Soft-Collinear Messengers: A New Mode in Soft-Collinear Effective Theory

TL;DR

The paper argues that soft-collinear interactions between soft and collinear sectors in SCETII require a new soft-collinear messenger mode. It derives the corresponding soft-collinear Lagrangian, establishes its scaling, and systematically constructs the leading and induced interactions with soft and collinear fields, including gauge-invariant building blocks. Through explicit diagrams and a detailed power-counting analysis, it shows that soft-collinear exchanges are essential to reproduce full-theory amplitudes and to account for long-distance effects in heavy-to-light form factors and endpoint regions. The work clarifies factorization properties in exclusive B decays by revealing non-local induced interactions that persist after integrating out hard and hard-collinear modes, and connects these insights to Sudakov-like dynamics. Overall, it refines SCET’s treatment of soft and collinear degrees of freedom and provides a framework to incorporate soft-collinear messenger effects in phenomenological applications.

Abstract

It is argued that soft-collinear effective theory for processes involving both soft and collinear partons, such as exclusive B-meson decays, should include a new mode in addition to soft and collinear fields. These "soft-collinear messengers" can interact with both soft and collinear particles without taking them far off-shell. They thus can communicate between the soft and collinear sectors of the theory. The relevance of the new mode is demonstrated with an explicit example, and the formalism incorporating the corresponding quark and gluon fields into the effective Lagrangian is developed.

Figures (5)

• Figure 1: Examples of interactions between two soft and two collinear fields induced by the exchange of a hard-collinear particle (left) and of a soft-collinear particle (right). Hard-collinear modes are integrated out in SCET$_{\rm II}$, while soft-collinear modes remain as low-energy degrees of freedom.
• Figure 2: Scalar triangle diagram with an external soft momentum $l$ and a collinear momentum $p$. The loop momentum is denoted by $k$.
• Figure 3: Effective field-theory graphs. Full lines denote soft fields, dashed lines collinear fields, and dotted lines soft-collinear fields.
• Figure 4: A QCD diagram contributing to the decay of a $B$ meson to an energetic light meson $M$. The relevant loop subgraph is a pentagon with two collinear external lines with momenta $p_1$, $p_2$, a soft line with momentum $l_1$, and a heavy-quark line with momentum $m_b v+l_2$.
• Figure 5: A QCD diagram contributing to the decay of a $B$ meson to an energetic light meson $M$, and its representation in the effective theory. Solid lines carry soft, dashed lines collinear momentum. External soft momenta are incoming and collinear ones outgoing. The last diagram, which involves the interaction (\ref{['Lsc2']}), vanishes.