de Sitter breaking through infrared divergences

Abstract

Just because the propagator of some field obeys a de Sitter invariant equation does not mean it possesses a de Sitter invariant solution. The classic example is the propagator of a massless, minimally coupled scalar. We show that the same thing happens for massive scalars with M 2 S < 0, and for massive transverse vectors with M 2 V ≤ −2(D − 1)H 2 , where D is the dimension of spacetime and H is the Hubble parameter. Although all masses in these ranges give infrared divergent mode sums, using dimensional regularization (or any other analytic continuation technique) to define the mode sums leads to the incorrect conclusion that de Sitter invariant solutions exist except at discrete values of the masses.

The graviton propagator in de Donder gauge on de Sitter background

“…These structure functions obey completely de Sitter invariant equations, but they fail to possess de Sitter invariant solutions on account of infrared divergences. The procedure is so general that we implement it as well for a vector particle of general mass M V , and check that it agrees with the known de Sitter invariant solutions [3] for M 2 V > −2(D − 1)H 2 in the transverse sector and M 2 V > 0 in the longitudinal sector. When de Sitter breaking must occur we have chosen to give explicit solutions which preserve the symmetries of homogeneity and isotropy that are relevant to cosmology.…”

“…These choices had been dismissed as unphysical, "singular gauges" which must simply be avoided [13]. However, we can now see that they are precisely the cases for which the order of the omnipresent infrared divergence in the formal, de Sitter invariant mode sum changes from power law to logarithmic [3]. The power law infrared divergences of other choices were automatically -but incorrectly -subtracted by analytic regularization techniques to produce solutions of the propagator equations that are not true propagators.…”

“…The graviton propagator equation will then be used to infer invariant equations for these scalar structure functions. The point of this section is to review and systematize previous work [1,3] about how to solve such equations. We begin giving a general scalar propagator equation and explaining why infrared divergences for M 2 S ≤ 0 preclude a de Sitter invariant solution.…”

“…Hence the right hand side of the equation is symmetric under interchange of x µ and z µ and can in fact be guessed [3],…”

“…• One can impose covariant gauges in which the field obeys exact conditions, but previous solutions employed analytic continuation techniques that incorrectly subtract off power law infrared divergences [3].…”