The $T^{μν}$ of the conformal scalars

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