Closed-Form Summation of RG-Accessible Logarithmic Contributions to Semileptonic B-Decays and Other Perturbative Processes

For any perturbative series that is known to $k$-subleading orders of perturbation theory, we utilise the process-appropriate renormalization-group (RG) equation in order to obtain all-orders summation of series terms proportional to ${\alpha }^n{\log }^{n-k}({\mu }^2)$ with $k=0,1,2,3$, corresponding to the summation to all orders of the leading and subsequent-three-subleading logarithmic contributions to the full perturbative series. These methods are applied to the perturbative series for semileptonic $b$-decays in both MS-bar and pole-mass schemes, and they result in RG-summed series for the decay rates which exhibit greatly reduced sensitivity to the renormalization scale $\mu$. Such summation via RG-methods of all logarithms accessible from known series terms is also applied to perturbative QCD series for vector- and scalar-current correlation functions, the perturbative static potential function, the (single-doublet standard-model) Higgs decay amplitude into two gluons, as well as the Higgs-mediated high-energy cross-section for $W^{+}{W}^{-}\to ZZ$ scattering. The resulting RG-summed expressions are also found to be much less sensitive to the renormalization scale than the original series for these processes.