  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.
