  We consider the 4d compact U(1) gauge theory with extended action
  S=-beta sum_P cos theta_P -gamma sum_P cos 2 theta_P
  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 \cite{Jose:1977gm}
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 beta_R. We measure beta^c_R at the transition point and give a
counterexample to its conjectured universal value \cite{Cardy:jg}.
