Quark mass effects in the soft-collinear effective theory and B -> X_s + photon in the endpoint region

TL;DR

This paper extends SCET to include a light quark mass and studies its impact on $\bar{B}\to X_s\gamma$ in the endpoint region. It shows an extended reparameterization invariance that relates mass-containing subleading operators to leading ones, with mass operators remaining non-renormalized at ${\cal O}(\alpha_s)$. The practical outcome is that strange-quark mass effects enter only through jet functions in the factorized rate $d\Gamma/dE_\gamma$, yielding $\mathcal{O}(\Lambda/m_b)$ corrections near the endpoint and allowing a systematic resummation in moments. Overall, the work provides a rigorous framework to incorporate $m$-dependent corrections in SCET and clarifies their size and structure in endpoint $B$ decays, with potential applications to SU(3) flavor breaking in related processes.

Abstract

We consider the effects of a light quark mass in the soft-collinear effective theory (SCET) and we apply them to B -> X_s gamma in the endpoint region. We find that the reparameterization invariance can be extended by including the collinear quark mass in the SCET Lagrangian. This symmetry constrains the theory with the quark mass terms, and we present explicit results at one loop. It also relates the Wilson coefficients of some mass operators to those of the leading operators, which are useful in organizing the subleading effects due to the quark mass in B -> X_s gamma. We present strange quark mass corrections to B -> X_s gamma in the endpoint region as an application. The forward scattering amplitude from the mass corrections is factorized, and it can be expressed as a convolution of the m_s^2/p_X^2-suppressed jet function and the leading-order shape function of the B meson. This contribution should be added to the existing subleading contributions from the B meson shape functions to obtain complete subleading corrections.

Figures (6)

• Figure 1: Feynman rules for the operators $O_m^{(1)}$ and $O_m^{(2)}$ with one or two collinear gluons.
• Figure 2: Feynman diagrams for the radiative corrections to $O_m^{(2)}$ at one loop.
• Figure 3: Feynman diagrams for one-loop corrections to $O_m^{(1)}$ in the background field gauge.
• Figure 4: Feynman diagrams for the mass correction to the heavy-to-light current with $j_i^{(m)}$ ($i=1,2,\cdots,5$) and $O_m^{(1)}$.
• Figure 5: Tree-level Feynman diagrams for the leading mass corrections to $\overline{B}\rightarrow X_s \gamma$ near the endpoint region in $\rm{SCET_I}$. The mirror image of the diagram (a) should be included. The intermediate hard-collinear strange quark is integrated out at $\mu\sim \sqrt{m_b \Lambda}$ to match onto $\rm{SCET_{II}}$.