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We study the "Higgs" amplitude mode in the relativistic quantum O($N$) modelin two space dimensions. Using the nonperturbative renormalization group andthe Blaizot--M\'endez-Galain--Wschebor approximation (which we generalize tocompute 4-point correlation functions), we compute the O($N$) invariant scalarsusceptibility at zero temperature in the vicinity of the quantum criticalpoint. 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 thresholdbehavior with no Higgs-like peak. We also show that for $N=2$ the Higgs modemanifests itself as a very broad peak in the longitudinal susceptibility inspite of the infrared divergence of the latter. We compare our findings withresults from quantum Monte Carlo simulations and $\epsilon=4-(d+1)$ expansionnear $d=3$.

Higgs amplitude mode in the vicinity of a $(2+1)$-dimensional quantum critical point: a nonperturbative renormalization-group approach