Conformal structure of massless scalar amplitudes beyond tree level

Nabamita Banerjee,Sayali Atul Bhatkar,Shamik Banerjee,Sachin Jain

doi:10.1007/jhep04(2018)039

Nabamita Banerjee, Sayali Atul Bhatkar + Show 2 more

Open Access

https://doi.org/10.1007/jhep04(2018)039

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Abstract

We show that the one-loop on-shell four-point scattering amplitude of massless ϕ4 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.

Highlights

The transformation properties of scattering amplitudes under SL(2, C) was first considered by Dirac [1]
We show that the one-loop on-shell four-point scattering amplitude of massless φ4 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
We have studied the conformal structure of the flat space QFT four-point amplitude of massless φ4 theory

Summary

Conformal basis for massless scalar fields

We briefly review how four dimensional scattering amplitudes of a QFT on a flat space can be recasted with manifest global conformal symmetry. In [17], the authors have defined a new basis for scalars (massive and massless) and spin one gluons, namely the conformal primary wave functions, that manifests the conformal structure of their corresponding 4D amplitudes These conformal primaries are characterized by their conformal dimensions and positions (w, w) on a 2dimensional space, that refer to the boundary of the on-shell three diemnsional momentum hyperboloid (H3). We see that the one-loop amplitude of flat space massless φ4 theory retains its conformal structure when expressed on the Celestial sphere. This is not a surprise, but, here we see that the Function of the cross rations f (z, z) is identical at tree level and loop-level amplitude. We shall comment more on this structure in the later section

Unitarity for Mellin amplitudes

Comments on higher order Mellin amplitudes

Conclusions and future directions

A Two-loop computation in momentum space