Renormalization of the radiative jet function

TL;DR

This paper develops a method to compute the renormalization of the subtracted radiative jet function in B and Higgs decays, revealing nonlocal renormalization effects due to soft contribution subtractions.

Contribution

It introduces a formalism for the nonlocal renormalization of the subtracted radiative jet function using zero-bin subtractions in soft-collinear effective theory.

Findings

Derived the order-$_s$ evolution kernel for the subtracted radiative jet function.

Confirmed the nonlocal renormalization structure through explicit calculation.

Aligned the results with previous factorization theorem in B decay processes.

Abstract

We show how to compute directly the renormalization/evolution of the radiative jet function that appears in the factorization theorems for $B\to \gamma\ell\nu$ and $H\to \gamma\gamma$ through a $b$-quark loop. We point out that, in order to avoid double counting of soft contributions, one should use in the factorization theorems a subtracted radiative jet function, from which soft contributions have been removed. The soft-contribution subtractions are zero-bin subtractions in the terminology of soft-collinear effective theory. We show that they can be factored from the radiative jet function and that the resulting soft-subtraction function gives rise to a nonlocal renormalization of the subtracted radiative jet function. This is a novel instance in which zero-bin subtractions lead to a nonlocality in the renormalization of a subtracted quantity that is not present in the renormalization of the unsubtracted quantity. We demonstrate the use of our formalism by computing the order-$\alpha_s$ evolution kernel for the subtracted radiative jet function. Our result is in agreement with the result that had been inferred previously by making use of the factorization theorem for $B\to \gamma\ell\nu$, but that had been ascribed to the unsubtracted radiative jet function.