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Recursion Relations in Witten Diagrams and Conformal Partial Waves

Xinan Zhou

TL;DR

This paper addresses the problem of decomposing exchange Witten diagrams in the crossed channel by uncovering infinite linear relations among the crossed-channel OPE coefficients, enabling a recursive solution from seed data. It leverages an equation-of-motion operator to relate exchange diagrams to sums of contact diagrams, and then uses conformal blocks to derive finite-term recursions for the crossed-channel coefficients in both 1D and higher dimensions. The main contributions are the explicit recursive algorithms for computing all crossed-channel double-trace coefficients, the identification of seed data (lowest-dimension or leading-twist double-trace OPEs), and the extension of these methods to conformal partial waves. This framework streamlines the extraction of CFT data from tree-level holographic correlators and has potential applications in Mellin bootstrap, crossing kernels, and BCFT generalizations, with possible extensions to spinning operators, loops, and supersymmetric backgrounds.

Abstract

We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.

Recursion Relations in Witten Diagrams and Conformal Partial Waves

TL;DR

This paper addresses the problem of decomposing exchange Witten diagrams in the crossed channel by uncovering infinite linear relations among the crossed-channel OPE coefficients, enabling a recursive solution from seed data. It leverages an equation-of-motion operator to relate exchange diagrams to sums of contact diagrams, and then uses conformal blocks to derive finite-term recursions for the crossed-channel coefficients in both 1D and higher dimensions. The main contributions are the explicit recursive algorithms for computing all crossed-channel double-trace coefficients, the identification of seed data (lowest-dimension or leading-twist double-trace OPEs), and the extension of these methods to conformal partial waves. This framework streamlines the extraction of CFT data from tree-level holographic correlators and has potential applications in Mellin bootstrap, crossing kernels, and BCFT generalizations, with possible extensions to spinning operators, loops, and supersymmetric backgrounds.

Abstract

We revisit the problem of performing conformal block decomposition of exchange Witten diagrams in the crossed channel. Using properties of conformal blocks and Witten diagrams, we discover infinitely many linear relations among the crossed channel decomposition coefficients. These relations allow us to formulate a recursive algorithm that solves the decomposition coefficients in terms of certain seed coefficients. In one dimensional CFTs, the seed coefficient is the decomposition coefficient of the double-trace operator with the lowest conformal dimension. In higher dimensions, the seed coefficients are the coefficients of the double-trace operators with the minimal conformal twist. We also discuss the conformal block decomposition of a generic contact Witten diagram with any number of derivatives. As a byproduct of our analysis, we obtain a similar recursive algorithm for decomposing conformal partial waves in the crossed channel.

Paper Structure

This paper contains 16 sections, 203 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Witten Diagrams and Conformal Partial Waves
  3. Witten Diagrams
  4. Relating Exchange Diagrams to Contact Diagrams
  5. Conformal Partial Waves
  6. Recursion Relations in Contact Witten Diagrams
  7. Conformal Block Decomposition of Contact Witten Diagrams
  8. Direct Channel Decomposition of Exchange Diagrams from Contact Diagrams
  9. Recursion Relations in the Crossed Channel
  10. The Strategy for Crossed Channel Decomposition
  11. One Dimension
  12. $d>1$ Dimensions
  13. Discussion and Outlook
  14. Contact Diagrams
  15. The Special Case of Equal Weights
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Illustration of the recursive algorithm for solving the OPE coefficient $\mathcal{B}^{12}_{n,\ell}$. The equation $Recur^{12}_{n-1,\ell+1}=\tilde{a}^{12}_{n-1,\ell+1}$ solves $\mathcal{B}^{12}_{n,\ell}$ in terms of $\mathcal{B}^{12}_{n-1,\ell}$, $\mathcal{B}^{12}_{n-1,\ell+1}$, $\mathcal{B}^{12}_{n-1,\ell+2}$ and $\mathcal{B}^{12}_{n-2,\ell+2}$ which have smaller values of $n$. This procedure is iterated until one reaches the $n=0$ data which are the seed coefficients one inputs.
  • Figure 2: Illustration of the constraints on $\mathcal{B}^{12}_{0,0}$, $\mathcal{B}^{12}_{0,1}$, …, $\mathcal{B}^{12}_{0,2n+1}$ imposed by the equation $Recur^{12}_{n,0}=\tilde{a}^{12}_{n,0}$.