COMPOSITENESS, ANOMALOUS DIMENSIONS AND RENORMALIZABILITY IN 4-FERMI THEORIES

We study the interplay of compositeness, nontriviality and hyperscaling in a four-Fermi theory with interaction (psi-psiBAR)2 in space-time dimension d, 2 < d less-than-or-equal-to 4. We establish that nonvanishing anomalous dimensions, compositeness and hyperscaling lead to renormalizability and a nontrivial continuum limit. We argue that the theory possesses a renormalizable 1/N expansion, where N is the number of fermion species, about a strongly-coupled fixed point in all dimensions 2 < d less-than-or-equal-to 4. Explicit calculations of the critical indices to O (1/N) illustrate these points. For d = 4 we confirm that the theory's cutoff can be removed only at the expense of making the continuum theory trivial and that this result prevails beyond leading order. This model illustrates that strongly-coupled theories can be renormalizable in a continuum of dimensions, rather than at just one upper critical dimension as occurs in weakly coupled theories.