Evolution equation for theB-meson distribution amplitude in the heavy-quark effective theory in coordinate space

Abstract

The $B$-meson distribution amplitude (DA) is defined as the matrix element of a quark-antiquark bilocal light-cone operator in the heavy-quark effective theory, corresponding to a long-distance component in the factorization formula for exclusive $B$-meson decays. The evolution equation for the $B$-meson DA is governed by the cusp anomalous dimension as well as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-type anomalous dimension, and these anomalous dimensions give the ``quasilocal'' kernel in the coordinate-space representation. We show that this evolution equation can be solved analytically in the coordinate space, accomplishing the relevant Sudakov resummation at the next-to-leading logarithmic accuracy. The quasilocal nature leads to a quite simple form of our solution which determines the $B$-meson DA with a quark-antiquark light-cone separation $t$ in terms of the DA at a lower renormalization scale $\ensuremath{\mu}$ with smaller interquark separations $zt$ ($z\ensuremath{\le}1$). This formula allows us to present rigorous calculation of the $B$-meson DA at the factorization scale $\ensuremath{\sim}\sqrt{{m}_{b}{\ensuremath{\Lambda}}_{\mathrm{QCD}}}$ for $t$ less than $\ensuremath{\sim}1\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}1}$, using the recently obtained operator product expansion of the DA as the input at $\ensuremath{\mu}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$. We also derive the master formula, which reexpresses the integrals of the DA at $\ensuremath{\mu}\ensuremath{\sim}\sqrt{{m}_{b}{\ensuremath{\Lambda}}_{\mathrm{QCD}}}$ for the factorization formula by the compact integrals of the DA at $\ensuremath{\mu}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$.