Region analysis of Hto γ γ via a bottom quark loop

The $H\to \gamma\gamma$ decay is an ideal process to study the structure of next-to-leading power logarithms induced by quarks due to its simple initial and final states. We perform a region analysis of this process up to two-loop level to inspect the origins of the logarithms. To deal with the endpoint singularities that are prevalent for the next-to-leading power logarithms, we have adopted two different kinds of regulators to exhibit the advantages and disadvantages of each regulator. In the analytic regulator we have chosen, the power of the propagator is changed by $\eta$. And the endpoint singularities are regulated in the form of $1/\eta$. These poles cancel between the collinear and anti-collinear sectors since there is no soft mode in such a regulator. In the $\Delta$ regulator, the soft sector is important. The leading and next-to-leading logarithms can be inferred from only this sector. Moreover, the symmetry between the collinear and anti-collinear sectors is preserved. After imposing a cut on the bottom quark transverse momentum, the leading order result is finite in each sector. We also discuss the next-to-next-to-leading power contributions and find that the potential factorization formulae involve two-dimensional endpoint singularities. Our region analysis could help to develop sophisticated factorization and resummation schemes beyond leading power.