Conformal Structure of Massless Scalar Amplitudes Beyond Tree level

TL;DR

This work studies the conformal structure of massless scalar scattering in flat space by mapping four-point amplitudes of massless $\phi^4$ theory to the Celestial sphere via a Mellin transform. It demonstrates $SL(2,\mathbb{C})$ covariance of the one-loop and two-loop Mellin amplitudes, with a universal cross-ratio dependence $[z(z-1)]^{2/3}$ and a delta-function pattern in the Mellin variable $\Lambda$, thereby connecting loop-level dynamics to a 2D conformal framework. The authors also formulate a Mellin-space unitarity constraint and verify that the optical theorem is satisfied in this basis, supporting a consistent celestial CFT interpretation. They propose a universal all-loop leading-log structure for the four-point amplitude and discuss extensions to other theories and a deeper celestial holography program.

Abstract

We show that the one-loop on-shell four-point scattering amplitude of massless $φ^4$ scalar field theory in 4D Minkowski space time, when Mellin transformed to the Celestial sphere at infinity, transforms covariantly under the global conformal group ($SL(2,C)$) on the sphere. The unitarity of the four-point scalar amplitudes is recast into this Mellin basis. We show that the same conformal structure also appears for the two-loop Mellin amplitude. Finally we comment on some universal structure for all loop four-point Mellin amplitudes specific to this theory.

Figures (4)

• Figure 1: Tree level diagram's. Cross-channel corresponds to two other channel of scattering.
• Figure 2: One loop diagram's. Cross-channel corresponds to two other channel of scattering.
• Figure 3: Leading log two loop diagram
• Figure 4: Two loop contribution.