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A compared analysis of the susceptibility in the O(N) theory

The longitudinal susceptibility $χ_L$ of the O(N) theory in the broken phase is analyzed by means of three different approaches, namely the leading contribution of the 1/N expansion, the Functional Renormalization Group flow in the Local Potential approximation and the improved effective potential via the Callan-Symanzik equations, properly extended to $d=4$ dimensions through the expansion in powers of $ε=4-d$. The findings of the three approaches are compared and their agreement in the large $N$ limit is shown. The numerical analysis of the Functional Renormalization Group flow equations at small $N$ supports the vanishing of $χ_L^{-1}$ in $d=3$ and $d=3.5$ but is not conclusive in $d=4$, where we have to resort to the Callan-Smanzik approach. At finite $N$ as well as in the limit $N\to\infty$, we find that $χ^{-1}_L$ vanishes with $J$ as $J^{ε/2}$ for $ε>0$ and as $(\ln (J))^{-1}$ in $d=4$.