Probing the $Wtb$ vertex structure in $t$-channel single-top-quark production and decay in $pp$ collisions at $\sqrt{\mathrm{s}}=8$ TeV with the ATLAS detector

TL;DR

This study measures top-quark and W-boson polarization in t-channel single-top production at 8 TeV to test the Standard Model and constrain anomalous couplings using ATLAS data.

Contribution

It provides the first detailed polarization measurements in t-channel single-top events at 8 TeV, including limits on anomalous Wtb couplings.

Findings

Measured top-quark and W-boson polarizations agree with Standard Model predictions.

Set limits on the imaginary part of the anomalous coupling g_R.

Performed angular asymmetry analysis at parton level after background subtraction.

Abstract

To probe the $Wtb$ vertex structure, top-quark and $W$-boson polarisation observables are measured from $t$-channel single-top-quark events produced in proton--proton collisions at a centre-of-mass energy of 8 TeV. The dataset corresponds to an integrated luminosity of 20.2 fb$^{-1}$, recorded with the ATLAS detector at the LHC. Selected events contain one isolated electron or muon, large missing transverse momentum and exactly two jets, with one of them identified as likely to contain a $b$-hadron. Stringent selection requirements are applied to discriminate $t$-channel single-top-quark events from background. The polarisation observables are extracted from asymmetries in angular distributions measured with respect to spin quantisation axes appropriately chosen for the top quark and the $W$ boson. The asymmetry measurements are performed at parton level by correcting the observed angular distributions for detector effects and hadronisation after subtracting the background contributions. The measured top-quark and $W$-boson polarisation values are in agreement with the Standard Model predictions. Limits on the imaginary part of the anomalous coupling $g_{\mathrm{R}}$ are also set from model-independent measurements.

Tables

Table 1. Table 1: Asymmetries with their associated angular observables and their relation to the top-quark polarisation and W 𝑊 W -boson spin observables. The values predicted by the Standard Model are also given. They are calculated using the predictions at NLO in QCD for P 𝑃 P and α ℓ subscript 𝛼 ℓ \alpha_{\ell} , the predictions at NNLO for the helicity fractions, and the predictions at LO for ⟨ S 1 , 2 ⟩ delimited-⟨⟩ subscript 𝑆 1 2 \langle S_{1,2}\rangle and ⟨ A 1 , 2 ⟩ delimited-⟨⟩ subscript 𝐴 1 2 \langle A_{1,2}\rangle . The uncertainties in these values are all lower than 0.01. They are estimated from the uncertainties in the top-quark, b 𝑏 b -quark and W 𝑊 W -boson masses, added in quadrature, including the uncertainty in α s subscript 𝛼 s \alpha_{\mathrm{s}} and an estimate of the higher-order effects for the asymmetries related to the W 𝑊 W -boson spin observables.

Asymmetry	Angular observable	Polarisation observable	SM prediction
$A_{\mathrm{FB}}^{\ell }$	$\cos {\theta }_{\ell }$	$\frac{1}{2}{\alpha }_{\ell }P$	0.45
$A_{\mathrm{FB}}^{tW}$	$\cos {\theta }_W\cos {\theta }_{\ell }^{\ast }$	$\frac{3}{8}P\left(F_R+F_L\right)$	0.10
$A_{\mathrm{FB}}$	$\cos {\theta }_{\ell }^{\ast }$	$\frac{3}{4}⟨S_3⟩=\frac{3}{4}\left(F_R-F_L\right)$	$-$0.23
$A_{\mathrm{EC}}$	$\cos {\theta }_{\ell }^{\ast }$	$\frac{3}{8}\sqrt{\frac{3}{2}}⟨T_0⟩=\frac{3}{16}\left(1-3F_0\right)$	$-$0.20
$A_{\mathrm{FB}}^T$	$\cos {\theta }_{\ell }^T$	$\frac{3}{4}⟨S_1⟩$	0.34
$A_{\mathrm{FB}}^N$	$\cos {\theta }_{\ell }^N$	$-\frac{3}{4}⟨S_2⟩$	0
$A_{\mathrm{FB}}^{T,\varphi }$	$\cos {\theta }_{\ell }^{\ast }\cos {\varphi }_T^{\ast }$	$-\frac{2}{\pi }⟨A_1⟩$	$-$0.14
$A_{\mathrm{FB}}^{N,\varphi }$	$\cos {\theta }_{\ell }^{\ast }\cos {\varphi }_N^{\ast }$	$\frac{2}{\pi }⟨A_2⟩$	0

Table 2. Table 2: Scale factors and uncertainties extracted for the signal and background processes from the simultaneous maximum-likelihood fit of the event yields in the signal, anti-signal and t ​ t ¯ 𝑡 ¯ 𝑡 t\bar{t}  regions. The quoted uncertainties are statistical only.

Process	Scale factor
$t$-channel	0.95$\pm$	0.02
$t\overline{t}$, $Wt$, $s$-channel	1.01$\pm$	0.01
$W$+jets	1.10$\pm$	0.01

Table 3. Table 3: Signal and background event yields in the signal region after scaling to the results of the maximum-likelihood fit. The quoted uncertainties add in quadrature the post-fit uncertainties and the uncertainties due to the limited size of the simulation samples, except for the data-driven multijet contribution to which the normalisation uncertainty of 70% is applied. The total expectation is compared to the observed number of events.

Process	Event yield
$t$-channel	5700$\pm$	110
$Wt$, $s$-channel	265$\pm$	12
$t\overline{t}$	1914$\pm$	15
$W$+jets	2044$\pm$	57
$Z$+jets, diboson	188$\pm$	9
Multijet	420$\pm$	290
Total expectation	10530$\pm$	320
Data	10527

Table 4. Table 4: Uncertainties contributing to the measurements of the A FB ℓ superscript subscript 𝐴 FB ℓ A_{\mathrm{FB}}^{\ell} , A FB t ​ W superscript subscript 𝐴 FB 𝑡 𝑊 A_{\mathrm{FB}}^{tW} , A FB subscript 𝐴 FB A_{\mathrm{FB}} and A EC subscript 𝐴 EC A_{\mathrm{EC}} asymmetries. For better readability the uncertainties are multiplied by 10 2 .

Uncertainty source	$\Delta A_{\mathrm{FB}}^{\ell }\times {10}^2$	$\Delta A_{\mathrm{FB}}^{tW}\times {10}^2$	$\Delta A_{\mathrm{FB}}\times {10}^2$	$\Delta A_{\mathrm{EC}}\times {10}^2$
Statistical uncertainty	$\pm$2.6	$\pm$3.1	$\pm$2.3	$\pm$2.8
Simulation statistics	$\pm$1.7	$\pm$1.9	$\pm$1.4	$\pm$1.7
Luminosity	0.1	0.1	0.1	0.1
Background normalisation	$\pm$0.5	$\pm$0.5	$\pm$0.9	$\pm$0.7
$E_{\text{T}}^{\text{miss}}$ reconstruction	${}_{-0.1}{}^{}{}_{}{}^{+0.9}$	${}_{-0.7}{}^{}{}_{}{}^{+0.4}$	${}_{-0.7}{}^{}{}_{}{}^{+1.1}$	${}_{-0.2}{}^{}{}_{}{}^{+0.8}$
Lepton reconstruction	${}_{-0.4}{}^{}{}_{}{}^{+1.0}$	${}_{-1.3}{}^{}{}_{}{}^{+0.1}$	$\pm$1.4	${}_{-0.3}{}^{}{}_{}{}^{+0.6}$
Jet reconstruction	$\pm$2.1	$\pm$2.5	$\pm$1.2	$\pm$1.8
Jet energy scale	${}_{-1.2}{}^{}{}_{}{}^{+1.3}$	${}_{-1.6}{}^{}{}_{}{}^{+2.0}$	${}_{-2.7}{}^{}{}_{}{}^{+3.4}$	${}_{-0.7}{}^{}{}_{}{}^{+2.0}$
Jet flavour tagging	$\pm$0.9	$\pm$0.3	$\pm$0.6	$\pm$0.4
PDF	$\pm$0.2	0.1	0.1	$\pm$0.2
$t\overline{t}$ generator	$\pm$2.3	$\pm$1.0	$\pm$0.2	$\pm$1.2
$t\overline{t}$ parton shower	$\pm$0.6	$\pm$0.5	$\pm$2.7	$\pm$0.3
$t\overline{t}$ scales	$\pm$0.2	$\pm$0.4	$\pm$1.2	$\pm$0.3
$Wt$, $s$-channel generator	$\pm$1.0	$\pm$1.1	$\pm$0.4	$\pm$0.3
$Wt$, $s$-channel scales	$\pm$0.9	$\pm$0.3	$\pm$0.3	$\pm$0.3
$t$-channel NLO generator	$\pm$1.4	$\pm$0.6	$\pm$0.6	$\pm$2.7
$t$-channel LO–NLO generator	$\pm$1.5	$\pm$2.0	$\pm$2.6	$\pm$1.8
$t$-channel parton shower	$\pm$0.5	$\pm$1.0	$\pm$3.5	$\pm$0.2
$t$-channel scales	$\pm$1.1	$\pm$2.0	$\pm$0.6	$\pm$1.6
$W$+jets, multijet modelling	${}_{-2.4}{}^{}{}_{}{}^{+1.9}$	${}_{-1.0}{}^{}{}_{}{}^{+0.9}$	${}_{-2.1}{}^{}{}_{}{}^{+2.2}$	${}_{-1.2}{}^{}{}_{}{}^{+1.3}$
Total systematic uncertainty	${}_{-5.4}{}^{}{}_{}{}^{+5.4}$	${}_{-5.3}{}^{}{}_{}{}^{+5.2}$	${}_{-6.9}{}^{}{}_{}{}^{+7.3}$	${}_{-4.8}{}^{}{}_{}{}^{+5.3}$

Table 5. Table 5: Uncertainties contributing to the measurements of the A FB N superscript subscript 𝐴 FB 𝑁 A_{\mathrm{FB}}^{N} , A FB T superscript subscript 𝐴 FB 𝑇 A_{\mathrm{FB}}^{T} , A FB N , ϕ superscript subscript 𝐴 FB 𝑁 italic-ϕ A_{\mathrm{FB}}^{N,\phi} and A FB T , ϕ superscript subscript 𝐴 FB 𝑇 italic-ϕ A_{\mathrm{FB}}^{T,\phi} asymmetries. For better readability the uncertainties are multiplied by 10 2 .

Uncertainty source	$\Delta A_{\mathrm{FB}}^N\times {10}^2$	$\Delta A_{\mathrm{FB}}^T\times {10}^2$	$\Delta A_{\mathrm{FB}}^{N,\varphi }\times {10}^2$	$\Delta A_{\mathrm{FB}}^{T,\varphi }\times {10}^2$
Statistical uncertainty	$\pm$2.2	$\pm$3.1	$\pm$3.0	$\pm$4.6
Simulation statistics	$\pm$1.3	$\pm$2.0	$\pm$1.8	$\pm$2.9
Luminosity	0.1	0.1	0.1	0.1
Background normalisation	$\pm$0.4	$\pm$1.1	$\pm$0.6	$\pm$1.1
$E_{\text{T}}^{\text{miss}}$ reconstruction	${}_{-0.4}{}^{}{}_{}{}^{+0.3}$	${}_{-0.3}{}^{}{}_{}{}^{+0.5}$	${}_{-0.8}{}^{}{}_{}{}^{+0.5}$	${}_{-1.3}{}^{}{}_{}{}^{+0.4}$
Lepton reconstruction	${}_{-0.2}{}^{}{}_{}{}^{+0.1}$	${}_{-1.5}{}^{}{}_{}{}^{+1.3}$	${}_{-0.5}{}^{}{}_{}{}^{+0.6}$	${}_{-0.6}{}^{}{}_{}{}^{+1.6}$
Jet reconstruction	$\pm$0.8	$\pm$0.5	$\pm$1.6	$\pm$1.3
Jet energy scale	${}_{-0.8}{}^{}{}_{}{}^{+0.9}$	${}_{-4.6}{}^{}{}_{}{}^{+3.9}$	${}_{-2.5}{}^{}{}_{}{}^{+0.6}$	${}_{-2.5}{}^{}{}_{}{}^{+4.5}$
Jet flavour tagging	$\pm$0.2	$\pm$0.6	$\pm$0.3	$\pm$0.6
PDF	$\pm$0.1	$\pm$0.1	$\pm$0.1	$\pm$0.4
$t\overline{t}$ generator	$\pm$0.2	$\pm$3.5	$\pm$1.7	$\pm$1.3
$t\overline{t}$ parton shower	$\pm$1.5	$\pm$1.0	$\pm$0.9	$\pm$1.6
$t\overline{t}$ scales	$\pm$0.3	$\pm$0.8	$\pm$0.3	$\pm$1.3
$Wt$, $s$-channel generator	$\pm$0.2	$\pm$0.8	$\pm$0.3	$\pm$1.4
$Wt$, $s$-channel scales	$\pm$0.6	$\pm$0.5	$\pm$0.4	$\pm$0.9
$t$-channel NLO generator	$\pm$0.3	$\pm$4.5	$\pm$2.6	$\pm$7.2
$t$-channel LO–NLO generator	$\pm$0.5	$\pm$1.9	$\pm$1.3	$\pm$3.2
$t$-channel parton shower	$\pm$0.7	$\pm$0.9	0.1	$\pm$1.1
$t$-channel scales	$\pm$0.9	$\pm$2.2	$\pm$1.4	$\pm$2.6
$W$+jets, multijet modelling	${}_{-0.6}{}^{}{}_{}{}^{+0.7}$	${}_{-1.7}{}^{}{}_{}{}^{+1.3}$	$\pm$0.6	${}_{-1.7}{}^{}{}_{}{}^{+2.3}$
Total systematic uncertainty	${}_{-2.9}{}^{}{}_{}{}^{+2.9}$	${}_{-8.8}{}^{}{}_{}{}^{+8.3}$	${}_{-5.4}{}^{}{}_{}{}^{+4.8}$	${}_{-10.1}{}^{}{}_{}{}^{+10.9}$

Equations

$${\mathcal{L}}_{Wtb}=-\frac{g}{\sqrt{2}}\overline{b}{\gamma }^{\mu }\left(V_{\mathrm{L}}{P}_{\text{L}}+V_{\mathrm{R}}{P}_{\text{R}}\right)t{W}_{\mu }^{-}-\frac{g}{\sqrt{2}}\overline{b}\frac{i{\sigma }^{\mu \nu }{q}_{\nu }}{m_W}\left(g_{\mathrm{L}}{P}_{\text{L}}+g_{\mathrm{R}}{P}_{\text{R}}\right)t{W}_{\mu }^{-}+\text{h.c.}$$

$$\frac{1}{\Gamma }\frac{\mathrm{d}\Gamma }{\mathrm{d}(\cos {\theta }_X)}=\frac{1}{2}\left(1+{\alpha }_{X}P\cos {\theta }_X\right),$$

$$\frac{1}{\Gamma }\frac{\mathrm{d}\Gamma }{\mathrm{d}(\cos {\theta }_{\ell }^{\ast })\mathrm{d}{\varphi }_{\ell }^{\ast }}$$

$$A_{\mathrm{FB}}=\frac{N(\cos \theta >0)-N(\cos \theta <0)}{N(\cos \theta >0)+N(\cos \theta <0)},$$

$$A_{\mathrm{EC}}=\frac{N(|\cos \theta |>\frac{1}{2})-N(|\cos \theta |<\frac{1}{2})}{N(|\cos \theta |>\frac{1}{2})+N(|\cos \theta |<\frac{1}{2})}.$$

$$N_j^{\text{unfolded}}=\frac{{\sum }_i{M}_{ji}^{-1}{N}_i^{\text{measured}}}{{ϵ}_j},$$

$$A_{\mathrm{FB}}^{\ell }$$

$$A_{\mathrm{FB}}^{tW}$$

$$A_{\mathrm{FB}}$$

$$A_{\mathrm{EC}}$$

$$A_{\mathrm{FB}}^T$$

$$A_{\mathrm{FB}}^{N,\varphi }$$

$$A_{\mathrm{FB}}^{T,\varphi }$$

$${\alpha }_{\ell }P$$

$$P(F_{\mathrm{R}}+F_{\mathrm{L}})$$

$$⟨S_3⟩$$

$$⟨T_0⟩$$

$$⟨S_1⟩$$

$$⟨A_2⟩$$

$$⟨A_1⟩$$

$$A_{\mathrm{FB}}^N$$

$$⟨S_2⟩$$

Full text

\AtlasTitle

Probing the $Wtb$ vertex structure in $t$-channel single-top-quark production and decay in $pp$ collisions at $\sqrt{\mathrm{s}}=8$ TeV with the ATLAS detector

\PreprintIdNumberCERN-EP-2017-011 \AtlasJournalJHEP \AtlasAbstractTo probe the $Wtb$ vertex structure, top-quark and $W$-boson polarisation observables are measured from $t$-channel single-top-quark events produced in proton–proton collisions at a centre-of-mass energy of 8 TeV. The dataset corresponds to an integrated luminosity of 20.2 fb*-1*, recorded with the ATLAS detector at the LHC. Selected events contain one isolated electron or muon, large missing transverse momentum and exactly two jets, with one of them identified as likely to contain a $b$-hadron. Stringent selection requirements are applied to discriminate $t$-channel single-top-quark events from background. The polarisation observables are extracted from asymmetries in angular distributions measured with respect to spin quantisation axes appropriately chosen for the top quark and the $W$ boson. The asymmetry measurements are performed at parton level by correcting the observed angular distributions for detector effects and hadronisation after subtracting the background contributions. The measured top-quark and $W$-boson polarisation values are in agreement with the Standard Model predictions. Limits on the imaginary part of the anomalous coupling $g_{\mathrm{R}}$ are also set from model-independent measurements.

\size@chapter\sectfont

Contents

@afterheading@starttoc

toc

1 Introduction

At hadron colliders, top quarks are predominantly produced in pairs ($t\bar{t}$) via the flavour-conserving strong interaction, but single top-quark production can occur via charged-current electroweak processes involving a $Wtb$ vertex. At leading order in QCD perturbation theory, three sub-processes contribute to single top-quark production: an exchange of a virtual $W$ boson either in the $t$-channel or in the $s$-channel, or the associated production of a top quark with an on-shell $W$ boson ($Wt$). The $t$-channel and $s$-channel processes do not interfere at next-to-leading-order in QCD and are thus well defined with that precision [1].

In proton–proton ($pp$) collisions, the $t$-channel exchange, depicted in Figure 1, is the dominant production process of single top quarks. The exchange of a space-like $W$ boson due to the interaction of a light quark with a $b$-quark produces a top quark and a forward light-quark (called the spectator quark) in the final state. Furthermore, as a consequence of the vector minus axial-vector (V–A) form of the $Wtb$ vertex in the Standard Model, the produced top quarks are highly polarised, in particular along the direction of the spectator-quark momentum [2, 3].

Within the Standard Model the top quark decays through the electroweak interaction into an on-shell $W$ boson and a $b$-quark, with a lifetime much shorter than the time scale necessary to depolarise the spin. The information on the top-quark spin can thus be obtained from its decay products. The produced real $W$ boson also possesses a polarisation (or helicity state), which can be extracted from angular distributions of its decay products through the measurement of spin-dependent observables [4].

Measuring the top-quark polarisation and the $W$-boson spin observables in $t$-channel single top-quark production provides a powerful probe for studying the $Wtb$ vertex in both top-quark production and decay. New physics effects resulting in corrections to the $Wtb$ vertex would affect the top-quark and $W$-boson polarisations. In the effective operator formalism the most general $Wtb$ Lagrangian can be written as [5]:

[TABLE]

In this expression $g$ is the weak coupling constant, $m_{W}$ and $q_{\nu}$ are the mass and the four-momentum of the $W$ boson, respectively, $P_{\text{{L,R}}}\equiv(1\mp\gamma^{5})/2$ are the left- and right-handed projection operators, and $\sigma^{\mu\nu}=[\gamma^{\mu},\gamma^{\nu}]/2$. The constants $V_{\text{L,R}}$ and $g_{\text{L,R}}$ are the left- and right-handed vector and tensor couplings, respectively. In the Standard Model at tree level the coupling $V_{\mathrm{L}}$ is the $V_{tb}$ element of the quark-mixing Cabibbo–Kobayashi–Maskawa (CKM) matrix that is close to one, while the anomalous couplings $V_{\mathrm{R}}$ and $g_{\text{L,R}}$ are all zero. Deviations from these values would provide hints of physics beyond the Standard Model, and complex values would imply that the top-quark decay has a CP-violating component [5]. The imaginary part of $g_{\mathrm{R}}$ (Im $g_{\mathrm{R}}$) can be probed with the best precision in the $t$-channel production of single top quarks through the measurement of polarisation observables [5]. Limits on Im $g_{\mathrm{R}}$ have been set at the LHC by the ATLAS Collaboration at a centre-of-mass energy of 7 TeV from the analysis of the double-differential angular decay rates of the produced $t$-channel single-top-quark events [6].

The top-quark polarisation and the $W$-boson spin observables can be extracted in an alternative way from the measurement of asymmetries in various angular distributions of the top-quark decay products [4, 5]. Firstly, this article reports a determination of the top-quark polarisation as well as the $W$-boson spin observables extracted from the measured angular asymmetries. Such measurements serve as a consistency check with the Standard Model predictions. Secondly, limits on Im $g_{\mathrm{R}}$ are presented from the measurement of the so-called normal forward-backward asymmetry, which is predicted to have the highest sensitivity to Im $g_{\mathrm{R}}$ [5], and the asymmetry related to the top-quark polarisation. Here Standard Model values are assumed for all other couplings.

The measurements reported in this article use 20.2 fb*-1* of data collected at a centre-of-mass energy of 8 TeV with the ATLAS detector at the LHC. Stringent selection requirements are applied in order to separate signal from background. The $W$ boson from the top-quark decay is identified through its decay modes leading to a final state with an electron or a muon, and missing transverse momentum for the neutrino. The measurement at parton level of the asymmetries is performed by unfolding the observed angular distributions from detector and physics effects after subtracting the background contributions. For all reported results the electron and muon channels are merged, and the analysis is carried out independently of the lepton charge, in order to measure the polarisation observables associated with the combined production and decay of top quarks and top antiquarks.

2 Polarisation observables and asymmetries

The top-quark polarisation is determined from angular distributions of the decay products reconstructed in the top-quark rest frame, while the $W$-boson spin observables are determined from angular distributions of the charged lepton reconstructed in the $W$-boson rest frame.

In the top-quark rest frame, the angular distribution of any decay product $X$ of the top quark is given by

[TABLE]

where $\theta_{X}$ is the angle between the top-quark spin axis and the direction of motion of the chosen decay product in the top-quark rest frame, $\Gamma$ is the total decay width of the top quark, $\alpha_{X}$ is the spin analysing power associated with $X$, and $P$ is the top-quark degree of polarisation. The charged lepton is the most sensitive spin analyser; at next-to-leading-order (NLO) precision in QCD its spin analysing power is $\alpha_{\ell^{\pm}}$ $=$ $\pm 0.998$ [7]. In the $t$-channel, single top quarks are produced with a large degree of polarisation in the direction of motion of the spectator quark [8, 3]. This direction is used to define the top-quark spin axis in this measurement. The corresponding degrees of polarisation calculated at NLO in QCD are $0.91$ and $-0.86$ for top-quark and top-antiquark production, respectively [3].

In the framework of a general formalism developed in Ref. [4], the spin-density matrix elements for the $W$-boson helicity components 0, $\pm$1, resulting from the decay of polarised top-quarks, can be parameterised in terms of expectation values of six independent spin observables: $\langle S_{1,2,3}\rangle$, $\langle T_{0}\rangle$ and $\langle A_{1,2}\rangle$. With ($\theta_{\ell}^{*},\phi_{\ell}^{*}$) denoting the polar and azimuthal angles of the charged-lepton momentum in the $W$-boson rest frame, the fully differential decay width of a $W$ boson can be written as

[TABLE]

In this formalism the $W$-boson spin axis is taken along the direction of the $W$-boson momentum in the top-quark rest frame, or equivalently along the direction opposite to the $b$-quark momentum in the $W$-boson rest frame. The coordinate system used and the various angles defined for the charged lepton in the $W$-boson rest frame are depicted in Figure 2.

The angular distribution expressed in Equation (3) implies an integration over all the possible directions of the top-quark spin relative to the $W$-boson spin axis. The top-quark polarisation is propagated to the spin observables $\langle S_{1,2}\rangle$ and $\langle A_{1,2}\rangle$, which depend in a proportional way on the value of $P$. The spin observables $\langle S_{3}\rangle$ and $\langle T_{0}\rangle$ do not depend on $P$, and are related to the $W$-boson helicity fractions $F_{\mathrm{R}}$, $F_{\mathrm{L}}$ and $F_{0}$ [4].

From the values of the helicity fractions predicted by the Standard Model at next-to-next-to-leading order (NNLO) in QCD assuming a top-quark mass of 172.5 GeV and a $b$-quark mass of 4.8 GeV [9], one obtains $\langle S_{3}\rangle$ $=-0.31$ and $\langle T_{0}\rangle$ $=-0.43$. The uncertainties in these predictions due to the theoretical uncertainties in the helicity fractions are lower than 0.01 for both $\langle S_{3}\rangle$ and $\langle T_{0}\rangle$. Combining the predicted degrees of polarisation $P_{t}=0.91$ and $P_{\bar{t}}=-0.86$ with the $t$-channel single-top cross-sections $\sigma_{t}=54.9$ pb and $\sigma_{\bar{t}}=29.7$ pb calculated at NLO in QCD for top-quark and top-antiquark production [10], the Standard Model predictions for $\langle S_{1,2}\rangle$ and $\langle A_{1,2}\rangle$ are: $\langle S_{1}\rangle$ $=0.46$, $\langle A_{1}\rangle$ $=0.23$ and $\langle S_{2}\rangle$ $=$ $\langle A_{2}\rangle$ $=0$. These values are calculated at leading order (LO) in QCD from the expressions of the spin-density matrix elements given in Refs. [5, 4]. The uncertainties in these predictions resulting from the uncertainties in the top-quark, $b$-quark and $W$-boson masses, and from higher-order effects [11], are all smaller than 0.01. Measured values not equal to zero for the $\langle S_{2}\rangle$ and $\langle A_{2}\rangle$ spin observables would signal the presence of an imaginary coupling in the $Wtb$ vertex, since $\langle S_{2}\rangle$ and $\langle A_{2}\rangle$ are only sensitive to Im $g_{\mathrm{R}}$ [4].111Including one-loop QCD and electroweak corrections the prediction for $g_{\mathrm{R}}$ in the Standard Model is $(-7.17-1.23\mathrm{i})\times 10^{-3}$ [12], leading to values of the order of $10^{-3}$ for the $\langle S_{2}\rangle$ and $\langle A_{2}\rangle$ spin observables. However, $\langle S_{2}\rangle$ is twice as sensitive as $\langle A_{2}\rangle$ to Im $g_{\mathrm{R}}$, making this observable more suitable for determining this coupling. The other four $W$-boson spin observables are mainly sensitive to Re $g_{\mathrm{R}}$, with a poor sensitivity to Im $g_{\mathrm{R}}$ [5, 4].

The top-quark polarisation and the $W$-boson spin observables can be extracted from asymmetries derived by integrating the angular distributions expressed in Equations (2) and (3). These asymmetries are based on single or combined angular observables. They are listed in Table 1, together with their associated angular observables and their relation to the polarisation observables.222The asymmetries used in this article and in Ref. [5] are related to the ones defined in Refs. [4, 13] through the equations $A_{\mathrm{FB}}^{T}=A_{\mathrm{FB}}^{x}$, $A_{\mathrm{FB}}^{N}=-A_{\mathrm{FB}}^{y}$, $A_{\mathrm{FB}}^{T,\phi}=A_{\mathrm{FB}}^{1}$, $A_{\mathrm{FB}}^{N,\phi}=-A_{\mathrm{FB}}^{2}$, $A_{\mathrm{FB}}=A_{\mathrm{FB}}^{z}$ . The asymmetry values predicted by the Standard Model are also reported in the table.

Most of the polarisation observables are based on a forward-backward asymmetry, which is generically defined as a function of a given angular observable $\cos\theta$ according to

[TABLE]

where $N$ is the number of events. One of the $W$-boson spin observables is determined from an asymmetry called edge-central and defined as follows

[TABLE]

The product $\alpha_{\ell}P$ is extracted from the forward-backward asymmetry $A_{\mathrm{FB}}^{\ell}$ of the $\cos\theta_{\ell}$ angular distribution, where $\theta_{\ell}$ is the angle between the lepton momentum in the top-quark rest frame and the top-quark spin axis. The measurement of $P$ can also be performed from the forward-backward asymmetry $A_{\mathrm{FB}}^{tW}$ defined with respect to the combined angular observable $\cos\theta_{W}\cos\theta_{\ell}^{*}$ [14], where $\theta_{W}$ is the angle between the $W$-boson momentum in the top-quark rest frame and the top-quark spin axis. This asymmetry is proportional to the product of $P$ and the sum of two $W$-boson helicity fractions, as reported in Table 1. The $W$-boson spin observables $\langle S_{3}\rangle$ and $\langle T_{0}\rangle$ are derived from the forward-backward asymmetry $A_{\mathrm{FB}}$ and from the edge-central asymmetry $A_{\mathrm{EC}}$ of the $\cos\theta_{\ell}^{*}$ angular distribution, respectively. Using the definition [5] of the normal axis ${\vec{N}}={\vec{s}}_{t}\times{\vec{q}}$ and transverse axis ${\vec{T}}={\vec{q}}\times{\vec{N}}$, as illustrated in Figure 2, $\langle S_{1}\rangle$ and $\langle S_{2}\rangle$ are determined from the forward-backward asymmetries $A_{\mathrm{FB}}^{T}$ and $A_{\mathrm{FB}}^{N}$ in the angular observables $\cos\theta_{\ell}^{T}$ and $\cos\theta_{\ell}^{N}$, respectively. The $\langle A_{1}\rangle$ and $\langle A_{2}\rangle$ spin observables are determined from the forward-backward asymmetries $A_{\mathrm{FB}}^{T,\phi}$ and $A_{\mathrm{FB}}^{N,\phi}$ based on the combination of $\cos\theta_{\ell}^{*}$ with the cosine of the azimuthal angles $\phi_{T}^{*}$ and $\phi_{N}^{*}$ defined relatively to ${\vec{T}}$ and ${\vec{N}}$, respectively.

Limits on Im $g_{\mathrm{R}}$ can be extracted from the measurement of the $A_{\mathrm{FB}}^{N}$ asymmetry, which has the highest sensitivity to this coupling. For small Im $g_{\mathrm{R}}$ values, taking $V_{\mathrm{L}}=1$ and $V_{\mathrm{R}}=g_{\mathrm{L}}=0$, a linear dependence on Im $g_{\mathrm{R}}$ is obtained for this asymmetry: $A_{\mathrm{FB}}^{N}=0.64\,P\,\mathrm{Im}\,g_{\mathrm{R}}$ [5]. In this relation the weak dependence of $P$ on Im $g_{\mathrm{R}}$, which is of quadratic form, is not included. As $A_{\mathrm{FB}}^{N}$ depends on $P$, the measured value of the $A_{\mathrm{FB}}^{\ell}$ asymmetry is required to constrain $P$ for the limit computation. The quadratic variation of $P$ and $\alpha_{\ell}$ as a function of Im $g_{\mathrm{R}}$ [5, 15] is taken into account when setting the limits through the procedure explained in

3 sec:Results

. The $A_{\mathrm{FB}}^{\ell}$ asymmetry is chosen to constrain $P$ because it is measured independently of Im $g_{\mathrm{R}}$; this is discussed in

4 sec:Unfolding

.

5 The ATLAS detector

The ATLAS detector [16] is a multi-purpose particle detector with a forward-backward symmetric, cylindrical geometry and a near $4\pi$ coverage in solid angle around the collision point.333ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point in the centre of the detector and the $z$-axis along the beam pipe. The $x$-axis points from the interaction point to the centre of the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer. The inner detector is immersed in a $2\text{\,}\mathrm{T}$ axial magnetic field, and provides charged-particle tracking in the pseudorapidity range $|\eta|<2.5$. It contains a high-granularity silicon pixel detector, a silicon microstrip tracker, and a straw-tube transition radiation tracker. Lead/liquid-argon sampling calorimeters provide electromagnetic energy measurements with high granularity in the pseudorapidity ranges $|\eta|<1.5$ (barrel region) and $1.4<|\eta|<3.2$ (endcap region). Hadronic energy measurements are provided by steel/scintillator-tile calorimeters in the central pseudorapidity range $|\eta|<1.7$ and by copper/liquid-argon calorimeters in the endcap region $1.5<|\eta|<3.2$. The forward region is instrumented with liquid-argon calorimeters for electromagnetic and hadronic energy measurements, extending the coverage to $|\eta|=4.9$. The muon spectrometer surrounds the calorimeters and incorporates three large air-core toroid superconducting magnets with eight coils each. It includes separate trigger detectors and high-precision tracking chambers, providing muon momentum measurement for $|\eta|<2.7$ and muon triggering up to $|\eta|=2.4$.

A three-level trigger system is used to select interesting events [17]. The first-level trigger is hardware-based and uses a subset of the detector information to reduce the accepted event rate to less than $75\text{\,}\mathrm{kHz}$. The second and third levels are software-based and together reduce the event rate to about $400\text{\,}\mathrm{Hz}$.

6 Data and simulation samples

The analysis is performed using $pp$ collision data collected in 2012 by the ATLAS detector at a centre-of-mass energy of $8\text{\,}\mathrm{TeV}$. The events are required to pass single-electron or single-muon triggers [17, 18], resulting, after detector and data-quality requirements, in a data sample corresponding to an integrated luminosity of $20.2\text{\,}{\mathrm{fb}}^{-1}$. The electron and muon triggers impose a threshold of $24\text{\,}\mathrm{GeV}$ on the transverse momentum ($p_{\text{T}}$), along with isolation requirements. To recover efficiency for higher-$p_{\text{T}}$ leptons, the isolated lepton triggers are complemented by triggers without isolation requirements, but with a threshold raised to $60\text{\,}\mathrm{GeV}$ for electrons and to $36\text{\,}\mathrm{GeV}$ for muons.

Samples of signal and background events are simulated using various Monte Carlo generators. The generated events are passed through a simulation of the ATLAS detector [19] based on the Geant4 framework [20]. For some samples a faster simulation (ATLFAST-II [21]), making use of a parameterised response of the electromagnetic and hadronic calorimeters, is performed instead. Minimum-bias events simulated with the Pythia (8.1) [22] generator are overlaid to model the pile-up effects from additional $pp$ collisions in the same and nearby bunch crossings. All simulated events are then processed using the same reconstruction and analysis chain as for data events.

Signal $t$-channel single-top-quark events are generated with the NLO Powheg-Box (r2556) [23, 24, 25] generator, which uses the four-flavour scheme (Figure 1) for the matrix-element calculations [26]. Events are generated with the CT10f4 [27] parton distribution functions (PDFs), and the renormalisation and factorisation scales are set to $\mu_{\text{R}}^{2}=\mu_{\text{F}}^{2}=16\left(m^{2}_{b}+p^{2}_{\text{T},b}\right)$, where $m_{b}$ is the mass of the $b$-quark and $p_{\text{T},b}$ is the transverse momentum of the $b$-quark from the initial gluon splitting (called the spectator $b$-quark) [26]. Additional $t$-channel samples are produced with the LO Protos (2.2) [28] generator using the CTEQ6L1 PDFs [29]. Protos events are generated using the four-flavour scheme, as well, and anomalous couplings are enabled in both the production and the decay vertices, varying Re $V_{\mathrm{L}}$ and Im $g_{\mathrm{R}}$ simultaneously to keep the top-quark width invariant. The factorisation scale is set to $\mu_{\text{F}}^{2}=-p^{2}_{W}$ for the light quark, where $p_{W}$ is the four-momentum of the exchanged $W$ boson, and to $\mu_{\text{F}}^{2}=m^{2}_{b}+p^{2}_{\text{T},b}$ for the gluon. Eight Protos samples generated with Im $g_{\mathrm{R}}$ in the range [$-$0.144, 0.144] and Re $V_{\mathrm{L}}$ in the range [0.982, 1] are used, including the Standard Model configuration Im $g_{\mathrm{R}}=0$ and Re $V_{\mathrm{L}}=1$. These Protos samples are used to compute the parton-level unfolding corrections and to check the reliability of the unfolding method, while the Powheg-Box sample is used to determine the expected event yields and template distributions.

Samples of $t\bar{t}$ [30], $s$-channel single-top-quark and $Wt$ [31] background events are produced using the Powheg-Box (r2819, r3026) generator with the CT10 PDFs. To generate the $t\bar{t}$ sample, the model parameter $h_{\text{damp}}$, which effectively regulates the high-$p_{\text{T}}$ gluon radiation, is set to the top-quark mass $m_{t}$ [32].

For the above samples, parton showering, hadronisation and the underlying event are simulated with Pythia (6.426) [33] using parameter values set to the Perugia 2011C tune [34], and the CTEQ6L1 PDFs.

To study the modelling uncertainties of all processes involving top quarks, either alternative generators or parameter variations in the Powheg-Box and Pythia settings are used. For the estimation of the uncertainty in the $t$-channel matrix-element calculation, a sample is produced using the MadGraph5_aMC@NLO (2.0) [35] generator, interfaced to Herwig (6.52) [36, 37] for parton showering and to Jimmy (4.31) [38] for the underlying-event modelling with the ATLAS AUET2 tuned parameter settings [39] and the CT10f4 PDFs. The events are generated using the four-flavour scheme. For the $t\bar{t}$, $s$-channel and $Wt$ processes, alternative samples are produced using the MC@NLO (4.03) [40, 41, 42, 43] generator interfaced to Herwig (6.52) for parton showering and Jimmy (4.31) for the underlying-event modelling with the ATLAS AUET2 tune and the CT10 PDFs. To specifically study the impact of the parton-shower modelling, a $t$-channel sample and a $Wt$ sample both generated with Powheg-Box and coupled to Herwig (6.52) and Jimmy (4.31) with the AUET2 tune are used. For the $t\bar{t}$ process, samples generated using Powheg-Box with the CT10 PDFs, interfaced to Herwig (6.52) with the AUET2 tune or to Pythia (6.426) with the AUET2B tune, are used. Effects of varying the amount of radiation are studied by changing the hard-process and parton-shower scales simultaneously in the Powheg-Box and Pythia (6.426, 6.427) simulations. In the single-top-quark samples the factorisation and renormalisation scales are increased or decreased by a factor of two or one-half, respectively, in combination with the Perugia 2012 radLo and radHi tunes [34]. In the $t\bar{t}$ samples, $h_{\text{damp}}$ is set to $m_{t}$ or $2m_{t}$ in combination with the radLo and radHi parameterisations, respectively.

All top-quark processes are simulated assuming a top-quark mass of 172.5 GeV, and the top-quark decay is assumed to proceed exclusively through t$\to$Wb. The baseline Powheg-Box samples are passed through the fully Geant4-based simulation of the ATLAS detector, while the Protos samples and all samples used in studies of modelling uncertainties are processed through the ATLFAST-II simulation.

Vector-boson production in association with jets is simulated using the multileg LO Sherpa (1.4.1) [44] generator with its own parameter tune and the CT10 PDFs. Sherpa is used not only to generate the hard process, but also for the parton shower and the modelling of the underlying event. $W$+jets and $Z$+jets events with up to four additional partons are generated. The CKKW method [45] is used to remove overlaps between the partonic configurations generated by the matrix element and by the parton showering. Diboson samples of $WW$, $WZ$ and $ZZ$ events are also produced, using the Sherpa (1.4.1) generator with the CT10 PDFs. All the generated Sherpa single-boson and diboson events are passed through the ATLFAST-II simulation of the detector.

7 Event reconstruction and selection

The analysis considers only $W$-boson decay modes to an electron or a muon. Events in which the $W$ boson decays to a $\tau$ lepton are thus included if the $\tau$ lepton subsequently decays to an electron or a muon.

The signal event candidates are selected by requiring a single isolated electron or muon, significant missing transverse momentum, and exactly two jets with one of them identified as likely to contain a $b$-hadron ($b$-tagged jet). In fact, the presence of a third jet is not required in the event selection. Indeed, the additional jet resulting from the spectator $b$-quark originating from the gluon splitting as shown in Figure 1 is expected to have a softer $p_{\text{T}}$ spectrum and a broader $|\eta|$ distribution than the $b$-tagged jet produced in the top-quark decay, and, therefore, is in general not detected.

Electron candidates are reconstructed from isolated energy deposits in the electromagnetic calorimeter which are associated with inner-detector tracks fulfilling strict quality requirements [46]. They are required to satisfy $p_{\text{T}}>$25\text{,}\mathrm{GeV} and $|\eta|<2.47$, excluding the barrel–endcap transition region, corresponding to $1.37<|\eta|<1.52$. Muon candidates are reconstructed using combined tracking information from the inner detector and the muon spectrometer [[47](#bib.bibx47)]. They are required to have $p_{\text{T}}>$25\text{\,}\mathrm{GeV} and $|\eta|<2.5$. The electron and muon candidates must fulfil additional isolation requirements, as described in Ref. [48], in order to reduce contributions from misidentified jets, non-prompt leptons from the decay of heavy-flavour quarks and electrons from photon conversions.

Jets are reconstructed using the anti-$k_{t}$ algorithm [49] with a radius parameter of 0.4, from topological clusters [50], calibrated with a local cluster weighting method [51]. Jets are calibrated using an energy- and $\eta$-dependent simulation-based scheme, with in situ corrections based on data. The jet energy is further corrected for the effect of multiple $pp$ interactions. To reject jets from pile-up events, a so-called jet-vertex-fraction criterion [52] is applied to the jets with $p_{\text{T}}<$50\text{,}\mathrm{GeV} and $|\eta|<2.4$: at least 50% of the scalar sum of the $p_{\text{T}}$ of the tracks associated with a jet is required to be from tracks compatible with the primary vertex.444A primary-vertex candidate is defined as a reconstructed vertex with at least five associated tracks with $p_{\text{T}}>400$ MeV. The primary vertex associated with the hard-scattering collision is the candidate with the largest sum of the squared $p_{\text{T}}$ of the associated tracks. Only events containing two reconstructed jets with $p_{\text{T}}>$30\text{\,}\mathrm{GeV} are selected. In addition, one of them must be $b$-tagged with $|\eta|<2.5$, while the second jet is required to be untagged and to have $|\eta|<4.5$. The $b$-tagging is performed using a neural network which combines three different algorithms exploiting the properties of a $b$-hadron decay in a jet [53]. The $b$-tagging algorithm is optimised to improve the rejection of $c$-quark jets, since $W$-boson production in association with $c$-quarks is a major background for the selected final state. The requirement applied to the neural-network discriminant corresponds to a $b$-tagging efficiency of 50%, and mistagging rates of 3.9% and 0.07% for $c$-quark jets and light-flavour jets, respectively, as predicted in simulated $t\bar{t}$ events [54, 55].

The missing transverse momentum, with magnitude $E_{\text{T}}^{\text{miss}}$, is reconstructed from the vector sum of energy deposits in the calorimeter projected onto the transverse plane [56]. All cluster energies are corrected using the local cluster weighting method. Clusters associated with high-$p_{\text{T}}$ jets and electrons are further calibrated using their respective energy corrections. Contributions from the $p_{\text{T}}$ of the selected muons are also included in the calculation.

Events are required to contain at least one good primary-vertex candidate, and no jets failing to satisfy reconstruction quality criteria. The magnitude of the missing transverse momentum is required to be larger than $30\text{\,}\mathrm{GeV}$. In addition, the transverse mass of the lepton–$E_{\text{T}}^{\text{miss}}$ system must be greater than $50\text{\,}\mathrm{GeV}$ in order to reduce the multijet background contribution. 555The transverse mass of the lepton–$E_{\text{T}}^{\text{miss}}$ system is defined as $m_{\mathrm{T}}(\ell,E_{\text{T}}^{\text{miss}})=\sqrt{2p_{\text{T}}(\ell)E_{\text{T}}^{\text{miss}}\left(1-\cos\Delta\phi(\ell,E_{\text{T}}^{\text{miss}})\right)}$, where $\Delta\phi(\ell,E_{\text{T}}^{\text{miss}})$ is the difference in azimuthal angle between the lepton transverse momentum and the missing transverse momentum. Further reduction of this background is achieved by imposing an additional requirement on events where the lepton and the leading jet in $p_{\text{T}}$ have opposite directions in the transverse plane [57]. To reduce the $t\bar{t}$ dilepton background, events containing an additional lepton, identified with less stringent criteria (referred to as a loose lepton) and with a $p_{\text{T}}$ threshold lowered to $10$ GeV, are rejected.

The lepton and neutrino four-momenta are used to reconstruct the $W$ boson. Since the neutrino escapes undetected, the $x$- and $y$-components of the missing transverse momentum are assumed to correspond to the transverse momentum of the neutrino. The unmeasured longitudinal component of the neutrino momentum is computed by imposing a $W$-boson mass constraint on the lepton–neutrino system. If there are two real solutions, the solution giving the smallest magnitude of the longitudinal neutrino momentum is taken. If there are complex solutions, the magnitude of the measured missing transverse momentum is rescaled in order to obtain a physical solution [6]. The top-quark candidate is reconstructed by combining the four-momenta of the reconstructed $W$ boson and the $b$-tagged jet.

Additional requirements, defining the signal region, are finally applied to the preselected events:

• •

The pseudorapidity of the untagged jet must satisfy $|\eta|>2.0$, since the spectator quark tends to be produced in the forward direction in the $t$-channel process.

• •

The separation in $\eta$ between the untagged jet and the $b$-tagged jet must be larger than 1.5, to reduce the contribution from $t\bar{t}$ background events.

• •

The mass of the reconstructed top quark is required to be between 130 GeV and 200 GeV, to reject background events from processes not involving top quarks.

• •

The scalar sum ($H_{\mathrm{T}}$) of the $p_{\text{T}}$ of the lepton, the $p_{\text{T}}$ of the jets and $E_{\text{T}}^{\text{miss}}$ must be larger than 195 GeV, to further reduce the number of background events, in particular the $W$+jets contribution.

Figure 3 shows the distributions of the four variables relevant for these requirements, comparing data to the predicted signal and background distributions normalised to the results of the maximum-likelihood fit described in

8 sec:Yields

. The cuts that define the signal region are indicated for each of the variables. The multijet background estimate shown in the figure is discussed in

9 sec:Backgrounds

.

10 Background normalisation and modelling

The largest background contributions to $t$-channel single top-quark production arise from $t\bar{t}$ and $W$+jets production. The former is difficult to distinguish from the signal since $t\bar{t}$ events contain real top quarks in the final state. The $W$+jets production contributes to the background if there is a $b$-quark in the final state or due to mistagging of jets containing other quark flavours. Multijet production via the strong interaction can contribute as well if, in addition to two reconstructed jets, an extra jet is misidentified as an isolated lepton, or if a non-prompt lepton appears to be isolated (both referred to as fake leptons). Other minor backgrounds originate from $Wt$, $s$-channel single top-quark, $Z$+jets and diboson production.

For all processes, except multijet production, the normalisation is initially estimated by using the Monte Carlo simulation scaled to the theoretical cross-section predictions, and the event distribution modelling is taken from simulation.

The $t\bar{t}$ production cross-section is calculated at NNLO in QCD including resummation of next-to-next-to-leading-logarithm (NNLL) soft gluon terms with Top++2.0 [58, 59, 60, 61, 62, 63]. Its predicted value is $253^{+13}_{-15}$ pb [58]. The quoted uncertainties include the PDF and $\alpha_{\mathrm{s}}$ uncertainties calculated according to the PDF4LHC prescription [64] with the MSTW2008 NNLO [65, 66], CT10 NNLO [27, 67] and NNPDF2.3 5f FFN [68] PDF sets, and the QCD scale uncertainty. The $t$-channel, $Wt$ and $s$-channel single-top-quark production cross-sections are calculated at NLO precision in QCD through NNLL resummation, leading to $87.7^{+3.4}_{-1.9}$ pb [69], $22.4\pm 1.5$ pb [70] and $5.6\pm 0.2$ pb [71], respectively. The calculations assume a top-quark mass of 172.5 GeV and use the MSTW2008 NNLO [65] PDFs. The quoted uncertainties include those due to the QCD scale uncertainty and the correlated PDF–$\alpha_{\mathrm{s}}$ uncertainty.

The cross-sections for inclusive $W$- and $Z$-boson production are estimated with NNLO precision using the FEWZ program [72, 73] and the MSTW2008 NNLO PDFs. The diboson samples are normalised to the NLO cross-section predictions calculated with MCFM [74]. A normalisation uncertainty of 20% is assigned to the $W$+jets background. This uncertainty is estimated from parameter variations of the Sherpa generator covering the measured $W$+jets cross-sections [75]. A normalisation uncertainty of 20% is also assumed for the $Z$+jets and diboson processes.

The normalisation as well as the event modelling of the multijet background is estimated from data using the matrix method [48, 76]. This method allows the derivation of the true composition of the data sample in terms of prompt (real) and fake leptons from its observed composition in terms of tight (signal selection) and loose leptons. An alternative normalisation and modelling based on the mixed data–simulation jet-electron method [77, 57] and the purely data-driven anti-muon selection [48] are used to estimate the systematic uncertainties. From the comparison an overall normalisation uncertainty of 70% is assigned to the multijet contribution.

To check the modelling of the $t\bar{t}$ and $W$+jets background contributions, the simulated events are compared to the data in two dedicated background-dominated regions. Samples enriched in $t\bar{t}$ events ($t\bar{t}$ control region) are defined by considering events preselected as explained in

11 sec:Selections

, but containing two additional jets that are required to be untagged. This control region is also used in the normalisation fit described in

12 sec:Yields

. Samples enriched in $W$+jets events ($W$+jets control region) are selected by applying a relaxed $b$-tagging requirement corresponding to an efficiency of 80%. In addition, all events satisfying the signal $b$-tagging requirement are excluded. For these two control regions the dilepton rejection and the four final selection cuts are not applied. An additional category of events is defined by selecting all events not passing the four signal selection cuts (anti-signal region). This region is only used in the normalisation fit, in combination with the $t\bar{t}$ control region. It is preferred to the $W$+jets control region to constrain the $W$+jets normalisation because it has a flavour composition more similar to that of the signal region. The predicted fraction of heavy-flavour events in the $W$+jets contribution is around 95% for both the signal and anti-signal selections, whereas it is 55% in the $W$+jets control region.

Good overall data–prediction agreement is found in the $t\bar{t}$   $W$+jets and anti-signal control regions for the relevant kinematic observables, as well as for the various angular observables used in the measurements. Figure 4 shows the distributions in the $t\bar{t}$  control region of the four variables used to define the final selections. The distributions obtained in the $W$+jets control region are displayed in Figure 5.

13 Signal and background event yields

The signal and background event yields are estimated through a simultaneous maximum-likelihood fit to the numbers of data events observed in the signal and anti-signal regions, and in the $t\bar{t}$ control region.

The likelihood function [57] is given by the product of Poisson probability terms associated with the fitted regions, combined with the product of Gaussian priors to constrain the background rates to their predictions within the associated uncertainties. In the fit the $t$-channel single-top-quark contribution is treated as unconstrained. The top-quark background contributions ($t\bar{t}$, $Wt$ and $s$-channel single top-quark production) are merged with their relative fractions taken from simulation, and the applied constraint is derived from the combination of their cross-section uncertainties presented in

14 sec:Backgrounds

. The flavour composition of the $W$+jets contribution is taken from simulation. In all fitted regions the production of a $W$ boson in association with heavy-flavour jets is the dominant contribution to the $W$+jets background, predicted to be around 95% in the three regions. The $Z$+jets and diboson contributions, which are very low in the signal region (2% of the total expectation), are merged and fixed to the predictions. The multijet contribution is kept fixed to its data-driven estimate.

The results of the maximum-likelihood fit together with the associated statistical uncertainties (referred to as statistical post-fit uncertainties) are shown in Table 2. They are presented as scale factors to be applied to the predicted event yields. The results are found to be stable when the constraints imposed on the top-quark and $W$+jets backgrounds are significantly relaxed. Table 3 provides the signal and background event yields in the signal region after scaling to the results of the fit to the data. The signal-to-background ratio is 1.2, the $t$-channel single top-quark production representing 54% of the total expectation. The two main background contributions come from $W$+jets (19%) and $t\bar{t}$ production (18%).

15 Angular distributions

The distributions observed at reconstruction level for the angular observables used to measure the various asymmetries are shown in Figures 6 and 7. They are compared to the predicted signal and background distributions, normalised to the results of the maximum-likelihood fit. To minimise the unfolding corrections that are applied after background subtraction, two bins are chosen for the angular distributions from which forward-backward asymmetries are extracted, while four bins are used for the angular distribution from which the $A_{\mathrm{EC}}$ asymmetry is determined.

Depending on the angular observable, as described in Section 2, the charged-lepton four-momentum is computed in the rest frame of the reconstructed top quark or in the rest frame of the reconstructed $W$ boson. The angular observables related to the top-quark polarisation are defined by taking the momentum of the untagged jet as the spectator-quark direction, whereas those related to the $W$-boson spin observables are defined by considering the reverse momentum of the $b$-tagged jet as the $W$-boson direction.

16 Unfolding

The measured angular distributions are unfolded to the parton level,666Partons are defined from the matrix-element hard process and immediate decays. so that the asymmetries extracted from the corrected angular distributions can be directly compared to theoretical calculations. The unfolding corrections account for distortions due to detector resolution, selection efficiencies, and reconstruction of the $W$ boson and top quark. They also include the effects due to hadronisation and parton showering.

The unfolding procedure is applied to the angular distributions after subtracting the background contributions, and is based on a matrix inversion combined with an efficiency correction. The number of unfolded signal events $N^{\text{unfolded}}_{j}$ in each bin $j$ of the parton-level distribution is obtained from the background-subtracted yields $N^{\text{measured}}_{i}$ measured in all bins $i$ of the reconstructed distribution, according to

[TABLE]

where $M_{ji}$ is the migration matrix which relates the parton-level and reconstructed values of the considered angular variable, and $\epsilon_{j}$ is the event selection efficiency. Both the migration matrix and the selection efficiency are computed using samples of $t$-channel events simulated with the Protos generator, as described below. For the chosen numbers of bins, the fractions of simulated events belonging to the diagonal elements of the migration matrices are found to be between 68% and 90%, depending on the angular observable. The selection efficiencies are between 0.6% and 1.6%, depending on the angular observable and on the bin range. The matrix inversion is performed by using the iterative Bayesian method [78] as implemented in the RooUnfold framework [79]. The number of iterations is chosen such that the absolute change in the extracted asymmetry between two successive steps becomes lower than 0.0005. The unfolding procedure has been validated through convergence and closure tests performed by using template distributions constructed from the $t$-channel Powheg-Box and Protos samples presented in Section 6. The closure tests showed that the residual bias induced by the unfolding method is negligible, whatever the measured asymmetry.

With the aim of testing their compatibility with the Standard Model predictions, all asymmetries described in

17 sec:Observables

, except $A_{\mathrm{FB}}^{N}$, are extracted using the Protos simulation generated with the Standard Model values of the $Wtb$ couplings to determine the migration matrix and the selection efficiency. For all the asymmetry measurements, the Standard Model $Wtb$ couplings, as implemented in the Powheg-Box generator, are considered for the subtracted top-quark backgrounds.

To constrain Im $g_{\mathrm{R}}$ using the method explained in

18 sec:Observables

, the $A_{\mathrm{FB}}^{N}$ and $A_{\mathrm{FB}}^{\ell}$ asymmetries must be measured without any assumption about Im $g_{\mathrm{R}}$. It is observed that the presence of anomalous couplings in general modifies the kinematics in such a way that the efficiency corrections are dependent on the $Wtb$ couplings. While the measurement of $A_{\mathrm{FB}}^{\ell}$ is found to be independent of the value of Im $g_{\mathrm{R}}$ assumed in the unfolding corrections, the measurement of $A_{\mathrm{FB}}^{N}$ is found to depend on the unfolding corrections used. By applying an interpolation technique it is possible to unfold the $\cos\theta_{\ell}^{N}$ angular distribution independently of any assumption about Im $g_{\mathrm{R}}$, so that the extracted $A_{\mathrm{FB}}^{N}$ asymmetry, combined with $A_{\mathrm{FB}}^{\ell}$, can be used to constrain this coupling.

The interpolation method is based on determining the unfolding corrections using a linear combination of the migration and efficiency corrections provided by five Protos samples in which Im $g_{\mathrm{R}}$ is varied (Im $g_{\mathrm{R}}$ $=0,\pm 0.094,\pm 0.23$). An iterative procedure is applied to determine the coefficients of the linear combination until convergence is reached in the extracted $A_{\mathrm{FB}}^{N}$ asymmetry. The method proceeds as follows. An initial value of $A_{\mathrm{FB}}^{N}$ is first extracted using the standard Protos unfolding corrections. This value is then used to determine, via a Lagrange interpolation, the weights to be applied to the five predicted corrections. A new value of $A_{\mathrm{FB}}^{N}$ is obtained after unfolding the $\cos\theta_{\ell}^{N}$ angular distribution with these corrections using the Bayesian method. The chosen convergence criterion for the interpolation procedure requires that the difference between the extracted $A_{\mathrm{FB}}^{N}$ from two successive steps is smaller than 0.0005. By using template distributions given by Protos samples not used in the linear combination of the unfolding corrections (Im $g_{\mathrm{R}}$ $=\pm 0.043,\pm 0.144$), it has been checked that this method recovers the generated asymmetries at parton level.

The sensitivity to Im $g_{\mathrm{R}}$ of the $\cos\theta_{\ell}$ and $\cos\theta_{\ell}^{N}$ distributions, which are used to set limits on this coupling, is illustrated in Figure 8. In this figure the observed distributions are compared to the signal-plus-background predictions built by adding the signal templates given by the Protos samples generated with Im $g_{\mathrm{R}}$ $=0$ (Standard Model parameterisation) and Im $g_{\mathrm{R}}$ $=\pm 0.23$, the latter corresponding to the maximum values considered in the interpolation method described above.

19 Systematic uncertainties

Several sources of systematic uncertainty affect the asymmetry measurements, modifying the signal and background event yields and angular distributions. To evaluate the impact of each source the asymmetries are extracted by unfolding the template distributions after varying them to reflect that source of uncertainty. In each case a new background estimation is performed before subtraction, using the fitting procedure described in

20 sec:Yields

. For all sources of systematic uncertainty other than those associated with the limited size of the simulation samples, the nominal unfolding corrections are considered. The systematic uncertainty is evaluated as the difference between the nominal asymmetry value and the one measured using the varied normalisations and shapes.

The sources of systematic uncertainty are split into the following categories:

Background normalisation: The uncertainties in the normalisation of the top-quark and $W$+jets background processes are determined from the maximum-likelihood fit. For the merged $Z$+jets and diboson processes the normalisation uncertainty of 20% introduced in

21 sec:Backgrounds

is applied to the predictions. For the data-driven normalisation of the multijet background the uncertainty of 70% estimated from the comparison of the matrix-method estimates with those given by the jet-electron and anti-muon methods is used.

The uncertainty in the integrated luminosity is 1.9% [80]. It is propagated to the asymmetry measurements through the normalisation of the simulated backgrounds.

Detector modelling: Systematic uncertainties in the reconstruction and energy calibration of jets, electrons and muons are propagated in the analysis through variations in the modelling of the detector response. For the jets, the main source of uncertainty is the energy scale, evaluated using a combination of in situ techniques [51]. Other jet-related uncertainty sources are the modelling of the energy resolution [81] and reconstruction efficiency [51] (both referred to as jet reconstruction uncertainties), and the modelling of the tagging efficiencies of $b$-quark jets, $c$-quark jets and light-flavour jets [54, 55]. Uncertainties related to leptons come from trigger, identification and isolation efficiencies, as well as from the energy scale and resolution [46, 47] (all referred to as lepton reconstruction uncertainties). The uncertainties in the energy scale and resolution corrections applied to leptons and jets are propagated to the computation of the missing transverse momentum. The scale and resolution uncertainties due to soft jets and to contributions of calorimeter energy deposits not associated with any reconstructed objects are also considered and evaluated independently (they are labelled $E_{\text{T}}^{\text{miss}}$ reconstruction uncertainties). For all detector modelling uncertainties, positive and negative uncertainties are estimated separately from the corresponding shifts.

Signal and background modelling: Systematic uncertainties associated with the signal and background modelling are estimated by comparing event samples from different generators and by varying parameters in the event generation.

The uncertainty in the matrix-element calculation in the simulation of the $t$-channel single-top-quark process is estimated by comparing MadGraph5_aMC@NLO+Herwig with Powheg-Box+Herwig. For the $t\bar{t}$ and $Wt$ processes, MC@NLO is compared with Powheg-Box, both generators interfaced to Herwig. The uncertainty in the parton shower is estimated by comparing Powheg-Box interfaced with Pythia and Herwig for the $t$-channel, $t\bar{t}$ and $Wt$ processes. For the $s$-channel single-top-quark contribution the uncertainty due to the choice of generator and parton shower is estimated in a combined way by comparing MC@NLO+Herwig with Powheg-Box+Pythia.

An additional modelling uncertainty is considered for the signal process by comparing the NLO Powheg-Box sample to the LO Protos sample implementing the Standard Model parameterisation of the $Wtb$ couplings. To estimate this uncertainty, only the shapes of the distributions are varied in order to assess the impact of using a LO generator to determine the unfolding corrections.

The uncertainty in the amount of QCD radiation is evaluated for all top-quark processes by comparing the Powheg-Box samples generated with the varied hard-process and parton-shower scales presented in

22 sec:Samples

. The largest shift in the measured asymmetries is taken as uncertainty.

The dependence of the measured asymmetries on the top-quark mass is estimated using Powheg-Box samples generated with different top-quark masses. Variations lower than 0.01 per GeV are found for the measured asymmetry values. Therefore, these variations are not included in the total systematic uncertainty.

The impact of the flavour composition on the modelling of the $W$+jets distributions is determined by propagating an uncertainty of 50% in the ratio of $W$+$b\bar{b}$ to $W$+$c\bar{c}$ events. As reported in

23 sec:Yields

, $W$+light-flavour jets events give a small contribution in the signal region and no associated modelling uncertainty is taken into account. An additional shape-modelling uncertainty is considered for the $W$+jets distributions. Indeed, in the $W$+jets control region a few kinematic variables are slightly mismodelled, and the impact of this mismodelling is evaluated by reweighting the $W$+jets angular distributions in the signal region. The applied event weights are derived from matching to data (after subtraction of all processes other than $W$+jets) the mismodelled kinematic variables in the $W$+jets control region. This procedure leads to a conservative estimate since it also accounts for mismodelling of the $W$+light-flavour jets events, which have a much more important contribution in the $W$+jets control region than in the signal region.

The systematic uncertainty associated with the data-driven shape modelling of the multijet events is estimated by comparing the shapes provided by the baseline matrix method and the alternative modelling given by the jet-electron and anti-muon methods.

All the signal and background modelling uncertainties, except that associated with the $W$+jets flavour composition, are symmetrised by taking the difference between the nominal and varied measurements as positive and negative uncertainties.

Systematic uncertainties related to the parton distribution functions are estimated for all processes, except for the multijet contribution. The uncertainty is estimated, following the PDF4LHC prescription [64], by calculating the envelope of the uncertainties at 68% confidence level of the CT10 [27], MSTW2008NLO [65] and NNPDF2.3 [68] sets.

Limited size of simulation samples: The uncertainty due to the limited size of the Monte Carlo samples is evaluated by varying the background normalisation and shape, as well as the unfolding corrections, through Gaussian fluctuations. The standard deviation of the distribution of the measured asymmetry provided by an ensemble test of pseudo-experiments built from these variations is taken as a systematic uncertainty.

Tables 4 and 5 show the contribution of each source of systematic uncertainty to the asymmetry measurements. The total uncertainties are obtained from the sum in quadrature of all contributions. Tables 4 and 5 also include the statistical uncertainty from the data sample. It is evaluated using a procedure similar to that used for the uncertainty associated with the size of the simulation samples, but varying the observed numbers of events and the shape of the angular distributions through Poisson fluctuations.

The asymmetry measurements are dominated by the systematic uncertainties. The largest contributions are from the uncertainties in the modelling of the $t$-channel and $t\bar{t}$ processes, and in the jet reconstruction and energy scale. Significant contributions also come from the uncertainty in the modelling of the multijet or $W$+jets events, depending on the measured asymmetry, and from the limited size of the simulation samples. The statistical uncertainty of the data sample, although lower than the systematic uncertainty, also has a sizeable impact on the measurement precision.

24 Results

The values of the asymmetries related to the top-quark polarisation and to the $W$-boson spin observables, measured using the Standard Model $Wtb$ couplings for the signal unfolding corrections and for the top-quark background modelling, are

[TABLE]

The values for the top-quark polarisation combined with the charged-lepton spin analysing power and with the sum of the $W$-boson helicity fractions, derived from the measured $A_{\mathrm{FB}}^{\ell}$ and $A_{\mathrm{FB}}^{tW}$ asymmetries using the relations given in Table 1, are

[TABLE]

The values of the $W$-boson spin observables derived from the measured $A_{\mathrm{FB}}$, $A_{\mathrm{EC}}$, $A_{\mathrm{FB}}^{T}$, $A_{\mathrm{FB}}^{N,\phi}$ and $A_{\mathrm{FB}}^{T,\phi}$ asymmetries through the relations given in Table 1 are

[TABLE]

The results for the $A_{\mathrm{FB}}^{N}$ asymmetry, which has the highest sensitivity to the anomalous $Wtb$ coupling Im $g_{\mathrm{R}}$, and for its associated $W$-boson spin observable are

[TABLE]

These observables are measured using the signal corrections interpolated with respect to Im $g_{\mathrm{R}}$ as explained in

25 sec:Unfolding

, and using the Standard Model couplings for the top-quark background modelling.

Figure 9 shows the measured and predicted values of all asymmetries, while Figure 10 compares the derived values for the six $W$-boson spin observables. Compatibility between the measurements and Standard Model predictions is observed.

The overall compatibility of the measurements with the Standard Model predictions is evaluated through the construction of a $\chi^{2}$ test statistic taking into account all measured quantities with their correlations. The theoretical uncertainties, which are negligible compared to the measurement uncertainties, are not taken into account in the $\chi^{2}$ calculation. The overall covariance matrix is computed from the sum of the covariance matrices associated with the various sources of statistical and systematic uncertainty. To calculate the covariance matrices associated with the detector-related and $W$+jets flavour composition uncertainties, the positive and negative uncertainties are symmetrised by taking the larger value. The overall $p$-value for the eight asymmetries is found to be 0.94, and it is 0.83 for the six $W$-boson spin observables.

Limits on the anomalous coupling Im $g_{\mathrm{R}}$ are extracted from the $A_{\mathrm{FB}}^{N}$ and $A_{\mathrm{FB}}^{\ell}$ asymmetries, which, as discussed in

26 sec:Unfolding

, are measured independently of any assumption about Im $g_{\mathrm{R}}$ in the unfolding procedure, but assuming the Standard Model couplings for the subtracted top-quark backgrounds. However, for the main $t\bar{t}$ background a negligible dependence on Im $g_{\mathrm{R}}$ is expected.

The limit extraction is based on the TopFit code [5, 82]. By taking into account the analytic expressions and parameterisations introduced in Refs. [5, 4, 15] for the $Wtb$ coupling dependence of $\langle S_{2}\rangle$, $\alpha_{\ell}$ and $P$, it is possible to determine the allowed region for Im $g_{\mathrm{R}}$ from the measured values of $A_{\mathrm{FB}}^{N}$ and $A_{\mathrm{FB}}^{\ell}$. The limit setting is based on the computation of the $\chi^{2}$ test statistic using the covariance matrix associated with the $A_{\mathrm{FB}}^{N}$ and $A_{\mathrm{FB}}^{\ell}$ measurements. An overall correlation coefficient of $-$0.05 is found.

Assuming $V_{\mathrm{L}}=1$ and that all anomalous couplings other than Im $g_{\mathrm{R}}$ vanish ($V_{\mathrm{R}}=g_{\mathrm{L}}=0$ and Re $g_{\mathrm{R}}=0$), the limits set at the 95% confidence level are Im $g_{\mathrm{R}}$ $\in$ [$-$0.18, 0.06]. The measured interval of allowed values slightly improves on the limits set at 7 TeV by the ATLAS Collaboration from the measurement of double-differential angular decay rates [6].

27 Conclusion

Measurements of the top-quark and $W$-boson polarisation observables in $t$-channel single top-quark production at $\sqrt{s}=8$ TeV with 20.2 fb*-1* of $pp$ collision data recorded with the ATLAS detector at the LHC are presented. The selected events contain one isolated electron or muon, large missing transverse momentum and exactly two jets, of which one is tagged as a $b$-jet. A cut-based analysis is used to discriminate the signal events from background, and the electron and muon channels are combined. The polarisation observables are measured from asymmetries in various angular distributions unfolded to the parton level. Unfolding corrections based on a Standard Model simulation of the $t$-channel process are used, as well as model-independent corrections derived through an interpolation method. The measured asymmetries and the measured polarisation observables are in agreement with the predictions of the Standard Model. Limits on the imaginary part of the anomalous coupling $g_{\mathrm{R}}$ are also set, giving Im $g_{\mathrm{R}}$ $\in$ [$-$0.18, 0.06] at the 95% confidence level. The extracted values improve on the most recently published limits for this coupling.

Acknowledgements

We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.

We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, the Canada Council, CANARIE, CRC, Compute Canada, FQRNT, and the Ontario Innovation Trust, Canada; EPLANET, ERC, ERDF, FP7, Horizon 2020 and Marie Skłodowska-Curie Actions, European Union; Investissements d’Avenir Labex and Idex, ANR, Région Auvergne and Fondation Partager le Savoir, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF; BSF, GIF and Minerva, Israel; BRF, Norway; CERCA Programme Generalitat de Catalunya, Generalitat Valenciana, Spain; the Royal Society and Leverhulme Trust, United Kingdom.

The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN, the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA), the Tier-2 facilities worldwide and large non-WLCG resource providers. Major contributors of computing resources are listed in Ref. [83].