Patrick Janot CERN, PH Department, Geneva, Switzerland

Abstract

An optimal-observable analysis of the lepton angular and energy distributions from top-quark pair production with semi-leptonic decays in $${\rm e^{+}e^{-}}$$ collisions is used to predict the potential sensitivity of the FCC-ee to the couplings of the top quark to the photon and the Z.

Introduction

\label{sec:Introduction}

The design study of the Future Circular Colliders (FCC) in a 100-km ring in the Geneva area has started at CERN at the beginning of 2014, as an option for post-LHC particle accelerators. The study has an emphasis on proton-proton and electron-positron high-energy frontier machines (FCC 2015). In the current plans, the first step of the FCC physics programme would exploit a high-luminosity $${\rm e^{+}e^{-}}$$ collider called FCC-ee, with centre-of-mass energies ranging from below the Z pole to the $${\rm t\bar{t}}$$ threshold and beyond. A first look at the physics case of the FCC-ee can be found in Ref. (Bicer 2014).

In this first look, the focus regarding top-quark physics was on precision measurements of the top-quark mass, width, and Yukawa coupling through a scan of the $${\rm t\bar{t}}$$ production threshold, with $$\sqrt{s}$$ comprised between 340 and 350 GeV. The expected precision on the top-quark mass was in turn used, together with the outstanding precisions on the Z peak observables and on the W mass, in a global electroweak fit to set constraints on weakly-coupled new physics up to a scale of 100 TeV. Although not studied in the first look, measurements of the top-quark electroweak couplings are of interest, as new physics might also show up via significant deviations of these couplings with respect to their standard-model predictions. Theories in which the top quark and the Higgs boson are composite lead to such deviations. The inclusion of a direct measurement of the ttZ coupling in the global electroweak fit is therefore likely to further constrain these theories.

It has been claimed that both a centre-of-mass energy well beyond the top-quark pair production threshold and a large longitudinal polarization of the incoming electron and positron beams are crucially needed to independently access the tt$$\gamma$$ and the ttZ couplings for both chirality states of the top quark. In Ref. (Baer 2013), it is shown that the measurements of the total event rate and the forward-backward asymmetry of the top quark, with 500 $${\rm fb}^{-1}$$ at $$\sqrt{s}=500$$ GeV and with beam polarizations of $${\cal P}=\pm 0.8$$, $${\cal P}^{\prime}=\mp 0.3$$, allow for this distinction.

The aforementioned claim is revisited in the present study. The sensitivity to the top-quark electroweak couplings is estimated here with an optimal-observable analysis of the lepton angular and energy distributions of over a million events from $${\rm t\bar{t}}$$ production at the FCC-ee, in the $$\ell\nu{\rm q\bar{q}b\bar{b}}$$ final states (with $$\ell={\rm e}$$ or $$\mu$$), without incoming beam polarization and with a centre-of-mass energy not significantly above the $${\rm t\bar{t}}$$ production threshold.

Such a sensitivity can be understood from the fact that the top-quark polarization arising from its coupling to the Z is maximally transferred to the final state particles via the weak top-quark decay $${\rm t\to Wb}$$ with a 100% branching fraction: the lack of initial polarization is compensated by the presence of substantial final state polarization, and by a larger integrated luminosity. A similar situation was encountered at LEP, where the measurement of the total rate of $${\rm Z}\to\tau^{+}\tau^{-}$$ events and of the tau polarization was sufficient to determine the tau couplings to the Z, regardless of initial state polarization (Jadach 1989, Schael 2006).

This letter is organized as follows. First, the reader is briefly reminded of the theoretical framework. Next, the statistical analysis of the optimal observables is described, and realistic estimates for the top-quark electroweak coupling sensitivities are obtained as a function of the centre-of-mass energy at the FCC-ee. Finally, the results are discussed, and prospects for further improvements are given.

Theoretical framework

\label{sec:theory}

The top-quark couplings to the photon and the Z can be parameterized in several ways. In Ref. (Baer 2013), for example, the analysis makes use of the usual form factors denoted $$F_{1}$$, $$F_{2}$$, defined in the following expression (with $$X=\gamma,Z$$):

$$\Gamma_{\mu}^{ttX}=-ie\left\{\gamma_{\mu}\left(F_{1V}^{X}+\gamma_{5}F_{1A}^{X}\right)+{\sigma_{\mu\nu}\over 2m_{\rm t}}(p_{t}+p_{\bar{t}})^{\nu}\left(iF_{2V}^{X}+\gamma_{5}F_{2A}^{X}\right)\right\},\\$$

with, in the standard model, vanishing $$F_{2}$$s and

\begin{aligned} F_{1V}^{\gamma}=-{2\over 3} & , & F_{1V}^{Z}={1\over 4\sin\theta_{W}\cos\theta_{W}}\left(1-{8\over 3}\sin^{2}\theta_{W}\right)\ , \\ F_{1A}^{\gamma}=0 & , & F_{1A}^{Z}={1\over 4\sin\theta_{W}\cos\theta_{W}}\ .\\ \end{aligned}

The sensitivities are expressed therein in terms of $$\tilde{F}_{1}$$, $$\tilde{F}_{2}$$ defined as

$$\tilde{F}_{1V}^{X}=-({F}_{1V}^{X}+{F}_{2V}^{X})\ ,\ \tilde{F}_{2V}^{X}={F}_{2V}^{X}\ ,\ \tilde{F}_{1A}^{X}=-{F}_{1A}^{X}\ ,\ \tilde{F}_{2A}^{X}=-i{F}_{2A}^{X}\ .\\$$

On the other hand, the optimal-observable statistical analysis presented in the next section, based on Ref. (Grzadkowski 2000), uses the following $$A,B,C,D$$ parameterization (with $$v=\gamma,Z$$):

$$\Gamma^{\mu}_{ttv}={g\over 2}\left[\gamma^{\mu}\left\{(A_{v}+\delta A_{v})-\gamma_{5}(B_{v}+\delta B_{v})\right\}+{(p_{t}-p_{\bar{t}})^{\mu}\over 2m_{\rm t}}\left(\delta C_{v}-\delta D_{v}\gamma_{5}\right)\right],\\$$

which easily relates to the previous parameterization with

\begin{aligned} A_{v}+\delta A_{v}=-2i\sin\theta_{W}\left(F_{1V}^{X}+F_{2V}^{X}\right) & \ , & B_{v}+\delta B_{v}=-2i\sin\theta_{W}F_{1A}^{X}\ , \\ \delta C_{v}=-2i\sin\theta_{W}F_{2V}^{X} & \ , & \delta D_{v}=-2\sin\theta_{W}F_{2A}^{X}\ .\\ \end{aligned}

The expected sensitivities on the anomalous top-quark couplings can be derived in any of these parameterizations. Although originally derived with that of Ref. (Grzadkowski 2000), the final estimates presented in this study, however, use the parameterization of Ref. (Baer 2013), for an easy comparison. For the same reason, \crossalthough it is not needed, the same restrictions as in Ref. (Baer 2013) are applied here: only the six CP conserving form factors are considered (i.e., the two $$F_{2A}^{X}$$ are both assumed to vanish), and either the four form factors $$F_{1V,A}^{X}$$ are varied simultaneously while the two $$F_{2V}^{X}$$ are fixed to their standard model values, or vice-versa. A careful reading of Ref. (Baer 2013) shows that the form factor $$F_{1A}^{\gamma}$$ was also kept to its standard model value, as a non-zero value would lead to gauge-invariance violation. It is straightforward to show that, under these restrictions, the three parameterizations lead to the same sensitivities on $$F_{i}$$, $$\tilde{F}_{i}$$ and $$A,B,C,D$$ (with a multiplicative factor $$2\sin\theta_{W}\sim 0.96$$ for the latter set).

The tree-level angular and energy distributions of the lepton arising from the $${\rm t\bar{t}}$$ semi-leptonic decays are known analytically as a function of the incoming beam polarizations and the centre-of-mass energy (Grzadkowski 2000):

$${{\rm d}^{2}\sigma\over{\rm d}x{\rm d}\cos\theta}={3\pi\beta\alpha^{2}(s)\over 2s}B_{\ell}S_{\ell}(x,\cos\theta),\\$$

where $$\beta$$ is the top velocity, $$s$$ is the centre-of-mass energy squared, $$\alpha(s)$$ is the QED running coupling constant, and $$B_{\ell}$$ is the fraction of $${\rm t\bar{t}}$$ events with at least one top quark decaying to either $${\rm e}\nu_{\rm e}{\rm b}$$ or $$\mu\nu_{\mu}{\rm b}$$ (about 44%). As the non-standard form factors $$\delta(A,B,C,D)_{v}\equiv\delta_{i}$$ are supposedly small, only the terms linear in $$\delta_{i}$$ are kept:

$$\label{eq:optimal} \label{eq:optimal}S(x,\theta)=S^{0}(x,\theta)+\sum_{i=1}^{8}\delta_{i}f_{i}(x,\cos\theta),\\$$

where $$x$$ and $$\theta$$ are the lepton (reduced) energy and polar angle, respectively, and $$S^{0}$$ is the standard-model contribution. The eight distributions $$f_{A,B,C,D}^{\gamma,Z}(x,\cos\theta)\equiv f_{i}(x,\cos\theta)$$ and the standard-model contribution $$S^{0}(x,\cos\theta)$$ are shown for $$\ell^{-}$$ in Fig. \ref{fig:distributions} at $$\sqrt{s}=365$$ GeV, with no incoming beam polarization.