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Testing the Higgs Sector of the

arXiv:hep-ph/9203223v1 29 Mar 1992CERN-TH.6150/91ETH-TH/91-7Testing the Higgs Sector of theMinimal Supersymmetric Standard Modelat Large Hadron CollidersZ. KunsztInstitute of Theoretical Physics, ETH,Zurich, SwitzerlandandF.

Zwirner1Theory Division, CERN,Geneva, SwitzerlandAbstractWe study the Higgs sector of the Minimal Supersymmetric StandardModel, in the context of proton-proton collisions at LHC and SSCenergies. We assume a relatively heavy supersymmetric particle spec-trum, and include recent results on one-loop radiative corrections toHiggs-boson masses and couplings.

We begin by discussing presentand future constraints from the LEP experiments. We then computebranching ratios and total widths for the neutral (h, H, A) and charged(H±) Higgs particles.

We present total cross-sections and event ratesfor the important discovery channels at the LHC and SSC. Promising1On leave from INFN, Sezione di Padova, Italy.

physics signatures are given by h →γγ, H →γγ or Z∗Z∗or τ +τ −,A →τ +τ −, and t →bH+ followed by H+ →τ +ντ, which shouldallow for an almost complete coverage of the parameter space of themodel.CERN-TH.6150/91ETH-TH/91-7December 19911

1IntroductionAll available experimental data in particle physics are consistent with theStandard Model (SM) of strong and electroweak interactions, provided [1]91 GeV < mt < 180 GeV(95% c.l. )(1)and57 GeV < mϕ(95% c.l.) ,(2)where mt and mϕ denote the masses of the top quark and of the SM Higgsboson, respectively.

The lower limits on mt and mϕ are obtained from un-successful direct searches at the Tevatron and LEP. The upper limit on mtis obtained as a consistency condition of the SM, after the inclusion of ra-diative corrections, with the high-precision data on electroweak phenomena.Strong evidence for the existence of the top quark, with the quantum num-bers predicted by the SM, is also provided by the precise measurements ofthe weak isospin of the b-quark.

In the case of the Higgs boson, the situationis radically different. There is no experimental evidence yet that the mini-mal SM Higgs mechanism is the correct description of electroweak symmetrybreaking.

Fortunately, present and future accelerators will give decisive con-tributions towards the experimental solution of this problem.If the SMdescription of the Higgs mechanism is correct, LEP or the LHC and SSCshould be able to find the SM Higgs boson and study its properties.Despite its remarkable successes, the SM can only be regarded as aneffective low-energy theory, valid up to some energy scale Λ at which it isreplaced by some more fundamental theory.Certainly Λ is less than thePlanck scale, MP ∼1019 GeV, since one needs a theory of quantum gravityto describe physics at these energies. However, there are also arguments,originating precisely from the study of the untested Higgs sector 2, whichsuggest that Λ should rather be close to the Fermi scale G−1/2F∼300 GeV.The essence of these arguments is the following.

Triviality of the λϕ4 theory,absence of Landau poles and perturbative unitarity imply that within theSM mϕ < 600–800 GeV. If one then tries to extend the validity of the SMto energy scales Λ ≫G−1/2F, one is faced with the fact that in the SM thereis no symmetry to justify the smallness of the Higgs mass with respect to2For reviews of Higgs boson physics see, e.g., refs.

[2,3]2

the (physical) cut-offΛ. This is apparent from the fact that in the SM one-loop radiative corrections to the Higgs mass are quadratically divergent; itis known as the naturalness (or hierarchy) problem of the SM.

Motivated bythis problem, much theoretical effort has been devoted to finding descriptionsof electroweak symmetry breaking which modify the SM at scales Λ ∼G−1/2F.The likely possibility of such modifications is the reason why, when discussingthe experimental study of electroweak symmetry breaking, one should not beconfined to the SM Higgs, but also consider alternatives to it, which mighthave radically different signatures, and in some cases be more difficult todetect than the SM Higgs. Only after a thorough study of these alternativescan one be definite about the validity of the so-called ‘no-lose theorems’,stating that the physics signatures of electroweak symmetry breaking cannotbe missed at LEP or the LHC and SSC.When considering alternatives to the minimal SM Higgs sector, it is natu-ral to concentrate on models which are theoretically motivated, phenomeno-logically acceptable and calculationally well-defined.The most attractivepossibility satisfying these criteria is the Minimal Supersymmetric StandardModel (MSSM) [4].

This possibility is theoretically motivated by the factthat low-energy supersymmetry, effectively broken in the vicinity of the elec-troweak scale, is the only theoretical framework that can naturally accommo-date elementary Higgs bosons. The simplest and most predictive realizationof low-energy supersymmetry is the MSSM, defined by 1) minimal gaugegroup: SU(3)C × SU(2)L × U(1)Y ; 2) minimal particle content: three gener-ations of quarks and leptons and two Higgs doublets, plus their superpartners;3) an exact discrete R-parity, which guarantees (perturbative) baryon- andlepton-number conservation: R = +1 for SM particles and Higgs bosons,R = −1 for their superpartners; 4) supersymmetry breaking parametrizedby explicit but soft breaking terms: gaugino and scalar masses and trilinearscalar couplings.Besides the solution of the naturalness problem, there are other virtuesof the MSSM which are not shared by many other alternatives to the SMHiggs and should also be recalled to further motivate our study.

The MSSMsuccessfully survives all the stringent phenomenological tests coming fromprecision measurements at LEP: in most of its parameter space, the MSSMpredictions for the LEP observables are extremely close to the SM predic-tions, evaluated for a relatively light SM Higgs [5]. This can be compared, forexample, with the simplest technicolor models, which are ruled out by the re-3

cent LEP data [6]. Again in contrast with models of dynamical electroweaksymmetry breaking, the MSSM has a high degree of predictivity, since allmasses and couplings of the Higgs boson sector can be computed, at thetree-level, in terms of only two parameters, and radiative corrections can bekept under control: in particular, cross-sections and branching ratios for theMSSM Higgs bosons can be reliably computed in perturbation theory.

Fur-thermore, it is intriguing that the idea of grand unification, which fails in itsminimal non-supersymmetric implementation, can be successfully combinedwith that of low-energy supersymmetry: minimal supersymmetric grand uni-fication predicts a value of sin2 θW(mZ) which is in good agreement with themeasured one, and a value of the grand-unification mass which could explainwhy proton decay has escaped detection so far [7]. Finally, as a consequenceof R-parity, the lightest supersymmetric particle, which is typically neutraland weakly interacting, is absolutely stable, and thus a natural candidate fordark matter.Any consistent supersymmetric extension of the SM requires at least twoHiggs doublets, in order to give masses to all charged quarks and leptons andto avoid gauge anomalies originated by the spin-1/2 higgsinos.

The MSSMhas just two complex Higgs doublets, with the following SU(3)C ×SU(2)L ×U(1)Y quantum numbers (Q = T3L + Y ):H1 ≡ H01H−1∼(1, 2, −1/2) ,H2 ≡ H+2H02∼(1, 2, +1/2) . (3)Other non-minimal models can be constructed, but they typically increasethe number of parameters without correspondingly increasing the physicalmotivation.

For example, the simplest non-minimal model [8] is constructedby adding a singlet Higgs field N and by requiring purely trilinear super-potential couplings. In this model, the Higgs sector has already two moreparameters than in the MSSM.

Folklore arguments in favour of this model arethat it avoids the introduction of a supersymmetry-preserving mass parame-ter µ ∼G−1/2Fand that the homogeneity properties of its superpotential recallthe structure of some superstring effective theories. A closer look, however,shows that these statements should be taken with a grain of salt.

First, inthe low-energy effective theory with softly broken global supersymmetry, thesupersymmetric mass µ ∼G−1/2Fcould well be a remnant of local supersym-metry breaking, if the underlying supergravity theory has a suitable structureof interactions [9]. Moreover, when embedded in a grand-unified theory, the4

non-minimal model with a singlet Higgs field might develop dangerous insta-bilities [10]. Also, the trilinear N3 superpotential coupling, which is usuallyinvoked to avoid a massless axion, is typically absent in string models.

Wetherefore concentrate in this paper on the MSSM only.The previous considerations should have convinced the reader that theHiggs sector of the MSSM is worth a systematic study in view of the forth-coming hadron colliders, the LHC and SSC. To perform such a study, one hasto deal with the rich particle spectrum of the MSSM.

As discussed in moredetail later, the Higgs sector contains one charged (H±) and three neutral(h, H, A) physical states. At the classical level, all Higgs boson masses andcouplings can be expressed in terms of two parameters only, for example mAand tan β ≡v2/v1.

This makes the discussion more complicated than in theSM, where the only free parameter in the Higgs sector is the Higgs mass,mϕ. In addition, when considering production and decay of Higgs bosons,the whole particle spectrum of the model has to be taken into account.

Asin the SM, the top-quark mass mt is an important parameter: barring thefine-tuned cases of a very light stop squark, or of charginos very close in massto mZ/2, the limits of eq. (1) are also valid in the MSSM [5].

In contrastwith the SM, also the supersymmetric R-odd particles (squarks, sleptons,gauginos, higgsinos) can play an important role in the production and decayof supersymmetric Higgs bosons [11]. Clearly, to keep track simultaneouslyof all supersymmetric-particle masses would be a difficult (and confusing)task.

We shall therefore concentrate, following the approach of ref. [12], onthe limiting case where all supersymmetric-particle masses are heavy enoughnot to play an important role in the phenomenology of supersymmetric Higgsbosons.

This is phenomenologically meaningful, since one can argue that arelatively light supersymmetric-particle spectrum is likely to give indepen-dent, detectable signatures at LEP or at the LHC and SSC.Another motivation for the present study is the recent realization [13] thattree-level formulae for Higgs-boson masses and couplings can receive largeradiative corrections, dominated by the exchange of virtual top and bottomquarks and squarks in loop diagrams. For example, tree-level formulae wouldpredict the existence of a neutral Higgs boson (h) lighter than the Z.

If thiswere true, there would be a chance of testing completely the MSSM Higgssector at LEP II, with no need for the LHC and SSC. However, mh can receivea large positive shift by radiative corrections, which can push h beyond theLEP II discovery reach.

This makes the LHC and SSC important, not only5

for a possible confirmation of a SUSY Higgs signal seen at LEP, but also forthe exploration of the parameter space inaccessible to LEP.The phenomenology of the SM Higgs at the LHC [14–16] and SSC [17,18]has been intensely studied over the last years: a lot of effort was requiredto prove [14,15], at least on paper, that the combination of LEP and theLHC/SSC is sufficient to explore the full theoretically allowed range of SMHiggs masses. However, those results cannot be directly applied to the neu-tral states of the MSSM, since there are important differences in the cou-plings, and of course one needs to analyse separately the case of the chargedHiggs.

Even in the case in which all the R-odd supersymmetric particles arevery heavy, the Higgs sector of the MSSM represents a non-trivial extensionof the SM case. Also several studies of the MSSM Higgs sector have alreadyappeared in the literature.

In particular, tree-level formulae for the MSSMHiggs boson masses and couplings are available, and they have already beenused to compute cross-sections and branching ratios for representative valuesof the MSSM parameters [3]. However, the existing analyses are not system-atic enough to allow for a definite conclusion concerning the discovery poten-tial of the LHC and SSC, even in the simple case of large sparticle masses.Also, they do not include radiative corrections to Higgs-boson masses andcouplings.

In this paper we plan to help filling these two gaps. The strategyfor a systematic study of neutral supersymmetric Higgs bosons at the LHCwas outlined in ref.

[12]: however, at that time radiative corrections werenot available, and also the γγ branching ratio was incorrectly encoded in thecomputer program. Our goal will be to see if LEP and the LHC/SSC can besensitive to supersymmetric Higgs bosons in the whole (mA, tan β) space.The structure of the paper is the following.In sect.

2 we review thetheoretical structure of the Higgs sector of the MSSM, including radiativelycorrected formulae for Higgs-boson masses and couplings. In sect.

3 we surveythe present LEP I limits, after the inclusion of radiative corrections, and theplausible sensitivity of LEP II. In sect.

4 we present branching ratios andwidths of neutral and charged supersymmetric Higgs bosons. In sect.

5 wecompute the relevant cross-sections at the LHC and SSC, and in sect. 6 weexamine in some detail the most promising signals for discovery.

Finally,sect. 7 contains a concluding discussion of our results and of prospects forfurther work.6

2Higgs masses and couplings in the MSSMFor a discussion of Higgs-boson masses and couplings in the MSSM, theobvious starting point is the tree-level Higgs potential [4]V0=m21 |H1|2 + m22 |H2|2 + m23 (H1H2 + h.c.)+18g2 H†2⃗σH2 + H†1⃗σH12 + 18g′2 |H2|2 −|H1|22 ,(4)where m21, m22, m23 are essentially arbitrary mass parameters, g and g′ arethe SU(2) and U(1) coupling constants, respectively, and ⃗σ are the Paulimatrices. SU(2) indices are left implicit and contracted in the obvious way.It is not restrictive to choose m23 real and negative, and then the vacuumexpectation values v1 ≡⟨H01⟩and v2 ≡⟨H02⟩real and positive.The physical states of the MSSM Higgs sector are three neutral bosons(two CP-even, h and H, and one CP-odd, A) and a charged boson, H±.A physical constraint comes from the fact that the combination (v21 + v22),which determines the W and Z boson masses, must reproduce their mea-sured values.

Once this constraint is imposed, in the Born approximationthe MSSM Higgs sector contains only two independent parameters. A con-venient choice, which will be adopted throughout this paper, is to take asindependent parameters mA, the physical mass of the CP-odd neutral bo-son, and tan β ≡v2/v1, where v1 gives mass to charged leptons and quarksof charge −1/3, v2 gives mass to quarks of charge 2/3.

The parameter mAis essentially unconstrained, even if naturalness arguments suggest that itshould be smaller than O(500 GeV), whereas for tan β the range permittedby model calculations is 1 ≤tan β <∼mtmb.At the classical level, the mass matrix of neutral CP-even Higgs bosonsreadsM0R2 =" cot β−1−1tan β m2Z2 + tan β−1−1cot β m2A2#sin 2β(5)and the charged-Higgs mass is given bym2H± = m2W + m2A. (6)From eq.

(5), one obtainsm2h,H = 12m2A + m2Z ∓q(m2A + m2Z)2 −4m2Am2Z cos2 2β,(7)7

dd, ss, bbe+e−, µ+µ−, τ +τ −uu, cc, ttW +W −, ZZh−sin α/ cos βcos α/ sin βsin (β −α)Hcos α/ cos βsin α/ sin βcos (β −α)A−iγ5 tan β−iγ5 cot β0Table 1: Correction factors for the couplings of the MSSM neutral Higgsbosons to fermion and vector boson pairs.and also celebrated inequalities such as mW, mA < mH±, mh < mZ < mH,mh < mA < mH.Similarly, one can easily compute all the Higgs-bosoncouplings by observing that the mixing angle α, required to diagonalize themass matrix (5), is given bycos 2α = −cos 2β m2A −m2Zm2H −m2h,−π2 < α ≤0. (8)For example, the couplings of the three neutral Higgs bosons are easily ob-tained from the SM Higgs couplings if one multiplies them by the α- andβ-dependent factors summarized in table 1.

The remaining tree-level Higgs-boson couplings in the MSSM can be easily computed and are summarized,for example, in ref. [3].

An important consequence of the structure of theclassical Higgs potential of eq. (4) is the existence of at least one neutralCP-even Higgs boson, weighing less than or about mZ and with approxi-mately standard couplings to the Z.

This raised the hope that the crucialexperiment on the MSSM Higgs sector could be entirely performed at LEPII (with sufficient centre-of-mass energy, luminosity and b-tagging efficiency),and took some interest away from the large hadron collider environment.8

However, it was recently pointed out [13] that the masses of the Higgsbosons in the MSSM are subject to large radiative corrections, associatedwith the top quark and its SU(2) and supersymmetric partners3. Severalpapers [20–23] have subsequently investigated various aspects of these cor-rections and their implications for experimental searches at LEP.

In the restof this section, we shall summarize and illustrate the main effects of radiativecorrections on Higgs-boson parameters.As far as Higgs-boson masses and self-couplings are concerned, a conve-nient approximate way of parametrizing one-loop radiative corrections is tosubstitute the tree-level Higgs potential of eq. (4) with the one-loop effectivepotential, and to identify Higgs-boson masses and self-couplings with theappropriate combinations of derivatives of the effective potential, evaluatedat the minimum.

The comparison with explicit diagrammatic calculationsshows that the effective potential approximation is more than adequate forour purposes. Also, inspection shows that the most important correctionsare due to loops of top and bottom quarks and squarks.

At the minimum⟨H01⟩= v1, ⟨H02⟩= v2, ⟨H−1 ⟩= ⟨H+2 ⟩= 0, and neglecting intergenerationalmixing, one obtains for the top and bottom quark and squark masses thefamiliar expressionsm2t = h2tv22 ,m2b = h2bv21 ,(9)m2˜t1,2=m2t + 12(m2Q + m2U) + 14m2Z cos 2β±s12(m2Q −m2U) + 112(8m2W −5m2Z) cos 2β2+ m2t (At + µ cot β)2,(10)m2˜b1,2=m2b + 12(m2Q + m2D) −14m2Z cos 2β±s12(m2Q −m2D) −112(4m2W −m2Z) cos 2β2+ m2b (Ab + µ tan β)2. (11)3Previous studies [19] either neglected the case of a heavy top quark, or concentratedon the violations of the neutral-Higgs mass sum rule without computing corrections toindividual Higgs masses.9

In eqs. (9) to (11), ht and hb are the top and bottom Yukawa couplings,and mQ, mU, mD are soft supersymmetry-breaking squark masses.

The pa-rameters At, Ab and µ, which determine the amount of mixing in the stopand sbottom mass matrices, are defined by the trilinear potential termshtAt(˜tL ˜tcLH02 −˜bL ˜tcLH+2 ) + h.c., hbAb(˜bL ˜bcLH01 −˜tL ˜bcLH−1 ) + h.c. and bythe superpotential mass term µ(H01H02 −H−1 H+2 ), respectively.To simplify the discussion, in the following we will take a universal softsupersymmetry-breaking squark mass,m2Q = m2U = m2D ≡m2˜q ,(12)and we will assume negligible mixing in the stop and sbottom mass matrices,At = Ab = µ = 0 . (13)Formulae valid for arbitrary values of the parameters can be found in refs.

[22,23], but the qualitative features corresponding to the parameter choiceof eqs. (12) and (13) are representative of a very large region of parameterspace.

In the case under consideration, and neglecting D-term contributionsto the field-dependent stop and sbottom masses, the neutral CP-even massmatrix is modified at one loop as followsM2R =M0R2 + ∆2100∆22,(14)where∆21 =3g2m4b16π2m2W cos2 β logm2˜b1m2˜b2m4b,(15)∆22 =3g2m4t16π2m2W sin2 β logm2˜t1m2˜t2m4t. (16)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 (14).

The one-loop-corrected charged Higgsmass is given instead bym2H± = m2W + m2A + ∆2,(17)10

where, including D-term contributions to stop and sbottom masses,∆2=3g264π2 sin2 β cos2 βm2W×(m2b −m2W cos2 β)(m2t −m2W sin2 β)m2˜t1 −m2˜b1hf(m2˜t1) −f(m2˜b1)i+m2tm2bm2˜t2 −m2˜b2hf(m2˜t2) −f(m2˜b2)i−2m2tm2bm2t −m2bhf(m2t) −f(m2b)i(18)andf(m2) = 2m2 log m2Q2 −1!. (19)The most striking fact in eqs.

(14)–(19) is that the correction ∆22 is propor-tional to (m4t/m2W). This implies that, for mt in the range of eq.

(1), thetree-level predictions for mh and mH can be badly violated, and so for therelated inequalities. The other free parameter is m˜q, but the dependence onit is much milder.

To illustrate the impact of these results, we display in fig. 1contours of the maximum allowed value of mh (reached for mA →∞), in the(mt, tan β) and (mt, m˜q) planes, fixing m˜q = 1 TeV and tan β = mt/mb, re-spectively.

In the following, when making numerical examples we shall alwayschoose the representative value m˜q = 1 TeV. To plot different quantities ofphysical interest in the (mA, tan β) plane, which is going to be the stage ofthe following phenomenological discussion, one needs to fix also the value ofmt.

In this paper, whenever an illustration of the mt dependence is needed,we work with the two representative values mt = 120, 160 GeV, which aresignificantly different but well within the range of eq. (1).

Otherwise, wework with the single representative value mt = 140 GeV. As an example, weshow in figs.

2–4 contours of constant mh, mH, and mH± in the (mA, tan β)plane. Here and in the following we vary mA and tan β in the ranges0 ≤mA ≤500 GeV,1 ≤tan β ≤50 .

(20)The effective-potential method allows us to compute also the leading cor-rections to the trilinear and quadrilinear Higgs self-couplings.A detaileddiscussion and the full diagrammatic calculation will be given elsewhere.11

Here we just give the form of the leading radiative corrections to the trilin-ear hAA, HAA, and Hhh couplings, which will play an important role inthe subsequent discussion of Higgs-boson branching ratios. One finds [23,24]λhAA = λ0hAA+∆λhAA ,λHAA = λ0HAA+∆λHAA ,λHhh = λ0Hhh+∆λHhh ,(21)whereλ0hAA = −igmZ2 cos θWcos 2β sin(β + α) ,(22)λ0HAA =igmZ2 cos θWcos 2β cos(β + α) ,(23)λ0Hhh = −igmZ2 cos θW[2 sin(β + α) sin 2α −cos(β + α) cos 2α] ,(24)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,(25)∆λHAA = −igmZ2 cos θW3g2 cos2 θW8π2sin α cos2 βsin3 βm4tm4Wlog m2˜q + m2tm2t,(26)∆λHhh = −igmZ2 cos θW3g2 cos2 θW8π2cos2 α sin αsin3 βm4tm4W 3 log m2˜q + m2tm2t−2m2˜qm2˜q + m2t!.

(27)Notice that, besides the obvious explicit dependence, in eqs. (21)–(27) there isalso an important implicit dependence on mt and m˜q, via the angle α, whichis determined from the mass matrix of eqs.

(14)–(16). We also emphasize thatneglecting the D-terms in the stop and sbottom mass matrices is guaranteedto give accurate results only for mt ≫mZ.

For mt ∼mZ, one should makesure that the inclusion of D-terms does not produce significant modificationsof our results. In the case of the h and H masses, and of the mixing angleα, complete formulae are available, and this check can be easily performed.In the case of the hAA, HAA and Hhh couplings, complete formulae arenot yet available.

For the phenomenologically most important coupling atthe LHC and SSC, λHhh, we have explicitly checked that the inclusion ofD-terms does not produce important modifications of our results.12

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 radiativecorrections is taken into account by using one-loop-corrected formulae to de-termine α from the input parameters.

Other corrections are at most of orderg2m2t/m2W and can be safely neglected for our purposes. Tree-level couplingsto fermions are expressed in terms of the fermion masses and of the anglesβ and α.

In this case, the leading radiative corrections can be taken into ac-count by using the one-loop-corrected expression for α and running fermionmasses, evaluated at the scale Q which characterizes the process under con-sideration.This brings us to the discussion of the renormalization groupevolution of the top and bottom Yukawa couplings in the MSSM. As bound-ary conditions, we assume as usual that mt(mt) = mt and mb(mb) = mb, withmb = 4.8 GeV and mt numerical input parameters.

As stated in the Intro-duction, we assume in this paper that all supersymmetric particles are heavy.Then, since we want to compute Higgs-boson production cross-sections andbranching ratios, we are interested in the standard renormalization groupevolution of ht(Q) [hb(Q)] from Q = mt [Q = mb] to Q ≃mH±, mH, whichis dominated by gluon loops.To illustrate the behaviour of the Higgs couplings to vector bosons andfermions, as functions of the input parameters, we show in figs. 5–7 contoursin the (mA, tan β) plane of some of the correction factors appearing in table 1.3LEP limits and implicationsIn this section, we briefly summarize the implications of the previous resultson MSSM Higgs boson searches at LEP I and LEP II.

Partial results werealready presented in refs. [21,22].As already clear from tree-level analyses, the relevant processes for MSSMHiggs 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 [23] is toallow, for some values of the parameters, the decay h →AA, which would bekinematically forbidden according to tree-level formulae. Experimental limitswhich take radiative corrections into account have by now been obtained bythe four LEP collaborations [25], using different methods to present and13

analyse the data, and different ranges of parameters in the evaluation ofradiative corrections. A schematic representation of the presently excludedregion of the (mA, tan β) plane, for the standard parameter choices discussedin sect.

2, is given in fig. 8, where the solid lines correspond to our na¨ıve4extrapolation of the exclusion contour given in the first of refs.

[25]. Fora discussion of the precise experimental bounds, we refer the reader to theabove-mentioned experimental publications.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 discov-ery potential can be made only if crucial theoretical parameters, such as thetop-quark mass and the various soft supersymmetry-breaking masses, and ex-perimental 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 kine-matical limit. Associated hA production could be a useful complementarysignal, but obviously only for mh + mA < √s.

Associated HA productionis typically negligible at these energies. To give a measure of the LEP IIsensitivity, we plot in fig.

8 contours associated with two benchmark val-ues of the total cross-section σ(e+e−→hZ∗, HZ∗, hA, HA). The dashedlines correspond to σ = 0.2 pb at √s = 175 GeV, which could be seen asa rather conservative estimate of the LEP II sensitivity.

The dash-dottedlines correspond to σ = 0.05 pb at √s = 190 GeV, which could be seen asa rather optimistic estimate of the LEP II sensitivity. In computing thesecross-sections, we have taken into account the finite Z width, but we have4 We fitted the experimental exclusion contours, corresponding to mt = 140 GeVand the other parameters as chosen here, with two numerical values for Γ(Z →hZ∗)and Γ(Z →hA).

We have then computed radiative corrections for the two values of mtconsidered here, assuming that the variations in experimental efficiencies are small enoughnot to affect our results significantly.14

neglected initial state radiation, which leads to suppression near threshold.A more accurate estimate of the LEP II sensitivity can be found in ref. [26].Of course, one should keep in mind that there is, at least in principle, thepossibility of further extending the maximum LEP energy up to values ashigh as √s ≃230–240 GeV, at the price of introducing more (and moreperforming) superconducting cavities into the LEP tunnel [27].In summary, a significant region of the parameter space for MSSM Higgsescould be beyond the reach of LEP II, at least if one sticks to the referencecentre-of-mass energy √s <∼190 GeV.

The precise knowledge of this regionis certainly important for assessing the combined discovery potential of LEPand LHC/SSC, but it does not affect the motivations and the techniques ofour study, devoted to LHC and SSC searches. Whether or not a Higgs bosonwill be found at LEP in the future, we want to investigate the possibilities ofsearching for all the Higgs states of the MSSM at large hadron colliders, inthe whole region of parameter space which is not already excluded at present.Even if a neutral Higgs boson is found at LEP, with properties compatiblewith the SM Higgs boson within the experimental errors, it will be impossibleto exclude that it belongs to the MSSM sector.

The LHC and SSC could thenplay a role in investigating its properties and in looking for the remainingstates of the MSSM.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 [23,20] that for generic values of the parametersthere are no large negative radiative corrections to the charged-Higgs massformula, eq. (6).

A comparison of figs. 4 and 8 indicates that there is verylittle hope of finding the charged Higgs boson of the MSSM at LEP II (or,stated differently, the discovery of a charged Higgs boson at LEP II wouldmost probably rule out the MSSM).4Branching ratios15

4.1Neutral Higgs bosonsThe branching ratios of the neutral Higgs bosons of the MSSM were sys-tematically studied in ref. [12], using the tree-level formulae for masses andcouplings available at that time5 (previous work on the subject is summa-rized in ref.

[3]). Here we present a systematic study which includes theradiative corrections described in sect.

2. As usually done for the SM Higgsboson, we consider the two-body decay channelsh, H, A −→cc, bb, tt, τ +τ −, gg, γγ, W ∗W ∗, Z∗Z∗, Zγ .

(28)For consistency, we must also consider decays with one or two Higgs bosonsin the final stateh →AA,H →hh, AA, ZA,A →Zh. (29)On the other hand, we neglect here possible decays of MSSM Higgs bosonsinto supersymmetric particles: as previously stated, we consistently assumea heavy spectrum of R-odd particles, so that only R-even ones can be kine-matically accessible in the decays of h, H, and A.

We perform our study inthe framework of MSSM parameter space, with the representative parameterchoices illustrated in sect. 2.

The effects of changing the mass of the topquark, and the sensitivity to squark masses in the high-mass region, will alsobe briefly discussed.The partial widths for the decays of eq. (28) that correspond to tree-level diagrams can be obtained from the corresponding formulae for the SMHiggs boson (for a summary, see ref.

[3]), by simply multiplying the vari-ous amplitudes by the supersymmetric correction factors listed in table 1.For decays that are described by loop diagrams, however, in the MSSMone has to include some contributions that are absent in the SM. Diagramscorresponding to the exchange of R-odd supersymmetric particles give negli-gible contributions to the corresponding partial widths, in the limit of heavysupersymmetric-particle masses that we have chosen for our analysis (in ac-cordance with intuitive ideas about decoupling).

One must also include thecharged-Higgs loop contributions to the γγ and Zγ final states. When con-sidering instead the processes of eq.

(29), we improve the tree-level formulae5Also, the partial widths for the decays h, H, A →γγ were affected by numerical errors.16

of ref. [3] not only with the self-energy corrections to the mixing angle α,but also with the vertex corrections of eqs.

(21)–(27).QCD [28] and electroweak [29] radiative corrections to the fermion-anti-fermion and the WW, ZZ channels have been recently computed for theSM Higgs boson, ϕ. They have been found to be small (less than ∼20%),with the exception of the QCD corrections to the decays into charm andbottom quark pairs, which are large because of running-quark-mass effects.We then included the QCD corrections as described in ref.

[14]. One mayalso wonder whether running-mass effects induced by the large top Yukawacoupling could give further important effects.

However, one can easily seethat these effects give corrections which are certainly less than 20%.The QCD correction to ϕ →γγ is also available, and known to be negli-gibly small [30]. Sizeable QCD corrections are found, however, for the decayϕ →gg [31].

Although this effect is not important for the branching ratiostudy, since ϕ →gg is neither the dominant decay mode nor a useful channelfor detection, it still has to be included in the production cross-section of hvia the two-gluon fusion mechanism.Another general and well-known property of the MSSM is that the self-interactions of the Higgs bosons are controlled, modulo the logarithmic cor-rections discussed in sect. 2, by the SU(2) and U(1) gauge couplings.

There-fore, the total widths of all MSSM Higgs bosons, displayed in fig. 9, staybelow 10 GeV in the whole parameter space we have considered.The most important branching ratios for the neutral MSSM Higgs bosonsare shown, as a function of the mass of the decaying particle, in figs.

10–12.To avoid excessive proliferation of figures, we consider the two representativevaluestan β = 1.5, 30,(30)and for each of these we vary mA between the experimental lower bound offig. 8 (mA ≃59 GeV for tan β = 1.5, mA ≃44 GeV for tan β = 30) and theupper bound of eq.

(20), assuming mt = 140 GeV and m˜q = 1 TeV.We consider first the branching ratios of h (fig. 10).

We can clearly seethe effect of radiative corrections on the allowed range of mh for the givenvalues of tan β. For mA <∼25 GeV, the decay h →AA can be kinematicallyallowed and even become the dominant mode.

This decay channel was im-portant at LEP I, but since the corresponding region of parameter space isalready excluded by experiment, this decay mode does not appear in fig. 10.17

The dominant decay mode is then h →b¯b, whereas the τ +τ −mode has abranching ratio of about 8% throughout the relevant part of the parameterspace. In fig.

10, one immediately notices the rather steep slopes for thec¯c and γγ branching ratios plotted versus mh, with larger effects for largervalues of tan β: their origin can be understood by looking at figs. 2 and 5–7,which show how mh and the h couplings to heavy fermions and vector bosonsvary in the (mA, tan β) plane.If the SM Higgs boson is in the intermediate mass region, mϕ = 70–140GeV, at large hadron colliders a measurable signal can be obtained via the γγmode.

Since the mass of the light Higgs h is indeed below or inside this region,the γγ mode is also crucial for the MSSM Higgs search. Furthermore, theγγ branching ratio as a function of the Higgs mass exhibits a rather peculiarbehaviour, not only for h but also for H and A, so a more detailed discussionis in order.

The partial width is given byΓ(φ →γγ) =α2g21024π3m3φm2WXiIφi (τ φi )2,τ φi = 4m2im2φ,(31)where φ = h, H, A and i = f, W, H, ˜f, ˜χ indicates the contributions fromordinary fermions, charged gauge bosons, charged Higgs bosons, sfermionsand charginos, respectively. The functions Iiφ(τ iφ) are given byIφf=F φ1/2(τ φf )Ncfe2f Rφf ,IφW=F1(τ φW)RφW ,IφH=F0(τ φH)RφHm2Wm2φ,Iφ˜f=F0(τ φ˜f )Ncfe2fRφ˜fm2Zm2˜f,Iφ˜χ=F φ1/2(τ φχ)Rφ˜χmWm˜χ,(32)where Ncf is 1 for (s)leptons and 3 for (s)quarks, and the subscripts of thecomplex functions F S1/2(τ), F P1/2(τ), F0(τ), and F1(τ), which were calculatedin ref.

[32], indicate the spin of the particles running in the loop. In the caseof spin-1/2 particles, the contribution is different for CP-even and CP-oddneutral Higgses.

The symbols Rφi denote the appropriate correction factors18

for the MSSM Higgs couplings: for i = f, W they are given in table 1, fori = H, ˜f, ˜χ they can be found, for example, in Appendix C of ref. [3].

The Wcontribution dominates the h →γγ decay rate. The function F1 is large atand above τ = 1.

For the W contribution τ = 4m2W/m2h > 1, and increasingmh gives increasing values of F1. The steep dependence of the branching ratioon mh is a consequence of the fast change of sin2(β −α) as mA is increasedfor fixed tan β.

This is further enhanced by the fact that the large interval100 GeV <∼mA ≤500 GeV is mapped into a very small interval (a few GeV)in mh. We elucidate this effect by plotting in fig.

13 the branching ratios ofh as a function of mA, for the same values of the parameters as in fig. 10.We can see that the tip of the gg, cc and γγ curves in fig.

10 is mapped intoa long plateau in fig. 13.

We can also observe that in a large region of theparameter space the h →γγ branching ratio has a value somewhat smallerthan (but comparable to) the corresponding branching ratio for a SM Higgsof mass mh. This is due to the fact that all the h couplings tend to the SMHiggs couplings for mA ≫mZ; however, for the h couplings to fermions theapproach to the asymptotic value is much slower than for the h couplingsto vector bosons, as can be seen from figs.

5 to 7. In fig.

13, the branchingratios for the W ∗W ∗and Z∗Z∗decays are also plotted, whereas they wereomitted in fig. 10 in order to avoid excessive crowding of curves.

However,for our parameter choice they have little interest at large hadron colliders,because of the small production rates and the large backgrounds.The branching ratios of the heavy Higgs boson H, depicted in fig. 11,have a rather complicated structure.

We make here four remarks.i) The γγ mode has a steeply decreasing branching ratio with increasingmH, except at small values of tan β and at the lower kinematical limit ofmH, where one or more of the AA, ZA and hh decay channels are open.The steep fall of the γγ branching ratio at large values of tan β can beeasily understood. The partial width Γ(H →γγ) is dominated by the Wcontribution, proportional to cos2(β −α).

As we can see in fig. 7, cos2(β −α) decreases very fast, for increasing mH, at fixed values of tan β.Thissteep decrease is slightly compensated by the increase of F1(τW) at τW ≤1,which has a peak at the W threshold mH = 2mW.

Another peak in the γγbranching ratio is obtained, for small values of tan β, at mH = 2mt, wherethe top-quark loop gives the dominant contribution.ii) The complicated structure in the H branching ratio curves is mainlydue to the H →hh channel.For mH < 2mt, and not too high values19

of tan β, this decay mode is dominant whenever kinematics allows.Thischannel is always open at the lower kinematical limit of MH. Increasing MHa little bit, however, it may become strongly suppressed, because for smallincreasing values of mA the value of mh rises faster than that of mH, so thatthe channel can become kinematically closed.

Obviously, for sufficiently highvalues of mH the channel is always open. At high values of tan β, the massregion at the lower kinematical limit where H →hh is open becomes smallerand smaller, explaining the presence of the almost vertical line in fig.

11. Afurther structure is present in this decay channel due to the coupling factorλHhh [see eqs.

(24) and (27)]. There are relatively small values of mH atwhich λHhh accidentally vanishes.

Furthermore, for very large values of mHand tan β one has α ≃0, β ≃π/2, and therefore λHhh ≃0. Unfortunately,even when it is dominant, this mode has very large backgrounds, so it seemsunlikely to give a measurable signal at large hadron colliders.

The H →AAmode is kinematically allowed only for values of mA below 50–60 GeV, inwhich case it can have a large branching ratio, competing with the one forH →hh. The H →ZA mode is kinematically allowed only in the region ofparameter space which is already excluded by the LEP I data.iii) H can decay at tree level into ZZ →l+l−l+l−, which is the ‘gold-plated’ signature for the SM Higgs boson.

Unfortunately, in the case of Hthe branching ratio is smaller, and it decreases fast with increasing tan βand/or mA. For small tan β and 2mh < mH < 2mt, this mode is suppressedby the competition with H →hh, and this effect is further enhanced bythe inclusion of the radiative correction of eq.

(27), which typically gives anadditional 50% suppression. Nevertheless, as we shall see in the next section,the four-lepton channel can give a measurable signal in some small region ofthe parameter space.iv) The decay into t¯t is dominant above threshold at moderate values oftan β.

But above tan β ∼8 or so the b¯b mode remains dominant and τ +τ −has the typical ∼10% branching ratio.Finally, we discuss the branching ratios of A, shown in fig. 12.

The γγ oneis always small, although at small tan β and slightly below the top threshold,mA ∼2mt, it reaches a value ∼8×10−4, which may give a measurable signalin a small island of the parameter space. The behaviour of the A →γγbranching ratio can be easily understood by taking into account that thepartial width is dominated by the top loop contribution.

Two features areimportant here. First, the function F1/2(τ) appearing in eq.

(32) has a strong20

enhancement at τ ∼1. Furthermore, the t¯tA coupling gives a suppressionfactor 1/(tan β)2 for increasing values of tan β.At smaller values of mA and tan β, there is a substantial branching ratioto Zh, which however does not look particularly promising for detection atlarge hadron colliders, because of the very large Zb¯b background.

We can seethat all the dominant decay modes of the A boson correspond to channelswhich are overwhelmed by very large background, except perhaps the τ +τ −mode, which, as we shall see in the next section, may give a detectable signalfor very high values of tan β.We have studied the neutral Higgs branching ratios also at mt = 120, 160, 180 GeV.Increasing the top mass has two major effects. First, the maximum value ofmh increases (see fig.

1).Next, owing to the increased value of the topthreshold, the structure generated by the opening of the top decay channelis shifted to higher mass values. We also note that varying m˜q in the range0.5–2 TeV has negligible effects on the branching ratio curves of figs.

10–12.Finally, if one chooses mt and m˜q so large that mh > 130 GeV, the W ∗W ∗and Z∗Z∗branching ratios can become relevant also for h.4.2Charged Higgs bosonIn the case of the charged Higgs boson, we considered only the two-bodydecay channelsH+ →cs , τ +ντ , tb , W +h . (33)Tree-level formulae for the corresponding decay rates can be found, for in-stance, in ref.

[3]. Loop-induced decays such as H+ →W +γ, W +Z have verysmall branching ratios [33] and are not relevant for experimental searches atthe LHC and SSC.

Radiative corrections to the charged-Higgs-boson massformula were included according to eqs. (17) and (18).

The H+W −h cou-pling, proportional to cos(β −α), was evaluated with the one-loop correctedvalue of α. The leading QCD corrections to the H+bt and H+sc vertices wereparametrized, following refs.

[34], by running quark masses evaluated at ascale Q ∼mH±. The resulting branching ratios for the charged Higgs bosonare displayed in fig.

14, for tan β = 1.5, 30 and the standard parameter choicemt = 140 GeV, m˜q = 1 TeV. One can see that the dominant factor affectingthe branching ratios is the mH± = mt + mb threshold.

Above threshold, thetb mode is dominant for any value of tan β within the bounds of eq. (20).21

Below threshold, the dominant mode is τ +ντ, with the competing mode csbecoming more suppressed for higher values of tan β. For small values oftan β, the W +h decay mode can also be important, and even dominate, ina limited mH± interval, if the W +h threshold opens up before the tb one.The exact position of the two thresholds on the mH± axis depends of courseon tan β, mt, and m˜q.

It is just a numerical coincidence that in fig. 14athe two thresholds correspond almost exactly.

For increasing values of mH±and tan β, the numerical relevance of the W +h branching ratio rapidly dis-appears, because of the cos2(β −α) suppression factor in the correspondingpartial width.The total charged Higgs boson width is shown, as a function of mH± andfor tan β = 1.5, 3, 10, 30, in fig. 9d.

Again one can see the effects of the tbthreshold, and also the tan β-dependence of the couplings to fermions. Inany case, the charged Higgs width remains smaller than 1 GeV for mH±

These processes are controlled by theHiggs couplings to heavy quarks and gauge bosons, whose essential fea-tures were summarized in table 1. We briefly discuss here the correspondingcross-sections and the status of their theoretical description, emphasizing thefeatures which are different from the SM case.

We shall always adopt theHMRSB structure functions [35] with Λ(4) = 190 MeV.22

Gluon fusion.In the SM, gg →φ [36] is the dominant productionmechanism, the most important diagram being the one associated with thetop-quark loop. In the MSSM, this is not always the case, since the correctionfactors of table 1 give in general suppression for the top contribution andenhancement for the bottom one, and stop and sbottom loops could alsoplay a role.The leading-order amplitudes for the gluon-fusion processes are deter-mined by the functions of eqs.

(32), with top, bottom, stop and sbottomintermediate states.For m˜q = 1 TeV, the squark contributions are verysmall, owing to the suppression factor m2Z/m2˜q in the corresponding I˜q func-tions. For large values of tan β, the bottom contribution can compete withthe top one and even become dominant.QCD corrections to the gluon-fusion cross-section were recently evaluatedin ref.

[31], for a SM Higgs in the mass region below the heavy-quark thresh-old. In this region, QCD corrections increase the top contribution by about50%.

To a good approximation, the bulk of QCD corrections can be takeninto account by performing the replacementσ0(gg →φ) −→σ0(gg →φ)1 +112 + π2 αSπ,(39)at the renormalization scale Q = mφ.This calculation, unfortunately, isnot valid above the heavy-quark threshold, a region which is relevant forthe bottom contribution and for the top contribution to H, A production,when mH, mA > 2mt. Even below the heavy-quark threshold, the SM QCDcorrections are applicable to h and H production, but not to A production,because of the additional γ5 factor appearing at the Aqq vertex.

In view ofthis not completely satisfactory status of QCD corrections, we calculate, con-servatively, the top contribution without QCD corrections. However, whendiscussing the detectability of the different physics signals, we shall take intoaccount the results of ref.

[31], when applicable. In the case of the bottomcontribution, we use the running mb, which leads to suppression.In fig.

15 we display cross-sections for gg →φ (φ = h, H, A), as functionsof mφ, for tan β = 1.5, 3, 10, 30 and for LHC and SSC energies. The SM Higgscross-section is also shown for comparison.

For large values of tan β and nottoo high values of mφ, the cross-sections can be enhanced with respect to theSM value. This effect is due to the enhanced bottom-quark contribution, asapparent from table 1 and fig.

5. The fast disappearance of this effect for23

increasing Higgs masses is due to the fast decrease of the function F φ1/2(τ φb )as τ φb →0. When the neutral Higgs couplings to fermions are SM-like, thegluon-fusion cross-sections approach the SM value, and are always dominatedby the top contribution.

The changes in the slopes of the curves in fig. 15 aredue to the competing top and bottom contributions.

In particular, one cannotice an important threshold effect, for mA ∼2mt, in the process gg →A,which can bring the corresponding cross-section above the SM value for lowtan β.As a final remark, we notice that the LHC and SSC curves in fig. 15 havevery similar shapes, with a scaling factor which is determined by the gluonluminosity and uniformly increases from ∼2.5 at mφ ∼100 GeV to ∼5 atmφ ∼500 GeV.W fusion.In the SM case, the W-fusion mechanism [37] can competewith the gluon-fusion mechanism only for a very heavy (mϕ >∼500 GeV)Higgs boson, owing to the enhanced WLWLϕ coupling and to the relativeincrease of the quark number-densities.

In the MSSM, the correction factorsfor the couplings to vector boson pairs (see table 1 and fig. 7) are alwayssmaller than 1, so that the MSSM W-fusion cross-sections are always smallerthan the SM one.We illustrate this in fig.

16, where W-fusion cross-sections for h and Hare displayed, for the same tan β and √s values as in fig. 15.For bothh and H, the SM cross-section is approached from below in the regions ofparameter space where sin2(β −α) →1 and sin2(β −α) →0, respectively.

Infigs. 16b and 16d, for mA →0 there is a positive lower bound on sin2(β −α)(see fig.

7), reflecting the fact that at the tree level α →−β in this limit,so the SM value is actually never reached.For increasing mH, one cannotice the fast decoupling of H from W-pairs, as already observed whendiscussing the total width. In leading order, A does not couple to W-pairs.A non-vanishing cross-section could be generated at one loop, but such acontribution is completely negligible, since even for h and H the W-fusioncross-section is small (<∼20%) compared with the gluon-fusion cross-section.Finally, we observe that the LHC and SSC cross-sections of fig.

16 differ byan overall factor ∼3 in the phenomenologically relevant region, mφ = 70–140GeV.24

Associated b¯bφ production.This mechanism is unimportant in theSM, since its cross-section is too small to give detectable signals [38]. Inthe MSSM model, however, for large values of tan β the b¯bφ couplings canbe strongly enhanced.

Then for not too high values of the Higgs masses, asignificant fraction of the total cross-section for neutral Higgs bosons can bedue to this mechanism.The associated bbφ production involves two rather different mass scales,mφ >> mb, therefore at higher orders large logarithmic corrections of orderαnslnnmφmbmay destroy the validity of the Born approximation, depending on the valueof mφ. One needs an improved treatment where these logarithms are re-summed to all orders.

The origin of these logarithms is well understood. Partof them come from configurations where the gluons are radiated collinearlyby nearly on-shell bottom quarks, which are obtained by splitting the initialgluons into a b¯b pair.

This type of logarithms are responsible for the QCDevolution of the effective b-quark density within the proton: they were care-fully analysed and resummed to all orders, and it was found [39] that thecorrections are positive and increase with the Higgs-boson mass. A secondsubset of logarithms lead to running quark mass effects.

An analysis whereboth effects are treated simultaneously is still missing. In view of this ambi-guity, we interpolated the existing results by using the Born approximationwith the bottom quark mass adjusted to the fixed value mb = 4 GeV.

How-ever, one should keep in mind that the theoretical estimate in this case hasa large (factor of 2) uncertainty.In fig. 17 we display cross-sections for associated b¯bφ production, for thesame tan β and √s values as in fig.

15.Comparing the cross-sections offigs. 15 and 17, we can see that the hb¯b cross-section can give at most a 20%correction to the total h cross-section.

The Hb¯b and Ab¯b cross-sections, how-ever, can be even larger than the corresponding gluon-fusion cross-sectionsfor tan β >∼10. Comparing the LHC and SSC curves of fig.

17, one can noticea rescaling factor varying from ∼3 to ∼8 in the mφ region from 60 to 500GeV.Associated Wφ (Zφ) production.This mechanism [40] is the hadroncollider analogue of the SM Higgs production mechanism at LEP, with the25

difference that at hadron colliders Wφ production is more important thanZφ production.In the Zφ case, the event rate at the LHC and SSC istoo low to give a detectable signal, both in the SM and (consequently) inthe MSSM. The Wφ mechanism has considerable importance at the LHCfor φ = h, H and in the Higgs mass range mφ = 70–140 GeV, where ameasurable signal may be obtained from final states consisting of two isolatedphotons and one isolated lepton.

The calculation of the cross-section is wellunderstood, including the QCD corrections, since it has a structure similar tothe Drell-Yan process, with the same next-to-leading-order corrections (fora recent study concerning the numerical importance of the QCD correctionssee ref. [41]).

The QCD corrections are positive, and amount to about 12%if one chooses Q2 = ˆs as the scale of Q2 evolution. The production cross-sections of h and H are obtained by rescaling the SM model cross-section bythe appropriate correction factors given in table 1.In fig.

18, cross-sections for Wh and WH are displayed, as functions ofcorresponding Higgs masses, for the same tan β and √s values as in fig. 15.Since the SUSY correction factors are the same, the approach to the SM caseand the irrelevance of WA production can be described in the same way asfor the W-fusion mechanism.In the phenomenologically relevant region, mφ = 70–140 GeV, the scalingfactor between the LHC and SSC curves is ∼2.5.Associated t¯tφ production.In the SM, the Born cross-sectionformula for this process is the same as for the bbϕ case [38].

In the MSSMcase, one just needs to insert the appropriate SUSY correction factors, asfrom table 1. Note, however, that the leading-order QCD calculation is morereliable in this case, since in the ttφ case one does not have two very differentphysical scales when mφ is in the intermediate mass region.

The next-to-leading QCD corrections are not known, therefore the Born cross-sectionstill suffers from a relatively large (∼50%) scale ambiguity.In fig. 19, the production cross-sections for t¯tφ (φ = h, H, A) are plotted,as functions of the corresponding Higgs mass, for the same tan β and √svalues as in fig.

15. In general, the MSSM cross-sections are smaller than theSM one, which is approached in the limit in which the t¯tφ coupling becomesSM-like.

A possible exception is the ttH cross-section for small values ofmA and tan β, since in this case the corresponding coupling can be slightlyenhanced with respect to the SM one.26

In the phenomenologically relevant range, mφ = 70–140 GeV, the rescal-ing factor between the LHC and SSC curves in fig. 19 varies from ∼6 to∼7.In the phenomenologically allowed range of eq.

(1), the top-mass depen-dence of the cross-sections of figs. 15–19 is not negligible, but it does notchange qualitatively the previous considerations.

The largest effect comesfrom the increase of the upper limit on mh for increasing top mass (seefig. 1).

This induces a shift in the limiting values for the h and H productioncross-sections. There are also obvious kinematical top-mass effects in thegluon-fusion mechanism and in the t¯tφ mechanism, which are well under-stood from SM studies [14].

In the MSSM, additional effects are given by theradiative corrections to the relevant Higgs couplings, which were discussedin sect. 2.6Physics signals6.1Neutral Higgs bosonsWe now calculate the rates for a number of processes that could provideevidence for one or more of the neutral MSSM Higgs bosons at the LHCand SSC, and we summarize our results with the help of contour plots inthe (mA, tan β) plane.

We consider production cross-sections, folded withbranching ratios, for the following signals:• two isolated photons;• one isolated lepton and two isolated photons;• four isolated charged leptons;• a pair of tau leptons.In the SM case, the first two signals are relevant for the region of intermediateHiggs mass, 70 GeV < mφ < 140 GeV, the third one is the so-called ‘gold-plated’ signal in the high-mass region 130 GeV < mφ < 800 GeV, and theτ +τ −signal appears to be hopelessly difficult.27

In a complete phenomenological study, one would like to determine pre-cisely the statistical significance of the different physics signals. This wouldrequire, besides the computation of total signal rates, the calculation of thebackgrounds, the determination of the efficiencies (for both signals and back-grounds) due to kinematical cuts and detector effects, the optimization ofthe kinematical cuts to achieve the best signal/background ratio, etc.

Sucha complete analysis would require the specification of several detector andmachine parameters, and goes beyond the aim of the present paper. Instead,we try here to present total rates for well-defined physics signals, in a formwhich should be useful as a starting point for dedicated experimental studies.As the only exception, to illustrate with an example how our results canbe used to establish the statistical significance of a given physics signal in agiven detector, we shall describe the case of the ‘two-isolated-photons’ signal,using the results of recent simulation works.

A similar procedure should beadopted for any other physics signal, detector, and collider, once completeresults of simulation works are available. In many cases, the existing resultsfrom previous background and simulation studies, carried out for the SM, canalso be used to draw conclusions concerning the MSSM case.

We mention,however, two important differences: 1) the total widths of H and A remainsmall even in the high-mass region; 2) for large tan β, the number of signalevents in the τ +τ −final state is significantly higher than in the SM case.Inclusive two-photon channel.In fig. 20 we display cross-sectionstimes branching ratios for the inclusive production of φ = h, H, A, followedby the decay φ →γγ, as functions of mφ, and for the same parameter choicesand energies as in fig.

15. We sum the contributions of the gluon-fusion, W-fusion, and b¯bφ mechanisms.

For comparison, the SM value is also indicated.The QCD corrections of ref. [31] are not included, for the reasons explainedin the previous section.

In the case of h and H, the signal rates are alwayssmaller than in the SM, and approach the SM values at the upper and loweredge of the allowed mh amd mH ranges, respectively. The rather steep slopecharacterizing the approach to the SM limit, for varying Higgs mass andfixed tan β, is a reflection of the property of the branching ratios discussedin sect.

4. Also the structure in figs.

20b and 20e can be attributed to thethreshold behaviour of the H →hh channel. The signal rate for the CP-oddA boson is extremely small for tan β >∼3.

However, we observe that in asmall region of the parameter space, for mA just below 2 mt and tan β <∼3,28

the rate can become larger than the SM one: nevertheless, in general it isstill too low to produce a detectable signal, unless one chooses tan β ∼1 andmA ∼2mt.For the inclusive two-photon channel, the results of detailed simulationsof signal and SM background are now available, for some of the LHC detectorconcepts [42,16,43,44]. For illustrative purposes, in the following example weshall follow the treatment of ref.

[43]. In the mass range mφ = 80–150 GeV,and assuming 105 pb−1 integrated luminosity, this LHC simulation considersa fairly wide range of detector performances, which affect the significanceof the signal.

For an energy resolution ∆E/E = [2%/qE(GeV)] + 0.5%,ref. [43] obtains a 104 efficiency for rejecting jets faking an isolated photonin the relevant pT region.

Applying standard kinematical cuts, this simula-tion finds ∼40–50% kinematical acceptance, with an additional ∼30–40%loss due to isolation cuts and reconstruction efficiency for the isolated pho-tons. Typically, for a SM Higgs with mϕ ∼100 GeV, one obtains ∼103signal events over ∼104 background events, corresponding to a statisticalsignificance S/√B ∼10.

More generally, ref. [43] determined the statisticalsignificance of the signal for given values of the generic Higgs mass mφ andof the signal rate σ · BR(φ →γγ) (see fig.

21). In our opinion, this is an ex-cellent way of summarizing the simulation work, since it gives the possibilityof studying alternatives to the SM case, and in particular the MSSM.

Thedashed line in fig. 21 corresponds to the signal for the SM Higgs, which in-cludes both the gluon-fusion and the W-fusion production mechanisms, andalso the QCD corrections of ref.

[31]. One can see from fig.

21 that for suchoptimistic detector parameters there is some margin for detecting smallerrates than in the SM. Clearly the SUSY Higgs search further enhances theneed for the best possible mγγ resolution and γ-jet rejection.In extending the SM analysis to the MSSM, one should pay attentionto the applicability of the QCD corrections of ref.

[31] to the gluon-fusioncross-section. We have checked that in the phenomenologically relevant re-gion, which corresponds to h or H in the intermediate mass range, and tosignal rates within an order of magnitude from the SM one, the gluon-fusionmechanism is dominated by the top-quark loop.Since in this region thegluon-fusion mechanism accounts for ∼80% of the total cross-section, andthe correction is roughly a multiplicative factor 1.5, as a rule of thumb wecan take it into account by multiplying the total cross-section by a factor29

∼1.4.In fig. 22 we show contour plots in the (mA, tan β) plane, correspondingto fixed values of σ · BR(φ →γγ) (φ = h, H).

QCD corrections have beenincluded according to ref. [31].

The region where the rate is large enough topromise a measurable signal is rather large for h, is concentrated in a smallstrip for H, and is possibly a very small area, just below mA = 2mt andjust above tan β = 1, for A. In our representative example [43], we can nowevaluate the statistical significance of the ‘two-isolated-photons’ signal at anypoint of the (mA, tan β) plane, by just combining the information containedin figs.

22, 2, 3, and 21. In the case of h searches, and for mh >∼90 GeV, asignal rate beyond 40 fb should give detectable signals.

A signal rate of 30fb is the borderline of detectability for one year of running, and signal ratesbelow 20 fb appear extremely difficult to detect. In the case of H, which hashigher mass, a signal rate of 20 fb appears to be the borderline of what canbe achieved in one year of running.

In the case of A, the interesting massregion is mA ∼2mt: for mA = 250 GeV, and taking σ · BR(A →γγ) = 3 fbas a plausible discovery limit at the LHC [43], a signal for A →γγ will befound only if tan β <∼1.5.One isolated lepton and two isolated photons.This signal cancome from either Wφ or t¯tφ production. In the latter case, two or moreisolated jets are also produced.

The physics signals from Wφ production areparticularly important at the LHC, and were studied in ref. [45].

The im-portance of the physics signals from ttφ production was recently emphasizedin ref. [46].

The production rates, multiplied by the φ →γγ branching ratio,are shown in fig. 23.

We can see that, similarly to the inclusive γγ channel,the rates for Wh, WH, t¯th, t¯tH are always smaller than the SM value, whichrepresents the boundary curve in the limit of large tan β and large (small)mA for h (H). From figs.

23g and 23j one can see that ttA production cangive a llγ signal larger than in the SM for small tan β and near the mA = 2mtthreshold, but even in this case the rate appears to be too small for detection.We emphasize that the production rates shown in fig. 23 do not include thebranching ratios of leptonic W and semileptonic t decays.

If top decays areas in the SM, one should still include a combinatorial factor of 2, comingfrom the fact that both top and antitop can decay semileptonically. On theother hand, in the MSSM there is the possibility of t →bH+ decays, wherethe subsequent H+ decay cannot produce a direct lepton l = e, µ.

We shall30

take this possibility into account in the following, but its impact on the de-tectability of the lγγ signal is rather small. The only case in which this effectis not completely negligible is for H, when mA <∼100 GeV and tan β <∼4 ortan β >∼10, in which case the t →bH+ branching ratio can play a role.From parton-level simulations [42,47,48], for a SM Higgs of about 100GeV, one typically obtains ∼(12 + 15) and ∼(3 + 11) lγγ signal eventsat the LHC and SSC, respectively.

Here we assumed 105 pb−1 of integratedluminosity for the LHC and 104 pb−1 for the SSC. The quoted numbers sep-arately show the contributions from Wϕ and ttϕ production.

Furthermore,they include losses due to acceptances (∼30%), and lepton and photon de-tection efficiencies [ǫ ∼(0.9)3]. The total background is roughly 20–30%of the signal and is dominated by the irreducible Wγγ and t¯tγγ contribu-tions.

There are many different contributions to the reducible background(bbg, bbγ, bbγγ, Wjγ, . .

. ).

Parton-level simulations indicate that they can besuppressed well below the irreducible background, provided one assumes, asfor the inclusive γγ case, excellent detector performances: a γ-jet rejectionfactor >∼3 × 103 and a suppression factor >∼7 for the leptons from b-decaysafter isolation cuts.Clearly, there is very little margin (a factor of 2?) to be sensitive to signalrates smaller than in the SM.

In fig. 24, we show contour plots correspondingto fixed values for the quantityLφ ≡[σ · BR (lγγ)]φ= [2σ(t¯tφ) · BR(t →Wb) + σ(Wφ)] · BR(φ →γγ) · BR(W →lν) ,(40)for the same choice of mt and m˜q as in fig.

15, and for LHC and SSC energies.In eq. (40), l = e, µ and we have not considered the strongly suppressedpossibility of getting a light charged lepton from both top and antitop.The four-lepton channel.The channel ϕ →Z∗Z∗→l+l−l+l−(l = e, µ) gives the so-called ‘gold-plated’ signal for the SM Higgs in themass range mϕ = 130–800 GeV.

Below mϕ ∼130 GeV, both the total rateand the acceptance decrease very rapidly, leading to too small a signal fordetection. For all three neutral Higgs bosons of the MSSM, the rates in thischannnel are always smaller than in the SM.

In the case of A, there is noAZZ coupling at tree level, and loop corrections cannot generate measurable31

rates in the four-lepton channel. As for h, if mt <∼180 GeV and m˜q <∼1 TeV,one can see from fig.

1 that mh <∼130 GeV. Therefore, the h →Z∗Z∗→4lsignal does not have chances of detection at the LHC and SSC, unless onechooses extremely high values for mt and m˜q or one has superb resolutionand acceptance for leptons.

The situation is somewhat better in the caseof H, despite the strong suppression with respect to the SM, due to thecompetition with the hh, bb, tt channels, as discussed in sect. 3.In fig.

25, we show signal rates for the SM Higgs boson and for H, forthe same choices of parameters as in fig. 15.

The threshold effects and thesuppression for large values of tan β are clearly visible.The LHC and SSC discovery potential can be estimated by using theresults of simulations carried out for the SM [15,49,50,16,18], taking also intoaccount that ΓH < 2 GeV all over the mass region of interest, mH <∼2mt.Assuming excellent lepton momentum resolution, in the mass range 2mZ

26, we show contour plots in the (mA, tan β) plane,corresponding to fixed values of σ·BR(H →4l). QCD corrections have beenincluded according to ref.

[31]. In view of the strong sensitivity to the value ofmt, we show contours for mt = 120, 140, 160 GeV, for LHC and SSC energies,and for m˜q = 1 TeV.

The two almost vertical dashed lines correspond tomH = 2mZ and to mH = 2mt. For mH > 2mZ, a detectable signal could beobtained up to tan β ∼5.

Notice that the experimental acceptances changewith mH; in particular, in the region mH < 2mZ they fastly decrease withdecreasing mH: for a realistic assessment of the discovery limits in this massregion, one should take this and other effects into account.Anyway, theprospects for detection for mH < 2mZ do not look good if mt <∼150 GeVand m˜q <∼1 TeV.The τ +τ −channel.For the SM Higgs boson, the τ +τ −decay channelhas been found hopelessly difficult for discovery [51,52], since this channel hasbad mass resolution and overwhelmingly large background coming from theproduction of t¯t, WW+ jets, Drell-Yan pairs, Z+ jets, b¯b+ jets, . .

. .

The badresolution is due to the fact that the tau-decay products always include one ormore neutrinos, which carry away energy; therefore one cannot reconstructthe signal as a resonance peak. The situation is improved if the Higgs isproduced with large transverse momentum that is balanced by a jet [53].

In32

this case one can use the approximationp(1)ν⃗p (1)l,TE(1)l+ p(2)ν⃗p (2)l,TE(2)l= ⃗p missT(41)to reconstruct the transverse momenta of the neutrinos and hence the in-variant mass of the tau pair. In the above equation, p(i)νdenotes the totaltransverse momentum of the neutrinos coming from the decay of τ (i), i = 1, 2,while ⃗p (i)l,T and E(i)ldenote the lepton momenta and energies, respectively.

Itwas shown in ref. [51] that, in the mass range mφ = 70–140 GeV, a massresolution of ∼13–17% can be achieved.

This method can also be used for thehadronic decay modes, taking advantage of the fact that the rate is higherby a factor of ∼3.5. When a tau decays hadronically, the hadrons have verylow multiplicity and invariant mass, and these properties might be used torecognize the ‘τ-jet’ [54].

There is a price for the better mass resolution.Tagging on a large-pT jet can reduce the rate by an order of magnitude. Fur-thermore, at 105 pb−1/year luminosity, the presence of pile-up deterioratessignificantly the measurement of ⃗p missT, and therefore the τ +τ −signal canonly be studied with this method at lower luminosities, ∼104 pb−1/year 6.While these difficulties appear prohibitive in the case of the SM, the situationis not entirely negative in the MSSM.In fig.

27, we display signal rates for φ →τ +τ −production (φ = h, H, A),for the same parameter choices as in fig. 15, together with the SM values.We can see that for large values of tan β the production rates can becomemuch larger than in the SM.

In the case of h production, for tan β = 30the enhancement can be more than one order of magnitude, and increaseswith decreasing values of mh (see figs. 27a and 27d).

Huge enhancementscan be obtained also for H and A, thanks to the properties of the τ +τ −branching ratios discussed in sect. 4.

Note for example that, at the LHC, formH, mA ∼500 GeV and tan β >∼10, we get σ · BR(H, A →τ +τ −) ≥20 pb,while for mA, mH ∼120 GeV and tan β >∼30, we obtain σ · BR(H, A →τ +τ −) ≥30 pb. The rates for the SSC are rescaled by the factor alreadydiscussed in the previous section.In order to assess, for any given mass, the cross-section values abovewhich one obtains a measurable signal over the large background, detailed6Alternatively, at high luminosity one may try to just search for an excess of events inthe e±µ∓or l±+ ‘τ-jet’ channels.33

simulations are needed. Preliminary studies have been reported for the leptonchannel e±µ∓in ref.

[51] and for the mixed channel l±+ ‘τ-jet’ in ref. [55].In the second case, the difficulty of recognizing a ‘τ-jet’ may be compensatedby the higher rate of this channel.

The preliminary analysis of ref. [55] findsfor the LHC sensitivity to values of σ · BR(φ →τ +τ −) down to ∼10 pbin the low-mass region mφ ∼100 GeV and ∼1 pb in the high-mass regionmφ ∼400 GeV.

This result cannot be easily rescaled to the SSC case, since alarge mass interval is involved and the SSC luminosity gives more favourableexperimental conditions for the srudy of this channel.In fig. 28, we show contour plots corresponding to fixed values of σ ·BR(φ →τ +τ −), for the same values of mt and m˜q as in fig.

15.6.2Charged Higgs bosonWe now move to the discussion of possible physics signals associated withthe charged Higgs boson. The phenomenology of the charged Higgs bosonat hadron colliders was previously discussed in refs.

[56]. The benchmarkmass value for charged-Higgs-boson searches at the LHC and SSC is mH± =mt −mb.

For lower values of mH±, the dominant production mechanismat large hadron colliders is gg →tt, followed by t →H+b.For highervalues of mH±, the dominant production mechanism is gb →tH+. As far asdetectable signals are concerned, this last case appears hopeless, in view ofthe suppressed cross-section and of the large backgrounds coming from QCDsubprocesses.

The first case appears instead to be experimentally viable overa non-negligible region of parameter space. Given the known tt productioncross-section, one can compute the t →bH+ branching ratio according towell-known formulae, parametrizing again the leading QCD corrections byrunning masses evaluated at a scale Q ∼mH±.

The charged Higgs branchingratios were discussed in the previous section, where it was found that the τ +ντmode dominates in the mass range under consideration. The experimentalsignal of a charged Higgs would then be a violation of lepton universalityin semileptonic top decays.

As a convenient indicator, one can consider theratioR = BR(t →τ +ντb)BR(t →µ+νµb) ≡1 + ∆R ,(42)34

with∆R = BR(t →H+b) · BR(H+ →τ +ντ)BR(t →W +b) · BR(W + →µ+νµ). (43)Preliminary investigations [56] show that the experimental sensitivity couldreach ∆R >∼0.15 at the LHC.

At the SSC the increased tt production cross-section is likely to give better sensitivity.In fig. 29, we display contourlines of ∆R in the (mA, tan β) plane, for the three representative valuesmt = 120, 140, 160 GeV.The dashed lines denote the kinematical limitmH± = mt −mb.

One can see that the most difficult values of tan β are thosebetween 3 and 10, and that the process under consideration could give accessto values of mA as high as 80–120 GeV for top-quark masses in the range120–160 GeV.7Conclusions and outlookIn this paper we carried out a systematic analysis of the possible physicssignals of the MSSM Higgs sector at the LHC and SSC, assuming that thesupersymmetric (R-odd) particles are heavy enough not to affect significantlythe production cross-sections and the branching ratios of the MSSM Higgsparticles. As independent parameters in the Higgs sector, we chose mA andtan β, and we considered the theoretically motivated region of the parameterspace0 ≤mA ≤500 GeV,1 ≤tan β ≤50 .We assumed m˜q = 1 TeV and negligible mixing in the squark sector.

Weincluded the most important radiative corrections to the Higgs masses mh,mH, mH±, and to the Higgs couplings to fermions and vector bosons. Wealso included the most important radiative corrections to the three-pointcouplings of the neutral Higgs bosons.We estimated the discovery potential of LEP I and LEP II, and we carriedout detailed cross-section calculations for the LHC and SSC.

We singled outfour classes of final states (γγ, l±γγ, l+l−l+l−, τ +τ −) which could providesignificant signals for neutral Higgs bosons at the LHC and SSC, and we alsoexamined possible signals of charged Higgs bosons in top decays.We calculated all the relevant branching ratios, and the cross-sections forall the relevant production mechanisms. We presented our results with thehelp of branching-ratio curves, cross-section curves, signal-rate curves and35

contour plots in the (mA, tan β) plane. We did not perform new backgroundstudies, but we pointed out that, using the results of our calculations andof the existing simulations carried out for the SM Higgs, supplemented byestimates of the acceptances and efficiencies of typical experiments, in manycases one can draw conclusions concerning the discovery range.

In some cases,as for the τ +τ −channel, further simulation work appears to be needed inorder to reach firm conclusions. Nevertheless, some preliminary conclusionscan already be drawn and will now be summarized.At large hadron colliders, the MSSM Higgs search is, in general, moredifficult than the SM Higgs search.

This is due to the fact that, in a largeregion of the parameter space, at least one of the MSSM neutral Higgs bosonsis in the intermediate mass region, 80 GeV <∼mφ <∼140 GeV, but with ratesin the γγ and lγγ channels which can be significantly suppressed with respectto the SM case. Similarly, neutral Higgs bosons with mφ >∼130 GeV havetypically strongly suppressed rates in the l+l−l+l−channel.

On the contrary,in the MSSM, for rather large values of tan β, one can obtain a much largersignal rate in the τ +τ −channel than in the SM. Finally, t →bH+ decays,followed by H+ →τ +ντ, can give detectable signals only in a rather restrictedregion of the parameter space.As an example, we now try to assess the discovery potential of the differentchannels for the representative parameter choice mt = 140 GeV, m˜q = 1 TeV,working as usual in the (mA, tan β) plane.To begin with, we recall the expectations for LEP II.

The size of theLEP II discovery region depends rather strongly on mt and m˜q, and on theassumed energy and luminosity. For standard machine parameters, LEP IIcannot test the whole parameter space allowed by the present data.

Lookingback at fig. 8, one may tentatively say that LEP II will give us (if no Higgsboson is discovered) lower limits of about mA >∼70–100 GeV and tan β >∼3–8 for mt = 120 GeV, tan β >∼1.5–3 for mt = 160 GeV.We observe that the LHC and SSC will test the MSSM Higgs sector in alargely complementary region of the (mA, tan β) plane.

A pictorial summaryof the discovery potential of the different channels is presented in fig. 30.We emphasize once again that the final discovery limits will depend on themachine and detector properties, as well as on the actual values of the topand the soft supersymmetry-breaking masses.We therefore drew fig.

30just for illustrative purposes, to exemplify a particularly convenient way ofconsidering all the discovery channels at once.36

The γγ and lγγ channels are important in approximately the same regionof the parameter space, mA ≫mZ for h and 50 GeV <∼mA <∼100 GeVfor H. Therefore, these two channels can be experimentally cross-checkedone against the other, reinforcing the significance of a possible signal. As anoptimistic discovery limit for h, we show in fig.

30 the contour line σ·BR(h →γγ) = 30 fb at the LHC, corresponding to mA >∼200 GeV. This contourline is shown only for mh >∼80 GeV.

In the region of the parameter spaceto the right of this line [indicated by the labels h →γγ and l + (h →γγ)],it is expected that measurable signals will be found, assuming detector andmachine parameters as discussed in refs. [16].Approximately the samecontour line is obtained by taking σ · BR(h →γγ) = 85 fb at the SSC.

Thisindicates that, in the inclusive γγ channel, the discovery range of the LHCand SSC will be the same if the luminosity at the LHC will be ∼3 timeshigher than at the SSC and if the detectors used at the two machines will havesimilar efficiencies in suppressing the backgrounds. Very similar discoverylines can be drawn by considering the lγγ channel and taking σ·BR[l+(h →γγ)] ∼0.8 fb for the LHC and ∼4 fb for the SSC.In fig.

30 we also show the contour line for σ · BR(H →γγ) = 20 fb atthe LHC, corresponding to ∼55 fb at the SSC. The slightly smaller signalrate was chosen to account for the improved efficiencies at higher Higgs-massvalues.

The contour line defines a narrow strip around mA ∼75 GeV (shadedin fig. 30), where the discovery of H is expected to be possible both in theγγ and in the lγγ channels [for lack of space the label l +(H →γγ) has beenomitted].The four-lepton channel is important mainly for H, in the mass region2mZ <∼mH <∼2mt, which translates into 150 GeV <∼mA <∼2mt, andfor relatively small tan β.

As a reference value for discovery in this massregion, we take σ · BR(H →4l) = 1 fb for the LHC, which corresponds toσ · BR(H →4l) ∼3 fb for the SSC. This contour defines the area in fig.

30indicated by the label H →4l. In a small part of this area, correspondingto mA ∼mt and tan β ∼1, A →γγ could also give a detectable signal.In the region of very large tan β, and moderately large mA, one could takeadvantage of the enhanced production cross-sections and of the unsuppresseddecays into τ +τ −to obtain a visible signal for one or more of the MSSMneutral Higgs bosons, and in particular for H and A, whose masses can besignificantly larger than 100 GeV.

The simulation work for this process isstill at a rather early stage [55], so that no definite conclusion can be drawn37

yet. For reference, the dotted line in fig.

30 corresponds to a (somewhatarbitrary) interpolation of σ · BR(φ →τ +τ −) ∼10 pb at mφ = 100 GeV andσ · BR(φ →τ +τ −) ∼1 pb at mφ = 400 GeV, for LHC energy and summingover the φ = H, A channels.Finally, in the region of parameter space corresponding to mA <∼100 GeV,a violation of lepton universality due to the decay chain t →bH+ →bτ +ντcould indicate the existence of the charged Higgs boson. This region is in-dicated by the label H± →τν in fig.

30. Its right border is defined by thecontour line of R = 1.15, where R was defined in eq.

(42).By definition, our contour lines do not take into account changes in theacceptances and efficiencies, which are expected in realistic experimental con-ditions, and depend on the Higgs mass and on the detector. We thereforeexpect some deformations of our contours once discovery lines are extractedfrom realistic experimental simulations [43,55].As a last piece of information we also display in fig.

30 the border ofthe expected discovery region at LEP II, which depends rather sensitively, asalready discussed, on the assumed values of the machine energy and luminos-ity. We then show two representative lines: the lower dashed line correspondsto σ(e+e−→hZ, HZ, hA, HA) = 0.2 pb at √s = 175 GeV, while the up-per dashed line corresponds to σ(e+e−→hZ, HZ, hA, HA) = 0.05 pb at√s = 190 GeV.Using the result summarized in fig.

30, we can draw several importantqualitative conclusions:• The discovery potentials of LEP and the LHC/SSC show a certaincomplementarity. The discovery region at LEP covers all tan β valuesat small values of mA, and all mA values at small values of tan β, whileat the LHC/SSC one should be sensitive to the large tan β, large mAregion.• One may ask whether the LHC and SSC, combined with LEP II, canexplore the full parameter space of the MSSM Higgs sector, being sen-sitive to at least one signal in each point of the (mA, tan β) plane, forall plausible values of mt and m˜q.

At present, this question cannot beanswered positively. The union of the regions where one should findsignals at least for one Higgs boson does not cover the whole parameterspace: the discovery region has a hole in the middle of the parameter38

space. For our parameter choice, the most difficult region appears tobe the cross-hatched area around mA = 150 GeV and tan β = 10.Therefore we cannot claim yet the existence of a ‘no-lose’ theorem forthe MSSM Higgs search.• One may also ask if there are regions of parameter space where one canfind more than one signal from the MSSM Higgs sector.

The answeris that around mA = 200 GeV and tan β < 5 one can discover h atLEP II and H at the LHC/SSC in the four-lepton channel. There isa somewhat smaller region above mA = 200 GeV where one can alsofind h in the γγ and lγγ channels.

Furthermore, at high values of tan β(>∼20) and at mA > 200 GeV one may discover A and H in the ττchannel and h in the γγ and lγγ channels, although the separation of Hand A appears to be impossible, due to their almost perfect degeneracyin mass. This part of the parameter space is inaccessible at LEP II.

Thediscovery region for H in the γγ and l, γγ channels, corresponding tolow values of mA, largely overlaps with the LEP II discovery region andwith the discovery region related to charged-Higgs production in topdecays. In the low tan β, for 80 GeV <∼mA <∼180 GeV and mA >∼2mtone has a signal at LEP II and no signal at the LHC and SSC, sincemh is too small for detection.Finally, we would like to make some comments on the theoretical uncer-tainties and on possible future studies.Our values for the signal rates depend on several phenomenological inputparameters, as the value of the bottom mass, the parton-number densitiesand the value of αS.The given cross-sections and branching ratios willchange if the input parameters are varied in their allowed range.

Also, forsome production mechanisms, only the Born cross-sections are known. Weestimate that the theoretical errors of the calculated rates vary from about30%, in the case of the γγ channel, up to about a factor of 2 when the bbφor ttφ production mechanisms are important.We did not study all effects associated to variations of the parameters inthe SUSY (R-odd) sector.

It would be interesting to examine the case whensome of the Higgs bosons are allowed to decay into R-odd SUSY particles,or the effects of squark mixing. More importantly, serious simulation workis still needed, in particular for the τ +τ −and the lγγ channel.39

Note addedAfter the completion of most of the work presented in this paper, which wasanticipated in many talks [57], we received a number of papers [58,59] dealingwith different subsets of the material presented here, and reaching similarconclusions. Reference [59] also contains the generalization of eqs.

(21–27)to the case of arbitrary mixing in the stop and sbottom mass matrices, butstill neglecting the D-term contributions.AcknowledgementsWe are grateful to G. Altarelli for discussions, and for insisting that we shouldcarry out this study. We also thank A. Brignole, D. Denegri, J. Ellis, L. Fa-yard, D. Froidevaux, J.-F. Grivaz, P. Janot, F. Pauss, G. Ridolfi, C. Seez,T.

Sj¨ostrand, D. Treille, J. Virdee and P. Zerwas for useful discussions andsuggestions.40

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Figure captionsFig.1: Contours of mmaxh(the maximum value of mh, reached for mA →∞):a) in the (mt, tan β) plane, for m˜q = 1 TeV; b) in the (mt, m˜q) plane,for tan β = mt/mb.Fig.2: Contours of mh in the (mA, tan β) plane, for m˜q = 1 TeV and a) mt =120 GeV, b) mt = 160 GeV.Fig.3: Contours of mH in the (mA, tan β) plane, for m˜q = 1 TeV and a) mt =120 GeV, b) mt = 160 GeV.Fig.4: Contours of mH± in the (mA, tan β) plane, for m˜q = 1 TeV. The solidlines correspond to mt = 120 GeV, the dashed ones to mt = 160 GeV.Fig.5: Contours of sin2 α/ cos2 β in the (mA, tan β) plane, for m˜q = 1 TeV.The solid lines correspond to mt = 120 GeV, the dashed ones to mt =160 GeV.Fig.6: Contours of cos2 α/ sin2 β in the (mA, tan β) plane, for m˜q = 1 TeV.The solid lines correspond to mt = 120 GeV, the dashed ones to mt =160 GeV.Fig.7: Contours of sin2(β −α) in the (mA, tan β) plane, for m˜q = 1 TeV.The solid lines correspond to mt = 120 GeV, the dashed ones to mt =160 GeV.Fig.8: 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 TeVand a) mt = 120 GeV, b) mt = 160 GeV.The solid lines corre-spond to the present LEP I limits.

The dashed lines correspond toσ(e+e−→hZ, HZ, hA, HA) = 0.2 pb at √s = 175 GeV, which couldbe seen as a rather conservative estimate of the LEP II sensitivity. Thedash-dotted lines correspond to σ(e+e−→hZ, HZ, hA, HA) = 0.05 pbat √s = 190 GeV, which could be seen as a rather optimistic estimateof the LEP II sensitivity.Fig.9: Total widths of the MSSM Higgs bosons, as functions of their respectivemasses, for mt = 140 GeV, m˜q = 1 TeV and tan β = 1.5, 3, 10, 30: a) h;b) H; c) A; d) H±.48

Fig.10: Branching ratios for h, as functions of mh, for mt = 140 GeV, m˜q =1 TeV and: a) tan β = 1.5; b) tan β = 30.Fig.11: Branching ratios for H, as functions of mH, for mt = 140 GeV, m˜q =1 TeV and: a) tan β = 1.5; b) tan β = 30.Fig.12: Branching ratios for A, as functions of mA, for mt = 140 GeV, m˜q =1 TeV and: a) tan β = 1.5; b) tan β = 30.Fig.13: Branching ratios for h, as a function of mA, for mt = 140 GeV, m˜q =1 TeV and: a) tan β = 1.5; b) tan β = 30.Fig.14: Branching ratios for H±, as functions of mH±, for mt = 140 GeV,m˜q = 1 TeV and: a) tan β = 1.5; b) tan β = 30.Fig.15: Cross-sections for neutral Higgs production, via the gluon-fusion mech-anism, as functions of the corresponding masses and for mt = 140 GeV,m˜q = 1 TeV, tan β = 1.5, 3, 10, 30: a) h, LHC; b) H, LHC; c) A, LHC;d) h, SSC; e) H, SSC; f) A, SSC. QCD corrections are not included.Fig.16: Cross-sections for h and H production, via the W-fusion mechanism, asfunctions of the corresponding masses, for the same parameter choicesas in fig.

15: a) h, LHC; b) H, LHC; c) h, SSC; d) H, SSC.Fig.17: Cross-sections for associated b¯bφ production, as functions of the corre-sponding Higgs masses, for the same parameter choices as in fig. 15:a) h, LHC; b) H, LHC; c) A, LHC; d) h, SSC; e) H, SSC; f) A, SSC.Fig.18: Cross-sections for associated Wφ production, as functions of the corre-sponding Higgs masses and for the same parameter choices as in fig.

15:a) h, LHC; b) H, LHC; c) h, SSC; d) H, SSC.Fig.19: Cross-sections for associated t¯tφ production, as functions of the corre-sponding Higgs masses and for the same parameter choices as in fig. 15:a) h, LHC; b) H, LHC; c) A, LHC; d) h, SSC; e) H, SSC; f) A, SSC.Fig.20: Cross-sections times branching ratios for inclusive Higgs production(the gluon-fusion, W-fusion, and b¯bφ contributions are summed) anddecay in the γγ channel, for the same parameter choices as in fig.

15:a) h, LHC; b) H, LHC; c) A, LHC; d) h, SSC; e) H, SSC; f) A, SSC.49

For the sake of comparison, the SM values are also indicated. QCDcorrections to the gluon-fusion mechanism are not included.Fig.21: Significance of the inclusive φ →γγ signal, in the plane defined by mφand σ · BR(φ →γγ), for the CMS detector proposal at the LHC, withan energy resolution ∆E/E = [2%/qE(GeV)] + 0.5%.

The solid linesare contours of constant S/√B, where S is the signal and B is thebackground. The dashed line corresponds to the SM Higgs, includingQCD corrections to the gluon-fusion mechanism.

Courtesy of C. Seez[43].Fig.22: Contours of constant cross-sections times branching ratios, in the (mA, tan β)plane, for the inclusive φ →γγ channel: a) h, LHC; b) H, LHC; c) h,SSC; d) H, SSC. The choice of mt and m˜q is the same as in fig.

15, andQCD corrections to the gluon-fusion mechanism are included.Fig.23: Cross-sections for associated Wφ and t¯tφ production, times branchingratios for the φ →γγ channel, for the same parameter choices as infig. 15: a) Wh, LHC; b) WH, LHC; c) Wh, SSC; d) WH, SSC; e) t¯th,LHC; f) t¯tH, LHC; g) t¯tA, LHC; h) t¯th, SSC; i) t¯tH, SSC; j) t¯tA, SSC.For the sake of comparison, the SM values are also indicated.Fig.24: Contours of constant Lφ = [2σ(t¯tφ) · BR(t →Wb) + σ(Wφ)] · BR(φ →γγ)·BR(W →lν), for the same choice of mt and m˜q as in fig.

15: a) h,LHC; b) H, LHC; c) h, SSC; d) H, SSC.Fig.25: Cross-sections for inclusive H production (the gluon-fusion, W-fusionand b¯bφ contributions are summed) and decay in the Z∗Z∗→4l±channel (l = e, µ), for the same parameter choices as in fig. 15: a) LHC;b) SSC.

For the sake of comparison, the SM values are also indicated.QCD corrections to the gluon-fusion mechanism are not included.Fig.26: Contours of constant cross-sections times branching ratios for H →Z∗Z∗→4l±, for the same choice of m˜q as in fig. 15 and: a) mt =140 GeV, LHC; b) mt = 140 GeV, SSC; c) mt = 120 GeV, LHC;d) mt = 120 GeV, SSC; e) mt = 160 GeV, LHC; f) mt = 160 GeV,SSC.

QCD corrections to the gluon-fusion mechanism are included.50

Fig.27: Cross-sections times branching ratios for φ →τ +τ −, for the same pa-rameter choices as in fig. 15: a) h, LHC; b) H, LHC; c) A, LHC; d) h,SSC; e) H, SSC; f) A, SSC.

For the sake of comparison, the SM valuesare also indicated.Fig.28: Contours of constant cross-sections times branching ratios for φ →τ +τ −, for the same choice of mt and m˜q as in fig. 15: a) h, LHC; b) H,LHC; c) A, LHC; d) h, SSC; e) H, SSC; f) A, SSC.

The vertical linesin c) and f) correspond to mA = 60 GeV.Fig.29: Contours of constant ∆R = [BR(t →H+b)·BR(H+ →τ +ντ)]/[BR(t →W +b) · BR(W + →µ+νµ)], for m˜q = 1 TeV and: a) mt = 140 GeV;b) mt = 120 GeV; c) mt = 160 GeV.Fig.30: Pictorial summary of the discovery potential of large hadron collidersfor m˜q = 1 TeV and mt = 140 GeV.51

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