Dispersion relation for CFT four-point functions

101

For a single free scalar field in d ≥ 2 dimensions, almost all the unitary conformal defects must be ‘trivial’ in the sense that they cannot hold interesting dynamics. The only possible exceptions are monodromy defects in d ≥ 4 and co-dimension three defects in d ≥ 5. As an intermediate result we show that the n-point correlation functions of a conformal theory with a generalized free spectrum must be those of the generalized free theory.

Section: Triviality Of Defects Of Dimension 2 and Highermentioning

confidence: 99%

“…Single-variable and two-variable dispersion relations in CFT were recently studied in[58] and[57] respectively.…”

mentioning

confidence: 99%

Line and surface defects for the free scalar field

Lauria

Liendo

Rees

et al. 2021

101

“…Note added. While this paper was being completed, the work [31] appeared on arxiv who introduced a single-variable dispersion relation that reconstructs correlators from a single-discontinuity. This appears to be quite distinct from the formulas considered here: the input in this case (to our knowledge) is neither sign-definite nor admits a physical interpretation as an absorptive part.…”

Section: Jhep09(2020)009mentioning

confidence: 99%

A conformal dispersion relation: correlations from absorption

Carmi

Caron-Huot

2020

157

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its "absorptive part", defined as a double discontinuity, times a theoryindependent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the "inverted" conformal block with the ordinary conformal block.

“…Once we have this double-discontinuity, we can perform the inversion integral in (2.51) to determine one-loop corrections to the OPE data. These can then be used to determine the full correlator at one-loop [60,61]. It is also useful to understand how this approach maps to the standard loop expansion for Witten diagrams.…”

Section: Brief Articlementioning

confidence: 99%

AdS/CFT unitarity at higher loops: high-energy string scattering

Meltzer

2020

What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large N , 4d CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in 1/N for large, but finite, coupling. In the language of Regge theory, we show Pomeron exchange naturally exponentiates in the 1/N expansion. To do this, we solve the bootstrap equations at tree-level and then use the Lorentzian inversion formula to find the one-loop correlator in the Regge limit. This is a unitarity method for AdS/CFT which can be repeated iteratively to make all orders statements. We also explain under what conditions the tree-level result exponentiates in the 1/N expansion at arbitrary coupling. Finally, we comment on further inelastic effects and show they give subleading contributions at large coupling. As a consistency check, we recover results from bulk Einstein gravity in the limit where all higher-spin particles decouple.