[Paper Review] The five-point bootstrap

David Poland, Valentina Prilepina+1 more|arXiv (Cornell University)|May 15, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper introduces a novel method to compute five-point conformal blocks for arbitrary spin exchanged operators in $d$-dimensional conformal field theories by generalizing radial coordinates and solving quadratic Casimir differential equations perturbatively. The approach enables the numerical computation of OPE coefficients involving one scalar and two spinning operators in the critical 3d Ising model, providing previously inaccessible inputs for non-perturbative methods like Hamiltonian truncation.

ABSTRACT

We study five-point correlation functions of scalar operators in d-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators $\langle σσεσσ angle$ in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff $Δ\leqslant Δ_{ m cutoff}$. We then compute several OPE coefficients involving $ε$ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.

Motivation & Objective

• To develop a tractable method for computing five-point conformal blocks with arbitrary spin exchanged operators in conformal field theories.
• To overcome the technical challenges of existing approaches that rely on recursive relations or complex summations over multiple variables.
• To apply the method to compute OPE coefficients involving one scalar and two spinning operators in the critical 3d Ising model.
• To provide numerically accessible OPE coefficients for use in non-perturbative frameworks such as Hamiltonian truncation.
• To demonstrate the utility of the method by truncating OPEs and minimizing a cost function to enforce crossing symmetry.

Proposed method

• Generalize radial coordinates used in four-point blocks to five-point functions to simplify the structure of conformal blocks.
• Introduce a series expansion in the generalized radial coordinates and determine coefficients by solving two quadratic Casimir differential equations perturbatively.
• Use the recursion relations from Poland et al. (2021) as a benchmark, but bypass their computational complexity by employing a coordinate-based ansatz.
• Impose Ward identities for conserved currents by applying specific sets of differential operators to the conformal blocks.
• Truncate the OPE expansion of the five-point function by including only primary operators with conformal dimension below a cutoff $\Delta_{\text{cutoff}}$.
• Minimize a cost function measuring violation of crossing symmetry to extract approximate OPE coefficients in the critical 3d Ising model.

Experimental results

Research questions

• RQ1 Can a simplified coordinate-based method be developed to compute five-point conformal blocks for arbitrary spin exchanged operators?
• RQ2 How can OPE coefficients involving two spinning operators be extracted from five-point correlators in strongly coupled CFTs?
• RQ3 What is the numerical accuracy and feasibility of computing such OPE coefficients in the critical 3d Ising model using truncated OPEs and crossing symmetry?
• RQ4 How do the resulting OPE coefficients compare to those in mean-field theory or free field theory?
• RQ5 Can this method provide reliable inputs for non-perturbative techniques such as Hamiltonian truncation?

Key findings

• The generalized radial coordinate approach yields a significantly simplified structure for five-point conformal blocks compared to previous recursive or sum-based methods.
• The method enables the computation of OPE coefficients involving one scalar and two spinning operators in the critical 3d Ising model, which were previously inaccessible via standard four-point bootstrap techniques.
• The OPE coefficients for spin-2 and spin-4 operators in the critical 3d Ising model were computed numerically by truncating the OPE and minimizing a crossing-violation cost function.
• The approach successfully enforces the Ward identity for conserved currents by applying specific sets of differential operators to the conformal blocks.
• The method is validated by reproducing known MFT results for two-spinning-operator OPEs, confirming consistency and numerical reliability.
• The sets of differential operators used to impose Ward identities are explicitly provided for spin-2 and spin-4 cases, enabling systematic application to other models.