An effective field theory for collinear and soft gluons: heavy to light decays

TL;DR

This work develops a collinear-soft effective field theory to describe heavy-to-light decays with energetic light degrees of freedom, incorporating both collinear and soft gluons and two low-energy scales. The authors construct the Lagrangian, perform one-loop matching for heavy-to-light currents, and derive a renormalization-group evolution that sums Sudakov logarithms between the hard scale $Q$ and the intermediate scale $Q\lambda$. They show that soft form-factor relations for exclusive heavy-to-light decays persist when collinear gluons are included, and they apply the framework to inclusive endpoint decays, where Sudakov resummation is determined by the Wilson coefficient running. Overall, the EFT provides infrared-safe matching and a unified approach to both inclusive and exclusive heavy-to-light processes, with potential extensions to hard spectator contributions.

Abstract

We construct the Lagrangian for an effective theory of highly energetic quarks with energy Q, interacting with collinear and soft gluons. This theory has two low energy scales, the transverse momentum of the collinear particles, p_perp, and the scale p_perp^2/Q. The heavy to light currents are matched onto operators in the effective theory at one-loop and the renormalization group equations for the corresponding Wilson coefficients are solved. This running is used to sum Sudakov logarithms in inclusive B -> X_s gamma and B -> X_u \ell nu decays. We also show that the interactions with collinear gluons preserve the relations for the soft part of the form factor for heavy to light decays found by Charles et al., establishing these relations in the large energy limit of QCD.

Figures (8)

• Figure 1: Order $\lambda^0$ Feynman rules: collinear quark propagator with label $\tilde{p}$ and residual momentum $k$, and collinear quark interactions with one soft gluon, one collinear gluon, and two collinear gluons respectively.
• Figure 2: Order $\alpha_s \lambda^0$ self energy diagrams for a collinear quark.
• Figure 3: Matching for the order $\lambda^0$ Feynman rule for the heavy to light current with $n$ collinear gluons. All permutations of crossed gluon lines are included on the left.
• Figure 4: Order $\lambda^0$ Feynman rule for the effective theory heavy to light current with $m$ collinear gluons. The sum is over permutations of $\{1,\ldots,m\}$ and the Wilson coefficient depends only on the sum of momenta in the jet, $P=p+\sum_{i=1}^m q_i$.
• Figure 5: Full theory one loop diagram for matching the heavy-light current (denoted by $\otimes$). The incoming line is a massive quark and the outgoing line is massless.