Conformal Blocks from Celestial Gluon Amplitudes II: Single-valued Correlators

TL;DR

This paper constructs a single-valued celestial four-gluon amplitude by removing branch points from the shadow correlator, resulting in a well-defined, crossing-symmetric amplitude with consistent OPE and bootstrap properties.

Contribution

It introduces a method to obtain a single-valued, well-defined celestial gluon amplitude from shadow correlators using integral representations inspired by minimal models.

Findings

The single-valued amplitude is defined over the entire complex plane.

It satisfies crossing symmetry, OPE, and bootstrap properties.

The conformal block spectrum includes primary fields with specific dimensions and integer spins.

Abstract

In a recent paper, here referred to as part I, we considered the celestial four-gluon amplitude with one gluon represented by the shadow transform of the corresponding primary field operator. This correlator is ill-defined because it contains branch points related to the presence of conformal blocks with complex spin. In this work, we adopt a procedure similar to minimal models and construct a single-valued completion of the shadow correlator, in the limit when the shadow is "soft." By following the approach of Dotsenko and Fateev, we obtain an integral representation of such a single-valued correlator. This allows inverting the shadow transform and constructing a single-valued celestial four-gluon amplitude. This amplitude is drastically different from the original Mellin amplitude. It is defined over the entire complex plane and has correct crossing symmetry, OPE and bootstrap properties. It agrees with all known OPEs of celestial gluon operators. The conformal block spectrum consists of primary fields with dimensions $\Delta=m+i \lambda$, with integer $m\geq 1$ and various, but always integer spin, in all group representations contained in the product of two adjoint representations.