Next-to-next-to-leading QCD corrections to the ${\mathbf{B}}^{+}$-${\mathbf{B}}_{\mathbf{d}}^0$, ${\mathbf{D}}^{+}$-${\mathbf{D}}^0$, and ${\mathbf{D}}_{\mathbf{s}}^{+}$-${\mathbf{D}}^0$ lifetime ratios

Abstract

The total decay widths of heavy mesons can be systematically calculated in terms of an expansion in the two parameters $1/m_Q$ and $\hat{I}{\pm }_s(m_Q)$, where $Q=c,b$ denotes the heavy quark. The dominant contributions to meson lifetime splittings stem from terms which are suppressed by $1/m_Q^3$ with respect to the leading universal contribution to the total decay width. We calculate three-loop contributions of order $\hat{I}{\pm }_s^2/m_q^3$ to the lifetime ratios $\ddot{I}„(B^{+})/\ddot{I}„(B_d^0)$, $\ddot{I}„(D^{+})/\ddot{I}„(D^0)$, and $\ddot{I}„(D_s^{+})/\ddot{I}„(D^0)$ in the limit of exact isospin and V-spin symmetry, respectively. Furthermore, we present new $\hat{I}{\pm }_s/m_q^3$ corrections to the Cabibbo-suppressed terms in $\ddot{I}„(B^{+})/\ddot{I}„(B_d^0)$. Combining our perturbative coefficients with hadronic matrix elements calculated from Heavy Quark Effective Theory sum rules, we find $\ddot{I}„(B^{+})/\ddot{I}„(B_d^0)=1.072\pm 0.024$. Using hadronic matrix elements from a recent lattice QCD calculation we find $\ddot{I}„(D^{+})/\ddot{I}„(D^0)=2.344\pm 0.170$ and $\ddot{I}„(D_s^{+})/\ddot{I}„(D^0)=1.289\pm 0.042$. We find good agreement of our predictions with experimental data, which constitutes a successful probe of the calculations of hadronic matrix elements and permits estimates of the unknown $1/m_Q^4$ contributions as well as the V-spin breaking terms in $\ddot{I}„(D_s^{+})/\ddot{I}„(D^0)$.