I read in this paper (page 11 first a few sentences) that 2nd-order time derivative in the action or the equation of dynamics is indispensable for the excitation of Higgs/amplitude mode. The paper provides no detail but mentions that Gross-Pitaevski theory as a 1st-order example without Higgs.
Why is this claim? It doesn't seem obvious to me from the standard derivation of the Higgs excitaion (e.g., in this answer of Higgs mode). The spatial part remains the same. The temporal part becomes something like $$(\psi+\delta\psi)[\partial_0\delta\psi-iq(A_0+\frac{1}{q}\partial_0\theta)(\psi+\delta\psi)],$$ in which $\delta\psi$ and $\theta$ are the amplitude and phase fluctuations of $\psi$, respectively.
