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The longitudinal susceptibility $\chi_L$ of the O(N) theory in the brokenphase is analyzed by means of three different approaches, namely the leadingcontribution of the 1/N expansion, the Functional Renormalization Group flow inthe Local Potential approximation and the improved effective potential via theCallan-Symanzik equations, properly extended to $d=4$ dimensions through theexpansion in powers of $\epsilon=4-d$. The findings of the three approaches arecompared and their agreement in the large $N$ limit is shown. The numericalanalysis of the Functional Renormalization Group flow equations at small $N$supports the vanishing of $\chi_L^{-1}$ in $d=3$ and $d=3.5$ but is notconclusive in $d=4$, where we have to resort to the Callan-Smanzik approach. Atfinite $N$ as well as in the limit $N\to\infty$, we find that $\chi^{-1}_L$vanishes with $J$ as $J^{\epsilon/2}$ for $\epsilon>0$ and as $(\ln (J))^{-1}$in $d=4$.

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