  We study the "Higgs" amplitude mode in the relativistic quantum O($N$) model
in two space dimensions. Using the nonperturbative renormalization group and
the Blaizot--M\'endez-Galain--Wschebor approximation (which we generalize to
compute 4-point correlation functions), we compute the O($N$) invariant scalar
susceptibility at zero temperature in the vicinity of the quantum critical
point. In the ordered phase, we find a well-defined Higgs resonance for $N=2$
and $N=3$ and determine its universal properties. No resonance is found for
$N\geq 4$. In the disordered phase, the spectral function exhibits a threshold
behavior with no Higgs-like peak. We also show that for $N=2$ the Higgs mode
manifests itself as a very broad peak in the longitudinal susceptibility in
spite of the infrared divergence of the latter. We compare our findings with
results from quantum Monte Carlo simulations and $\epsilon=4-(d+1)$ expansion
near $d=3$.
