Loops in Anti de Sitter space

Abstract

We discuss general one and two-loop banana diagrams and one-loop diagrams with external lines with arbitrary masses on the anti de Sitter spacetime by using methods of AdS quantum field theory in the dimensional regularization approach. The banana diagrams explicitly computed in this paper are indeed the necessary ingredients for the evaluation of the two-loop effective potential of the Standard Model and can be used to extend the flat space results in in presence of a negative cosmological constant. In the one-loop case we also compute the effective potential for an $O(N)$ model in $d=4$ dimension as an explicit function of the cosmological constant $\hat{I}›$, both exactly and perturbatively up to order $\hat{I}›$. In the two-loop case we show the explicit calculation is possible thanks to a remarkable discrete Källén-Lehmann formula which we found and proved sometimes ago and whose domain of applicability we extend in the present paper.

New introduction. Extended appendix A and new appendix B extending the Kallen-Lehmann formula generalizing our original formula and other subsequent results. Extended Section 5 including a general formula for the bubble in any background geometry. New section 6 with some comments about the relation among bulk diagrams and Witten diagrams. New section 8 including the tadpole and the sunset