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Both macroscopic Ginzburg-Landau Lagrangian and microscopic gauge-invariantkinetic equation suggest a finite Higgs-mode generation in the second-orderoptical response of superconductors at clean limit, whereas the previousderivations through the path-integral approach and Eilenberger equation withinthe Matsubara formalism failed to give such generation. The crucial treatmentleading to this controversy lies at an artificial scheme that whether theexternal optical frequency is taken as continuous variable or bosonic Matsubarafrequency to handle the gap dynamics within the Matsubara formalism. To resolvethis issue, we derive the effective action of the superconducting gap near$T_c$ in the presence of the vector potential through the path-integralapproach, to fill the long missing blank of the microscopic derivation of theGinzburg-Landau Lagrangian in superconductors. It is shown that only by takingoptical frequency as continuous variable within the Matsubara formalism, canone achieve the fundamental Ginzburg-Landau Lagrangian, and in particular, thefinite Ginzburg-Landau kinetic term leads to a finite Higgs-mode generation atclean limit. To further eliminate the confusion of the Matsubara frequencythrough a separate framework, we apply the Eilenberger equation within theKeldysh formalism, which is totally irrelevant to the Matsubara space. Bycalculating the gap dynamics in the second-order response, it is analyticallyproved that the involved optical frequency is a continuous variable rather thanbosonic Matsubara frequency, causing a finite Higgs-mode generation at cleanlimit.

Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg-Landau Lagrangian