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

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 $μ$ with smaller interquark separations $zt$ ($z \leq 1$). This formula allows us to present rigorous calculation of the B-meson DA at the factorization scale $\sim \sqrt{m_b Λ_{\rm QCD}}$ for $t$ less than $\sim 1$ GeV^{-1}, using the recently obtained operator product expansion of the DA as the input at $μ\sim 1$ GeV. We also derive the master formula, which reexpresses the integrals of the DA at $μ\sim \sqrt{m_b Λ_{\rm QCD}}$ for the factorization formula by the compact integrals of the DA at $μ\sim 1$ GeV.