Abstract
Applying the dispersion approach we compute perturbative QCD corrections to the power suppressed soft contribution of $B \to \gamma \ell \nu$ at leading twist. QCD factorization for the $B \to \gamma^{\ast}$ form factors is demonstrated explicitly for the hard-collinear transverse polarized photon at one loop, with the aid of the method of regions. While the one-loop hard function is identical to the matching coefficient of the QCD weak current $\bar u \, \gamma_{\mu \perp} \, (1- \gamma_5) \, b$ in soft-collinear effective theory, the jet function from integrating out the hard-collinear fluctuations differs from the corresponding one entering the factorization formula of $B \to \gamma \ell \nu$. Furthermore, we evaluate the sub-leading power contribution to the $B \to \gamma$ form factors from the three-particle $B$-meson distribution amplitudes (DAs) at tree level. The soft contribution to the $B \to \gamma$ form factors from the three-particle $B$-meson DAs is shown to be of the same power compared with the corresponding hard correction, in contrast to the two-particle counterparts. Numerically the next-to-leading-order QCD correction to the soft two-particle contribution in $B \to \gamma$ form factors will induce an approximately $\left (10 \sim 20 \right) \%$ shift to the tree-level contribution at $\lambda_B(\mu_0)=354 \, {\rm MeV}$. Albeit of power suppression parametrically, the soft two-particle correction can decrease the leading power predictions for the $B \to \gamma$ form factors by an amount of $\left (10 \sim 30 \right) \%$ with the same value of $\lambda_B(\mu_0)$. Employing the phenomenological model of the three-particle $B$-meson DAs inspired by a QCD sum rule analysis, the three-particle contribution to the $B \to \gamma$ form factors is predicted to be of ${\cal O} (1 \%)$, at leading order in $\alpha_s$, with the default theory inputs.
Highlights
Whether ξ(Eγ) can be computed straightforwardly in QCD factorization without encountering rapidity divergences
Applying the dispersion approach developed in the context of the pion-photon transition form factor, we computed perturbative QCD corrections to the subleading power soft twoparticle contribution of the B → γ transition form factors, which cannot be addressed directly with the QCD factorization approach due to the breakdown of light-cone operator product expansion (OPE) in the end-point region
We first demonstrated QCD factorization for the generalized B → γ∗ form factors with a hard-collinear photon at leading power in Λ/mb using the diagrammatic factorization approach. Both the hard coefficient and jet function entering the factorization formulae for the B → γ∗ form factors were determined at one loop explicitly based upon the method of regions
Summary
The one-loop correction to the weak vertex diagram displayed in figure 2(a) can be readily computed as. It is straightforward to identify that the leading-power contributions of Tν(μ1),weak arise from the hard, hard-collinear and soft regions of the loop momentum. It is evident that C⊥,weak(n · p) is precisely the same as the hard contribution to the weak vertex diagram for the vacuum-to-Λb-baryon correlation function at one loop [12]. Setting n ·p → 0, our result of J⊥,weak reproduces the hard-collinear contribution to the weak vertex diagram in the B → γ ν decay (see (69) in [2]). One concludes that soft dynamics of the weak vertex diagram in figure 2(a) can be parametrized by the B-meson DAs in the framework of perturbative QCD
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