Systematic approach to exclusive B ->V l^+l^-, V gamma decays

TL;DR

The paper establishes a systematic QCD factorization framework for exclusive radiative B decays in the heavy-quark limit, enabling the calculation of non-factorizable corrections to $B\to K^*\gamma$ and $B\to K^*\ell^+\ell^-$ amplitudes. It disentangles hard spectator, form-factor, and annihilation effects, providing NNLL-level control over Wilson-like coefficients ${\cal C}_7$, ${\cal C}_{9,\perp}(q^2)$, and ${\cal C}_{9,\parallel}(q^2)$ and applying the formalism to predict the lepton spectrum and, crucially, the forward-backward asymmetry zero. The analysis reveals a large NLO enhancement of the $B\to K^*\gamma$ rate and a significant shift in the FB zero position for $B\to K^*\ell^+\ell^-$, enabling a clean determination of $C_9$ from future measurements while highlighting the remaining uncertainties from form factors and $1/m_b$ corrections. The approach also sets the stage for precise tests of the Standard Model and potential new physics in $b\to s$ and $b\to d$ transitions.

Abstract

We show -- by explicit computation of first-order corrections -- that the QCD factorization approach previously applied to hadronic two-body decays and to form factor ratios also allows us to compute non-factorizable corrections to exclusive, radiative B meson decays in the heavy quark mass limit. This removes a major part of the theoretical uncertainty in the region of small invariant mass of the photon. We discuss in particular the decays B\to K^* γand B\to K^* l^+l^- and complete the calculation of corrections to the forward-backward asymmetry zero. The new correction shifts the asymmetry zero by 30%, but the result confirms our previous conclusion that the asymmetry zero provides a clean phenomenological determination of the Wilson coefficient C_9.

Figures (10)

• Figure 1: Leading contributions to $\langle \gamma^*\bar{K}^*| H_{\rm eff} | \bar{B}\rangle$. The circled cross marks the possible insertions of the virtual photon line.
• Figure 2: Non-factorizable contributions to $\langle \gamma^*\bar{K}^*| H_{\rm eff} | \bar{B}\rangle$. The circled cross marks the possible insertions of the virtual photon line. Diagrams that follow from (c) and (e) by symmetry are not shown. Upper line: hard spectator scattering. Lower line: diagrams involving a $B\to K^*$ form factor (the spectator quark line is not drawn for these diagrams).
• Figure 3: Vertex corrections to the weak annihilation diagram in Figure \ref{['fig2']}c. Only the photon coupling to the spectator quark line contributes at leading order in $1/m_b$ as indicated by the circled cross.
• Figure 4: The absolute value of the ratio of $B$-meson moments $|\lambda_{B,-}^{-1}(q^2)|/\lambda_{B,+}^{-1}$ as a function of $q^2$. The $B$-meson distribution amplitudes are taken as in (\ref{['GN_WF']}) with $\omega_0^{-1}=(3 \pm 1)$ GeV${}^{-1}$.
• Figure 5: Renormalization scale-dependence of $|{\cal C}_7|^2$ at leading (LO) and next-to-leading order (NLO). The curve "NLO${}_1$" shows the NLO result without the spectator scattering correction.