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Radiative corrections to the Higgs boson decay rate

arXiv:hep-ph/9206257v1 26 Jun 1992UCB-PTH-92/23LBL-32498June, 1992Radiative corrections to the Higgs boson decay rateΓ(H →ZZ) in the minimal supersymmetric modelDamien Pierce and Aris PapadopoulosDepartment of Physics, University of California, BerkeleyandLawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720.We consider radiative corrections to the decay rate Γ(H →ZZ) of the heavyCP-even Higgs boson of the minimal supersymmetric model to two Z bosons. Weperform a one loop Feynman diagram calculation in the on-mass-shell renormalizationscheme, and include the third generation of quarks and squarks.

The tree level rateis suppressed by a mixing angle factor and decreases as 1/MH for large MH. Thecorrected rate overcomes this suppression and increases with MH for MH>∼500 GeV.The corrections can be very large and depend in detail on the top squark masses andA-term, as well as the supersymmetric Higgs mass parameter µ.Typeset in REVTEXThis work was supported by the Director, Office of Energy Research, Office of High Energy and NuclearPhysics, Division of High Energy Physics of the U.S. Department of Energy under contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY–90–21139.

I. INTRODUCTIONOne of the least attractive features of the standard model (SM) is the existence of thenaturalness problem. Roughly speaking this means that when one computes corrections tothe Higgs boson mass one finds quadratically divergent contributions.

This situation impliesthat input parameters must be extremely fine-tuned at high energies to yield the low energyphysics that we observe, a situation that is unappealing especially in connection with GUTs.One way to control the naturalness problem is to consider supersymmetric (SUSY) exten-sions of the standard model. Here the quadratic divergences are cancelled by loop diagramsinvolving the superpartners of the SM particles.

We know that SUSY must be broken inthe real world, and yet the scale of supersymmetry breaking must not be too large or thehierarchy problem will be reintroduced. Thus, although superparticles must be sufficientlyheavy to have avoided detection at present colliders, they cannot be much heavier than afew TeV if we are to meet the naturalness criterion.In this work we will be concerned with the simplest supersymmetric extension of thestandard model (MSSM) [1,2].

In the MSSM we need two Higgs doublets H1 and H2 to givemasses to up and down type fermions and to assure cancellation of anomalies. The neutralHiggs spectrum consists of two CP-even Higgs scalar particles H and h (where MH > Mh),one CP-odd particle A, and a Goldstone boson G which is “eaten by” and gives mass tothe Z boson.

The Higgs sector of the MSSM is highly constrained. At tree level the Higgsboson masses and couplings are determined by two input parameters.

We take these to bethe mass of the CP-odd Higgs boson MA and an angle β which at tree level is given bytan β = v2/v1 where v2 and v1 are the vacuum expectation values of the two Higgs bosonfields H2 and H1. The tree level masses of the CP-even Higgs bosons are then given byM2H,h = 12M2A + M2Z ±q(M2A + M2Z)2 −4M2ZM2A cos2(2β).

(1)The above equation implies the inequalities Mh < MZ,MH > MZ and the sum ruleM2H + M2h = M2Z + M2A.1

Recently it was shown that one loop corrections involving top-quark and squark loopscan significantly modify the sum rule [3] and also violate the bound Mh < MZ [4,5]. For1 TeV squark masses the correction to the light Higgs boson mass is of the order 20 (50) GeVfor a top mass of 150 (200) GeV.

Corrections to the neutral Higgs boson mass sum rule dueto the gauge-Higgs and gaugino-higgsino sectors were considered earlier [6] and were foundto be generically small.In this work we consider corrections to the decay rate Γ(H →ZZ) which is relevantfor the detection of the heavy Higgs boson at a proton supercollider such as the SSC viathe “gold-plated” mode H →ZZ →ℓ+ℓ−ℓ+ℓ−, where ℓis e or µ. We confine ourselvesto corrections due to third family (top and bottom) quark and squark loops.Previouswork on this subject has appeared in Ref.’s [7] and [8] where the effective potential andthe renormalization group methods are used.

We perform a Feynman diagram calculationutilizing the on-mass-shell renormalization scheme, and present explicit analytic results. Thestructure of the paper is as follows: in Section 2 we present our renormalization procedure, inSection 3 we discuss our results, Section 4 lists briefly our conclusions, and in the Appendixwe present the necessary explicit formulas.II.

FORMALISM FOR RADIATIVE CORRECTIONSDue to the presence of mixing in the CP-even and CP-odd sectors the renormalization ofthe Higgs sector of the MSSM presents a few complications when compared to the standardmodel. Therefore, in this section we present in detail our renormalization procedure.

Wefollow the approach of Aoki et. al.

[9] adapted to the MSSM.The Higgs potential in the MSSM isV = g2 + g′28Hi∗1 Hi1 −Hi∗2 Hi22 + g′22 |H∗i1 Hi2|2(2)+(m21 + µ2)Hi∗1 Hi1 + (m22 + µ2)Hi∗2 Hi2 −m23ǫijHi1Hj2 + h.c.,where g(g′) is the SU(2)L(U(1)Y ) gauge coupling, the mi’s, (i = 1,2,3) are the soft super-symmetry breaking Higgs sector mass parameters, and µ is the supersymmetric Higgs mass2

parameter. We can absorb µ2 in Eq.

(1) by redefining m21 + µ2 →m21 and similarly for m22.H1 and H2 are given in terms of the shifted (but unrotated) fields byH1 = 1√2v1 + S1 −iP1√2H−,H2 = 1√2√2H+v2 + S2 + iP2.In order to discuss the tadpole and mixing structure of the theory we need the terms thatare linear and quadratic in S1, S2 and quadratic in P1, P2. These are given byVs = g2 + g′28(v21 −v22)v1 + m21v1 −m23v2!S1 + g2 + g′28(v22 −v21)v2 + m22v2 −m23v1!S2+ g2 + g′216(3v21 −v22) + m212!S21 + g2 + g′216(3v22 −v21) + m222!S22− g2 + g′24v1v2 + m23!S1S2(3a)Vp = g2 + g′216(v21 −v22) + m212!P 21 + g2 + g′216(v22 −v21) + m222!P 22 −m23P1P2.

(3b)We now define the coefficients of S1 and S2 in Eq. (3a) to beT1 = g2+g′28(v21 −v22)v1 + m21v1 −m23v2(4a)T2 = g2+g′28(v22 −v21)v2 + m22v2 −m23v1.

(4b)Eliminating m21, m22 in favor of T1, T2 from Eqs. (4) and substituting back in Eqs.

(3) weobtain, using a matrix notationVs = (S1 S2)T1T2+ 12(S1 S2)T1v100T2v2S1S2(5a)+12(S1 S2)g2+g′24v21 + m23v2v1−g2+g′24v1v2 −m23−g2+g′24v1v2 −m23g2+g′24v22 + m23v1v2S1S2Vp = 12(P1 P2)T1v100T2v2P1P2+ 12(P1 P2)m23v2v1−m23−m23m23v1v2P1P2. (5b)The next step is to introduce rotation matrices O(α) and O(β) such that the part of the CP-even and CP-odd mass matrices that does not depend on T1, T2 is diagonalized.

Specifically,by definingS1S2= O(α)Hh=cos α−sin αsin αcos αHh3

andP1P2= O(β)GA=cos β−sin βsin βcos βGAwe find thatVs = (H h)THTh+ 12(H h)O(−α)T1v100T2v2O(α)Hh+ 12(H h)M2H00M2hHh(6a)Vp = 12(G A)O(−β)T1v100T2v2O(β)GA+ 12(G A)000M2AGA. (6b)Here we have definedT1T2= O(α)THTh.The parameters β, α, MH, Mh and MA are related to the original fundamental parametersv1, v2 and m23 by the following formulastan β = v2v1,M2A = m23 (tan β + cot β) ,tan 2α = M2A + M2ZM2A −M2Ztan 2β,(7)as well as Eq.(1).

Here we used M2Z = g2+g′24(v21 + v22). Carrying out the remaining matrixmultiplications involving the tadpole contributions to the mass matrices we obtain the finalresultVs = HTH + hTh + 12(H h)M2H + bHHbHhbHhM2h + bhhHh(8a)Vp = 12(G A)bGGbGAbGAM2A + bAAGA(8b)withbHH =2v sin 2βTH(cos3 α sin β + sin3 α cos β) +Th sin α cos α sin(α −β))bHh = sin 2αv sin 2β (TH sin(α −β) + Th cos(α −β))(8c)bhh =2v sin 2β (TH cos α sin α cos(α −β) +Th(cos3 α cos β −sin3 α sin β)4

andbGG = 1v (TH cos(α −β) −Th sin(α −β))bGA = 1v (TH sin(α −β) + Th cos(α −β))(8d)bAA =2v sin 2βTH(sin3 β cos α + cos3 β sin α) + Th(cos3 β cos α −sin3 β sin α)The terms linear in H and h are to be thought of as counterterms for the tadpoles. To eachorder in the loop expansion we require that the total tadpole contribution vanishes.

At treelevel this implies −iTH = 0 = −iTh. This then gives the conventional tree level masses.

Atone loop −iTH (−iTh) must cancel the one loop H (h) tadpole diagrams iτH (iτh) (Fig.1).These conditions determine TH and Th and Eqs. (8) determine their contribution to the oneloop mass matrices.Taking as renormalized inputs tan β and MA we calculate the physical masses MH, Mhand the decay rate Γ(H →ZZ) at one loop.

It follows that the measurement of any two ofthe physical quantities MA, MH, Mh and Γ(H →ZZ) will allow us to make a prediction forthe other two. We stress that β is only to be viewed as a useful parametrization of physicalobservables.

Since by itself β has no physical meaning we can renormalize it in any suitablyconvenient way. We explain our renormalization prescription for β below.From this point on we adopt the following notation conventions: a quantity such as afield, coupling, or mass with a subscript 0 indicates a bare quantity, renormalized quantitieshave a subscript r, and physical observables such as the pole of a propagator do not havesubscripts.

The bare tree Lagrangian containsL ⊃12∂µH0∂µH0 + 12∂µh0∂µh0(9)−12(M2H0 + bHH)H20 −12(M2h0 + bhh)h20 −bHhH0h0where M2H0 and M2h0 are taken to be functions of MA0, β0 and MZ0 as given by equation(1). We now write the bare parameters in terms of renormalized parameters and shiftsβ0 = βr + δβ,M2A0 = M2Ar + δM2A,M2Z0 = M2Zr + δM2Z(10)5

and also introduce wave function renormalizationH0 = Z12HHHr + Z12Hhhr,h0 = Z12hhhr + Z12hHHr. (11)Note that Z12HH = 1 + O(α), Z12hh = 1 + O(α) while Z12Hh, Z12hH, bHH, bHh, and bhh are allO(α).

Substituting equations (10) and (11) into (9) we obtain the one loop renormalizedtwo-point functionsiΓHH(p2) = (Z12HH)2(p2 −M2Hr) −∂M2Hr∂xirδxi −bHH + ΠHH(p2)iΓhh(p2) = (Z12hh)2(p2 −M2hr) −∂M2hr∂xirδxi −bhh + Πhh(p2)(12)iΓHh(p2) = Z12Hh(p2 −M2Hr) + Z12hH(p2 −M2hr) −bHh + ΠHh(p2),where xir = M2Ar, M2Zr, βr and the Π’s are the scalar self-energies (Fig.2). The on-shellrenormalization conditions are [9]iΓHH(M2H) = iΓHh(M2h) = iΓHh(M2H) = iΓhh(M2h) = 0i ∂ΓHH∂p2p2=M2H= 1 = i ∂Γhh∂p2p2=M2h(13)Here MH and Mh are the physical masses of H and h. Making the definitions δM2H =ΠHH(M2Hr) −bHH and similarly for δM2h, we obtain from Eqs.

(12) and (13)M2H = M2Hr + ∂M2Hr∂xirδxi −δM2H(14a)M2h = M2hr + ∂M2hr∂xirδxi −δM2h(14b)Z12HH = 1 −12Π′HH(M2Hr)(14c)Z12hh = 1 −12Π′hh(M2hr)(14d)Z12hH =1M2Hr −M2hr−ΠHh(M2Hr) + bHh(14e)6

Z12Hh =1M2hr −M2Hr−ΠHh(M2hr) + bHh,(14f)where the prime in Eqs. (14c,d) indicates differentiation with respect to p2.

Note that M2Hrand M2hr have the same functional form as in Eq. (1) except that they are functions ofrenormalized quantities, i.e.M2Hr,hr = 12M2Ar + M2Zr ±q(M2Ar + M2Zr)2 −4M2ArM2Zr cos2(2βr).

(15)We now drop the subscript r on MZr, MAr and βr. Eqs.

(14a,b) determine the physicalCP-even Higgs boson masses in terms of self energies, tadpole contributions, and shifts ofthe inputs parameters δxi. We now determine the shifts.

The shift δM2A is defined so thatMA is equal to the physical A mass. An analysis similar to that of the CP-even sector yieldsδM2A = ΠAA(M2A) −bAA.

(16)Additionally, we find for the shift in the Z-boson massδM2Z = ΠTZZ(M2Z)(17)where ΠTZZ is the transverse part of the Z boson self energy, ΠµνZZ = gµνΠTZZ + pµpνp2 ΠLZZ. Atthis point it is worth noting that if we are only interested in the sum M2H + M2h we do notneed a specification for δβ .

When Eqs. (14a) and (14b) are added the terms proportionalto δβ cancel leavingM2H + M2h = M2A + M2Z−ΠHH(M2H) −Πhh(M2h) + ΠAA(M2A) + ΠTZZ(M2Z)+bHH + bhh −bAA(18)This is just the renormalization of the neutral Higgs boson mass sum rule and the divergencesin Eq.

(18) implicit in the Π’s and b’s cancel leaving behind a finite correction. Since wedemand that MH and Mh are physical masses they must be individually finite.

Equivalently,since M2H + M2h is finite we must have that M2H −M2h is also free of divergences. This latterrequirement gives7

∂∆∂β δβ + ∂∆∂M2ZδM2Z + ∂∆∂M2AδM2A −δM2H + δM2h = finite(19)where ∆=q(M2A + M2Z)2 −4M2AM2Z cos2(2β). The above equation clearly determines onlythe “infinite” part of δβ.

By “infinite” we mean the part that is proportional to CUV =1ǫ −γ+log 4π in dimensional regularization. To fully specify δβ we take a MS-type approachand define δβ to be purely “infinite” so that Eq.

(19) becomes∂∆∂β δβ = − ∂∆∂M2ZδM2Z + ∂∆∂M2AδM2A −δM2H + δM2h!∞(20)where the subscript ∞on a quantity indicates the “infinite” part of that quantity. Eq.

(20)impliesδβ =12M2AM2Z sin(4β)(21)×M2Z δM2A + M2A δM2Zcos2(2β) −M2H δM2h −M2h δM2H∞.This definition of β at one loop gives renormalized CP-even Higgs boson masses in closeagreement with those obtained using the effective potential [4]. This shift in δβ induces ashift in α through equation (7)δα = sin(4α) δβsin(4β) + M2A δM2Z −M2Z δM2A2(M4A −M4Z)!.

(22)We now come to the renormalization of the HZZ coupling.The bare HZZ and hZZcouplings are given byλHZZ0=e0M3Z0MW0(M2Z0 −M2W0)12 cos(β0 −α0),λhZZ0= λHZZ0tan(β0 −α0).Defininge0 = er + δe,Z12HH = 1 + δZ12HH,(Z12ZZ)2 = 1 + δZZZ(here Zµ0 = Z12ZZZµr + Z12ZAAµr where Zµ0(Zµr) is the bare (renormalized) Z boson fieldand Aµr is the renormalized photon field) we obtain for the renormalized one loop 3-pointfunctionΓHZZµν= (λHZZr+ λHZZCT )gµν + ∆ΓHZZµν(23)8

where λHZZr=erM3Zr cos(βr−αr)MW r(M2Zr −M2W r)12 andλHZZCT= λHZZrδee +32δM2ZM2Z−12δM2Z −δM2WM2Z −M2W−12δM2WM2W(24)−tan(βr −αr)(δβ −δα) + δZ12HH + δZZZ + Z12hH tan(βr −αr)and ∆ΓHZZµνis the explicit one loop Feynman diagram contribution (Fig.3).The angleαr is defined as in Eq. (7), but with the right hand side written in terms of renormalizedquantities.

The expressions for δM2Z, δβ, δα, Z12HH and Z12hH in terms of self energies andtadpole contributions are given in Eqs. (17), (21), (22) and (14c,e).

We simply state theresults for the remaining shifts δe, δM2W and δZZZ. We haveδee = 12ΠTγγ′(0) + 4c2W −34sWcW!

ΠTZγ(0)M2Z,(25)δM2W = ΠTW W(M2W),δZZZ = −ΠTZZ′(M2Z)where cW = MW/MZ and sW =q1 −c2W. We note that ΠTZγ(0) vanishes in our case.

The H−hmixing gives a contribution to ΓµνHZZ through the term proportional to Z12hH. The quantity onthe R.H.S.

of Eq. (23) is given as a sum of terms which are individually divergent.

In the fullsum the divergences must of course cancel. We checked both analytically and numericallythat this is indeed the case.

The renormalizability of the theory requires that the definitionof δβ which renders the CP-even Higgs boson masses finite also gives finite couplings.The explicit one loop Feynman diagrams shown in Fig.3 give a contribution to the three-point function which can be expanded in terms of form factors as∆ΓµνHZZ = D0gµν + D1pµ1pν1 + D2pµ2pν2 + D3pµ1pν2 + D4pµ2pν1(26)(a form factor proportional to ǫµναβp1αp2β vanishes by CP invariance). The formula for thedecay rate at one loop isΓ =√1 −4r128πr2MH((1 −4r + 12r2)(λHZZr)2 + 2λHZZrRe(λHZZCT+ D0) + |λHZZCT+ D0|2+ M2H(1 −2r)(1 −4r)λHZZrRe(D4) + Re[(λHZZCT+ D0)D∗4]+ M4H12 −2r2|D4|2)(27)9

where r = M2Z/M2H and we list D0 and D4 in the Appendix.We note that the terms in the above expression which do not involve λHZZrare formallyof O(g6). Nevertheless we find that for large Higgs boson mass (MH ≫MZ) they are numer-ically important.

This is because λHZZris proportional to cos(α −β) which is proportionalto 1/M2H for large MH and hence small. Keeping these O(g6) terms is consistent: the termsin the amplitude that are of O(g5) which arise at two loop level also give a contribution ofO(g6) in the decay rate, but these two loop O(g6) terms are proportional to cos(α −β) andare thus suppressed when MH ≫MZ, in precisely the region where the O(g6) terms in ourone loop expression become large.III.

RESULTSIn the MSSM at tree level the decay rate Γ(H →ZZ) is suppressed relative to the samedecay rate in the standard model by the factor cos2(α −β). The “gold-plated” decay modeH →ZZ →4ℓhas great discovery potential for a standard model Higgs boson at a protonsuper collider such as the SSC for Higgs boson masses 130 GeV <∼Mφ <∼800 GeV [10].

Thediscovery potential for the heavy Higgs boson of the MSSM in this mode is not as promisingdue to the above mentioned suppression factor. However, the “gold-plated” mode may bethe only discovery mode for the heavy Higgs boson at a hadron collider [11].

The discoverypotential is improved when radiative corrections are taken into account.We discuss our numerical results below. We have checked our numerics in a number ofways.

First, we checked the cancellation of divergences as mentioned in the last section.Second, we found our result for the correction to the neutral Higgs boson mass sum ruleagreed very closely with that of Ref. [3].

Lastly, we checked that our calculation, whenmodified to give the correction to the standard model Higgs boson decay rate to two Z’sdue to an extra heavy fermion doublet, agrees with the results of Ref. [14].In Fig.4a we show the tree level and radiatively corrected decay rate versus the heavyHiggs mass for tan β=5 and a top quark mass of 160 GeV.

In this figure we have not included10

mixing effects, i.e. At = Ab = µ = 0 and the squark masses are all equal.

We show thecorrected rate for the two squark mass choices Msq = 300 GeV and Msq = 1000 GeV. We seein Fig.4a the importance of keeping corrections which are of O(g6) in the rate.

The one loopcorrections which contribute O(g4) to the rate fall with MH (as they multiply the tree levelcoupling). However, the one loop corrections which contribute O(g6) to the rate increaseas MH increases.

Hence, these terms eventually dominate the rate as MH becomes large.In Fig.4a the corrected rate is dominated by the O(g4) terms for small MH, and hence itinitially falls as MH increases beyond the kinematic suppression. Eventually, however, theterms of order O(g6) become larger than the O(g4) terms and the rate then rises with MH.This begins to occur for values of MH of about 500 GeV.In Fig.4b we show the rate versus tan β for a Higgs boson mass of 300 GeV, a top quarkmass of 160 GeV, a squark mass of 1 TeV, and again for no mixing.

We see that the correctedrate is approximately twice as large as the tree level value, almost independent of tan β. Aswe will discuss below, the rate depends dramatically on tan β once mixing is included.In Fig.5 the ratio of the radiatively corrected rate to the tree level rate is shown versusthe top quark mass, for the same set of parameters as Fig.4b, and tan β=5. Fig.5 illustratesthat the corrected rate depends strongly on two parameters in the case of no mixing.

Clearlythe rate depends on the value of the top quark mass. But note for MH=1 TeV that even fora top quark mass as small as 100 GeV the corrected rate is still over a factor of two largerthan at tree level.

Thus the relative size of the correction depends greatly on the value ofMH as well. Note, however, that when the top quark mass is less than around 120 GeV weexpect that the corrections from other sectors will be of the same order of magnitude as thecorrection due to the quark/squark sector included here.When mixing is included the parameter space increases.

We will choose a point in mixingspace and examine the effect of mixing in deviations from that point. We choose A−termsAt = Ab = 600 GeV and squark masses ˜mt1 = ˜mb1=600 GeV, and ˜mt2 = ˜mb2=300 GeV.Additionally, we will consider the two cases µ = ±400 GeV.

In all three of the figures 6,7 and 8 the heavy Higgs boson mass is set to 300 GeV and the top quark mass is 16011

GeV. In order to isolate the effect of mixing we will plot the ratio of the corrected rateincluding mixing to the corrected rate with no mixing (where the common squark mass isset to 600 GeV).

In Figs.6 we plot this ratio vs. the squark mass ˜mt1. We find that the effectdue to mixing is strongly dependent on tan β and µ.

For large values of tan β the effects ofmixing are greatly enhanced. As shown in Figs.6, the inclusion of mixing can change therate by a factor 1.3 for tan β=2 and for tan β=20 by a factor 2.7 or 0.3, for µ=-400 GeV orµ=+400 GeV, respectively.Similar ratios are seen in Figs.7, where the ratio of the corrected rate including mixing tothe corrected rate with no mixing is shown vs. At, the top squark mixing parameter.

As inFigs.6 the two curves for µ = ±400 GeV are similar when tan β=2; the rate can be increasedby 50% or decreased by 25%. If tan β=20 the effects of mixing are more pronounced and theratio varies between roughly 1/3 and 3.

The µ=400 GeV curve in Fig.7a (and the tan β=2curve in Fig.8) does not span the entire ordinate axis shown because an unphysical regionof the squark mixing parameter space is encountered.In Fig.8 we plot the (mixing) to(no mixing) ratio vs. the supersymmetric Higgs mass parameter µ. We see there is littledependence on µ for small tan β, while for larger values of tan β the dependence is quitesignificant.

If tan β=20 the ratio varies between 4 and 1/36 as µ varies from -750 to 750GeV. Finally, we note that there is very little dependence on the bottom squark masses andA−term Ab for the mixing configurations considered.IV.

CONCLUSIONSTo summarize, we have computed the one loop corrections to the decay rate Γ(H →ZZ)in the MSSM including third family quark and squark loops. We perform a Feynman diagramcalculation in the on-mass-shell renormalization scheme.

As the tree level rate falls like 1/MHfor large MH and we find corrections that grow with MH, the corrected rate may be manytimes the tree level rate. For example, at MH = 1 TeV the corrected rate may be 13 timesthe uncorrected rate for mt=200 GeV (with no squark mixing).

The corrected rate depends12

very strongly on the squark mixing parameters. For example, for the mixing configurationconsidered here, the rate varies by two orders of magnitude as the Higgs mass parameter µvaries between ±750 GeV.

Indeed, the squark mixing parameters µ, At, and the top squarkmasses, in addition to the top quark mass, must be measured in order to test the Higgssector of the MSSM.ACKNOWLEDGMENTSWe would like to thank Mary K. Gaillard for many useful discussions. One of us (D.P.

)acknowledges support from a University of California at Berkeley Department of Educationfellowship.In this Appendix we give explicit analytic expressions for the self energies, tadpoles, andform factors introduced in the text. Our expressions are given in terms of the standardA, B, C functions introduced by Passarino and Veltman [12] which appear in one loop cal-culations.

We adopt the metric (1,-1,-1,-1), which is different than that of Ref. [12].

Explicitanalytic formulas for these functions appear in Ref. [13].To make the equations more concise we adopt the following conventions.

Nc denotesthe number of quark colors. The index α runs over the top and bottom sectors while theindices i, j, and k run over squark mass eigenstates.Thus, mα denotes a quark masswhile ˜mαi denotes a squark mass.For the A and B functions we define Aα = A(m2α),˜Aαi = A( ˜m2αi), B0α = B0(p2, m2α, m2α), ˜B0αij = B0(p2, ˜m2αi, ˜m2αj) and similarly for the restof the B’s.A C function has six arguments: three external squared momenta and thethree squared masses of the particles which appear in loop of the 3-point diagram.

We thusdefine ˜C0αijk = C0(M2Z, M2Z, M2H, ˜m2αi, ˜m2αj, ˜m2αk) and C0α = C0(M2Z, M2Z, M2H, m2α, m2α, m2α)with analogous definitions for the rest of the C’s.First we give expressions for the Higgs boson self energies.13

ΠHH(p2) = NcXαij( ˜V Hαij)2 ˜B0αij −NcXαi˜UHHαii ˜Aαi(A1)−12NcXα(V Hα )2m2αB0α + p2(B21α −B1α) +148π2(m2α −p26 ).The various V and U vertex factors are shown in Fig.9 and explicit expressions appear inRefs. [1,10].

However, the H −h −˜qkL −˜qkL and H −h −˜qkR −˜qkR vertices givenin Ref. [10] are incorrect.

In the notation of Ref. [10] the above couplings areig2 sin 2α4 2T3k −ek sin2 θWcos2 θW−m2qM2WDk!and ig2 sin 2α4 2ek tan2 θW −m2qM2WDk!respectively (Dup = 1/ sin2 β, Ddown = −1/ cos2 β).Πhh is given as ΠHH with ˜V Hαij →˜V hαij,˜UHHαii→˜Uhhαii, and V Hα →V hα .

ΠHh is given asΠHH with ( ˜V Hαij)2 →˜V hαij ˜V Hαij,˜UHHαii→˜UHhαii , and (V Hα )2 →V Hα V hα . ΠAA is given as ΠHHwith ˜V Hαij →˜V Aαij,˜UHHαii →˜UAAαii , V Hα →V Aα , and B0α →13B0α.

Next we list the transversepart of the gauge boson self energies.ΠTZZ(p2) = NcXαi˜UZZαii ˜Aαi(A2)−2NcXαij( ˜V Zαij)2˜m2αi ˜B0αij −( ˜m2αi −˜m2αj + p2) ˜B1αij + p2 ˜B21αij +116π2( ˜m2αi + ˜m2αj2−p26 )−8NcXα(V Z5α)2m2αB0α +(V Zα )2 + (V Z5α)2p2(B21α −B1α)ΠTW W(p2) = NcXαi˜UW Wαii˜Aαi(A3)−2NcXαij( ˜V Wij )2˜m2ti ˜BW0 −( ˜m2ti −˜m2bi + p2) ˜BW1 + p2 ˜BW21 +116π2( ˜m2ti + ˜m2bi2−p26 )−8Nc(V W)2m2tBW0 −(m2t −m2b + 2p2)BW1 + 2p2BW21where ˜BW = B(p2, ˜m2t, ˜m2b) and BW = B(p2, m2t, m2b). ΠTγγ is given as ΠTZZ with ˜V Zαij →˜V γαij, ˜UZZαii →˜Uγγαii, V Zα →V γα ,V Z5α →0.

The heavy Higgs boson tadpole contribution isgiven byTH = Nc4XαV Hα mαAα −Xαi˜V Hαii ˜Aαi. (A4)Th is given as TH with ˜V Hαii →˜V hαii and V Hα→V hα .

Lastly, the two three-point Feynmandiagram form factors which are relevant for calculating the Higgs boson decay rate are14

D0 = 8NcXαmαV Hα ((V Zα )2 + (V Z5α)2)×4C24α + (M2H −2M2Z)C12α + 2M2ZC11α −M2H2 C0α −B0(M2Z, m2α, m2α)+ 8NcXαmαV Hα (V Z5α)2M2H −4m2αC0α −2B0(M2Z, m2α, m2α)(A5)−8NcXαijk˜V Hαki ˜V Zαij ˜V Zαjk ˜C24αijk+ NcXαij˜V Hαij ˜UZZαij B0(M2H, ˜m2αi, ˜m2αj)andD4 = 8NcXαmαV Hα(V Zα )2 + (V Z5α)2(4C23α + C0α −4C12α) −2(V Z5α)2 (C11α −C12α)−8NcXαijk˜V Hαki ˜V Zαij ˜V Zαjk ˜C23αijk −˜C12αijk. (A6)15

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FIGURESFIG. 1.

The Higgs boson tadpole diagram.FIG. 2.

The Higgs boson and gauge boson self energy diagrams.FIG. 3.

The one loop H →ZZ Feynman diagrams included in our calculation. The dashedloops represent squarks, the solid loop represents quarks.FIG.

4. The tree level and one loop corrected decay rate Γ(H →ZZ).

Fig.4a shows the ratesvs. the heavy Higgs boson mass for tan β =5, mt=160 GeV, and no mixing.

Fig.4b shows thedecay rates vs. tan β for the same parameters as in Fig.4a, with MH=300 GeV.FIG. 5.

The ratio of the corrected decay rate to the tree level rate vs. the top quark mass. Theparameters are the same as in Fig.4a, with the Higgs boson mass set to 300 GeV and 1 TeV.FIG.

6. The ratio of the corrected rate including mixing to the corrected rate without mixingvs.

the top squark mass ˜mt1 as explained in the text. The top quark mass is 160 GeV.FIG.

7. The ratio of the corrected rate including mixing to the corrected rate without mixingvs.

the top mixing parameter At, as explained in the text. The top quark mass is 160 GeV.

InFig.7a the point where the µ=400 GeV curve stops corresponds to an unphysical point in squarkmixing parameter space.FIG. 8.

The ratio of the corrected rate including mixing to the corrected rate without mixing,as explained in the text, vs. µ. The heavy Higgs boson mass is 300 GeV, and the top quark massis 160 GeV.

The curve for tan β = 2 stops at an unphysical point in the squark mixing parameterspace.FIG. 9.

The vertices used in the Appendix are displayed. The values of the vertex factorsmay be found in Refs.

[1,10]. Note that the value of U Hhαij listed in Ref.

[10] is incorrect (see theAppendix for the correct values of these vertices).18

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