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Momentum space parity-odd CFT 3-point functions

Sachin Jain, Renjan Rajan John, Abhishek Mehta, Amin A. Nizami, Adithya Suresh

TL;DR

This work advances the understanding of parity-odd three-point functions in momentum-space conformal field theories by developing and cross-validating two complementary methods: direct solution of momentum-space conformal Ward identities and construction via parity-odd spin-raising and weight-shifting operators. The authors systematically classify allowed parity-odd structures across dimensions, solve for explicit form factors (often in terms of triple-$K$ integrals), and address divergences with carefully derived renormalization counter-terms. They provide concrete results for key correlators in three dimensions, including ⟨JJO⟩, ⟨JJJ⟩, and ⟨TTO⟩, and extend the parity-odd construction to four dimensions, notably obtaining ⟨JJJ⟩_odd with explicit operator methods. The discussion highlights remaining challenges (e.g., Schouten-identity complications for higher-spin cases) and outlines promising directions such as parity-odd conformal blocks, double-copy structures, and extensions to supersymmetric theories.

Abstract

We study the parity-odd sector of 3-point functions comprising of scalar operators and conserved currents in conformal field theories in momentum space. We use momentum space conformal Ward identities as well as spin-raising and weight-shifting operators to fix the form of these correlators. We discuss in detail the regularisation of divergences and their renormalisation using specific counter-terms.

Momentum space parity-odd CFT 3-point functions

TL;DR

This work advances the understanding of parity-odd three-point functions in momentum-space conformal field theories by developing and cross-validating two complementary methods: direct solution of momentum-space conformal Ward identities and construction via parity-odd spin-raising and weight-shifting operators. The authors systematically classify allowed parity-odd structures across dimensions, solve for explicit form factors (often in terms of triple- integrals), and address divergences with carefully derived renormalization counter-terms. They provide concrete results for key correlators in three dimensions, including ⟨JJO⟩, ⟨JJJ⟩, and ⟨TTO⟩, and extend the parity-odd construction to four dimensions, notably obtaining ⟨JJJ⟩_odd with explicit operator methods. The discussion highlights remaining challenges (e.g., Schouten-identity complications for higher-spin cases) and outlines promising directions such as parity-odd conformal blocks, double-copy structures, and extensions to supersymmetric theories.

Abstract

We study the parity-odd sector of 3-point functions comprising of scalar operators and conserved currents in conformal field theories in momentum space. We use momentum space conformal Ward identities as well as spin-raising and weight-shifting operators to fix the form of these correlators. We discuss in detail the regularisation of divergences and their renormalisation using specific counter-terms.

Paper Structure

This paper contains 44 sections, 193 equations.

Table of Contents

  1. Introduction
  2. Two approaches to determining momentum space correlators
  3. Using Conformal Ward identity : Strategy
  4. Dilatation and Special Conformal Ward identities
  5. Divergences
  6. Counter-terms
  7. Using Weight-shifting and Spin-raising operators
  8. Subtleties with the weight-shifting and spin-raising operator approach
  9. Allowed parity-odd 3-point functions in various dimensions
  10. Three-dimensions
  11. Four-dimensions
  12. Five-dimensions and above
  13. Momentum space parity-odd 3-point functions in three-dimensions using conformal Ward identity
  14. $\langle J^{\mu} O O \rangle_{\text{odd}}$
  15. $\langle J^{\mu} J^{\nu} O\rangle_{\text{odd}}$
  16. ...and 29 more sections