Free □ k scalar conformal field theory

bstract We consider the generalizations of the free U( N ) and O ( N ) scalar conformal field theories to actions with higher powers of the Laplacian □ k , in general dimension d . We study the spectra, Verma modules, anomalies and OPE of these theories. We argue that in certain d and k , the spectrum contains zero norm operators which are both primary and descendant, as well as extension operators which are neither primary nor descendant. In addition, we argue that in even dimensions d ≤ 2 k , there are well-defined operator algebras which are related to the □ k theories and are novel in that they have a finite number of single-trace states.