  The line of phase transitions, separating the confinement and the Coulomb
phases in the four-dimensional pure compact U(1) gauge theory with extended
Wilson action, is reconsidered. We present new numerical evidence that a part
of this line, including the original Wilson action, is of second order. By
means of a high precision simulation on homogeneous lattices on a sphere we
find that along this line the scaling behavior is determined by one fixed point
with distinctly non-Gaussian critical exponent nu = 0.365(8). This makes the
existence of a nontrivial and nonasymptotically free four-dimensional pure U(1)
gauge theory in the continuum very probable. The universality and duality
arguments suggest that this conclusion holds also for the monopole loop gas,
for the noncompact abelian Higgs model at large negative squared bare mass, and
for the corresponding effective string theory.
