Analysis of the $Wtb$ vertex from the measurement of triple-differential angular decay rates of single top quarks produced in the $t$-channel at $\sqrt{s}$ = 8 TeV with the ATLAS detector

The electroweak production and subsequent decay of single top quarks in the $t$-channel is determined by the properties of the $Wtb$ vertex, which can be described by the complex parameters of an effective Lagrangian. An analysis of a triple-differential decay rate in $t$-channel production is used to simultaneously determine five generalised helicity fractions and phases, as well as the polarisation of the produced top quark. The complex parameters are then constrained. This analysis is based on 20.2 fb$^{-1}$ of proton-proton collision data at a centre-of-mass energy of 8 TeV collected with the ATLAS detector at the LHC. The fraction of decays containing transversely polarised $W$ bosons is measured to be $f_1=0.30 \pm 0.05$. The phase between amplitudes for transversely and longitudinally polarised $W$ bosons recoiling against left-handed $b$-quarks, is measured to be $\delta_- = 0.002\pi^{+0.016\pi}_{-0.017\pi}$, giving no indication of CP violation. The fraction of longitudinal or transverse $W$ bosons accompanied by right-handed $b$-quarks are also constrained. Based on these measurements, limits are placed at 95% C.L. on the ratio of the complex coupling parameters Re$[g_\mathrm{R}/V_\mathrm{L}] \in [-0.12,0.17]$ and Im$[g_\mathrm{R}/V_\mathrm{L}] \in [-0.07,0.06]$. Constraints are also placed on the ratios $|V_\mathrm{R}/V_\mathrm{L}|$ and $|g_\mathrm{L}/V_\mathrm{L}|$. In addition, the polarisation of single top quarks in the $t$-channel is constrained to be $P > 0.72$ (95% C.L.). None of the above measurements make assumptions on the value of any of the other parameters or couplings and all of them are in agreement with the Standard Model.