Singularity Structure of the Four Point Celestial Leaf Amplitudes

TL;DR

This paper analyzes the singularity structure of four-point celestial leaf amplitudes for massless scalars and gluons, revealing a simple pole at $z = ar z$ and connecting leaf amplitudes to the full celestial amplitudes through spacetime wedge decompositions.

Contribution

It introduces the concept of leaf amplitudes as non-distributional components of celestial amplitudes and demonstrates their singularity structure and differential equation properties.

Findings

Leaf amplitudes have a simple pole at $z = ar z$.

Adding leaf amplitudes from different spacetime wedges recovers the distributional celestial amplitudes.

MHV gluon leaf amplitudes satisfy known differential equations related to soft theorems.

Abstract

In this paper, we study the four-point celestial leaf amplitudes of massless scalar and MHV gluon scattering. These leaf amplitudes are non-distributional decompositions of the celestial amplitudes associated with a hyperbolic foliation of the Klein spacetime. Bulk scale invariance imposes constraints on the total conformal weights of the massless scalars or gluons. Using this constraint we show that the four-point leaf amplitudes have a \textit {simple pole singularity at $ z = \bar z $}, where, $ z,\bar z $ are two real independent conformal cross ratios. The distributional nature of the four-point celestial amplitudes is recovered by adding the leaf amplitudes in the timelike and spacelike wedges of the spacetime. We also verify that the MHV gluon leaf amplitudes satisfy a set of differential equations previously obtained for celestial MHV gluon amplitudes by considering the soft gluon theorems and the subleading terms in the OPE expansion between two positive helicity gluons.