Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory

A rigorous lower bound is obtained for the Wilson loop expectation value $〈A\left[C\right]\mathrm{}〉$ in the four-dimensional U(1) lattice gauge theory with an action of the Villain form. The bound, which holds for $g^2<0.168\mathrm{}$, has the form of a Coulomb interaction and guarantees that electric charges are not confined. Using the strong-coupling results of Osterwalder and Seiler, one concludes that this model has at least one phase transition. Using the work of Elitzur, Pearson, and Shigemitsu, one also concludes that the four-dimensional Villain $Z\left(N\right)\mathrm{}$ lattice gauge theory has at least three phases if $N>\mathrm{}37\mathrm{}$. Finally, it is also shown that the infinite-volume limit of $〈A\left[C\right]\mathrm{}〉$ exists.