  We investigate the continuum limit of the gauge-ball spectrum in the
four-dimensional pure U(1) lattice gauge theory. In the confinement phase we
identify various states scaling with the correlation length exponent $\nu
\simeq 0.35$. The square root of the string tension also scales with this
exponent, which agrees with the non-Gaussian fixed point exponent recently
found in the finite size studies of this theory. Possible scenarios for
constructing a non-Gaussian continuum theory with the observed gauge-ball
spectrum are discussed. The $0^{++}$ state, however, scales with a Gaussian
value $\nu \simeq 0.5$. This suggests the existence of a second, Gaussian
continuum limit in the confinement phase and also the presence of a light or
possibly massless scalar in the non-Gaussian continuum theory. In the Coulomb
phase we find evidence for a few gauge-balls, being resonances in multi-photon
channels; they seem to approach the continuum limit with as yet unknown
critical exponents. The maximal value of the renormalized coupling in this
phase is determined and its universality confirmed.
