Physics Letters B 642 (2006) 85–92

www.elsevier.com/locate/physletb

Probing CP-violating contact interactions in e+e− → HZ

with polarized beams

Kumar Rao, Saurabh D. Rindani ∗

Theory Group, Physical Research Laboratory Navrangpura, Ahmedabad 380009, India

Received 6 June 2006; received in revised form 7 July 2006; accepted 7 July 2006

Available online 20 September 2006

Editor: M. Cvetic

Abstract

We examine very general four-point interactions arising due to new physics contributing to the Higgs production process e+e− → HZ. Wewrite all possible forms for these interactions consistent with Lorentz invariance. We allow the possibility of CP violation. Contributions to theprocess from anomalous ZZH and γZH interactions studied earlier arise as a special case of our four-point amplitude. Expressions for polarand azimuthal angular distributions of Z arising from the interference of the four-point contribution with the standard-model contribution in thepresence of longitudinal and transverse beam polarization are obtained. An interesting CP-odd and T-odd contribution is found to be presentonly when both electron and positron beams are transversely polarized. Such a contribution is absent when only anomalous ZZH and γZH

interactions are considered. We show how angular asymmetries can be used to constrain CP-odd interactions at a linear collider operating at acentre-of-mass energy of 500 GeV with transverse beam polarization.© 2006 Elsevier B.V. All rights reserved.

1. Introduction

Despite the dramatic success of the standard model (SM), anessential component of SM responsible for generating massesin the theory, viz., the Higgs mechanism, as yet remainsuntested. The SM Higgs boson, signalling symmetry breakingin SM by means of one scalar doublet of SU(2), is yet to bediscovered. A scalar boson with the properties of the SM Higgsboson is likely to be discovered at the Large Hadron Collider(LHC). However, there are a number of scenarios beyond thestandard model for spontaneous symmetry breaking, and as-certaining the mass and other properties of the scalar boson orbosons is an important task. This task would prove extremelydifficult for LHC. However, scenarios beyond SM, with morethan just one Higgs doublet, as in the case of minimal super-symmetric standard model (MSSM), would be more amenableto discovery at a linear e+e− collider operating at a centre-of-mass (cm) energy of 500 GeV. We are at a stage when such a

* Corresponding author.E-mail addresses: kumar@prl.res.in (K. Rao), saurabh@prl.res.in

(S.D. Rindani).

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.07.072

linear collider, currently called the International Linear Collider(ILC), seems poised to become a reality [1].

Scenarios going beyond the SM mechanism of symmetrybreaking, and incorporating new mechanisms of CP violationhave also become a necessity in order to understand baryo-genesis which resulted in the present-day baryon–antibaryonasymmetry in the universe. In a theory with an extended Higgssector and new mechanisms of CP violation, the physical Higgsbosons are not necessarily eigenstates of CP [2,3]. In such acase, the production of a physical Higgs can proceed throughmore than one channel, and the interference between two chan-nels can give rise to a CP-violating signal in the production.

Here we consider in a general model-independent way theproduction of a Higgs mass eigenstate H through the processe+e− → HZ. This is an important mechanism for the pro-duction of the Higgs, the other important mechanisms beinge+e− → e+e−H and e+e− → ννH proceeding via vector–boson fusion. e+e− → HZ is generally assumed to get a con-tribution from a diagram with an s-channel exchange of Z.At the lowest order, the ZZH vertex in this diagram wouldbe simply a point-like coupling (left panel of Fig. 1). Interac-tions beyond SM can modify this point-like vertex by means

86 K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92

Fig. 1. Higgs production diagram with an s-channel exchange of Z with point-like ZZH coupling (left panel) and with anomalous ZZH coupling (right panel).

Fig. 2. Higgs production diagram with a four-point coupling.

of a momentum-dependent form factor, as well as by addingmore complicated momentum-dependent forms of anomalousinteractions considered in [4–9]. The corresponding diagram isshown in the right panel of Fig. 1, where the anomalous ZZH

vertex is denoted by a blob. There could also be a diagram witha photon propagator and an anomalous γZH vertex, which wedo not show separately. We consider here a beyond-SM con-tribution represented by a four-point coupling shown in Fig. 2.This is general enough to include the effects of the diagram inthe right panel of Fig. 1. Such a discussion would be relevantin studying effects of box diagrams with new particles, or dia-grams with t -channel exchange of new particles, in addition tos-channel diagrams.

We write down the most general form for the four-point cou-pling consistent with Lorentz invariance. We do not assumeCP conservation. We then obtain angular distributions for Z

(and therefore for H ) arising from the square of amplitude M1for the diagram in Fig. 1 with a point-like ZZH coupling,together with the cross term between M1 and the amplitudeM2 for the diagram Fig. 2. We neglect the square of M2, as-suming that this new physics contribution is small comparedto the dominant contribution |M1|2. We include the possibilitythat the beams have polarization, either longitudinal or trans-verse. While we have restricted the actual calculation to SMcouplings in calculating M1, it should be borne in mind that inmodels with more than one Higgs doublet this amplitude woulddiffer by an overall factor depending on the mixing amongthe Higgs doublets. Thus our results are trivially applicable tosuch extensions of SM, by an appropriate rescaling of the cou-pling.

We are thus addressing the question of how well the formfactors for the four-point e+e−HZ coupling can be determinedfrom the observation of Z angular distributions in the pres-ence of unpolarized beams or beams with either longitudinalor transverse polarizations. A similar question taking into ac-

count a new-physics contribution which merely modifies theform of the ZZH vertex has been addressed before in severalworks [4–9]. Those works which do take into account four-point couplings, do not do so in all generality, but stop at thelowest-dimension operators [6]. Studies which include beampolarization in the context of a general V V H vertex are [4,7,9]. The approach we adopt here has been used for the processe+e− → γZ in [10,11] and for the process Z → bbγ in [12].A more general analysis of a one-particle inclusive final state iscarried out in [13].

The four-point couplings, in the limit of vanishing elec-tron mass, can be neatly divided into two types—chirality-conserving (CC) ones and chirality-violating (CV) ones. TheCC couplings involve an odd number of Dirac γ matrices sand-wiched between the electron and positron spinors, whereas theCV ones come from an even number of Dirac γ matrices. Inthis work, we obtain angular distributions for both CC and CVcouplings. However, since in practice, CV couplings are usu-ally proportional to the fermionic mass (in this case the electronmass), we concentrate on the CC ones (see, however, [14]).

Polarized beams are likely to be available at a linear col-lider, and several studies have shown the importance of linearpolarization in reducing backgrounds and improving the sen-sitivity to new effects [15]. The question of whether transversebeam polarization, which could be obtained with the use of spinrotators, would be useful in probing new physics, has been ad-dressed in recent times in the context of the ILC [10,14–17].In earlier work, it has been observed that polarization doesnot give any new information about the anomalous ZZH cou-plings when they are assumed real [9]. However, in our work,we find that there are terms in the differential cross sectionwhich are absent unless both electron and positron beams aretransversely polarized. Thus, transverse polarization, if avail-able at ILC, would be most useful in isolating such terms.This is particularly significant because these terms are CP vi-olating. Moreover, one of them is even under naive CPT, andthus would survive even when no imaginary part is presentin the amplitude. We discuss the ramifications of this in duecourse.

In the next section we write down the possible model-independent four-point e+e−HZ couplings. In Section 3, weobtain the angular distributions arising from these couplings inthe presence of beam polarization. Section 4 deals with angu-lar asymmetries which can be used for separating various formfactors and Section 5 describes the numerical results. Section 6contains our conclusions and a discussion.

K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92 87

2. Form factors for the process e+e− → HZ

The most general four-point vertex for the process

(1)e−(p1) + e+(p2) → Zα(q) + H(k)

consistent with Lorentz invariance can be written as

(2)Γ α4pt = Γ α

CC + Γ αCV,

where the chirality-conserving part Γ αCC containing an odd

number of Dirac γ matrices is

Γ αCC = − 1

Mγ α(V1 + γ5A1) + 1

M3/q(V2 + γ5A2)k

α

(3)− i

M3/q(V3 + γ5A3)(p2 − p1)

α,

and the chirality violating part containing an even number ofDirac γ matrices is

Γ αCV = i

M2

[−(S1 + iγ5P1)kα − (S2 + iγ5P2)(p2 − p1)

α]

(4)− 1

M4εμναβp2μp1νkβ(S3 + iγ5P3).

In the above expressions, Vi , Ai , Si and Pi are form factors,and are Lorentz-scalar functions of the Mandelstam variabless and t for the process Eq. (1). For simplicity, we will onlyconsider the case here when the form factors are constants. M isa parameter with dimensions of mass, put in to render the formfactors dimensionless.

It may be appropriate to contrast our approach with the usualeffective Lagrangian approach. In the latter approach, it is as-sumed that SM is an effective theory which is valid up to acut-off scale Λ. The new physics occurring above the scale ofthe cut-off may be parametrized by higher-dimensional oper-ators, appearing with powers of Λ in the denominator. Thesewhen added to the SM Lagrangian give an effective low-energyLagrangian where, depending on the scale of the momenta in-volved, one includes a range of higher-dimensional operatorsup to a certain maximum dimension. Our effective theory is nota low-energy limit, so that the form factors we use are func-tions of momentum not restricted to low powers. Thus, the M

we introduce is not a cut-off scale, but an arbitrary parameter,introduced just to make the form factors dimensionless.

We thus find that there are 6 independent form factors in thechirality conserving case, and 6 in the chirality violating case.An alternative form for the Γ above would be using Levi-Civitaε tensors whenever a γ5 occurs. The independent form fac-tors then are then some linear combinations of the form factorsgiven above. However, the total number of independent formfactors remains the same.

Note that we have not imposed CP conservation in the above.As a consequence, the terms corresponding to V3, A3, S1, P2and S3 are CP violating, whereas the remaining are CP conserv-ing. This conclusion assumes that the form factors are arbitraryfunctions of s and even functions of t −u ≡ √

s|�q| cos θ , whereθ is the angle between �q and �p1 (or constants). This is becausein momentum space, s ≡ (p1 + p2)

2 is even under CP, whereas

t − u ≡ √s|�q| cos θ is odd under C and even under P, and thus

odd under CP.The expression for the amplitude for (1) arising from the SM

diagram of Fig. 1 with a point-like ZZH vertex corresponds tothe special case with the following form factors nonzero:

V1 = e2

4 sin2 θW cos2 θW

MmZ

s − m2Z

gV ,

(5)A1 = − e2

4 sin2 θW cos2 θW

MmZ

s − m2Z

gA,

where the vector and axial-vector couplings of the Z to elec-trons are given by

(6)gV = −1 + 4 sin2 θW , gA = −1,

and θW is the weak mixing angle. As mentioned earlier, in othermodels with extra scalar doublets, the above expressions wouldbe modified simply by a factor depending on the mixing amongthe doublets.

3. Angular distributions

We now calculate the angular distribution arising from thesquare of the SM amplitude and from the interference betweenthe SM amplitude and the amplitude arising from the four-pointcouplings of (3) or (4). We ignore terms bilinear in the four-point couplings, assuming that the new-physics contribution issmall. We treat the two cases of longitudinal and transverse po-larizations for the electron and positron beams separately.

We choose the z axis to be the direction of the e− momen-tum, and the xz plane to coincide with the production plane.The positive x axis is chosen, in the case of tranvserse polariza-tion, to be along the direction of the e− polarization. We thendefine θ and φ to be the polar and azimuthal angles of the mo-mentum �q of the Z.

We obtain, for the differential cross section with longitudinalpolarization, the expression

(7)dσL

dΩ= dσ SM

L

dΩ+ dσ CC

L

dΩ,

where

dσ SML

dΩ= λ1/2

64π2s(1 − PLPL)F 2[g2

V + g2A − 2gV gAP eff

L

](8)×

[1 + |�q|2 sin2 θ

2m2Z

]

is the SM contribution, and

dσ CCL

dΩ= λ1/2

64π2s

2F

M(1 − PLPL)

×[{(

gV − P effL gA

)ReV1 + (

P effL gV − gA

)ReA1

}

×(

1 + |�q|2 sin2 θ

2m2Z

)

−√

sq02

{[(gV − P eff

L gA

)ReV2

M

88 K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92

+ (P eff

L gV − gA

)ReA2

] + [(gV − P eff

L gA

)ImV3

(9)+ (P eff

L gV − gA

)ImA3

]βq cos θ

} |�q|22m2

Z

sin2 θ

]

is the contribution of the chirality-conserving couplings. Thereis no contribution from the chirality-violating couplings for un-polarized or longitudinally polarized beams. In the above, wehave used

(10)F = mZ

s − m2Z

(e

2 sin θW cos θW

)2

,

λ = 4|�q|2s = (s − m2

H − m2Z

)2 − 4m2H m2

Z,

(11)βq = |�q|q0

,

and

(12)P effL = PL − PL

1 − PLPL

.

PL and PL are the degrees of longitudinal polarization respec-tively of the e− and e+ beams, with the convention that in eachcase, a positive value denotes net right-handed polarization.

For the case of transverse polarization, we assume that thespins of the electron and positron are both perpendicular to thebeam direction, and also that they are parallel (or antiparallel)to each other. When the beams are transversely polarized weobtain the differential cross section as

(13)dσT

dΩ= dσ SM

T

dΩ+ dσ CC

T

dΩ+ dσ CV

T

dΩ,

where

dσ SMT

dΩ= λ1/2F 2

64π2s

[(g2

V + g2A

)(1 + |�q|2

2m2Z

sin2 θ

)

(14)+PT PT

(g2

V − g2A

) |�q|22m2

Z

sin2 θ cos 2φ

]

is the SM contribution,

dσ CCT

dΩ= λ1/2

64π2s

2F

M

{[(gV ReV1 − gA ReA1)

+ |�q|22m2

Z

sin2 θ[(gV ReV1 − gA ReA1)

+ PT PT

[(gV ReV1 + gA ReA1) cos 2φ

− (gV ImA1 + gA ImV1) sin 2φ]]]

− s1/2q0

M2

|�q|22m2

Z

sin2 θ[(gV ReV2 − gA ReA2)

+ PT PT

[(gV ReV2 + gA ReA2) cos 2φ

− (gV ImA2 + gA ImV2) sin 2φ]]

− s1/2|�q|M2

|�q|22m2

Z

cos θ sin2 θ[(gV ImV3 − gA ImA3)

+ PT PT

[(gV ImV3 + gA ImA3) cos 2φ

(15)+ (gV ReA3 + gA ReV3) sin 2φ]]}

is the contribution from the chirality-conserving couplings, and

dσ CVT

dΩ= λ1/2

64π2s

F s1/2

M4|�q| sin θ

× (q0

[−{gV ReS1 sinφ + gA ImS1 cosφ}(PT − PT )

+ {−gV ReP1 cosφ + gA ImP1 sinφ}(PT + PT )]

+ |�q| cos θ[{gA ReS2 cosφ − gV ImS2 sinφ}

× (PT − PT )

− {gA ReP2 sinφ + gV ImP2 cosφ}(PT + PT )]

+ 1

2s1/2[{gA ReS3 sinφ + gV ImS3 cosφ}(PT − PT )

(16)

+ {gA ReP3 cosφ − gV ImP3 sinφ}(PT + PT )])

is the contribution from the chirality-violating couplings. In theabove, PT and PT denote the degrees of transverse polarizationrespectively of the e− and e+ beams, with the convention thatPT > 0 denotes a net polarization along the positive x direction,whereas PT > 0 denotes a net polarization along the negative x

direction.We now examine how the angular distributions in the pres-

ence of polarization may be used to determine the various formfactors.

4. Polarization and angular asymmetries

The parametrizations we use for the new-physics interac-tions have 6 complex couplings (form factors) in the CC andcase, and 6 in the CV case. Thus, there are 12 real parametersto be determined in each case. We start by making a simplifyingassumption that the form factors we have written down are onlyfunctions of s and not of t − u (or equivalently cos θ ). In thatcase, using the unpolarized distributions, which have approxi-mately the same form as the SM distribution, viz., A+B sin2 θ ,except for the V3 and A3 terms, which have a sin2 θ cos θ depen-dence, it is not possible to determine separately all the terms.The terms proportional to sin2 θ cos θ can be determined usinga simple forward–backward asymmetry:

(17)AFB(θ0) = 1

σ(θ0)

[ π/2∫θ0

dσ

dθdθ −

π−θ0∫π/2

dσ

dθdθ

],

where

(18)σ(θ0) =π−θ0∫θ0

dσ

dθdθ,

and θ0 is a cut-off in the forward and backward directionsneeded to keep away from the beam pipe, which could nev-ertheless be chosen to optimize the sensitivity. This asymme-try is odd under CP and is proportional to the combinationgV ImV3 − gA ImA3. An observation of AFB(θ0) can thus de-termine that combination of parameters. It should be noted thatonly imaginary parts of V3 and A3 enter. This can be related tothe fact that the CP-violating asymmetry AFB(θ0) is odd under

K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92 89

naive CPT. It follows that for it to have a non-zero value, theamplitude should have an absorptive part.

We now treat the cases of longitudinally and transverselypolarized beams.

Case A. Longitudinal polarization:The forward–backward asymmetry of Eq. (17) in the pres-

ence of longitudinal polarization, which we denote by ALFB(θ0),

determines a different combination of the same couplings ImV3and ImA3. Thus observing asymmetries with and without po-larization, the two imaginary parts can be determined indepen-dently.

In the same way, a combination of the cross section for theunpolarized and longitudinally polarized beams can be usedto determine two different combinations of the remaining cou-plings which appear in (9). However, one can get informationonly on the real parts of V1,A1,V2 and A2, not on their imagi-nary parts.

With unpolarized or longitudinally polarized beams, it is notpossible to get any information of the chirality-violating cou-plings, as they do not contribute.

Case B. Transverse polarization:In the case of the angular distribution with transversely po-

larized beams, there is a dependence on the azimuthal angleφ of the Z. Thus, in addition to φ-independent terms whichare the same as those in the unpolarized case, there are termswith factors sin2 θ cos 2φ, sin2 θ sin 2φ, sin2 θ cos θ cos 2φ andsin2 θ cos θ sin 2φ in the case of CC couplings, and factorssin θ cosφ, sin θ sinφ, sin θ cos θ cosφ, sin θ cos θ sinφ in thecase of CV couplings. The φ-dependent terms in the CC caseoccur with the factor of PT PT and in the CV case with a fac-tor of PT + PT or PT − PT . Thus, in the CC case, both beamsneed to have transverse polarization for a nontrivial azimuthaldependence. In the CV case, it is possible to have φ dependencewith either the electron or the positron beam polarized. We findthat with the possibility of flipping transverse polarization ofone beam, it is possible to examine 4 types of angular asym-metries in each of CC and CV cases. Each angular asymmetrywould enable the determination of a different combination ofcouplings.

We will concentrate on the CC case, as most theories permitonly CC couplings, at least in the limit of me = 0. We furtherrestrict ourselves here only to terms which involve a cos θ fac-tor, which gives rise to a forward–backward asymmetry, due tothe fact that cos θ changes sign under θ → π − θ . These corre-spond to the case of CP violation.

We can then define two different asymmetries, which serveto measure two different combinations of CP-violating cou-plings:

ATFB(θ0) = 1

σ(θ0)

[ π/2∫θ0

dθ

( π/2∫0

dφ −π∫

π/2

dφ

+3π/2∫π

dφ −2π∫

3π/2

dφ

)

−π−θ0∫π/2

dθ

( π/2∫0

dφ −π∫

π/2

dφ

(19)+3π/2∫π

dφ −2π∫

3π/2

dφ

)]dσ

dθ dφ,

and

A′TFB(θ0) = 1

σ(θ0)

[ π/2∫θ0

dθ

( π/4∫−π/4

dφ −3π/4∫

π/4

dφ

+5π/4∫

3π/4

dφ −7π/4∫

5π/4

dφ

)

−π−θ0∫π/2

dθ

( π/4∫−π/4

dφ −3π/4∫

π/4

dφ

(20)+5π/4∫

3π/4

dφ −7π/4∫

5π/4

dφ

)]dσ

dθdφ.

The former is odd under naive time reversal, whereas the lat-ter is even. The CPT theorem then implies that these would berespectively dependent on real and imaginary parts of form fac-tors. The integrals in the above may be evaluated to yield

ATFB(θ0)

= 3PT PT

(21)

× |�q|3s1/2(gV ReA3 + gA ReV3) cos θ0(cos(2θ0) − 3)

F (g2A + g2

V )M3π(12m2Z + 5|�q|2 − |�q|2 cos(2θ0))

,

and

A′TFB(θ0)

= 3PT PT

(22)

× |�q|3s1/2(gA ImA3 + gV ImV3) cos θ0(cos(2θ0) − 3)

F (g2A + g2

V )M3π(12m2Z + 5|�q|2 − |�q|2 cos(2θ0))

.

We see that the two asymmetries ATFB and A′T

FB can mea-sure, respectively, the combinations gV ReA3 + gA ReV3 andgA ImA3 + gV ImV3. The latter is dominated by ImA3 whichmay also be determined using unpolarized beams. The formerrequires transverse polarization to measure.

It can be checked that if one considers only contributionsfrom a modification of the ZZH vertex as in [4–9] AT

FB andA′T

FB vanish. This result for ATFB is obtained in [9]. Thus, obser-

vation of a nonzero asymmetry would signal the presence of theCP-violating four-point interaction.

90 K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92

5. Numerical results

We now obtain numerical results for the polarized crosssections, the asymmetries and the sensitivities of these asym-metries for a definite configuration of the linear collider. Forour numerical calculations, we have made use of the followingvalues of parameters: mZ = 91.19 GeV, α = 1/128, sin2 θW =0.22, M = 1 TeV. It should be noted that the particular choiceof M is simply for convenience, and is not simply related toany assumption about the scale of new physics—a change in M

can always be compensated by corresponding changes in theform factors. For the parameters of the linear collider, we haveassumed

√s = 500 GeV, PL = 0.8, PL = −0.6, PT = 0.8,

PT = 0.6, and an integrated luminosity∫Ldt = 500 fb−1. For

most of our calculations we choose three values of the Higgsmass, mH = 150 GeV, 200 GeV and 300 GeV.

We have assumed that the contribution of the Z exchangediagram with a point-like ZZH coupling of Fig. 1 is the sameas that in SM. Since we are keeping open the possibility thatthe Higgs boson we are dealing with is not an SM Higgs, thisassumption may not be correct. However, the modification fora Higgs of a different model will be multiplication by a certainoverall factor depending on the mixing of the different Higgsbosons in the model. This can easily be taken care of whileinterpreting our results for such a model.

In the left panel of Fig. 3 is plotted the forward–backwardasymmetry AL

FB(θ0) with longitudinal polarization as a functionof the cut-off θ0. Only the parameter ImV3 is chosen nonzero,

be zero.

and to have the value 0.1. This choice is for illustration. Theright panel of Fig. 3 shows the same asymmetry for a fixed valueof mH = 150 GeV, for the combinations ImV3 = 0.1, ImA3 =0 and ImV3 = 0, ImA3 = 0.1, and for values of PL differing insign. We find that the asymmetry depends on the relative signsof PL and PL and is larger in magnitude when the relative signsare opposite.

In the case of transverse polarization, the two asymmetriesAT

FB and A′TFB are shown as functions of θ0 in the left and

right panels, respectively, of Fig. 4 for values of polarizationPT = 0.8 and PT = 0.6. In the first case, the only nonzero pa-rameter is ReV3 = 0.1, whereas in the second case, the onlynonzero parameter is ImV3 = 0.1. In all the above figures, thedependence on the cut-off θ0 is mild for small values of θ0.Hence the results will not be sensitive to the choice of θ0, if itis small.

We now examine the accuracy to which each of the cou-plings can be determined for linear collider operating at

√s =

500 GeV and with an integrated luminosity of 500 fb−1. At the90% confidence level (CL), the limit that can be placed on a pa-rameter contributing linearly to a certain asymmetry A is givenby 1.64/(A1

√NSM ), where A1 is the asymmetry for unit value

of the parameter.We first consider the determination of the parameter ReV3

from a measurement of the asymmetry ATFB for a typical value

of θ0 = 45◦. If the asymmetry is not observed, we find thatthe limit placed on ReV3 is 3.9 × 10−2 for mH = 150 GeV,5.1 × 10−2 for mH = 200 GeV, and 1.3 × 10−1 for mH =

Fig. 3. Left panel: The forward–backward asymmetry for longitudinally polarized beams, ALFB(θ0), for ImV3 = 0.1 as a function of θ0. All other couplings are

taken to be zero. Right panel: ALFB(θ0) for mH = 150 GeV, PL = 0.8, for two combinations of couplings, and for two values of PL differing in sign.

Fig. 4. Left panel: The asymmetry ATFB for transverse polarizations PT = 0.8 and PT = 0.6 plotted against θ0 for ReV3 = 0.1. All other couplings are taken to be

zero. Right panel: The asymmetry A′TFB for transversely polarizations PT = 0.8 and PT = 0.6 plotted against θ0 for ImV3 = 0.1. All other couplings are taken to

K. Rao, S.D. Rindani / Physics Letters B 642 (2006) 85–92 91

300 GeV. Since the combination which appears in the asym-metry is gV ReA3 + gA ReV3, it implies that the correspondinglimits on ReA3 will be a factor |gA/gV | ≈ 8.3 higher. Thus,the asymmetry is more sensitive to ReV3 because of a largercoupling gA multiplying it.

The expression for A′TFB is identical to that for AT

FB, exceptthat the factor gV ReA3 + gA ReV3 is replaced by gV ImV3 +gA ImA3. Thus, now it will be ImA3 which will have the limitsmentioned above for ReV3, and ImV3 will have limits whichare a factor of about 8.3 larger.

It should be borne in mind that the definition of the couplings(form factors) are dependent on the value of the scale parameterM which is chosen. Thus changing the value of M will changethe limits on the form factors.

6. Conclusions and discussion

We have parametrized the amplitude for the process e+e− →HZ using only Lorentz invariance by means of form fac-tors, treating separately the chirality-conserving and chirality-violating cases. We then calculated the differential cross sectionfor the process e+e− → HZ in terms of these form factors forpolarized beams. The motivation was to determine the extent towhich longitudinal and transverse polarizations can help in anindependent determination of the various form factors.

We found that in the presence of transverse polarization,there is a CP-odd and T-odd contribution to the angular dis-tribution. The coupling combinations this term depends on can-not be determined using longitudinally polarized beams. More-over, this transverse-polarization dependent contribution doesnot arise when only V V H type of couplings are considered.Hence such a term, if observed, would be a unique signal ofCP-violating four-point interaction.

It should be emphasized that our results and conclusions aredependent on the assumption that the form factors are inde-pendent of t and u. In particular, the CP property of a giventerm in the distribution would change if the correspondingform factor is an odd function of cos θ . The reason is thatcos θ ≡ q · (p2 − p1)/(|�q|s1/2) is odd under CP.

We have discussed limits on the couplings that would be ex-pected from a definite configuration of the linear collider. Asfor the CP-conserving couplings, limits may be obtained evenfrom the existing LEP data, which has excluded SM Higgs upto mass of about 114 GeV. However, we have concentrated onlyon the limits on the CP-violating couplings. It should be bornein mind that the limits on these depend on the choice of M , thearbitrary parameter of dimension of mass that we introduced.

Though we have used SM couplings for the leading contri-bution of Fig. 1, as mentioned earlier, the analysis needs onlytrivial modification when applied to a model like MSSM ora multi-Higgs-doublet model, and will be useful in such ex-tensions of SM. It is likely that such models will give riseto four-point contributions through box diagrams or loop dia-grams with a t -channel exchange of particles. However, to ourknowledge, such calculations are not available for CP-violatingmodels. The interesting effects we have discussed would makeit useful to carry out such calculations.

We have discussed the angular distribution of the Z in theprocess e+e− → HZ. Clearly, for the discussion to be of prac-tical use, one has to include the means of detection of Z and H .Thus, it is important to include decays of Z and H and to seewhat our analysis implies for the decay products. In particular,one has to answer the question as to how leptons or jets fromthe decay of the Z can be used to measure the asymmetries wediscuss, and with what efficiency. To the extent that the sum ofthe four-momenta of the charged lepton pair or the jet pair canbe a measure of the Z four-momentum, it should be possible toreconstruct the asymmetries discussed here with reasonable ac-curacy. One should also investigate the effect experimental cutswould have on the accuracy of the determination of the cou-plings. One should keep in mind the possibility that radiativecorrections can lead to quantitative changes in the above results(see, for example, [18]). While these practical questions are notaddressed in this work, we feel that the interesting new featureswe found would make it worthwhile to address them in future.

Acknowledgements

This work was partly supported by the IFCPAR projectNo. 3004-2. We thank Rohini Godbole and B. Ananthanarayanfor discussions and for comments on the manuscript.

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