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We apply large $N$ diagrammatic techniques for theories with double-traceinteractions to the leading corrections to $C_J$, the coefficient of aconserved current two-point function, and $C_T$, the coefficient of thestress-energy tensor two-point function. We study in detail two famousconformal field theories in continuous dimensions, the scalar $O(N)$ model andthe Gross-Neveu model. For the $O(N)$ model, where the answers for the leadinglarge $N$ corrections to $C_J$ and $C_T$ were derived long ago using analyticbootstrap, we show that the diagrammatic approach reproduces them correctly. Wealso carry out a new perturbative test of these results using the $O(N)$symmetric cubic scalar theory in $6-\epsilon$ dimensions. We go on to apply thediagrammatic method to the Gross-Neveu model, finding explicit formulae for theleading corrections to $C_J$ and $C_T$ as a function of dimension. We checkthese large $N$ results using regular perturbation theory for the Gross-Neveumodel in $2+\epsilon$ dimensions and the Gross-Neveu-Yukawa model in$4-\epsilon$ dimensions. For small values of $N$, we use Pade approximantsbased on the $4-\epsilon$ and $2+\epsilon$ expansions to estimate the values of$C_J$ and $C_T$ in $d=3$. For the $O(N)$ model our estimates are close to thosefound using the conformal bootstrap. For the GN model, our estimates suggestthat, even when $N$ is small, $C_T$ differs by no more than $2\%$ from that inthe theory of free fermions. We find that the inequality $C_T^{\textrm{UV}} >C_T^{\textrm{IR}}$ applies both to the GN and the scalar $O(N)$ models in$d=3$.

On $C_J$ and $C_T$ in the Gross-Neveu and $O(N)$ Models