A measurement is presented of differential cross sections for $t$-channel single top quark and antiquark production in proton-proton collisions at a centre-of-mass energy of 13 TeV by the CMS experiment at the LHC. From a data set corresponding to an integrated luminosity of 35.9 fb$^{-1}$, events containing one muon or electron and two or three jets are analysed. The cross section is measured as a function of the top quark transverse momentum ($p_\mathrm{T}$), rapidity, and polarisation angle, the charged lepton $p_\mathrm{T}$ and rapidity, and the $p_\mathrm{T}$ of the W boson from the top quark decay. In addition, the charge ratio is measured differentially as a function of the top quark, charged lepton, and W boson kinematic observables. The results are found to be in agreement with standard model predictions using various next-to-leading-order event generators and sets of parton distribution functions. Additionally, the spin asymmetry, sensitive to the top quark polarisation, is determined from the differential distribution of the polarisation angle at parton level to be 0.440 $\pm$ 0.070, in agreement with the standard model prediction.
The measurement of the charge asymmetry in top quark pair events with highly Lorentz-boosted top quarks decaying to a single lepton and jets is presented. The analysis is performed using proton-proton collisions at $\sqrt{s}$ = 13 TeV with the CMS detector at the LHC and corresponding to an integrated luminosity of 138 fb$^{-1}$. The selection is optimized for top quarks produced with large Lorentz boosts, resulting in nonisolated leptons and overlapping jets. The top quark charge asymmetry is measured for events with a $\mathrm{t\bar{t}}$ invariant mass larger than 750 GeV and corrected for detector and acceptance effects using a binned maximum likelihood fit. The measured top quark charge asymmetry of (0.42 $_{-0.69}^{+0.64}$)% is in good agreement with the standard model prediction at next-to-next-to-leading order in quantum chromodynamic perturbation theory with next-to-leading-order electroweak corrections. The result is also presented for two invariant mass ranges, 750-900 and $\gt$ 900 GeV.
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