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Higgs sector CP violation in the

arXiv:hep-ph/9205247v1 30 May 1992SCIPP-92/19May, 1992Higgs sector CP violation in theMinimal Supersymmetric ModelAlex PomarolSanta Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064AbstractWe study the possibility that CP is spontaneously broken in the Minimal Su-persymmetric Model when radiative corrections to the Higgs potential are included.We show that this can only occur if a light Higgs boson exists. Considering therecent ALEPH Higgs search, we exclude most of the parameter space of the model.The possibility of explicit CP violation in the model is also briefly discussed.

It has been known for a long time that when supersymmetry (SUSY) is im-posed on the two Higgs doublet model (THDM), tree-level flavor changing neutralcurrents and CP violation are simultaneously avoided in the Higgs sector[1]. Nev-ertheless, since SUSY must be softly broken, new terms in the Higgs potential canbe induced by radiative corrections and CP non-conserving effects could show upin the Higgs sector.

CP may be broken in two different ways: explicitly and spon-taneously. In the first case, CP violation derives from complex scalar self-couplingsinduced by radiative corrections by sectors of the theory which violate CP.

In thesecond case, a relative phase between the vacuum expectation values (VEVs) ofthe two Higgs doublets arises which spontaneously breaks the CP symmetry[2].The purpose of this paper is to study the possibility that CP violation appearsin one of these ways in the Higgs sector of the minimal supersymmetric model(MSSM). In such a case the CP-even and CP-odd neutral scalars would mix witheach other giving rise to important phenomenological consequences[3].

It was re-cently pointed out[4] that spontaneous CP violation (SCPV) in the MSSM canoccur. However, it is known that in order that radiative corrections can cause aspontaneous broken vacuum, a light scalar is required[5,6].

Therefore, an analysis ofthe physical spectrum, not carried out in ref. [4], is necessary in order to determinethe viability of this model.Let Φ1 and Φ2 denote two Higgs doublets with hypercharges Y = 1.Themost general renormalizable SU(2)L × U(1)Y gauge invariant two Higgs doubletpotential is given byV (Φ1, Φ2) = m21Φ†1Φ1 + m22Φ†2Φ2 −(m212Φ†1Φ2 + h.c.)+ λ1(Φ†1Φ1)2 + λ2(Φ†2Φ2)2 + λ3(Φ†1Φ1)(Φ†2Φ2) + λ4(Φ†1Φ2)(Φ†2Φ1)+ 12hλ5(Φ†1Φ2)2 + h.c.i+ 12hΦ†1Φ2{λ6(Φ†1Φ1) + λ7(Φ†2Φ2)} + h.c.i,(1)where by hermiticity only m212, λ5, λ6 and λ7 can be complex.

Let us first considerthe case where these parameters are real, ie. CP is not explicitly violated.

Afterspontaneous symmetry breaking, the VEVs of the neutral components of the Higgs2

doublets are given by< φ01 >= v1 ,< φ02 >= v2eiξ .In order to have SCPV, ie. ξ ̸= nπ2 (n ∈Z), we needλ5 > 0 ,(2)2m212−λ6v21−λ7v224λ5v1v2 < 1 .

(3)In this case, at the minimum of the potential,cos ξ = 2m212−λ6v21−λ7v224λ5v1v2.When SUSY is imposed on the two Higgs doublet potential we have[1] ,λ1 = λ2 = 18(g2 + g′2) ,λ3 = 14(g2 −g′2) ,λ4 = −12g2 ,λ5 = λ6 = λ7 = 0 .Thus, eq. (3) does not hold and ξ must be 0 or π.

When radiative corrections areconsidered, new terms in the Higgs potential are induced. In the limit where theSUSY scale is large, MSUSY ≫mW , only terms of dimension less than or equalto 4 are not suppressed by inverse powers of MSUSY .

In this limit, the effectivelow-energy Higgs potential of the MSSM is given by eq. (1).In order to know whether eqs.

(2) and (3) hold, we must calculate the inducedλ5 parameter. The λ6 and λ7 parameters are in fact not relevant because m212 isa free parameter.

The only contribution that generates a positive λ5 comes fromdiagrams involving loops of charginos and neutralinos (fig. 1).

Squark and Higgsloops give a negative contribution to λ5 but they can be neglected in the case of3

small ˜qR −˜qL mixing and small m212 respectively. Quarks and gauge bosons do notcontribute.

In the limit of equal mass charginos and neutralinos, we findλ5 =g432π2 ∼5 · 10−4.Therefore, we see from eq. (3) that, in order that SCPV occur, the tree levelparameter m212 must be of O(λ5v1v2) ∼(3 GeV)2 .

This seems to contradict theGeorgi–Pais theorem[6] which says that SCPV can only be generated by radiativecorrections when a tree-level massless scalar field, other than the Goldstone boson,exists⋆.Notice, however, that this theorem is strictly true only for first ordercorrections to the effective potential. When two-loop corrections are considered,it is easy to see that the scalar can have a tree-level mass whose magnitude isof one-loop order[6].

Of course, the theorem can only be applied when the trueminimum is close to the tree-level minimum.To analyze the physical spectrum, let us make the following rotationΦ′1 = cosβ Φ1 + sinβ e−iξΦ2 = G+v +1√2h0 + iG0!,Φ′2 = −sinβ Φ1 + cosβ e−iξΦ2 = H+1√2H0 + iA0!.where tan β = v2/v1, v =qv21 + v22, G+ and G0 are the goldstone bosons, h0 andH0 are CP-even fields and A0 is a CP-odd field[7]. The three physical neutral Higgsboson mass eigenstates are mixtures of h0, H0 and A0.

The relevant elements ofthe neutral scalar mass matrix are given byM2h0A0 = −2m212 sin ξ ,M2H0A0 =λ5(v22 −v21) cos ξ + (λ6 −λ7)v1v2sin ξ ,M2A0A0 = 2λ5(v21 + v22) sin2 ξ . (4)It is clear from eq.

(4) that there is a light Higgs boson for any value of ξ andtan β.⋆We must have m212 = 0 in order to have a massless Higgs boson (A0) at tree-level.4

Let us first consider the case where the other neutral Higgs bosons are muchheavier. These will be predominantly CP-even states with a small admixture ofA0.

In this case our model will be similar to the MSSM without CP violation andwith a light A0:m2A0 ≃M2A0A0 <∼(6 GeV)2 .Since the recent limit from ALEPH Collaboration[8] implies a lower bound of 20GeV for the CP-odd scalar mass, this possibility is ruled out⋆.A second possibility is that the mass of one of the CP-even scalars is alsosmall and mixes substantially with the A0. In this case, the ALEPH data mustbe carefully examined to determine if this possibility is excluded.

In particular,the lower scalar mass limits from ALEPH are not valid in a CP violating THDM.To see why this is so, let us denote by h01 and h02 the two lightest Higgs bosons,and by g(p1+p2)µ2 cos θW Θh01h02Z andigmZcos θW gµνΘh0i ZZ the Feynman rules for the tree levelh01h02Z and h0i ZZ (i = 1, 2) couplings respectively. For a CP conserving Higgssector (h02 ≡A0),Θ2h01ZZ + Θ2h01h02Z = 1 .

(5)This relation, which plays a crucial role in inferring lower mass limits for the Higgsbosons, need not be satisfied when CP is violated in the Higgs sector. Nevertheless,a sum rule similar to eq.

(5) can be also derived for a CP violating THDM. It isgiven by[7]Θ2h01ZZ + Θ2h02ZZ + Θ2h01h02Z = 1 .

(6)On the other hand, assuming that mh01 ≃mh02 <∼20 GeV, the ALEPH Higgs⋆The ALEPH limit is only valid in the region tan β > 1. For tan β < 1 there exists a region ofthe MSSM parameter space in which a light A0 is not excluded by ALEPH.

However, regionsof parameter space where tan β < 1 are strongly disfavored in low-energy supersymmetricmodels[9].5

search[8] implies the following limits on the Θ’s:Θ2h0i ZZ <∼0.1 ,Θ2h01h02Z <∼0.7 . (7)Combining eq.

(6) and eq. (7) we can rule out the possibility of two light Higgsbosons of indefinite CP.

If the h01 and h02 were light enough, they would decayoutside the detector and the constraints of eq. (7) could not be deduced.

Nev-ertheless, since any contribution to the Z width from non-standard processes islimited to less than 0.26 Γν¯ν[10], bounds on the h01h02Z and h0i ZZ couplings canalso be inferred[8,11], which turn out to be in contradiction with eq. (6).Finally, let us briefly consider the case when other sectors of the theory violateCP.

In that case, the induced couplings λ5, λ6 and λ7 can be complex† and wehave a CP violating Higgs sector even for real VEVs. In supersymmetric theoriesthere are a number of new sources of CP violation from the various supersymmetricsectors.

Nevertheless, experimental limits on the neutron electric dipole momentrequire any such CP violating phases, ϕ, to be less than 10−2 [12]. As a result,explicit CP violation effects in the Higgs potential will be of orderλ5,6,7 · ϕ ∼10−5 .These effects are too small to have any significant phenomenological implications.Summarizing, we have seen that the MSSM with SCPV require the existence ofa Higgs boson with a mass of the order of a few GeV.

Based on the recent ALEPHHiggs search[8], we have seen that this model is easily ruled out (except perhaps fora small disfavored region of parameter space where tan β < 1). Although explicitCP violation in the Higgs sector is in principle possible, it turns out to be too smallto be phenomenologicaly relevant.† m212 can be made real by a redefinition of the Higgs doublets.6

AcknowledgementsI would like to thank Howard Haber for helpful conversations and for a criticalreading of the manuscript.This work was supported by a fellowship of MEC(Spain).7

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