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On the associativity of one-loop corrections to the celestial OPE

Kevin Costello, Natalie M. Paquette

TL;DR

This work shows that one-loop collinear singularities in a self-dual gauge-theory realization of celestial holography do not form an associative chiral algebra due to a twistor-space anomaly. The obstruction is cured by introducing an axion via a Green-Schwarz mechanism, which yields an associative chiral algebra and a compact, chiral-algebraic derivation of the standard one-loop QCD splitting amplitudes. The axion-extended framework also provides a simple formula for one-loop form factors of self-dual Yang-Mills with the operator (B^2)/2, matching QCD amplitudes with an axion and offering insights for special matter content and gauge groups. Overall, the paper connects twistor-space anomalies, Green-Schwarz cancelation, and celestial OPE associativity to produce concrete one-loop results and streamlined amplitude expressions.

Abstract

There has been recent interest in the question of whether QCD collinear singularities can be viewed as the OPE of a two-dimensional CFT. We analyze a version of this question for the self-dual limit of pure gauge theory (incorporating states of both helicities). We show that the known one-loop collinear singulaties do not form an associative chiral algebra. The failure of associativity can be traced to a novel gauge anomaly on twistor space. We find that associativity can be restored for certain gauge groups if we introduce an unusual axion, which cancels the twistor space anomaly by a Green-Schwarz mechanism. Alternatively, associativity can be restored for some gauge groups with carefully chosen matter.

On the associativity of one-loop corrections to the celestial OPE

TL;DR

This work shows that one-loop collinear singularities in a self-dual gauge-theory realization of celestial holography do not form an associative chiral algebra due to a twistor-space anomaly. The obstruction is cured by introducing an axion via a Green-Schwarz mechanism, which yields an associative chiral algebra and a compact, chiral-algebraic derivation of the standard one-loop QCD splitting amplitudes. The axion-extended framework also provides a simple formula for one-loop form factors of self-dual Yang-Mills with the operator (B^2)/2, matching QCD amplitudes with an axion and offering insights for special matter content and gauge groups. Overall, the paper connects twistor-space anomalies, Green-Schwarz cancelation, and celestial OPE associativity to produce concrete one-loop results and streamlined amplitude expressions.

Abstract

There has been recent interest in the question of whether QCD collinear singularities can be viewed as the OPE of a two-dimensional CFT. We analyze a version of this question for the self-dual limit of pure gauge theory (incorporating states of both helicities). We show that the known one-loop collinear singulaties do not form an associative chiral algebra. The failure of associativity can be traced to a novel gauge anomaly on twistor space. We find that associativity can be restored for certain gauge groups if we introduce an unusual axion, which cancels the twistor space anomaly by a Green-Schwarz mechanism. Alternatively, associativity can be restored for some gauge groups with carefully chosen matter.
Paper Structure (16 sections, 1 theorem, 84 equations, 1 table)

This paper contains 16 sections, 1 theorem, 84 equations, 1 table.

Table of Contents

  1. Introduction
  2. The chiral algebra
  3. One loop corrections
  4. Failure of associativity of the corrected OPE
  5. Twistor space anomalies and chiral algebra associativity
  6. Chiral algebra including the axion
  7. One loop corrections
  8. Normalizing the quantum corrections using associativity
  9. Matching coefficients
  10. Amplitudes
  11. Normalization of states on twistor space
  12. Proof of Proposition (\ref{['prop:OPEconstraint']}).
  13. The right-hand side of \ref{['eqn:associativity']}.
  14. The left-hand side of \ref{['eqn:associativity']}.
  15. Comparing the left and right hand sides
  16. ...and 1 more sections

Key Result

Proposition 1

The OPE associativity identity eqn:associativity holds if and only if we have the trace identity and we take the constants $C,D$ to be

Theorems & Definitions (1)

  • Proposition 1