Factorization and dispersion relations for radiative leptonic $B$ decay

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 ${\lambda }_B({\mu }_0)=354\mathrm{MeV}$. 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 $\mathcal{O}(1\%)$, at leading order in ${\alpha }_s$, with the default theory inputs.

32 pages, 9 figures

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