The $T^{\mu\nu}$ of the conformal scalars

TL;DR

This paper constructs a unique primary energy-momentum tensor for conformal free scalars with arbitrary scaling dimension, using Gegenbauer polynomials, and confirms its consistency with known two-point functions and GJMS operators.

Contribution

It provides a unified construction of the energy-momentum tensor for conformal scalars across integer and non-integer dimensions, extending previous results and clarifying nonlocal geometric couplings.

Findings

Reproduces known two-point functions for integer $$

Matches $T^{}$ with GJMS operators from Juhl's formulae

Provides a Gegenbauer polynomial representation of $T^{}$

Abstract

We construct the unique primary energy-momentum tensor $T^{\mu\nu}$ for the conformal free scalar with scaling dimension $\Delta=d/2-\zeta$ as a sum of Gegenbauer polynomials. For integer $\zeta$, the sum truncates at order $\zeta$, compactly reproducing all known results; for the nonlocal case of real $\zeta$, it is an infinite sum, with a two-parameter extension that reflects the nonuniqueness of the nonlocal geometric coupling. We find $T^{\mu\nu}$ by imposing off-shell conservation and tracelessness, and then directly solving the primary condition in momentum space. In the integer $\zeta$ case, we reproduce the known two-point function, and confirm the match with the $T^{\mu\nu}$ computed from Juhl's formulae for the GJMS operators (the Weyl-covariant upgrades of $(-\partial^2)^\zeta$), an equality following from the descent of Weyl covariance to conformal invariance.