

^L3&n5gLx. ~A42!

SU~3! D flatness impliesb25a2. There is no imposition ofD flatness in the U~1! and SU~2! sectors. The potential nowappears as

V5yt2~21a2!a2x41@a2~mQ3

2 1mu3

2 !1g2m12

1gL2mL3

2 1m22#x212~c1T1a

21c2T2a2ugu1c3T3ugu!x2

1 14 g

2~12a22g22gL2!x4, ~A43!

where

T15uAtytxu, ~A44!

T25uytmxu, ~A45!

T35umBu, ~A46!

M25mQ3

2 1mu3

2 2mL32 , ~A47!

m125mFd

2 1m2, ~A48!

m225mFu

2 1m2. ~A49!

Minimization with respect tog L2 yields

gL2512a22g22

2mL32

g2x2. ~A50!

If g L2.0, then minimization with respect toa2 and ugu gives

ugu5c2T2a

21c3T3mL32 2m1

2 , ~A51!

a252yt2x21M2/21c1T11c2T2ugu

yt2x2

. ~A52!

If g L2,0, then it must be set to zero, and minimization with

respect toa2 and ugu yields

2yt2x4~11a2!2 1

2 g2x4~12a22g2!1~mQ3

2 1mu3

2 !x2

12~c1T11c2T2ugu!x250, ~A53!

2g2x4~12a22g2!ugu12ugux2m1212~c2T2a

21c3T3!x2

50. ~A54!

Substituting forugu yields a cubic equation fora2. It muststill be checked that bothugu.0 anda2.0. In this case thetop quark mass isMt5ytx, and the top squark mass matrixentries are

MLL2 5mQ3

2 1yt2x2~11a2!1 1

12g82@12 23a22g22gL

2#x2

2 14g2

2@123a22g22gL2#x21 1

3g32@a2x2#, ~A55!

MRR2 5mu3

2 1yt2x2~11a2!2 1

3g82@12 73a22g22gL

2#x2

1 13g3

2@a2x2#, ~A56!

MLR2 5yt~mg2At!x1@yt

22 13 ~ 1

3g821g23!#x2ab.

~A57!

CCB~a!-ELECTRON. In the CCB~a! case for the electrontrilinear, the shifted fields are

^Fd&5x,

^Fu&50,

^L1&e5ax,

^e1&5bx,

^Q3&u25^u3&

25u25Umxyt U. ~A58!

D flatness impliesa25b2 anda2x25x21u2. The scalar po-tential appears as

V5ye2~21a2!a2x41@mFd

2 1a2~mL12 1me1

2 !#x2

1~mQ3

2 1mu3

2 !u222T1a2x2, ~A59!

whereT15uAeyexu. In this case we use the bottom quark orbottom squark contribution. The bottom quark mass isMb5ybx, and the bottom squark mass matrix entries are

MLL2 5mQ3

2 1yb2x21~yt

21 12g2

2!u2, ~A60!

d

6954 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

MRR2 5m

d3

21yb

2~x21u2!, ~A61!

MLR2 5Abybx. ~A62!

CCB~b!-ELECTRON. In the CCB~b! case for the electrontrilinear, the shifted fields are

^Fd&5x,

^Fu&5gx,

^L1&e5ax,

^e1&5bx. ~A63!

D flatness again impliesb25a2 anda2512g2. The potentialappears as

V5ye2~21a2!a2x41@m1

21g2m221a2~mL1

2 1me 12!#x2

12~c1T1a21c2T2a

2ugu1c3T3ugu!x2, ~A64!

where

T15uAeyexu, ~A65!

T25uyemxu, ~A66!

T35umBu. ~A67!

Substituting fora2 and minimizing with respect tougu givesthe cubic equation

ugu3@2ye2x2#2g2@3c2T2#1ugu@~m2

22mL12 2me1

2 !

24ye2x222c1T1#1@c2T21c3T3#50. ~A68!

It must be checked that bothugu.0 anda2.0. In this case thetop quark mass isMt5gytx, and the top squark mass matrentries are

MLL2 5mQ3

2 1g2yt2x2, ~A69!

MRR2 5mu3

2 1g2yt2x2, ~A70!

MLR2 5yt~m2gAt!x. ~A71!

If g50, then we use the bottom quark or bottom squacontribution. The bottom quark mass isMb5ybx, and thebottom squark mass matrix entries are

MLL2 5mQ3

2 1yb2x2, ~A72!

MRR2 5m

d3

21yb

2x2, ~A73!

MLR2 5Abybx. ~A74!

APPENDIX B

We find that Eq.~2.3! cannot be reliably calculated usinour computer for large~.106! values of the VEV’s. Wetherefore use a limiting form of this expression in the lar

ix

rk

g

ge

VEV limit. This limiting form of DV1 is presented in thisappendix. We begin with some definitions:

mt5ytx, ~B1!

MLL2 5mt

21mL21MLx1GLx

2, ~B2!

MRR2 5mt

21mR21MRx1GRx

2, ~B3!

MLR2 5Mx1Gx2. ~B4!

These expressions cover all the cases we have analyzed, anit is a simple matter to identify the various coefficients foreach case. Note that in the CCB-ELECTRON cases,mtshould be substituted bymb . In terms of these definitions,the two top squark masses are

Mt 6

25 1

2 @2mt21~mL

21mR2 !1~ML1MR!x1~GL1GR!x2#

6$@~mL22mR

2 !1~ML2MR!1~GL2GR!x2#2

14@Mx1Gx2#2%1/2. ~B5!

To simplify the notation some new definitions are used, andthe top squark masses rewritten in terms of these:

Mt 6

25mt

2F11S G1

2yt2D 1

1

mtSM1

2ytD 1

1

mt2 Sm1

2

2 D G6 1

2mt2Fa21

1

mtb21

1

mt2 g21

1

mt3 d21

1

mt4 e2G ,1/2

~B6!

where

G65GL6GR , ~B7!

M65ML6MR , ~B8!

m62 5mL

26mR2, ~B9!

a25~G22 14G2!/yt

4, ~B10!

b252~G2M214MG!/yt3, ~B11!

g25~M22 12m2

2 G214M2!/yt2, ~B12!

d252m22 M2 /yt , ~B13!

e25m24 . ~B14!

Modulo overall factors, the one-loop correction is

DV1522mt4ln$mt

2/Q2%1Mt 1

4ln$M t 1

2/Q2%

1Mt 2

4ln$M t 2

2/Q2%, ~B15!

whereQ5Qe3/4. There are three cases to be considered.

54 6955CONSTRAINTS ON THE MINIMAL SUPERGRAVITY . . .

Case~a!: a2Þ0,

A6511G1

2yt6

a

2, ~B16!

B65M1

2yt6

b2

4a, ~B17!

C65m12

264a2g22b4

16a3 . ~B18!

Case~b!: a250 ~⇒b250!, g2Þ0,

A6511G1

2yt, ~B19!

B65M1

2yt6

g

2, ~B20!

C65m12

26

d2

4g. ~B21!

Case~c!: a25b25g25d 250, e2Þ0,

A6511G1

2yt, ~B22!

B65M1

2yt, ~B23!

C65m12

26

e

2. ~B24!

Finally, the form of the one-loop correction in the large-mtlimit is

DV15mt4H @2~a11a2!1~a1

2 1a22 !#L1~A1

2 lnA1

1A22 lnA2!1

2

mt@A1B1~1/21L1 lnA1!

1A2B2~1/21L1 lnA2!#11

mt2 @B1

2 ~3/21L1 lnA1!

1B22 ~3/21L1 lnA2!12A1C1~1/21L1 lnA1!

12A2C2~1/21L1 lnA2!#J , ~B25!

where L5ln$mt2/Q2% and a65A621. To arrive atDV1,

multiply the above expression by 3/32p2.

r

.

d

@1# H. Haber, inRecent Directions in Particle Theory: From Su-perstrings and Black Holes to the Standard Model, Proceed-ings of the Theoretical Advanced Study Institute, BouldeColorado, 1992, edited by J. Harvey and J. Polchinski~WorldScientific, Singapore, 1993!, Report No. hep-ph/9306207~un-published!; X. Tata, in The Standard Model and Beyond, Pro-ceedings of the Symposium, Mt. Sorak, Korea, 1990, edited bJ. E. Kim ~World Scientific, Singapore, 1991!, p. 304.

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y

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.

.

I. Hinchliffe, ibid. B228, 501~1983!; M. Drees, M. Gluck, andK. Grassie, Phys. Lett.157B, 164~1985!; J. Gunion, H. Haber,and M. Sher, Nucl. Phys.B306, 1 ~1988!; H. Komatsu, Phys.Lett. B 215, 323 ~1988!.

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@10# H. Baer and M. Brhlik, Phys. Rev. D53, 597 ~1996!.@11# G. W. Anderson and D. J. Castan˜o, Phys. Rev. D53, 2403

~1996!.@12# H. Baer, M. Brhlik, R. Munroe, and X. Tata, Phys. Rev. D52,

5031 ~1995!.@13# H. Baer, C. H. Chen, C. Kao, and X. Tata, Phys. Rev. D52,

1565~1995!; H. Baer, C. H. Chen, F. Paige, and X. Tata,ibid.~to be published!.

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6956 54HOWARD BAER, MICHAL BRHLIK, AND DIEGO CASTANO

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