Conformal Structure of Massless Scalar Amplitudes Beyond Tree level

Abstract

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

15 pages, 4 figures, V2: Reference Added

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