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Large Spin Systematics: Patterns from Reciprocity for Multiple Spinning Operators

Pulkit Agarwal

TL;DR

This work extends the lightcone bootstrap program to higher-point conformal correlators in $d=4$ by deriving the large-spin expansion of five-point conformal blocks and translating spin sums into integrals over rescaled spins. By imposing analyticity in cross ratios, it obtains an infinite set of reciprocity-type constraints on OPE coefficients involving multiple spinning operators for all tensor structures $l$, with a notable result that, for $l=0$, all odd powers in the $1/J$ expansion vanish to all orders. A refined ansatz reveals a universal trivialisation of these constraints at $l=0$, while the leading relations for nonzero $l$ survive, providing concrete, algorithmically computable relations among OPE data. The methods rely on a Bessel-function expansion of large-spin blocks in the double lightcone limit and offer a framework to constrain higher-spin OPE data in 4D CFTs, with potential connections to Casimir-based approaches and integrability-inspired bases for simplifying the tensor structures involved.

Abstract

We study the behaviour of the conformal block expansions of scalar fivepoint Lorentzian conformal correlators in the limit where multiple cross ratios approach zero. Since this limit is controlled by intermediate operators with large spin, we use it to study the large spin expansion of the OPE coefficients involving these operators. By imposing bootstrap assumptions such as analyticity of the correlators, we derive an infinite set of new constraints on the large spin behaviour of OPE coefficients involving multiple spinning operators. We also show that for the case of $l=0$, these constraints can be trivialised to all orders in $1/J$ by identifying a pattern in the coefficients.

Large Spin Systematics: Patterns from Reciprocity for Multiple Spinning Operators

TL;DR

This work extends the lightcone bootstrap program to higher-point conformal correlators in by deriving the large-spin expansion of five-point conformal blocks and translating spin sums into integrals over rescaled spins. By imposing analyticity in cross ratios, it obtains an infinite set of reciprocity-type constraints on OPE coefficients involving multiple spinning operators for all tensor structures , with a notable result that, for , all odd powers in the expansion vanish to all orders. A refined ansatz reveals a universal trivialisation of these constraints at , while the leading relations for nonzero survive, providing concrete, algorithmically computable relations among OPE data. The methods rely on a Bessel-function expansion of large-spin blocks in the double lightcone limit and offer a framework to constrain higher-spin OPE data in 4D CFTs, with potential connections to Casimir-based approaches and integrability-inspired bases for simplifying the tensor structures involved.

Abstract

We study the behaviour of the conformal block expansions of scalar fivepoint Lorentzian conformal correlators in the limit where multiple cross ratios approach zero. Since this limit is controlled by intermediate operators with large spin, we use it to study the large spin expansion of the OPE coefficients involving these operators. By imposing bootstrap assumptions such as analyticity of the correlators, we derive an infinite set of new constraints on the large spin behaviour of OPE coefficients involving multiple spinning operators. We also show that for the case of , these constraints can be trivialised to all orders in by identifying a pattern in the coefficients.
Paper Structure (7 sections, 21 equations, 1 figure)

This paper contains 7 sections, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kinematics
  3. Large Spin Analysis
  4. Constraints around tree level OPE coefficients
  5. Trivialising Constraints at $l = 0$
  6. Conclusion
  7. Bessel limit of Conformal Blocks

Figures (1)

  • Figure 1: Figure borrowed from Bercini:2020msp. Fivepoint conformal block in the lightcone limit with two large spin operators $\ell_1,\ell_2$ exchanged. The integer $l$ counts the tensor structures in the threepoint function involving these two operators and the external scalar.