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New-Physics Search through gamma gamma ->t tbar ->lX/bX

We probe optimal beam polarizations for new-physics search in top-quark and Higgs-boson sectors at Photon Linear Colliders (PLC). Expressing possible non-standard effects generated by SU(2)xU(1) gauge-invariant dimension-6 effective operators as anomalous top and Higgs couplings, we estimate expected statistical sensitivities of these couplings in gamma gamma ->t tbar ->lX/bX, using the optimal-observable method.


Recent neutrino experiments strongly indicate that the Standard Model (SM) has
to be modified to explain ν-oscillation phenomena [1]. However this does not necessarily require any new particles and we have no information yet on SM behavior in much higher-energy region where even new heavy particles could produce some effects directly or indirectly. Since top-quark/Higgs-boson interactions might receive corrections via such new-physics effects, precise studies of their reactions are expected to be a window on possible beyond-the-SM physics.
Current measurements of top quark at Fermilab Tevatron, which is the unique facility for top-quark production at present, are not sufficient yet to study if topquark can be fully described in the framework of SM [2], but in the near future Large Hadron Collider (LHC) [3] and International Linear Collider (ILC) [4] will work as top-quark factories and realize its precise studies. Considering that the top-Higgs coupling is proportional to m t , these colliders will also give us an opportunity to study the Higgs sector.
Focusing on ILC, we have been carrying out those studies intensively. Here we would like to show some of our results about drawing possible anomalous top and Higgs couplings from the processes γγ → tt → ℓ/bX at Photon Linear Collider (PLC), an interesting option of ILC, and discuss optimal beam polarizations that could lead us to the smallest statistical uncertainties. This report is organized as follows; We briefly review the framework of the analysis in section 2. Our numerical results are presented in section 3, and then a summary and some discussions are contained in section.4.

FRAMEWORK
In this section, we briefly explain the effective-Lagrangian approach [5] and optimalobservable method [6] used in our analysis. In addition, polarization parameters of PLC are reviewed.

Effective Lagrangian
We assumed that all non-standard particles are heavier than the lowest new-physics scale Λ and decouple in lower-energy processes. In the effective-Lagrangian ap-proach with this assumption, effects from such heavy particles can be parameterized as coupling constants of effective operators which are composed of standard fields alone.
Let us describe this scenario more concretely. The SM Lagrangian L SM is modified by the addition of a series of SU(3) × SU(2) × U(1) gauge-invariant operators with coefficients suppressed by inverse powers of Λ. Among those operators the largest contribution comes from dimension-6 operators, denoted as O i , since dimension-5 operators violate lepton number [5] and are irrelevant for the processes considered here. Consequently we have as the basis of our analysis.
The operators relevant here lead to the following non-standard top-quark-and Higgs-boson-couplings: (i) CP -conserving and CP -violating ttγ vertices, (ii) CPconserving and CP -violating γγH vertices, and (iii) the anomalous tbW vertex.
The corresponding coupling constants are denoted respectively as α γ1 , α γ2 , α h1 , α h2 and α d . The explicit expressions for these couplings in terms of the coefficients of dimension-6 operators are to be found in Ref. [7,8].

Optimal-observable method
The optimal-observable method [6] is a useful tool for estimating expected statistical uncertainties in various coupling measurements, which we summarize below.
Suppose we have a cross section where f i (φ) are known functions of the final-state phase-space variables φ and c i 's are model-dependent coefficients. The goal is to determine the c i 's. This can be done by using appropriate weighting functions In general, different choices for w i (φ) are possible, but there is a unique choice for which the resultant statistical error is minimized. Such functions are given by where X ij is the inverse matrix of M ij which is defined as When we use these weighting functions, the statistical uncertainty of c i becomes where σ T ≡ (dσ/dφ)dφ and N is the total number of events.
In order to adapt this method to our processes, we decompose cross sections into a sum of several functions which are proportional to each non-standard couplings ♯1 .
In this way, the angular and energy distributions of the secondary fermions ℓ/b in the eē CM frame are expressed as where f SM and f i are calculable functions: f SM denotes the SM contribution, f γ1,γ2 describe the anomalous CP -conserving and CP -violating ttγ-vertices contributions respectively, f h1,h2 those generated by the anomalous CP -conserving and CP -violating γγH-vertices, and f d that by the anomalous tbW -vertex (see, e.g., Ref. [7] for detailed calculations).

Polarization parameters of PLC
At PLC, a colliding photon beam originates as a laser beam back-scattered off an electron (e) or positron (ē) beam. The polarizations of the initial state are characterized by the electron and positron longitudinal polarizations P e and Pē, the maximum average linear polarizations P t and Pt of the laser photons with the azimuthal angles ϕ andφ (defined in the same way as in [9,10]), and their average helicities P γ and Pγ.
The photon polarizations P t,γ and Pt ,γ satisfy For the linear polarization, we fixed the relative azimuthal angle χ ≡ ϕ −φ to be π/4, because we found it the optimal value to study α γ2 and α h2 through checking χ dependence of σ(γγ → tt). ♯1 We kept only linear terms in non-standard couplings.
We did not probe the Higgs-resonance region, which had been extensively studied previously (see, for example, [11]). This means that the non-standard corrections are expected to be moderate and we may compute the number of secondary fermions, N ℓ/b , from the SM total cross section multiplied by the lepton/b-quark detection efficiency ǫ ℓ/b and the integrated eē luminosity L eē ; this leads to N ℓ/b independent of m H .
Before showing the corresponding results, we have to explain one serious problem we encountered in Ref. [7]. We have noticed that the numerical results for X ij are often unstable when inverting the matrix M ij : even a tiny variation of M ij changes X ij significantly. This indicates that some of f i have similar shapes and therefore their coefficients cannot be disentangled easily. The presence of such instability has forced us to refrain from determining all the couplings at once through this process alone. That is, we have assumed that some of α i 's had been measured in other processes (e.g., in eē → tt → ℓ ± X), and we performed an analysis with smaller number of independent parameters.
When estimating the statistical uncertainty in simultaneous measurements, e.g., of α γ1 and α h1 (assuming all other coefficients are known), we need only the components with indices 1, 2 and 4. In such a "reduced analysis", we still encountered the instability problem, and we selected "stable solutions" according to the following criterion: Let us express the resultant uncertainties as ∆α [3] γ1 and ∆α [3] h1 , where "3" shows that we use the input M ij , keeping three decimal places. In addition, we also compute ∆α [2] γ1 and ∆α [2] h1 by rounding M ij off to two decimal places. Then, we accept the result as a stable solution if both of the deviations |∆α [3] γ1,h1 − ∆α [2] γ1,h1 |/∆α [3] γ1,h1 are less than 10 %. This is what we faced in [7], and we found that we are not free from this problem even for the wider range of polarization parameters given at the beginning of this section. That is, we did not find again any stable solution in the four-and fiveparameter analysis. Fortunately, however, we did find some solutions not only in the two-but also in the three-parameter analysis [8]. In this report, we show the results of two parameter analysis as examples below. We did not fix the detection efficiencies ǫ ℓ/b since they depend on detector parameters and will get better with development of detection technology. On the other hand, the N ℓ/b were estimated for L eē = 500 fb −1 .

Final charged-lepton detection
This result is free from m H dependence since the Higgs-exchange diagrams do not contribute to α γ1 and α d determination within our approximation.
Note that it is difficult to determine α γ1 and α γ2 together in this analysis.
Although we have found some stable solutions that would allow for a determination of α γ1 in the lepton analysis, which we did not find in [7], the expected precision is rather low. Nevertheless this is telling us that the use of purely linear polarization for the laser is crucial for measuring α γ1 . Unfortunately, the statistical uncertainty of α γ2 is still large even in this analysis, so we did not list it in the solutions which gave us good statistical precisions. Therefore, we have to look for other suitable processes to determine this parameter, for a review see [12].
On the other hand, it was found that there are many combinations of polarization parameters that make uncertainties of α h1,h2 and α d relatively small. For instance, analyzing the b-quark final state with the polarization P e = Pē = −1, P t = Pt = 1/ √ 2, P γ = −Pγ = −1/ √ 2 enables us to probe the properties of Higgs bosons whose mass is around 300 GeV through the determination of α h1 and α h2 .
As mentioned, the results were obtained for Λ = 1 TeV. If one assumes the new-physics scale to be Λ = λ TeV, then all the above results (∆α i ) are replaced with ∆α i /λ 2 , which means that the right-hand sides of eqs.

SUMMARY
Studying γγ → tt → ℓX/bX in detail, we derived optimal beam polarizations that minimize uncertainties in the determination of ttγ, tbW and γγH coupling parameters. To perform this analysis, we have applied the optimal-observable method to the final lepton/b-quark angular and energy distributions.
We showed a number of two-parameter solutions, most of which allow for the γγH-and tbW -couplings determination. The expected precision of the measurement of the Higgs coupling is of the order of 10 −2 (for the scale of new physics Λ = 1 TeV). This shows that the γγ collider is going to be useful for testing the Higgs sector of the SM.
Let us consider the top-quark-coupling determination in an ideal case such that the beam polarizations could be easily tuned and that the energy is sufficient for the on-shell Higgs boson production. Then the best strategy would be to adjust polarizations to construct semi-monochromatic γγ beams such that √ s γγ ≃ m H and on-shell Higgs bosons are produced. This would allow for precise α h1,h2 measurement, so the virtual Higgs effects in γγ → tt would be calculable. Unfortunately, as we have shown earlier, it is difficult to measure α γ2 by looking just at ℓX/bX final states from γγ → tt. Therefore to fix α γ2 , one should, e.g., measure the asymmetries adopted in [13] to determine the top-quark electric dipole moment which is proportional to α γ2 . Then, following the analysis presented here, one can determine α γ1 and α d .
Finally, one must not forget that it is necessary to take into account carefully the Standard Model contribution with radiative corrections when trying to determine the anomalous couplings in a fully realistic analysis. In particular this is significant when we are interested in CP -conserving couplings since the SM contributions there are not suppressed unlike the CP -violating terms. On this subject, see for instance [14].