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Constraints from Global Symmetries on

arXiv:hep-ph/9207267v1 29 Jul 1992SCIPP 92/29July, 1992Constraints from Global Symmetries onRadiative Corrections to the Higgs SectorHoward E. Haber and Alex PomarolSanta Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064AbstractWe discuss the implications of global symmetries on the radiative correctionsto the Higgs sector. We focus on two examples: the charged Higgs mass in theminimal supersymmetric model and the Higgs couplings to vector boson pairs.

Inthe first case, we find that in the absence of squark mixing a global SU(2)×SU(2)symmetry protects the charged Higgs mass from corrections of O(g2m4t/m2W ). Inthe second case, it is the custodial symmetry which plays an analogous role inconstraining the fermion-mass dependence of the radiative corrections.

1. IntroductionGlobal symmetries play an important role in analyzing the radiative correc-tions of the tree-level parameters of a theory.

Often, a theory will possess a “nat-ural” tree-level relation – i.e., a relation among tree-level parameters which isattributable to some underlying symmetry. In this case, radiative corrections tothis relation must be finite; moreover, the nature of the underlying symmetry canprovide information of the order of magnitude of these corrections.

As an example,in the Standard Model (SM) the so-called global custodial SU(2) symmetry[1] playsa crucial role in the analysis of the radiative corrections to the ρ-parameter. Oneof the most important implications of this global SU(2) symmetry is the screeningtheorem of the Higgs boson[2].The purpose of this paper is to make use of global symmetries in the analy-sis of the radiative corrections to the Higgs sector.

The study of such radiativecorrections in the minimal supersymmetric model (MSSM) has recently receivedmuch attention. One-loop effects have been found which significantly modify thetree-level predictions for the Higgs masses of the MSSM and give rise to importantphenomenological consequences[3,4].For the light neutral CP-even Higgs mass,radiative corrections involving loop contributions from top quarks and their su-persymmetric partners induce a substantial squared mass shift of O(g2m4t/m2W ).However, for the charged Higgs squared mass, the radiative corrections are not soimportant because (in the absence of squark mixing) only one-loop corrections ofO(g2m2t ) are induced.

In this paper, we shall show that these results follow eas-ily by studying the implications of the underlying global symmetries of the Higgspotential.In section 2, we make use of the global symmetries of the Higgs potentialto analyze the radiative corrections to the charged Higgs mass in the MSSM. Inparticular, we will see that due to an approximate extended custodial symmetry ofthe Higgs potential, radiative corrections of O(g2m4t /m2W) never arise.

In section 3,we study in a similar way the one-loop effects to the couplings of the Higgs bosons2

to a pair of vector bosons. We shall demonstrate that the custodial SU(2) symmetryplays a similar role to that in the radiative corrections to the ρ-parameter.2.

Radiative corrections to the charged Higgs massOne of the relations that supersymmetry (SUSY) imposes on the Higgs poten-tial is the mass sum-rule[5]m2H± = m2A0 + m2W . (1)Because SUSY is not an exact symmetry of nature, eq.

(1) only holds at tree-level and is modified by radiative corrections. On dimensional grounds, one mightnaively expect that the radiative corrections to eq.

(1) should depend quadraticallyon some large mass scale in the problem.Specifically, the largest contributionexpected would come from loops of superpartners whose masses are of the order ofthe SUSY breaking scale, MSUSY . However, such contributions are certainly absentat one-loop for physical observables[6].

Specifically, all one-loop corrections thatgrow as M2SUSY can be absorbed in the redefinition of the mass-squared parametersof the Higgs potential. In contrast, whereas these mass-squared parameters are allindependent, the scalar self-couplings are related by SUSY.

Therefore, we do nothave enough freedom to absorb all the effects of the superpartners. Since theseeffects can only show up in dimensionless parameters, these will depend at mostlogarithmically on MSUSY .

Note that decoupling does not apply when the mass ofa heavy particle, M, can be made large by increasing a dimensionless parameter(e.g., the masses of the fermions and the Higgs boson in the SM). In that case,one-loop corrections to eq.

(1) of O(M2) can show up.The recent experimental result that mt > 91 GeV[7] suggests that radiativecorrections to eq. (1) due to the loop contributions of top quarks and top squarksshould be the dominant corrections.

Naively, one expects one-loop corrections oforder h2t m2t ∼g2m4t /m2W where ht is the top Yukawa coupling. Nevertheless, ex-plicit calculation shows that in the absence of squark mixing, the leading radiativecorrections are only of order g2m2t[4].3

To exlain this result, we first analyze the two-doublet Higgs potential beforeimposing SUSY. Let Φ1 and Φ2 denote two Higgs doublets with hyperchargesY = 1.The most general renormalizable and SU(2)L×U(1)Y gauge invariantHiggs potential 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,(2)where a discrete symmetry Φ2 →−Φ2 has been imposed on the dimension-fourterms.

This discrete symmetry guarantees the absence of flavor changing neutralcurrent[8]. It will be convenient to represent Φ1 by a real four vector, i.e.,Φ1 = φ+1φ01!= φ3 + iφ4φ1 + iφ2!→Φ1 = (φ1, φ2, φ3, φ4) .

(3)Since the MSSM Higgs sector automatically conserves CP at tree-level, we hence-forth make this assumption. The physical spectrum of the model consists in twocharged Higgs bosons (H±) and three neutral ones: two CP-even (h0 and H0) andone CP-odd (A0).

The masses of the A0 and H± are related bym2H± = m2A0 + 2m2Wg2(λ5 −λ4) . (4)Consider the limit where m12 = λ4 = λ5 = 0.

In this limit the global sym-metry of the Higgs potential of eq. (2) is enlarged to O(4)1×O(4)2.Here, wefind convenient to choose the symmetry transformations such that Φ1 transformsas a 4-vector under both O(4)1 and O(4)2, whereas Φ2 transforms as a 4-vectorunder O(4)1 and as a singlet under O(4)2.

When the scalar fields develop vac-uum expectation values (VEVs), ⟨Φi⟩= (vi, 0, 0, 0), the O(4)1×O(4)2 symmetrybreaks down to O(3)1×O(3)2 which is locally isomorphic to SU(2)×SU(2). Threeof the six Goldstone bosons produced can be associated with the breakdown ofSU(2)L×U(1)Y →U(1)EM; these will be “eaten” when the Z and W ± bosons ac-quire mass.

The other three Goldstone bosons are the A0 and the H±. If the4

broken symmetries corresponding to the A0 and H± Goldstone bosons are sym-metries of the full theory (prior to symmetry breaking), then it would follow thatmA0 = mH± = 0 to all orders in perturbation theory. In general, this will not bethe case, in which case the A0 and H± are pseudo-Goldstone bosons (i.e., theywould acquire a calculable mass due to radiative corrections).Consider first the coupling of Higgs bosons to third generation quarks.Insupersymmetric theories, the coupling quark doublets to Higgs doublets is suchthat Φ1 couples exclusively to bR and Φ2 couples exclusively to tR.

We assumethis coupling pattern in the following. In the limit hb = 0,LY = −ht¯tL ¯bLiτ2Φ∗2tR + h.c.(5)Since the global symmetry of this term is SU(2)L×U(1)Y × O(4)2, t-loop radiativecorrections will not induce a mass terms for the A0 and the H±.

In the case of theMSSM, SUSY imposes the following condition on the parameters of the two Higgsdoublet potential of eq. (2)[5]λ1 = λ2 = 18(g2 + g′2) ,λ3 = 14(g2 −g′2) ,λ4 = −12g2 ,λ5 = 0 .

(6)According to the above argument, the A0 and H± must be massless to all ordersof perturbation theory in the limit of m12 = g = 0. That is, t-loop corrections tothe mass sum-rule [eq.

(1)] must go to zero in this limit. It follows that correctionsof O(h2t m2t) to eq.

(1) must cancel out. In fact, each term in the one-loop radiativecorrections to eq.

(1) must depend quadratically on either mA0, mW or mb.However, in the SUSY model, we must also consider the squark sector sinceradiative corrections of O(h2t m2t ) can also arise from top squark loops. Assumingthat there is no ˜tL −˜tR mixing, we find in the limit of hb = g = 0Lstop = L(Φ†1Φ1, Φ†2Φ2, ˜Q† ˜Q, ˜U∗˜U) + h2t ˜Q†iτ2Φ∗22,(7)5

where˜Q = ˜tL˜bL!and˜U = ˜t∗R . (8)These terms are also SU(2)L×U(1)Y ×O(4)2 invariant and, therefore, correctionsof O(h2t m2t ) cannot arise from this sector either.

Finally, if ˜tL −˜tR is present, wehave new terms given byLmix = −µht ˜Q†(iτ2Φ∗1) ˜U∗+ htAU ˜Q†(iτ2Φ∗2) ˜U∗+ h.c.(9)which are not invariant under the global O(4)2 symmetry. Thus, top squark loopsinvolving the interactions of eq.

(9) can induce corrections to eq. (1) of O(h2t m2t ).⋆Notice that in the limit µ = 0 the terms in eq.

(9) restore the O(4)2 symmetry and,although we still have a ˜tL −˜tR mixing (AU ̸= 0), no corrections of O(h2tm2t ) canarise. This results are in agreement with the explicit one-loop radiatively correctedcharged Higgs mass obtained in the literature[4].Let us now analyze the corrections to eq.

(1) from other sectors of the theory.First, we consider the two Higgs doublet potential [eq. (2)] before imposing SUSY,where now we take the limit λ4 = λ5.

In this limit the Higgs potential is only O(4)1invariant. After spontaneous symmetry breaking (SSB), the residual symmetry,O(3)1 ∼SU(2), is the so-called custodial symmetry[1] which is responsible for therelation m2W = m2Z cos2 θW .

Setting λ4 = λ5 in eq. (4) yieldsm2H± = m2A0 .

(10)Radiative corrections to this relation will only come from sectors of the theory notinvariant under the global custodial symmetry. The custodial SU(2) symmetry is⋆The one-loop top squark corrections for large MSUSY are in fact of Ohµ2h2t(m2˜tL −m2˜bL)/m2˜tLi.However, if µ and the diagonal soft-supersymmetry breaking squark masses are of the sameorder, we have that µ2(m2˜tL −m2˜bL)/m2˜tL ∼m2t resulting in a top squark correction ofO(h2t m2t).6

an approximate symmetry of the minimal supersymmetric Higgs potential (λ4 =−g2/2 ∼λ5 = 0). In the limit g →0, i.e., λ4 = λ5, eq.

(10) must hold to all ordersin the Higgs self-interactions. We conclude that the only non-vanishing correctionto eq.

(1) from Higgs self-interactions must be proportional to g2.Finally, let us consider the Higgs-gauge boson interactions. They derive fromthe scalar kinetic termLkin =2Xi=112 trn(DµMi)†(DµMi)o,(11)whereDµMi = ∂µMi + 12igτ·WµMi −12ig′BµMiτ3 ,(12)andMi = (iτ2Φ∗i Φi) ≡ φ0∗iφ+i−φ−iφ0i!.

(13)In the limit g′ = 0, the kinetic term is invariant under the global SU(2)L×SU(2)R ∼O(4)1 transformation,Mi →L MiR† ,τ·W →Lτ·WL† . (14)After the neutral Higgs fields acquire VEVs the residual symmetry of eq.

(11) [forg′ = 0] is SU(2)L+R which is the custodial symmetry described above. Therefore,corrections to eq.

(4) are expected to be of O[m2W (λ4 −λ5)] for small custodialbreaking. In the MSSM it means corrections to m2H± of O(g2m2W ).

When thefactor U(1)Y is gauged, the presence of τ3 in eq. (12) explicitily breaks the custodialsymmetry and corrections to m2H± of O(g′2m2H), where mH is the largest Higgsmass, can be generated.

However, in the MSSM, the Higgs masses can only be madesubstantially larger than mZ increasing the soft m212 mass-squared parameter.†Therefore, one-loop corrections of O(g′2m2H) must cancel in the large mH limit.† This is not the case of the SM or a non-supersymmetric two Higgs doublet model. In thesecases, the Higgs masses can be made large by increasing the self-couplings λi.7

We end this section with a comment concerning the natural relation given ineq. (4) which relates Higgs masses and the combination (λ4 −λ5) of Higgs self-couplings.

In principle, (λ4 −λ5) can be measured independently of the masses.Then, one can discuss finite radiative corrections to eq. (4).

The analysis is iden-tical to the one presented above in the case of the MSSM. Speciflcally, one-loopcorrections terms to eq.

(4) can be of O(g2m2t ) or O[m2W (λ4 −λ5)]. In particular,no O(g2m4t /m2W ) corrections can be generated at one-loop.

Since these correctionsarise from the violation of custodial symmetry, the size of these corrections canbe constrained by the ρ-parameter (ρ ≡m2W /m2Z cos2 θW ) whose deviation from 1also reflects the presence of custodial symmetry violating terms. It is well knownthat the size of mt (or ht) is limited via this constraint.

However, the dependenceof the ρ-parameter on (λ4 −λ5) can only occur at the two-loop level and probablycannot provide a useful constraint.3. One-loop effective HV VHV VHV V verticesThe trilinear HV V vertices, where H refers generically to any Higgs bosonand V to any vector boson, are of interest for the phenomenology of the Higgsbosons.

The HV V vertices can provide an important production mechanism forHiggs bosons at future colliders. Furthermore, the decay H →V V can be usedas a clear signature of the H. In Higgs sectors consisting in only doublets, HV Vvertices are absent at tree-level for the CP-odd and charged Higgs bosons[9].

Thisis the primary reason why the A0 and H± may be difficult to find at future hadroncolliders. The one-loop induced H±W ∓Z and A0V V vertices in the MSSM havebeen calculated in ref.

[10] and refs. [11,12] respectively.

The primary contributionsto the respective amplitudes arise from a virtual heavy quark pair. In the caseof the H±W ∓Z vertex, the contribution of a heavy quark doublet (u, d) growsquadratically with the quark mass for mu ̸= md.

However, this leading contributionvanishes exactly if the heavy quarks in the doublet are mass-degenerate. Scalar andgauge bosons contributions are found to be rather small due to large cancellationsamong different diagrams.

As we shall see, such results are a consequence of the8

global custodial symmetry. For simplicity, we will consider a sector with only twoHiggs doublets.

The analysis, however, can be easily generalize to multi-doubletmodels.Let us begin by assuming that the global SU(2)L+R symmetry defined byeqs. (14) with L = R is an exact symmetry of our theory even after SSB.

Let usalso work in the limit g′ = 0. In this case, the most general form for the one-loopeffective HV V vertices is given byLHV V =XjOµνj2Xi=1µij tr {Miτ·Wµτ·Wν} + h.c. ,(15)where Oµνj= (gµν, ∂µ∂ν, ∂ν∂µ, ǫµνρσ∂ρ∂σ) and µij are complex constants of di-mension d = 4 −dim{Oµνj }.

Using eq. (13), eq.

(15) can be written asLHV V ∝XjOµνj2Xi=1Re µijW 3µW 3ν + W +µ W −νReφ0i . (16)It is then clear that only the CP-even fields h0 and H0 couple to a pair of gaugebosons.

Thus the A0V V and H±V V vertices will only be generated if the custodialsymmetry is violated.Let us analyze the quark-Yukawa sector and, in particular, its custodial limit.In a general model with two Higgs doublets, there are two possible ways to couplethe Higgs to the quarks in a manner consistent with the discrete symmetry Φ2 →−Φ2:Case I: Quarks couple only to the first Higgs doublet Φ1.Case II: Φ2 couples to uR and Φ1 couples to dR.In case I, the quark-Yukawa interactions are SU(2)L×SU(2)R invariant if hu =hd ≡h,LY = −h¯uL ¯dLM1 uRdR!+ h.c. ,(17)9

where the relevant transformation laws areΨL ≡ uLdL!→LΨL ,ΨR ≡ uRdR!→RΨR ,(18)M1 →L M1R† . (19)When the neutral scalars develop VEVs, the symmetry is broken down to SU(2)L+R.The custodial limit, therefore, corresponds to the limit mu = md.

Thus, we shallneed a large mass splitting within the quark doublet to generate A0V V and H±V Vvertices that are phenomenologically relevant.We now turn to case II (which is the quark-Higgs interactions required bythe MSSM). If we define the transformation law of the scalar fields according toeq.

(14), we find that the quark Yukawa sector is not SU(2)L×SU(2)R invariant,even in the limit hu = hd. However, by making the following redefinition,Φ1 →1hdΦ1 ,Φ2 →1huΦ2 ,(20)the quark Yukawa sector can be written byLY = −¯uL ¯dL(iτ2Φ∗2 Φ1) uRdR!+ h.c. ,(21)which is SU(2)L×SU(2)R invariant if the scalar fields transform asM21 ≡(iτ2Φ∗2 Φ1) →L M21R† .

(22)After SSB the quark mass term is given byLm = −¯uL ¯dL huv200hdv1! uRdR!+ h.c.(23)This term is SU(2)L+R invariant only if huv2 = hdv1, i.e., mu = md.

Then, theeffective HV V vertices in the SU(2)L+R custodial limit (i.e., take L = R) are given10

byLHV V =XjµjOµνj tr {M21τ·Wµτ·Wν} + h.c.∝XjOµνjW 3µW 3ν + W +µ W −ν Im µjIm(φ02 −φ01) + Re µjRe(φ02 + φ01). (24)Note that this differs from eq.

(16) due to the new scalar transformation law[eq. (22) instead of eq.

(14)]. From eq.

(24), we see that the A0V V vertex canbe generated even in the custodial limit. Note however that the H±W ∓Z vertex isstill absent in the same limit.

When we turn on the U(1)Y gauge interactions, newtrilinear HWB and HBB vertices can be generated.⋆Notice, however, that theabove conclusions are still valid up to terms of O(m2W /E2) where E is the energyof the vector bosons. This can be seen using the equivalence theorem[13] whichstates that the vector bosons can be replaced by their correponding Goldstonebosons (G) in processes with E ≫mW .

Proceeding as before, it is possible toshow that demanding custodial invariance in the quark-Yukawa interactions, theHGG vertices (H = A0, H± for the case I and H = H± for the case II) are zero.In order to estimate the contribution of the Higgs self-interactions and Higgs-gauge interactions to the HV V vertices,† we can make use of the same argumentsof the previous section, i.e., these contributions are expected to be small in theMSSM where the custodial SU(2) symmetry is slightly violated. Moreover, thecontribution of the Higgs-gauge interactions must vanish in the limit mh0 = mH0.This can be seen by noting that the kinetic scalar term is invariant under therotation of the Higgs doublets:Lkin =2Xi=1(DµΦi)† (DµΦi) =2Xi=1DµΦ′i† DµΦ′i,(25)⋆In the case of the H±W ∓B vertex, the virtual quark-loop contribution does not yield aterm that grows quadratically in the quark mass at one-loop[10].† In fact, gauge and Higgs loops do not contribute to A0V V vertices to all orders in pertur-bation theory[12].11

where Φ′i are defined such as ⟨Φ′1⟩= v ≡qv21 + v22 and ⟨Φ′2⟩= 0. In the limitmh0 = mH0, we haveΦ′1 = G+v +1√2h0 + iG0!Φ′2 = H+1√2H0 + iA0!,(26)so that Φ′1 and Φ′2 do not mix with each other.

Since ⟨Φ′2⟩= 0, the kinetic termof Φ′2 is still SU(2)L×U(1)Y invariant after SSB. Thus, Φ′2V V vertices cannot begenerated.4.

ConclusionsWe have analyzed the radiative corrections to the charged Higgs mass in theMSSM and to the HV V vertices by making use of approximate global symmetriesof the theory with two Higgs doublets.In the analysis of the charged Higgs mass, we have shown that one-loop radia-tive corrections from top quarks and top squarks cannot be of O(g2m4t /m2W ) inthe absence of ˜tL −˜tR mixing. This has been accomplished by analyzing the limitof g = m12 = hb = 0 where the Higgs potential possesses a global O(4) ×O(4)symmetry.In this limit the charged Higgs boson is a pseudo-Goldstone bosonassociated with the breakdown O(4)×O(4) →O(3)×O(3).

By studing the globalsymmetry properties of the other sectors of the theory, the dependence of m2H± onthe model parameters can be ascertained.In the analysis of the one-loop effects to the trilinear HV V vertices, we haveshown that a custodial SU(2) symmetry plays a crucial role. The appropriate def-inition of the SU(2) symmetry depends on two possible choices for the pattern ofHiggs-fermion couplings.

In the first case, the CP-odd Higgs and charged Higgscouplings to V V generated at one-loop are zero if the theory is custodial invariant.In the second case only the charged Higgs couplings to V V are zero in this limit.To evaluate the order of magnitude of the radiative corrections to such A0V V andH±W ∓Z vertices, we have studied the custodial limit of the different sectors of the12

theory. In the MSSM, one learns why in the limit g′ = 0 the H±W ∓Z vertex isthe only HV V vertex that does not receive contributions from a heavy degeneratefermion doublet.

Moreover, due to the approximate invariance of the Higgs po-tential and the Higgs-gauge interactions under the custodial SU(2), contributionsfrom the gauge/Higgs sectors of the theory to the H±W ∓Z vertex must be verysmall.AcknowledgementsWe gratefully acknowledge conversations with Marco D´ıaz, Ralf Hempfling andGraham Ross. This work was supported in part by the U.S. Department of Energy.The work of A.P.

was supported by a fellowship from the MEC (Spain).13

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