Renormalization of the next-to-leading-power $\gamma\gamma \to h $ and $gg\to h$ soft quark functions

Abstract

We calculate directly in position space the one-loop renormalization kernels of the soft operators $O_\gamma$ and $O_g$ that appear in the soft-quark contributions to, respectively, the subleading-power $\gamma\gamma\to h$ and $gg\to h$ form factors mediated by the $b$-quark. We present an IR/rapidity divergence-free definition for $O_g$ and demonstrate that with a correspondent definition of the collinear function, a consistent factorization theorem is recovered. Using conformal symmetry techniques, we establish a relation between the evolution kernels of the leading-twist heavy-light light-ray operator, whose matrix element defines the $B$-meson light-cone distribution amplitude (LCDA), and $O_\gamma$ to all orders in perturbation theory. Application of this relation allows us to bootstrap the kernel of $O_\gamma$ to the two-loop level. We construct an ansatz for the kernel of $O_g$ at higher orders. We test this ansatz against the consistency requirement at two-loop and find they differ only by a particular constant.