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We investigate the spectral functions of the Higgs mode in $O(2)$ model,which can be experimentally realized in a 2D Bose gas. Zero temperature limitis considered. Our calculation fully includes the 2-loop contributions. Peaksshow up in the spectral functions of both the longitudinal and the scalarsusceptibilities. Thus this model cannot explain the disappearance of theresponse at the weak interaction limit. Neither it can explain the similaritybetween the longitudinal and the scalar susceptibilities in the visibility ofthe Higgs mode. A possible lower peak at about $2m_{\sigma}$ is also noted.

Higgs mode in $2+1$ dimensional $O(2)$ model