Verify beta-gauge independence of the de Sitter scalar self-mass

Verify that the gauge-independent one-loop self-mass of a massless, minimally coupled scalar on de Sitter space remains independent under variations of the gauge-fixing parameter beta in the two-parameter de Sitter covariant gauge family.

Background

The paper reports that the additional external-leg diagrams and wavefunction corrections suffice to remove gauge dependence associated with variations of the gauge parameter alpha. However, the corresponding check for the second gauge parameter beta has not been carried out.

The authors expect that the beta check will not require additional diagrams, but they leave the verification unresolved because it also tests the de Sitter generalizations of the Donoghue identities used to reduce three-point and four-point amplitudes to two-point contributions.