Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals

Abstract

We investigate the structure of the constraints on three-point correlation functions emerging when conformal invariance is imposed in momentum space and in arbitrary space-time dimensions, presenting a derivation of their solutions for arbitrary scalar operators. We show that the differential equations generated by the requirement of symmetry under special conformal transformations coincide with those satisfied by generalized hypergeometric functions (Appell's functions). Combined with the position space expression of this correlator, whose Fourier transform is given by a family of generalized Feynman (master) integrals, the method allows to derive the expression of such integrals in a completely independent way, bypassing the use of Mellin-Barnes techniques, which have been used in the past. The application of the special conformal constraints generates a new recursion relation for this family of integrals.

Recurrence relations for finite-temperature correlators via AdS2/CFT1

“…Indeed, in spite of its simplicity in coordinate space, three-point functions in momentum space are known to be very complicated. (For recent studies on conformal constraints in momentum-space three-point functions, see [3,4].) Since retarded correlators in momentum space, for example, are directly related to physical observables such as spectral density and/or conductivities within the linear response approximation, it would be desirable to understand how directly conformal symmetry restricts the possible forms of momentum-space correlators.…”

de Sitter symmetries and inflationary correlators in parity violating scalar-vector models

“…In an Euclidean space, the conformal group is defined as the group of transformations that leave invariant the metric up to a factor, this is, 5) where Ω(x) is an arbitrary function of the coordinates. It is easy to see that the transformations (2.2)-(2.4) act as conformal transformations on R 3 for |τ | | x| with the euclidean metric g ij = δ ij .…”

“…5 Reference [38] study in detail the features of this model and describe analytically its asymptotic behavior, here, we recall some basics about the solutions of this system. The equation (3.14) can be solved analytically in terms of irregular G and irregular F Coulomb functions:w…”

“…According with (3.18) and (3.19), the σ = +1 mode is enhanced by the exponential factor e πξ 2 [38,63], while the σ = −1 mode is suppressed by the factor e − πξ 2 . This 5 The choice of the sign of γ determines which one of the two helicities is the growing one. If γ > 0 the increasing mode is denoted with a subscript "+" and, if γ < 0 the growing will be denoted with a "−".…”

“…During inflation, the relevant symmetry group of the space time is the de Sitter group, so, we expect that this group gives us hints about the structure of the correlation functions of the different fields present during the inflationary epoch [1][2][3][4][5]. Over the years, several studies had explored the roll of the de Sitter group during inflation [1][2][3][4][5][6][7][8][9][10]. A crucial point in this study is the fact that, for super horizon scales, the de Sitter symmetry group approaches the behavior of the conformal group in one dimension below, this is, the de Sitter group acting on four dimensional de Sitter space, acts as the conformal group in the euclidean three dimensional space.…”