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The role of leading twist operators in the Regge and Lorentzian OPE limits

Miguel S. Costa, James Drummond, Vasco Goncalves, Joao Penedones

TL;DR

This work analyzes the weak-coupling Regge and Lorentzian OPE limits of the stress-tensor four-point function in $\mathcal{N}=4$ SYM, showing that both limits are governed by leading-twist operators. By leveraging the known three-loop correlator and the pomeron residue at next-to-leading order, the authors derive leading-log behaviors at arbitrary loop order and the next-to-leading logs up to four loops, establishing consistency with conformal Regge theory. They relate Regge data to OPE coefficients via an all-loop Regge-OPE dictionary, enabling predictions for the behavior of OPE coefficients near $J=1$ and providing nontrivial cross-checks of the approach through maximal transcendentality. The results constrain the higher-loop structure of the four-point function and offer inputs for a perturbative bootstrap program in planar $\mathcal{N}=4$ SYM.

Abstract

We study two kinematical limits, the Regge limit and the Lorentzian OPE limit, of the four-point function of the stress-tensor multiplet in Super Yang-Mills at weak coupling. We explain how both kinematical limits are controlled by the leading twist operators. We use the known expression of the four-point function up to three loops, to extract the pomeron residue at next-to-leading order. Using this data and the known form of pomeron spin up to next-to-leading order, we predict the behaviour of the four-point function in the Regge limit at higher loops. Specifically, we determine the leading log behaviour at any loop order and the next-to-leading log at four loops. Finally, we check the consistency of our results with conformal Regge theory. This leads us to predict the behaviour around $J=1$ of the OPE coefficient of the spin $J$ leading twist operator in the OPE of two chiral primary operators.

The role of leading twist operators in the Regge and Lorentzian OPE limits

TL;DR

This work analyzes the weak-coupling Regge and Lorentzian OPE limits of the stress-tensor four-point function in SYM, showing that both limits are governed by leading-twist operators. By leveraging the known three-loop correlator and the pomeron residue at next-to-leading order, the authors derive leading-log behaviors at arbitrary loop order and the next-to-leading logs up to four loops, establishing consistency with conformal Regge theory. They relate Regge data to OPE coefficients via an all-loop Regge-OPE dictionary, enabling predictions for the behavior of OPE coefficients near and providing nontrivial cross-checks of the approach through maximal transcendentality. The results constrain the higher-loop structure of the four-point function and offer inputs for a perturbative bootstrap program in planar SYM.

Abstract

We study two kinematical limits, the Regge limit and the Lorentzian OPE limit, of the four-point function of the stress-tensor multiplet in Super Yang-Mills at weak coupling. We explain how both kinematical limits are controlled by the leading twist operators. We use the known expression of the four-point function up to three loops, to extract the pomeron residue at next-to-leading order. Using this data and the known form of pomeron spin up to next-to-leading order, we predict the behaviour of the four-point function in the Regge limit at higher loops. Specifically, we determine the leading log behaviour at any loop order and the next-to-leading log at four loops. Finally, we check the consistency of our results with conformal Regge theory. This leads us to predict the behaviour around of the OPE coefficient of the spin leading twist operator in the OPE of two chiral primary operators.

Paper Structure

This paper contains 23 sections, 129 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Four-point function of stress-energy multiplets
  3. Weak coupling expansion
  4. Conformal block expansion
  5. Lorentzian OPE limit
  6. Weak coupling expansion
  7. Anomalous dimensions and OPE coefficients
  8. Higher loop prediction for leading logs
  9. Regge limit
  10. Weak coupling expansion
  11. Pomeron spin and residue
  12. Higher-loop prediction for leading logs
  13. Next-to-leading log at four loops
  14. Conformal Regge Theory
  15. Prediction for OPE coefficients
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Shape of the leading Regge trajectory with vacuum quantum numbers in a CFT.
  • Figure 2: Leading Regge trajectory with vacuum quantum numbers in SYM. At weak coupling $g\to 0$, the trajectory breaks up into three branches.
  • Figure 3: (a) Regge limit shown in a conformal compactification of the light cone plane. The positions of the operators $x_i$ go to null infinity as shown. (b) The analytic continuation in $z$ and $\overline{z}$, starting from the Euclidean correlation function.