Factorisation theorem and pole cancellation in the resolved $Q_1^q$-$Q_{7γ}$ contribution to $\bar{B}\to X_sγ$ at NLO

Abstract

We study the subleading resolved $Q_1^q$-$Q_{7γ}$ contribution to the inclusive decay $\bar{B}\to X_sγ$ at next-to-leading order (NLO) in $α_s$. We present a formulation of the factorisation theorem which is valid beyond the leading order (LO). On this basis, we explicitly demonstrate the cancellation of all $1/\varepsilon$ singularities between the hard, hard-collinear, anti-hard-collinear and soft contributions. For this purpose, a central result of this work is the computation of the $u$-integrated massless two-loop anti-hard-collinear jet function, which determines the structure of the $1/\varepsilon$ singularities. In addition, we compute the hard function at the required order, combine our results with the hard-collinear jet function known from the literature, and employ the shape function whose renormalisation was determined in our previous work. There, we showed, for inclusive as well as for exclusive modes, that certain contributions to these multi-light-cone soft functions are physically irrelevant and can be consistently omitted; the explicit cancellation of the corresponding divergences provides a non-trivial confirmation of this statement. Overall, our findings constitute a non-trivial consistency check of the factorisation framework for this subleading resolved contribution at NLO and provide the basis for a full phenomenological analysis.