A Complete Basis for Power Suppressed Collinear-Ultrasoft Operators

TL;DR

This work develops a complete, gauge-invariant basis for power-suppressed mixed collinear–ultrasoft operators in SCET and extends the heavy-to-light current sector to include novel three-body structures. By enforcing gauge invariance, spin-structure reduction, and full reparameterization invariance (RPI), the authors constrain the allowed operators and fix most Wilson-coefficient relations across orders in the power expansion, leaving only a small set of undetermined coefficients. They construct the most general ultrasoft–collinear Lagrangian ${ m L}_{uc}$ up to ${ m O}(\lambda^2)$ and perform a detailed derivation of the heavy-to-light currents up to ${ m O}(\lambda)$, including a new class of three-body currents and their RPI-imposed constraints, along with explicit Feynman rules. The results underpin a universal factorization framework for heavy-to-light form factors, with jet functions that are process- and state-independent, and provide concrete tree-level matching and frame-specific reductions that facilitate phenomenological applications to $B$ decays to light hadrons. Overall, the paper delivers a systematic, symmetry-driven method to enumerate and constrain power-suppressed SCET operators and clarifies their role in factorization theorems for heavy-to-light processes.

Abstract

We construct operators that describe power corrections in mixed collinear-ultrasoft processes in QCD. We treat the ultrasoft-collinear Lagrangian to O(lambda^2), and heavy-to-light currents involving collinear quarks to O(λ) including new three body currents. A complete gauge invariant basis is derived which has a full reduction in Dirac structures and is valid for matching at any order in alpha_s. The full set of reparameterization invariance (RPI) constraints are included, and are found to restrict the number of parameters appearing in Wilson coefficients and rule out some classes of operators. The QCD ultrasoft-collinear Lagrangian has two O(lambda^2) operators in its gauge invariant form. For the O(lambda) heavy-to-light currents there are (4,4,14,14,21) subleading (scalar, pseudo-scalar, vector, axial-vector, tensor) currents, where (1,1,4,4,7) have coefficients that are not determined by RPI. In a frame where v_perp=0 and n.v =1 the total number of currents reduces to (2,2,8,8,13), but the number of undetermined coefficients is the same. The role of these operators and universality of jet functions in the factorization theorem for heavy-to-light form factors is discussed.

Figures (5)

• Figure 1: Feynman rules for the $O(\lambda)$ currents $J^{(1a)}$ in Eq. (\ref{['J1aJ1b']}) with zero and one gluon (the fermion spinors are suppressed). For the collinear particles we show their (label,residual) momenta, where label momenta are $p,q\sim \lambda^{0,1}$ and residual momenta are $k,t\sim \lambda^2$. Momenta with a hat are normalized to $m_b$, $\hat{p} = p/m_b$ etc.
• Figure 2: Feynman rules for the $O(\lambda)$ currents $J^{(1b)}$ in Eq. (\ref{['J1aJ1b']}) with zero and one gluon. For the collinear particles we show their (label,residual) momenta, where label momenta are $p,q,q_i\sim \lambda^{0,1}$ and residual momenta are $k,t\sim \lambda^2$. Momenta with a hat are normalized to $m_b$, $\hat{p} = p/m_b$ etc.
• Figure 3: Feynman rules for the subleading usoft-collinear Lagrangian ${\cal L}_{\xi q}^{(1)}$ with one and two collinear gluons (springs with lines through them). The solid lines are usoft quarks while dashed lines are collinear quarks. For the collinear particles we show their (label,residual) momenta. (The fermion spinors are suppressed.)
• Figure 4: Feynman rules for the $O(\lambda^2)$ usoft-collinear Lagrangian ${\cal L}_{\xi q}^{(2a)}$ with one and two gluons. The spring without a line through it is an usoft gluon. For the collinear particles we show their (label,residual) momenta, where label momenta are $p,q,q_i\sim \lambda^{0,1}$ and residual momenta are $k,t,t_i\sim \lambda^2$.
• Figure 5: Feynman rules for the $O(\lambda^2)$ usoft-collinear Lagrangian ${\cal L}_{\xi q}^{(2b)}$ with one and two gluons. For the collinear particles we show their (label,residual) momenta, where label momenta are $p,q,q_i\sim \lambda^{0,1}$ and residual momenta are $k,t,t_i\sim \lambda^2$.