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

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.

39 + 11 pages