Page 1

arXiv:hep-ph/0005313v1 31 May 2000

KEK–TH–698

KIAS–P00028

OCHA–PP–160

hep–ph/0005313

February 1, 2008

Measuring the Higgs CP Property through Top Quark

Pair Production at Photon Linear Colliders

Eri Asakawa1, S.Y. Choi2, Kaoru Hagiwara3and Jae Sik Lee4

1Graduate School of Humanities and Sciences, Ochanomizu University,

1–1 Otsuka 2–chome, Bunkyo, Tokyo 112-8610, Japan

2Department of Physics, Chonbuk National University, Chonju 561–756, Korea

3Theory Group, KEK, Tsukuba, Ibaraki 305-0801, Japan

4Korea Institute for Advanced Study, Seoul 130–012, Korea

Abstract

We present a model–independent study of the effects of a neutral Higgs boson without

definite CP–parity in the process γγ → t¯t around the mass pole of the Higgs boson. Near the

resonance pole, the interference between the Higgs–exchange and the continuum amplitudes

can be sizable if the photon beams are polarized and helicities of the top and anti–top quarks

are measured. Study of these interference effects enables one to determine the CP property

of the Higgs boson completely. An example of the complete determination is demonstrated

in the context of the minimal supersymmetric standard model.

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1Introduction

Search for Higgs bosons and precise measurements of their properties such as their masses, the

decay widths and the decay branching ratios [1] are among the most important subjects in our

study of the electroweak symmetry breaking. In the standard model (SM), only one physical

neutral Higgs boson appears and its couplings to all the massive particles are uniquely deter-

mined. On the other hand, models with multiple Higgs doublets have neutral Higgs bosons of

definite CP parity as well as charged Higgs bosons, if CP is a good symmetry. If CP is not a

good symmetry of the symmetry breaking physics, these neutral Higgs bosons do not necessarily

carry definite CP parity.

The CP–violating interactions beyond the Kobayashi–Maskawa mechanism and their conse-

quences at high–energy colliders have been intensively studied. This is motivated in part by the

search of an efficient mechanisms of generating the cosmological baryon asymmetry at the elec-

troweak scale [2]. An extended Higgs–boson sector, as predicted by many extensions of the SM

such as the minimal supersymmetric SM (MSSM), can provide such CP–violating interactions

in a natural way. One attractive scenario is to make use of explicit CP violations in the MSSM

Higgs sector [3, 4] which are induced through loop corrections with complex supersymmetric pa-

rameters in the mass matrices of the third generation squarks. Such interactions cause mixings

among CP–even and CP–odd Higgs bosons.

It is of great interest to examine the possibility of studying these Higgs bosons in detail in

CP non–invariant theories. A two–photon collision option [5] of the future linear e+e−colliders

offers one of the ideal places to look for such Higgs signals [6]. There, neutral Higgs bosons can

be produced via loop diagrams of charged particles. If the Higgs boson is lighter than about

140 GeV, its two-photon decay width can be measured accurately by looking for its main de-

cay mode, which is usually the bb mode [7]. When the Higgs boson is heavy, the processes

γγ → W+W−,ZZ and t¯t can be useful to detect its signal [8]. In theories with weakly coupled

Higgs sector, such as the MSSM, heavier Higgs bosons have suppressed branching fractions to

the WW/ZZ modes [9]. Furthermore, CP-odd Higgs bosons do not have the ZZ decay modes

in the tree level. On the other hand, the tt decay mode can be significant irrespective of the CP

property of the Higgs boson φ. In this case, it is expected that the heavy Higgs boson contributes

to the process γγ → tt significantly around its mass pole.

The s–channel resonant amplitude of γγ → φ(∗)→ tt can interfere with the tree–level t– and

u–channel continuum amplitudes, if the resonant and the continuum amplitudes have comparable

magnitudes near the resonance pole [10]. This often happens for heavy Higgs bosons where both

the peak resonant amplitudes and their total decay widths are large enough to make the interfer-

ence effects significant. In this article, we study the contribution of a heavy Higgs boson without

definite CP–parity to the process γγ → tt and describe an efficient method to determine its CP

property completely by making use of the photon and t¯t polarizations; the procedure is crucially

2

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based upon the interference effects among various helicity amplitudes. We also demonstrate the

feasibility of determining the heavy Higgs–boson contribution in the context of the MSSM.

The remainder of this article is organized as follows. In Sect. 2, the helicity amplitudes of the

process γγ → t¯t are calculated with model–independent parametrizations of the couplings of the

Higgs boson φ to a photon pair and a top–quark pair. In Sect. 3 we present all the observables

constructed by use of photon polarizations and by measuring the helicities of the final top and

anti–top quarks. Section 4 gives the description of the procedure to determine the CP property of

the Higgs boson completely. In Sect. 5 we study the properties of polarized photon beams through

the Compton laser backscattering. In Sect. 6 we demonstrate the complete determination of the

CP property of the heaviest neutral Higgs boson in the MSSM. Finally, we give conclusions in

Sect. 7.

2Helicity amplitudes

In CP non–invariant theories, the lowest–dimensional interaction of the Higgs boson φ with a

top–quark pair can be described in a model–independent way by the vertex:

Vφtt= −iemt

mW

(St+ iγ5Pt) ,(1)

and the loop–induced interaction of the Higgs boson with a photon pair is parameterized in a

model–independent form as follows:

Vγγφ =

√sα

4π

?

Sγ(s)

?

ǫ1· ǫ2−2

s(ǫ1· k2)(ǫ2· k1)

?

− Pγ(s)2

s?ǫ1ǫ2k1k2?

?

,(2)

where the form factors depend on the c.m. energy squared s of two colliding photons, ǫ1,2stand

for the wave vectors of the two photons, k1,2are the four–momenta of the two photons, and

?ǫ1ǫ2k1k2? = ǫµναβǫµ

1ǫν

2kα

1kβ

2,(3)

with ǫ0123= 1. Since we are interested in the s–channel virtual (and real) Higgs–boson exchange,

for the sake of consistency the s–dependence of the form factors Sγand Pγis exhibited explicitly

in Eq. (2). We note that a simultaneous presence of {Sγ,Pγ} or/and {St,Pt} implies CP non–

invariance of the theory.

In the two–photon c.m coordinate system with?k1along the positive z direction and?k2along

the negative z direction, the wave vectors ǫ1,2of two photons are given by

ǫµ

1(λ) = ǫµ∗

2(λ) =−λ

√2

?

0,1,iλ,0

?

.(4)

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where λ = ±1 denote the right and left photon helicities, respectively. Using Eqs. (1), (2) and

(4) one can derive the explicit forms of the helicity amplitudes for the process γγ → tt which con-

sist of two parts; the s–channel Higgs–boson exchange and the tree–level continuum contributions:

(i) Higgs–exchange contribution:

M(λ1λ2:σσ)

φ

=eαmt

4πmW

s

s − m2

φ+ imφΓφ[Sγ(s) + iλ1Pγ(s)][σβSt− iPt]δλ1,λ2δσσ.(5)

(ii) Tree–level continuum contribution:

M(λ1λ2:σσ)

cont

=

4παQ2

1 − β2cos2Θ

−4mt

t

?4mt

√s(λ1+ σβ)δλ1λ2δσσ

√sσβ sin2Θδλ1,−λ2δσσ− 2β(cosΘ + λ1σ)sinΘδλ1,−λ2δσ,−σ

?

,(6)

where Θ is the scattering angle of the top quark with respect to the positive z direction,

β =

1 − 4m2

with its three–momentum along the positive (negative) direction, respectively, in units of ¯ h, and

σ = ±1 (σ = ±1) denote the right and left helicities of t (t), respectively, in units of ¯ h/2. There

exists the interference between the Higgs–boson and the continuum contributions for equal t and

¯t helicities which can be used to observe the CP property of the Higgs boson. On the other hand,

the continuum contribution with opposite t and¯t helicities dominates the γγ → t¯t events at

high energies so that it is important to distinguish events with equal t and¯t helicities from those

with opposite t and¯t helicities. With the above points in mind, we concentrate on the helicity

amplitudes of equal t and¯t helicities in the present work.

?

t/s, λ1= ±1 (λ2= ±1) denote the right and left helicities of the incident photon

When the helicities of the top and anti–top quarks are equal, i.e. σ = σ, the helicity ampli-

tudes for the process γγ → tt can be rewritten in the following simple form

M(λλ:σσ)

M(λ,−λ:σσ)

where for the sake of brevity the explicit s–dependence of the form factors Sγ and Pγ is not

denoted and two s–dependent functions Acont(s) and Aφ(s) are introduced:

= Acont(s)(λ + σβ) + Aφ(s)[Sγ+ iλPγ][σβSt− iPt] ,

= −Acont(s)σβ sin2Θ, (7)

Acont(s) =

16παQ2

√s(1 − β2cos2Θ),

tmt

Aφ(s) =eα

4π

mt

mW

Dφ(s). (8)

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Here Dφ(s) is the s–channel propagator of the Higgs boson

Dφ(s) =

s

s − m2

φ+ imφΓφ. (9)

A few important comments concerning the helicity amplitudes (7) are in order:

• Only the amplitudes with equal photon helicities have contributions from the Higgs–boson

exchange.

• The spin–0 Higgs–boson contributions are independent of the scattering angle Θ.

• The s–dependent form factors Sγ and Pγ as well as Aφare in general complex while the

other form factors are real in the leading order.

• The continuum part is CP–preserving while the Higgs–exchange part can be CP–violating

if the ‘scalar’ (CP–even) form factors, Sγand St, and ‘pseudoscalar’ (CP–odd) form factors,

Pγand Pt, are present simultaneously.

3Polarized cross sections

3.1 Equal photon helicities and top–quark helicities

In this section, we investigate what physics information on the Higgs–boson contribution can be

extracted from the cross section with equal photon and top–quark helicities. First of all, we can

construct four independent squared amplitudes:

???M(++:++)???

???M(−−:−−)???

???M(++:−−)???

???M(−−:++)???

2= |M|2

2= |M|2

2= |M|2

2= |M|2

0

?

?

?

?

1 + A0+ A1− (1 + β)(A2− A3)

?

?

?

?

,

0

1 + A0− A1+ (1 + β)(A2− A3),

0

1 − A0+ A1+ (1 − β)(A2+ A3),

0

1 − A0− A1− (1 − β)(A2+ A3),(10)

where |M|2

0is the unpolarized squared amplitude, i.e. the average

|M|2

0=1

4

????M(++:++)???

2+

???M(−−:−−)???

2+

???M(++:−−)???

2+

???M(−−:++)???

2?

.(11)

The explicit form (7) of the helicity amplitudes then leads to the following expressions for |M|2

and the quantities Ai(i = 0,1,2,3):

|M|2

0

0

= (1 + β2)A2

cont+ (β2S2

t+ P2

t)(|Sγ|2+ |Pγ|2)|Aφ|2

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+2Acont

?

β2StR(AφSγ) + PtR(AφPγ)

Acont+ [StR(AφSγ) + PtR(AφPγ)]

(β2S2

?

,

A0 = 2βAcont

A1 = 2|Aφ|2

?

?

?

?

? ?

|M|2

0,

t+ P2

t)I(SγP∗

? ?

? ?

γ)

? ?

|M|2

0,

A2 = 2βAcont

StI(AφPγ)

|M|2

|M|2

0,

A3 = 2Acont

PtI(AφSγ)

0, (12)

where |M|2

they can be non–vanishing only in CP non–invariant theories. More explicitly, we can exploit

the CP–odd combinations

0and A0are CP–even, but the three asymmetries A1,2,3are CP-odd, that is to say,

?

?

λ

λ

????M(++:λλ)???

????M(σσ:++)???

2+

???M(−−:λλ)???

???M(σσ:−−)???

2? ?

2? ?

4|M|2

0

= −A2+ βA3,

σ

σ

2+

4|M|2

0

= A1− βA2+ A3, (13)

in extracting the CP–odd asymmetries A1,2,3. It is however clear that we need to exploit more

observables to determine all the form factors, Sγ, St, Pγ and Ptcompletely. This can be done

by using linear photon polarization as well as circular photon polarization as shown in the next

section.

3.2Two photon spin correlations

Taking into account the general polarization configuration of two photon beams and taking the

sum over the final polarization configuration with equal t and¯t helicities, we obtain the polarized

squared amplitude as

|M|2

= |M|2

0

′??

+ sin2Θ

1 + ζ2˜ζ2

?

−C0

+ C3

+ B1

?

?

ζ2+˜ζ2

?

+ B2

?

?

?

?

ζ1˜ζ3+ ζ3˜ζ1

?

− B3

?

???

?

ζ1˜ζ1− ζ3˜ζ3

ζ3+˜ζ3

?

?

ζ2˜ζ2− ζ3˜ζ3

?

+ C1

+ C4

ζ1+˜ζ1

?

+ C2

?

ζ1˜ζ2+ ζ2˜ζ1

?

ζ2˜ζ3+ ζ3˜ζ2

, (14)

where the newly–introduced unpolarized squared amplitude |M|2

|M|2

The second term above comes from the continuum contributions with opposite photon helici-

ties. The parameters {ζi} and {˜ζi} (i = 1,2,3) are the Stokes parameters of two photon beams,

0

′is given by

0

′= |M|2

0+ β2sin4ΘA2

cont. (15)

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respectively, which should be initially prepared. In Sect. 5 we will give a brief description of gen-

erating energetic photon beams and controlling their polarizations through the Compton laser

backscattering off the electron or positron beams.

The observables Bi(i = 1,2,3) in Eq. (14) are due to the interference of the continuum and

Higgs–boson parts in the helicity amplitudes M(λλ:σσ)of Eq. (7) and they are explicitly given by

?

B2= 2

?

+ 2Acont

β2StR(AφSγ) − PtR(AφPγ)

B1=A1− βA2+ A3

?

(−1 + β2− β2sin4Θ)A2

?

?

|M|2

0

?

|M|2

0

′,

|Aφ|2(β2S2

t+ P2

t)R(SγP∗

γ) + Acont

?

β2StR(AφPγ) + PtR(AφSγ)

?? ?

|M|2

0

′,

B3=

cont+ (β2S2

t+ P2

t)(|Sγ|2− |Pγ|2)|Aφ|2

?? ?

|M|2

0

′.(16)

The observable B3is CP–even, while the other two observables B1,2are CP–odd. We note that

the latter CP–odd observable B2corresponds to the so–called T–odd triple product of one photon

momentum and two photon polarization vectors.

On the other hand, the five additional observables Ci(i = 0 to 4) are due to the interference

between the helicity amplitudes with equal photon helicities and those with opposite photon

helicities and they are explicitly given by

C0= 2β2sin2ΘA2

C1= 2β2Acont

C2= 2β2Acont

C3= 2β2Acont

C4= −2β2Acont

cont

?

|M|2

0

′,

?

?

?

StR(AφPγ)

? ?

|M|2

0

′,

Acont+ StR(AφSγ)

? ?

′,

|M|2

0

′,

StI(AφSγ)

?

? ?

? ?

|M|2

|M|2

0

StI(AφPγ)

0

′. (17)

Among these polarization asymmetries, the observables C0,2,3are CP–even, while the observables

C1,4are CP–odd.

3.3 Top and anti–top quark polarizations

At asymptotically high energies the chirality conservation of the gauge–interactions leads to the

dominance of tt–pair production with opposite helicity in the continuum amplitudes. However,

near the threshold, there is also substantial production of tt–pairs of the same helicity. The

tt states with the same helicity transform to each other under CP, so any asymmetry in their

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production rates can provide a useful tool for studying CP violation.

Along with the initial two–photon polarizations we consider the final polarization configura-

tion with equal t¯t helicities to construct the polarization asymmetry

∆ =|M|2(σ = ¯ σ = +) − |M|2(σ = ¯ σ = −)

|M|2

0

′

. (18)

Depending on the photon spin–spin correlations, the observable ∆ is decomposed as follows:

∆ = D1

?

1 + ζ2˜ζ2

?

ζ1+˜ζ1

+ D2

?

ζ2+˜ζ2

?

+ D3

?

+ E3

ζ1˜ζ3+ ζ3˜ζ1

?

− D4

?

+ E4

ζ1˜ζ1− ζ3˜ζ3

?

?

+ sin2Θ

?

E1

??

+ E2

?

ζ3+˜ζ3

??

ζ1˜ζ2+ ζ2˜ζ1

?

ζ2˜ζ3+ ζ3˜ζ2

??

.(19)

The observables Di(i = 1 to 4) are due to the interference of the continuum and Higgs–boson

parts in the helicity amplitudes M(λλ:σσ)and they are explicitly given by

?

D2= 2βAcont

?

D4= 2βAcont

D1= 2βAcont

− StI(AφPγ) + PtI(AφSγ)

??

|M|2

0

′,

?

Acont+ [StR(AφSγ) + PtR(AφPγ)]

? ?

′,

|M|2

0

′,

D3= 2βAcont

− StI(AφSγ) + PtI(AφPγ)

??

|M|2

|M|2

′.

0

?

StI(AφPγ) + PtI(AφSγ)

? ?

0

(20)

The observables D1,4are CP–odd and the observables D2,3are CP–even.

On the other hand, the four additional observables Ei(i = 1 to 4) come from the interference

between the helicity amplitudes with equal photon helicities and those with opposite photon

helicities. Their explicit forms are

E1= 2βAcont

?

?

PtI(AφPγ)

? ?

? ?

|M|2

|M|2

? ?

0

′,

E2= 2βAcont

PtI(AφSγ)

?

?

0

′,

E3= −2βAcont

PtR(AφSγ)

|M|2

??

0

′,

E4= 2βAcont

Acont+ PtR(AφPγ)

|M|2

0

′. (21)

The observables E1,4are CP–even and the observables E2,3are CP–odd.

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4 Complete measurements of the Higgs–boson CP prop-

erty

In order to completely determine the CP parity of the Higgs boson we need to measure the

following six quantities (see Eq. (5)):

{mφ,Γφ,Sγ,Pγ,St,Pt} ,(22)

among which the one–loop induced γγφ form factors Sγand Pγare in general complex while the

others are real in the leading order. However, the helicity amplitudes are determined by helicity–

dependent multiplications of those quantities so that it is necessary to measure the following

quantities:

StR(AφSγ), StR(AφPγ), StI(AφSγ), StI(AφPγ),

PtR(AφSγ), PtR(AφPγ), PtI(AφSγ), PtI(AφPγ).

The above 8 quantities are not completely independent and satisfy, for example, the following

relations at all s ;

(23)

PtR(AφSγ) · StR(AφPγ) = PtR(AφPγ) · StR(AφSγ),

PtI(AφSγ) · StI(AφPγ) = PtI(AφPγ) · StI(AφSγ).

On the other hand, in principle 22 observables are available as shown in the previous section;

|M|2

parameters are completely determined and that they are over–constrained.

(24)

0, |M|2

0

′, 4 A’s, 3 B’s, 5 C’s, 4 D’s and 4 E’s. Therefore, it is expected that the Higgs–boson

Assuming that each observable is measured with a reasonable efficiency, we provide a straight-

forward procedure to determine the quantities listed in Eq. (23):

(1) The first four quantities in Eq. (23) can be determined directly through four observables

Ci(i = 1 to 4) even without measuring top and anti-top helicities.

(2) The remaining four quantities in Eq. (23) can be determined directly through four observ-

ables Ei(i = 1 to 4).

(3) The constraints of Eq. (24), the observables Di(i = 1 to 4), and also Ai(i = 0 to 3) and

Bi(i = 1 to 3) can be used to test and improve the above measurements.

To recapitulate, the Higgs–exchange contribution can be completely determined by a judicious

use of photon polarizations and t¯t helicity measurements.

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5 Polarized high energy photon beams

The observed cross section is a convoluted one of the parton–level cross section with a (polarized)

γγ luminosity function describing the spread of the γγ collision energy. A detailed study of the

possible luminosity and polarization distributions at future γγ colliders has been performed by

the simulation program CAIN [11]. However, these quantities are strongly dependent on the

machine design of the colliders. Thus we adopt an ideal situation of the beam conversion that

the photon beam is generated by the tree–level formula of the Compton backward–scattering and

that the effect of the finite scattering angle is negligible [12].

High energy colliding beams of polarized photons can be generated by Compton backscat-

tering of polarized laser light on (polarized) electron/positron bunches of e+e−linear collidersa.

The polarization transfer from the laser light to the high energy photons is described by three

Stokes parameters ζ1,2,3; ζ2is the degree of circular polarization and {ζ3,ζ1} the degree of linear

polarization transverse and normal to the plane defined by the electron direction and the direc-

tion of the maximal linear polarization of the initial laser light. Explicitly, the Stokes parameters

take the form [12]:

ζ1=f3(y)

f0(y)Ptsin2κ,ζ2= −f2(y)

f0(y)Pc,ζ3=f3(y)

f0(y)Ptcos2κ,(25)

where y is the energy fraction of the back–scattered photon with respect to the initial electron

energy Ee, {Pc,Pt} are the degrees of circular and transverse polarization of the initial laser

light, and κ is the azimuthal angle between the directions of initial photon and its maximum

linear polarization. Similar relations can be obtained for the Stokes parameters˜ζ of the opposite

high energy photon beam by replacing (Pc,Pt,κ) with (˜Pc,˜Pt,−˜ κ). The functions f0, f2, and f3

determining the photon energy spectrum and the Stokes parameters are given by

f0(y) =

1

1 − y+ 1 − y − 4r(1 − r),

f2(y) = (2r − 1)

f3(y) = 2r2,

?

1

1 − y+ 1 − y

?

,

(26)

with r = y/x(1 − y) and

x =4Eeω0

m2

e

≈ 15.4

?

Ee

TeV

??ω0

eV

?

(27)

for the initial laser energy ω0. We note from Eq. (25) that the linear polarization of the high

energy photon beam is proportional to Ptwhereas the circular polarization is proportional to Pc.

aIn the present work the electron and positron beams are assumed to be unpolarized. It is however straight-

forward to take into account polarized electron and positron beams.

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Thus it is necessary to have both circularly and linearly polarized initial laser beams so as to

measure all the polarization asymmetries Bi’s and Ci’s through the distribution (14).

After folding the luminosity spectra of two photon beams, the event rate of the process

γγ → tt is given by

d3N

dτ dΦdcosΘdτ

dcosΘ

?

−sin2Θ

?

+?23?τ

=

dLγγ

dˆ σ0

?

sin2(κ − ˜ κ)B2+ cos2(κ − ˜ κ)B3

1 + ?22?τPc˜Pc− ?02?τ

?

Pc+˜Pc

?

?

B1

+?33?τPt˜Pt

??

{Ptsin2κ −˜Ptsin2˜ κ}C1+ {Ptcos2κ +˜Ptcos2˜ κ}C2

?

?22?τPc˜Pc− ?33?τPt˜Ptcos2κcos2˜ κ

?

C0

−?03?τ

?

{Pt˜Pcsin2κ −˜PtPcsin2˜ κ}C3+{Pt˜Pccos2κ +˜PtPccos2˜ κ}C4

???

, (28)

where Φ is an azimuthal angle to be identifiable with κ, the function dLγγ/dτ is the two–photon

luminosity function depending on the details such as the e-γ conversion factor and the shape of

the electron/positron bunches [12], and τ ≡ s/see. The differential cross–section is then given by

dˆ σ0

dcosΘ=

β

32πs|M|2

0

′. (29)

The correlation ratios ?ij?τ(i,j = 0 to 3) are defined as

?ij?τ≡?fi∗ fj?τ

?f0∗ f0?τ

, (30)

where the correlation function ?fi∗ fj?τis given by the integrated function

?fi∗ fj?τ=

?ymax

τ/ymax

dy

yfi(y)fj(τ/y), (31)

with ymax= x/(1 + x). The difference κ − ˜ κ of two azimuthal angles κ and ˜ κ is independent

of the azimuthal angle Φ while each of them is linearly dependent on the angle Φ. This implies

that the measurements of the observables Cirequire the reconstruction of the scattering plane,

which can be done statistically.

Figure 1 shows the unpolarized correlation function ?f0∗ f0?τ and the five correlation ratios

{?02?τ,?03?τ,?22?τ,?23?τ,?33?τ} appearing in Eq. (28) for x = 0.5 (solid line), 1.0 (dashed line),

and 4.83 (dotted line). For a larger value of x, the correlation function ?f0∗ f0?τ, to which

dLγγ/dτ is proportional in the ideal situation of the beam conversion, becomes more flat and

the maximally–obtainable photon energy fraction becomes closer to the electron beam energy.

Exploiting this feature appropriately could facilitate Higgs–boson searches at the photon collider.

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τ

<f0*f0>τ

√τ

<22>τ

√τ

<02>τ

√τ

<33>τ

√τ

<03>τ

√τ

<23>τ

10

-2

10

-1

1

10

10

-1

-1

-0.5

0

0.5

1

0.20.4 0.60.8

-1

-0.5

0

0.5

1

0.2 0.40.6 0.8

0

0.2

0.4

0.6

0.8

1

0.2 0.40.60.8

0

0.2

0.4

0.6

0.8

1

0.20.40.6 0.8

-1

-0.5

0

0.5

1

0.20.40.60.8

Figure 1: The unpolarized correlation function ?f0∗f0?τand the five ratios ?ij?τof the correlation

functions for three x values; x = 0.5 (solid line), 1.0 (dashed line), and 4.83 (dotted line).

12

Page 13

The five figures for the correlation ratios clearly show that the maximal sensitivity to each

polarization asymmetry of Bi and Ci can be acquired near the maximal value of τ = y2

Therefore, once the Higgs–boson mass is known, one can obtain the maximal sensitivities by

tuning the initial electron energy to be

max.

Ee=

?1 + x

2x

?

mφ. (32)

On the other hand, the ratios ?33?τ, ?03?τand ?23?τare larger for a smaller value of x and for a

given x the maximum value of the ratio ?33?τis given by

?33?τmax=

?

2(1 + x)

1 + (1 + x)2

?2

. (33)

Consequently, it is necessary to take a small x and a high Eeby changing the laser beam energy

ω0so as to acquire the highest sensitivity to CP violation in the neutral Higgs sector.

6 An example in the MSSM

The MSSM Higgs sector constitutes a typical two–Higgs-doublet model in which CP violation

can be induced at the loop level from the stop and sbottom sectors through the complex trilinear

parameters At,band the higgsino mass parameter µ [3, 4]. Although there exist three neutral

Higgs bosons, we consider only the heaviest Higgs boson so as to estimate the unpolarized parton–

level cross section and the polarization asymmetries, which will allow us to completely determine

the CP property of the Higgs boson following the procedure described in Sect. 4.

Table 1:

MSSM Higgs boson for the parameter set (34) and tanβ = 3,10.

The mass and width {mφ,Γφ} and the four form factors {Sγ,Pγ,St,Pt} of the heaviest

tanβ

3

10

mφ[GeV]

500

500

Γφ[GeV]

1.9

1.1

Sγ

Pγ

St

0.33

0.11

Pt

0.15

0.02

−1.3 − 1.2i

−0.39 − 0.35i

−0.51 + 1.1i

−0.06 + 0.14i

In the MSSM the form factors Sγand Pγ, which describe the coupling of the Higgs boson to

two photons, have the loop contributions from the bottom and top quarks, the charged Higgs

boson, the W boson, and the lighter top and bottom squarks as well as other charged particles

such as charginos and heavier top and bottom squarks. The contributions from the charginos and

heavier top and bottom squarks are neglected in the present work by taking them to be heavy.

For our numerical example based on the work [4], we assume a universal trilinear parameter

A = At= Aband take the physically–invariant phase ΦAµ= ΦA+ Φµto be π/2, leading to the

13

Page 14

(almost) maximal CP violation. Then, we take for the remaining dimensionful parameters the

parameter set:

|A| = 1.0TeV, |µ| = 2.0TeV, M2

˜ QL= M2

˜tR= M2

˜bR= (0.5TeV)2, MH± = 0.5TeV (34)

where M2

MH± is the charged Higgs boson mass. In addition, we take two values of tanβ, tanβ = 3 and

10, so as to obtain a crude estimate of the dependence of the form factors on tanβ. For the

above MSSM parameters, the mass, the width, and the four form factors, {mφ,Γφ,Sγ,Pγ,St,Pt},

of the heaviest MSSM Higgs boson on the mass pole√s = mφare presented in Table 1. Several

comments on our results are in order:

˜ QL, M2

˜tR, and M2

˜bRare the soft SUSY breaking top/bottom squark masses squared and

• The Higgs–boson width is reduced for large tanβ. This is due to the suppression of the

dominant partial decay width Γ(φ → t¯t).

• The absolute values of all the form factors are very small for large tanβ, leading to a

strong suppression of the Higgs–boson contribution to the process γγ → t¯t. In particular,

the ‘pseudoscalar’ couplings, Ptand Pγ, are very small, which implies an (almost) CP–even

heavy Higgs boson.

One natural consequence from the tanβ dependence of the form factors is that all the CP–odd

polarization asymmetries are strongly suppressed for large tanβ.

We integrate the polarized distributions over the angular variables so as to obtain the unpo-

larized parton–level cross section ˆ σ0and the averaged polarization asymmetries:

Ai≡ ?Ai?,Bi≡ ?Bi?′,Ci≡ ?Cisin2Θ?′,Di≡ ?Di?′,Ei≡ ?Eisin2Θ?′,

′). We do not present

(35)

where ?X? (?X?′) denotes the average over the distribution |M|2

the polarized distributions folded with the photon luminosity spectrum explicitly because those

distributions can be obtained in a rather straightforward way from the parton–level cross sec-

tions. We do not take into account the QCD radiative corrections either, but for the details we

refer to Ref. [13].

0( |M|2

0

Table 2 shows the parton–level unpolarized cross section ˆ σ0 and polarization asymmetries

A ’s, which are constructed with equal photon helicities and equal t¯t helicities, in the MSSM

parameter set (34) for the CP phase ΦAµ= π/2. Each sign ± in the square brackets is for the

CP–parity of the observable. Note that the CP-odd observables A1, A2and A3are significantly

suppressed for tanβ = 10 as expected.

In Tables 3 and 4, we show the polarization asymmetries {Bi, Ci} constructed with general

two–photon spin correlations, and the polarization asymmetries D’s and E’s with general two–

photon spin correlations and equal t and¯t helicities, in the MSSM parameter set (34). The

14

Page 15

Table 2:

structed with equal photon and top–pair helicities, in the MSSM parameter set (34) for the CP

phase ΦAµ= π/2. Each sign ± in the square brackets is for the CP–parity of the observable.

The parton–level cross section and polarization asymmetries A ’s, which are con-

tanβ

3

10

ˆ σ0[+]

0.88 pb

0.62 pb

A0[+]

0.45

0.91

A1[−]

0.13

0.00

A2[−]

−0.17

−0.02

A3[−]

0.26

0.03

Table 3:

general two–photon spin correlations, in the MSSM parameter set (34) for the CP phase Φ = π/2.

Each sign ± in the square brackets is for the CP–parity of the observable.

The values of the polarization asymmetries {Biand Ci}, which are constructed with

tanβ

3

10

B1[−]

0.46

0.03

B2[−]

−0.27

0.00

B3[+]

−0.60

−0.47

C0[+]

0.17

0.24

C1[−]

0.13

0.01

C2[+]

0.09

0.30

C3[+]

0.17

0.04

C4[−]

0.06

0.00

re-phasing–invariant phase ΦAµis taken to be π/2 and each signature ± in the square brackets

is for the CP–parity of the observable as in Table 2. The CP–odd polarization asymmetries are

sizable for tanβ = 3, but they are strongly suppressed for tanβ = 10. In addition, the CP–even

observables which are dependent only on the products SγPγand StPtof the form factors are also

suppressed for large tanβ. In particular, we note that the polarization asymmetry E1is deter-

mined by only the CP-odd (‘pseudoscalar’) form factors Pγand Ptso that its strong suppression

implies an almost CP–even Higgs boson.

Table 4:

photon spin correlations and equal top–pair helicities, in the MSSM parameter set (34). The re-

phasing–invariant phase ΦAµis taken to be π/2 and only the interference between the continuum

and the heaviest Higgs boson in the MSSM is taken into account. The signature ± in the square

brackets denotes the CP–parity of the corresponding observable.

The polarization asymmetries D’s and E’s, which are constructed with general two–

tanβ

3

10

D1[−]

0.32

0.03

D2[+]

0.41

0.80

D3[+]

−0.45

−0.09

D4[−]

0.03

0.00

E1[+]

−0.04

−0.00

E2[−]

0.10

0.01

E3[−]

0.09

0.01

E4[+]

0.40

0.46

15

Page 16

7 Conclusions

In this article, we have studied the effects of a neutral Higgs boson without definite CP–parity

in the process γγ → t¯t in a model–independent way.

We have found that the interference between the Higgs–boson–exchange and the continuum

amplitudes with polarized photon beams and with the top and anti–top helicity measurements

enables us to determine the CP property of a neutral Higgs boson completely even in CP non–

invariant theories. We have classified the physical observables such as the unpolarized cross

sections and polarization asymmetries depending on the polarization configuration of the initial

two–photon beams and the helicity configuration of the final t¯t system. As an demonstration

of the procedure, the contribution of the heaviest neutral Higgs boson in the MSSM has been

investigated quantitatively.

Certainly, the precision with which the Higgs–boson contribution is determined depends on

the background processes as well as the efficiencies for controlling beam polarization and mea-

suring the t and¯t helicities. Nevertheless, the algorithm presented in the present work with more

than 20 polarization observables will be helpful in determining the s–channel Higgs contribution

to the process γγ → t¯t efficiently even in the CP non–invariant theories.

Acknowledgements

S.Y.C. wishes to acknowledge financial support of the 1997 Sughak program of the Korea Research

Foundation.

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