Decay of Superconducting Correlations for Gauged Electrons in Dimensions $D\le 4$

We study lattice superconductors coupled to gauge fields, such as an attractive Hubbard model in electromagnetic fields, with a standard gauge fixing. We prove upper bounds for a two-point Cooper pair correlation at finite temperatures in spatial dimensions $D\le 4$. The upper bounds decay exponentially in three dimensions, and by power law in four dimensions. These imply absence of the superconducting long-range order for the Cooper pair amplitude as a consequence of fluctuations of the gauge fields. Since our results hold for the gauge fixing Hamiltonian, they cannot be obtained as a corollary of Elitzur's theorem.