Higgs boson decay to two photons and dispersion relations

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 FW(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 FW(s) is obtained. However, the FW(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 FW(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.