Aspects of radiative electroweak breaking in supergravity models

NUCLEARNuclearPhysicsB398 (1993)3—51 P H Y S I CS BNorth-Holland ________________

Aspectsof radiativeelectroweakbreakingin supergravitymodels

S. Kelley a,b,1 JorgeL. Lopeza,b D.V. Nanopoulosa,b,c H Pois a,b

andKajia Yuand,2

“ Centerfor TheoreticalPhysics,Departmentof Physics,TexasA&M Unitersity, CollegeStation,TX77843-4242,USA

h AstroparticlePhysicsGroup, HoustonAdt’ancedResearchCenter (HARC), The Woodlands,

TX 77381, USAC CERNTheoryDivision, CH-1211Geneva23, Switzerland

d Departmentof Physicsand Astronomy,Unii ‘ersity ofAlabama,Box 870324, Tuscaloosa,

AL 35487-0324,USA

Received17 June1992(Revised14 January1993)

Acceptedfor publication 15 January1993

We discussseveralaspectsof state-of-the-artcalculationsof radiativeelectroweaksymmetry

breakingin supergravitymodels. These models have a five-dimensionalparameterspace incontrastwith the 21-dimensionalone of the MSSM. We examinethe 1-Iiggs one-loopeffectivepotential ~ = V

0 + zlV, in particularhow its renormalization-scale(Q) independenceis affectedby the approximationusedto calculate iIV and by the presenceof a Higgs-field-independentterm which makes l/~(0)~ 0. We show that the latter must be subtractedOut to achieveQ-independence.We alsodiscussour own approachto the explorationof the five-dimensionalparameterspaceandthe tine-tuningconstraintswithin this approach.We applyour methodstothe determinationof theallowedregionin parameterspaceof two modelswhichwe argueto bethe prototypesfor conventional(SSM) andstring (SISM) unified models.To this endwe imposethe electroweakbreakingconstraintby minimizing theone-loopeffectivepotentialandstudytheshifts in jx and B relativeto thevaluesobtainedusingthetree-level potential.Theseshifts aremost significantfor small valuesof ~sand B, andinduce correspondingshifts on the lightest jx-and/or B-dependentparticle masses,i.e., thoseof the lightest stau,neutralino,chargino,andHiggs bosonstates.Finally, we discuss the predictions for the squark, slepton, and one-loopcorrectedHiggs bosonmasses.

1. Introduction

The standardmodel of electroweakand strong interactionsis well establishedby now. In fact, the effects of the top quark in one-loopelectroweakprocesses

Presentaddress:Departmentof Physics,MaharishiInternationalUniversity,Fairfield, IA 52557-1069,

USA.2 Presentaddress:AstroparticlePhysicsGroup, HARC, TheWoodlands,TX 77381,USA.

0550-3213/93/$06.00© 1993 — Elsevier SciencePublishersB.V. All rights reserved

4 S. Kelley et al. / Radiativeelectroweakbreaking

predict its mass(within 20%)centeredaround 140 GeV [1]. Therefore,itsexpecteddirect experimentaldetectionin the nearfuturewill completethe set ofstandardmodel predictionsfor the vectorandfermion sectors.The scalarsectorisanotherstory. The simplestelectroweaksymmetrybreakingscenariowith a single

Higgs boson is only mildly constrainedexperimentally,with a lower bound ofmH> 57 GeV [2] and no firm indirect experimentalupperbound, althoughthissituationwill changeoncethe top quarkmassis measured[31.On the other hand,interestingupperboundson m~follow from varioustheoreticalassumptions,suchas perturbativeunitarity at tree- (mH~ 700 GeV) [4] and one-loop (mH ~ 400GeV) [5] levels, and the stability of the Higgspotential (mH~ 500 GeV) [6,7]. Inpractice,with the adventof the SSCandLHC, experimentalinformation abouttheTeV scale is likely to clarify the compositionof the Higgs sector. Nevertheless,despiteall theseefforts the structureof the standardmodel andits correspondingHiggssectorwill remainbasicallyunexplained.

It hasthereforebecomecustomaryto turn to the physicsat very highenergiestosearchfor answersto thesetheoreticalquestions.The mostpromising theoriesofthis kind contain two new ingredients:supersymmetryand unification. Togetherthesecan explain the origin of the weak scale(i.e., the gaugehierarchyproblem)relative to the very high energyunification (Me) or Planck (M~1)scales[8—11].Furthermore,this class of theoriespredict a new set of relatively light (� 0(1TeV)) particlesconsistingof partnersfor the standardmodel particlesbut withspin offset by ~ unit. In fact, the new set of particlesappearsevermore likely tooverlap little with the mass scales of the standardones, thus their presentunobservedstatus.Moreover, thestandardmodel Higgsbosonwill thenappearasone of the new particlesbut with massclose to M~,thus avoiding naturally thetheoreticalproblemsmentionedabove.

Unfortunately, the introduction of supersymmetryalso increasessignificantlythe numberof unknownparametersin the theory, mainly becausethis symmetrymust be softly brokenat low energies.Indeed,to describea genericlow-energysupersymmetrymodel (the so-called minimal supersymmetricstandardmodel

(MSSM)) neglectingthe first- and second-generationYukawa couplings, the KMangles,andpossibleCP violating phases,we needthe following set of parameters(the valuesof sin

2O~,a3, ae, M~are takenas measuredparameters):

(a) The Yukawa (Ar, Ab, A~)and Higgs mixing (jx) superpotentialcouplings.(We can tradethe Yukawacouplingsfor m~,tan f3; mb, mr, with tan /3 =

the ratio of Higgsvacuumexpectationvalues,and mb, m~given.).(b) The soft-supersymmetrybreakingtrilinear (Ar, Ab, AT) and bilinear (B)

scalarcouplings(correspondingto the superpotentialcouplingsin (a).)(c) The soft-supersymmetrybreaking left—left and right—right entries in the

squarkand sleptonmassmatricesfor the first and second(mQUCDC, mLEC), andthird (mQ3U~D~,mLE~) generations.

(d) The soft-supersymmetrybreakinggauginomassesmg, mW, m~.

S. Kelleyetal. / Radiativeelectroweakbreaking 5

(e) The Higgs sector parameter(at tree-level), e.g., the pseudoscalarHiggsbosonmassm~.The above 21 unknown parametersmake any thoroughanalysisof this classofmodelsratherimpractical, and haveallowed in the pastonly limited explorationsof thisparameterspace.If we now addthe gaugeunification constraint(a,(Mu) =

a~,i = 1, 2, 3), the assumptionof universalsoft-supersymmetrybreakingat a scale

= M~,and high-energy dynamics (in the form of renormalization groupequations(RGEs) for all the parametersinvolved), the set of parametersin (b)

reducesto A =A~=Ab =A~and B, those in (c) to m0= mOUCDC= mQ =

mLE1, andthose in (d) to rn1,,2 = mg= m~,= rn0 theserelationsare valid only atthe scale~ The numberof parametershasbeendramaticallyreduceddown toeight.

Let us now add low-energy dynamicsby demandingradiativebreakingof theelectroweaksymmetry.The tree-levelHiggspotential is given by

~~0= (m~+~2)~H

12+ ~

~ H1 2)2, (11)

where H1 (~)and H2 (~)are the two complex Higgs doublet fields, g’= -t/~7ig1 and g2 are the U(1~~andSU(

2)L gaugecouplings,and Bp. is takentobe real and negative.This potential has a minimum * if 3V

0/~1= 0, with 4~denoting the eight real degreesof freedom of H1 and H2. In particular, for

= Re H1°oneobtainstwo constraintswhich allow the determinationof x andB,

m~—m~ tan

2/3tan2/3—1 ~ (1.2a)

~ (1.2b)

up to the sign of ~. In these expressions,m~,m~ are soft-supersymmetrybreakingmassesequal to m~at ~ Since the whole set of Higgs massesandcouplings(at tree-level)follows from m~(andtan /3), andonecaneasily showthatm~= —2Bjt/sin 2f3, the parameterin (e) is also determined.(This result alsoholds at one-loop although the expressionfor rn~ is more complicatedin thiscase.)

The final parametercount in the classof modelsweconsideris then just five:

m~,tan J3, rnl/2, rn0, A (plus the sign of p.). Note also that sin

2O~(aswell as M~and a~)gets determined(from a

3 and ae) by the gaugeunification condition.

* The parametersin V0 must satisfy further consistencyconstraintsto insure that this is a true

minimum of the tree-level Higgs potential. As discussedin ref. 1121 (and below), the one-loopeffectivepotentialsatisfiesmost of theseconstraintsautomatically.


What are the a priori expectedvalues of rnl/2, m0, A? In principle choosinga

suitablesupergravitymodel (i.e., suitablehiddensector)one could havearbitraryvalues for theseparameters.The so-called “minimal” supergravitymodels [13]predict A = B + m0 (at ~ with m0, ml/2 ‘/‘ 0 in general.Supergravitymodelswhich explain naturally the vanishingof the cosmologicalconstant,the so-calledno-scalemodels[14,151,typically requirern0 =A = 0 and rnl/2 * 0, althoughmod-

elswith m0, A * 0 arepossiblealso.With this scenarioin mind we haveundertakena systematicstudyof supersym-

metric unified modelswith universalsoft-supersymmetrybreaking.We pursuethisobjective in threesteps:(i) determinationof the parameterspaceallowedby allconsistencyandexperimentalconstraintsusingthe Higgsone-loopeffectivepoten-tial; (ii) calculationof particlemasses,upperandlower bounds,correlationsamongthem, and discoverylimits; and (iii) study of specific reactionssuch as colliderprocessesrelevant for particle detection, rare decays, neutralino dark matter,supersymmetricloop correctionsto sin

2O~,etc. In this paperwe addressthe firsttwo points.

Analysesof this naturein unified modelsexist in the literature(althoughtheir

numberis small relative to those in generic low-energysupersymmetricmodels)and date back one full decade(for a recent review see ref. [16]). The recentinterest in this type of analyseshas been spurred by the experimental dataavailable from LEP, and it hasmostly beenconcernedwith systematicstudiesof

variousaspectsof unified supersymmetricmodels,suchasgaugecoupling unifica-tion [17—21],the Yukawa sector[20,22], radiativeelectroweaksymmetrybreakingusing the tree-level[23—25,21]andone-loop [12,26,27]Higgs potentials,the one-loop correctedHiggsbosonmasses[28,29,27],supersymmetryloop effectson thep-parameterand sin2O~[30,21], neutralinodark matter [26,31—33],proton decay[34,35], etc.

The purposeof this paper is to sharpenthe determinationof the allowedparameterspaceof this class of models by using the one-loop Higgs effective

potential. In sect.2 we comparebothapproximationsanddiscussseveraltheoreti-cal andpractical issuesrelated to the useof the one-loop effectivepotential. Wepresenta numericalstudyof theeffect of the usualapproximationsto the one-loopeffectivepotential (e.g.,keepingonly the top—stopcontributions)on its renormal-ization-scale(Q) independenceproperties.We also quantify the phrase“up totwo-loop effects”,clearly identifying theone-and two-loop leading-andnon-lead-ing-log contributions,and obtain a Q-independentone-loop potential. We showthat in problemswherethevalueof the one-loopeffectivepotential(asopposedtoany of its field derivatives) is relevant ~, one must define a new “subtracted”potentialwhich vanishesat the origin of field spaceandis manifestlyrenormaliza-

* These problems include studies of the cosmologicalconstantand dynamical determinationsof

supersymmetrybreakingparametersin no-scalesupergravitymodels.

S. Kelley Ct al. / Radiativeelectroweakbreaking 7

tion-scaleindependent.We also provide details of our numerical and analyticaltechniquesusedto calculatethe one-loopvaluesof p. and B andto demonstrateone-loopQ-independence.

In sect. 3 we discuss the “fine-tuning” constraint [36,37], which limits theparameterspaceandyieldsan allowedregionboundedin all five variables.In sect.4 we show that the class of models we considerhere constitute a good firstapproximationto any realistic traditional (SSM) or string (SISM) supersymmetryunified modelbelow theunification scale.In sect.5 we discussthe consistencyandphenomenologicalconstraintson the parameterspaceand the ensuingboundedregionsin the (rn~,tan /3) plane.We thenstudythe shift in p. and B obtainedwith

the one-looppotential relativeto their tree-levelcounterparts,andfind them to bemost significant for small values of p. and B. We also explore the allowed

parameterspaceextendingour previousanalysisto nonvanishingvaluesof rn0 andA, and usingthe one-loopeffective potential. In sect. 6 we studythe squarkand

sleptonmassesand the effect of the shifted valuesof p. and B on the massesofthe lightest stau, the lightest neutralino,andthe lightestcharginostates.We also

study the one-loop correctedHiggs boson massesand present tables of massrangesfor all particlespeciesfor typical valuesof the parameters.Finally, in sect.7we summarizeour conclusionsand in the appendixwe presentsome detailsof thecomputationof the one-loopeffectivepotential.

2. The effective potential

As a practicalmatter,in studiesof radiativeelectroweaksymmetrybreakingit isimportant to determinewhether the tree-levelapproximationto the scalarHiggspotential (l/~)is adequatefor the purposesat hand.There are two observations

which make it apparentthat this approximationmay not be accurateenoughnorreliable(at least in some regionsof parameterspace):(i) It had beennoted longago[11] (andit hasbeenemphasizedrecently[12]) that the low-energyscale Q atwhich the RGEs connectingphysics at very high energieswith physics at theelectroweakscale are stopped,influences the (tree-level)physical predictionsofthis classof modelsin significant ways, such as in the determinationof minima ofthe tree-level Higgs potential. Equivalently put, V~does not obey the RGEdV0/dt= 0 (with t = In Q) in any sensibleapproximation.(ii) It hasbeenrecentlyobservedthat the one-loopcorrectionsto the Higgsbosonmassesarenon-negligi-

ble in certainregionson parameterspace(e.g., for largem~)[38,28,39].It hasbeenargued[12] that the one-loopeffectivepotential~ = V0 + ~~1Vsolves

the problem in (i) sinceit is presumedto satisfy the correspondingRGE (dV1/dt= 0) automatically and therefore yields one-loop Q-independentpredictions.Infact, this propertyof V1 is routinely usedto derive one-loopRGEs[40]. However,the one-loopQ-independenceof V1 hasbeenexploredonly in a limited way in ref.


[12] andthe detailsby which it works haveso far remainedunclearor at leastnot

explicitly described.Additionally, it hasbeenshown that the naive expectationofusingthe one-loopHiggspotential to obtain one-loopcorrectedHiggsmassesis infactaccurateto 0 (few GeV) [41,42],andthereforeone-loopcorrectionsto V0 areexpectedto affect the valuesof p. and B calculated(asdescribedin the introduc-tion) from the corresponding minimization conditions 3V1/ö Re H1°=

~I/1/0 Re H2°= 0. The expressionfor iW [7],

1 ~ ~STr 4~ in — — (2.1)

64ir2 Q2 2

where STr f(4’2) = ~( — 1)~’(2j+ 1)Tr f(4.2) and 4,2 are the field-dependentspin-i massmatrices,receivescontributionsfrom all particle species.However, in

analysesof this kind it is customaryto includeonly the “dominant” contributionsto LW, i.e., those from the top—stop (and sometimesalso bottom—sbottom)system(s).This statementis indeedaccurateprovidedone is only interestedin thefield derivativesof V

1. However, it is notclear to what extentthis approximationto~V will affect the Q-independenceof V1. We will show that this questionis notonly of conceptualbut also of practical importance. In the remainderof thissectionwe review the derivationof V1, studyits formal Q-independenceproperties(in a A/~ theory and the MSSM), and discussthe numericalmethodsused to

obtain the one-loopvaluesof p. and B.

2.1. GENERAL REMARKS

The first calculationof the one-loop effective potential in field theorieswas

performedby Coleman and Weinberg[43]. Their method consistedbasically ofsumming the infinite seriesof one-loop diagramswith all possiblenumbersofzero-momentumexternallines. This procedurebecomesverycumbersomefor two-and higher-loopcalculationsand is replacedin practice(even at one-loop)by theso-called“tadpolemethod” first noticedby Weinberg[44] and laterdevelopedbyLee and Sciaccaluga[45] and extendedto supersymmetrictheoriesby Miller [461.This methodis derived[7] by expandingthe effective action aroundshiftedvaluesof the fields. The first derivative of the n-loop contribution to the effectivepotential(V~’°)is simply given by the n-loop tadpole(Ta)) diagramscalculatedinthe shifted theory and with zero externalmomentum~. For example,consideramasslessA~

4theory with scalarpotential V0(~)= (A/4!)4.

4. Shifting 4 —‘ — wgives a linear term in ~, —(A/6)4w3,a “mass term” (A/4)w242,andcubic term—(A/6)w43. At tree-level we have = (A/6)w3 and therefore V~°k4)=

(A/4!)~4as expected. The constant of integration has been fixed such that

* In what follows we denoteV,= V(©+ V~1~+ ... + vt”~.Note that V0 = V~°~and ~1V= Vtl).

S. Kelley etal. / Radiativeelectroweakbreaking 9

Vt°)(0)= 0. At one-loop we have a single diagram due to the induced cubiccoupling,giving

~(i) 2f(2~)~k2~Aw2’ (2.2)

where the ~ is the symmetry factor for the one-loop diagram. This integral isdivergentandcan be regulatedby introducinga cut-off or more convenientlybyusingdimensionalregularization.The resultusing the latter is

dV~~ A2w3 1 Aw2= = ______ — + ln — 1 (2.3)

dw (1) 4(4)2 �~ 2Q2

where D = 4 — 2�, 1/ern = — 1/c + YE — In 4ir, and Q is an arbitraryscale intro-duced for dimensional reasons.In the M3~renormalization schemewe simplydiscard1/~and obtain the finite result immediatelyby integratingeq. (2.3),

rn~(~) m2(~) 3V(’)(Q, ~ = 64~2 ln( Q2 ) — , (2.4)

where we have defined a field-dependentmass rn2(4) -~A~to make theconnectionwith eq. (2.1) more apparent.The arbitrary scaleQ appearingin V~1~can be specified in connectionwith physically measured(or “renormalized”)quantities.One usually sets a2V

1/3~2= 0 to preservethe “masslessness”of the

theory and 04V1/0q~

4(4= M) = AR. The latter relation implies that the quarticcouplingtakesthe valueAR at scale Q = M. In this caseone gets

24A2 AM2 8A =A+ ln -~ +— (2.5)R (16~)2 2Q2 3

and

AR A~44 ~2 25V

1 = V~°~+ Vtl) = + (j~)2 ln(~-~)— -i-- . (2.6)

Alternatively, one mayjust leave Q unspecifiedas in eq.(2.4). This is much moreconvenientin complicatedtheories(like the MSSM) but hasthe drawbackthat itdoesnot involve true physical parametersmeasuredat a specific scale. Until thesparticlespectrumis detected,high accuracypredictionsare not required(unlikee.g., the electroweak sector in the SM), and this approximation is perfectlyadequate.

The calculation of V~1~sketchedabove can be easily generalizedto gauge

theorieswith fermions [7]. The final result for the one-loopcontribution to the


effective potential is as given in eq. (2.1) with Vt1~= ~1V.We should remark thatthis result has beenobtainedin the convenientLandaugauge(andusing the 1J~renormalizationscheme[47], i.e., the supersymmetry-preservingcounterpartof the

usualMS schemewherein dimensionalregularizationis replacedby dimensionalreduction)and is otherwisegaugedependent[44,48].However, physicalquantitiesextractedfrom V~1~shouldbe gaugeindependent.This was explicitly verified toone-loopin one examplein ref. [49] and showngenerallyto hold to all ordersinperturbationtheory in ref. [50]. Thereforethe Landaugaugechoiceis as good asany otherone.

2.2. Q-(IN)DEPENDENCE

In perturbativeexpansionsinvolving the renormalizationgroup,it is well knownthat cancellationof the Q-dependencetakesplace acrossdifferent orders. Forexample, the tree-levelQCD crosssection for jet productionis highly Q-depen-dent,whereasthe one-loopcorrectedexpressionis Q-independentup to two-loopeffects. We now study the analogouseffect for the scalarHiggs potential. It isinstructiveto studya particularly simplefield theory in detail to seein what senseandto what degreeof approximationis V

1 = V~°~+ V~1~Q-independent.A calcula-

tion analogousto the one sketchedabovegives for the massiveA44 theory

A rn4(q~) rn2(çb) 3VI(Q, ~)= ~p.2~2+ + 64~ ln( Q2 ) - , (2.7)

with rn2(4) p.2+ ~A42. To study the Q-dependenceof V1 we take the total

derivative dV1/dt and make use of the one-loop RGEs dA/dt = 3A2/(4ir)2,

dp.2/dt = p.2A/(4~r)2,and d4/dt = 0, to obtain

dV1 p.

4= — + “two-loop” (2.8)

2ut 2(4w)

where “two-loop” denotescontributionswhere the derivativehas actedon piecesin V(l) otherthan ln Q2. Using the RGEsonecan easily seethat thesepiecesare

of higherorder (e.g., a 1/(4~)~asopposedto a1/(4ir)2 andwill be cancelledbytwo-loop contributionsin ~ Eq.(2.8)indicatesthat V

1 doesnot obeythe RGEdV1/dt = 0 in one-loopapproximation.This problemcan be easily tracedbacktothe constantpiece V1(Q, 0) = V~’~(Q,0) which may be subtractedfrom V1 suchthat

12

1(Q’ 4) V1(Q, 4) — V1(Q, 0) (2.9a)

= V~°~(4)+ V”)(Q, 4.) — V(1~(Q,0), (2.9b)

S. Kelley et al. / Radiativeelectroweakbreaking 11

satisfies 121(Q, 0) = 0. Indeed, in this case V,(Q, 0) = (p.

4/64~-2)[ln(p.2/Q2)—and

dV1(Q,0) p.

4________ = — + “two-loop” (2.10)

dt 2(4~~)2

such that dV1/dt = 0 + “two-loop” obeys the one-loop RGE as it should. The

aboveprescription(eq. (2.9)) can be justified in severalways (see also the recentdiscussionin ref. [51]). First of all, from the practical point of view, adding afield-independentpiece to V is perfectly harmlessin problemswhere only fieldderivativesof V are of interest.From the requirementof renormalizabilityweknow that the all-orderssolution to the RGE dV/dt = 0 must resembleV~°~butwith suitablymodified coefficients and wave-function renormalizations,and thisexpressionclearly vanishesat the origin of field space.Yet anotherway of seeingthis is in the MS cancellationof infinities process.The countertermsa ~2/~ anda 4~/~that need to be addedto the lagrangianwill not absorb the ap.

4/idivergence(see e.g., eq.(2.7) with 1/�~addedinside the squarebrackets)andoneis forced to add a “constant” counterterm a1/~to remove it. This occursnaturally for V~.In the contextof supersymmetrictheories,the fact that V(0) = 0to all ordersis equivalent[46] to the statementthat unbroken(global) supersym-metry at tree-levelcannotbe brokenby radiativecorrections.

Does this “subtraction” proceduresurvive higher-loopcorrections?To study

this questionwe considerthe two-loop effective potential V2 = V

t0~+ Vt’) + Vt2),with Vt2~given by [51]

1 1 rn2 1 m24 {4Am4[~ln2(~) — ~ln(~) + a

200]

1 m2 1 rn2

+2A2~2m2{~j~ln2(_-~)— ~ln(_-~) +a2io]}~ (2.11)

wherern2 p.2 + -~-A~2and a

200, a210 aresomenumericalcoefficients.We also usethe following two-loop expressionsfor the relevantRGEs [52]:

dA 3A

2

____ — , (2.12a)dt (4)2 (4)4

1 dp.2 A____ — , (2.12b)

p.2 dt (4~.)2 (4~)4

1d4 ___

= - 4• (2.12c)


After somestraightforwardalgebraicmanipulationsandkeepingonly termsup totwo-loop order we obtain

dV~°~ rn4 p.4 A242rn2

= 2 2 4’ (2.13a)dt 2(4w) 2(4ir) 2(4ir)

dVtU m4 Arn2 rn2____ = — + (rn2 + A42) ln — — 1 , (2.13b)

dt 2(4~r)2 2(4~)~ Q2

dV~2~ Arn2 rn2 A2~2m2= — (rn2 + A42) ln — 1 + , (2.13c)

dt 2(4~)~ 2(4~)~

andtherefore

d p.4—(V~°~+ V11~+ V~2~)= — + “three-loop”. (2.14)dt 2(4~~)2

That is, the one-loop subtraction in eq. (2.9) is still neededat two loops asanticipated.However, an analogoustwo-loopsubtractionis not necessarysincethefield-independentpiecesin dVW/dt and dVt2~/dt(aAp.4[ln(p.2/Q2) — 1]/(4ir)4)cancelamongthemselves.This is a ratherinterestingresult whichmay be not thatsurprising if onerealizesthat in the cancellationsof the field-independentpiecesin dV~/dt and dV~°~’~/dt,thereis a mismatchfor n = 0 sincedV~°~/dthasnofield-independentpiece. It is not clear to uswhetherthis phenomenonpersistsathigherordersor in morecomplicatedtheories.

We now considerthe case of the MSSM (our unification constraintsdo not

affect the presentdiscussion).Analogousto the massiveA44 theory, we define apotential V

1 = V~+ ~V— ~.lV(0)for the MSSM usingeqs.(1.1) and(2.1). It is thennot hard to seethat the Q-dependenceof V, can be studiedto one-loop orderfrom the following expression:

dv, dV0 1= — 2STr{~~’

4(h1,h2) —%‘~(O,0)) + “two-loop”, (2.15)dt dt 32ir

whereh12 Re H~2.This equationevidencesthe fact that it is the STr .E” termin z.~1Vwhich cancelsthe running with Q of the parametersin V0. It is also clearthat an incomplete set of contributions to the supertracewill make for an“incomplete” Q-independenceof 17,. Below we quantify this statement,but firstwe considera particularly simple limit of the MSSM where the derivative in eq.(2.15) canbe carriedout explicitly, andjustifies our useof thesubtractedone-looppotential.

S. Kelley et a!. / Radiativeelectroweakbreaking 13

Let ustakethe limit of vanishinggaugecouplingsand p. = A t,b,T = A b,T 0 (seeref. [28]). This allows us to set v, = 0 and the tree-level potential reducestoV0 = rn~2h~.The relevantRGEsfor the left-over tree-levelparametersare

drn~ 3

dt 2 = ~—~A~(2rn~+ m~2), (2.16a)

dh2 3h2

= — ~ (2.16b)

once we take all squark mass parametersdegenerate(rn0 = ~ = ... = m4).

Therefore

dV 3m2 3rn~m2H 3m~m2

— ~j 2 2’l 2 ____ 217-a--- — -~---~-~ rn

4 rn11~- 8~2 — 4~2

(Note the role played in the result by the often-neglected“VEV running”.) With

our approximationsonly the top-stopcontribution(m~= ~ rn~= rn~+ A~h~)toeq. (2.15) survives,since any field-independentcontribution is subtractedout andwith vanishing gauge couplings this is the case for the gauge/gaugino andHiggs/higgsinosectors.We then get

STr{4’4(h~,h

2) —%~(0,0)} —o 12{[(m~+ m~)2— in~]— rn~}= 24rn~rn~.

(2.18)

Substitutingeqs.(2.17) and(2.18) into (2.15) finally gives d121/dt= 0 as expected.

Note that had we not used the subtractedone-loop potential we would haveobtaineddV1/dt = —3m~j/8ir

2,i.e., a huge Q-dependence(seebelow).Let us now take a morequantitativelook at the Q-dependenceof the various

approximationsto the effectivepotential.To this endwe expandV0 andV1 around

t = 0 (t ln(Q/M~)),as follows

d’1 V

0V0(t, ~)=a0 + a,t + a2t

2+ a3t

3 + . . ., a~ —(t = 0, ~), (2.19)

and

(2.20)

with

1 g2 3~V(t, 4) = 64

2Str %~“ In ~ — ~ (t, 4). (2.21)


0,3

0.2 — V1 ....... —

0.1-

0.0- •..•~ -

-0.1

—0.21 2 3 4 5

Q/MZ

Fig. 1. Evolutionwith the renormalizationscaleQ/MZ of theHiggs potential for a particularpoint inparameterspace(with m1 = 100 0eV, tan /3 = 3, ml/2 = 150 GeV, m0= A = 0, and ~> 0). Thevariousapproximationsusedare: V0 tree-level,V1 one-loopsubtracted,and V1 one-loopunsubtracted.Only V,

is t-independentto one-loop.

To one-loop order we only need to include in LIV(t, 41) the tree-level massesevaluatedat t = 0 (Q = M’). In fig. 1 we plot the the tree-level(V0) andone-loop~ Higgs potentials (following a numericalproceduredescribedin sect. 2.3) as afunction of t = ln(Q/M~) for a particularpoint in parameterspace(with m~= 100

GeV, tan /3 = 3, ml/2 = 150 GeV, rn0 =A = 0, p. > 0). (Note the large t-depen-dence of the unsubtractedone-loop effective potential V1, as anticipated). IncalculatingStr ~2,4 we have included the full spectrum.The curves shown arewell fit by third-degreepolynomials in t, as follows *:

a0= —0.0795, a1=0.2751, a2= —0.0315, a3=0.0027, (2.22a)

b0= —0.0895, b1= —0.0007, b2= —0.0322, b3=0.0027. (2.22b)

(Thenon-vanishinga23 coefficientsaredueto higher-ordereffects in the runningof the parametersin V0 (since the RGEsare solved numerically) and are smallrelativeto thelinear term.)The expectedresult is evident: I b1/a1 = 0.0025 =

that is, 1/~is Q-independentto one-looporder (i.e., “flat” at t = 0). It is importantto realizethat the only testof the calculationis b1/a1 ~ 1, sinceb23 aresubjecttotwo- and higher-loopcorrections.

* Thevaluesfor thevariouspotentialshave beenresealedby (m,/2)4.

S. Kelley etal. / Radiativeelectroweakbreaking 15

0.3 I .• I

linear approx.

0,2 — ..... —

V0

0.1- ...-.... -

1*0~

0.0- •~..•• —

~, (full)

-0.1 -

—0 2

1 2 3 4 5

Q/M2

Fig. 2. Detail of the evolution of the minimum of the potentials shown in fig. 1 in the_linearapproximationin t = ln(Q/Mz). Note the null slopeof thefull one-loopsubtractedpotential(V1) andthe non-zeroslopewhenonly thedominantcontributionsto iV (i.e., t, 1, b, b) are included.Notealso

thenon-zeroslopeof V, when themassesin ~1Vareallowed to runwith t.

To study the relativeimportanceof the variouscontributionsto STr ~ in Eq.(2.21), in fig. 2 we show the linear (in t) contributionsto V0, and to 1~in twosteps:(a) only t, i, b, b, and (b) all contributions. The correspondingcoefficientsare

~ —0.0224, b~~=—0.0007, (2.23)

which give b~”~/a1 ~, ~, showing an explicit convergence to the correctresult. It is clear that the “dominant” contributions to ~iV (i.e., t, 1, b, b) areindeedthe largestones.However, fig. 2 shows that thesemay not be enoughinapplicationswhere the Q-independenceof the one-loop potential over a severalhundredGeVrangeis essential.

Let us now commenton what happensif we allow the massesthat enterinto~iV(t, 41) to run with t. To this endwe rewrite z.IV(t, 41) as follows:

1~V(t, 41) = — 2t Str .~

4(t, 41)32rr

1 ~-2 ~+

64~2ST~~K4in ~ — ~ (t, 41). (2.24)

16 5. Kelley et a!. / Radiativeelectroweakbreaking

The leading-log piece(a t) displaysthe expectedone-loop t-dependence.How-ever, the non-leading-logresiduealso has a linear term in t which is of two-looporder (i.e., a 1/(4ir)4) andwhich thereforespoils thevanishingof the linear termin the potential. The magnitudeof this effect is shown in fig. 2 as the V

1’ line,which clearly deviatesfrom the V1 (full) line. Note that the potential is formallyQ-independentto one-loop no matter where the massesin the supertracearerenormalized.However, only when thesemassesare renormalizedat t = 0 do thetwo-loop subleading-logterms and the linear term in the potentialvanish.

In sum, we haveshown that the subtractedone-loop effective potential V1 is

Q-independentto one loop and should be used in calculationsto this order.Explicit runningof the parametersin ~V leads to a residual Q-dependence due tothe introductionof spurioustwo-loop contributionswhich affect the quadraticaswell as the linear t-dependenceof V1. Note howeverthat two- and higher-loopeffects (i.e., the sourcefor the curvatureof V1 in fig. 1) increaselogarithmicallywith Q/MZ and signala progressivedeteriorationof the one-loopapproximation.It is thereforenot advisableto use V1 to studyeffectsover scalesgreaterthan 1TeV. A renormalization-group-improvedone-loop effectivepotential [7] (whereinall powersof t are summedup) would be the propertool for thispurpose.

2.3. NUMERICAL METHODS

In this section we discussthe numerical methodsand assumptionsused to

explore the Q-(in)dependenceof the tree-leveland the one-loopHiggspotentialsas well as the one-loop minimization. As we havedemonstratedin the previoussection, this procedure involves the following basic steps: (i) the full scalarfield-dependents/particlespectrummustbe definedat a fixed scale,in order tocalculate~.4V(see appendixfor details),(ii) a_particularchoice in the scalar fieldspacemust be madein orderto calculateV0, ~ and(iii) the parametersandfieldsin V0, namely g1(Q), g2(Q), p.(Q), B(Q), rn~Q),m~(Q),H1(Q), I-12(Q) mustbe

RG evolved to the new scale in question. As we have shown, the issue ofQ-independencefor V1 thus becomesa numerical test as to whether the implicitleading-logcorrectionsto the tree-levelparametersconspireto cancelthe explicitleading-log Q-dependencein i.IV (eq. (2.1)).

For step(i), in order to solve for the “physical” theoryat a fixed scaleQ = M~

and define ~V, we begin by making a choice for the initial set of independentparametersand integratethe RGEsto this scale. In order to calculate ~1V,thecompletescalar field-dependents/particle spectrummust be defined. We thenminimize the tree-levelor one-looppotential to obtain a consistent,completeset

of parametersthat representthe groundstateof the theory. Oneoption would beto choosep., B, m0, rnl/2, A, at M~,evolveto Q = M~andthenminimize V1(M~)or V0(M~)to determinee1(M~), s’2(M~). From the experimentalconstraintsforM~,M~,the original set of high-energyparametersareeither allowed or ruled

S. Kelley Ct al. / Radiativeelectroweakbreaking 17

out. While this approachinvolvesthe “direct” calculationof v1(M~),v2(M~),the

original parameterspaceis at M~,andthe connectionto low-energyphysicswhichinclude the constraintsfor rnb, m~mustbe donevia an iterativeprocedure.

Our calculationprocedurefor minimization of the one-looppotential is quitedifferent. For a given point in the five-dimensional parameter space rn,(rn~),tan I3(M~),rnl/2, rn0, A we begin by integratingthe RGEsfor the gaugeandYukawacouplingsup to Q = M~in order to specifythe completeset of boundaryconditions at this scale. We then evolve back down to Q = M~but this timeincluding the RGEs for the scalar masses as well. The feasibility of this approachrelieson the basicfact that thevaluesof p., B decouplefrom the full set of RGEs.Thus,initial specificationof p.(M~),B(M~) is not required.Now armedwith a setof low-energyparameters(except for p., B, for which we must make an initialguess),we can calculatethe s/particlespectrumwhich entersinto the supertracein zilV. We then solve for p.(M~) and B(M~)via the minimization conditions for

the one-loopscalarpotential. Specifically, we numericallydeterminethe valuesofp. and B which solve the following conditions:

=03>~l,<47>2,(4l2,4,568>=0

where V= V(v1, v2, p., B, m1/2, rn0, A, m~)is the scalarpotential (tree-level orone-loop),andthe 41, describethe eight real degreesof freedomof the two Higgs

doublets(in the notation of ref. [53]). At tree-level V= V0 and the conditionsabovecan be solved analytically for p. and B (see eq. (1.2)). For the one-looppotential, weemploy a two-dimensionalNewtonmethodwhich quickly locatestheextremalvaluesfor p., B. We begin by making a guessfor p., B values,and thenallow the systemto self-consistencyrelax to the extremump., B values.Note thatsincewe havedemonstratedthe one-loop Q-independenceof V~(Q),this impliesthat if we were to minimize V1(Q) for Q * M~, the values of p.(Q) and B(Q)would just bethe one-loopRG-evolvedp.(M~) andB(M~)obtainedby minimizingV1(M~).We can thus derive the boundaryconditionsfor p., B at any scale; we

conveniently choose Q = M~such that M~= ~[g~(M~) + g’2(M~)][v~(M~)+

v~(M~)]correspondsto the physicalZ mass.In principle, therecould be more than one point in the p., B parameterspace

which is consistent with the above conditions. We have searchedthe p., Bparameterspacevia a Monte Carlo analysis and find, howeverthat in all casesconsidered,thereis only one allowedpoint for p., B. This simplifies the searchofthe parameterspaceenormously.

As for step(ii), to evaluatethevalue of the potential itself, the discussionabovemakes clear the natural choice for the configuration of the scalar field space: theminimum of the potentialat Q = M~.In general,it is not necessarynor essentialto choosethe minimization configuration; due to the non-negligible two-loop

18 S. Kelley etal. / Radiativeelectroweakbreaking

effectsthatwe havedescribedin sect.2.2, it is likely that this initial choice in field

spacewill no longer representthe minimumfor Q > M~anyhow. Finally, regard-ing the tree-level parametersin step (iii), we simply evolve the Q-dependentparameters g1, g2, p., B, rn~,, rn~2, v1, v2 from M~to variable Q, using thestandardone-loop RGEs. These running parameterscontribute to the implicitQ-dependenceof V1 andappearsolely in V0.

3. The fine-tuning problem

Even though supersymmetrictheories technically solve the gauge hierarchyproblem, it is well known that this problem may be re-introducedin a differentguise if the splitting of the supersymmetric multiplets exceeds 0(1 TeV). Aquantification of this “fine-tuning” effect was proposedin ref. [36] and laterelaborated on in ref. [37]. The purpose of this section is to explore, in the light ofour own approach,the consequencesof the proposed“naturalness”cut on theparameter space.

3.1. GENERAL REMARKS-

The basic ideacan be easily graspedby studying eq. (1.2a),which we triviallyrewrite as follows:

1 2 rnH rnH tan2/3 — 2=X 2+ 2 — 2 1

— tan2/3 — 1 p. —

0rn0 ,~2m1~2p. . ( . )

Clearly, for increasinglylargervaluesof the dimensionalsupersymmetrybreakingparameters,the renormalization-groupevolved massm~ tends to increase,forc-ing p.

2 to larger values as well. It can also happenthat the value of m~ isfine-tunedsuchthat p. remainssmall evenif m,~

2and m0 grow large. Therefore,the measuredvalue of M~becomesthe result of increasinglymore “fine-tuned”cancellations among the supersymmetric masses.

Before proceedingto the specifics, we would like to point out a differencebetweenour approachand the standardone to the explorationof the parameterspacein this classof models.This differenceis in the treatmentof thevariablesp.and m~.We take rn~ as an inputandobtain p.

2 (which is relatedto p.0 = p.(M~) via

RG-evolution)directly usingeq.(3.1). In the standardapproachon the otherhand,onegives p.0 ~-‘ p. anduseseq.(3.1) asa constraint to determinern~ implicitly. Theimplied numericalinversion becomesharder (i.e., more time-consuming)as thefine-tuning (e.g., rn1~2)grows. For example, to obtain p.

2 + ~ with a relative

S. Kelleyetal. / Radiativeelectroweakbreaking 19

accuracyof 0.001 (for p. = 150 GeV, rn0 = rnl/2, A = 0, tan /3 = 5) one needstosolve for rn~ to the first/second/third decimal place for m,/2 = 150/300/600GeV (one obtains: m~=120.8/97.45/91.061 GeV). A variant of the standard

approach,wherein rn~ and p. are given as inputs and m0 is solvedfor using eq.(3.1), hasan analogousproblem involving the determinationof rn0.

3.2. THE FINE-TUNING COEFFICIENTS

The usualdefinition of the fine-tuningparametersc• is given by [36,37]

a- I9M~c,-= ~ a1=p., rn~,rn1~2,rn~, (3.2)

where the a, are the relevantparametersof the theory. It is then arguedthat ifc• <4, then cancellations among the parameters of at most log 4-orders ofmagnitude occur. The various expressions for the c1 scalewith (rn1,,2/M~)

2andtherefore we define scaledcoefficients c’~, cl/(m,/

2/MZ)2, and obtain (with

M~~Mz/rni,,2and ~0~m0/rn~,,2)

= 2p.2/m~~

2= 2(Xl/2 +X0~— ~ (3.3a)

aX0 aX1 2am~ (3.3b)

= 2 X,/2 I, (3.3c)

ê0=2IX0j~. (3.3d)

The importantpoint is that sincethe ê1 scalewith ~ then an upperbound onrn1~2(for a given ~) results for a given choiceof 4, i.e., rn1~2<m’~’~(~0,4), and

theseboundsscalewith ~ZTo facilitate the subsequentdiscussion,we now give analyticalexpressionsfor

the quantitiesX0, X1/2 [37,34] whicharevalid in the limit of vanishingAb, A~,or

* Thereis of coursealsoanupperboundon m0= f0m,,2 <~0m~(~0,~i) (unlessX0 = 3X0/öm~= 0;

seebelow).

20 5. Kelleyet a!. / Radiativeelectroweakbreaking

equivalentlyfor not too largetan 13 (i.e., tan 13 <8 [34])

3 tan2/3+1 rn, 2

X0 1+ 2tan

2/3—1(~), (3.4a)

tan2/3+1 m1 2

X 2=—K1+1/ tan

2/3 — 1 rn~

m 2X ~ (1+tan2$), (3.4b)

where 4A ~A/m1,,2, rn~= 192 GeV, and K1, a, b aresome (positive)numerical

coefficients.The simplest fine-tuning parameteris ~, which basically imposesan upper

boundon p.,

Ip.I<p.m~~=~_Mz (3.5)

However, since in our approachp. is a derivedquantity, the constraintson the

parametersof the theory are less transparent.In fig. 3a we show ê~(calculatedexactlyand for tan /3 = 5) as a function of ~ (for ~ = 0) andfor threevaluesofrn~. From eq. (3.4a)we seethat X0 has a zero at rn~ 151 GeV (for tan /3 = 5)and therefore will be quadraticin ~ but with negative(positive)curvaturefor

rn, < 151 GeV (rn,> 151 GeV), as observedin the figure. The magnitudeof thecurvaturegrows with rn,, also as anticipated. In fig. 3b we show the case where

= ~ which hasa similarbehavior,althoughthe zeroof X0 is not the relevantturningpoint anymoresince XI/2 also depends on ~ through ~A•

From the expressionfor one canseethat if X0 = 0, theneventhough m,~2would still need to be boundedabove, m0 would not. Analogously,if X1/2 = 0,then rn,/2 could grow indefinitely without affecting ç. Thesepeculiar points inparameter space [37] are isolated and are not stable in perturbation theory.Moreover,in the secondinstance,they correspondto smallvaluesof p. which areexcluded on phenomenologicalgrounds.In the figures, the first caseis seento

occur in the rn~= 150 GeV curve in fig. 3a,whereasthe secondcaseis approachedby the ~o= 7 point on the rn, = 145 GeV curve in fig. 3b.

The ê1 fine-tuning coefficientis shownin fig. 4 in analogyto fig. 3. Largevaluesof this coefficient correspond to instances in which rn, is fine-tunedto give smallvaluesof p. in eq. (3.1), eventhough ml/2 and rn0 grow large. This coefficientgives qualitatively similar constraintssince it is now quadratic in ~. From theapproximateexpressionfor X0 onecaneasily show that rn~öX0/9m~= X0 + 1 > 0,

S. Kelleyeta!. / Radiativeelectroweakbreaking 21

and thereforewe expectê~to grow with ~ for all allowedvaluesof rn~, andalso

~ > ê~,.Thesefactsareevidentin the figures.The effect of the coefficientcanbe studiedfrom ê~by setting ~ = 0 (c.f.

50 I I I I I I I I I

(a) /40— EA—0 —

~

5: 068 10

(b) /40 — —

30 — m~=l?OGeV —

//20— 160 —

1:

~0~A

Fig. 3. The scaledfine-tuningcoefficient i~as a function of ~ for variousvaluesof m, (andtan/3 = 5,> 0) for (a) f.., = 0, and (b) ~., = ~. Values of cc = êc(m1/2 /Mz)2 above ~1= 100 correspondto

fine-tuningof parametersto morethantwo ordersof magnitude.

22 S. Kelleyet a!. / Radiativeelectroweakbreaking

eqs.(3.3a),(3.3c)). As seenfrom fig. 3 (for ~o= 0), c1~2 dependsvery weaklyon rn,and therefore gives a direct upper bound on rnl/2, independently of ~, ~A’

althoughthis bound is not as strongas the onesobtainedfrom ê~,,ê~.Finally, theê0 coefficient hasthe same~ dependenceas doesê~,althoughits magnitudeisshifteddown by X,/2.

50

(a) EA=O ///40 — —

mt=l’7O GeV

30— 150 —

Ct

20 — —

10 — —

I I I I I I I I I I I

0 1 2 3 4 5

50 - I I I f~T~I~I~I I I I I ThT~[~/ I/I /(b)

40— -

- m~=l’7O GeV

30 — 150 —

CL

20 —

10 — —

0 I I I~I I I I I I I I I 1111

0 1 2 3 4 5

Fig. 4. Sameas fig. 3 but for the scaledfine-tuning coefficient ê,. The middle curve correspondstom,= 160 0eV.

S. Kelley etal, / Radiativeelectroweakbreaking 23

3.3. SOME SAMPLE BOUNDS

It is not our intentionto quoteconcreteupperboundson the variousparame-ters in the model, sincethesewould dependon the chosenvaluefor the cutoff 4(althoughtheseboundswould scalewith ~ For example,from eq. (3.5) taking

4 = 10 (as done in previousanalyses[37,32])gives p. I ~ 200 GeV, and p. I� 2.2. This bound appearsunnecessarilystringent.A more reasonablecutoff of

I p. I � 450 (650) GeV is obtained for 4 = 50 (100). It is also not clear to uswhetherthe abovedefinition of the c1 is the only possibleone,or rather whetheralternativedefinitions would yield quantitativelysimilar results (qualitatively they

must all agree).Our purposehereis to determinewhetherthe valuesof ml/2, ~ ~A which we

will examine later give values of the c1__below a ‘reasonable’ cutoff of say

4 = 50—100. In general we have rn,/2 < ~ From the figures it is clearthat for ~ � 1 onegets ê~ 3—6, and thereforern,/2 ~ 370—265(525—370) GeVfor 4 = 50 (100). Theseestimateshold for tan /3 = 5 anddecreasewith decreasingtan /3: for tan /3 = 2, e~(~0~ 1) 9, ml/2 ~ 220 (310) GeV for 4 = 50 (100).Larger values of ~ strengthenthesebound rapidly ~.We thus see that if werestrict rnl/2 to ml/2 � 400 GeV or equivalently mg ~ 1 TeV (and take ~ ~ 1)then the resulting class of models will presumablyremain in the reasonablefine-tuningregime. Since the squarkmassescanbe approximatedby m~= rnl/2(c+ ~ with c 6, we also seethat m4 < 1 TeV.

Eventhoughthesefine-tuning “bounds” on ml/2 and ~ arenot very precise

andin fact do not necessarilyhaveto hold in therealworld, it is useful to haveanestimateof where thingsstart to become“un-natural”.

4. The models and their motivation

For the purposesof this paper we will consider two SU(3) X SU(2) X U(1)

supersymmetricmodels. The first is the supersymmetricstandardmodel (SSM)with the minimal threegenerationsand two Higgs doubletsof matterrepresenta-tions,andwhich is assumedto unify into a largergaugegroupat a unification massof M~ 1016 GeV. The secondis the string-inspiredstandardmodel (SISM) [54]with the additional rninimal set of vector-like Q and D’ matter representationswith massesset to obtainsin

2O~= 0.233and astringunification scaleof M~= 1018

GeV. For attemptsat obtaining string-derivedmodelsof this type seeref. [55].However, it is widely believedthat theremust be more than this to nature,in

particular grand-unificationand string unification. Surprisingly, many granduni-

* Note that larger valuesof f~are fine as long as ~~1/2 is low enough,e.g., for m,/2 ~ 100 GeV,

c, <50 (100) for ~ <4 (6).

24 S. Kelley et a!. / Radiativeelectroweakbreaking

TABLE 1Valuesfor thefine-tuningcoefficientscorrespondingto typical points in parameterspace(with

tan /3 = 1.73, m,= 125 0eV, j.~>0, A, = — 0.6m0[35])which satisfyproton decayboundsin minimal

SU(5)supergravitymodelsfor M0 3 M,~.Theml/2 valuesarein GeV

m,/2 ~A c~ c,

74 8.1 —5.4 39 113122 6.5 —3.2 67 204187 5.4 —1.6 107 342267 4.5 —0.41 155 503364 3.8 + 0.55 217 699

fied and string models can reduce (below the unification scale) to the SSMorSISM with a few minor alterations.Thusthesesimpler modelscangive a goodfirstapproximationfor the low-energypredictionsof morerealisticmodels,andprovidethe groundworkupon which the extradetails of more complicatedmodelsmay be

added.Take for exampleminimal supersymmetricSU(5) [56,13]. After the GUT is

broken, the light contentof the model is exactlythat of the SSM. Probably themost important difference are the dimension-fiveproton-decay-mediatingopera-tors [57] resultingfrom integratingoutheavyGUT Higgstriplet fields of massM11.The presenceof theseoperatorsimposesstrong constraintson the parametersofthe model [58,34,35].Indeed,in ref. [35] it is shownthat the currentexperimentallower boundon the decaymodep —p 1K~can betransformedinto an upperboundon a quantity P (called B in ref. [35]) which encompassesthe sparticle massdependencesof the decaywidth (T a 1/P

2). These authors find P < (103 ±15)(MH/MU). The function P is complicatedbut it is arguedto be reducedby smallgluino massesand large scalarmasses,i.e., by largevalues of ~o. It also has the

following explicit dependenceon tan /3, P a (1 + tan2/3)/tan/3, which then re-duces P for small values of tan /3. How large does ~ need to be to obtainacceptableproton decayrates? In table 1 we show the valuesof the fine-tuningcoefficients, c,,, and c

1 which follow from the choices made in fig. 2 of ref. [35] forthe model parameters(tan /3 = 1.73, rn, = 125 GeV, p. > 0, and A, = —0.6rn0).

The minimumvaluesof ~ are thoserequired to obtain a value of P equal to itsexperimentalupperbound andwith the assumptionM11 = 3Mg. We see that thefine-tuning coefficient c~exceeds4 = 100 in all casesand therefore compatibilitywith proton decay experiments drives the minimal model into the fine-tunedregime. This was also observed in ref. [35]. Our point here is to be morequantitative in the light of our studiesof the fine-tuning constraint.Note thatincreasingthe value of tan /3 increasesP and therefore makesthe experimental

bound harder to satisfy. Note also that had we chosen M11 = M~instead,then theminimum ~ values get increased significantly (refer to table 1): for m1/2 = 74

S. Kelley eta!. / Radiativeelectroweakbreaking 25

(122) GeV, ~“ goes from 8.1 (6.5) to 16.2 (11.5) and c~from 113 (204) to 432(603).

Clearly, the minimal SU(5) supergravity GUT is a rather constrained model,which may not survive proton decayboundsif the fine-tuning constraintshold atface value ‘~. Therefore, the SSM is probably best thought of as the low-energylimit of a unified model where the strict proton decay constraints are naturallyavoided. One strategy to remedy the ailments of the minimal SU(5) model is tochangethe gaugegroup to flipped SU(5) [61]. After GUT breaking,the resultingeffective theoryis the SSMwith the Higgsmixing term p.hh provided elegantlybythe superpotentialterms A7hh41 + A841

3 involving a singlet field [62]. This isqualitatively the same as the usual Higgs mixing term once the singlet gets a VEV.However, thereis a crucial differencerelative to the minimal SU(5) model sincenow the Higgs triplet mixing term (a (41)) is naturallyof O(M~)and leads to a

suppression of the dangerous dimension-five proton decay operators ofO(M~/M~)relative to the minimal SU(5) case, thus making theseoperatorsinnocuous[63]. Quantitatively,therearedifferencesin the two models,thoughthedimensionof the solution spaceis the same.The tree-levelcasemay be solvedby

noting that the dimensional parameters in the tree-levelpotential scalewith m1/2,

allowing one to minimize a dimensionlesspotential and then fix m1/2 from themeasuredvalue of M~[63]. This approach does not work in the one-loop casebecausethe spectrum does not scale with rnl/2, and A7 and A8, unlike p. and B,feed into the other RGEs. Solving the one-loop problem requires a search overparametersat the unification scalefor the subspacewhich gives the correct M~,the needlein the haystackapproach.In string-derivedflipped models,the role ofthe singletVEV is played by a hidden sector condensate in a non-renormalizablesuperpotentialterm which effectively reproduces the usual Higgs mixing term inthe SSM [64].

Anotherpossibility is that thereis no GUT. The requirementof gaugecouplingunification when thereis no GUT is unmotivatedin field theory but completelynatural in string theory, andmodelsexist in which the string directly gives rise tothe standardmodel gaugegroup [65,66,67].String models predict a unificationscaleof about1018 GeV [68,70]andusuallycontainseveralextravector-like matterrepresentations. This motivates the SISM, in which the extra vector masses arechosenso that the gaugeunification scalegivesthe required 1018 GeVresult [54].Among the many possible sets of extra vector representations which might arisefrom the stringandrealizethecorrectunification scale,we havechosenthe uniqueminimal set for the SISM to analyzein this work. Theseare an extra vector-like

* Theminimal model with a Higgs singletmustalso contendwith thedoublet—tripletsplitting problem

[59].The missingpartnermechanism[60] solves this potential problem at the expenseof introducingnew largeGUT massiverepresentations.Nevertheless,the light contentof the model is still theSSM.


quarkdoubletwith massm0 3 x 10~~GeV and an extravector-like charge—

quarksingletwith mass mDc 3 X i0~GeV [54].One concernis that the effectiveSSMor SISM from morecomplicatedmodels

might not result in universalsoft supersymmetrybreakingat the gaugeunificationscale.If a GUT hasuniversalsoft supersymmetrybreakingat a scalemuch largerthanthe gaugeunificationscale,differencesin the supersymmetrybreakingmassesbetweenthe different group representationswould exist at the gaugeunificationscale. However, most scenarios of supersymmetry breaking predict a scale wheresupersymmetrybreaksnear the gaugeunification scaleso thesemodificationsofuniversalsoft supersymmetrybreakingareexpectedto be generallysmall.

In the flipped modelit can be shown that the onsetof supersymmetrybreaking,

the SU(5)x U(1) and the SU(5)x U(1) —‘ SU(3)x SU(2) x U(1) scalesmust bewithin aboutan order of magnitudeof eachother [63]. The two gaugeunification

scalesarealmost identicalbecauseof couplingconstantunification andexperimen-tal boundson the low-energygaugecouplings[71]. The GUT breakingis along adirection of the potentialwhich is flat in the absenceof supersymmetrybreaking.Upon the introduction of soft supersymmetrybreakingterms,dynamical calcula-tions reveala minimumjust slightly below the scalewhere supersymmetrybreakingis introduced[63].

In summary,the simpleSSM and SISM modelsnot only havethe advantageof

ignoring the complicatedandmodel-dependentdetails of morerealistic GUT andstring models,but also capturethe essenceof many of thesemorerealisticmodelsas theseextradetails,in manycases,are irrelevantor give small correctionsto theresultsof the simpler models.

5. The allowed parameter space

5.1. CONSTRAINTSON THE PARAMETER SPACE

We now presentthe several consistencyand experimentalconstraintson this

classof modelsandlater discusshow they restrictthe allowedparameterspace~.

(1) Consistencyconstraints.(i) Perturbativeunification: the valuesof the t, b, and T Yukawa couplings

should remain in the perturbativeregime(andcertainly finite) all the way up toM~.Tree-levelpartial wave unitarity is violated if thesecouplingsexceedA 5 atany scale[72]. We apply theserelationsat M~where theyaremost constraining.

* For reference,the values of the measuredparameterswhich have been used are: a3= 0.113,

mh = 4.9 GeV, a~= 127.9, Mz = 91.2 GeV, andm~= 1.784 0eV. Theallowedregionsin parameterspacehave a small sensitivityto ba3= 0.004.

S. Ke!leyeta!. / Radiativee!ectroweakbreaking 27

At low energiestheseupperboundson the Yukawacouplingsget transmittedinto

upperboundson m~and tan /3 as follows,

rn~ = 174A, sin /3 < 174A~/1/i i7t~13~ 190 ±1 GeV, (5.1)

for a3 = 0.113±0.004, and

rnb = 174Ab/i/1 + tan2/3 <174A~/~/1+ tan2/3 =~ tan /3 ~ 47 ±2 (5.2)

for mb(rnb) = 4.9~ 0.1 GeV. Since dynamicallyonegetstan /3> 1 (seebelow)andexperimentally rn~>90 GeV, we get a completely bounded region in the(rn~, tan /3) plane.

(ii) Electroweakbreaking:as discussedin sect.2.3, we solve for p. and B usingthe one-loop effective potential. This implies [12] that the set of subsidiaryconditionswhich areusually imposedto obtain a good minimum of the tree-levelpotential (i.e., boundedness,stability, and avoidanceof electric chargeand colorbreakingminima)areautomaticallysatisfiedby the one-looppotentialand do notneed to be (and have not been) imposed~. It is however necessaryto demandboundednessof the potential at the unification scale,i.e.,

~ =m~,1+rn~, + 2p.

2+ 2Bp. —s 2(m~+p.~+B0p.0) >0. (5.3)

We also demandthat all squaredsquark,slepton,andHiggsmassesbe positive. Inparticular, this must hold for the pseudoscalarHiggsmass,rn~> 0. At tree-levelrn~= — 2Bp./sin2/3 and therefore(Bp.),~~~<0. However, this doesnot necessar-ily hold at one-loopsince there are additional contributionsto m~which allowbothsignsof (Bp.),000.

(iii) Cosmology: astrophysicalconsiderationsindicate that the lightest super-symmetric particle (LSP) must be neutral and colorless [73]. This leaves twocandidates:the sneutrinoand the lightest neutralino.As discussedin ref. [25], inmost of the parametersspaceit is the lightestneutralinowhich is the LSP,and issensibleto neglectthe small regionsof parameterspacewherethe sneutrinois theLSP (i.e., rn~ 42—46 GeV **)~ In what follows we exclude points in parametersspacewhere the lightestneutralinois not the LSP.

* Actually, the constraintsto avoid electric charge and color breakingminima as derivedfrom the

tree-level potential need to be imposed in a less direct way when using the one-loop effectivepotential [12]. When appliedproperly, theseconstraintsgenerallylose much of their effectivenessand in a numberof calculations[12,26] theyhavebeenfound to becompletelyineffective. Thereforewe chooseto neglectthem here.

** If thesneutrinosmake up thedark matterin thegalacticehalo,this massrangeis eliminatedby LEPmeasurements[74].


(iv) Naturalness:as discussedin sect. 3 to avoid re-introducingthe fine-tuningproblem, we require rnl/2 ~ 400 GeV (i.e., rng � 1 TeV) and ~o~ 1 (or moreproperly, ~ ~ 100).

(2) Experimentalconstraints.We imposethe following cuts on the sparticlemassesand/orcouplings:

(i) The LEP lower boundon the charginomassm~±>45 0eV [751.(ii) The CDF lower boundson the gluino (mg> 150 0eV) and the squark

(m4> 100 0eV, except11) masses[761.Theseboundsare actually correlatedand

are subjectto numerousassumptions.Fortunately,they aresatisfiedautomaticallyin this classof modelsoncethe otherconstraintsare imposed.

(iii) The LEP lower boundon the chargedsleptonmassmj> 43 GeV[77].(iv) The CDF lower bound on the top quark mass rn, � 90 GeV [781.We do

not expecta significantweakeningof this SM bound due to potentialnon-SM topdecayprocesses,since theseare rather suppressed(e.g., t —s bH~,rnH±>M~,t —b t~A~’,rn1 + rnX~>72 GeV in thesemodels).

(v) The contributionsto the invisible and new Z widths from Z decay intoneutralinopairs,i.e., F(Z —~x,°x,°)<Frn” < 18 MeV and F(Z —sxi°x~)<F~~ew< 28MeV (i =1 = 1 excluded)at 95% CL [79].

(vi) The contributionto the invisible Z width from Z —* i~5for rn~~ ~ i.e.,

F(Z —~ i3~)< 18 MeV, since we expect the invisible decay mode i —~ r’A’1° todominatein this m~range.For threedegeneratesneutrinos,as is the casein thetwo models we consider,we get rn~� 42 GeV. Realistically this bound will befurtherpushedtowards~-M~since the F(Z —sxi°x~)also contributesto F~~Iin the

sameregionof parameterspace,thuspossiblyclosing up the window for sneutrinoLSP.

(vii) The experimentalconstraintson the lighter Higgs bosonsh and A, asfollows: we require F(Z —~ hZ* ~ hp.~p.)/F(Z—s p.’7s) < 5 X iO~,and F(Z—s hA)/F(Z — p.~p.)<0.11; thesevalueswere obtainedfrom a graphical fit to

the experimentalresults[80,79].The Higgsbosonmasseshavebeencalculatedtoone-loopaccuracyas describedin ref. [27].

5.2. ANATOMY OFTHE ALLOWED REGIONSIN (m,, tan 13) SPACE

The consistencyconstraintsdescribedabove allow us to obtain a bounded

region in (rn~, tan /3) spacefor all valuesof rnl/2, ~ ~. These areas are furtherconstrainedby the phenomenologicalcuts given above.It is howevermoreillumi-nating to understandthe shapeof the boundedregionprior to thesecuts.This isparticularly simplein the tree-levelcasewherep. and B areobtaineddirectly fromeq. (1.2). In this caseone has to demandtwo subsidiaryconditionsto ensureagoodsymmetrybreakingminimum, namelythat the tree-levelpotentialbe boundedfrom below

~=m~,+rn~2+2p.2+2Bp.>0, (5.4)

S. Kelley ci a!. / Radiativeelectroweakbreaking 29

andthat it possessesa minimum away from the origin of field space,

5°= (rn~i, + p.2)(m2~+ p.2) — B2p.2 <0. (5.5)

In the one-loop case p. and B need to be solved for numerically and theseconditionsareenforcedautomatically[12]. But then,when is an (rn~, tan /3) pointnot allowed at one-loop?This happenswhen the searchfor a p., B pair failsbecauseB is driven to very large values (i.e., p. —~ 0) or when an otherwiseacceptablepair neverthelessgives m~<0.

To illustrate our remarks,in fig. 5a andfig. 5b we show the resulting allowedregionsfor ~ = 0, 1 and ~ = 0 for in

17.2 = 150, 250 0eV andbothsignsof p. for

the SSM prior to the applicationof any phenomenologicalcuts. (The followingdiscussionappliesqualitatively to the SISM casealso.)The solid (dashed)bound-ariesare those calculatedusing the tree-level(one-loop) potential. The dottedlines are one-loop contours of p. (for p. > 0 boundaries) and B (for p. <0

boundaries).The tree-level and one-loopboundarieshavefive distinct portions.Thesearemost easilyunderstoodin tree-levelapproximationas follows:

150 GeV E000, EAO.O m17.2= 250 GeV50 III~II~IjIIII ~~II~IIII IL>

0 so

50 75 100 125 150 175 200 50 75 100 125 150 175 200

m~(GeV) m~(GeV)

150 GeV p.<0 m1/2= 250 GeV

50 ~ 50 ‘~‘‘T II~IIIIIIIII[III III

tO~..•1~

50 75 100 125 150 175 200 50 75 100 125 150 175 200

(a) m~(GeV) SSM mt(GeV)Fig. 5. Tree-level(solid) andone-loop(dashed)boundarieswith no phenomenologicalcutsimposedfor(a) f~= 0, (b) ~ = 1, and ~A = 0, rn1,,2 = 150, 250 0eV, and both sign of iz for theSSM. The dotted

lines areone-loopcontoursof ~ (z > 0 boundaries)andB (~z<0 boundaries)in GeV.

30 S. Kelley eta!. / Radiativeelectroweakbreaking

150 GeV ~0~°’ EA°° m112= 250 GeV

~50 75 100 125 150 175 200 50 75 tOO 125 150 175 200

mt(GeV) mt(GeV)

150 GeV p.<0 m17.2= 250 GeV~ 50 IIIIIIIIIIIIIIIIIIIIIIIIIIII

ØI±l.III

50 75 tOO 12S 150 175 200 50 75 100 125 150 175 200

(b) mt(GeV) SSM mt(GeV)Fig. 5 (continued).

(i) The top boundary: this is the positively sloped line restricting the values oftan /3. For points abovethis line £~‘<0. For pointson or below this line we canuseeq. (1.2) to obtain

tan

2f3 + 1= (1 — sin 213) tan2/3— 1 (rn~, — rn~,

2) — M~. (5.6)

Clearly sincernH2 scalewith m1,/2, for rn17.2 >> M~an asymptoticstateis obtainedin which the explicit M~dependencebecomesirrelevant,whereasany finite valueof ml/2 helps drive ..~/-~3’ to negative values and therefore more restrictive topboundariesresult, as the figures show. Note though that rn1,,2 = 250 0eV isalreadyquite asymptotic.For what valuesof rn,, tan /3 is this constraintrelevant(or where in the planedoesthis line lie)? Considerthe following ROE:

d 1— rnH) = —~(3A~,Fb— 3A~F,), (5.7)

dt 8~

S. Kelley ci a!. / Radiativee!ectroweakbreaking 31

where Fbt are linear combinationsof squaredsupersymmetrybreakingmasses.AtM~, rn~, = rn~2 and F,, = F, and therefore for A,, <A,, m~,— m~2grows as tdecreases.OneneedsAb A, to turn this aroundandhave rn~,, — m~2<0 at M~,otherwise~ > 0 automatically.From eqs. (5.1) and (5.2) we get for tan /3>> 1,A, m,/174 and A,, tan /3 mb(MZ)/l

74, where rnb(Mz) = 3.77 GeV (forrnb(mb) = 4.9 GeV). Therefore the .~ 0 line occurs for tan /3 m,/3.77, asobservedin the figures.This result is independentof ~o’ ~ For later reference,from eqs.(1.2b) and(5.4) it is easyto seethat rn~= —Bp./sin 2/3 0 is equivalentto .~ 0, i.e., on the top boundarym~= 0.

(ii) The uppercorner: the perturbativeunification constraint in (5.2) cuts off

the growth of the top boundary.The roundedportion at the top andtowardstheside of increasingvaluesof rn~ resultsfrom the strengtheningof the perturbativecut on tan /3 dueto the increasingvalueof A, [72].

(iii) The right boundary: this originatesfrom the perturbativecut on in, in eq.(5.1). The curved portion for small tan /3 also follows from (5.1) for the corre-

spondingvaluesof tan /3, e.g.,we get m~”= 135, 171, 181 GeV for tan /3 = 1, 2,3.

(iv) Thebottomboundary:From eq.(1.2a)weseethat if tan /3 <1 then p.2 <0since rn~, — m~, > 0 for A, >> A,,, as is true for tan /3 ~ I (see discussionfollowingeq. (5.7)).Thereforetan /3> 1 always~.

(v) The left boundary:(a) p. > 0: pointsto the left of this line havep.2 <0 at tree-level.From eq.(3.1)

we have

p.2=rn~/2(Xl/2+xO~_ ~Mz), (5.8)

with X017.2 given in eq. (3.4) in the limiting case of small tan /3. For the valuesofm~where this boundaryoccurs X0 is negative(it turns positive for m, = 97, 121,

140 GeV for tan /3 = 1.5, 2, 3) and thereforeincreasing~o tends to drive p.2 to

negativevalues, unless this is compensatedby an increasein X1/2, i.e., by

increasingm~.Thereforeraising ~o shifts the left boundaryto the right, as shownin the figures.The effect of ~A is lesspronounced.For fixed ~, eq. (3.4b) showsthat ~A > 0 ~ <0) decreases(increases)X,/2 since b > 0. That is, a larger(smaller) value of rn, is neededto obtain the same p.

2 and therefore ~A > 0

(~A<0) shifts the left boundaryto the right (left).(b) p. <0: for ~ � 0.1 the sameremarksas the p. > 0 caseapply.For ~ ~ 0.1 a

peculiarityoccursdue to the boundednessat the unificationscaleconstraintin eq.

* In the tan~3= 1 caseat tree-levelone gets rn~= ~ B1.~= —(rn~,1+ ~~2) and thereforeP1= 0,

5” = 0, i.e., a ratherspecial scenariowhich alsogives rnh = 0. One-loopcorrectionsto V0 are mostimportant in this case[11].

32 S. Ke!!ey ci a!. / Radiativeelectroweakbreaking

TABLE 2Tree-levelandone-loopvaluesfor ~ contoursin fig. 5a(~= = 0) for e> 0 (similar resultsfor

j~<0). All massesin 0eV

rn,/2=15OGeV rn1~2—250GeV

P’tree P’ioop M’tree I~1oop

75 25 —50 150 60 —90100 65 —35 200 130 —70150 120 —30 300 235 —65200 170 —30 350 285 —65240 210 —30 400 335 —65

(5.3). In this case .s?~’o 2(p.~+ B0p.0) and for p.0 <0 (and therefore p. <0) toosmall valuesof p. can give 5lI’,,~<0. This results in the peculiar shapeof the p. <0

boundariesin fig. 5a.As seenfrom fig. 5a and fig. 5b, the one-loop minimizationyields very similar

boundaryconfigurations. In fact, only the left and (to a lesserextent the) topboundariesshow any shift relative to their tree-level counterparts,as easilyinferred from the above discussionon the origins of the variousportionsof theboundaries.Justoutsidethe left boundaryonefinds that B is driven to very largevalues(i.e., p. —* 0), andjust outsidethe top boundarym~< 0. Thesearepreciselythe sameconditionsdefining theseboundariesat tree-level.

The shapeof the p. contourscanbededucedfrom eq.(5.8), at leastat tree-levelandfor smalltan /3. For fixed rn1,,2 and ~o’ the value of p. becomesindependentof tan /3 for large tan /3 (see eq.(3.4)) and increaseswith rn, (at least as long asX1/2 dominatesoverX0~).This behavioris indeedrealizednumerically.Surpris-ingly, it is found to persistfor largevaluesof tan /3 and at the one-looplevel, asthe figuresshow.In fact, for not too small valuesof tan /3, the tree-levelcontourshave a “shadow” one-loop contour running on top and viceversa.The relationbetweenthesecontoursis shownin table2 for the typical caseof ~ = = 0 in fig.5a.Onecanseethat the shift in p. is largestfor small p. andthenasymptotesto a

constantvaluewhich growswith ml/2. (Similar resultsareobtainedfor the ~ = 1,

= 0 casein fig. 5b.)In fact, the shift is maximurn for (mi, tan /3) points on the one-loop left

boundary,since therep.1,~,,,,= 0 by definition. This meansthat on that boundary

the relative changein p. is 100%,anddecreasesas onemovesaway towardslargervaluesof rn,. The fact that the one-loopcorrectedpotential inducesa 100% shifton p. indicatesthat the underlyingperturbativeapproachto the determinationofp. breaksdown in this regionof parameterspace.Fortunately,thesepoints (where

loop 0) are ruled out on phenomenologicalgrounds:(i) for p. = 0 the LSP is amasslesshiggsinostatewhich gives F(Z —s xx) = F(Z —s vi)[(tan

2/3 — 1)/(tan2/3+

1)12 < 10 MeV for tan /3 < 1.4 only; (ii) for largervalues of p. (but still close tozero) the lightestcharginofalls below the LEP lower bound.Thereforeone does

S. Ke!!ey ci a!. / Radiativee!ectroweakbreaking 33

not have to worry too much about the possibleeffect of two- and higher-loopcontributionsto p..

The figures also show (althoughperhapsnot very clearly) that the valuesof p.scale with m1/2. At tree-level, the — p. tree I and I p.tree contours overlap in(rn,, tan /3) space.Eventhoughthis is not supposedto be necessarilythe casefortheir one-loopcounterparts,our resultsindicate that this is veryapproximatelythecasetoo. Therefore,the I ii I contourson the p. > 0 boundariesin fig. 5 can bemappedontothe p. <0 boundariesas — I p. I contourson the samepositionsin the(rn~, tan /3) plane.

A similar analysisfor the B contours(shown at one-loopon the p. <0 bound-ariesin fig. 5) revealsthat “shadow” contoursexist only for the largervaluesof B.Note also that contrary to the tree-level casewhere Btree vanishesat the topboundary, B100~ does not. This is becauseon the top boundary rn~= 0 and(rn~)tree = —(Bp.)~~e~/sin2/3 = 0 for Btree = 0, whereas (rn~)~00~=sin 2/3 + = 0 for B1,,,, * 0.

5.3. EFFECTSOFPHENOMENOLOGICALCUTS ON THE PARAMETER SPACE

We have explored the five-dimensionalparameterspaceby determiningtheslices in (rn,, tan /3) which areallowedfor bothsignsof p., rnl/2 = 150, 250 0eV,and severalchoicesof ~, ~, which respectthe fine-tuning constraints.Herewehaveimposedall the phenomenologicalcuts discussedabove(seefig. 6). (However,for clarity of presentation,in thefigureswe do not restrictthe valuesof rn,.) Thesediscretechoicesfor rn17.2, ~ ~A representa good sampleof the full spaceandenough to infer the shape of the (m,, tan /3) region for other values of theparameters.

The phenomenologicalcuts on the parameterspacehavevarious degreesof

effectiveness.For ~ ~ 0.1 theneutralLSP cut is mostconstraining.This is becausefor increasingrn,12 the LSP tendsto becomethe lightest ‘i state.Indeed,sincewehave not neglectedA7 in the calculation, the off-diagonal elements in the ~

squaredmassmatrix (a rn7(A7 + p. tan j3)) grow with p. (which grows with m,)andpushdown the i~mass ~. To avoid a chargedLSP the rangeof m, has to becut off as shown in fig. 6a. In particular, fig. 6a (solid line) shouldbe comparedwith fig. 5a whereno phenomenologicalcutswere imposed;the severityof the cutis stunning.The value of ~A has a small effecton the magnitudeof thiscut as faras the direct effect of A,. is concerned(for largevaluesof tan ~3).However, ~A

influencesthe value of p. also (see eqs.(5.8) and(3.4b)). In fact, ~A > 0 (~A<0)decreases(increases)X,/2 and therefore p. (for fixed ~ Thus ~A > 0 (~A<0)weakens(strengthens)the LSP cut by shifting m~upwards(downwards), as isevident in fig. 6 (i.e., comparethe dashed(~A = — 1), solid (~A= 0), and dotted

* Had we set = 0, a similar effect would still be presentwith ~R insteadof ~, although with no

significant rn, dependence.

34 S. Kel!eyci a!. / Radiativee!ectroweakbreaking

(~A= 1) boundaries).Note that the p. <0 figuressuffer from a lesseffective LSPcut. This is becausefor the same I p. I and ml/

2 values (and low tan f3), m~islower for p. <0. The LSP cut becomesineffective for ~ � 0.1 sinceby then thesleptonsget an additional significant contribution to their squaredmasses,i.e.,in,. TtZl/2(C( + ~ This canbe seenfor exampleby comparingfig. 6b (solid line)with fig. Sb (dashedline).

There are two phenomenologicalcuts which affect the left boundary: thecharginocut comesfirst and rulesout small valuesof p. (recall that p. = 0 on theleft boundary),effectively shifting the left boundaryto the right by a significantamount for small ml/2, for example,comparefig. 6a (solid) with fig 5a (dashed)andfig. 6b (solid) with fig. Sb (dashed).Oncethis is satisfied,theZ-width cut shiftsthe left boundaryslightly further to the right. The topboundary(where rn~= 0) isconstraineda little by the one-loop Higgsmasscut. This cut also constraintsthebottom boundaryfor valuesof tan /3 very close to 1. For ~ = 0 the neutralLSPcut is very effective on the bottom boundary, basically eliminating the range1 <, tan /3 � 2.

150 GeV ~0_°’°~ ~=O,+l m17.2= 250 GeV4OrrIIIII~~~~ ~ 40

:/1~~50 75 100 125 150 175 200 50 75 100 125 150 175 200

m,(GeV) mt(GeV)

m17.2= 150 GeV m17.2= 250 GeV

40 ~ 40 ~ [rrrr

30 — — 30 — —

I: __050 75 100 125 150 175 200 50 75 100 125 150 175 200

(a) m~(GeV) SSM mt(GeV)

Fig. 6. One-loopboundariesfor the SSM with all consistencyand phenomenologicalcuts imposedforrn172= 150, 250 0eV, both signs of ~ and the following (fo, ~ values: (a) (0, — 1) (dashed),(0, 0)(solid), (0, 1) (dotted); (b) (1, —1) (dashed), (1, 0), (solid), (1, 1) (dotted). The figures show the

progressionof theleft boundaryto highervaluesof rn, dueto theunit variationsof ~ and~A’

S. Ke!!ey eta!. / Radiativee!ectroweakbreaking 35

m1/2= 150 GeV E01,0, EA~°~ m17.2= 250 GeV50 r-rr, ‘‘‘‘l’’’’I’’’’I’’H’’’’ p.>0 so

___ ri50 75 100 125 150 175 200 50 75 [00 t25 150 175 200

mt(GeV) mt(GeV)

m17.2= 150 GeV m,/2— 250 GeV

~‘~‘~‘I”’’’’’ .

50 75 100 125 150 175 200 SO 75 100 125 150 175 200

(b) mt(GeV) SSM m,(GeV)

Fig. 6 (continued).

The boundariesshownin fig. 6 appearin thesequence:(0, — 1) —~ (0, 0) —p (0, 1)

and (1, — 1) —b (1, 0) —~ (1, 1) exemplifying the motion of the left boundaryto theright dueto the effectsof ~ and ~A’ as describedin sect.5.2. Othervaluesof ~

~A yield boundarieswhoseshapecan beinferred from the given ones.The one-loopSISM boundariesdiffer from the SSMonesin two respects:(i) the

LSP cut is lesseffectivewhenapplicable,and(ii) the left boundariesareshiftedtothe right relativeto thosefor the SSM. In fig. 7 we show two illustrativecasesplusone-loopcontoursof p. (for p. > 0). (The

4p. shiftshere follow the samepatternasin the SSM case.)Both of theseeffectscan be tracedbackto a lowervalue of p.for correspondingpointsin parameterspace(e.g.,comparethe p. contoursin fig. 7with those in fig. 5). Indeed,if p. is lower then the ~ masswill not be shifteddownwards as much and the LSP cut will be lesseffective. Also, p.2 is driven to

negativevaluessoonerandthereforeoneneedsto increasein, to compensate,i.e.,the left boundaryshifts to the right.

6. Predictions for particle masses

The spectrumof sparticleandHiggsbosonmassesin the SSM/SISMdependson the particularpoint in the five-dimensionalparameterspace,andthereforeit is

36 S. Kelicyeta!. / Radiativeelectroweakbreaking

a complicatedmatter to give the valuesof thesemassesfor all allowedpoints inthis space.However, not all massesdependon all parametersand someof theirdependencescan be workedout analytically.We now discussthe predictionsforthe squark, slepton, neutralino, chargino, and one-loop correctedHiggs bosonmasses.Whenrelevant,we also commenton the shiftson thep.- and/orB-depen-dentmasseswhich result from the minimization of the one-loopversustree-level

Higgspotentials.

6.1. SQUARKS AND SLEPTONS

The first andsecondgeneration(andthird generationfor thesneutrino)squarkand sleptonmassescanbe determinedanalyticallyas follows (seee.g., ref. [15])

tan2/3 — 12 21 2\ ______ 2rn=rn

12uc+~0I—u lvlwz ~ / ‘tan2f3+1

with d, = (T3, — Q) tan

2O~+ T3, (e.g., d4 = — + ~tan

2O~and dR = — tan2O~)

150 GeV ~O°~ E~00 m112= 250 GeV

40 j”,j’j”j’j,, p.>0

________ ~2OL 300

50 75 tOO 125 150 175 200 50 75 100 125 150 175 200

m,(GeV) m~(GeV)

150 GeV p.<0 250 GeV40 ‘‘j’’’~’’~’’’~’’’~ 40

30h —

I - çN~i~ I - ~~~~~

0 110

50 75 tOO 125 150 175 200 50 75 tOO 125 150 175 200

SISM(a) mt(GeV) mt(GeV)

Fig. 7. One-loop boundariesfor the SISM with all consistencyand phenomenologicalconstraintsimposedfor rn

172 = 150, 250 0eV, both signsof ~ andtwo illustrative valuesof ~ fA): (a) (0,0), (b)(1, 0). The dottedlines areone-loopcontoursof j.~(hr> 0 boundaries)andthe lightest Higgs mass rnh

~ <0 boundaries)in GeV.

S. Keilcyct a!. / Radiativeelectroweakbreaking 37

150 GeV ~0”’°’ E~0.0 m112= 250 GeV50 ‘‘‘i’’’’i’’’’r’’’i’’’i’~ p.>0

‘E~ ~Eij50.:~~~~

50 75 tOO t25 t5O 175 200 50 75 tOO 125 t5O 175 200

mt(GeV) m~(GeV)

m1/2= 150 GeV /~<0 m

17.2= 250 GeV

50

~O5~ I~ ~

50 75 100 125 150 175 200 50 75 100 125 150 175 200

(b) mt(GeV) SISM mt(GeV)

Fig. 7 (continued).

and the c, are coefficientsdirectly calculablein termsof the gaugecouplings(seee.g., [11]). These are listed in table 3 for both SSM and SISM. In detailedcalculationsinvolving the sparticlemasses,the c’s shouldbe renormalizedat thephysical sparticle mass [81]. However, for the purposesof this paper,we haverenormalizedall the c’s at M~for simplicity. We shouldpoint out that the squark

(slepton) coefficients have a 10% (~2%) uncertainty due to the presentuncertaintieson the low-energygaugecouplings[54]. The gluino massis analo-

TABLE 3Valuesof the c, coefficientsin eq. (6.1) for thefirst andsecondgenerationfermions (i.e., c55 = c,,,,, and

so on) for the SSMandSISM. Also shown is cg m~/rn,/2

c~ SISM SSM

cULd 3.91 6.283.60 5.87

cdR 3.55 5.82cOLe 0.402 0.512c- 0.143 0.149

2.01 2.77

38 S. Kelley eta!. / Radiativec!ectroweakbreaking

P (dots); eL (solid); eR (dashes)1.25 ~ ~ I’’ I,,,,I,,

3 ~

100 — —

0 75 — 2 ~ —

E

0 50 — 1

0,25 — —

0

(a)

0.00 I I I -0 200 400 600 800 1000

m~(GeV)

u1 (solid); 1,, (dashes);

2L (dotdash); ~,, (dots)

lhII~!1 I

—

1.4 3 ~

1.2— . —

2 ~

1.0 — — — . — — —

08I I I

0 200 400 600 800 1000

m~ (GeV)

Fig. 8. The ratios of the first and second generation (a) slepton and (b) squark massesrelative to thegluino mass as a function of the gluino mass in the SSM. The various lines represent the following: (a)

eL (solid), ~R (dashed),i (dotted); (b) 11L (solid), 0,, (dashed), dL (dotdash), dR (dotted). When a

splitting of lines of the sameparticlespeciesis noticeable(for the same~ this reflects the effect ofthe D-term contributionsto thesparticlemasses(for tan 13 = 2—10).

S. Ke!!cyet a!. / Radiativeelectroweakbreaking 39

gously given by m4= c4mt7.2,with 8% uncertaintyon c4. In fig. 8 we have

plotted the ratio of thesemassesto the gluino massas a function of the gluinomass for severalvalues of ~, in the SSM. Thesemassesdependon the other

parametersof the model and/or tree-levelversusone-loopminimization only tothe extent that specific valuesof theseparametersmay exclude certain valuesof

tan /3, ~ and rn17.2. In fact, in fig. 8 (where we have taken4A = p. > 0, and

= 125 GeV) not all linesstartat rn~ 150 GeV or continueup to in4 = I TeV,

due to the various cuts on the parameterspace.This dependenceis particularlyimportantfor the ~ = 0 case,where the upperboundon the massesvariesquite a

bit with in,. Note that the near proportionality to rn4 is only broken by smallD-term (tan /3-dependent)effects. In fact, to a good approximation(which im-proveswith increasingrn4) theseratios aresimply given by

- ~ci+~ (6.2)

m4

with the c, given in table 3. Both squarksand sleptonscan be lighter or heavier

150 GeV ~o=0.0, E~00 m1/2= 250 GeV40 ‘I’’I’’’l’’L’T’’’ p.>0

- t~,:290±20 r, 520±25

~ ~ ~: ~

50 76 tOO 125 150 175 200 50 75 100 125 150 175 200

m~(GeV) m~(GeV)

150 GeV p.<0 m1/2= 250 GeV40 ~ 40

t~,: 263±2t t,’ 472±48

!:OI~.~TO.~O5I I~95IOOh~

50 75 100 125 150 175 200 50 75 100 125 150 175 200

(a) m,(GeV) SSM m,(GeV)

Fig. 9. One-loop boundaries for the SSM with all consistency and phenomenological constraints

imposed for m~72= 150, 250 GeV, both signs of ~ and two valuesof (f,), fA): (a) (0, 0), (b) (1, 0). Thedotted lines are one-loop contours of rn~ (se> 0 boundaries) and the lightest Higgs mass rn~~ <0

boundaries) in 0eV. Also shown are the corresponding 1, mass ranges in 0eV.

40 5. Kc!ley ct a!. / Radiativeclectroweakbreaking

mt/2= 150 GeV E010, ~A°° m1/2= 250 GeV50 ~IIIIIIIIIIIII p.>0 50

50 75 tOO 125 150 176 200 50 75 100 125 150 175 200

m~(GeV) mt(GeV)

m17.2= 150 GeV ~<0 m

112= 250 GeV50 IIIIIIIII~ 50 ~

I:~1OOhb0I __50 75 tOO 125 150 175 200 50 75 100 125 150 175 200

(b) mt(GeV) SSM mt(GeV)

Fig. 9 (continued).

than the gluino, dependingon the value of ~. However, squarksare alwaysheavierthan sleptons.

All the aboveremarksapplyto the SISM aswell. With the help of eq. (6.2) andtable 3 onecan easily reproducethe analogof fig. 8 for the SISM. Note that thegluino andall of the squarksandsleptonsarelighter in the SISM than in the SSMfor the samepoint in parameterspace[54], althoughthe ratios rn,/mg are largerin the SISM than in the SSM.

The r12, b12, t,2 mass eigenstatesreceive additional contributions from theoff-diagonalleft—right mixing term(plus the correspondingfermion masses).In thecaseof b12 we haveverified that theseeigenstateshavemassesvery approximatelyequal to that of dR and dL respectively.This also happensfor the stau masseigenstateswhich arecloseto eRL, although the discrepancygrowswith ~ sincethen largervaluesof p. can occur. A rather interestingeffect occurs in this casewhich is worth pointing out. From eq. (6.1), d0,, < 0 andthereforem4,, growswithincreasingtan /3. In fig. 9 (for p. > 0) we show contoursof m.~1,whichexhibit justthe oppositetan /3 dependence,due to the off-diagonal rn,.(A,. + p. tan (3) term.

Thus,the effectof the D-term is completelywashedout by the A,. contribution.Thei~masseigenstatecanbesignificantly lighter thanthe averagesquarkmass.

In fig. 9 we indicatethe rangesof m1, throughouteachof the boundariesshown.

S. Kc!leyeta!. / Radiativee!cctroweakbreaking 41

For example,for rn4 = 415(690) 0eV (i.e., inl/2 = 150 (250)0eV) and ~ = = 0,the u~squarkmassis 360 (600) 0eV whereas~ 270—310(495—545)GeV forp. > 0 and 240—285 (425—520)GeV for p. <0. This discrepancygrows with rn,,,2

since p. scaleswith rn1,,2. The lowest value of m11 occurs for ~ 1 since thensmallervaluesof rn1,,2 are allowed.For rn~= 150 0eV and ~A = —1, —2, —3, weobtain rn~> 100, 80, 45 GeV. However,this effect occursonly for m~ 110 GeVand tan /3 2, i.e., a ratherfine-tunedandvery small regionof parameterspace.A more typical lower boundwould be in1 � 100 0eV.

6.2. NEUTRALINOS AND CHARGINOS

The neutralino,i~,°(i = 1, 2, 3, 4) and chargino~ (i = 1, 2) massesdependonjust threeparameters:inl/2, p., tan /3. We haveexploredthe massesof the lightest

of eachkind (using p.~00~)andfound that they get shifted relativeto what would beobtainedusing p.tree~As discussedin sect.5.2, the zip. shift is largestfor small p..This reflectsitself on the shifts of the ,~ and ~ massesas well. For example,forthe ~ = = 0 casefor m1/2 = 150 GeV we get rn’~r= 50 (55) GeV—s rn~°”=4 1—66 (50—53) GeV and rn~ = 75 (100) GeV —sm~~0±l~= 58—64 (92—93) GeV.

Since zip. <0, we expectthe shifts to be negative,asobserved.Note that the massshifts are rather small (~few GeV) unlessone is veryclose to the left boundary.Furthermore,as onemovesawayfrom this boundaryandp. increases,its effect onthe massesdecreasesconsiderably,making the zip. shift irrelevant.

6.3. HIGGS BOSONS

In both the SSM and SISM models we considerhere, the Higgs sector iscomprisedof two complex Higgs doublets.After electroweaksymmetry breakingthere remain three neutral (two scalarsh, H and a pseudoscalarA) and onecharged(H ~) physical Higgs bosons. Dependingon the specific point in theallowed parameterspace,the mass of h (and to a lesserextent H, A, H ±)mayreceivesignificantone-loopradiativecorrections[38,28,29,27].Herewe extendourprevious “no-scale” analysis(~= = 0) [27] of the one-loop correctedHiggsbosonmassesin the allowed(m,, tan /3, rn0, rn,,,2, A) parameterspaceby consid-

ering other ~ ~A options as well. As a fine-tuning constraint (see sect. 3.3), werestrict rn17.2 <400 0eV which effectively boundsthe variousHiggs massesfrom

above.The analysisof the MSSM Higgssectoris usuallyparameterizedby tan /3and mA; in contrast,for the SSM/SISM unified models inA is redundant,sinceinA (aswell as the otherHiggsmasses)are predicted from the original set of fiveparameters.

In order to include radiativecorrectionsto the Higgsbosonmasses,simplifying

choicesfor the many low-energyparametersare usuallymade(noneof which aremade in this paper). For example, it is typical to assume,A1 =Ab = p. and

rn~ = m~= m~= rn~. However, as discussedin the introduction, the assumption

42 5. Kc!lcy eta!. / Radiativec!ectrowcakbreaking

of universalsoft supersymmetrybreakingat M~in the SSM andSISM leadsto acorrelationof the many low-energyparameters,and thesecorrelationsare essen-tial to the overall predictivenessof the models. The Higgs spectrumwe presenthere incorporatesthesecrucial low-energycorrelations.

As expected, the Higgs massesscalewith rn,12, and the one-loop radiativecorrectionsdependstronglyon the valueof m~.In fig. 7 (SISM) and fig. 9 (SSM)we give contoursof h (p. <0 boundaries;contoursfor p. > 0 areverysimilar). Onecan seethat h is driven quickly to values above M~,although small corners ofparametersspaceexist where it can be as light as ‘~ ~ For the “no-scale”scenario(fig. 7a and fig. 9a), the maximal valuesfor h are rn~” 120 (130)GeVfor the SSM(SISM). The largerrn~”°’ valuefor SISM is due to the lessrestrictiveLSP cut as discussedin sect. 5.3. For a givenpoint in the parameterspacewe findthat overall thereis a slight systematicdownwardshift (<5%) in inh in the SISMcomparedto the SSM. This is probablya consequenceof the slightly smallervaluesof p. in the SISM relativeto the SSM (seesect. 5.3). For increasingvaluesof ~,

the valuesof rn~°’ rise and saturatewith the fine-tuningor “naturalness”value ofrn17.2 = 400 0eV; thevalue of rn~ — 135 (130) 0eV for SSM(SISM) andfor both

(~,~) = (1, 1), (1, — 1) choices. The values of rn~’°’ for the p. <0 casesaresimilar.

From the fine-tuning constraintsandthe various constraintson the (rn~, tan f3)parameterspacediscussedat length in sect.5.2, we canmakethe prediction thatin,, < 135 (130) 0eV in “natural” SSM (SISM) scenariosirrespective of thesupersymmetrybreaking choice; this is a ratherrestrictive andstriking prediction.This result agreeswith the previousanalysisof DreesandNojiri [29], eventhough

they madethe additional assumptionof A = B + in0. In comparison,the MSSManalysis(seefig. la of ref. [82]) showsthat rn~’°’ — 132(for rn~ = I TeV), againincloseagreementwith our result.

Due to the more severeLSP phenomenologicalcuts, the “no-scale”SSMseemsto be rather unique in requiring a “light” h (mh < 120 GeV). Of course,if the

fine-tuning constraintwere relaxedand in17.2 increased,then rn~” would rise aswell. Thesemassconstraintssuggestthat if h isnot foundat LEP, theWhX —‘ lyyXfinal statesat SSC/LHCmay be the only hopeof discoveringthe h Higgs. For amoredetailedphenomenologicaldiscussionof the h Higgsboson,including radia-

tive corrections,seerefs. [82,83].As for theh Higgs, as in1/2 increases,the massesfor H, A, H ± rise aswell, and

become increasingly degenerate.For rn17.2 = 400 0eV, in fig. 10 we presentcontoursof inA for the ~ ~A) = (1, ±1) casesfor both SSM and SISM, andp. > 0. Due to the highdegreeof degeneracy,the contourscanbe assumedto alsorepresentcontoursof inH and inH to within 5%.

In summary, the Higgs sector of “natural” supergravitymodels genericallypredict a relatively light Higgs scalar,

1~h< 135 0eV possibly detectableat the

SSC/LHC. This limit takesinto accountnaturalness,the full set of constraintson

S. Kel!eyet aL / Radiativecicciroweakbreaking 43

rn17.2— 400 GeV~Ø

10~ 1.0 E010, ~A~’°

50 ~ p.>0 50 ,,~,, ~ 11111111

IIIIIIIIi1~,I IIIiL~iIII~

50 75 tOO 125 150 175 200 50 75 100 125 150 175 200

mt(GeV) SISM mt(GeV)

~ ~A’° p.>0 ~ ~A’°

50 IIIIII~~II~II J-~-,-~’~-~111111 50 ~ 1111111 III

1 L 10 10 ~0O

50 75 100 125 150 175 200 50 75 100 125 150 175 200

mt(GeV) SSM mt(GeV)

Fig. 10. One-loop boundariesfor the SSM and SISM with all consistencyand phenomenologicalconstraints imposed for ~e> 0, mI/2 = 400 0eV (i.e., the fine-tuning upper bound), and ~,, = 1,

= ±~ The dottedlines are one-loopcontoursof mA in GeV. Within 5% these can be also be takenascontoursof m~and mH±.

the allowedparameterspace,and is maintainedovera wide rangeof supersymme-try breaking scenarios.In this limit, the H, A, H ± Higgs massesare nearlydegenerate,but could be anywhere from 100 0eV—i TeV. Due to the neardegeneracyof the SSM/SISM Higgs sectors, it is unlikely that discovery of aminimal supersymmetryHiggs sectorwill allow for an immediate discriminationbetweenthesetwo models.

6.4. SAMPLE PARTICLE MASS RANGES

In tables4 and5 we havecollectedthe massrangesfor all theparticlespeciesin

the SSM (resultsfor the SISM are similar) when one sweepsthe entire one-loopallowed regionin (m,, tan 13) for the given ~ ~, and rn~7.2 values.The valuesofrn17.2 havebeenchosensuch that the correspondingregions(in (mi, tan /3)) arenot too restrictive.For example,for too small valuesof rn17.2 the regionstend to beconfinedto small values of tan /3. Thus, we avoid “small” regionsof parameterspacein thesetwo variables.For the ~ = = 0 case(in table4), inl/2 = 110 GeVis closeto the lowest possiblyobtainablevalue (~re100 GeV). The massrangesfor

44 S.Kelley eta!. / Radiativec!cctroweakbreaking

TABLE 4Particle massrangesin theSSM(throughoutthe allowed(m,, tan j3) region) for f,~= = 0 andasampleof m,/

2 values.The ±indicatethesign of ~.t. Thelowestvalueof ml/2 is closeits absoluteminimum. All massesin GeV

m1/2

110~’ 110 150” 150— 250’~ 250

34— 50 22— 38 36— 67 32— 58 37—104 60—102

55—103 62— 69 73—133 78—104 76—175 112—18893—204 123—196 87—274 104—269 116—234 122—440

,~ 136—220 168—240 161—287 179—305 231—269 242—466

53—103 45—8w065 56—133 52—102 53—176 81—187j2~ 137—220 170—235 161—287 180—301 230—272 242—462

P 45— 60 45— 60 86— 94 86— 94 167—171 167—171eL 87— 92 87— 92 114—117 114—117 183—185 183—185eR 55— 61 55— 61 68— 73 68— 73 103—106 103—106

43— 59 43— 57 53— 72 49— 71 91—105 89—10487—100 88—101 114—124 114—126 183—187 183—188

g 300 300 415 415 690 69011L 271—272 271—272 372—374 372—374 624—625 624—625UR 264—265 264—265 362—362 362—362 605—605 605—605

dL 280—282 280—282 379—381 379—381 628—629 628—629

dR 266—267 266—267 362—363 362—363 604—604 604—604

b, 247—266 242—258 336—362 328—358 565—603 558—597

b2 266—273 268—275 361—371 364—370 592—614 594—608

1, 189—252 165—220 269—316 242—284 494—545 425—521313—346 338—360 391—425 417—440 643—651 638—656

h 42—102 66—104 46—109 46—112 46— 97 46—121A 48—232 86—229 49—325 49—321 53—278 54—539H 96—244 99—240 93—332 94—329 93—277 94—542

H ± 94—244 118—241 94—333 95—330 97—289 98—545

this caseshouldbe comparedwith their tree-levelcounterpartsgiven in ref. [25].

The rangeshavesome differencesbut they are quite close, except for the Higgsmasseswhichwerenot givenin ref. [25]. Forthe ~o= 1, ~ = 0 case(in table5) thelowest valueof int/2 for which “large” regionsstill existdependson the sign of p.,

as indicatedin the table. The most striking differencebetweenthis case(and ingeneralfor any valueof ~o~ 0) is that valuesof the squarkandgluino massesaslow as their currentexperimentallower boundsareattainable(althoughnot shown

explicitly in the table since thesecorrespondto “small” regions of parameterspace). Note also that a light chargino () is always allowed, although thefraction of the parameterspacefor which this happensdecreaseswith increasingrnl/2.

7. Conclusions

The study of low-energy supersymmetricmodelsis a subject of greattopicalinterestsincethis is the physicsbeyondthe standardmodel which is most likely to

S. Kel!eyci aL / Radiativee!cctrowcakbreaking 45

TABLE 5Particlemassrangesin the SSM(throughouttheallowed(m,, tan 13) region) for f,, = 1, ~A = 0 anda

sampleof m1/2 values.The ±indicatethesign of ~e.The lowestvalueof ml/2 (for thegiven signof ~,)

is closeits absoluteminimum. All massesin GeV

m1/2

70’~ 85 150” 150’ 250k 250”

27— 34 26— 29 36— 67 23— 60 38—108 31—10440— 75 50— 52 73—133 78—108 78—212 82—201

101—147 136—150 84—877 104—304 115—568 119—332113—168 165—179 162—885 178—336 232—573 238—340

45— 76 45— 48 56—133 46—107 55—212 50—201119—168 169—182 161—883 179—331 232—571 238—338

P 58— 69 82— 83 173—183 173—177 301—307 301—307

94— 98 115—115 185—190 188—190 308—311 308—31183— 87 101—101 161—167 165—167 268—272 268—27279— 85 45— 91 85—166 85—165 172—271 172—27194—103 121—135 186—208 188—208 303—318 303—318

195 235 415 415 690 690

181—184 223—223 401—404 401—403 673—675 673—675180—181 220—220 392—393 392—392 654—655 654—655

dL 195—198 237—237 405—409 408—409 675—677 675—677

184—185 223—223 392—393 392—393 653—653 653—653168—183 143—197 264—390 260—385 464—648 460—647184—188 230—234 371—394 374—394 578—654 581—654

ii 140—204 182—197 204—324 245—303 385—567 310—559243—279 300—312 401—453 416—452 608—688 610—743

h 47— 92 46—100 41—112 45—113 38—124 39—125A 88—170 48—152 50—244 50—397 55—240 55—897H 108—187 99—152 97—234 98—402 99—214 99—883

H’ 118—187 97—171 97—246 98—404 100—241 100—899

exist on theoreticalgrounds.The minimal supersymmetricextensionof the stan-dard model (MSSM) by itself contains so many parameters(21) that comprehen-sivestudiesareimpractical. In fact, to alleviatethis problempeopleroutinely makesimplifying assumptionsabout the various parametersin the MSSM, typicallyassuminga sort of “low-energy unification” of the sparticlemasses.On the otherhand, the approachfollowed in this paper does away with ad-hoc low-energyansätzeby invoking the structureof unified modelswherewell motivatedtheoreti-cal constraintsreducethe dimensionof the parameterspacedown to eight. Thisapproachhas becomevery popular over the last decade.However, even theseconstraintsare not enoughto producea consistentpictureof low-energyphysicssince the electroweaksymmetry must be broken as well. This last constraintdeterminesp. and B and generallycorrelatesthem with the dominantsourceof

supersymmetrybreaking. The major advantageof the above constraints is toreducethe dimensionof the parameterspacedown to just five.

Our presentstudyutilizes the one-loopeffective Higgspotential to executethe

46 S. Kc!ley et a!. / Radiativec!ectrowcakbreaking

last step of radiative electroweaksymmetry breaking. This makes the resultslargely independentof the renormalizationscaleQ used(i.e., the scalewhere theRGEsarestoppedand theHiggspotential is minimized).The useof this potentialis a technical advancementwhich hasyet to catch up in the literature, and webelieveit to be essentialin the determinationof the groundstateof this classofsupersymmetricunified models.We haveexploredthe virtuesof this more sophis-ticated approachand in the processhaverealizedthat a Higgs-field-independentterm in ztV ruins the Q-independenceof V1. For fixed-Q calculationsthis term isirrelevantsince it dropsout in the minimizationprocess.However,thewhole pointof usingV1 is preciselyto obtain Q-independence.We haveshownexplicitly that ifthis piece is subtractedout, then V1 is indeedQ-independentto one-looporder.

We havealso discussedthe “fine-tuning constraint” in the light of our analysis

of the parameterspace.We found that a good measureof this effect is given bycoefficients c a rn~7.2(a + b~) which can easily surpassa “limit” zi = 50—100 ifrn17.2> 400 GeV for all valuesof ~. Our results here are not new, we still getrn4~~ 1 TeV. However, thesecoefficientshelp quantify the level of “un-natural-ness” in a model. For example, we have found that in the minimal SU(5)supergravityGUT, proton decayconstraintscanbe satisfiedonly for valuesof theparameterswhich give c ~ 100. This indicates a possiblefine-tuning problem inthis model.

Our studyhasbeenbasedon two SU(3)x SU(2) x U(1) supersymmetrymodelswhich exemplify the low-energy limits of traditional (SSM) and string-inspired(515M) unified models. For thesemodelswe havedescribedin detail the com-pletely boundedregionsin (in,, tan (3) spaceand their dependenceon the otherparametersof the model. We have also comparedthe tree-level and one-loopvaluesof p. and concludedthat I zip. I = I p. loop — p.tree I is largestfor small p., asexpected.In fact, the relative shift in p. is maximal (—100%) at the one-loopleftboundaryanddecreasesas mt increases.Fortunately,thissmall regionof parame-ter space,where one-loop perturbationtheory is unreliable,is excludedon phe-nomenologicalgrounds.

We havestudiedthe sparticlespectrumand havegiven useful plots of the first

and secondgenerationsquark and slepton masses.We have shown that theinclusionof A,. ~ 0 is relevantsincep. effectsdominateover the D-termcontribu-

tions to the r12 masses.The lighteststopeigenstateL is generallylighter thantheaveragesquarkmass,and can evenbe as light as ~ although only for a verysmall regionof parameterspace;a more typical lower bound is m11 ~ 100 GeV.We have also studiedthe one-loop correctedHiggsbosonmassesand concludedthat for “natural” values of the parameters(i.e., rn~< 1 TeV) one must havein,, <135 0eV. Finally, we have given mass rangesfor all particle speciesfortypical valuesof the parameters.If supergravityis indeedrealizedin Nature,thenthe correlationsamongthe many sparticle andHiggsbosonmassesthat we havepresentedhereshouldbe dramaticallyrevealedin the nearfuture.

S. Ke!!cyci a!. / Radiativee!eciroweakbreaking 47

This work has beensupportedin part by DOE grant DE-FGO5-91-ER-40633.The work of J.L. has been supported in part by an ICSC-World LaboratoryScholarship.The work of D.V.N. has been supported in part by a grant fromConoco Inc. The work of K.Y. was supported in part by the Texas NationalLaboratoryResearchCommissionunderGrantNo. RCFY9155,andin part by theUS Departmentof EnergyunderGrantNo. DE-FGO5-84ER40141.We would liketo thank Doni Branchand the HARC SupercomputerCenterfor the use of theirNEC SX-3 supercomputer.

Noteadded in proof

Since the completion of this paper, some of us have explored in detail theparameterspaceof the minimal SU(5) supergravitymodel alluded to in sect. 4.This explorationhasyieldeda small but acceptableregionof parameterspace,asfar as the proton decayandfine-tuning constraintsare concerned[85,86]. More-

over, the additional cosmologicalconstraintof a not too young Universehasbeenfound to be quite restrictive [87,85],implying interestingphenomenologicalcorre-lations amongthe sparticlemasseswhich may allow for a completeexperimentalscrutinyof this model in the nearfuture.

Appendix A. Calculation of 4V

In this appendixwe presentsome details of the calculation of ziV usedthroughout our work. Theevaluationof ziV proceedsfrom thefollowing definition

1 4~2 3ziV= —Str ,J(~ln — — — (A.1)

64rr2 Q2 2

whereSTr f(4’2) = ~~C~(— 1)2~’(2j~ + 1)f(rn~)and rn~ are the Higgs-field-depen-dent mass-squaredeigenstateswith spin j,. We refer the readerto the literaturefor the necessarysquark[15,84],slepton [15,84], Higgs [53], andgaugino/higgsino

[32] mass formulas in the MSSM. The factor C, that entersinto the supertraceaccountsfor the color degreesof freedom, and we define a spin factor Si =

(—1)2~’(2j~ + 1), where 1, is the spin of the ith particle. The appropriateS,, C,factorsfor the particleswhich contributenonnegligiblyto ziV aresummarizedintableA.1

In calculatingchargedHiggsmasses,scalarfield valuesaway from the minimum(andwhich thereforebreakUem(1)invariance)are needed.In this casecharge-con-jugate particlesare no longerdegeneratein massand we treat eachcharge-con-

48 S.KcI!cy ci a!. / Radiativec!cctrowcakbreaking

TABLE A.1Valuesfor themultiplicity, spin,andC, and S factorsfor eachof theparticlesof eachspeciesthat

havebeen includedin thesupertracefor ztV. The index i runsfrom 110 n. We havenot includedtheu,d, c, s, e, ~.t, T fermions sincetheir contributionis negligible

Particle n Spin (j) C~ _______ 5,

24 0 3 118 0 1 1

H 8 0 14 3 —28 1 —2

W 2 1 1 3Z 1 1 1 3

jugate stateas a separatemass-eigenstatein the supertrace.For example,for thesquarksEl, the completeset of masseigenstates(at the minimumof the potential)would be

= (u~~U~, UiR, UjR, djL, djL, dIR, d~,i = 1, 2, ~ ~ ~2’ ~2’ b1’ b1, b2,

(A.2)

where i = 1, 2 labels the first andsecondgenerationsquarks.The contributionsof the u, d, c, s, e, p., T fermionsto ziV are utterly negligible

(andwould vanishidentically in the masslesslimit), andwe do not include them inour calculations.Dependingon whetherthe scalar field configurationis chosentobeat (i) the origin of field spaceor (ii) away from it, the contributionsto ziV fromthe two Higgsdoubletsand the W ± and Z fields are included as follows: (i) theW ± and Z fields are masslessand all eight Higgs mass eigenstatesmust beincluded(i.e., the “Goldstonebosons”arenot massless);(ii) the W ± andZ fieldsaremassiveandonly the Higgsphysical statesh, A, H, H ±are included(i.e., theGoldstonebosonsaremassless).In certaincases,it is possiblefor rn~< 0, andinthis eventwe makethe replacementrn~ —s I m~I [12].

We thuscalculateziV usingthe following completesupertraceformula:

STr f(~2)=3 ~ f(m~,)-6 ~ f(in~,)+ ~ f(rn~,)

i=1,24 i= 1,4 i= 1,18

+ ~ f(rn~.) -2 ~ f(rn~,)+ 3 ~ f(m~,). (A.3)i=1,8 i= 1,8 i= 1,3

S. Kc!!cy ci a!. / Radiativecleciroweakbreaking 49

References

[1] J. Carter, in Proc. Joint Intern. Lepton—photonSymp. and EurophysicsConf. on High energyphysics(Geneva,Switzerland,25 July—i August 1991),eds.S. Hegarty,K. PotterandE. Quercigh,Vol. 2, P. 1

[2] M. Davier, in Proc. Joint Intern. Lepton—photonSymp. and EurophysicsConf. on High energyphysics(Geneva,Switzerland,25 July—l August 1991),eds.S. Hegarty,K. PotterandE. Quercigh,Vol. 2, p. 151

[3] Seee.g.,J. Ellis, G.L. Fogli andE. Lisi, Phys.Lett. B 274 (1992)456;F. Halzen,B. Kniehl and ML. Stong,University of Wisconsinpreprint MAD/PH/658 (Septem-ber 1991);F. del Aguila, M. Martinezand M. Quiros, Nucl. Phys.B 381 (1992)451

[4] M. Veltmann,ActaPhys.Pol. B 8 (1977)475;C. Vayonakis,Lett. NuovoCimento 17 (1976) 383;B.W. Lee, C. Quigg and H.B. Thacker, Phys.Rev. D 16 (1977) 1519;M. Lüscherand P. Weisz,Phys.Lett. B 212 (1989)472;W. Marciano,0. ValenciaandS. Willenbrock, Phys.Rev. D 40 (1989) 1725

[5] L. Durand,J. Johnsonand J.L. Lopez,Phys. Rev. Lett. 64 (1990) 1215; Phys.Rev. D 45 (1992)3112

[6] M. Lindner, M. SherandH.W. Zagluer, Phys.Lett. B 228 (1989) 139[7] For a reviewsee,M. Sher,Phys.Rep.179 (1989) 273[8] L. Ibáñezand0. Ross,Phys.Lett. B 110 (1982)215;

K. Inoueet al., Prog. Theor. Phys.68 (1982)927[9] L. Ibáñez,Nucl. Phys.B 218 (1983)514; Phys. Lett. B118 (1982)73;

H.P. Nilles, Nucl. Phys.B 217 (1983)366[10] J. Ellis, D.V. NanopoulosandK. Tamvakis,Phys.Lett. B 121 (1983) 123;

J. Ellis, J. Hagelin, D.V. Nanopoulosand K. Tamvakis,Phys.Lett. B 125 (1983)275;L. Alvarez-Gaumé,J. Polchinskiand M. Wise, Nucl. Phys.B 221 (1983)495;L. Ibáñezand C. Lopez,Phys.Left. B 126 (1983)54; NucI. Phys.B 233 (1984) 545;C. Kounnas,A. Lahanas,DV. Nanopoulosand M. Quirós,Phys.Lett. B 132 (1983) 95

[Ii] C. Kounnas,A. Lahanas,D.V. Nanopoulosand M. Quirós,NucI. Phys.B 236 (1984)438[12] 0. Gamberini, 0. Ridolfi andF. Zwirner, NucI. Phys.B 331 (1990)331[13] For reviewssee: R. Arnowitt and P. Nath, Applied N = 1 supergravity(World Scientific, Singa-

pore, 1983);H.P. Nilles, Phys.Rep.110 (1984) 1

[14] E. Cremmer,S. Ferrara, C. KounnasandDV. Nanopoulos,Phys.Lett. B 133 (1983)61;J. Ellis, A.B. Lahanas,DV. Nanopoulosand K. Tamvakis,Phys.Left. B 134(1984)429;J. Ellis, C. KounnasandDV. Nanopoulos,NucI. Phys.B 241 (1984) 406; B 247 (1984) 373

[15] For a reviewseeA.B. Lahanasand D.V. Nanopoulos,Phys.Rep.145 (1987) 1[16] L. Ibáñezand G.G. Ross,in Perspectives in Higgs Physics: reviewsandspeculations,ed. 0. Kane

(World Scientific, Singapore,1992)[17] U. Ajnaldi etal., Phys.Rev. D 36 (1987)1385;

G. Costaetal., NucI. Phys.B 297 (1988)244[18] J. Ellis, S. Kelley and DV. Nanopoulos,Phys.Left. B 249 (1990)441; B 260 (1991)131; 287 (1992)

95; Nucl. Phys. B 373 (1992) 55[19] P. Langackerand M.-X. Luo, Phys.Rev. D 44 (1991)817;

U. Amaldi, W. de Boer and H. Fürstenau,Phys.Lett. B 260 (1991)447;F. Anselmo,L. Cifarelli, A. Petermanand A. Zichichi, Nuovo Cimento 104A (1991) 1817; 105A(1992) 581;R. Barbieri and L. Hall, Phys.Rev.Lett. 68 (1992)752

[20] H. Arason et al., Phys.Rev. Left. 67 (1991)2933;A. Giveon, L. Hall andU. Sand,Phys.Lett. B 271 (1991) 138

[211G.G. Rossand R.G. Roberts,NucI. Phys.B 377 (1992)571

50 S. Kc!lcycta!. / Radiativee!ectroweakbreaking

[22] B. Ananthanarayan,0. Lazaridesand 0. Shafi, Phys.Rev. D 44 (1991)1613;S. Kelley, J.L. LopezandDV. Nanopoulos,Phys. Lett. B 274 (1992)387

[23] M. DreesandMM. Nojini, Nucl. Phys.B 369 (1992)54

[241 K. Inoue, M. Kawasaki,M. YamaguchiandT. Yanagida,Phys.Rev. D 45 (1992)328[25] S. Kelley, J.L. Lopez,DV. Nanopoulos,H. Poisand K. Yuan, Phys.Lett. B 273 (1991) 423[26] J. Ellis and F. Zwirner,Nucl. Phys.B 338 (1990) 317[27] S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Poisand K. Yuan, Phys.Letf. B 285 (1992)61[28] J. Ellis, 0. Ridolfi andF. Zwirner, Phys.Left. B 257 (1991)83[29] M. DreesandM.M. Nojiri, Phys.Rev. D 45 (1992)2482[30] M. DreesandK. Hagiwara,Phys.Rev. D 42 (1990) 1709;

M. Drees,K. Hagiwanaand A. Yamada,Phys.Rev. D 45 (1992) 1725[31] MM. Noun, Phys.Lett. B 261 (i991) 76[32] J.L. Lopez, DV. Nanopoulosand K. Yuan, Nucl. Phys.B 370 (1992)445; Phys.Lett. B 267 (1991)

219[33] J. Ellis and L. Roszkowski,Phys.Lett. B 283 (1992)252[34] M. Matsumoto, J. Arafune,Fl. TanakaandK. Shiraishi,Phys.Rev.D 46 (1992)3966[35] R. Arnowitt andP. Nath, Phys.Rev. Left. 69 (1992) 725[36] J. Ellis, K. Enqvist, D.V. NanopoulosandF. Zwirner, Mod. Phys.Lett. A 1(1986) 57[37] R. Barbieri and 0. Guidice, NucI. Phys.B 306 (1988)63[38] Y. Okada, M. Yamaguchiand T. Yanagida,Prog. Theor. Phys.85 (1991) 1; Phys. Lett. B 262

(1991) 54[39] H. Haberand R. Hempfling, Phys.Rev. Lett. 66 (1991) 1815[40] Seee.g., N.K. Faick, Z. Phys.C 30 (1986)247, andreferencestherein[41] P. Chankowski,S. Pokorski andJ. Rosiek, Phys.Left. B 274 (1992) 191;

A. Yamada,Phys.Lett. B 263 (1991)233[42] A. Brignole, Phys.Lett. B 277 (1992)313; B 281 (1992)284[43] S. Colemanand E. Weinberg,Phys.Rev. D 7 (1973)1888[44] S. Weinberg,Phys.Rev. D 7 (1973)2887[45] S.Y. Lee andAM. Sciaccaluga,NucI. Phys.B 96 (1975) 435[46] R.D.C. Miller, Phys.Lett. B 124 (1983)59; NucI. Phys.B 228 (1983)316[47] W. Siegel,Phys.Lett. B 84 (1979) 193;

D.M. Capper,D.R.T. JonesandP. Van Nieuwenhuizen,NucI. Phys.B 167 (1980)479[48] R. Jackiw, Phys.Rev. D 9 (1974) 1686[49] J.S. Kang, Phys.Rev.D 10 (1974)3455[50] J. Iliopoulos andN. Papanicolaou,NucI. Phys.B 105 (1976)77[511 B. Kastening,Phys.Lett. B 283 (1992) 287[52] Seee.g., K.G. Chetyrkin et al., Phys.Left. B 132 (1983)351[53] J. Gunion andH. Haber,Nuci. Phys.B 272 (1986) 1[54]S. Kelley, J.L. Lopez and DV. Nanopoulos, Phys. Left. B 278 (1992) 140[55] J.L. Lopez, DV. Nanopoulosand K. Yuan, Texas A&M University preprint CTP-TAMU-l1/92

(to appearin NucI. Phys.B)[56] A. Chamseddine,R. Arnowift andP. Nath, Phys.Rev. Left. 49 (i982) 970[57] S. Weinberg,Phys.Rev. D 26 (1982) 287;

N. Sakaiand T. Yanagida,Nucl. Phys.B 197 (1982)533[58] J. Ellis, D.V. Nanopoulosand S. Rudaz,Nucl. Phys.B 202 (1982)43;

B. Campbell,J. Ellis andDV. Nanopoulos,Phys.Left. B i41 (1984)229;K. Enqvist, A. Masiero and DV. Nanopoulos,Phys.Lett. B 156 (1985)209;P. Nath, A. ChamseddineandR. Arnowiff, Phys.Rev.D 32 (1985)2348;P. Nath and R. Arnowitt, Phys.Rev.D 38 (1988) 1479

[59] H.P. Nilles, M. Srednickiand D. Wyler, Phys.Lett. B 124 (1983)337;A.B. Lahanas,Phys.Lett. B 124 (1983)341;J. Kubo andS. Sasakibara,Phys.Left. B 140 (1984) 223

[60] 5. Dimopoulosand F. Wilczek, Proc. Enice SummerSchool(1981);A. Masiero, DV. Nanopoulos,K. Tamvakisand T. Yanagida,Phys.Lett. B 115 (1982) 380;B. Grinstein,NucI. Phys.B 206 (1982) 387

S. Kcl!eyci a!. / Radiativeelectroweakbreaking 51

[61] S. Barr, Phys.Left. B 1i2 (1982) 219; Phys.Rev. D 40 (1989)2457;J. Derendinger,J. Kim and DV. Nanopoulos,Phys. Left. B 139 (i984) 170

[62] I. Anfoniadis, J. Ellis, J. Hagelin andDV. Nanopoulos,Phys.Lett. B 194 (1987)231[63] J. Ellis, J. Hagelin,S. Kelley andD.V. Nanopoulos,NucI. Phys.B 311 (1988/89)1[64] I. Antoniadis,J. Ellis, J. Hagelin andDV. Nanopoulos,Phys.Lett. B 231 (1989)65;

J.L. LopezandDV. Nanopoulos,Phys.Left. B 251 (1990)73; B 268 (1991)359[65] B. Greene,K.H. Kirklin, P.J. Miron and0G. Ross,Phys.Lett. B 180 (1986)69; NucI. Phys.B 278

(1986) 667; B 292 (1987)606;S. Kalaraand R.N. Mohapatra,Phys.Rev. D 36 (1987)3474;J. Ellis, K. Enqvisf, D.V. Nanopoulosand K. Olive, NucI. Phys.B 297 (1988) 103;R. Arnowitf andP. Nath, Phys.Rev. D 39 (1989)2006

[66] L. Ibáflez, H. Nilles and F. Quevedo,Phys.Letf. B 187 (1987)25;L. Ibáñez,J. Mas,H. Nilles and F. Quevedo,NucI. Phys.B 301 (1988) 157;A. Font, L. Ibáñez,F. Quevedoand A. Sierra, NucI. Phys.B 331 (1990)421

[67] A. Faraggi,DV. Nanopoulosand K. Yuan, NucI. Phys.B 335 (1990)347;A. Faraggi,Phys.Left. B 278 (1992) 131;K. Yuan, PhD ThesisTexasA&M University (1991)

[68] V. Kaplunovsky, Nucl. Phys. B 307 (1988) 145;J. Derendinger,S. Ferrara,C. Kounnasand F. Zwirner, Nucl. Phys.B 372 (1992) 145

[69] J. Ellis and DV. Nanopoulos,Phys.Lett. B 110 (1982)44[70] I. Antoniadis, J. Ellis, R. LacazeandDV. Nanopoulos,Phys.Left. B 268 (i99i) 188;

S. Kalara, J.L. Lopezand DV. Nanopoulos,Phys.Lett. B 269 (1991)84[71] I. Anfoniadis, J. Ellis, S. Kelley andDV. Nanopoulos,Phys.Letf. B 272 (1991)31[72] L. DurandandJ.L. Lopez,Phys.Lett. B 217 (1989) 463; Phys.Rev. D 40 (i989) 207[73] J. Ellis, J.S. Hagelin, DV. Nanopoulos,K.A. Olive andM. Srednicki,NucI. Phys. B 238(1984)453[74] K. Griest andJ. Silk, Nature343 (1990) 26;

L. Krauss,Phys. Rev.Left. 64 (1990)999;J. Ellis, DV. Nanopoulos,L. RoszkowskiandD. Schramm,Phys. Lett. B 245 (1990) 251

[75] L3 Collab., B. Adevaet al., Phys Lett. B 233 (1989)530;ALEPH Collab., D. Decampef al., Phys.Lett. B 236 (1990)86;OPAL Collab., M.Z. Akrawy et al., Phys. Left. B 240 (1990) 261

[76] CDF Collab., F. Abe etal., Phys.Rev.Letf. 68 (1992)447[77] Particle Dafa Group,J.J. Hernándezef al., Reviewof particleproperties,Phys.Left. B 239 (1990)

[78] CDF Collaboration,F. Abe ef. al., Phys.Rev. D 43 (1991)664[79] 0. Croseffi andV. Ruhlmann,in Proc.Joint Intern. Lepton--photonSymp.andEurophysicsConf.

on High energyphysics(Geneva,Switzerland, 25 July—l August, 1991), eds.S. Hegarty,K. Potterand E. Quercigh,Vol. 1, p. 353

[80] ALEPH Collab., D. Decamp et al., Phys. Letf. B 265 (1991) 475;P. Igo-Kemenes(OPAL Collab.), in Proc. Joint Intern. Lepton—photonSymp. and EurophysicsConf. on High energyphysics(Geneva,Switzerland,25 July—i August, 1991), eds.S. Hegarty,K.Potterand E. Quercigh,Vol. 1, p. 342;L. Banone and S. Shevchenko(L3 Collab.), in Proc. Joint Intern. Lepfon—photonSymp. andEurophysicsConf. on High energyphysics(Geneva,Switzerland, 25 July—i August, 1991), eds.S.Hegarty,K. Potter andE. Quercigh,Vol. 1, p. 348

[81] A. Faraggi,J. Hagelin, S. Kelley and DV. Nanopoulos,Phys. Rev.D 45 (1992)3272[821 Z. KunsztandF. Zwirner, NucI. Phys.B 385 (1992) 3[83] V. Barger,MS. Berger,AL. Stangeand R. Phillips, Phys. Rev. D 45 (1992) 4128;

H. Baer, M. Bissef, C. Kao and X. Tata, Phys.Rev. D 46 (1992)1067;R. Bork, J.F.Gunion, HE. Haberand A. Seiden,Phys.Rev. D 46 (1992) 2040;J.F. Gunion andL.H. Orr, Phys.Rev. D 46 (1992)2052

[84] A. Bnignole,J. Ellis, 0. Ridolfi andF. Zwirner, Phys.Lett. B 271 (199i) 123[85]J.L. Lopez,DV. NanopoulosandH. Pois, Phys.Rev. D 47 (1993) 2468[86] J.L. Lopez,DV. Nanopoulos,H. Pois andA. Zichichi, Phys.Lett. B 299 (1993)262[87] J.L. Lopez,DV. NanopoulosandA. Zichichi, Phys.Lett. B 291 (1992) 255

Aspects of radiative electroweak breaking in supergravity models

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