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≤1). This formula allows us to present rigorous calculation of the B-meson DA at the factorization scale ∼√(mbΛQCD) for t less than ∼1 GeV-1, using the recently obtained operator product expansion of the DA as the input at μ∼1 GeV. We also derive the master formula, which reexpresses the integrals of the DA at μ∼√(mbΛQCD) for the factorization formula by the compact integrals of the DA at μ∼1 GeV.