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

TL;DR

This work constructs a single-valued completion of the shadow-transformed celestial four-gluon amplitude, addressing monodromies and complex-spin blocks that plagued the part I formulation. By mimicking minimal-model techniques and employing Dotsenko–Fateev integral representations plus single-valued projections, the authors produce a conformally invariant, crossing-symmetric, and bootstrap-consistent four-point correlator defined on the entire complex plane. The resulting spectrum features primary operators with dimensions $\Delta=m+i\lambda$ ($m\ge1$), with two spin values at each level, and includes a leading OPE consistent with known CCFT data; SU(2) color factors decompose cleanly into representations and CLEBSCH–GORDAN structures. Inverting the shadow transform yields a four-gluon single-valued celestial amplitude with Parke–Taylor denominators dressed by conformal factors, suggesting a robust CCFT underpining and opening paths toward connections with minimal models, Coulomb gas, and twistor/string frameworks.

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 $Δ=m+i λ$, with integer $m\geq 1$ and various, but always integer spin, in all group representations contained in the product of two adjoint representations.