Abstract
Using the approach based on conformal symmetry we calculate the three-loop (NNLO) contribution to the evolution equation for flavor-nonsinglet leading twist operators in the overline{mathrm{MS}} scheme. The explicit expression for the three-loop kernel is derived for the corresponding light-ray operator in coordinate space. The expansion in local operators is performed and explicit results are given for the matrix of the anomalous dimensions for the operators up to seven covariant derivatives. The results are directly applicable to the renormalization of the pion light-cone distribution amplitude and flavor-nonsinglet generalized parton distributions.
Highlights
Local operators one has to deal with a triangular mixing matrix where the diagonal entries are the anomalous dimensions, the same as in deep-inelastic scattering, but the off-diagonal elements require a separate calculation
The NNLO analysis of parton distributions and fragmentation functions is becoming the standard in this field [4], so that the NNLO evolution equations for off-forward distributions are appearing on the agenda
Using the two-loop result for the conformal anomaly obtained in ref. [16] we have completed here the calculation of the three-loop evolution kernel for the flavor-nonsinglet leading-twist operators in off-forward kinematics
Summary
Where the Wilson line is implied between the quark fields on the light-cone, is defined as the generating function for renormalized local operators:. The eigenvalues γN (a) correspond to moments of the evolution kernel in the representation (2.9), γN = dα dβ (1 − α − β)N h(α, β) = aγN(1) + a2γN(2) + a3γN(3) + They define the anomalous dimensions of leading-twist local operators in eq (2.2) where N = m + k is the total number of covariant derivatives acting either on the quark or the antiquark field. The second equation, eq (2.23b), is, technically, a first-order inhomogeneous differential equation on the NLO evolution kernel H(2) To solve this equation one needs to find a particular solution with the given inhomogeneity (the expression on the r.h.s.), and add a solution of the homogeneous equation [S+(0), H(2)] such that the sum reproduces the known NLO anomalous dimensions. The basic idea is to simplify the structure as much as possible by separating parts of the three-loop kernel that can be written as a product of simpler kernels
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