Analytical evaluation of AdS4 Witten diagrams as flat space multi-loop Feynman integrals

Abstract

We describe a systematic approach for the evaluation of Witten diagrams for multi-loop scattering amplitudes of a conformally coupled scalar ϕ4-theory in Euclidean AdS4, by recasting the Witten diagrams as flat space Feynman integrals. We derive closed form expressions for the anomalous dimensions for all double-trace operators up to the second order in the coupling constant. We explain the relation between the flat space unitarity methods and the discontinuities of the short distance expansion on the boundary of Witten diagrams.

Analytical evaluation of cosmological correlation functions

“…We will then be able to use the formalism of [30] for evaluating the Witten diagrams. From now, we specialize to the case of d = 3.…”

“…Using the normalization of the fields and the coupling constant introduced in (2.46) and the conformal mappings as described in [30], we can write a generic EAdS four-point Witten diagram with equal external dimensions ∆ as…”

“…where W ∆∆∆∆,D L is the corresponding Witten diagram in EAdS with standard normalization of the propagator as defined in [30]. The four-point function with mixed boundary conditions will be given by acting with the differential operator, defined in section 3.2 of [30], onto the corresponding legs of the ∆ = 1 Witten diagrams. All calculations will be done in the loop dependent dimensional regularisation scheme introduced and described in section 3.1.2 of [30].…”

“…The four-point function with mixed boundary conditions will be given by acting with the differential operator, defined in section 3.2 of [30], onto the corresponding legs of the ∆ = 1 Witten diagrams. All calculations will be done in the loop dependent dimensional regularisation scheme introduced and described in section 3.1.2 of [30].…”

“…We are going to take advantage of progress made in [26][27][28] to express the cosmological correlation functions in the Schwinger-Keldysh formalism as a sum over euclidean AdS (EAdS) Witten diagrams which was expressed in [1] as an auxiliary EAdS action. Here we calculate the four-point functions up to one-loop order by direct integration in position space, applying the formalism developed in [30] to evaluate EAdS Witten diagrams. Interestingly, the Witten diagrams up to this order do not contain any elliptic integrals, in contrast to EAdS, and therefore can all be expressed in terms of single-valued multiple polylogarithms.…”