H → γγ, gauge invariance, and the hierarchy problem

Abstract

The calculation of [Formula: see text] displays interesting behavior which depends on the regulator used in the integration over loop momenta. If calculated using a gauge-invariant regulator, such as dimensional regularization, the calculation yields a unique, finite, gauge-invariant result. If four-dimensional symmetric regulation is used without finite subtractions, additional pieces occur which spoil QED gauge invariance. In both cases, a finite result is obtained, but the particular finite result depends on the regulator utilized in the calculation. While gauge-invariant regulators such as dimensional regularization are normally used, four-dimensional symmetric integration is also physically motivated. Also, the gauge-invariance-violating terms that arise using four-dimensional symmetric integration are of the same form for the fermionic, scalar, and the SM [Formula: see text] loop calculated in renormalizable gauge. This presents an interesting possibility. Inspired by anomaly cancellation, we ask if it is possible that these gauge-invariance-violating terms may cancel in certain models when contributions from all diagrams are included. Here, we calculate the regulator-dependent contributions to [Formula: see text] arising from generic fermion and scalar loops, as well as the Standard Model [Formula: see text] loop contribution, which we evaluate in renormalizable gauge for general [Formula: see text]. We find that a cancellation between such terms is possible, and derive the cancellation condition. Additionally, we find that this cancellation condition ensures QED gauge invariance without finite subtractions for any regulator used, not just for four-dimensional symmetric integration. We additionally relate the regulator-dependent terms in [Formula: see text] to the behavior of quadratically-divergent Higgs tadpole diagrams under shifts of internal loop momentum. Thus, the cancellation condition for the gauge-invariance-violating terms in [Formula: see text] implies a relation between the quadratic divergences in Higgs tadpole diagrams; this has consequences for hypothesized solutions to the hierarchy problem. Lastly, we find that the MSSM obeys our cancellation condition.