Nonleptonic B decays into two light mesons in soft-collinear effective theory

TL;DR

This work establishes, within soft-collinear effective theory, that nonleptonic $B$ decays to two light mesons factorize at leading power and to all orders in $\alpha_s$. It develops a two-scale framework with $\mathrm{SCET}_{\mathrm{I}}$ and $\mathrm{SCET}_{\mathrm{II}}$, constructs gauge-invariant four-quark operators via Wilson lines, and computes Wilson coefficients through full-theory to SCET matching, including spectator effects. The formalism yields factorized expressions for decay amplitudes as convolutions of short-distance coefficients with meson light-cone distribution amplitudes, and explicitly applies the results to $\overline B \to \pi\pi$, recovering known results at $\mathcal{O}(\alpha_s)$ while extending to all orders in $\alpha_s$ with finite jet-function integrals. The approach provides a rigorous foundation for naive factorization and sets the stage for systematic subleading corrections and chirally enhanced contributions.

Abstract

We consider nonleptonic B decays into two light mesons at leading order in soft-collinear effective theory, and show that the decay amplitudes are factorized to all orders in alpha_s. The operators for nonleptonic B decays in the full theory are first matched to the operators in SCET_I, which is the effective theory appropriate for sqrt{m_b Lambda} <mu <m_b with Lambda~0.5 GeV. We evolve the operators and the relevant time-ordered products in SCET_I to SCET_II, which is appropriate for mu < sqrt{m_b Lambda}. Using the gauge-invariant operators in SCET_II, we compute nonleptonic B decays in SCET, including the nonfactorizable spectator contributions and spectator contributions to the heavy-to-light form factor. As an application, we present the decay amplitudes for B ->pi,pi in soft-collinear effective theory.

Figures (5)

• Figure 1: QCD diagrams attaching two gluons to external fermions to integrate out off-shell modes. The external gluons $A_n$, $A_{\overline{n}}$ or $A_s$ make the intermediate states off the mass shell. Diagrams with gluons attached to other fermions are omitted.
• Figure 2: Radiative corrections at one loop in the full theory. The momenta $p_1$, $p_2$, $p$ are outgoing with $p_b = p +p_1+p_2$. Infrared divergences exist in diagrams (a)--(f). Diagrams (g) and (h) are infrared finite. In (h), the square is the chromomagnetic operator $O_8$.
• Figure 3: Radiative corrections at one loop in SCET. Curly lines with a line represent collinear gluons, and curly lines represent soft gluons.
• Figure 4: A Feynman diagram for nonfactorizable spectator contributions from the subleading operator in the light-to-light current. The soft momentum $k$ is incoming, and $p_i$ ($i=1,2,3,4$) are outgoing.
• Figure 5: Tree-level graphs in $\mathrm{SCET}_{\mathrm{I}}$ for the spectator contribution to the heavy-to-light form factor. The first diagram contributes to $T_{1,2,4}$, and the second diagram contributes to $T_{0,1,3,4,5,6}$.