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0911

3 v2

27

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200

6hep-ph/0609113, IISc-CHEP/4/06, CERN-PH-TH/2006-117, LPT-ORSAY-06-56

Probing CP-violating Higgs contributions in γγ → f f through fermion polarization

Rohini M. Godbole∗

Center for High Energy Physics, IISc, Bangalore 560 012, India

Sabine Kraml†

Theory Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland

Saurabh D. Rindani‡

Physical Research Laboratory, Navrangapura, Ahmedabad 380 009, India

Ritesh K. Singh§

Laboratoire de Physique Theorique, 91405 Orsay Cedex, France

We discuss the use of fermion polarization for studying neutral Higgs bosons at a photon collider.To this aim we construct polarization asymmetries which can isolate the contribution of a Higgsboson φ in γγ → ff , f = τ/t, from that of the QED continuum. This can help in getting informationon the γγφ coupling in case φ is a CP eigenstate. We also construct CP-violating asymmetries whichcan probe CP mixing in case φ has indeterminate CP. Furthermore, we take the MSSM with CPviolation as an example to demonstrate the potential of these asymmetries in a numerical analysis.We find that these asymmetries are sensitive to the presence of a Higgs boson as well as its CPproperties over a wide range of MSSM parameters. In particular, the method suggested can coverthe region where a light Higgs boson may have been missed by LEP due to CP violation in theHiggs sector, and may be missed as well at the LHC.

PACS numbers: 14.80.Cp, 14.60.Fg, 12.60.Jv

I. INTRODUCTION

The Higgs boson is the only particle of the StandardModel (SM) to have eluded experimental discovery so far.The discovery of the Higgs boson and the subsequentstudy of electroweak symmetry breaking is one of theprime aims of all the current and next generation colliders[1]. Electroweak precision measurements indicate, in theSM, the existence of a Higgs boson lighter than 204 GeVat 95% C.L. [2]. A Higgs boson with SM couplings lighterthan 114.4 GeV is ruled out by direct searches at LEP [3].Thus one expects to find the SM Higgs boson with a massin this range. In models with an extended (and possiblyCP-violating) Higgs sector, the couplings of electroweakvector bosons to the lightest Higgs can be suppressed [4].In such a case direct searches allow the existence of aHiggs boson much lighter than 114.4 GeV [5].

The Large Hadron Collider (LHC), scheduled to goin operation in 2007, is expected [6] to be capable ofsearching for the SM Higgs boson in the entire mass rangeexpected theoretically and still allowed experimentally.The International Linear Collider (ILC) [7], currently inplanning, is expected to be capable of profiling the Higgsboson very accurately, again for the entire mass rangementioned above. Determination of the CP properties ofthe spin-0 particle, which we hope will be discovered andstudied at the LHC and the ILC, is an important part ofthis project of profiling the Higgs boson, see e.g. [8]. The

∗Electronic address: rohini@cts.iisc.ernet.in†Electronic address: Sabine.Kraml@cern.ch‡Electronic address: saurabh@prl.res.in§Electronic address: Ritesh.Singh@th.u-psud.fr

Higgs couplings with a pair of electroweak gauge bosons(V = W/Z) and those with a pair of heavy fermions(f = t/τ) are the ones that prove the most useful in thiscontext. These couplings, for a neutral Higgs boson φ,which may or may not be a CP eigenstate, can be writtenas

φff :−ig mf

2 MW

(vf + iaf γ5) (1)

φV V :ig M2

V

MW

(

AV gµν + BV

pµpν

M2Z

+ i CV ǫµνρσ

pρqσ

M2Z

)

(2)

where p = PV1+PV2

, q = PV1−PV2

and PV1, PV2

are thefour momenta of the two massive vector bosons. In theSM, vf = AV = 1 and af = BV = CV = 0. At the LHC,the tt final state produced in the decay of an inclusivelyproduced Higgs can provide knowledge of the CP na-ture of the ttφ coupling through spin-spin correlations [9]whereas ttφ production can allow a determination of vf

and af [10]. It should also be possible to exploit the φZZcoupling via φ → ZZ → l+l−l′+l′− [11], and the vectorboson fusion mode [12]. At the ILC [7], a rather cleandetermination of the CP of the Higgs boson should bepossible using the Higgsstrahlung process [13]. Angularcorrelations of the decay products of the φ, in particularφ → f f , may be used effectively at an e+e− collider todistinguish between vf = 1, af = 0 and vf = 0, af = 1 aswell as to get information on CP mixing [14]. A remarkis in order here. The different methods which exploit theV V φ coupling test the terms with tensor structure gµν

and ǫµνρσ in Eq. (2); actually in most cases the CP-evenpart ∝ gµν is projected out since in most CP violating(CPV) extensions of the SM one has AV ≫ BV , CV . Thef fφ coupling, on the other hand, allows equal sensitivity

2

γ

γ

(a)

f

f

f

γ

γ

(b)

f

f

f

γ

γ

(c)

f

f

Vγγφ Vf fφφ

FIG. 1: Feynman diagrams contributing to γγ → ff produc-tion.

to the CP-even and CP-odd parts. At both colliders, theLHC and the ILC, the determination of the CP quantumnumber of the Higgs boson seems feasible, while deter-mination of CP mixing seems difficult; the best chancefor the latter being offered by the exploitation of the f fφcoupling [15].

A more sensitive laboratory for studying the CP prop-erties of a neutral Higgs boson is the photon collider op-tion [16] of the ILC. At a photon collider, the Higgs bo-son is produced resonantly in the s-channel and thus onewill be able to produce it copiously even if the γγφ cou-pling is small. In presence of CP violation (CPV), theCP-even and CP-odd components of the Higgs coupleto photons with comparable strength. A study of theproduction rates with linearly- and circularly-polarizedphotons can help determine the CP quantum numbers ifconserved, as well as the CP mixing phases in the caseof CP violation. Angular distributions of decay products(V V, bb) of the produced φ along with measurements ofΓ(φ → bb), Γ(φ → γγ) may also allow the determinationof a CP-mixing phase [17]; the φ → tt decay can alsobe used [18, 19] if it is kinematically allowed. While acomplete reconstruction of couplings would require bothcircularly and linearly polarized photons [18], substan-tial information can already be obtained using circularlypolarized photons alone [19]. Polarization of the finalstate fermions, which are pair produced, can be a probeof CP violation in the production process [18, 19, 20].Among SM fermions, the polarization can be measuredonly for t-quarks and tau-leptons. The t-quark decaysbefore it hadronizes and hence the polarization informa-tion gets reflected in the angular distributions of decayproducts, whereas for the τ lepton one can extract polar-ization information by looking at the energy distributionof pions [21, 22]. For a Higgs mass larger than 2mt,top polarization is the best probe of the Higgs inter-action. However, for a CP-violating light Higgs boson,which could have escaped detection at LEP, the decayinto tt is not possible. In this case the τ+τ− final statemight provide a useful probe. We therefore study thecontribution of a light Higgs boson to τ production viaγγ → φ → τ+τ− and that of heavier Higgs bosons to tproduction via γγ → φ → tt. Higgs contribution to τpolarization has also been studied recently for the LHC[23], for a photon collider [24] and for an e+e− linearcollider [25] in the context of resonant CP violation inthe MSSM. The effect of neutral Higgs boson exchangein γγ → tt has been addressed in Ref. [18, 19, 26], in

terms of various correlations and asymmetries for a pho-ton collider.

In this paper, we investigate the possibility of usingtau/top polarization to get information on the φff andφγγ couplings, as well as to probe possible CP violationin them in a generic scenario. First, we formulate themethod in a model-independent way. We then apply itin a numerical analysis to the minimal supersymmetricstandard model with CP violation (CPV-MSSM). Theneutral Higgs sector of the CPV-MSSM has been studiedin detail theoretically [27], and constraints from LEP areavailable [5]. In this work, we study τ/t polarization inthe CPX scenario [28] over regions of the CPV-MSSMparameter space that are allowed by the current data,and assess the feasibility of using it to probe the Higgscontribution to f f production. In the numerical anal-ysis, we use CPsuperH [29] and FeynHiggs 2.1 [30] forcalculating the masses, the decay widths and the relevantcouplings of the Higgs bosons.

The rest of the paper is organized as follows. In sectionII, we describe fermion pair production at a γγ collider ina model-independent way. Section III deals with polar-ization observables as potential probes of the Higgs con-tribution and its CP structure. Numerical results withinthe CPV-MSSM are presented in section IV, and in sec-tion V we present our conclusions.

II. FERMION PAIR PRODUCTION IN γγCOLLISION

At a photon collider, the production of a fermion pairinvolves the γff , γγφ and φff vertices as shown inFig. 1. We take the γff vertex to be the standard QEDone, while the vertices involving Higgs bosons are takento be the most general allowed. The model independentvertex for Higgs interactions with fermions is given inEq. (1) and with a pair of photons, allowing for CP vio-lation, can be written as :

Vµνγγφ =

−i√

sα

4π

[

Aγ(s)

(

gµν − 2

skν1 kµ

2

)

− Bγ(s)2

sǫµναβk1αk2β

]

. (3)

Here k1 and k2 are the four-momenta of the collidingphotons. The helicity amplitudes for fermion pair pro-duction in the s- and the t/u-channels can be obtainedfrom those given in [18, 19] :

Mφ(λ1, λ2; λf , λf ) =−igαmf

8πMW

s

s − m2φ + imφΓφ

[Aγ(s) + iλ1Bγ(s)] [λfβvf − iaf ] δλ1,λ2δλf ,λf

, (4)

MQED(λ1, λ2; λf , λf ) =−i4παQ2

1 − β2 cos2 θf[

4mf√s

(λ1 + λfβ) δλ1,λ2δλf ,λf

−4mf√s

λfβ sin2 θf δλ1,−λ2δλf ,λf

−2β (cos θf + λ1λf ) sin θf δλ1,−λ2δλf ,−λf

]

. (5)

3

TABLE I: Combinations of form factors vf , af , Aγ and Bγ

that occur in the helicity amplitudes of Eqs. (4) and (5).

Combination Alias CP Combination Alias CPvfℜ(Aγ) x1 even vfℑ(Aγ) x2 evenvfℜ(Bγ) y1 odd vfℑ(Bγ) y2 oddafℜ(Aγ) y3 odd afℑ(Aγ) y4 oddafℜ(Bγ) x3 even afℑ(Bγ) x4 even

The form factors Aγ and Bγ are complex whereas vf , af

can be taken to be real without loss of generality. Thenon-standard vertices given by Eqs. (1) and (3) involvefour independent form factors: vf , af , Aγ , Bγ . In theMSSM with CPV, these form factors are functions of var-ious model parameters: tanβ; mH+ ; (|µ|, Φµ); (|Af |, Φf );

(|Mi|, Φi), i = 1, 2, 3; mq,l; etc., where (|x|, Φx) denotes

x = |x|eiΦx .

The helicity amplitudes of Eqs. (4) and (5) involve onlycertain combinations of the form factors which are listedin Table I. Only five of these eight combinations areindependent, the other three can be obtained by inter-relations such as x1x3 = y1y3, etc. In all the extensionsof the SM, Aγ and Bγ are generated at the one-loop level.Simultaneous existence of vf and af , or Aγ and Bγ vio-lates CP, i.e. non-vanishing values of yi, (i = 1, ..., 4) im-ply CP violation. Even in case of CP invariance, whereonly the xi’s are non-zero, the Higgs contribution can al-ter the polarization of the fermions f from that predictedby pure QED. CP violation, giving rise to non-zero yi’s,gives an additional contribution to the fermion polariza-tion.

It should be noted that the Higgs-mediated diagramcontributes only when the helicities of the colliding pho-tons are equal. The helicities of f and f are also equalin this case. The QED contribution for this helicitycombination is proportional to the fermion mass. Boththese facts indicate that one should choose equal pho-ton helicities to enhance the effect of the Higgs medi-ated diagram. The contribution with opposite helicitiesof photons comes from QED diagrams alone and it islarge as compared to that of equal photon helicities for√

s ≫ 4mf . Thus with unpolarized photons the net con-tribution from Higgs exchange will be relatively small.Hence one expects poor sensitivity to the Higgs contri-bution with unpolarized initial state photons.

Among the vertices contributing to fermion-pair pro-duction, the standard γff vertex conserves chirality,while the φff vertex mixes different chiralities. Owingto the finite mass mf of the fermion, there is a chirality-mixing contribution even for the pure QED diagrams(Figs. 1a,b). The presence of the Higgs boson exchange,Fig. 1c, provides an additional, polarization dependent,spatially isotropic, chirality-mixing contribution. Theproperty of spatial isotropy is unique to Higgs exchangecontribution, in contrast to other means of chirality mix-ing, such as the finite mass effect. With unpolarizedinitial-state photons, the QED as well as a CP-conservingHiggs contribution lead to unpolarized fermions in thefinal state; CP violation in the Higgs sector leads toa net, though very small, fermion polarization. With

polarized initial-state photons, already pure QED leadsto a finite polarization. The additional chirality mixingfrom the Higgs exchange causes a change in this polar-ization in both the CP-conserving and the CP-violatingcase. It is thus possible to construct observables relat-ing initial-state photon and final-state τ/t polarizationswhich probe the Higgs couplings as well as possible CPviolation in the Higgs sector.

III. FERMION POLARIZATION IN γγCOLLISION

At a γγ collider, Compton back-scattering of a laserfrom e−/e+ is used to produce high-energy photons[31].The energy spectrum of the back-scattered photons de-pends upon the polarizations of the e−/e+ beams andthe laser as shown in Fig. 2. This figure shows the dis-tribution of the reduced invariant mass of the γγ systemz =

√ω1ω2/Eb, where ω1,2 are the energies of the two

colliding photons in the lab frame, for a laser energy ω0

corresponding to

xc =4Ebω0

m2e

= 4.8. (6)

The spectrum is peaked in the high energy region foropposite polarizations of the electron and laser beams.Further, most of these high-energy photons have a largedegree of polarization. By choosing appropriate polariza-tions for the e−/e+ and laser beams, one can thus obtaina peaked and highly polarized spectrum for the collid-ing photons. We shall use this fact when constructingvarious observables with τ polarization.

The polarization of fermions is defined as the fractionalsurplus of positive helicity fermions over negative helicityones, i.e.

P ijf =

N ij+ − N ij

−

N ij+ + N ij

−

, (7)

where the superscript ij stands for the polarizations ofthe parent e−, e+ beams of the ILC (with P++

f meaning

100% right polarized electrons and 100% right polarizedpositrons); N+ and N− stands for the number of fermionswith positive and negative helicities, respectively. Anal-ogously, P ij

f is the polarization of the anti-fermion. Dueto the left-chiral nature of the W–boson interaction, thepositively and negatively polarized f ’s lead to differentdistributions of the decay products. For τ ’s, by lookingat the net energy distribution of decay π one can get in-formation on its polarization [21]. On the other hand,for t-quarks, it is the energy distribution of b-quarks orthe angular distribution of decay leptons.

The various N ij± , and hence P ij

f , with different polar-izations of initial-state photons are related to each othervia discrete symmetry transformations, such as C, P andCP, if these symmetries are respected by the underlyingdynamics. Thus any deviation from these relations canbe a probe of the violation of the corresponding discretesymmetry.

For unpolarized initial-state photons the polarization,PU

f , is zero for the QED diagrams (Figs. 1a,b). Even in

4

TABLE II: Polarization observables and interactions andcombinations that they can probe.

Observables Interactions Combinationsprobed probed

P Uf P/CP violating yi’s

δP+

f = P++

f − (P++

f )QED Chirality-mixing xi’s, yi’s

δP−f = P−−

f − (P−−f )QED Chirality-mixing xi’s, yi’s

δPCPf = P++

f + P−−f P/CP violating yi’s

the presence of a CP-conserving Higgs PUf is zero. This is

because the left chiral and the right chiral components offermions couple to the Higgs boson with equal strength.Thus the net polarization of the fermions, if any, will sig-nal CP violation in the Higgs sector. In general it is asignal of P violation, but in the process under consider-ation f couples only to self-conjugate neutral particles,γ and φ. Hence PU

f 6= 0 is also a signal of CP violation.

We note that PUf is a pure but a poor probe of CPV.

For polarized initial state photons, QED predicts anon-zero value of Pf and this prediction is modified bythe presence of the Higgs exchange diagram. The devi-ation from the QED prediction is a probe of the Higgscontribution and hence its couplings. We define

δP+

f = P++

f − (P++

f )QED, (8)

δP−f = P−−

f − (P−−f )QED. (9)

Such a deviation does not have any definite P or CPproperty, thus it can be non-zero even when the Higgsboson is a CP eigenstate. This allows us to detect thepresence of a Higgs over a large range of xi’s and yi’s(and hence a large range of model parameters, as we willsee later) by measuring the fermion (t/τ) polarization.We choose equal polarization of e+ and e− beams so asto have equal helicities for the colliding photons; thisenhances the chirality-mixing Higgs contribution as dis-cussed in the previous section, c.f. Eqs. (4) and (5).

In QED, the polarization of f flips its sign if we changethe signs of the polarizations of the initial state photons(actually those of the electron and positron of the parentcollider), i.e. P++

f = −P−−f . This is due to P invariance

of QED. Also, due to the self-conjugate nature of theneutral particles involved, we have P ii

f = P iif . In CP-

violating models, however, we expect P++

f + P−−f 6= 0.

Although it is a probe of P violation in general, it willbe a probe of CP violation in our process. In Table IIwe list all the observables and their potentials. Thedefinitions in Eqs. (8) and (9), along with the fact that(P++

f )QED = −(P−−f )QED, implies

δPCPf = δP+

f + δP−f . (10)

For polarized photons we therefore have two independentobservables, δP−

f and δPCPf . These can be sizable over a

large range of, for instance, MSSM parameters and hencecan be used to probe the Higgs interactions. As we havealready mentioned, the expected polarization PU

f for un-polarized photons is very small. In Fig. 3 we show, asan example, expected values of δP±

τ and δPCPτ as func-

tions of Eb for a Higgs mass of 54 GeV and vτ = 2.0,

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

(1/L

ee)

dLγγ

/dz

z

FIG. 2: Luminosity distribution plotted against z (whichis related to the γγ invariant mass W = 2

√ω1ω2 via z =

W/(2Eb)) for xc = 4.8. The solid line corresponds to λeλl =−1, the short-dashed line is for λeλl = 1, and the long-dashedone for λeλl = 0. The conversion distance is taken to be zero.The grey patch highlights the region 0.75 < z < 0.83.

-12

-10

-8

-6

-4

-2

0

2

4

24 26 28 30 32 34 36 38 40 42

δPτ

X (

106 )

Eb (GeV)

δPτCP

δPτ+

δPτ-

FIG. 3: Variation of δP+, δP− and δPCP as a function of Eb

for a Higgs boson of mass 54 GeV. For a value of Eb in thegrey patch, the mass of the Higgs boson matches with the γγinvariant mass in the grey band shown in Fig. 2.

aτ = −2.3, Aγ = −0.76 + 0.032i, Bγ = −0.13 + 0.039i.The peak in δP occurs when the Higgs mass matcheswith the γγ invariant mass corresponding to the peakof photon spectrum in Fig. 2. By adjusting the beamenergy Eb, we can thus maximize the sensitivity of thepolarization observables defined above. Details of the se-lection criteria for Eb for a scan over the parameters of amodel are discussed in Section V.

IV. THE CPV-MSSM HIGGS SECTOR

We choose the MSSM as an example for demonstrat-ing the potential of the observables constructed in theprevious section to isolate the Higgs boson contribution

5

CPsuperH FeynHiggs

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200

tanβ

mH+ (GeV)

(a)

gVVH1

0.9

0.9

0.1

0.10.2

0.2

0.3

0.3

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200ta

nβmH+ (GeV)

(a)

gVVH1

0.9

0.9

0.1

0.1

0.2

0.2

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200

tanβ

mH+ (GeV)

(b)mφ1

(GeV)1007550

4030

20

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200

tanβ

mH+ (GeV)

(b)

mφ1 (GeV)

20

4050

75 100

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200

tanβ

mH+ (GeV)

(c)

(mφ2-mφ1

)/(Γφ1+Γφ2

)

100

100

50 50

30

30

20

2

4

6

8

10

12

14

16

18

20

100 120 140 160 180 200

tanβ

mH+ (GeV)

(c)

(mφ1-mφ1

)/(Γφ1+ Γφ2

)

10075

5040

30

20

FIG. 4: CPX scenario: contours of (a) AV (see Eq.(2)) forV = W, Z, (b) mφ1

and (c) (mφ2− mφ1

)/(Γφ1+ Γφ2

) com-puted with CPsuperH as well as FeynHiggs.

TABLE III: List of MSSM parameters for the CPX scenarioused as input for the programs CPsuperH and FeynHiggs.

MSSM Value MSSM Valueparam. param.tanβ 3 – 40 (for scan) mH+ 150 – 500 GeV (for scan)

µ 2 TeV, Φµ = 0 M1, M2 200 GeV, Φ1,2 = 0M3 1 TeV, Φ3 = 90◦ mq,l 500 GeVAt,b 1 TeV, Φt,b = 90◦ Aτ 500 GeV, Φτ = 90◦

and to probe the CP properties of its couplings. In theCP-conserving MSSM, there exist three neutral Higgsbosons: the CP-even h, H and the CP-odd A. CP-violating phases of the MSSM parameters such as the hig-gsino mass parameter µ, gaugino masses Mi (i = 1, 2, 3)and trilinear couplings Af (f = t, b, τ), can induce CPviolation in the Higgs sector via loops. This allows Higgsstates with different CP to mix; the three mass eigen-states hence do not have definite CP. These states aredenoted by φ1, φ2 and φ3 with their masses in increasingorder. For the numerical analysis, we choose the so-calledCPX [28] scenario with parameters as listed in Table III.In this scenario, one can have large CP-violating effectsin the Higgs sector depending upon the size of the phasesof the trilinear couplings At,b,τ . Due to this large CP vi-olation, the coupling of the lightest state φ1 to vector

bosons can go down drastically for some values of mH+

and tanβ, see Fig. 4(a). This results in highly suppressedproduction rates for the lightest Higgs boson; the lowerbound on the mass of such a Higgs boson from directsearches at LEP can be as low as 10–50 GeV [5]. Wetake into account all three neutral Higgs bosons in thecalculation of the fermion polarization by adding theirs-channel diagrams. This may, however, not be valid insome regions of the CPV-MSSM parameter space wherethe Higgs masses become nearly degenerate, i.e. the massdifference of two Higgs bosons is smaller than sum of theirwidths and hence mixing between these states is reso-nantly enhanced. In this case one should do a coupledchannel analyze [26, 32] of the degenerate states. For-tunately, in the region of parameter space we consider,the mass differences are always much larger than the sumof the decay widths, see Fig. 4(c). Thus our analysis iscomplementary to that of Refs. [26, 32].

V. NUMERICAL RESULTS

At a photon collider, the center of mass energy of thecolliding photons is not fixed but has a wide spectrum.The shape of this spectrum depends upon the polariza-tions of the laser and the e+/e− beams of the parentcollider. For the numerical calculation of cross sections,we use the ideal Compton back-scattered spectrum [31]with polarizations of the electron beam and the laser cho-sen such that one obtains the hard spectrum in Fig 2.The parent electron beam energy, Eb, can be chosen tomaximize the deviation from the QED prediction, cf.Fig. 3. Our observables are maximized when the Higgsmass matches the value of the γγ invariant mass at thepeak of the polarized photon spectrum. This happens forEb = (mφi

/2)/z, where z (the scaled γγ invariant mass,z =

√ω1ω2/Eb) takes a value between 0.75 to 0.83 for

xc = 4.8. This corresponds to the case where the scaledHiggs mass, mφi

/2Eb, lies in the grey band of Fig. 2.We can therefore pursue two different strategies for

choosing Eb :

1. Parameterizing the relationship between mφiand

Eb in terms of z0 as Eb = (mφi/2)/z0, we choose

an optimal value of z0, say z0 = 0.80, for each pointin the scan such that δP is maximized. This givesa very good estimate of the ultimate potential ofthe particular observable used in adaptation. Wecall this the “peak Eb” choice.

2. Fixing Eb such that the relevant Higgs mass (mφ1

for τ+τ− and mφ2,φ3for tt production) matches

approximately with the γγ invariant mass corre-sponding to z values within the peak of the photonspectrum (the grey band in Fig. 2). Though thischoice does not exploit the observable optimally, itis closer to what will be done in a realistic experi-ment. We call this the “fixed Eb” choice.

In the case of τ polarization, due to the small valuesof mτ and Γφ1

, the absolute values of the polarizationobservables are <∼ 10−5. These can be enhanced by

6

CPsuperH, Φt,b,τ = 0◦FeynHiggs, Φt,b,τ = 0◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

0.5

1

1.5

2

345

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

2.0

1.8

1.6 1.4

1.2

1.00.80.6

CPsuperH, Φt,b,τ = 90◦ FeynHiggs, Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

dEm = 1 GeV

0.5

1

1.5

234

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

dEm = 1 GeV

2.2

2.0

1.8

1.6

1.4

1.21.00.2

FIG. 5: Contours of constant δP−τ in units of 10−2 in the (tan β–mH+) plane for the CPX scenario with Φt,b,τ = 0◦ (top panels)

and Φt,b,τ = 90◦ (bottom panels) for the “peak Eb” choice with dEm = 1 GeV. The left panels show the results obtained withCPsuperH, the rights panels those obtained with FeynHiggs.

putting a cut on the invariant mass of the τ+τ− pair toselect the ones coming from φ1 decay [33] :

|mττ − mφ1| ≤ max(dEm, 5 Γφ1

), (11)

where dEm is the minimum resolution of mττ reconstruc-tion. We use dEm = 1 GeV for purposes of illustrationin this paper. For the case of top production such a cutis not necessary.

In the following, we perform a scan over the MSSMparameters as given in Table III and calculate the τ andt polarization observables for both the peak and the fixedEb choices using both CPsuperH and FeynHiggs for cal-culating the Higgs masses, couplings, and widths. Thestatistical fluctuation in the value of the fermion polar-

ization is given by

∆Pf =

√

1 − P 2f

√

N+ + N−

. (12)

The typical value of ∆Pf for an integrated luminosityof 100 fb−1 is about 0.003 for a total rate of 1 pb.The typical τ -pair production rate with the above cutis about 1–10 pb over the (tanβ–mH+) plane. τ decaysinto hadronic channels reduce the useful cross section andhence cause this error to increase. As a conservative mea-sure we take δPτ ≥ 0.01 in order to be measurable. Thett production rate, on the other hand, is less than 1 pb forthe energies considered, and it can be as low as 8 fb withthe “peak Eb” choice in some regions of the (tan β–mH+)plane. Thus the statistical error in the polarization mea-

7

CPsuperH, Φt,b,τ = 0◦FeynHiggs, Φt,b,τ = 0◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−t

1

23

5

9

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP+t

-1

-2

-3-5-10-13

CPsuperH, Φt,b,τ = 90◦ FeynHiggs, Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−t

-1-2

0

1

2

3

5

1011

40

30

20

15

10

8

65

4

3 150 200 250 300 350 400 450 500

tanβ

mH+

δP+t

2

0

-1

-2

-4-4

-8-10

-13

FIG. 6: Contours of constant δP±t in units of 10−2 in the (tan β–mH+) plane for the CPX scenario with Φt,b,τ = 0◦ (top

panels) and Φt,b,τ = 90◦ (bottom panels) for the “peak Eb” choice. The left panels show the results for δP−t obtained with

CPsuperH and the rights panels those for δP+t obtained with FeynHiggs, see text.

surement goes up and the sensitivity goes down in thiscase, even if the polarization asymmetry is large.

A. Peak Eb scan

We first discuss the case of τ -pair production throughφ1 exchange for the “peak Eb” choice. Figure 5 showscontours of constant δP−

τ as obtained with CPsuperH andFeynHiggs in the (tanβ−mH+) plane for Φt,b,τ = 0◦ andΦt,b,τ = 90◦. For each point in the scan, the beam en-ergy is set to Eb = mφ1

/(2z0) to maximize δP±τ . In the

CP-conserving case, Φt,b,τ = 0◦, δP−τ should be measur-

able for tanβ ≥ 4 and mH+ ≥ 200 GeV; smaller val-ues of mH+ require somewhat higher tanβ to achieve

δP−τ ≥ 0.01. In the case of maximal CPV phases,

Φt,b,τ = 90◦, δP−τ > 0.01 holds over practically the entire

parameter range, including the region where a very lightHiggs boson may have been missed at LEP2 [5]. Such alight CPV Higgs boson will also be difficult to discoverat the LHC [34]. The process γγ → τ+τ− may hence of-fer a unique possibility for this case. While δP−

τ covers alarge part of the parameter space, the CP-odd observableδPCP

τ is very small, δPCPτ ∼ 10−5, and hence below the

limit of measurability even if the CPV phases are max-imal. This means in turn that δP+

τ ≈ −δP−τ . It is also

worth noting that the size of the observable is rather sen-sitive to the value of dEm : increasing for instance dEm

from 1 GeV to 2 GeV, δP−τ goes down by about a factor

of 2 over most of the parameter space in Fig. 5.

8

CPsuperH, Φt,b,τ = 90◦FeynHiggs Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δPCPt

-8

-6

-4

-2

0

2

4

63

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δPCPt

10864

2

0

-2

-4-6-8-10

FIG. 7: Contours of constant δPCPt in units of 10−2 in the (tanβ–mH+) plane for the CPX scenario with Φt,b,τ = 90◦ and

“peak Eb” choice. The left panel shows the results obtained with CPsuperH, the right panel those obtained with FeynHiggs.

Let us now turn to the top polarization in γγ → tt.Due to the large top quark mass, here only the heav-ier Higgs bosons φ2,3 contribute. For the compara-tively large mH+ values needed to obtain mφ2,3

≥ 2mt,the mass difference between φ2 and φ3 is usually sosmall that both their scaled masses can be within thepeak of the hard photon spectrum of Fig. 2, althoughmφ3

− mφ2≫ Γ2,3 is still maintained. We therefore

choose

Epeakb = max

[

E0b , (mφ2

+ mφ3)/(4z0)

]

, (13)

where E0b = 220 GeV and z0 = 0.8. Figure 6 shows con-

tours of constant δP±t in the (tanβ − mH+) plane anal-

ogous to Fig. 5. Owing to different sign conventions thetwo programs, δP±

t of CPsuperH corresponds to −δP∓t in

FeynHiggs. In Fig. 6 we therefore plot δP−t for CPsuperH

and δP+t for FeynHiggs. The deviation from the pure

QED prediction (i.e. the Higgs contribution) is sizable fortan β <∼ 10. Note that the CP-even polarization observ-ables for tt can reach much larger values than those forτ+τ−. Moreover, in the case of γγ → tt also the CP-oddobservable, δPCP

t , may be large enough to be measurable.This is shown in Fig. 7 for Φt,b,τ = 90◦. Because of thelow cross sections, ranging from 8 fb to 150 fb in Fig. 7,as compared to 1–10 pb for the τ case, the statistical fluc-tuations are large, about ∆Pt ∼ 0.10 − 0.03. Thereforethe region of sensitivity to the Higgs-boson contributionsis restricted to low tan β values.

B. Fixed Eb scan

The “peak Eb” choice discussed above gives an esti-mate of the ultimate potential of our polarization ob-servables. In reality, however, one will have a colliderrunning at some fixed beam energy. Obviously it will beof advantage to set Eb such that one is sensitive to the

Higgs contributions over a large part of the parameterspace.

In the CPX scenario, CPsuperH predicts mφ1<∼ 123

GeV. Thus Eb = 77 GeV leads to a good sensitivity overmost of the parameter space. With FeynHiggs, how-ever, the maximum value of mφ1

considerably changeswith Φt,b,τ in the scan; we obtain maximum values ofmφ1

= 123 GeV for Φt,b,τ = 0◦ and mφ1= 131 GeV for

Φt,b,τ = 90◦, respectively. Hence we choose Eb = 77 GeVfor Φt,b,τ = 0◦ and Eb = 82 GeV for Φt,b,τ = 90◦ inthe computation with FeynHiggs. The polarization ob-servable δP−

τ for this fixed Eb choice is shown in the(tanβ − mH+) plane in Fig. 8. We observe that forΦt,b,τ = 90◦, δP−

τ ≥ 0.01 unless tan β is very small. ForΦt,b,τ = 0◦, on the other hand, observability of δP−

τ islimited to tanβ >∼ 8–10. It is apparent that δPτ willbe mainly useful if tanβ is large. To explicitly see thephase dependence we show in Fig. 9 contours of constantδP−

τ in the (mH+ − Φ) plane for Eb = 77 GeV. There isa rather large difference in the results of the two codes,which is also apparent in the other figures, due to differ-ences in the implementation of radiative corrections inthe two programs [35]. It is clear that for analyses assuggested in this paper, more precise computations willbe necessary.

We next turn to tt production with fixed Eb. In thiscase, as for the peak Eb choice, it is the mean mass ofφ2 and φ3 that should be within the peak of the pho-ton spectrum. However, since mφ2,3

change linearly withmH+ , one cannot have optimal sensitivity over the wholeparameter space with fixed Eb. We hence take Eb = 300GeV as a good compromise. For this choice one hascomparatively large rates while the scaled masses of φ2,3

still lie within the peak of the photon spectrum for asizable portion of the (tanβ − mH+) plane. The re-sults for δP±

t obtained with CPsuperH and FeynHiggs

for Φt,b,τ = 0◦ and Φt,b,τ = 90◦ are shown in Fig. 10.

9

CPsuperH Φt,b,τ = 0◦FeynHiggs Φt,b,τ = 0◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

0.5

1

1.5

2

34

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

1.8

1.6 1.4

1.2

1

0.2

0

CPsuperH Φt,b,τ = 90◦FeynHiggs Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

0 0.5

1

1.5

2

3

4

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

δP−τ

2

1.8

1.6

1.41.2

1

0.2

0

FIG. 8: Contours of constant δP−τ in units of 10−2 for fixed Eb and Φt,b,τ = 0◦ and 90◦ using CPsuperH (left panels) and

FeynHiggs (right panels) to compute the Higgs masses, couplings and widths. Eb = 77 GeV, except for the lower-right plotwhere Eb = 82 GeV, see text.

Again the role of δP+t and δP−

t is interchanged in the twocodes. In Fig. 11, we show contours of constant δPCP

t inthe (Φ − mH+) plane for tanβ = 4 and in Fig. 12 in the(Φ − tanβ) plane for mH+ = 475 GeV. As one can see,there is sensitivity to CP violation if tanβ is small. More-over, there is rather good agreement between CPsuperH

and FeynHiggs in δPCPt (up to a sign). Note also that

the signal can be enhanced by tuning Eb.

Comparing these results with the “peak Eb” choice, wesee that for γγ → ττ one can be sensitive to as large aregion of the parameter space if Eb is chosen carefully.For γγ → tt, on the other hand, we loose sensitivity topart of the parameters space with fixed Eb.

C. Lepton asymmetries

The polarization of τ leptons can be measured usingthe energy distribution of the decay pions [22, 36, 37, 38,39, 40]. The polarization of top quarks can be measuredusing energy distribution of b quarks [41] or the angu-lar distribution of decay leptons [42, 43, 44, 45]. Thiskind of analysis requires the full reconstruction of thetop momentum. Such a reconstruction may not alwaysbe possible for the semi-leptonic decay of the t (or t)quark. On the other hand, it is possible to construct sim-ple asymmetries involving the polarization of the initial-state e± (and hence of the photons) and the charge ofthe final-state lepton, which are sensitive to CP viola-tion. We denote the integrated cross section for the pro-cess γγ → tt → l+νbt (tl−νb) by σ(λe− , Ql), where λe−

10

CPsuperH FeynHiggs

150

175

200

225

250

275

300

325

350

0 10 20 30 40 50 60 70 80 90

mH

+ (

GeV

)

Φ (in degrees)

δP-τ

(tanβ=30)

3.5

3

3

2.52

0.5 150

175

200

225

250

275

300

325

350

0 10 20 30 40 50 60 70 80 90

mH

+ (

GeV

)

Φ (in degrees)

δP-τ

(tanβ=30)

0

0.5

1

1.5

2

2.5

FIG. 9: Contours of constant δP−τ in units of 10−2 in the (Φt,b,τ −mH+) plane for tan β = 30 and Eb = 77 GeV with CPsuperH

(left panel) and FeynHiggs (right panel).

is the polarization of the electron beam in the parent col-lider and Ql the charge of the secondary lepton comingfrom the t/(t) decay. The polarizations of all the otherbeams are adjusted to get a peaked spectrum and equalhelicities for the incident photons. With this setup, wecan define the following asymmetries [19]:

A1 =σ(+, +) − σ(−,−)

σ(+, +) + σ(−,−),

A2 =σ(+,−) − σ(−, +)

σ(+,−) + σ(−, +),

A3 =σ(+, +) − σ(−, +)

σ(+, +) + σ(−, +),

A4 =σ(+,−) − σ(−,−)

σ(+,−) + σ(−,−), (14)

Only one of the above asymmetries is independent [19] ifno cut is put on the lepton’s polar angle in the labora-tory frame. Even with a finite cut on the polar angle, theA1...4 have almost identical sensitivities to the Higgs cou-plings. We use a 20◦ beam-pipe cut on the lepton. Thecontours of constant A3 for Φt,b,τ = 30◦ and 90◦, usingCPsuperH, are shown in Fig. 13 for the “peak Eb” choice.Analogously, Fig. 14 shows A3 for fixed Eb = 300 GeVand Φt,b,τ = 30◦ and 90◦. The asymmetries are sizablefor Φt,b,τ = 90◦ and decrease rapidly as Φt,b,τ decreases.For Φt,b,τ = 0◦ the only source of CPV is the phase ofM3, Φ3 = 90◦, in our scenario. All the yi’s are then negli-gibly small as compared to the xi’s, leading to very smallvalues of the Ai. Here note that, as shown in Ref. [19],the lepton asymmetries of Eq. (14) are sensitive only toCP-odd combinations of the form factors, i.e. the yi’s.This should be contrasted with the polarization observ-ables, which are sensitive to both the CP-odd and CP-even combinations.

VI. CONCLUSIONS

We have investigated the use of fermion polarizationin the process γγ → f f (with f = t or τ) for studyingneutral Higgs bosons at a photon collider. To this aim wehave constructed polarization asymmetries δP±

f , whichare sensitive to the Higgs exchange contributions. Wehave also constructed a CP-odd asymmetry δPCP

f whichis sensitive to CP violation in the Higgs sector. All theseasymmetries are constructed in a model-independent wayand can be used to study Higgs bosons in various modelsbeyond the SM.

We have applied this in a numerical analysis to the caseof the MSSM with explicit CP violation. In particular wehave evaluated our asymmetries for the CPX scenario,using the two public codes CPsuperH and FeynHiggs tocalculate the Higgs masses, couplings and widths.

Scanning the (tanβ − mH+) plane for various phasesΦt,b,τ , we found that δP±

τ is sensitive to a light Higgs,especially if tanβ is large. Assuming a measurement ac-curacy of 10−2 for τ polarization, δP±

τ can in fact probea large part of the CPV-MSSM parameter space. A cuton the ττ invariant mass is, however, necessary to en-hance the signal. The CP-odd asymmetry δPCP

τ , on theother hand, is always very small, well below measurabil-ity. While δP±

τ can be enhanced by the cut mentionedabove, this does not work for δPCP

τ .This is complemented by the top polarization in tt pro-

duction, which is sensitive to the heavier neutral Higgsbosons φ2,3, and also to CP mixing between them, formφ2,3

≥ 2mt and small tanβ. A similar region is coveredby the lepton asymmetries constructed from top decays,which are a pure measure of CP violation. These leptonasymmetries are large for large Φt,b,τ but quickly decreaseas this phase decreases.

To conclude, the top and tau polarization asymme-tries presented in this paper may prove useful to studythe effects of a CP-violating Higgs sector at a photon col-lider. They may in particular cover the parameter region

11

CPsuperH Φt,b,τ = 0◦FeynHiggs Φt,b,τ = 0◦

15

10987

6

5

4

3 300 350 400 450 500

tanβ

mH+

δP−t

1

1.5

2

3.5

15

10987

6

5

4

3 300 350 400 450 500

tanβ

mH+

δP+t

-0.5

-1

-1.5

-2

-3

-4

CPsuperH Φt,b,τ = 90◦FeynHiggs Φt,b,τ = 90◦

15

10987

6

5

4

3 300 350 400 450 500

tanβ

mH+

δP−t

1 2

3

5

10

15

10987

6

5

4

3 300 350 400 450 500

tanβ

mH+

δP+t

-1

-2

-3

-4

-5

-10

-12

FIG. 10: Contours of constant δP±t in units of 10−2 for fixed Eb = 300 GeV and Φt,b,τ = 0◦ (top) and 90◦ (bottom) using

CPsuperH (left) and FeynHiggs (right) to compute the Higgs masses, couplings and widths.

where a light CP-violating Higgs may have been missedat LEP, and may be missed as well at LHC. We found,however, large quantitative differences between the re-sults obtained with CPsuperH and FeynHiggs. In thisregard we emphasize the need for a standardization ofthese tools.

Note added

While this paper was in preparation, at a pointwhere the numerical analysis was already finished,a new version of FeynHiggs was released, seehttp://www.feynhiggs.de and the contribution onFeynHiggs in [8]. This version contains new radiativecorrections also for the CP-violating case. It will be inter-

esting to see their effect on the polarization observablesdiscussed in this paper.

Acknowledgments

We thank S. Heinemeyer for discussions regardingFeynHiggs. R.M.G., S.D.R. and R.K.S. acknowledgethe support of the Department of Science and Technol-ogy, India, under project no. SP/S2/K-01/2000-II and ofthe Indo-French Center for the Promotion of AdvancedResearch under IFCPAR project no. IFC/3004-B/2004.The work of S.K. is financed by an APART (AustrianProgram for Advanced Research and Technology) grantof the Austrian Academy of Sciences.

12

CPsuperH FeynHiggs

325

350

375

400

425

450

475

500

0 10 20 30 40 50 60 70 80 90

mH

+ (

GeV

)

Φ (in degrees)

δPtCP

(tanβ=4)

5432

1

0

0 350

375

400

425

450

475

500

0 10 20 30 40 50 60 70 80 90

mH

+ (

GeV

)

Φ (in degrees)

δPtCP

(tanβ=4)

0

-1

-2

-3

-4-5

FIG. 11: Contours of constant δPCPt in units of 10−2 in the (Φt,bτ −mH+) plane for tanβ = 4 and Eb = 300 GeV with CPsuperH

(left panel) and FeynHiggs (right panel).

CPsuperH FeynHiggs

15

1098

7

6

5

4

3 0 10 20 30 40 50 60 70 80 90

tanβ

Φ (in degrees)

δPtCP

(mH+ = 475 GeV)

0

1

2

3

4 5

15

1098

7

6

5

4

3 0 10 20 30 40 50 60 70 80 90

tanβ

Φ (in degrees)

δPtCP

(mH+ = 475 GeV) -0.5

-1

-1.5

-2-3 -3.5

FIG. 12: Contours of constant δPCPt in units of 10−2 in the (Φt,bτ − tan β) plane for mH+ = 475 GeV and Eb = 300 GeV with

CPsuperH (left panel) and FeynHiggs (right panel).

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Φt,b,τ = 30◦ Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

21

0.5

00

-0.5

-1

-1.5

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mH+

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-8-10

FIG. 13: Contours of constant A3 in units of 10−2 for “peak Eb” and Φt,b,τ = 30◦ and 90◦; Higgs masses, couplings and widthscomputed with CPsuperH.

Φt,b,τ = 30◦ Φt,b,τ = 90◦

3

4

56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

0

-0.2

-0.4

-0.6

-1

-1.8

3

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56

8

10

15

20

30

40

150 200 250 300 350 400 450 500

tanβ

mH+

0

-1

-2

-3

-5

-10

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