Evidence of the Cooper-Pair Field with Gaussian Memory Kernel in Unconventional Superconductors

We develop a dynamical description of the superconducting pair field in which the Cooper-channel Hubbard--Stratonovich field $\Delta$ is treated as a memory-dressed Bogoliubov pair field rather than as a purely static order parameter. Starting from the standard pair-field effective action, we couple $\Delta$ to antinode-selected collective or self-generated fields. An ensemble of such modes produces a distribution of local Bogoliubov frequencies; when this distribution is approximately Gaussian, ensemble averaging gives the memory factor $\exp[-t^2/(2\tau_g^2)]$. In cuprate superconductors, the antinodal gap or pseudogap restricts the active electronic phase space and acts as a momentum-space spectral cavity. It selects fluctuation wavevectors $\mathbf Q_a$ that may become charge-density-wave-like instabilities in an ordered limit, but behave as a reservoir of local collective fields in the fluctuating regime. The same framework admits resonant algebraic prefactors, so that threshold and forced-oscillator responses generate the hierarchy $p=-1/2,1/2,1,3/2,\ldots$, while the Gaussian envelope cuts off secular growth and converts these branches into finite spectral components. The resulting picture contains a robust pseudogap memory channel and, below $T_c$, an additional condensate-assisted coherent channel proportional to $|\Delta_0(T)|^2$. Thus the superconducting transition primarily reorganizes pair-field spectral weight between incoherent pseudogap memory and coherent Bogoliubov memory. The frequency-domain response is expressed in terms of parabolic-cylinder functions, and comparisons with Raman, ARPES, tunneling, and doping-dependent ARPES scaling suggest that these probes are complementary projections of the same Gaussian-memory pair continuum. We compare our numerical results with the recent experimental data on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.