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We analyze generalizations of the Schwinger model with more massless fermionsand more vector fields. We focus on models with the gauge structure of``diagonal color $SU(n)$'' but unlike previous investigators, we do not assumethat all the gauge boson masses are the same. Unlike the Schwinger model, theseare Banks-Zaks models with conformal sectors that survive at long distances. Inaddition to local operators that go to ``unparticle operators'' with non-zeroanomalous dimensions at long distances, they contain local operators like the$\bar\psi_L\psi_R$ operator in the Schwinger model which go to constants atlong distances. These operators have calculable vacuum expectation values (upto phases). Cluster decomposition applied to correlation functions involvingthese operators yields nontrivial and calculable non-perturbative constraintson correlation functions. One consequence is ``conformal coalescence'' in whichlinear combinations of short distance operators disappear from thelong-distance theory, leaving only one kind of unparticle stuff in thelow-energy theory. We believe that our detailed analysis of diagonal color$SU(n)$ paints an appealing picture of unparticle operators as the result of anincomplete binding of the massless fermions. We complete the picture (and thebinding) by analyzing the diagonal color $U(n)$ model with a very small $U(1)$coupling and thus a gauge boson with a dynamical mass much smaller than theother masses in the model. This model has a mass gap and we can see explicitlythe transition from free-fermion behavior at short distances to unparticlephysics at intermediate distances to the physics of massive particles at longdistances.

Non-perturbative Effects and Unparticle Physics in Generalized Schwinger Models