Electromagnetic fluxes, monopoles, and the order of 4d compact U(1) phase transition

We consider 4d compact U(1) gauge theory with extended action S=−β∑PcosθP−γ∑Pcos2θP. We give a full characterization of the phase diagram of this model using the notion of flux. The relation with the usual monopole picture is discussed. In analogy with the XY model we consider the helicity modulus [Phys. Rev. B 16 (1977) 1217] for this theory, and show that it is an order parameter. Analyzing the finite-size effects of the helicity modulus we conclude that the transition is first-order. The value of this order parameter is related to the renormalized coupling βR. We measure βcR at the transition point and give a counterexample to its conjectured universal value [Nucl. Phys. B 170 (1980) 369].