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

TL;DR

The paper develops and applies a unified large-$N$ diagrammatic framework for double-trace perturbations to compute leading $1/N$ corrections to $C_J$ and $C_T$ in the scalar $O(N)$ model and the Gross-Neveu family of models across continuous dimensions. By constructing and renormalizing the relevant diagrams, it reproduces known analytic bootstrap results for $O(N)$, provides new perturbative checks in $d=6-oldsymbol{ m olinebreak ef{epsilon}}$, and yields explicit $1/N$ corrections in the GN model that agree with $2+oldsymbol{ m olinebreak ef{epsilon}}$ and $4-oldsymbol{ m olinebreak ef{epsilon}}$ expansions and with GNY. Padé extrapolations are used to estimate $C_J$ and $C_T$ in $d=3$ for small $N$, revealing $C_T$ closely tracks free-fermion values in GN and confirming $C_T^{ m UV} > C_T^{ m IR}$ in $d=3$ for both scalar and GN theories. The results provide robust cross-checks among diagrammatic large-$N$, epsilon expansions, and conformal bootstrap, and offer precise predictions for how conserved-current and stress-tensor two-point-function normalizations evolve along RG flows in these models.

Abstract

We apply large $N$ diagrammatic techniques for theories with double-trace interactions to the leading corrections to $C_J$, the coefficient of a conserved current two-point function, and $C_T$, the coefficient of the stress-energy tensor two-point function. We study in detail two famous conformal field theories in continuous dimensions, the scalar $O(N)$ model and the Gross-Neveu model. For the $O(N)$ model, where the answers for the leading large $N$ corrections to $C_J$ and $C_T$ were derived long ago using analytic bootstrap, we show that the diagrammatic approach reproduces them correctly. We also carry out a new perturbative test of these results using the $O(N)$ symmetric cubic scalar theory in $6-ε$ dimensions. We go on to apply the diagrammatic method to the Gross-Neveu model, finding explicit formulae for the leading corrections to $C_J$ and $C_T$ as a function of dimension. We check these large $N$ results using regular perturbation theory for the Gross-Neveu model in $2+ε$ dimensions and the Gross-Neveu-Yukawa model in $4-ε$ dimensions. For small values of $N$, we use Pade approximants based on the $4-ε$ and $2+ε$ expansions to estimate the values of $C_J$ and $C_T$ in $d=3$. For the $O(N)$ model our estimates are close to those found using the conformal bootstrap. For the GN model, our estimates suggest that, even when $N$ is small, $C_T$ differs by no more than $2\%$ from that in the 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 $d=3$.

Figures (34)

• Figure 3.1: Diagrams for $C_{J}$ up to order $\epsilon$.
• Figure 3.2: Diagrams for $C_{T}$ up to order $\epsilon$.
• Figure 3.3: Momentum space Feynman rules for $T(p)$ and $J^{a}(p)$.
• Figure 3.4: One loop correction to the $\langle \phi^{i}(p) \phi^{j}(-p)\rangle$ propagator.
• Figure 3.5: Diagrams contributing to $\langle T(0) \phi_{0}(p)\phi_{0}(-p) \rangle$ up to order $1/N$.