  We give a non-trivially interacting field theory example of scale invariant
but non-conformal field theory. The model is based on the exactly solvable
Liouville field theory coupled with free scalars deformed by an exactly
marginal operator. We show non-vanishing of the trace of the energy-momentum
tensor by using the quantum Schwinger-Dyson equation for the Liouville field
theory, which is a sophistication of the quantum higher equations of motion for
the Liouville field theory introduced by Alyosha Zamolodchikov. Possibly
dangerous implications for the super-critical string theory will be discussed.
