Notes on weight-shifting operators and unifying relations for cosmological correlators

TL;DR

This paper investigates inverse formulas for cosmological relations between gluons and scalars, showing success at 3-point correlators using weight-shifting operators and proposing a new method for 4-point correlators.

Contribution

It demonstrates the use of conformal symmetry-derived weight-shifting operators as inverse tools for 3-point correlators and introduces a novel 'weight-shifting uplifting' method for 4-point correlators.

Findings

Inverse formulas work for 3-point correlators using weight-shifting operators.

Inverse formulas cannot be constructed for 4-point correlators from these operators.

Proposes a new 'weight-shifting uplifting' method for 4-point gluon correlators.

Abstract

We seek the inverse formulas for the cosmological unifying relation between gluons and conformally coupled scalars. We demonstrate that the weight-shifting operators derived from the conformal symmetry at the dS late-time boundary can serve as the inverse operators for the 3-point cosmological correlators. However, in the case of the 4-point cosmological correlator, we observe that the inverse of the unifying relation cannot be constructed from the weight-shifting operators. Despite this failure, we are inspired to propose a "weight-shifting uplifting" method for the 4-point gluon correlator.