Conformal moments of the two-loop coefficient functions in DVCS

We develop a new technique and calculate conformal (Gegenbauer) moments of the two-loop coefficient functions in Deeply Virtual Compton Scattering (DVCS). These results are necessary for the extraction of the generalized parton distributions from the experimental data to the NNLO accuracy within the Mellin-Barnes approach.

💡 Research Summary

The paper addresses a crucial bottleneck in the extraction of Generalized Parton Distributions (GPDs) from Deeply Virtual Compton Scattering (DVCS) data at next‑to‑next‑to‑leading order (NNLO). While the two‑loop coefficient functions (CFs) for DVCS have been derived in the literature, their use in the Mellin‑Barnes representation—required for efficient GPD evolution and global fitting—demands the calculation of conformal (Gegenbauer) moments. These moments are defined as analytic continuations of integrals of the momentum‑fraction‑space CFs against Gegenbauer polynomials. At two loops the CFs are expressed through generalized polylogarithms of high transcendentality, making direct integration extremely cumbersome.

The authors introduce a novel technique based on the SL(2,R) conformal symmetry of the collinear sector. They construct a set of SL(2,R)‑invariant integral operators Hα with simple position‑space kernels h(τ). The eigenfunctions of these operators are precisely the Gegenbauer polynomials G_{3/2}^N(z) (for quarks) and G_{5/2}^N(z) (for gluons). Crucially, for any function f(z) the conformal moment of the transformed function f_H(z)=∫dα dβ h(τ) f(z \barα+ \bar z β) satisfies M_N