Search for anomalous couplings in the $Wtb$ vertex from the measurement of double differential angular decay rates of single top quarks produced in the $t$-channel with the ATLAS detector

The electroweak production and subsequent decay of single top quarks is determined by the properties of the $Wtb$ vertex. This vertex can be described by the complex parameters of an effective Lagrangian. An analysis of angular distributions of the decay products of single top quarks produced in the $t$-channel constrains these parameters simultaneously. The analysis described in this paper uses 4.6 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}$ = 7 TeV collected with the ATLAS detector at the LHC.Two parameters are measured simultaneously in this analysis. The fraction $f_1$ of decays containing transversely polarised $W$ bosons is measured to be $0.37 \pm 0.07$ (stat.$\oplus$syst.). The phase $\delta_{-}$ between amplitudes for transversely and longitudinally polarised $W$ bosons recoiling against left-handed $b$-quarks is measured to be $-0.14\pi \pm 0.036\pi$ (stat.$\oplus$syst.).The correlation in the measurement of these parameters is $0.15$. These values result in two-dimensional limits at the 95% confidence level on the ratio of the complex coupling parameters $g_\mathrm{R}$ and $V_\mathrm{L}$, yielding Re [$g_\mathrm{R}/V_\mathrm{L}$] $\in [-0.36,0.10]$ and Im [$g_\mathrm{R}/V_\mathrm{L}$] $\in [-0.17,0.23]$ with a correlation of 0.11. The results are in good agreement with the predictions of the Standard Model.