Branching Ratios for $B \to K^* γ$ and $B \to ργ$ Decays in Next-to-Leading Order in the Large Eneregy Effective Theory

TL;DR

The paper develops a next-to-leading order LEET-based analysis of exclusive radiative B decays, computing hard spectator corrections at $O(\alpha_s)$ and combining them with vertex and annihilation contributions to predict ${\cal B}(B \to K^*\gamma)$ and ${\cal B}(B \to \rho\gamma)$ at leading power in $\Lambda_{\rm QCD}/M_B$. It finds large corrections and a notable mismatch between LEET-form-factor predictions and LC-QCD/Lattice results for $T_1^{(K^*)}(0)$, yielding $\xi_\perp^{(K^*)}(0) \approx 0.25$ and $T_1^{(K^*)}(0) \approx 0.27$, while ratios such as ${\cal B}(B \to \rho\gamma)/{\cal B}(B \to K^*\gamma)$ help constrain $|V_{td}/V_{ts}|$ and the CKM angle $\alpha$ with reduced hadronic uncertainties. The analysis also shows that hard spectator effects reduce CP asymmetries and that isospin-violating and annihilation contributions introduce sizeable parameter sensitivities, underscoring the need for improved nonperturbative inputs and more precise data. Overall, the work tests LEET factorization in exclusive radiative decays and highlights both its predictive power and its current theoretical limitations.

Abstract

We calculate the so-called hard spectator corrections in ${\cal O} (α_s)$ in the leading-twist approximation to the decay widths for $B \to K^{*} γ$ and $B \to ργ$ decays and their charge conjugates, using the Large Energy Effective Theory (LEET) techniques. Combined with the hard vertex and annihilation contributions, they are used to compute the branching ratios for these decays in the next-to-leading order (NLO) in the strong coupling $α_s$ and in leading power in $Λ_{\rm QCD}/M_B$. These corrections are found to be large, leading to the inference that the theoretical branching ratios for the decays $B \to K^* γ$ in the LEET approach can be reconciled with current data only for significantly lower values of the form factors than their estimates in the QCD sum rule and Lattice QCD approaches. However, the form factor related uncertainties mostly cancel in the ratios ${\cal B}(B \to ργ)/{\cal B}(B \to K^* γ)$ and $Δ= (Δ^{+0}+ Δ^{-0})/2$, where $Δ^{\pm 0} = Γ(B^\pm \to ρ^\pm γ)/ [2 Γ(B^0 (\bar B^0)\to ρ^0 γ)] - 1$, and hence their measurements will provide quantitative information on the standard model parameters, in particular the ratio of the CKM matrix elements $| V_{td}/V_{ts}|$ and the inner angle $α$ of the CKM-unitarity triangle. We also calculate direct CP asymmetries for the decays $B^\pm \to ρ^\pm γ$ and $B^0/\bar B^0 \to ρ^0 γ$ and find, in conformity with the observations made in the existing literature, that the hard spectator contributions significantly reduce the asymmetries arising from the vertex corrections.

Figures (18)

• Figure 1: Feynman diagrams contributing to the spectator corrections involving the ${\cal O}_7$ operator in the decay $B \to \rho \gamma$. The curly (dashed) line here and in subsequent figures represents a gluon (photon).
• Figure 2: Feynman diagrams contributing to the spectator corrections involving the ${\cal O}_8$ operator in the decays $B \to V \gamma$. Row $a$: photon is emitted from the flavour-changing quark line; Row $b$: photon radiation off the spectator quark line.
• Figure 3: Feynman diagrams contributing to the spectator corrections in $B \to V \gamma$ decays involving the ${\cal O}_2$ operator. Row $a$: photon emission from the flavour-changing quark line; Row $b$: photon radiation off the spectator quark line.
• Figure 4: Feynman diagrams contributing to the spectator corrections in $B \to V \gamma$ decays involving the ${\cal O}_2$ operator for the case when both the photon and the virtual gluon are emitted from the internal (loop) quark line.
• Figure 5: Feynman diagrams contributing to the spectator corrections in $B \to V \gamma$ decays involving the ${\cal O}_2$ operator for the case when only the photon is emitted from the internal (loop) quark line in the $b s (d) \gamma$ vertex.