Abstract
We derive the two-loop evolution equation of the B-meson light-cone distribution amplitude which is the last missing element for the next-to-next-to-leading logarithmic resummation of QCD corrections to B decays in QCD factorization. We argue that the evolution kernel to all orders in perturbation theory can be written as a logarithm of the generator of special conformal transformations times the cusp anomalous dimension, up to a scheme-dependent overall constant. Up to this constant term, the evolution kernel to a given order in perturbation theory can be obtained from the calculation of special conformal anomaly at one order less.
Highlights
The B-meson light-cone distribution amplitude (LCDA) [1] is the crucial nonperturbative quantity in the description of charmless hadronic B-decays and studies of direct CP violation in the framework of QCD factorization [2,3,4] and the “perturbative QCD” factorization [5,6,7]
We argue that the evolution kernel to all orders in perturbation theory can be written as a logarithm of the generator of special conformal transformations times the cusp anomalous dimension, up to a scheme-dependent overall constant
In this work we argue that the structure found in Ref. [17] holds to all orders in perturbation theory: The evolution kernel HðaÞ, a 1⁄4 αs=ð4πÞ can be written as a logarithm of the generator of special conformal transformation KðaÞ times the cusp anomalous dimension ΓcuspðaÞ, up to an overall additive constant
Summary
We remind that the QCD Lagrangian is conformally invariant at the classical level, and as a consequence one-loop evolution kernels for composite operators built from light quarks commute with the generators of conformal transformations. It is, possible to write these kernels as functions of the quadratic Casimir operator of the collinear subgroup [19]. For the heavy-light operators considered here the conformal symmetry is lost because the effective heavyquark field hv is essentially a nonlocal object—it can be replaced by the Wilson line going from zero to infinity along the velocity vector vμ [20]—and it does not transform covariantly under the Poincaregroup. The coefficient in front of ln iμz is called cusp anomalous dimension [21] and is known at NNLO [22], ΓcuspðaÞ 1⁄4 aΓðc1uÞsp þ a2Γðc2uÞsp þ Á Á Á
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