Approximate N$^2$LO and N$^3$LO QCD Predictions for $tW$ Production

We report a calculation of approximate next-to-next-to-leading-order (N$^2$LO) and next-to-N$^2$LO (N$^3$LO) QCD corrections to associated $tW$ production at the LHC, which constitute the dominant contributions to full perturbative predictions. The approximate N$^2$LO corrections consist of the large logarithmic terms $\ln^n (1-Q^2/\hat{s})$ (with $\sqrt{\hat{s}}$ being the partonic center-of-mass energy and $Q$ the invariant mass of the $tW$ system) and the terms proportional to $\delta(1-Q^2/\hat{s})$ at $\mathcal{O}(\alpha_s^2)$, which are obtained by utilizing the newly obtained two-loop hard and soft functions. The approximate N$^3$LO corrections further include the large logarithms at $\mathcal{O}(\alpha_s^3)$ by using renormalization group evolution equations and the three-loop soft anomalous dimension, while the $\delta(1-Q^2/\hat{s})$ term is only partially accurate at this order. Numerical evaluation reveals that they increase the NLO cross section by more than $10\%$. The inclusion of these higher-order corrections leads to improved agreement with the experimental data at the LHC, resulting in a direct determination of the CKM matrix element $|V_{tb}|=0.99\pm 0.03({\rm exp.})\pm 0.03({\rm theo.})$ without assuming unitarity of the matrix.