  We study the "Higgs" amplitude mode in the relativistic quantum O($N$) model
in two space dimensions. Using the nonperturbative renormalization group we
compute the O($N$)-invariant scalar susceptibility in the vicinity of the
zero-temperature quantum critical point. In the zero-temperature ordered phase,
we find a well defined Higgs resonance for $N=2$ with universal properties in
agreement with quantum Monte Carlo simulations. The resonance persists at
finite temperature below the Berezinskii-Kosterlitz-Thouless transition
temperature. In the zero-temperature disordered phase, we find a maximum in the
spectral function which is however not related to a putative Higgs resonance.
Furthermore we show that the resonance is strongly suppressed for $N\geq 3$.
