The $T^{μν}$ of the conformal scalars

TL;DR

The paper derives the unique primary energy-momentum tensor $T^{\mu\nu}$ for a conformal free scalar with dimension $\Delta=\tfrac{d}{2}-\zeta$, expressing it as a Gegenbauer-polynomial expansion. For integer $\zeta$, the construction yields a local, known EM tensor and reproduces the two-point function, while noninteger $\zeta$ leads to a nonlocal, two-parameter family consistent with a Weyl-covariant interpretation of the GJMS operators; the authors also establish the equivalence with Juhl’s recursive GJMS framework by matching the linearized variation of the corresponding metrical operator. The strategy proceeds by enforcing off-shell conservation and tracelessness, then solving the four conformal constraints in momentum space, with the primary condition forcing a Gegenbauer-structured fixed part and exposing possible nonlocal primaries in the noninteger case. These results connect conformal invariance, Weyl covariance, and explicit EM-tensor realizations in generalized free field theories, offering a platform for perturbations around nonlocal CFTs and potential extensions to fermions and gauge fields.

Abstract

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.