The Simplest B Decay, Precisely

TL;DR

The paper derives and implements a comprehensive multi-scale factorization framework for the leptonic B decay $B^-\to \mu^-\bar{\nu}_\mu(\gamma)$, combining LEFT, SCET-1, SCET-2, HQET, and HH$\chi$PT to separate electroweak, hard, and soft dynamics across scales from $m_Z$ down to $\Lambda_{\rm QCD}$. It introduces a complete NLP SCET-1 operator basis, treats endpoint divergences via refactorization-based subtraction, and develops a two-scale RG structure that resums large logarithms; below $\Lambda_{\rm QCD}$ the decay is matched onto HH$\chi$PT with a boosted lepton EFT for the soft photons. The analysis yields a state-of-the-art, percent-level prediction for the photon-vetoed rate, including structure-dependent QED effects and a nonperturbative parameter $F_-(\Lambda,\mu)$, with a detailed phenomenology of direct and indirect contributions (e.g., through $B^*$ intermediate states) and the impact of the chiral anomaly. The results enable a robust extraction of $|V_{ub}|$ and provide a framework for future precision tests of the SM and searches for new physics, particularly as experimental capabilities grow at Belle II and future colliders. The methodology also clarifies the role of endpoint divergences in NLP factorization and demonstrates how to control them in a systematic EFT approach.

Abstract

We derive the QCD$\times$QED factorization theorem governing the leptonic decay $B^-\toμ^-\barν_μ(γ)$ at all orders in $α_s$ and $α$. Electromagnetic corrections to this decay probe multiple scales, which we disentangle through a sequence of effective field theories (EFTs). The resulting state-of-the-art prediction for the photon-vetoed rate includes the complete structure-dependent component and is accurate at the percent level, establishing the theoretical framework required for future high-precision measurements of this channel, which will allow for a clean determination of $|V_{ub}|$ and powerful tests of new physics. Our work presents the first complete study of QED effects to an exclusive $B$-meson decay at next-to-leading power (NLP) in the heavy-quark expansion. Important milestones are (i) the construction of the complete NLP operator basis in soft-collinear effective theory (SCET); (ii) the proposal of a "SCET-friendly" reduction scheme for the Dirac structures of four-fermion operators in dimensional regularization, which avoids power-enhanced evanescent operators; (iii) the consistent refactorization of endpoint-divergent convolution integrals and the first complete resummation of "rapidity logarithms" arising at the boundary between the contributions involving soft and hard-collinear quarks; (iv) the systematic discussion of the EFT below the scale of QCD confinement and the non-perturbative matching of SCET onto this low-energy theory; (v) the decoupling of pseudoscalar mesons in the context of heavy-hadron chiral perturbation theory, so that they can be integrated out for processes in which they do not appear as external particles. We perform a phenomenological analysis of direct and indirect contributions to the decay rate and radiation-energy spectrum, highlighting the importance of the chiral anomaly.

Figures (20)

• Figure 1: Illustration of the appropriate effective theories below the electroweak scale to treat the $B^- \to \ell^- \bar{\nu}$ process for the cases $\ell=\mu,\tau,e$. In the case of the $\tau$ lepton, two options are shown, in which the lepton is either treated as a hard-collinear or a hard particle. The dots in the electron case indicate the scales $m_e$ and $E_{\rm cut}{\space}(m_e/m_B)$, which are both much smaller than $E_{\rm cut}$.
• Figure 2: Ingredients of the $B^-\to\mu^-{\space}\bar{\nu}_\mu$ factorization formula and their respective natural scales. Solid arrows denote RG evolution to a common renormalization scale $\mu_0$. Dashed arrows indicate that the corresponding component functions are evaluated at $\mu=\mu_0$ without resummation.
• Figure 3: Tree-level and one-loop Feynman diagrams for the LEFT to SCET-1 matching for the type-$A$ four-fermion operators. There is also a contribution from the wave-function renormalization of the $b$ quark.
• Figure 4: Hard two-loop diagrams which can generate the operator $O_4^E$ in SCET-1. The light quark, the charged lepton and the external gluons carry hard-collinear momenta.
• Figure 5: Left: One-loop diagrams leading to a mixing of the operators $O_i^B$ and $O_i^A$ (with $i=1,2$). The lepton mass insertion is not shown explicitly. Right: One-loop diagram leading to a mixing of the operators $O_i^C$ and $O_i^A$ (with $i=1,2$). Here and in later figures, red lines refer to hard-collinear fields, gray lines to soft fields, and light blue lines to collinear or anti-collinear fields. The soft heavy-quark is represented by a double line.