Spectator Interactions in Soft-Collinear Effective Theory

TL;DR

This work extends soft-collinear effective theory (SCET) to include soft massless quarks, enabling systematic treatment of spectator interactions in exclusive B decays. It develops the SCET framework with detailed field content, power counting, and gauge-invariant operator construction, revealing that soft-collinear interactions vanish at leading power but give rise to non-local operators at intermediate scales. The authors analyze current and four-quark operator matching, showing two leading operator structures and potential higher-twist contributions that can affect factorization theorems, and they introduce gauge-invariant building blocks to simplify operator construction. The results establish a foundation for two-step matching and Sudakov resummation in heavy-to-light processes and highlight when standard factorization may need generalization to accommodate soft-quark effects.

Abstract

Soft-collinear effective theory is generalized to include soft massless quarks in addition to collinear fields. This extension is necessary for the treatment of interactions with the soft spectator quark in a heavy meson. The power counting of the relevant fields and the construction of the effective Lagrangian are discussed at leading order in Lambda/m_b. Several novel effects occur in the matching of full-theory amplitudes onto effective-theory operators containing soft light quarks, such as the appearance of an intermediate mass scale and large non-localities of operators on scales of order 1/Lambda. Important examples of effective-theory operators with soft light quarks are studied and their renormalization properties explored. The formalism presented here forms the basis for a systematic analysis of factorization and power corrections for any exclusive B-meson decay into light particles.

Figures (7)

• Figure 1: Effective four-gluon coupling obtained after integrating out the off-shell gluon in the first diagram and adding the corresponding local QCD vertex. The resulting SCET operator is denoted by a crossed circle. Collinear gluons in SCET are drawn as springs with a line. Soft and collinear gluons in full QCD are labeled by $S$ and $C$.
• Figure 2: Example of a long-distance soft-collinear interaction induced by the exchange of a soft messenger gluon.
• Figure 3: Attachment of a collinear gluon to a soft light quark.
• Figure 4: Diagrams contributing to the matching calculation for the soft-collinear current for the case of one soft and one collinear external gluon, and the resulting non-local operator in the SCET. The dashed line represents the current insertion.
• Figure 5: Multiple attachments of collinear gluons to a soft light quark, and the resulting non-local interaction in SCET.