Collinear effective theory at subleading order and its application to heavy-light currents

TL;DR

This paper develops the collinear effective theory (CET) for highly energetic light quarks interacting with collinear and soft gluons, and systematically expands the EFT to subleading order in λ. It establishes the CET Lagrangian with collinear gauge and reparameterization invariance, performs one-loop matching of heavy-light currents from QCD to CET+HQET, and derives the RG evolution of the Wilson coefficients, enabling Sudakov resummation. The approach yields explicit expressions for heavy-to-light form factors at order λ, separating perturbative Wilson coefficients from nonperturbative light-cone functions and providing a structured framework for B→light-meson transitions in the large-energy regime. The results clarify symmetry constraints and offer a roadmap for incorporating hard spectator effects and extending to nonleptonic decays in future work.

Abstract

We consider a collinear effective theory of highly energetic quarks with energy E, interacting with collinear and soft gluons by integrating out collinear degrees of freedom to subleading order. The collinear effective theory offers a systematic expansion in power series of a small parameter lambda ~ p_{\perp}/E, where p_{\perp} is the transverse momentum of a collinear particle. We construct the effective Lagrangian to first order in $λ$, and discuss its features including additional symmetries such as collinear gauge invariance and reparameterization invariance. Heavy-light currents can be matched from the full theory onto the operators in the collinear effective theory at one loop and to order lambda. We obtain heavy-light current operators in the effective theory, calculate their Wilson coefficients at this order, and the renormalization group equations for the Wilson coefficients are solved. As an application, we calculate the form factors for decays of B mesons to light energetic mesons to order lambda and at leading-logarithmic order in alpha_s.

Figures (5)

• Figure 1: Feynman rules for $\mathcal{L}_0$ to order $g$ in the collinear effective theory: (a) collinear quark propagator with label $\tilde{p}$ and residual momentum $k$, (b) collinear quark interaction with one soft gluon, and (c) collinear quark interaction with one collinear gluon, respectively.
• Figure 2: Feynman rules for the operator $O_i^{\mu}$ ($i=1,2,3$) containing a collinear gluon at order $\lambda$. Here $\Gamma_i^{\mu}=\gamma^{\mu}$, $v^{\mu}$ and $n^{\mu}$ for $i=1,2,3$ respectively. The momentum of the gluon is outgoing.
• Figure 3: Feynman diagrams for the renormalization of $O_i^{\mu}$ ($i=1,2,3$) at one loop.
• Figure 4: Feynman rules for the effective Lagrangian $\mathcal{L}_1$ to order $g$: (a) collinear quark without an external gluon, (b) collinear quark interaction with a soft gluon, and (c) collinear quark interaction with a collinear gluon, and $k^{\mu}$ denotes residual momentum of order $\lambda^2$.
• Figure 5: Feynman diagrams for the renormalization of $\mathcal{L}_1$ at one loop.