Nonperturbative Mellin amplitudes: existence, properties, applications

Abstract

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.

Landau diagrams in AdS and S-matrices from conformal correlators

“…In agreement with the general discussion earlier in this section, but in marked contrast with the flat-space equations, we managed to find solutions to the AdS Landau equations everywhere we looked in the complex s plane. 19 For each value we computed the on-shell action and checked the sign of Re(f t ). The region where Re(f t ) > 0 is the problematic region because the Landau diagram dominates over the correct saddle point.…”

“…On the other hand, the Mellin approach involves integral transforms of some correlators which make it hard to discuss their analytic properties at the nonperturbative level. In fact the existence of the Mellin-space representation of the correlators was established only quite recently in [19] and yet, its analyticity is not fully understood. Another, more technical issue is that there are singularities of the flat-space amplitudes which are not well-understood in the Mellin approach.…”

“…Although the Mellin space representation works very well for Witten diagrams[45,46], it is more general and according to recent work[19] can also be used for certain non-perturbative CFT correlation functions. It would be extremely interesting to further explore the applicability of Mellin space because of the immediate implication on our conjectures as outlined below.…”

A scattering amplitude in Conformal Field Theory

“…where the Mellin variables satisfy the constraints 2) and the integration contours (over the two independent Mellin variables) run parallel to the imaginary axis. More precisely, as explained in detail in [44], the integration contours are usually deformed to pass on the appropriate side of poles of the integrand. 13 In appendix E, we show that Fourier transforming and taking the limit p 2 j → 0 for j = 1, 2, 3 leads to the following Mellin representation of the form factor (1.2),…”

The unique Polyakov blocks

“…A better dispersion relation which takes into account the behaviour of the Mellin amplitude at infinity [5] can be obtained considering subtractions which improve the behaviour of the Mellin amplitude at infinity. A clever subtraction can be performed recalling that the Mellin amplitude has to have a zero at the location of the leading Regge trajectory [17,18]. In this case indeed it is natural to write down the following subtracted dispersion relation:…”