Optimal-Observable Analysis of Top Production/Decay at Photon-Photon Colliders Univ. of Tokushima Zenrō HIOKI 1. Introduction 2. Basic Framework 3. Optimal-Observable Analyses 4. Summary 1. Introduction Discovery of Top-quark We have now all the SM-fermions, but Is the 3rd generation a copy of the 1st & 2nd.? Isn t there any New-Physics in top-quark couplings? Based on collaboration with B. Grza ς dkowski (Warsaw U), K.Ohkuma (Fukui U. Thechnology) and J. Wudka (UC Riverside). 1
So far, we have studied e + e t t to explore possible non-sm top-quark couplings. Here we perform a similar analysis in γγ t t 2. Basic Framework What we aim to do is A Model-Independent Analysis. For this purpose, what terms must be taken into account? In the case of eē t t: We are able to write down the most general covariant t tγ/z amplitude. Γ v µ = g [ 2ū(p t) γ µ (A v B v γ 5 )+ (p t p t) µ ] (C v D v γ 5 ) v(p t ). (1) 2m t (v = γ or Z) However in γγ t t, t or t in t tγ coupling is virtual. Those which were dropped in eē t t thanks to the on-shell condition can contribute 2
Take ψψ as an example. If ψ and ψ are both on-shell, ψf( )ψ = ψf(m 2 )ψ So they are all equivalent to ψψ. However if ψ is virtual, we have infinite numbers of ūf (q 2 )S F (q) in an amplitude since F can be arbitrary. We decided to perform an analysis in the framework of Effective Operator Approach à la Buchmüller & Wyler Basic assumption New Physics with Energy scale Λ Below Λ, we have only SM particles The leading non-sm interactions are dim.-6 operators: O ub = iūγ µ D ν ub µν O qb = i qγ µ D ν qb µν O qw = i qτ i γ µ D ν qw iµν O ub =( qσ µν u) ϕb µν O uw =( qσ µν τ i u) ϕw iµν O ϕ W =(ϕ ϕ) W µνw i iµν O ϕ B =(ϕ ϕ) B µν B µν O WB =(ϕ τ i ϕ) W µνb i µν O ϕw =(ϕ ϕ)wµνw i iµν O ϕb =(ϕ ϕ)b µν B µν O WB =(ϕ τ i ϕ)wµνb i µν L = L SM + [ 1 α Λ 2 i O i +(h.c.) ] i (2) 3
One new discovery: are not independent of the others O ub, O qb, O qw O ub = ig u O ub +[ ig u O ub ] + O qb = ig u O ub +[ig u O ub ] + O qw = ig u O ub +[ig u O ub ] + via some equations of motion Independent operators lead to the following Feynman rules. (1) CP -conserving t tγ vertex 2 Λ vα 2 γ1 k/γ µ, (3) (2) CP -violating t tγ vertex (3) CP -conserving γγh vertex i 2 Λ 2 vα γ2 k/γ µ γ 5, (4) 4 Λ 2 vα h1 [(k 1 k 2 )g µν k 1ν k 2µ ], (5) (4) CP -violating γγh vertex 8 Λ 2 vα h2 k ρ 1k σ 2 ɛ ρσµν, (6) 4
Here k & k 1,2 are incoming photon momenta, α γ1,γ2,h1,h2 are defined as α γ1 sin θ W Re(α uw ) + cos θ W Re(α ub), (7) α γ2 sin θ W Im(α uw ) + cos θ W Im(α ub), (8) α h1 sin 2 θ W Re(α ϕw ) + cos 2 θ W Re(α ϕb ) 2 sin θ W cos θ W Re(α WB ), (9) α h1 sin 2 θ W Re(α ϕ W ) + cos 2 θ W Re(α ϕ B) sin θ W cos θ W Re(α WB ). (10) On the other hand, the general amplitude for t bw can be written as Γ µ Wtb = g [ ū(p b ) 2 ] γ µ P L iσµν k ν f2 R P R u(p t ), (11) M W where P L,R (1 ± γ 5 )/2 and f R 2 is f R 2 = 1 Λ 2 [ 4M W v g α uw M W v ] α Du. (12) 2 for m b = 0 and on-shell W approximation. Using them, we calculated the cross section of γγ l ± X via FORM. HOWEVER, the result is too long to show here. Sorry! 5
3. Optimal-Observable Analysis How can we determine several unknown parameters simultaneously? = Optimal-observable method Brief summary of this method: Suppose we have a distribution dσ dφ ( Σ(φ)) = i c i f i (φ) where f i (φ) are calculable functions, and c i are the parameters we try to determine. Determining c i = We make weighting functions w i (φ) which satisfies w i (φ)σ(φ)dφ = c i The one which minimizes the statistical uncertainty of c i is w i (φ) = j X ij f j (φ)/σ(φ), where X is the inverse matrix of M ij fi (φ)f j (φ) dφ Σ(φ) This X gives c i = X ii σ T /N, 6
where σ T (dσ/dφ)dφ N = L eff σ T is the total number of the events, L eff is the product of the integrated luminosity and detection efficiency. We applied this procedure to the angular & energy distribution of γγ t t l + X : dσ de l d cos θ l = f SM (E l, cos θ l )+α γ1 f γ1 (E l, cos θ l )+α γ2 f γ2 (E l, cos θ l ) + α h1 f h1 (E l, cos θ l )+α h2 f h2 (E l, cos θ l )+α d f d (E l, cos θ l ) (13) in eē-cm frame, where f SM is the SM contribution, f γ1,γ2 are CP -conserving- & CP -violating-t tγ-vertices contribution, f h1,h2 are CP -conserving- & CP -violating-γγh-vertices contribution, and f d is from the anomalous tbw-vertex α d =Re(f R 2 ). 7
Parameters Higgs mass : m H = 100 300 500 GeV Polarizations of e &ē : P e = Pē =1 Polarization of the Laser: (1) Linear Polarization P e = Pē =1,P t = P t = P γ = P γ =1/ 2 and χ( ϕ 1 ϕ 2 )=π/4 (2) Circular Polarization P e = Pē = P γ = P γ =1. Problems It turned out that our results for X ij are very unstable: even a tiny fluctuation of M ij changes X ij significantly. Some of f i have similar shapes? The only option in such a case is to refrain from determining all the couplings at once through this process alone. We assumed some parameters can be determined in other processes: Result We found several sets of solution in two-parameter analysis 8
1) Linear polarization Independent of m H α γ2 =73/ N l, α d =1.9/ N l, (14) m H = 100 GeV α h2 = 107/ N l, α d =1.6/ N l, (15) m H = 300 GeV α h1 =3.4/ N l, α d =3.2/ N l, (16) Here N l 63 for L eff eē = 500 fb 1. 2) Circular polarization m H = 100 GeV α h1 =9.0/ N l, α d =3.0/ N l, (17) m H = 300 GeV m H = 500 GeV α h1 =3.5/ N l, α d =3.0/ N l, (18) α h2 =35/ N l, α d =3.1/ N l, (19) α h1 =7.7/ N l, α d =2.8/ N l, (20) 9
α h2 =10/ N l, α d =2.8/ N l, (21) Here N l 48 for L eff eē = 500 fb 1. ********** ********** ********** We also performed a similar analysis using γγ t t bx Isn t it harder to study b-quark distribution? b-quark tagging must be done to distinguish t t events from possible background (WW production). 1) Linear polarization Independent of m H α γ2 =29/ N b, α d =2.6/ N b, (22) m H = 100 GeV α h2 =38/ N b, α d =2.4/ N b, (23) 10
m H = 300 GeV m H = 500 GeV α γ2 =24/ N b, α h1 =2.4/ N b, (24) α h1 =5.4/ N b, α d =4.9/ N b, (25) α γ2 =23/ N b, α h1 =5.0/ N b, (26) α h1 =18/ N b, α h2 =22/ N b, (27) α h1 =8.0/ N b, α d =3.3/ N b, (28) where N b 140 for L eff eē = 500 fb 1. 2) Circular polarization m H = 100 GeV α h1 =14/ N b, α d =5.2/ N b, (29) m H = 500 GeV α h1 =10/ N b, α d =4.2/ N b, (30) where N b 100 for L eff eē = 500 fb 1. 1 The above results are for Λ = 1 TeV. When one takes the new-physics 1 We used the tree-level SM formula for computing N b, so that we have the same N b for different m H. 11
scale to be Λ = λλ, then all the above results ( α i ) are replaced with α i /λ 2, which means that the right-hand sides of eqs. (14) (30) are multiplied by λ 2. Comparing two results The following parameter sets are measurable in (1) Lepton analysis (α γ2,α d ), (α h1,α d ), (α h2,α d ) (2) b-quark analysis (α γ2,α h1 ), (α γ2,α d ), (α h1,α h2 ), (α h1,α d ), (α h2,α d ) 4. Summary In order to explore possible anomalous top-quark couplings, we studied t t production/decay in γγ collisions. We assumed a New-Physics with an energy-scale Λ, and we have only the SM particles below Λ. All leading non-sm interactions are given in terms of dimension-6 effective operators à la Buchmüller & Wyler. We found some new Equation-of-motion relations among several operators, which reduced the number of operators necessary in our 12
analysis. We found it impossible to determine all the parameters in this process alone, but also found some stable solutions in two-parameter analysis. If we encounter phenomena which cannot be described in our framework, it will be an indication of some New-Physics beyond B& W scenario. References (1) B.Grzadkowski, Z.Hioki, K.Ohkuma and J.Wudka, Nucl. Phys. B689 (2004) 108 (hep-ph/0310159). (2) B.Grzadkowski, Z.Hioki, K.Ohkuma and J.Wudka, Phys. Lett. B593 (2004) 189 (hep-ph/0403174). 13