Final charged-lepton angular distribution and possible anomalous top-quark couplings in p p ->t tbar X ->l^+ X'

Possible anomalous (or nonstandard) top-quark interactions with the gluon and those with the W boson induced by SU(3) x SU(2) x U(1) gauge-invariant dimension-6 effective operators are studied in p p ->t tbar X ->l^+ X' (l=e or mu) at the Large Hadron Collider (LHC). The final charged-lepton (l^+) angular distribution is first computed for nonvanishing nonstandard top-gluon and top-W couplings with a cut on its transverse momentum. The optimal-observable procedure is then applied to this distribution in order to estimate the expected statistical uncertainties in measurements of those couplings that contribute to this process in the leading order.


Introduction
The Large Hadron Collider (LHC) at CERN has been presenting us fruitful experimental data on various particles/processes ever since it started operating, of course including the historic discovery of the/a Higgs boson [1]. Exploring possible new physics beyond the standard model (BSM) is also an important mission of the LHC. Although they have not found so far any exciting signals indicating BSM yet, this fact never means that there do not exist exotic particles since their masses might be too high to be directly produced there.
Even in such a case, we still would be able to investigate certain new-physics effects indirectly, using data from the LHC. For example, we have studied possible nonstandard chromomagnetic and chromoelectric dipole moments of the top-quark (denoted as d V and d A respectively) in Refs. [2]- [4], and obtained much stronger restrictions on them than before ♯1 by adding the data on the tt total cross sections from the LHC to those from the Tevatron. We then carried out an optimalobservable analysis (OOA) to show how precisely we could determine those nonstandard couplings in pp → ttX → ℓ + X ′ (ℓ = e or µ) under a linear approximation by using the ℓ + angular and energy distributions, where we also took into account possible nonstandard top-W coupling (denoted as d R ) [5]. There, however, we were not able to study the d R contribution through the angular distribution due to the decoupling theorem [6]- [8].
The d R dependence of this distribution recovers if we perform the energy integration necessary to derive it in some limited range, as will be discussed later. The purpose of this article is to study if we could thereby draw any new information on d R via a similar OOA: After summarizing our calculational framework, we are going to clarify to what extent the distribution becomes dependent of this parameter by computing it for some different d R values with a ℓ + transverse-momentum (p ℓ T ) cut. Then we apply the optimal-observable procedure to this distribution with and without the d V -term contribution. Concerning the ℓ + energy distribution, on the other hand, we do not re-study it here because that distribution is d R -dependent from the beginning and therefore adding the p ℓ T cut does not bring us anything ♯1 As for the preceding analyses, see the reference lists of [2]- [4]. essentially-new in comparison with what we have done in [5].

Framework
The framework of our model-independent analyses is based on an effective-Lagrangian whose low-energy form reproduces the standard-model (SM) interactions. This is one of the most promising methods to describe new-physics phenomena when the energy of our experimental facility is not high enough to produce new particles. Assuming any non-SM particles too heavy to appear as real ones, we take the following effective Lagrangian: where operators of mass-dimension 6 involving only the SM fields and their coefficients C i parameterize virtual effects of new particles at an energy less than the assumed new-physics scale Λ. Note here that the dimension-6 operators give the largest contributions in relevant processes as long as we assume the lepton-number conservation. In this framework, all the form factors related to C i are dealt with as constant parameters, without supposing any specific new-physics models.
All those dimension-6 operators have been arranged in Refs. [9]- [12]. Following the notation of [11], the effective Lagrangian for the parton-level process qq/gg → tt → bbW + W − is given in [3] as where g s and g are the SU(3) and SU(2) coupling constants, P L/R ≡ (1 ∓ γ 5 )/2, with v being the Higgs vacuum expectation value and V tb being the (tb) element of Kobayashi-Maskawa matrix. Among those unknown parameters, d V and d A are respectively the top-quark chromomagnetic and chromoelectric dipole moments, and we use d R defined as instead of f R 2 in order to make our formulas a little bit simpler. In the following work, we use the above effective Lagrangian for top-quark interactions, and adopt the linear approximation for those nonstandard parameters as in [5], where d V and d R come into our analyses (note that d A terms do not contribute to qq/gg → tt in the leading order because of their CP -odd property).
We assume the other interactions, e.g. the one for W + → ℓ + ν, are described by the usual SM Lagrangian, and all the fermions lighter than the top quark are treated as massless particles. Concerning the parton distribution functions, we have been using CTEQ6.6M (NNLO approximation) [13].

Lepton angular distribution and decoupling theorem
What we call "the decoupling theorem" is a theorem which states that the leading contribution of the anomalous top-decay couplings, d R in our case, to final-particle angular distributions vanishes when only a few conditions are satisfied [6]- [8]. In terms of the ℓ + angular distribution under consideration, this theorem holds if we assume the standard V − A structure for the νℓW coupling and perform the lepton-energy integration fully over the kinematically-allowed range. As a result, this distribution becomes exclusively dependent of d V . That is, we can no longer get any information thereby on the nonstandard top-decay coupling d R .
Although it is not possible to cover the full phase space of the final-lepton momentum in actual experiments, we could carry out the above energy integration using the energy distribution reconstructed through a proper extrapolation. Therefore the above-mentioned full integration is not unrealistic. This however tells us that we might be able to draw certain new information on d R by using the angular distribution with some cut on the lepton momentum.
Let us calculate the ℓ + angular distribution with a ℓ + transverse-momentum (p ℓ T ) cut as a typical and realistic experimental condition. We first take one of the proton beams as the base axis and express the differential cross section of pp → ttX → ℓ + X ′ (the angular and energy distribution of ℓ + ) in the proton-proton CM frame as follows: where E ℓ is the lepton energy, θ ℓ is the lepton scattering angle, i.e., the angle formed by the ℓ + momentum and the above-mentioned base axis, f SM (E ℓ , cos θ ℓ ) denotes the SM contribution, and the other two f I (E ℓ , cos θ ℓ ) describe the non-SM terms corresponding to their coefficients. The explicit forms of f I (E ℓ , cos θ ℓ ) at the parton level are easily found in the relevant formulas in [3]. Then, the ℓ + angular distribution is written as where g i (cos θ ℓ ) are given by with i = 1, 2 and 3 corresponding to I = SM, d V and d R , respectively. In the above E ℓ integration, the kinematically-allowed range is with β ≡ 1 − 4m 2 t /s. As mentioned, g 3 (cos θ ℓ ) disappears if we perform the integration fully over this range due to the decoupling theorem.
We compute this angular distribution for √ s = 14 TeV ♯2 and p ℓ T ≥ p min ℓ T , the latter of which leads to the lower bound of E ℓ as ♯2 We performed analyses for √ s = 7, 8, 10 and 14 TeV in [5], but we here focus on 14 TeV since the LHC is now being upgraded toward this energy. and Eqs. (10,11) require Practically, however, this restriction on cos θ ℓ affects its range only a little, e.g., the right-hand side of this inequality is 0.999898 even for p min ℓ T = 100 GeV. We show the d R dependence of the angular distribution within the range |d R | ≤ 0.1 [14,15] in Figs.1-3 4. Optimal-observable analysis with p ℓ T cut The optimal-observable analysis (OOA) is a way that could systematically estimate the expected statistical uncertainties of measurable parameters. Here we apply this procedure to the ℓ + angular distribution studied in the preceding section.
Leaving its detailed and specific description to [16]- [19], let us show how to compute the uncertainties thereby: What we have to do first is to calculate the following 3 × 3 matrix using g 1,2,3 defined in Eqs. (8,9), and next its inverse matrix X c ij , both of which are apparently symmetric. ♯3 This integration is to be performed over the range given by Eq. (12). Then the statistical uncertainties for the measurements of couplings d V and d R could be estimated by where σ ℓ , N ℓ and L denote the total cross section, the number of events and the integrated luminosity for the process pp → ttX → ℓ + X ′ , respectively.
We are now ready to carry out necessary numerical computations. Below we show the elements of M c computed for √ s =14 TeV: (1) p min ℓ T = 20 GeV Here all these results were derived from the cross section in [pb] unit. Using the inverse matrices calculated from these elements, we can estimate the statistical uncertainties of the relevant couplings δd V and δd R according to Eqs. (15,16) (Twoparameter analysis).
The set of M c ij (17)- (19) also enables us to give another numerical results. That is, we can do a similar analysis but assuming only d R is unknown. This