Integral Transformations for Conformally Invariant Celestial Gluon Amplitudes

TL;DR

This work introduces a string-inspired integral transformation that maps celestial amplitudes from Cartesian celestial coordinates $(z_i,\bar z_i)$ to a new pair of complex variables $(s_i,\bar s_i)$ and derives a well-defined inverse by regularizing translational redundancies. When applied to celestial MHV gluon amplitudes, the transform yields explicit sum-rule constraints on $(s_i,\bar s_i)$ for 3-, 4-, and general $n$-point functions, ensuring invariance under global conformal transformations provided the total energies satisfy $\sum_j \lambda_j=0$. The approach mirrors worldsheet integrals in closed-string theory and suggests avenues to import holomorphic factorization techniques, albeit with interdependent holomorphic and antiholomorphic sectors due to momentum conservation. These results offer a new framework for constructing conformally invariant celestial amplitudes and motivate further connections between celestial holography and string-theoretic methods.

Abstract

We propose an integral transformation for celestial gluon amplitudes that maps the celestial coordinates \((z_i,\bar z_i)\) to a new set of complex variables \((s_i,\bar s_i)\), inspired by the structure of closed string scattering amplitudes. A consistent inverse transformation is constructed by regulating a divergence associated with translational redundancy and absorbing it into an overall normalization. Applying this transformation to celestial MHV amplitudes, we derive constraints on \((s_i,\bar s_i)\) for three-, four-, and general \(n\)-point amplitudes, and show that these conditions are necessary for invariance under global conformal transformations.

Figures (1)

• Figure 1: Contour of integration for $s_1$ in the complex plane.