Higgs boson decay to two photons and the dispersion relations

TL;DR

This paper investigates the computation of the Higgs to two photons decay amplitude using dispersion relations, highlighting the necessity of subtractions and the role of regulators or unphysical degrees of freedom in obtaining consistent results.

Contribution

It clarifies why a subtraction is needed in dispersion relation calculations of the Higgs decay amplitude and explores methods to recover the correct constant term.

Findings

Unsubtracted dispersion integrals differ by a constant from diagrammatic results.

Introducing a subtraction or regulator aligns the dispersive result with diagrammatic calculations.

Unregulated computations in the unitary gauge are insufficient for correct amplitude determination.

Abstract

We discuss the computation of the Higgs boson decay amplitude to two photons through the W-loop using dispersion relations. The imaginary part of the form factor F_W(s) that parametrizes this decay is unambiguous in four dimensions. When it is used to calculate the unsubtracted dispersion integral, the finite result for the form factor F_W(s) is obtained. However, the F_W(s) obtained in this way differs by a constant term from the result of a diagrammatic computation, based on dimensional regularization. It is easy to accommodate the missing constant by writing a once-subtracted dispersion relation for F_W(s) but it is unclear why the subtraction needs to be done. The goal of this paper is to investigate this question in detail. We show that the correct constant can be recovered within a dispersive approach in a number of ways that, however, either require an introduction of an ultraviolet regulator or unphysical degrees of freedom; unregulated and unsubtracted computations in the unitary gauge are insufficient, in spite of the fact that such computations give a finite result.