$B\to X_{c(u)}\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}\gamma$ and determination of non-perturbative parameters

This article explores the calculation of non-perturbative parameters present in the matrix element of semileptonic inclusive $B$ decays. These are important for the inclusive determination of the Cabbibo-Kobayashi-Maskawa (CKM) parameters in the standard model. We focus on calculating the rate for radiative inclusive decay, $B\to X_c\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}\gamma$, where $\gamma$ is hard. Both the radiative and non-radiative ($B\to X_c\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}$) modes are found to require the same non-perturbative parameters. We propose forming ratios from $B\to X_c\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}\gamma$ and $B\to X_c\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}$ widths in different lepton energy ranges. Our results provide an efficient way to unambiguously calculate the non-perturbative parameters, especially when combined with total width calculations of $B\to X_c\mathrm{\ell }{\bar{\nu }}_{\mathrm{\ell }}$. This is shown explicitly by working to $\mathcal{O}(1\mathrm{/}{m}_b)$ and determining ${\lambda }_1$ and ${\lambda }_2$.