---

The quest for low-energy supersymmetry

arXiv:hep-ph/9203204v1 9 Mar 1992CERN-TH.6357/91The quest for low-energy supersymmetryand the role of high-energy e+e−collidersF. Zwirner1Theory Division, CERN,Geneva, SwitzerlandAbstractThe motivations for low-energy supersymmetry and the main featuresof the minimal supersymmetric extension of the Standard Model arereviewed.

Possible non-minimal models and the issue of gauge cou-pling unification are also discussed. Theoretical results relevant for su-persymmetric particle searches at present and future accelerators arepresented, with emphasis on the role of a proposed e+e−collider with√s = 500 GeV.

In particular, recent results on radiative correctionsto supersymmetric Higgs boson masses and couplings are summarized,and their implications for experimental searches are discussed in somedetail.Plenary talk at the Workshop on Physics and Experiments with LinearColliders, Saariselk¨a, Lapland, Finland, 9–14 September 1991CERN-TH.6357/91December 19911On leave from INFN, Sezione di Padova, Italy.

1IntroductionRealistic models of low-energy supersymmetry have been studied for about15 years, starting with the pioneering works of Fayet [1] and continuing withmore and more systematic investigations [2], but there is no decisive experi-mental evidence yet either in favour of or against this idea. It is then almosta duty for the theoretical speaker on the subject (the experimental aspectsare discussed in ref.

[3]) to argue in favour of the following two statements:• Low-energy supersymmetry is, today more than ever, a phenomeno-logically viable and theoretically motivated extension of the StandardModel.• High-energy e+e−colliders can play a crucial role in testing it experi-mentally.With the above two goals in mind, the discussion will be organized as follows.This introduction will end with a brief reminder of the motivations for low-energy supersymmetry. Sect.

2 will introduce the Minimal Supersymmetricextension of the Standard Model (MSSM), and its possible non-minimal al-ternatives. Plausible theoretical constraints on the MSSM, including the onescoming from gauge coupling unification, will be also discussed.

Sects. 3 and4 will take a closer look at the particle spectrum of the MSSM, thus provid-ing an introduction to the experimental discussion of ref.

[3]. The presentlimits from LEP I and Tevatron and the expected sensitivity of LEP II andLHC/SSC will be reviewed, followed by some theoretical considerations onthe potential of a 500 GeV e+e−collider (EE500).

Sect. 3 will discuss re-cent results on radiative corrections to Higgs boson masses and couplings,and their implications for experimental searches.

Sect. 4 will deal with su-persymmetric partners of quarks, leptons, gauge and Higgs bosons.

Finally,sect. 5 will contain some concluding remarks.1.1Motivations for low-energy supersymmetryThere are many good reasons to believe that supersymmetry [4] and its localversion, supergravity [5], could be relevant in a fundamental theory of par-ticle interactions.

Symmetries, even when broken, have been very important1

in establishing modern particle theory as we know it today: supersymme-try is the most general symmetry of the S-matrix consistent with relativisticquantum field theory [6], so it is not inconceivable that Nature might makesome use of it. Also, superstrings [7] are the present best candidates fora consistent quantum theory unifying gravity with all the other fundamen-tal interactions, and supersymmetry appears to play a very important rolefor the quantum stability of superstring solutions in four-dimensional space-time.

Experimental data, however, tell us that supersymmetry is not realizedexactly, and none of the above motivations gives us any insight about thescale of supersymmetry breaking.The only motivation for low-energy supersymmetry, i.e. supersymmetryeffectively broken around the electroweak scale, comes from the naturalnessor hierarchy problem [8] of the Standard Model (SM), whose formulation willnow be sketched.

Despite its remarkable phenomenological success [9], it isimpossible not to regard the SM as an effective low-energy theory, valid upto some energy scale Λ, at which it is replaced by some more fundamentaltheory. Certainly Λ is less than the Planck scale MP ∼1019 GeV, sinceone needs a theory of quantum gravity to describe physics at these energies.However, the study of the Higgs sector of the SM suggests that Λ shouldrather be close to the Fermi scale, G−1/2F∼300 GeV.

The argument goes asfollows. Consistency of the SM requires the SM Higgs mass to be less thanO(1 TeV).

If one then tries to extend the validity of the SM to energy scalesΛ ≫G−1/2F, one is faced with the fact that in the SM there is no symme-try to justify the smallness of the Higgs mass with respect to the (physical)cut-offΛ. This is apparent from the fact that in the SM one-loop radiativecorrections to the Higgs mass are quadratically divergent.

Motivated by thisproblem, much theoretical effort has been devoted to finding descriptions ofelectroweak symmetry breaking which modify the SM at scales Λ ∼G−1/2F.Here supersymmetry comes into play because of its improved ultraviolet be-haviour with respect to ordinary quantum field theories [10], due to cancel-lations between bosonic and fermionic loop diagrams. If one wants to havea low-energy effective Lagrangian valid up to scales Λ ≫G−1/2F, with oneor more elementary scalar fields, kept light without unnatural fine-tuningsof parameters, the solution [11] is to introduce supersymmetry, effectivelybroken in the vicinity of the electroweak scale.

This does not yet explainwhy the scale MSUSY of supersymmetry breaking is much smaller than Λ,2

but at least links the Fermi scale G−1/2Fto the supersymmetry-breaking scaleMSUSY, and makes the hierarchy G−1/2F∼MSUSY << Λ stable against radia-tive corrections.2The MSSMThe most economical realization of low-energy supersymmetry is the Min-imal Supersymmetric extension of the Standard Model [2], whose definingassumptions are listed below.1: Minimal gauge group.In the MSSM, the gauge group is just G = SU(3)C × SU(2)L × U(1)Y ,as in the SM. Supersymmetry then implies that spin-1 gauge bosonsbelong to vector superfields, together with their spin- 12 superpartners,the gauginos.2: Minimal particle content.The MSSM contains just three generations of quark and lepton spin- 12fields, as does the SM, but embedded in chiral superfields together withtheir spin-0 superpartners, the squarks and the sleptons.

In addition, togive masses to all charged fermions and to avoid chiral anomalies, one isforced to introduce two more chiral superfields, containing two complexspin-0 Higgs doublets and their spin- 12 superpartners, the higgsinos.3: Exact R-parity.Once the gauge group and the particle content are given, to determinea globally supersymmetric Lagrangian, LSUSY, one must specify ananalytic function of the chiral superfields, the superpotential. To enforcebaryon and lepton number conservation in renormalizable interactions,in the MSSM one imposes a discrete, multiplicative symmetry calledR-parity, defined asR = (−1)2s+3B+L,(1)where s is the spin quantum number.

In practice, the R-parity assign-ments are R = +1 for all ordinary particles (quarks, leptons, gaugeand Higgs bosons), R = −1 for their superpartners (squarks, sleptons,3

gauginos and higgsinos). The most general superpotential compatiblewith gauge invariance, renormalizability and R-parity isf = hUQUcH2 + hDQDcH1 + hELEcH1 + µH1H2 ,(2)where Q, Uc, Dc, L, Ec are the chiral superfields containing the left-handed components of ordinary quarks and leptons, H1 and H2 arethe two Higgs chiral superfields, and family and group indices havebeen left implicit for notational simplicity.

The first three terms arenothing else than the supersymmetric generalization of the SM Yukawacouplings, whereas the fourth one is a globally supersymmetric Higgsmass term. Exact R-parity has very important phenomenological con-sequences: (R-odd) supersymmetric particles are always produced inpairs, their decays always involve an odd number of supersymmetricparticles in the final state, and the lightest supersymmetric particle(LSP) is absolutely stable.4: Soft supersymmetry breaking.The above three assumptions are sufficient to completely determinea globally supersymmetric renormalizable Lagrangian, LSUSY.TheMSSM Lagrangian is obtained by adding to LSUSY a collection Lsoft ofexplicit but soft supersymmetry-breaking terms, which preserve thegood ultraviolet properties of supersymmetric theories.In general,Lsoft contains [12] mass terms for scalar fields and gauginos, as wellas a restricted set of scalar interaction terms proportional to the cor-responding superpotential couplings−Lsoft=Pi ˜m2i |ϕi|2 + 12PA MAλAλA +hUAUQUcH2+hDADQDcH1 + hEAELEcH1 + m23H1H2 + h.c.,(3)where ϕi (i = H1, H2, Q, Uc, Dc, L, Ec) denotes the generic spin-0 field,and λA (A = 1, 2, 3) the generic gaugino field.Observe that, sinceAU, AD and AE are matrices in generation space, the most generalform of Lsoft contains in principle a huge number of free parameters.Moreover, for generic values of these parameters one encounters phe-nomenological problems with flavour-changing neutral currents [13],4

with new sources of CP-violation 1 [15] and with charge- and colour-breaking vacua.5: Unification assumptions.All the above problems can be solved at once if one assumes that therunning MSSM parameters, defined at the one-loop level and in a mass-independent renormalization scheme, obey a certain number of bound-ary conditions at some grand-unification scale MU. First of all, oneassumes grand unification of the gauge couplingsg3(MU) = g2(MU) = g1(MU) ≡gU,(4)where g1 =q3/5·g′ as in most grand-unified models.

Furthermore, oneassumes that all soft supersymmetry-breaking terms can be parametrized,at the scale MU, by a universal gaugino massM3(MU) = M2(MU) = M1(MU) ≡m1/2 ,(5)a universal scalar mass˜m2H1(MU) = ˜m2H2(MU) = ˜m2Q(MU) = . .

. = ˜m2Ec(MU) ≡m20 ,(6)and a universal trilinear scalar couplingAU(MU) = AD(MU) = AE(MU) ≡A ,(7)whereas m23 remains in general an independent parameter.

In addition,all possible CP-violating phases besides the Kobayashi-Maskawa oneare set to zero at the scale MU.2.1Non-minimal alternatives to the MSSMThe above assumptions, which define the MSSM, are plausible but not com-pulsory. Relaxing them leads to non-minimal supersymmetric extensions ofthe SM, which typically increase the number of free parameters without acorresponding increase of physical motivation.1 The phenomenology of CP violation in supersymmetric models has been discussedrecently, in connection with high-energy e+e−colliders, in ref.

[14].5

For example, relaxing assumption 1, a low-energy gauge group largerthan the SM one could be considered, as is possible in non-minimal grand-unification schemes and in some string compactifications, and as was origi-nally suggested in some models for spontaneous breaking of global supersym-metry. However, the present limits on the masses and mixing of extra gaugebosons are so stringent that such a departure is certainly not motivated bynow.Similarly, there are various possibilities to enlarge the particle contentof the MSSM, relaxing assumption 2.

One possibility is the introductionof additional chiral superfields with the quantum numbers of exotic statescontained in the fundamental 27 representation of E6: under assumption 1,however, these states have naturally superheavy masses and decouple fromthe low-energy effective theory. A particularly popular variation, which cor-responds to the simplest non-minimal model [1,16], is constructed by addinga gauge-singlet Higgs superfield N and by requiring purely trilinear superpo-tential couplings.

Without unification assumptions, this model has alreadytwo more parameters than the MSSM, but with an assumption analogous toeq. (7) the number of free parameters remains the same as in the MSSM.Folklore arguments in favour of this model are that it avoids the introduc-tion of a supersymmetry-preserving mass parameter µ ∼G−1/2F, and thatthe homogeneity properties of its superpotential recall the structure of somesuperstring effective theories.

A closer look, however, shows that these state-ments should be taken with a grain of salt. First, in the effective low-energytheory with softly broken global supersymmetry, the supersymmetric massµ ∼G−1/2Fcould well be a remnant of local supersymmetry breaking, if theunderlying supergravity theory has a suitable structure of interactions [17].Moreover, when embedded in a grand-unified theory, the non-minimal modelwith a singlet Higgs field might develop dangerous instabilities [18].

Also,the trilinear N3 superpotential coupling, which is usually invoked to avoida massless axion, is typically absent in string models.Phenomenologicalaspects of the non-minimal model with an extra singlet have been studiedrecently, in connection with high-energy e+e−colliders, in ref. [19], and willnot be discussed here.Assumption 3 is of crucial importance, since relaxing it can drasticallymodify the phenomenological signatures of supersymmetry.

If one does notimpose R-parity, the most general superpotential compatible with gauge in-variance and renormalizability contains, besides the terms of eq. (2), also6

the following ones:∆f = λQDcL + λ′LLEc + µ′LH2 + λ′′UcDcDc. (8)The first three terms on the right-hand side of eq.

(8) obey the selectionrule ∆B = 0, |∆L| = 1, and the last one the selection rule ∆L = 0, |∆B| =1. Their simultaneous presence would be phenomenologically unacceptable,since they could induce, for example, fast proton decay mediated by ˜dcsquarks.

However, imposing discrete symmetries weaker than R-parity onecan allow for some of the terms in eq. (8), and therefore for explicit R-paritybreaking, in a phenomenologically acceptable way [20].

Another possibility[21] is that R-parity is spontaneously broken by the VEV of a sneutrino field,but it is by now experimentally ruled out by LEP data. In order to obtainacceptable models with spontaneously broken R-parity, one would need tointroduce several extra fields and parameters.

The phenomenology of modelswith broken R-parity at high-energy e+e−colliders has been recently studiedin ref. [22], and will not be discussed here.To comment assumption 4, one has to discuss the problem of supersymme-try breaking.

Models with spontaneously broken global supersymmetry haveto face several phenomenological difficulties, which can be solved only at theprice of introducing rather baroque constructions. Present theoretical ideas,however, favour the possibility that supersymmetry is spontaneously brokenin the hidden sector of some underlying supergravity (or superstring) model,communicating with the observable sector (the one containing the states ofthe MSSM) only via gravitational interactions.

As for the precise mecha-nism of spontaneous supersymmetry breaking, there are several suggestions,among which non-perturbative phenomena such as gaugino condensation [23]and string constructions such as coordinate-dependent compactifications [24],but none of them has yet reached a fully satisfactory formulation. It thenappears to be a sensible choice to parametrize supersymmetry breaking inthe low-energy effective theory by a collection of soft terms, without strongassumptions on the underlying mechanism for spontaneous supersymmetrybreaking.Besides solving naturally the phenomenological problems connected withflavour-changing neutral currents, new sources of CP violation, charge andcolour breaking vacua, and proliferation of free parameters, assumption 5 isstrongly suggested by ideas about grand unification and spontaneous break-ing of local supersymmetry in a hidden sector; it receives further support by7

the present indications on the structure of the low-energy effective supergrav-ity theories of string models. We shall discuss later other phenomenologicaland theoretical facets of the unification assumptions.2.2Supersymmetric grand-unificationStarting from the boundary condition of eq.

(4), one can solve the appro-priate renormalization group equations (RGE) to obtain the running gaugecoupling constants gA(Q) (A = 1, 2, 3) at scales Q << MU. At the one-looplevel, and assuming that there are no new physics thresholds between MUand Q, one finds [25]1g2A(Q) = 1g2U+ bA8π2 log MUQ(A = 1, 2, 3) ,(9)where the one-loop beta-function coefficients bA depend only on the SU(3)C×SU(2)L × U(1)Y quantum numbers of the light particle spectrum.

In theMSSMb3 = −3,b2 = 1,b1 = 335 ,(10)whereas in the SMb03 = −7,b02 = −196 ,b01 = 4110. (11)Starting from three input data at the electroweak scale, for example [9]α3(mZ) = 0.118 ± 0.008,(12)α−1em(mZ) = 127.9 ± 0.2,(13)sin2 θW(mZ) = 0.2327 ± 0.0008,(14)where αA = g2A/(4π), sin2 θW = g′2/(g2+g′2), αem = α2 ·sin2 θW, and all run-ning parameters are defined in the modified minimal subtraction scheme MS[26], one can perform consistency checks of the grand-unification hypothesisin different models.In the minimal SU(5) model [27], and indeed in any other model whereeq.

(4) holds and the light-particle content is just that of the SM (withno intermediate mass scales between mZ and MU), eqs. (9) and (11) are8

incompatible with experimental data.This was first realized by noticingthat the prediction MU ≃1014−15 GeV, obtained by using as inputs eqs. (12) and (13), is incompatible with experimental data on nucleon decay [28].Subsequently, also the prediction sin2 θW ≃0.21 was shown to be in conflictwith experimental data [29], and this conflict became even more significant[30] after the recent LEP precision measurements.In the MSSM, assuming for simplicity that all supersymmetric particleshave masses of order mZ, one obtains [31] MU ≃1016 GeV (which increasesthe proton lifetime for gauge-boson-mediated processes beyond the presentexperimental limits) and sin2 θW ≃0.23.

At the time of refs. [31], when datawere pointing towards a significantly smaller value of sin2 θW, this was consid-ered by some a potential phenomenological shortcoming of the MSSM.

Thehigh degree of compatibility between data and supersymmetric grand unifica-tion became manifest [29] only later, after improved data on neutrino-nucleondeep inelastic scattering were obtained, and was recently re-emphasized, af-ter the LEP precision measurements, in refs. [32,30].

One should not forget,however, that unification of the MSSM is not the only solution which can fitthe data of eqs. (12)–(14): for example, non-supersymmetric models with adhoc light exotic particles or intermediate symmetry-breaking scales [33] couldalso do the job.

The MSSM, however, stands out as the simplest physicallymotivated solution.If one wants to make the comparison between low-energy data and thepredictions of specific grand-unified models more precise, there are severalfactors that should be further taken into account.After the inclusion ofhigher-loop corrections and threshold effects, eq. (9) is modified as follows1g2A(Q) = 1g2U+ bA8π2 log MUQ + ∆thA + ∆l>1A(A = 1, 2, 3) .

(15)In eq. (15), ∆thA represents the so-called threshold effects, which arise when-ever the RGE are integrated across a particle threshold [34], and ∆l>1Arepre-sents the corrections due to two- and higher-loop contributions to the RGE[35].

Both ∆thA and ∆l>1Aare scheme-dependent, so one should be carefulto compare data and predictions within the same renormalization scheme.∆thA receives contributions both from thresholds around the electroweak scale(top quark, Higgs boson, and in SUSY-GUTs also the additional particlesof the MSSM spectrum), and from thresholds around the grand-unificationscale (superheavy gauge and Higgs bosons, and in SUSY-GUTs also their9

superpartners). Needless to say, these last threshold effects can be computedonly in the framework of a specific grand-unified model, and typically de-pend on a number of free parameters.

Besides the effects of gauge couplings,∆l>1Amust include also the effects of Yukawa couplings, since, even in thesimplest mass-independent renormalization schemes, gauge and Yukawa cou-plings mix beyond the one-loop order. In minimal SU(5) grand unification,and for sensible values of the top and Higgs masses, all these corrections aresmall and do not affect substantially the conclusions derived from the na¨ıveone-loop analysis.

This is no longer the case, however, for supersymmetricgrand unification. First of all, one should notice that the MSSM by itselfdoes not uniquely define a SUSY-GUT, whereas threshold effects and eventhe proton lifetime (due to a new class of diagrams [36] which can be origi-nated in SUSY-GUTs) become strongly model-dependent.

Furthermore, thesimplest SUSY-GUT [37], containing only chiral Higgs superfields in the 24,5 and 5 representations of SU(5), has a severe problem in accounting forthe huge mass splitting between the SU(2) doublets and the SU(3) tripletssitting together in the 5 and 5 Higgs supermultiplets. Threshold effects aretypically larger than in ordinary GUTs, because of the much larger numberof particles in the spectrum, and in any given model they depend on sev-eral unknown parameters.

Also two-loop effects of Yukawa couplings can bequantitatively important in SUSY-GUTs, since they depend not only on thetop-quark mass, but also on the ratio tan β = v2/v1 of the VEVs of the twoneutral Higgs fields: as will be made clearer below, these effects become largefor mt >∼140 GeV and tan β ∼1, which correspond to a strongly interactingtop Yukawa coupling. All these effects have been recently re-evaluated [38]after the enthusiasm created by refs.

[30]. The conclusion is that, even imag-ining a further reduction in the experimental errors of eqs.

(12)–(14), it isimpossible to claim indirect evidence for supersymmetry and to predict theMSSM spectrum with any significant accuracy. The only safe statement is[32] that, at the level of precision corresponding to the na¨ıve one-loop approx-imation, there is a remarkable consistency between experimental data andthe prediction of supersymmetric grand unification, with the MSSM R-oddparticles roughly at the electroweak scale.To conclude the discussion of supersymmetric grand unification, it isworth mentioning how the unification constraints can be applied to the low-energy effective theories of four-dimensional heterotic string models.Thebasic fact to realize is that the only free parameter of these models is the10

string tension, which fixes the unit of measure of the massive string exci-tations. All the other scales and parameters are related to VEVs of scalarfields, the so-called moduli, corresponding to flat directions of the scalar po-tential.

In particular, there is a relation among the string mass MS ∼α′−1/2,the Planck mass MP ∼G−1/2N, and the unified string coupling constant gstring,which reflects unification with gravity and implies that in any string vacuumone has one more prediction than in ordinary field-theoretical grand unifica-tion. In a large class of string models, one can write down an equation of thesame form as (15), and compute gU, MU, ∆thA , .

. .

in terms of the relevantVEVs [39]. In the DR scheme [40], which is the most appropriate for super-symmetric models, one finds MU ≃0.7×gU ×1018 GeV, more than one orderof magnitude higher than the na¨ıve extrapolations from low-energy data il-lustrated before.

This means that significant threshold effects are needed inorder to reconcile string unification with low-energy data: for example, theminimal version of the flipped-SU(5) model [41] is by now ruled out [42]. Toget agreement, one needs some more structure in the spectrum, either at thecompactification scale or in the form of light exotics [43].

Once the presentstring calculations will be sufficiently generalized, unification constraints willprovide a very important phenomenological test of realistic string models.2.3More on the MSSMIt is perhaps useful, at this point of the discussion, to remind the readerof some other phenomenological virtues and theoretical constraints of theMSSM, besides the solution of the ‘technical’ part of the hierarchy problemand the grand unification of gauge couplings.It was already said that, because of R-parity, the LSP is absolutely sta-ble. In most of the otherwise acceptable parameter space, the LSP is neutraland weakly interacting, rarely a sneutrino and typically the lightest, ˜χ, ofthe neutralinos (the mass eigenstates of the neutral gaugino-higgsino sector).Then the LSP is a natural candidate for cold dark matter [44,45].

In par-ticular, for generic values of parameters one naturally avoids an excessive ˜χrelic density, but one often obtains cosmologically interesting values for it.This should also be considered an important consistency check of the MSSM,since a coloured or electrically charged LSP would be in conflict with astro-physical observations [45]. Recent analyses of supersymmetric dark matter,taking into account the LEP limits, can be found in ref.

[46].11

Another remarkable fact to be noticed is that LEP precision measure-ments of the Z properties put little indirect constraints, via radiative cor-rections [47], on the MSSM parameters. This is not the case, for example,of technicolour and extended technicolour models, which are severely con-strained by the recent LEP data [48,49].

In the MSSM, the most importanteffect could be given by additional contributions to the effective ρ parametercoming from the stop-sbottom sector: these can be sizeable only in the caseof large mass splittings in the stop-sbottom sector, in which case the upperbound on the top-quark mass, mt <∼180 GeV, obtained in the SM by fittingthe electroweak precision data, can be further strenghtened. However, devi-ations from the SM predictions due to loops of supersymmetric particles aretypically small for generic values of the parameters.A further predictive aspect of the MSSM is the possibility of comput-ing low-energy parameters, in particular the soft supersymmetry-breakingmasses, in terms of the few parameters assigned as boundary conditions atthe unification scale.

To do this, it is sufficient to solve the correspondingRGE [50], analogous to the ones given above for the gauge couplings. Forthe gaugino masses, one findsMA(Q) = g2A(Q)g2Um1/2(A = 1, 2, 3) .

(16)For the top Yukawa coupling, neglecting mixing and the Yukawa couplingsof the remaining fermions, one gets (t ≡log Q)dhtdt = ht8π2−83g23 −32g22 −1318g′2 + 3h2t. (17)A close look at eq.

(17) can give us some important information aboutthe top-quark mass in the MSSM. The important thing to realize is thatthe running top Yukawa coupling has an effective infrared fixed point [51],smaller than in the SM case [52].

However high the value one assigns to itat the unification scale, ht evaluated at the electroweak scale never exceedsa certain maximum value hmaxt≃1. This implies that, for any given valueof tan β, there is a corresponding maximum value for the top quark mass.

Anaive one-loop calculation givestan β :1248∞mmaxt(GeV ) :139176191195196 . (18)12

For the soft supersymmetry-breaking scalar masses, under the same assump-tions as above, and considering for the moment the sfermions of the thirdfamily, one findsd ˜m2idt=18π2−XA=1,2,3cA(i)g2AM2A + ct(i)h2tFt,(19)where i = H1, H2, Q, Uc, Dc, L, Ec,Ft ≡˜m2Q + ˜m2Uc + ˜m2H2 + A2t,(20)and the cA(i), ct(i) coefficients are given byi :H1H2QUcDcLEcc3(i) :0016316316300c2(i) :3330030c1(i) :3535115161541535125ct(i) :0312000(21)Similar equations can be derived for the remaining soft supersymmetry-breaking parameters and for the superpotential Higgs mass µ. Also, the inclu-sion of the complete set of Yukawa couplings, including mixing, is straightfor-ward.

In general, the RGEs for superpotential couplings and soft supersymmetry-breaking parameters have to be solved by numerical methods. Analyticalsolutions can be obtained for the soft squark and slepton masses when thecorresponding Yukawa couplings are negligible:˜m2i = m20 + m21/23XA=1cA(i)2bA 1 −1F 2A!,(22)whereFA = 1 + bA8π2g2U log MUQ .

(23)For example, one gets ˜m2Q, ˜m2Uc, ˜m2Dc ∼m20+(5−8) m21/2, ˜m2L ∼m20+0.5 m21/2,˜m2Ec ∼m20 + 0.15 m21/2. It should be stressed that also eqs.

(16) and (22), inanalogy with eq. (9), are valid up to higher-order corrections and thresholdeffects, so their accuracy should not be overestimated.One of the most attractive features of the MSSM is the possibility ofdescribing the spontaneous breaking of the electroweak gauge symmetry as13

an effect of radiative corrections [53], via a generalization of the mechanismdiscussed first by Coleman and E. Weinberg [54] in the context of the SM.It is remarkable that, starting from universal boundary conditions at theunification scale, it is possible to explain naturally why fields carrying colouror electric charge do not acquire non-vanishing VEVs, whereas the neutralcomponents of the Higgs doublets do. We give here a simplified description ofthe mechanism in which the physical content is transparent, and we commentlater on the importance of a more refined treatment.

The starting point isa set of boundary conditions on the independent model parameters at theunification scale Q = MU. One then evolves all the running parameters fromthe grand-unification scale to a low scale Q ∼mZ, according to the RGE,and considers the renormalization-group-improved tree-level potentialV0(Q)=m21 |H1|2 + m22 |H2|2 + m23 (H1H2 + h.c.)+18g2 H†2⃗σH2 + H†1⃗σH12 + 18g′2 |H2|2 −|H1|22 ,(24)wherem21 ≡˜m2H1 + µ2,m22 ≡˜m2H2 + µ2,(25)and it is not restrictive to choose a field basis such that m23 ≤0.

All massesand coupling constants in V0(Q) are running parameters, evaluated at thescale Q. The minimization of the potential in eq.

(24) is straightforward. Togenerate non-vanishing VEVs v1 ≡⟨H01⟩and v2 ≡⟨H02⟩, one needsB ≡m21m22 −m43 < 0 .

(26)In addition, a certain number of conditions have to be satisfied to have astable minimum with the correct amount of symmetry breaking and withunbroken colour, electric charge, baryon and lepton number: for example, allthe running squark and slepton masses have to be positive. A crucial rolein the whole process is played by the top Yukawa coupling, which stronglyinfluences the RGE for ˜m2H2, as should be clear from eqs.

(19)–(21). Forappropriate boundary conditions, the RGE drive B < 0 at scales Q ∼mZ,whereas all the squark and slepton masses remain positive as desired, to givea phenomenologically acceptable breaking of the electroweak symmetry.The use of V0(Q) is very practical for a qualitative discussion as the onegiven above, but it relies on the assumption that, once the leading logarithms14

have been included in the running parameters, all the remaining one-loopcorrections to the scalar potential can be neglected at the scale Q ∼mZ.However, as shown for example in ref. [32], for a quantitative discussion ofgauge symmetry breaking it is necessary to use the full one-loop effectivepotential, which in the Landau gauge and in the DR renormalization scheme[40] is given byV1(Q) = V0(Q) +164π2 Str(M4(Q)"log M2(Q)Q2−32#).

(27)In eq. (27), Str f(M2) =Pi(−1)2Ji(2Ji + 1)f(m2i ) denotes the conventionalsupertrace, where m2i is the field-dependent mass eigenvalue of the i-th par-ticle of spin Ji, and field-independent terms have been neglected.

To givean example, the VEVs determined from V0(Q) are strongly scale-dependent,whereas the ones determined from V1(Q) are not, as it should be. Only at ascale ˆQ, of the order of the stop masses, is the use of V0(Q) a good approx-imation.

This is a result of the fact that mass-independent renormalizationschemes, like MS or DR, do not automatically include decoupling: since themost important contributions to V1(Q) come from the stop sector, the opti-mal scale at which to freeze the evolution of the running parameters turnsout to be of the order of the stop masses. Another aspect of this effect, withimportant phenomenological consequences, are the radiative corrections toHiggs boson masses and couplings, which will be discussed in the followingsection.To conclude the discussion of radiative symmetry breaking, we show nowthat in the MSSM, with universal boundary conditions, one expects1 <∼tan β <∼mtmb.

(28)The simplest proof relies on the relation, derived from the minimization ofV0(Q)v2v1= m21 + m2Z/2m22 + m2Z/2 . (29)The boundary conditions at the unification scale is m21(MU) = m22(MU),and, neglecting as before all Yukawa couplings except ht, the RGE for thedifference m21 −m22 readsd(m21 −m22)dt= −38π2h2tFt .

(30)15

Imagine now that tan β < 1, and observe that the top and bottom massesare given by m2t = h2tv22 and m2b = h2bv21, respectively. Then mt ≫mb impliesht ≫hb, which makes eq.

(30) a good approximation. Solving now eq.

(30)at the scale ˆQ, where the use of V0(Q) is justified, and observing that it isalways Ft > 0, one finds m21 > m22. But eq.

(29) then tells us that tan β > 1,in contradiction with the starting assumption.Similarly, including in eq. (30) the contributions of the bottom and τ Yukawa couplings, one can provethat tan β <∼mt/mb.3Higgs bosonsWe begin the discussion of the MSSM particle spectrum with the (R-even)Higgs boson sector.

As explained in the previous section, the MSSM con-tains two complex Higgs doublets of opposite hypercharge, H1 ≡(H01, H−1 )and H2 ≡(H+2 , H02). After their neutral components develop non-vanishingVEVs, v1 and v2, which can be taken to be real and positive without loss ofgenerality, one is left with five physical degrees of freedom.

Three of theseare neutral (two CP-even, h and H, and one CP-odd, A) and two are charged(H±). The starting point for a discussion of Higgs-boson masses and cou-plings in the MSSM is the potential of eq.

(24). Besides the minimizationconditions, which relate v1 and v2 with the potential parameters, a physicalconstraint comes from the fact that the combination (v21 + v22), which deter-mines the W- and Z-boson masses, must reproduce their measured values.Once this constraint is imposed, in the approximation of eq.

(24) the MSSMHiggs sector contains only two independent parameters. A convenient choice,which will be adopted here, is to take as independent parameters mA, thephysical mass of the CP-odd neutral boson, and tan β ≡v2/v1.

The param-eter mA is essentially unconstrained, even if naturalness arguments suggestthat it should be smaller than O(500 GeV), whereas for tan β the rangepermitted in the MSSM is given by formula (28).In the approximation of eq. (24), the mass matrix of neutral CP-evenHiggs bosons readsM0R2 =" cot β−1−1tan β m2Z2 + tan β−1−1cot β m2A2#sin 2β(31)16

and the charged Higgs mass is given bym2H± = m2W + m2A . (32)From eq.

(31), one obtainsm2h,H = 12m2A + m2Z ∓q(m2A + m2Z)2 −4m2Am2Z cos2 2β,(33)and also celebrated inequalities as mW, mA < mH±, mh < mZ < mH, mh

(34)For example, the tree-level couplings of the neutral Higgs bosons are easilyobtained from the standard model Higgs couplings if one multiplies themby some appropriate α- and β-dependent factors [55]. An important conse-quence of the structure of the Higgs potential (24) is the existence of at leastone neutral CP-even Higgs boson, h, weighing less than mZ.

This raised thehope that the crucial experiment on the MSSM Higgs sector could be en-tirely performed at LEP II (with sufficient centre-of-mass energy, luminosityand b-tagging efficiency), and took some interest away from higher energycolliders. However, it was recently pointed out [56–58] that the Higgs-bosonmasses are subject to large radiative corrections, associated with the topquark and its SU(2) and supersymmetric partners2.

Several papers [62–75]have subsequently investigated various aspects of these corrections and theirimplications for experimental searches at LEP and LHC-SSC. In the follow-ing subsection, we shall summarize the main effects of radiative correctionson Higgs-boson parameters.2Previous studies [59–61] either neglected the case of a heavy top quark [59,60], orconcentrated on the violations of the neutral Higgs-mass sum rule [61] without computingcorrections to individual Higgs masses.17

3.1Radiative corrections to Higgs boson-masses andcouplingsAs far as Higgs-boson masses and self-couplings are concerned, a convenientapproximate way of parametrizing one-loop radiative corrections is to sub-stitute the tree-level Higgs potential of eq. (24) with the one-loop effectivepotential of eq.

(27), and to identify masses and self-couplings with theappropriate combinations of derivatives evaluated at the minimum.Thecomparison with explicit diagrammatic calculations [58,66,69,70,75] showsthat the effective potential approximation is more than adequate for ourpurposes. Also, inspection shows that the most important contributions toeq.

(27) come from the field-dependent mass matrices of the top and bot-tom quarks and squarks, whose explicit expressions depend on a numberof parameters and can be found in ref. [68].

To simplify the discussion, inthe following we will take a universal soft supersymmetry-breaking squarkmass, ˜m2Q = ˜m2Uc = ˜m2Dc ≡m2˜q, and we will assume negligible mixing inthe stop and sbottom mass matrices, At = Ab = µ = 0. More completeformulae for arbitrary values of the parameters can be found in refs.

[65,68],but the qualitative features corresponding to the above parameter choices arerepresentative of a very large region of parameter space. In the case underconsideration, and neglecting D-term contributions to the squark masses, theneutral CP-even mass matrix is modified at one loop as followsM2R =M0R2 + ∆2100∆22,(35)where∆21 =3g2m4b16π2m2W cos2 β logm2˜b1m2˜b2m4b,(36)∆22 =3g2m4t16π2m2W sin2 β log m2˜t1m2˜t2m4t.

(37)From the above expressions one can easily derive the one-loop-correctedeigenvalues mh and mH, as well as the mixing angle α associated with theone-loop-corrected mass matrix (35). The most striking fact in eqs.

(35)–(37) is that the correction ∆22 is proportional to (m4t/m2W).This impliesthat, for mt in the range allowed by experimental limits and by eq. (18), thetree-level predictions for mh and mH can be badly violated, and so for the18

related inequalities. The other free parameter is m˜q, but the dependence onit is much milder.

In the following, when making numerical examples, weshall always choose the representative value m˜q = 1 TeV. The reader caneasily rescale the displayed results to different values of m˜q.

To illustrate theimpact of these results, we display in fig. 1 [73] contours of the maximumallowed value of mh (reached for mA →∞), in the (mt, tan β) plane.

Toplot different quantities of physical interest in the (mA, tan β) plane, whichis going to be the stage of the following phenomenological discussion, oneneeds to fix also the value of mt. In the following, we shall work with therepresentative value mt = 140 GeV, which is near the centre of the presentlyallowed range.

As an example, we show in fig. 2 contours of constant mhand mH in the (mA, tan β) plane.

One-loop corrections to the charged Higgsmass have also been computed in refs. [68–71], and found to be small, atmost a few GeV, for generic values of the parameters.The effective potential method allows also the computation of the leadingcorrections to the trilinear and quadrilinear Higgs self-couplings.

For exam-ple, the leading radiative correction to the trilinear hAA coupling, whichplays a major role in the determination of the h branching ratios, is [68]λhAA = λ0hAA + ∆λhAA ,(38)whereλ0hAA = −igmZ2 cos θWcos 2β sin(β + α) ,(39)and, neglecting the bottom Yukawa coupling and the D-term contributionsto squark masses∆λhAA = −igmZ2 cos θW3g2 cos2 θW8π2cos α cos2 βsin3 βm4tm4Wlog m2˜q + m2tm2t. (40)Similarly, also the other Higgs self-couplings receive large corrections O(m4t/m4W).Finally, one should consider Higgs couplings to vector bosons and fermions.Tree-level couplings to vector bosons are expressed in terms of gauge cou-plings and of the angles β and α.

The most important part of the radia-tive corrections is taken into account by using one-loop-corrected formulaeto determine α from the input parameters. Other corrections are at mostO(m2t/m2W) and can be safely neglected for our purposes.

Tree-level cou-plings to fermions are expressed in terms of the fermion masses and of the19

angles β and α. In this case, the leading radiative corrections can be takeninto account by using the one-loop-corrected expression for α and runningfermion masses, evaluated at the scale Q which characterizes the processunder consideration.3.2The discovery potential of LEP and LHC-SSCIn this section, we briefly summarize the implications of the previous resultson MSSM Higgs-boson searches at LEP [64,65,68] and the LHC-SSC [73,74].As already clear from tree-level analyses [55], the relevant processes forMSSM Higgs boson searches at LEP I are Z →hZ∗and Z →hA, which playa complementary role since their rates are proportional to sin2(β −α) andcos2(β −α), respectively.

An important effect of radiative corrections [68] isto render possible, for some values of the parameters, the decay h →AA,which would be kinematically forbidden according to tree-level formulae. Ex-perimental limits which take radiative corrections into account have by nowbeen obtained by the four LEP collaborations [76], using different methodsto present and analyse the data, and different ranges of parameters in theevaluation of radiative corrections.The presently excluded region of the(mA, tan β) plane, for our standard parameter choice, is given in fig.

3 [73],where the solid line corresponds to the exclusion contour given in the first ofrefs. [76].The situation in which the impact of radiative corrections is most dra-matic is the search for MSSM Higgs bosons at LEP II.

At the time whenonly tree-level formulae were available, there was hope that LEP could com-pletely test the MSSM Higgs sector. According to tree-level formulae, in fact,there should always be a CP-even Higgs boson with mass smaller than (h)or very close to (H) mZ, and significantly coupled to the Z boson.

However,as should be clear from the previous section, this result can be completelyupset by radiative corrections. A detailed evaluation of the LEP II discoverypotential can be made only if crucial theoretical parameters, such as the top-quark mass and the various soft supersymmetry-breaking masses, and exper-imental parameters, such as the centre-of-mass energy, the luminosity, andthe b-tagging efficiency, are specified.

Taking for example √s = 190 GeV,mt >∼110 GeV, and our standard values for the soft supersymmetry-breakingparameters, in the region of tan β significantly greater than 1 the associatedproduction of a Z and a CP-even Higgs can be pushed beyond the kinemati-20

cal limit. Associated hA production could be a useful complementary signal,but obviously only for mh + mA < √s.

Associated HA production is typi-cally negligible at these energies. To give a measure of the LEP II sensitivity,we plot in fig.

3 contours associated to two benchmark values of the totalcross-section σ(e+e−→hZ, HZ, hA, HA). The dashed lines correspond toσ = 0.2 pb at √s = 175 GeV, which could be seen as a rather conserva-tive estimate of the LEP II sensitivity.

The dash-dotted lines correspond toσ = 0.05 pb at √s = 190 GeV, which could be seen as a rather optimisticestimate of the LEP II sensitivity. Of course, one should keep in mind thatthere is, at least in principle, the possibility of further extending the maxi-mum LEP energy up to values as high as √s ≃230 −240 GeV, at the priceof introducing more (and more performing) superconducting cavities into theLEP tunnel [77].Similar considerations can be made for charged Higgs searches at LEP IIwith √s <∼190 GeV.

In view of the β3 threshold factor in σ(e+e−→H+H−),and of the large background from e+e−→W +W −, it will be difficult tofind the H± at LEP II unless mH± <∼mW, and certainly impossible unlessmH± < √s/2. We also know [68–71] that for generic values of the parametersthere are no large negative radiative corrections to the charged Higgs massformula, eq.

(32). Thus there is very little hope of finding the charged Higgsboson of the MSSM at LEP II (or, to put it differently, the discovery of acharged Higgs boson at LEP II would most probably rule out the MSSM).The next question is then whether the LHC and SSC can explore thefull parameter space of the MSSM Higgs bosons.A systematic study ofthis problem, including radiative corrections, has been recently performedin ref.

[73] (see also [74]), following the strategy outlined in ref. [78].

Theanalysis is complicated by the fact that the R-odd particles could play arole both in the production (via loop diagrams) and in the decay (via loopdiagrams and as final states) of the MSSM Higgs bosons. For simplicity, onecan concentrate on the most conservative case in which all R-odd particlesare heavy enough not to play any significant role.

Still, one has to perform aseparate analysis for each (mA, tan β) point, to include radiative corrections(depending on the parameters of the top-stop-bottom-sbottom system), andto consider Higgs boson decays involving other Higgs bosons. We make hereonly a few general remarks on the LHC case, for the representative parameterchoice mt = 140 GeV, m˜q = 1 TeV, At = Ab = µ = 0, sending the reader toref.

[73] for a more complete discussion, and to ref. [79] for a review of recent21

simulation work.Beginning with the neutral states, when h or H are in the intermediatemass range (80–130 GeV) and have approximately SM couplings, the bestprospects for detection are offered, as in the SM, by their γγ decay mode.In general, however, σ · BR(h, H →γγ) is smaller than for a SM Higgsboson of the same mass. As a rather optimistic estimate of the possible LHCsensitivity, we display, in fig.

4, lines corresponding to σ · BR(h, H →γγ) ∼2/5 of the corresponding value for a SM Higgs of 100 GeV. Only in theregion of the (mA, tan β) plane to the right of the line denoted by ‘a’ (in thecase of h) and above the line denoted by ‘b’ (in the case of H) the γγ signalexceeds the chosen reference value.

Almost identical considerations can bemade for the production of h or H, decaying into γγ, in association with aW boson or with a tt pair. When H and A are heavy, in general one cannotrely on the ZZ →4l± (l = e, µ) decay mode, which gives the ‘gold-plated’Higgs signature in the SM case, since H and A couplings to vector-bosonpairs are strongly suppressed: only for small tan β and 2mZ <∼mH <∼2mtmight the decay mode H →ZZ →4l± still be viable despite the suppressedbranching ratio.

Again, as an estimate of the possible LHC sensitivity, weshow in fig. 4, under the line denoted by ‘c’, the region of the (mA, tan β)plane corresponding to σ · BR(H →4l±) > 10−3 pb (l = e, µ).

For verylarge values of tan β, and moderately large mA, the unsuppressed decaysH, A →τ +τ −could give visible signals, in contrast to the SM case.Asa very optimistic estimate (especially in the small mA region! )we showin fig.

4, above the line denoted by ‘d’, the region of the parameter spacecorresponding to σ · BR(H, A →τ +τ −) > 1 pb. Finally, in the region ofparameter space corresponding to mA <∼mZ, the charged Higgs could bediscovered via the decay chain t →bH+ →bτ +ντ, which competes with thestandard channel t →bW + →bl±νl (l = e, µ, τ).

A convenient parameteris the ratio R ≡BR(t →τ +ντb)/BR(t →µ+νµb). As a very optimisticestimate of the LHC sensitivity, the line of fig.

4 denoted by ‘e’ delimits fromthe right the region of the (mA, tan β) plane corresponding to R > 1.1. Forall processes considered above, similar remarks apply also to the SSC.In summary, a global look at figs.

3 and 4 shows that there is a highdegree of complementarity between the regions of parameter space accessibleto LEP II and to the LHC-SSC. However, for our representative choice ofparameters, there is a non-negligible region of the (mA, tan β) plane thatis presumably beyond the reach of both LEP II and the LHC-SSC.

This22

potential problem could be solved, as we said before, by a further increaseof the LEP II energy beyond the reference value √s <∼190 GeV. Otherwise,this is the ideal case for a 500 GeV (or even less) e+e−collider, as we shallsee below.

Even if in the future a Higgs boson will be found at LEP or theLHC-SSC, with properties compatible with those of a MSSM Higgs boson,it appears difficult to search effectively for all the Higgs states of the MSSMat the above machines. Again, as we shall see below, EE500 could play animportant role in investigating the properties of the newly discovered Higgsboson and in looking for the remaining states of the MSSM.3.3Production mechanisms at high-energy e+e−col-lidersWe now present, following ref.

[80], cross-sections for the main productionmechanisms of neutral susy Higgses in e+e−collisions at √s = 500 GeV,namely:e+e−→hZ[σ ∝sin2(β −α)] ,e+e−→HZ[σ ∝cos2(β −α)] ,e+e−→hA[σ ∝cos2(β −α)] ,e+e−→HA[σ ∝sin2(β −α)] ,e+e−→hνν[σ ∝sin2(β −α)] ,e+e−→Hνν[σ ∝cos2(β −α)] ,e+e−→he+e−[σ ∝sin2(β −α)] ,e+e−→He+e−[σ ∝cos2(β −α)] .Other production mechanisms of interest are discussed in refs. [80,81], anddetails about experimental searches can be found in refs.

[82,83]. We haveincluded radiative corrections to the masses mh, mH and to the mixing angleα for our standard parameter choice.We have neglected loops from thegauge-gaugino-Higgs-higgsino sector, which are known to give correctionssmaller than the ones we have included.We have also neglected propervertex corrections to vector boson-Higgs boson couplings and initial-stateradiation.In discussing our results, it is useful to estimate the cross-section for whichwe believe that any of the listed processes will be detectable.

A cross-sectionof 0.01 pb will lead to 25 events for an integrated luminosity of 10 fb−1 aftermultiplying by an efficiency of 25%; the latter is a crude estimate of the23

impact of detector efficiencies, cuts, and branching ratios to usable decaychannels. It will be helpful to keep this benchmark cross-section value inmind as a rough criterion for where in parameter space a particular reactioncan be useful.Fig.

5 shows contours of σ(e+e−→hZ) and σ(e+e−→HZ), respec-tively, in the (mA, tan β) plane. Observe that the two processes are trulycomplementary, in the sense that everywhere in the (mA, tan β) plane thereis a substantial cross-section for at least one of them (σ > 0.01 pb).

Thisshould be an excellent starting point for experimental searches. Similar con-siderations hold for hA, HA production, whose cross-sections are shown infig.

6. As long as one of the two channels is kinematically accessible, theinclusive cross-section is large enough to provide a substantial event rate.Even in this case the two processes are complementary, and together shouldbe able to probe the region of parameter space corresponding to mA <∼200GeV.

We now move to single Higgs production via vector-boson fusion. Thecross-sections for h,H production via WW and ZZ fusion are given in figs.

7and 8, respectively: they have been obtained using exact analytical formu-lae, rescaled from ref. [84].

Obviously, since the AWW and AZZ verticesare absent at tree level, one cannot get substantial A production with thismechanism for sensible values of the parameters. The ZZ fusion processesare suppressed by an order of magnitude with respect to the WW fusionones, but could still be useful for experimental searches.The global picture which emerges from our results is the following.

Ifno neutral Higgs boson is previously discovered, at EE500 one must find atleast one neutral susy Higgs, otherwise the MSSM is ruled out. If mA isnot too large, at EE500 there is the possibility of discovering all of the Higgsstates of the MSSM via a variety of processes, including charged-Higgs-bosonproduction, which has not been discussed here.

In the event that a neutralHiggs boson is discovered previously at LEP or the LHC-SSC, with propertiescompatible with one of the MSSM Higgs states, EE500 would still be a veryuseful instrument to investigate in detail the spectroscopy of the Higgs sector,for example to distinguish between the SM, the MSSM and possibly othernon-minimal supersymmetric extensions.24

4R-odd particlesWe now briefly review the R-odd spectrum of the MSSM, to introduce thediscussion of supersymmetric particle searches at e+e−and hadron colliders.In the spin-0 sector, one has sleptons and squarks, ˜f ≡(˜νL, ˜eL, ˜ecL ≡˜e∗R, ˜uL, ˜ucL ≡˜u∗R, ˜dL, ˜dcL ≡˜d∗R), with generation indices left implicit as usual.Neglecting intergenerational mixing, their diagonal mass terms are given bym2˜f = ˜m2 + m2f + m2D ,(41)where ˜m is the soft supersymmetry-breaking mass, mf is the correspondingfermion mass, andm2D = m2Ztan2 β −1tan2 β + 1(Y sin2 θW −T3L cos2 θW) . (42)For the sfermions of the first two generations, ˜fL- ˜fR mixing is negligible andthe soft masses are given by eq.

(22), so one can express m ˜f in terms of thebasic parameters m1/2, m0 and tan β. Notice for example that, neglecting thelepton masses, SU(2) invariance alone requires m2˜ν = m2˜eL −m2W[(tan2 β −1)/(tan2 β + 1)].

For the sfermions of the third generation, the off-diagonalterm in the ˜fL- ˜fR mass matrixm2˜fLR = mf ×(Af + µ · tan β(f = e, d)Af + µ/ tan β(f = u),(43)might be non-negligible, so that the mass eigenstates ( ˜f1, ˜f2) are non-trivialadmixtures of the interaction eigenstates ( ˜fL, ˜fR). Also, to compute the softcontributions to the masses in terms of the basic parameters one has to solvenumerically the associated RGE.In the spin- 12 sector, one has the strongly interacting gluinos, ˜g, with massm˜g ≡M3 directly given by eq.

(16). In addition, one has the weakly interact-ing charginos and neutralinos, i.e.

the charged and neutral mass eigenstatescorresponding to electroweak gauginos and higgsinos. Charginos ( ˜W ±, ˜H±)mix via the 2 × 2 matrix M2√2mW sin β√2mW cos βµ!,(44)25

whose mass eigenstates are denoted by ˜χ±k (k = 1, 2), and neutralinos ( ˜B, ˜W3, ˜H01, ˜H02)mix via the 4 × 4 matrixM10−mZ cos β sin θWmZ sin β sin θW0M2mZ cos β cos θW−mZ sin β cos θW−mZ cos β sin θWmZ cos β cos θW0−µmZ sin β sin θW−mZ sin β cos θW−µ0,(45)whose mass eigenstates are denoted by ˜χ0i (i = 1, 2, 3, 4). Notice that thelightest neutralino ˜χ ≡˜χ01, which in most of the acceptable parameter spaceis the LSP, is in general a non-trivial admixture of gauginos and higgsinos,and not just a pure photino ˜γ ≡cos θW ˜B + sin θW ˜W3 as often assumed inphenomenological studies.

In the MSSM, all the masses and couplings inthe chargino-neutralino sector can be characterized by the three parametersm1/2, µ, and tan β.To give an idea of the structure of the MSSM R-odd spectrum, we showin figs. 9 and 10 (updated from ref.

[85]) contours of some selected sparticlemasses in the (m0, m1/2) and in the (µ, m1/2) planes, respectively, for therepresentative values tan β = 2 and tan β = 10.4.1Searches for sleptonsThe most stringent limits on sleptons come from unsuccessful searches forthe processes Z →˜l+˜l−and Z →˜ν˜ν at LEP I. In the mass range of interest,and assuming that ˜χ is the LSP, the main decay modes are ˜l± →l± ˜χ and˜ν →ν ˜χ.

Indirect but powerful information can be extracted from the precisemeasurements of the total and partial Z widths. Direct searches are sensitiveto charged sleptons only, and look for acoplanar lepton pairs with missingtransverse momentum.

Experimental details on slepton searches at LEP I canbe found in refs. [86,87].

Crudely speaking, one can summarize the presentlimits by m˜l, m˜ν >∼mZ/2. In the future, LEP II will be sensitive to chargedsleptons up to m˜l ≃80–90 GeV, whereas the limits on m˜ν are not expectedto improve.

At large hadron colliders like the LHC-SSC, slepton searchesappear problematic [88], since the Drell-Yan production cross-sections aresmall and the backgrounds are large. It is then clear that high-energy e+e−colliders can play a very important role in slepton searches, as will be nowoutlined.26

Theoretical aspects of slepton production and decay at EE500 have beenrecently investigated in ref. [89].

The production mechanisms considered inthis study aree+e−→˜e+L ˜e−L, ˜e+R˜e−R, ˜e±L ˜e∓R,(46)e+e−→˜µ+L ˜µ−L, ˜µ+R˜µ−R,(47)e+e−→˜ν˜ν. (48)The first two processes in (46) occur via (γ, Z) exchange in the s-channel and˜χ0i exchange in the t-channel.

The last process in (46) receives only t-channelcontributions, the two processes in (47) only s-channel contributions. Theprocesses in (48) occur via Z exchange in the s-channel, with ˜χ±k exchangein the t-channel also contributing in the case of ˜νe.

In general, then, theproduction cross-sections depend not only on the slepton masses, but also onthe parameters of the chargino-neutralino sector.As far as decay modes are concerned, one has to take into account thepossibility of cascade decays, ˜l±L,R →l± ˜χ0i̸=1 →. .

., ˜l±L →ν ˜χ±k →. .

., ˜ν →ν ˜χ0i̸=1 →. .

., ˜ν →l± ˜χ∓k →. .

., in addition to the direct decays ˜l±L,R →l± ˜χ,˜ν →ν ˜χ. Also the relevant branching ratios depend on the parameters of thechargino-neutralino sector.A detailed analysis of the whole parameter space will not be attemptedhere.The most likely case, in view of the theoretical constraints on theMSSM, seems to be the one in which the lightest sleptons are ˜l±R (l = e, µ, τ).In this case one obtains [89] sizeable cross-sections, O(10 fb) or more, upto slepton masses of 80–90 % of the beam energy, which should allow for arelatively easy detection if the mass difference (m˜eR −m˜χ) is not too small[3].4.2Searches for squarks and gluinosBeing strongly interacting sparticles, squarks and gluinos are best searchedfor at hadron colliders.

Production cross-sections for ˜g˜g, ˜g˜q, ˜q˜q pair-productionin pp or pp collisions are relatively model-independent functions of m˜g andm˜q. As far as signatures are concerned, one has to distinguish two mainpossibilities: if m˜g < m˜q, then ˜q →q˜g immediately after production, and thefinal state is determined by ˜g decays; if m˜q < m˜g, then ˜g →˜qq immediatelyafter production, and the final state is determined by ˜q decays.

The first case27

is favoured by the theoretical constraints of the MSSM. In old experimentalanalyses, it was customary to work under a certain set of assumptions: 1)five or six (˜qL, ˜qR) mass-degenerate squark flavours; 2) LSP ≡˜γ, with massnegligible with respect to m˜q, m˜g; 3) the dominant decay modes of squarksand gluinos are the direct ones, ˜g →qq˜γ if m˜g < m˜q and ˜q →q˜γ if m˜q < m˜g.The signals to be looked for are then multijet events with a large amount ofmissing transverse momentum.

To derive reliable limits, however, one hasto take into account that the above assumptions are in general incorrect.For example, one can have cascade decays ˜g →qq˜χ0i̸=1, q′qχ±k →. .

. and˜q →q˜χ0i̸=1, q′ ˜χ±k →.

. ..

The effects of these cascade decays become moreand more important as one moves to higher and higher squark and gluinomasses. Taking all this into account, the present limits from the Tevatroncollider are roughly m˜q >∼150 GeV, m˜g >∼135 GeV [90].

At the LHC andSSC, one should be able to explore squark and gluino masses up to 1 TeVand probably more [91]. In general, therefore, EE500 will not be competitivefor squark and gluino searches.

Its cleaner environment, however, could beexploited for a detailed study of squark properties if they are discovered atsufficiently low mass. Also, there are special situations which might be diffi-cult to study at large hadron colliders: for example, the case of a stop squarksignificantly lighter than all the other squarks.

The threshold behaviour forstop production in e+e−collisions has been recently studied in ref. [92].4.3Searches for charginos and neutralinosThe most stringent limits on charginos and neutralinos come [93,86,87] fromunsuccessful LEP I searches for the processes Z →˜χ˜χ (contributing to theinvisible Z width), Z →˜χ˜χ0i̸=1 (originating spectacular one-sided events) andZ →˜χ+1 ˜χ−1 , ˜χ0i̸=1 ˜χ0j̸=1 (originating acoplanar leptons or jets accompaniedby missing energy).

The presently excluded region of the (µ, m1/2) planeis shown, for the two representative values tan β = 2 and tan β = 10, infig. 10.

As a crude summary, one could say that all states different from ˜χhave to be heavier than mZ/2, whereas LEP data alone would still allow forarbitrarily light ˜χ. For LEP II searches, the most effective process shouldbe ˜χ+1 ˜χ−1 pair production, with ˜χ˜χ0i̸=1 pair production slightly less effectivein probing parameter space because of the smaller cross-section.

At largehadron colliders [94], it seems very difficult to improve the LEP II sensitivitysignificantly, especially if the top quark mass is significantly smaller than 20028

GeV, as now favoured by radiative-correction analyses.Theoretical aspects of chargino and neutralino production and decays atEE500 have been recently studied in ref. [89], and experimental simulationsare reported in ref.

[3]. In the case of charginos, the most important pro-duction diagrams involve the s-channel exchange of (γ, Z) and the t-channelexchange of ˜νe.

The cross-section then depends not only on the parameters ofthe chargino-neutralino sector, but also on the sneutrino mass, and there canbe significant destructive interference between the two classes of diagrams.As for chargino decays, if the sneutrino is light enough the dominant decaymode is ˜χ±1 →l±˜ν, whereas in the case of a heavy sneutrino the dominantdecay modes are ˜χ±1 →qq′ ˜χ, l±ν ˜χ. Experimental analyses show that atEE500 one can enormously extend the parameter space accessible to LEP II,and reach chargino masses of the order of 80–90 % of the beam energy, pro-vided that the mass difference m˜χ±1 −m˜χ is not too small: this unfortunatesituation could occur in the region of parameter space where |µ| << m1/2.5ConclusionsIn summary, in this talk we have argued that the MSSM is a calculable,theoretically motivated and phenomenologically acceptable extension of theSM.

Of course, only experiment can tell if low-energy supersymmetry is ac-tually realized in Nature, but, to use the words of one of the speakers at thisWorkshop, searching for supersymmetry does not look like fishing in a deadsea. For a global view of the present limits and of the discovery potentialof future machines, including EE500, it is useful to look again at the mostimportant parameters of the MSSMmA,tan β ,m0 ,m1/2 ,µ ,(49)which, together with mt, determine the main features of its particle spectrum.The Higgs sector mainly depends on (mA, tan β), but also mt and (to alesser extent) the other parameters play a role via the large radiative correc-tions.

As an example, we have considered the case mt = 140 GeV, m˜q =1 TeV, At = Ab = µ = 0, summarized in figs. 3–8.

Fig. 3 shows that LEP I,despite its remarkable achievements, has explored only a small part, roughlymA <∼45 GeV, of the natural parameter space for the MSSM Higgs bosons.A much greater sensitivity will be achieved at LEP II, where, for standard29

values of the machine parameters (√s = 190 GeV,R Ldt = 500 pb−1), oneshould be able to test mA <∼80 GeV, tan β <∼3. However, as a result ofthe large radiative corrections, the rest of the (mA, tan β) parameter spacewill not be accessible to LEP II.

The LHC-SSC can greatly improve overLEP II, as shown in fig. 4.

The most promising experimental signatures areh, H →γγ (inclusive or in association with W or tt), H →ZZ →4l±,H, A →τ +τ −, t →bH+ →bτ +ντ. Combined, they might be able to probethe whole (mA, tan β) plane, with the exception of mZ <∼mA <∼200 GeV,2 <∼tan β <∼10.

For our choice of parameters, then, the overlap betweenLEP II and the LHC-SSC is likely not to be complete, giving rise to a pos-sible violation of the so-called ‘no-lose theorem’. A further increase of theLEP II energy might save the day.

On the other hand, as shown in figs. 5–8,EE500 is guaranteed to observe at least one neutral Higgs boson or to ruleout the MSSM.

In particular, in the region of the parameter space which ismost difficult for the LHC-SSC, EE500 can perform a detailed spectroscopyof the MSSM Higgs sector, observing all its physical states.As for the (R-odd) supersymmetric particles, the situation is summarizedin figs. 9 and 10.

Again, we can see that the already impressive limits ob-tained by LEP I and Tevatron have ruled out only a small part of the naturalparameter space. LEP II and the upgraded Tevatron will provide higher butstill limited sensitivities, corresponding roughly to m˜l, m˜χ± <∼80–90 GeV,m˜g, m˜q <∼200 GeV.

The LHC-SSC should definitely cover the rest of theparameter space, via gluino and squark searches up to masses of 1 TeV orhigher.However, a comparable sensitivity can be reached by EE500 viachargino and slepton searches up to masses of 200 GeV or even higher. Ifno signal of supersymmetry is found at the LHC-SSC, EE500 can providethe definitive confirmation that the MSSM is ruled out, in a cleaner environ-ment and in a more model-independent way.

In the optimistic case that asignal is found at the LHC-SSC, EE500 would constitute a unique facility forthe direct production of weakly interacting supersymmetric particles, whichshould allow for a detailed spectroscopy of the MSSM.In conclusion, for what concerns supersymmetry, EE500 is the ideal com-plement to the LHC-SSC, and the case for it could not be stronger.30

AcknowledgementsI am grateful to my collaborators A. Brignole, J. Ellis, Z. Kunszt and G.Ridolfifor their contributions to our common work on supersymetric Higgsbosons, and to them and F. Pauss for discussions concerning the presentpaper.31

References1. P. Fayet, Nucl.

Phys. B90 (1975) 104, Phys.

Lett. 64B (1976) 159 and69B (1977) 489.2.

For reviews and references, see, e.g.:H.-P. Nilles, Phys.

Rep. 110 (1984) 1;H.E. Haber and G.L.

Kane, Phys. Rep. 117 (1985) 75;S. Ferrara, ed., ‘Supersymmetry’ (North-Holland, Amsterdam, 1987);R. Barbieri, Riv.

Nuovo Cimento 11 (1988) 1.3. J.-F. Grivaz, these Proceedings.4.

Yu.A. Gol’fand and E.P.

Likhtman, JETP Lett. 13 (1971) 323;D.V.

Volkov and V.P. Akulov, Phys.

Lett. 46B (1973) 109;J. Wess and B. Zumino, Nucl.

Phys. B70 (1974) 39.5.

D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys.

Rev. B13(1976) 3214;S. Deser and B. Zumino, Phys.

Lett. 62B (1976) 335.6.

R. Haag, J. Lopuszanski and M. Sohnius, Nucl. Phys.

B88 (1975) 257.7. For reviews and references see, e.g.:M.B.

Green, J.H. Schwarz and E. Witten, ‘Superstring Theory’ (Uni-versity Press, Cambridge, 1987);B. Schellekens, ed., ‘Superstring construction’ (North-Holland, Ams-terdam, 1989).8.

K. Wilson, as quoted in L. Susskind, Phys. Rev.

D20 (1979) 2619;E. Gildener and S. Weinberg, Phys. Rev.

D13 (1976) 3333;G. ’t Hooft, in ‘Recent developments in gauge theories’, Cargese Lec-tures 1979 (Plenum Press, New York, 1980).9. For reviews and references, see, e.g.:J.R.

Carter, J. Ellis and T. Hebbeker, Rapporteur’s talks given at theLP-HEP ’91 Conference, Geneva, 1991, to appear in the Proceedings,and references therein.32

10. J. Wess and B. Zumino, Phys.

Lett. B49 (1974) 52;J. Iliopoulos and B. Zumino, Nucl.

Phys. B76 (1974) 310;S. Ferrara, J. Iliopoulos and B. Zumino, Nucl.

Phys. B77 (1974) 413;M.T.

Grisaru, W. Siegel and M. Rocek, Nucl. Phys.

B159 (1979) 429;S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev.

D20 (1979) 403.11. L. Maiani, Proc.

Summer School on Particle Physics, Gif-sur-Yvette,1979 (IN2P3, Paris, 1980), p. 1;M. Veltman, Acta Phys. Polon.

B12 (1981) 437;E. Witten, Nucl. Phys.

B188 (1981) 513;see also:S. Weinberg, Phys. Lett.

82B (1979) 387.12. L. Girardello and M.T.

Grisaru, Nucl. Phys.

B194 (1982) 65.13. J. Ellis and D.V.

Nanopoulos, Phys. Lett.

B110 (1982) 44;R. Barbieri and R. Gatto, Phys. Lett.

B110 (1982) 211.14. Y. Kizukuri and N. Oshimo, contribution to the Proceedings of theWorkshop ‘e+e−Linear Colliders at 500 GeV: the Physics Potential’,Hamburg, 1991, and references therein.15.

J. Ellis, S. Ferrara and D.V. Nanopoulos, Phys.

Lett. B114 (1982) 231;W. Buchm¨uller and D. Wyler, Phys.

Lett. B121 (1983) 321;J. Polchinski and M.B.

Wise, Phys. Lett.

B125 (1983) 393;F. del Aguila, J.A. Grifols, A. Mendez, D.V.

Nanopoulos and M. Sred-nicki, Phys. Lett.

B129 (1983) 77;J.-M. Fr`ere and M.B. Gavela, Phys.

Lett. B132 (1983) 107.16.

R.K. Kaul and P. Majumdar, Nucl. Phys.

B199 (1982) 36;R. Barbieri, S. Ferrara and C.A. Savoy, Phys.

Lett. B119 (1982) 36;H.P.

Nilles, M. Srednicki and D. Wyler, Phys. Lett.

B120 (1983) 346;J.M. Fr`ere, D.R.T.

Jones and S. Raby, Nucl. Phys.

B222 (1983) 11;J.-P. Derendinger and C. Savoy, Nucl. Phys.

B237 (1984) 307;J. Ellis, J.F. Gunion, H.E.

Haber, L. Roszkowski and F. Zwirner, Phys.Rev. D39 (1989) 844.17.

J. Polchinski and L. Susskind, Phys. Rev.

D26 (1982) 3661;J.E. Kim and H.-P. Nilles, Phys.

Lett. B138 (1984) 150;33

L. Hall, J. Lykken and S. Weinberg, Phys. Rev.

D27 (1983) 2359;G. F. Giudice and A. Masiero, Phys. Lett.

206B (1988) 480;K. Inoue, M. Kawasaki, M. Yamaguchi and T. Yanagida, preprint TU-373 (1991);J.E. Kim and H.-P. Nilles, Phys.

Lett. B263 (1991) 79;E.J.

Chun, J.E. Kim and H.-P. Nilles, preprint SNUTP-91-25.18.

H.-P. Nilles, M. Srednicki and D. Wyler, Phys. Lett.

B124 (1983) 337;A.B. Lahanas, Phys.

Lett. B124 (1983) 341;L. Alvarez-Gaum´e, J. Polchinski and M.B.

Wise, Nucl. Phys.

B221(1983) 495;A. Sen, Phys. Rev.

D30 (1984) 2608 and D32 (1985) 411.19. U. Ellwanger and M. Rausch de Traubenberg, contribution to sameProc.

as ref. [14], and references therein;B.R.

Kim, S.K. Oh and A. Stephan, ibid., and references therein.20.

L.J. Hall and M. Suzuki, Nucl.

Phys. B231 (1984) 419;F. Zwirner, Phys.

Lett. B132 (1983) 103.21.

C. Aulakh and R.N. Mohapatra, Phys.

Lett. B119 (1983) 136;G.G.

Ross and J.W.F. Valle, Phys.

Lett. B151 (1985) 375;J. Ellis, G. Gelmini, C. Jarlskog, G.G.

Ross and J.W.F. Valle, Phys.Lett.

B150 (1985) 142.22. H. Dreiner and S. Lola, contribution to same Proc.

as ref. [14], andreferences therein.23.

H.P. Nilles, Phys.

Lett. 115B (1982) 193 and Nucl.

Phys. B217 (1983)366;S. Ferrara, L. Girardello and H.P.

Nilles, Phys. Lett.

125B (1983) 457;J.-P. Derendinger, L.E. Ib´a˜nez and H.P.

Nilles, Phys.Lett.155B(1985) 65;M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett.

156B (1985)55.24. J. Scherk and J.H.

Schwarz, Phys.Lett.B82 (1979) 60 and Nucl.Phys. B153 (1979) 61;R. Rohm, Nucl.

Phys. B237 (1984) 553;34

C. Kounnas and M. Porrati, Nucl. Phys.

B310 (1988) 355;S. Ferrara, C. Kounnas, M. Porrati and F. Zwirner, Nucl. Phys.

B318(1989) 75;C. Kounnas and B. Rostand, Nucl. Phys.

B341 (1990) 641;I. Antoniadis, Phys. Lett.

B246 (1990) 377.25. H. Georgi, H.R.

Quinn and S. Weinberg, Phys. Rev.

Lett. 33 (1974)451.26.

W.A. Bardeen, A. Buras, D. Duke and T. Muta, Phys.Rev.D18(1978) 3998.27.

H. Georgi and S.L. Glashow, Phys.

Rev. Lett.

32 (1974) 438.28. For reviews and references, see, e.g.

:G. Costa and F. Zwirner, Riv. Nuovo Cimento 9 (1986) 1.29.

U. Amaldi, A. B¨ohm, L.S. Durkin, P. Langacker, A.K.

Mann, W.J.Marciano, A. Sirlin and H.H. Williams, Phys.

Rev. D36 (1987) 1385;G. Costa, J. Ellis, G.L.

Fogli, D.V. Nanopoulos and F. Zwirner, Nucl.Phys.

B297 (1988) 244.30. J. Ellis, S. Kelley and D.V.

Nanopoulos, Phys. Lett.

B249 (1990) 442and B260 (1991) 131;P. Langacker and M. Luo, Phys. Rev.

D44 (1991) 817;U. Amaldi, W. de Boer and H. F¨urstenau, Phys. Lett.

B260 (1991) 447;F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, preprint CERN-PPE/91-123.31. S. Dimopoulos, S. Raby and F. Wilczek, Phys.

Rev. D24 (1981) 1681;L.E.

Ib´a˜nez and G.G. Ross, Phys.

Lett. 105B (1981) 439.32.

G. Gamberini, G. Ridolfiand F. Zwirner, Nucl. Phys.

B331 (1990)331.33. H. Georgi and D.V.

Nanopoulos, Nucl. Phys.

B159 (1979) 16;F. del Aguila and L.E. Ib´a˜nez, Nucl.

Phys. B177 (1981) 60;L.E.

Ib´a˜nez, Nucl. Phys.

B181 (1981) 105;P. Frampton and S.L. Glashow, Phys.

Lett. 131B (1983) 340;35

U. Amaldi, W. de Boer, P.H. Frampton, H. F¨urstenau and J.T.

Liu,preprint CERN-PPE/91-233.34. S. Weinberg, Phys.

Lett. B91 (1980) 51;L.J.

Hall, Nucl. Phys.

B178 (1980) 75;T.J. Goldman and D.A. Ross, Nucl.

Phys. B171 (1980) 273;P. Bin´etruy and T. Sch¨ucker, Nucl.

Phys. B178 (1981) 293, 307;I. Antoniadis, C. Kounnas and C. Roiesnel, Nucl.

Phys. B198 (1982)317.35.

M.B. Einhorn and D.R.T.

Jones, Nucl. Phys.

B196 (1982) 475;M.E. Machacek and M.T.

Vaughn, Nucl. Phys.

B236 (1984) 221.36. S. Weinberg, Phys.

Rev. D26 (1982) 287;N. Sakai and T. Yanagida, Nucl.

Phys. B197 (1982) 533.37.

S. Dimopoulos and H. Georgi, Nucl. Phys.

B193 (1981) 150;N. Sakai, Z. Phys. C11 (1982) 153.38.

R. Barbieri and L.J. Hall, preprint LBL-31238 (1991);J. Ellis, S. Kelley and D.V.

Nanopoulos, preprint CERN-TH.6140/91;G.G. Ross and R.G.

Roberts, preprint RAL-92-005.39. V.S.

Kaplunovsky, Nucl. Phys.

B307 (1988) 145;L. Dixon, V.S. Kaplunovsky and J. Louis, Nucl.

Phys. B355 (1991)649;J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, preprintCERN-TH.6004/91, to appear in Nucl.

Phys. B;G. Lopez-Cardoso and B.A.

Ovrut, preprint UPR-0464T (1991);J. Louis, preprint SLAC-PUB-5527 (1991);I. Antoniadis, K.S. Narain and T. Taylor, Phys.

Lett. B267 (1991) 37;G. Lopez-Cardoso and B.A.

Ovrut, preprint UPR-0481T (1991);J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett.B271 (1991) 307.40.

W. Siegel, Phys. Lett.

B84 (1979) 193;D.M. Capper, D.R.T.

Jones and P. van Nieuwenhuizen, Nucl. Phys.B167 (1980) 479;36

I. Antoniadis, C. Kounnas and K. Tamvakis, Phys. Lett.

B119 (1982)377.41. I. Antoniadis, J. Ellis, J. Hagelin and D.V.

Nanopoulos, Phys. Lett.B231 (1989) 65, and references therein.42.

I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys.

Lett.B268 (1991) 188.43. I. Antoniadis, J. Ellis, S. Kelley and D.V.

Nanopoulos, preprint CERN-TH.6169/91;L.E. Ib´a˜nez, D. L¨ust and G.G.

Ross, Phys. Lett.

B272 (1991) 251.44. S. Weinberg, Phys.

Rev. Lett.

50 (1983) 387;H. Goldberg, Phys. Rev.

Lett. 50 (1983) 1419;L.M.

Krauss, Nucl. Phys.

B227 (1983) 556.45. J. Ellis, J.S.

Hagelin, D.V. Nanopoulos, K.A.

Olive and M. Srednicki,Nucl. Phys.

B238 (1984) 453.46. K.A.

Olive and M. Srednicki, Nucl. Phys.

B355 (1991) 208;J. Ellis, D.V. Nanopoulos, L. Roszkowski and D.N.

Schramm, Phys.Lett. B245 (1990) 251;L. Roszkowski, Phys.

Lett. B262 (1991) 59;J.L.

Lopez, K. Yuan and D.V. Nanopoulos, Phys.

Lett. B267 (1991)219;J. Ellis and L. Roszkowski, preprint CERN-TH.6260/91, UM-TH-91-25.47.

T.K. Kuo and N. Nakagawa, Nuovo Cimento Lett.

36 (1983) 560;L. Alvarez-Gaum´e, J. Polchinski and M.B. Wise, Nucl.

Phys. B221(1983) 495;R. Barbieri and L. Maiani, Nucl.

Phys. B224 (1983) 32;C.S.

Lim, T. Inami and N. Sakai, Phys. Rev.

D29 (1984) 1488;E. Eliasson, Phys. Lett.

B147 (1984) 65;Z. Hioki, Progr. Theor.

Phys. 73 (1985) 1283;J.A.

Grifols and J. Sol`a, Phys. Lett.

B137 (1984) 257 and Nucl. Phys.B253 (1985) 47;R. Barbieri, M. Frigeni, F. Giuliani and H.E.

Haber, Nucl. Phys.

B34137

(1990) 309;A. Bilal, J. Ellis and G.L. Fogli, Phys.

Lett. B246 (1990) 459;A. Djouadi, G. Girardi, C. Verzegnassi, W. Hollik and F.M.

Renard,Nucl. Phys.

B249 (1991) 48;M. Drees and K. Hagiwara, Phys. Rev.

D42 (1990) 1709;M. Boulware and D. Finnell, Phys. Rev.

D44 (1991) 2054;M. Drees, K. Hagiwara and A. Yamada, preprint DTP/91/34.48. M.E.

Peskin and T. Takeuchi, Phys. Rev.

Lett. 65 (1990) 964;M. Golden and L. Randall, Nucl.

Phys. B361 (1991) 3;B. Holdom and J. Terning, Phys.

Lett. B247 (1990) 88.49.

G. Altarelli, preprint CERN-TH.6245/91, talk given at the LP-HEP’91 Conference, Geneva, July 1991, to appear in the Proceedings, andreferences therein.50. R. Barbieri, S. Ferrara, L. Maiani, F. Palumbo and C.A.

Savoy, Phys.Lett. B115 (1982) 212;K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Progr.

Theor.Phys. 68 (1982) 927 and 71 (1984) 413.51.

J. Bagger, S. Dimopoulos and E. Mass´o, Phys. Rev.

Lett. 55 (1985)920.52.

B. Pendleton and G.G. Ross, Phys.

Lett. B98 (1981) 291;C. Hill, Phys.

Rev. D24 (1981) 691.53.

L.E. Ib´a˜nez and G.G.

Ross, Phys. Lett.

110B (1982) 215;K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, as in ref. [50].L.

Alvarez-Gaum´e, M. Claudson and M.B. Wise, Nucl.

Phys. B207(1982) 96;J. Ellis, D.V.

Nanopoulos and K. Tamvakis, Phys. Lett.

B121 (1983)123.54. S. Coleman and E. Weinberg, Phys.

Rev. D7 (1973) 1888;S. Weinberg, Phys.

Rev. D7 (1973) 2887.55.

For reviews and references see, e.g.:J.F. Gunion, H.E.

Haber, G.L. Kane and S. Dawson, ‘The Higgs Hunter’sGuide’ (Addison-Wesley, New York, 1990).38

56. Y. Okada, M. Yamaguchi and T. Yanagida, Prog.

Theor. Phys.

Lett.85 (1991) 1.57. J. Ellis, G. Ridolfiand F. Zwirner, Phys.

Lett. B257 (1991) 83.58.

H.E. Haber and R. Hempfling, Phys.

Rev. Lett.

66 (1991) 1815.59. S.P.

Li and M. Sher, Phys. Lett.

B140 (1984) 339.60. J.F.

Gunion and A. Turski, Phys.Rev. D39 (1989) 2701 and D40(1989) 2333.61.

M. Berger, Phys. Rev.

D41 (1990) 225.62. R. Barbieri, M. Frigeni and M. Caravaglios, Phys.

Lett. B258 (1991)167.63.

Y. Okada, M. Yamaguchi and T. Yanagida, Phys. Lett.

B262 (1991)54.64. R. Barbieri and M. Frigeni, Phys.

Lett. B258 (1991) 395.65.

J. Ellis, G. Ridolfiand F. Zwirner, Phys. Lett.

B262 (1991) 477.66. A. Yamada, Phys.

Lett. B263 (1991) 233.67.

J.R. Espinosa and M. Quir´os, Phys. Lett.

B266 (1991) 389.68. A. Brignole, J. Ellis, G. Ridolfiand F. Zwirner, Phys.Lett.B271(1991) 123.69.

P.H. Chankowski, S. Pokorski and J. Rosiek, Phys.

Lett. B274 (1992)191.70.

A. Brignole, preprint DFPD/91/TH/28 (1991), to appear in Phys.Lett. B.71.

M. Drees and M.N. Nojiri, preprint KEK-TH-305 (1991).72.

D.M. Pierce, A. Papadopoulos and S. Johnson, preprint LBL-31416,UCB-PTH-91/58.39

73. Z. Kunszt and F. Zwirner, preprint CERN-TH.6150/91, ETH-TH/91-7.74.

J.F. Gunion, R. Bork, H.E.

Haber and A. Seiden, preprint UCD-91-29,SCIPP-91/34;H. Baer, M. Bisset, C. Kao and X. Tata, preprint FSU-HEP-911104,UH-511-732-91;J.F. Gunion and L.H.

Orr, preprint UCD-91-15;V. Barger, M.S. Berger, A.L.

Stange and R.J.N. Phillips, preprintMAD-PH-680 (1991).75.

A. Brignole, preprint CERN-TH.6366/92.76. D. D´ecamp et al.

(ALEPH Collaboration), Phys. Lett.

B265 (1991)475;P. Igo-Kemenes (OPAL Collaboration), L. Barone (L3 Collaboration),W. Ruhlmann (DELPHI Collaboration), talks given at the LP-HEP’91 Conference, Geneva, 1991, to appear in the Proceedings;M. Davier, Rapporteur’s talk at the same Conference, and referencestherein.77. D. Treille, private communication;C. Rubbia, Rapporteur’s talk given at the LP-HEP ’91 Conference,Geneva, 1991, to appear in the Proceedings;U. Amaldi, these Proceedings.78.

Z. Kunszt and F. Zwirner, Proc. Large Hadron Collider Workshop,Aachen, 1990 (G. Jarlskog and D. Rein, eds.) (CERN 90-10, ECFA90-133, Geneva, 1990), Vol.II, p. 578.79.

F. Pauss, lectures given in the CERN Academic Training Programme,December 1991, and references therein.80. A. Brignole, J. Ellis, J.F.

Gunion, M. Guzzo, F. Olness, G. Ridolfi,L. Roszkowski and F. Zwirner, contribution to the Workshop ‘e+e−Linear Colliders at 500 GeV: the Physics Potential’, Hamburg, 1991,to appear in the Proceedings.81.

H.E. Haber, these Proceedings.40

82. P. Janot, contribution to the same Workshop as ref.

[80].83. S. Komamiya, these Proceedings.84.

D.R.T. Jones and S.T.

Petcov, Phys. Lett.

B84 (1979) 440;K. Hikasa, Phys. Lett.

B164 (1985) 385;R. Cahn, Nucl. Phys.

B255 (1985) 341;G. Altarelli, B. Mele and F. Pitolli, Nucl. Phys.

B287 (1987) 285.85. G. Ridolfi, G.G.

Ross and F. Zwirner, same Proc. as ref.

[78], Vol.II,p. 605.86.

D. D´ecamp et al. (ALEPH Collaboration), Phys.

Lett. B236 (1990)86;P. Abreu et al.

(DELPHI Collaboration), Phys. Lett.

B247 (1990)157;B. Adeva et al. (L3 Collaboration), Phys.

Lett. B233 (1989) 530;M.Z.

Akrawy et al. (OPAL Collaboration), Phys.

Lett. B240 (1990)261.87.

D. D´ecamp et al. (ALEPH Collaboration), preprint CERN-PPE/91-149, submitted to Physics Reports.88.

F. del Aguila, L. Ametller amd M. Quir´os, same Proc. as ref.

[78],Vol.II, p. 663;F. del Aguila and L. Ametller, Phys. Lett.

B261 (1991) 326.89. A. Bartl, W. Majerotto and B. M¨osslacher, contribution to the sameWorkshop as ref.

[80].90. H. Baer, X. Tata and J. Woodside, Phys.

Rev. D44 (1991) 207.91.

C. Albajar, C. Fuglesang, S. Hellman, E. Nagy, F. Pauss, G. Poleselloand P. Sphicas, same Proc. as ref.

[78], Vol.II, p. 621;H. Baer et al., preprint FSU-HEP-901110, to be published in the Pro-ceedings of the 1990 DPF Summer Study on High Energy Physics,Snowmass, CO, June 25-July 13, 1990.92. I.I.

Bigi, V.S. Fadin and V. Khoze, preprint UND-HEP-91-BIG03.41

93. D. D´ecamp et al.

(ALEPH Collaboration), Phys. Lett.

B244 (1990)541;M.Z. Akrawy et al.

(OPAL Collaboration), Phys. Lett.

B248 (1990)211.94. R. Barbieri, F. Caravaglios, M. Frigeni and M. Mangano, same Proc.as ref.

[78], Vol.II, p. 658; Nucl. Phys.

B367 (1991) 28.42

Figure captionsFig.1: Contours of mmaxh(the maximum value of mh, reached for mA →∞)in the (mt, tan β) plane, for m˜q = 1 TeV.Fig.2: Contours of a) mh and b) mH, in the (mA, tan β) plane, for m˜q = 1 TeVand mt = 140 GeV.Fig.3: Schematic representation of the present LEP I limits and of the fu-ture LEP II sensitivity in the (mA, tan β) plane, for m˜q = 1 TeV andmt = 140 GeV. The solid lines correspond to the present LEP I limits.The dashed lines correspond to σ(e+e−→hZ, HZ, hA, HA) = 0.2 pbat √s = 175 GeV, which could be seen as a rather conservative esti-mate of the LEP II sensitivity.

The dashed-dotted lines correspond toσ(e+e−→hZ, HZ, hA, HA) = 0.05 pb at √s = 190 GeV, which couldbe seen as a rather optimistic estimate of the LEP II sensitivity.Fig.4: Pictorial representation of the future LHC sensitivity in the (mA, tan β)plane, for m˜q = 1 TeV and mt = 140 GeV.Fig.5: Contours of a) σ(e+e−→hZ) and b) σ(e+e−→HZ), in the (mA, tan β)plane, for √s = 500 GeV.Fig.6: Contours of a) σ(e+e−→hA) and b) σ(e+e−→HA), in the (mA, tan β)plane, for √s = 500 GeV.Fig.7: Contours of a) σ(e+e−→hνν) and b) σ(e+e−→Hνν), in the (mA, tan β)plane, for √s = 500 GeV.Fig.8: Contours of a) σ(e+e−→he+e−) and b) σ(e+e−→He+e−), via ZZ-fusion, in the (mA, tan β) plane, for √s = 500 GeV.Fig.9: Present limits and future sensitivity in the (m0, m1/2) plane, for the rep-resentative values a) tan β = 2, b) tan β = 10 and using µ-independentconstraints. The shaded area is excluded by the present data, whereasthe solid lines correspond to the estimated discovery potential of thecomplete LEP and Tevatron programs.Dashed lines correspond tofixed values of an ‘average’ squark mass, defined by the relation m˜q =qm20 + 5.5m21/2.

Dotted lines correspond to fixed values of the mass of43

the lightest charged slepton ( ˜ec), as given in the text. The values ofthe gluino mass as given by eq.

(16) are also shown.Fig.10: Present limits and future sensitivity in the (µ, m1/2) plane, for the repre-sentative values a) tan β = 2, b) tan β = 10 and using m0-independentconstraints. The shaded area is excluded by the present data, whereasthe solid lines correspond to the estimated discovery potential of thecomplete LEP and Tevatron programs.

Dashed and dotted lines cor-respond to fixed values of the lighest chargino and neutralino mass,respectively. The values of the gluino mass as given by eq.

(16) arealso shown.44

---

출처: arXiv:9203.204 • 원문 보기