Conformal Blocks from Celestial Gluon Amplitudes

TL;DR

The paper develops conformal blocks for celestial gluon amplitudes by replacing one gluon with its shadow to circumvent 4D kinematic constraints. Using radial quantization and shadow techniques, it constructs a CCFT four-point correlator and derives its conformal-block decomposition, obtaining primary fields with dimensions $\Delta=2+M+i(\lambda_2-\lambda_1)$ and integer spins in compatible channels, while incompatible channels introduce a spectrum with complex spins. In the soft limit $\lambda_1=0$, the blocks simplify to a tower with weights $(h,\bar h)=(m+\frac{i\lambda_2}{2},1+\frac{i\lambda_2}{2})$, and the gauge structure is encoded in color coefficients from products of two adjoint representations; the general case involves Appell functions and Burchnall–Chaundy expansions yielding a double-sum block structure. The results imply an infinite CCFT tower beyond traditional BMS symmetries, relate to four-dimensional soft and supertranslation modes, and motivate a CCFT bootstrap program to explore the 4D-2D correspondence in gauge theories.

Abstract

In celestial conformal field theory, gluons are represented by primary fields with dimensions $Δ=1+iλ$, $λ\in\mathbb{R}$ and spin $J=\pm 1$, in the adjoint representation of the gauge group. All two- and three-point correlation functions of these fields are zero as a consequence of four-dimensional kinematic constraints. Four-point correlation functions contain delta-function singularities enforcing planarity of four-particle scattering events. We relax these constraints by taking a shadow transform of one field and perform conformal block decomposition of the corresponding correlators. We compute the conformal block coefficients. When decomposed in channels that are "compatible" in two and four dimensions, such four-point correlators contain conformal blocks of primary fields with dimensions $Δ=2+M+iλ$, where $M\ge 0$ is an integer, with integer spin $J=-M,-M+2,\dots,M-2,M$. They appear in all gauge group representations obtained from a tensor product of two adjoint representations. When decomposed in incompatible channels, they also contain primary fields with continuous complex spin, but with positive integer dimensions.