Summing Sudakov logarithms in B -> X_s + gamma in effective field theory

TL;DR

The paper develops a two-stage effective field theory to sum Sudakov logarithms in B -> X_s gamma near the endpoint. It first introduces a collinear-soft EFT that includes both soft and collinear degrees of freedom and demonstrates a clean one-loop matching with no large logs and a calculable RG evolution. It then integrates out collinear modes at a lower scale to obtain a nonlocal bilocal operator in LEET, whose coefficients run with a continuous kernel; resumming via moments reproduces known leading Sudakov results and aligns with factorization-based approaches. The framework provides a general strategy for summing Sudakov logs in processes with multiple scale separations and can be extended to exclusive decays and other high-energy hadronic processes.

Abstract

We construct an effective field theory valid for processes in which highly energetic light-like particles interact with collinear and soft degrees of freedom, using the decay B -> X_s + gamma near the endpoint of the photon spectrum, x = 2 E_gamma / m_b -> 1, as an example. Below the scale mu=m_b both soft and collinear degrees of freedom are included in the effective theory, while below the scale mu=m_b sqrt{x-y}, where 1-y is the lightcone momentum fraction of the b quark in the B meson, we match onto a theory of bilocal operators. We show that at one loop large logarithms cancel in the matching conditions, and that we recover the well known renormalization group equations that sum leading Sudakov logarithms.

Figures (11)

• Figure 1: The OPE for $B\rightarrow X_s\gamma$.
• Figure 2: One-loop corrections to the matrix element of $\hat{O}_7$ in QCD.
• Figure 3: One-loop correction to the $bs\gamma$ vertex in LEET.
• Figure 4: Propagators in the collinear-soft effective theory.
• Figure 5: Leading order quark-gluon interactions in the collinear-soft effective theory: (a) collinear-collinear, (b) collinear-soft, and (c) heavy-soft. Applying the rules from Table \ref{['tableone']}, the vertices scale as (a) $\lambda^{-1}$, (b) $\lambda^0$, and (c) $\lambda^0$.