Spectral function of the Higgs mode in4−εdimensions

We investigate the amplitude (Higgs) mode of the relativistic $O\left(N\right)$ model in the vicinity of the Wilson-Fisher quantum critical point in $D=4\ensuremath{-}\ensuremath{\varepsilon}$ space-time dimensions. We compute the universal part of the scalar spectral function near the transition, to leading nontrivial order in the ordered phase, and to next to leading order in both the disordered phase and the quantum critical regime. We find that, in the disordered phase, the spectral function has a threshold behavior with no Higgs-like peak, whereas in the ordered phase, the Higgs mode appears as a well defined resonance. The pole associated with this resonance is purely real in the $D\ensuremath{\rightarrow}3+1$ limit, evolving smoothly with dimensionality to become purely imaginary at $D=2+1$ in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. Our results complement previous studies of the scalar spectral function, and demonstrate that the resonance found in these studies can indeed be directly identified with the Higgs mode.