Techniques for computing two-loop QCD corrections tob→ctransitions

We have recently presented the complete $O\left({\alpha }_s^2\right)$ corrections to the semileptonic decay width of the $b$ quark at maximal recoil. Here we discuss various technical aspects of that calculation and further applications of similar methods. In particular, we describe an expansion which facilitates the phase space integrations and the treatment of the mixed real-virtual corrections, for which Taylor expansion does not work and the so-called eikonal expansion must be employed. Several terms of the expansion are given for the $O\left({\alpha }_s^2\right)$ QCD corrections to the differential semileptonic decay width of the $b$ quark at maximal recoil. We also demonstrate how the light quark loop corrections to the top quark decay rate can be obtained using the same methods. We briefly discuss the application of these techniques to the calculation of the $O\left({\alpha }_s^2\right)$ correction to zero recoil sum rules for heavy flavor transitions.