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Effective Conformal Theory and the Flat-Space Limit of AdS

A. Liam Fitzpatrick, Emanuel Katz, David Poland, David Simmons-Duffin

TL;DR

This work develops an effective conformal theory (ECT) framework to describe the low-lying dilatation spectrum of a CFT with a hierarchy of operator dimensions, using global AdS perturbations to preserve conformal invariance. A perturbative unitarity bound |γ(n,l)|<4 emerges, constraining the growth of anomalous dimensions and tying the breakdown of perturbation theory to the need for heavy states or integrating out them in the CFT. By analyzing heavy scalar exchange and EFT-like bulk interactions, the authors show how non-renormalizable AdS terms induce growth in γ(n,l) that is tamed by integrating in a heavy operator, producing resonance-like behavior. In the large-n limit, the anomalous dimensions reproduce the flat-space S-matrix, providing a direct link between d-dimensional CFT data and (d+1)-dimensional flat-space scattering, and illustrating how flat-space locality can emerge from conformal dynamics.

Abstract

We develop the idea of an effective conformal theory describing the low-lying spectrum of the dilatation operator in a CFT. Such an effective theory is useful when the spectrum contains a hierarchy in the dimension of operators, and a small parameter whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that there is a regime where the dilatation operator is modified perturbatively. Global AdS is the natural framework for perturbations of the dilatation operator respecting conformal invariance, much as Minkowski space naturally describes Lorentz invariant perturbations of the Hamiltonian. Assuming that the lowest-dimension single-trace operator is a scalar, O, we consider the anomalous dimensions, gamma(n,l), of the double-trace operators of the form O (del^2)^n (del)^l O. Purely from the CFT we find that perturbative unitarity places a bound on these dimensions of |gamma(n,l)|<4. Non-renormalizable AdS interactions lead to violations of the bound at large values of n. We also consider the case that these interactions are generated by integrating out a heavy scalar field in AdS. We show that the presence of the heavy field "unitarizes" the growth in the anomalous dimensions, and leads to a resonance-like behavior in gamma(n,l) when n is close to the dimension of the CFT operator dual to the heavy field. Finally, we demonstrate that bulk flat-space S-matrix elements can be extracted from the large n behavior of the anomalous dimensions. This leads to a direct connection between the spectrum of anomalous dimensions in d-dimensional CFTs and flat-space S-matrix elements in d+1 dimensions. We comment on the emergence of flat-space locality from the CFT perspective.

Effective Conformal Theory and the Flat-Space Limit of AdS

TL;DR

This work develops an effective conformal theory (ECT) framework to describe the low-lying dilatation spectrum of a CFT with a hierarchy of operator dimensions, using global AdS perturbations to preserve conformal invariance. A perturbative unitarity bound |γ(n,l)|<4 emerges, constraining the growth of anomalous dimensions and tying the breakdown of perturbation theory to the need for heavy states or integrating out them in the CFT. By analyzing heavy scalar exchange and EFT-like bulk interactions, the authors show how non-renormalizable AdS terms induce growth in γ(n,l) that is tamed by integrating in a heavy operator, producing resonance-like behavior. In the large-n limit, the anomalous dimensions reproduce the flat-space S-matrix, providing a direct link between d-dimensional CFT data and (d+1)-dimensional flat-space scattering, and illustrating how flat-space locality can emerge from conformal dynamics.

Abstract

We develop the idea of an effective conformal theory describing the low-lying spectrum of the dilatation operator in a CFT. Such an effective theory is useful when the spectrum contains a hierarchy in the dimension of operators, and a small parameter whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that there is a regime where the dilatation operator is modified perturbatively. Global AdS is the natural framework for perturbations of the dilatation operator respecting conformal invariance, much as Minkowski space naturally describes Lorentz invariant perturbations of the Hamiltonian. Assuming that the lowest-dimension single-trace operator is a scalar, O, we consider the anomalous dimensions, gamma(n,l), of the double-trace operators of the form O (del^2)^n (del)^l O. Purely from the CFT we find that perturbative unitarity places a bound on these dimensions of |gamma(n,l)|<4. Non-renormalizable AdS interactions lead to violations of the bound at large values of n. We also consider the case that these interactions are generated by integrating out a heavy scalar field in AdS. We show that the presence of the heavy field "unitarizes" the growth in the anomalous dimensions, and leads to a resonance-like behavior in gamma(n,l) when n is close to the dimension of the CFT operator dual to the heavy field. Finally, we demonstrate that bulk flat-space S-matrix elements can be extracted from the large n behavior of the anomalous dimensions. This leads to a direct connection between the spectrum of anomalous dimensions in d-dimensional CFTs and flat-space S-matrix elements in d+1 dimensions. We comment on the emergence of flat-space locality from the CFT perspective.

Paper Structure

This paper contains 25 sections, 115 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. Algebra Constraints
  4. Unitarity Limit
  5. Review of AdS Global Coordinate Wavefunctions
  6. Locality and Microcausality in AdS
  7. Dilatation Matrix Elements in Low-energy ECT
  8. Primary Wavefunctions
  9. Normalization of Primary Two-particle Wavefunctions
  10. Example Calculation of $V_{nm}$
  11. Dimensional Analysis with $n$
  12. Heavy field Exchange
  13. S-channel Scalar Exchange
  14. Matching Between Low and High Energies
  15. T- and U-channels
  16. ...and 10 more sections

Figures (2)

  • Figure 1: One-particle (left) and five-particle (right) intermediate state diagrams contributing to the "s-channel".
  • Figure 2: Plotted are the contributions to $|\gamma(n,0)|$ from s-channel scalar exchange (solid line) and s-channel contractions of the low-energy $\phi^4$ interaction (dashed line) in AdS$_5$ with $\Delta_{\chi}=100.1$ and $\Delta=2.2$.