Renormalisation group evolution of the shape function g17 in $$ \overline{B}\to {X}_s\gamma $$ and $$ \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} $$ at subleading power

A bstract We derive and solve the renormalisation-group (RG) equation of the shape function g 17 ( ω, ω 1 ; μ ), which appears at subleading power in the factorisation of the inclusive decays $$ \overline{B}\to {X}_s\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mi>γ</mml:mi> </mml:math> and $$ \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>−</mml:mo> </mml:msup> </mml:math> . Our results provide the first ingredient for a next-to-leading order analysis of the respective resolved-photon $$ {Q}_1^c-{Q}_{7\gamma } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> <mml:mi>c</mml:mi> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mn>7</mml:mn> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msub> </mml:math> interference contribution, whose current uncertainties are among the largest ones in both inclusive penguin modes. As a by-product of our study, we find that the analytic properties of the soft anomalous dimension as well as the jet functions in a factorisation theorem allow for a simplified renormalisation of operators in Heavy-Quark Effective Theory that are composed of fields smeared along distinct light-cone directions. Their hadronic matrix elements become relevant in various inclusive and exclusive B -decays beyond leading power in the heavy-quark and large-energy expansion. Using these insights, we derive and solve a simpler “reduced” RG equation of an amplitude-level soft function that describes the long-distance QCD dynamics for penguin contributions to exclusive $$ {\overline{B}}_{d,s}\to \gamma \gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>γγ</mml:mi> </mml:math> decays, and which has recently been discussed in the literature.