  The phenomenological Ginzburg-Landau theory and the charge conservation
directly lead to the finite Higgs-mode generation and vanishing charge-density
fluctuation in the second-order optical response of superconductors at clean
limit. Nevertheless, recent microscopic theoretical studies of the second-order
optical response, apart from the one through the gauge-invariant kinetic
equation [Yang and Wu, Phys. Rev. B {\bf 100}, 104513 (2019)], have derived a
vanishing Higgs-mode generation but finite charge-density fluctuation at clean
limit. We resolve this controversy by re-examining the previous derivations
with the vector potential alone within the path-integral and
Eilenberger-equation approaches, and show that both previous derivations
contain mathematical flaws. After fixing these flaws, a finite Higgs-mode
generation through the drive effect of vector potential is derived at clean
limit, exactly recovering the previous result from the gauge-invariant kinetic
equation as well as Ginzburg-Landau theory. By further extending the
path-integral approach to include electromagnetic effects from the scalar
potential and phase mode, in the second-order response, a finite contribution
from the drive effect of scalar potential to the Higgs-mode generation at clean
limit as well as the vanishing charge-density fluctuation are derived, also
recovering the results from the gauge-invariant kinetic equation. Particularly,
we show that the phase mode is excited in the second-order response, and
exactly cancels the previously reported unphysical excitation of the
charge-density fluctuation, guaranteeing the charge conservation.

Superconductivity;Strongly Correlated Electrons