Measurement of the polarisation of single top quarks and antiquarks produced in the $t$-channel collected with the ATLAS detector at $\sqrt{s}=$ 13 TeV and bounds on the $tWb$ dipole operator

A simultaneous measurement of the three components of the top-quark and top-antiquark polarisation vectors in $t$-channel single-top-quark production is presented. This analysis is based on data from proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 139 fb$^{-1}$, collected with the ATLAS detector at the LHC. Selected events contain exactly one isolated electron or muon, large missing transverse momentum and exactly two jets, one being $b$ tagged. Stringent selection requirements are applied to discriminate $t$-channel single-top-quark events from the background contributions. The top-quark and top-antiquark polarisation vectors are measured from the distributions of the direction cosines of the charged lepton momentum in the top-quark rest frame. The three components of the polarisation vector for the selected top-quark event sample are $P_x =0.01\pm0.18$, $P_y = -0.029\pm0.027$, $P_z =0.91\pm0.10$ and for the top-antiquark event sample they are $P_x = -0.02\pm0.20$, $P_y = -0.007\pm0.051$, $P_z = -0.79\pm 0.16$. Normalised differential cross-sections corrected to a fiducial region at the stable-particle level are presented as a function of the charged lepton angles for top-quark and top-antiquark events inclusively. These measurements are in agreement with Standard Model predictions. The angular differential cross-sections are used to derive bounds on the complex Wilson coefficient of the dimension-six $\mathcal{O}_{tW}$ operator in the framework of an effective field theory. The obtained bounds are $C_{tW}\in\ [-0.7,1.5]$ and $C_{itW} \in [-0.7,0.2]$, both at 95% CL.