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We study lattice superconductors coupled to gauge fields, such as anattractive Hubbard model in electromagnetic fields, with a standard gaugefixing. We prove upper bounds for a two-point Cooper pair correlation at finitetemperatures in spatial dimensions $D\le 4$. The upper bounds decayexponentially in three dimensions, and by power law in four dimensions. Theseimply absence of the superconducting long-range order for the Cooper pairamplitude as a consequence of fluctuations of the gauge fields. Since ourresults hold for the gauge fixing Hamiltonian, they cannot be obtained as acorollary of Elitzur's theorem.

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