Conformal Structure of Massless Scalar Amplitudes Beyond Tree level

TL;DR

This paper demonstrates that one-loop and two-loop massless scalar four-point amplitudes in 4D Minkowski space, when Mellin transformed to the celestial sphere, exhibit conformal covariance under the global conformal group, revealing universal structures across loops.

Contribution

It shows that loop-level scalar amplitudes transform covariantly under conformal symmetry in Mellin space, extending previous tree-level results and suggesting universal features.

Findings

Conformal covariance of one-loop four-point amplitudes.

Extension of conformal structure to two-loop amplitudes.

Identification of universal features in loop Mellin amplitudes.

Abstract

We show that the one-loop on-shell four-point scattering amplitude of massless $\phi^4$ 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.