Abstract
We discuss the generalization of the leading-twist light-cone distribution amplitude for light mesons including QED effects. This generalization was introduced to describe virtual collinear photon exchanges at the strong-interaction scale ΛQCD in the factorization of QED effects in non-leptonic B-meson decays. In this paper we study the renormalization group evolution of this non-perturbative function. For charged mesons, in particular, this exhibits qualitative differences with respect to the well-known scale evolution in QCD only, especially regarding the endpoint-behaviour. We analytically solve the evolution equation to first order in the electromagnetic coupling αem, which resums large logarithms in QCD on top of a fixed-order expansion in αem. We further provide numerical estimates for QED corrections to Gegenbauer coefficients as well as inverse moments relevant to (QED-generalized) factorization theorems for hard exclusive processes.
Highlights
This generalized factorization theorem expresses the matrix elements of weak effective operators Qi in the heavy-quark limit as convolutions of hard-scattering kernels Ti with light-cone distribution amplitudes (LCDAs) Φ of heavy and light mesons
We do not aim at an analytic solution of the full QCD×QED renormalization group equation (RGE), we provide numerical solutions and an analytic expression at O(αem) that resums large logarithms in QCD on top of a fixed-order expansion in the electromagnetic coupling
To qualitatively visualize the asymmetry of the QCD×QED LCDA generated by evolution, we show in figure 3 a numerical solution of the RGE obtained from solving the integro-differential evolution equation (3.6) by discretization for fictitious values of the electromagnetic coupling constant
Summary
In QCD, light-cone distribution amplitudes for light mesons M are well-established nonperturbative but universal objects, which appear in the theoretical description of hard exclusive processes at large energies. We first notice that gauge invariance dictates the operator to extend on the infinite lightray, instead of being localized on a finite interval [tn+, 0] This can be seen by combining the Wilson lines associated with the quark fields of different electric charge to. As discussed in detail in [2, 3], due to the non-decoupling of soft photons from electrically charged mesons, the operator (2.4) itself is no longer renormalizable in the sense that its anomalous dimension is IR divergent This is due to a non-trivial overlap between the soft and collinear sector in QED-generalized collinear factorization theorems. We defined the subtraction factors Rc and Rcin (2.6) by implicitly assuming one additional anti-collinear direction nμ+, ΦM (u; μ) remains a universal object relevant to various twoand multi-body exclusive processes This can be seen by performing the soft rearrangement for a generic n-jet SCET operator: Oeff. We do not compute this mixing in this paper, because we are mainly interested in electrically charged mesons, where the non-decoupling of soft photons is important
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