  The longitudinal susceptibility $\chi_L$ 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 $\epsilon=4-d$. 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 $\chi_L^{-1}$ 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 $\chi^{-1}_L$
vanishes with $J$ as $J^{\epsilon/2}$ for $\epsilon>0$ and as $(\ln (J))^{-1}$
in $d=4$.
