NNLO QCD corrections to the polarized top quark decay $t(\uparrow )\to X_b+W^{+}$

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the decay $t(\uparrow )\to X_b+W^{+}$ of a polarized top quark. The spin-momentum correlation in this quasi two-body decay is described by the polar angle distribution $$\mathrm{d}$\Gamma/$\mathrm{d}$\cos\theta_P=\frac{\Gamma}{2}(1+P_t, \alpha_P, \cos\theta_P)$where$P_t$isthepolarizationofthetopquarkand$\alpha_P$denotestheasymmetryparameterofthedecay\mathrm{.}Forthelatterwefind$\alpha^{$\mathrm{N}\mathrm{N}\mathrm{L}\mathrm{O}$}_P=0.3792\pm 0.0037$.