Renormalization of the next-to-leading-power γγ → h and gg → h soft quark functions

We calculate directly in position space the one-loop renormalization kernels of the soft operators O_γ and O_g that appear in the soft-quark contributions to, respectively, the subleading-power γγ → h and gg → 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_γ to all orders in perturbation theory. Application of this relation allows us to bootstrap the kernel of O_γ 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.