  The critical behavior of a non-local scalar field theory is studied. This
theory has a non-local quartic interaction term which involves a real power
-\beta of the Laplacian. The parameter \beta can be tuned so as to make that
interaction marginal for any dimension. The lowest order Feynman diagrams
corresponding to coupling constant renormalization, mass renormalization, and
field renormalization are computed. In all cases a non-trivial IR fixed point
is obtained. Remarkably, for dimensions different from 4, field renormalization
is required at the one-loop level. For d=4, the theory reduces to the usual
local \phi^{4} field theory and field renormalization is required starting at
the the two-loop level. The critical exponents \nu and \eta are computed for
dimensions 2,3,4 and 5. For dimensions greater than four, the critical exponent
\eta turns out to be negative for \epsilon>0, which indicates a violation of
the unitarity bounds.
