Finite Predictions from a Simple Nonrenormalizable Field Theory

Abstract

A simple, nonrenormalizable quantum field theory containing a scalar field derivative-coupled to a nonconserved vector current of massive fermions is investigated. A method is presented which permits the calculation of $S$-matrix elements as a power series in a symmetry-violating mass difference $\ensuremath{\Delta}{m}_{0}$ instead of the usual series expansion in the coupling strength $g$: $〈\ensuremath{\alpha}|S|\ensuremath{\beta}〉=\ensuremath{\Sigma}{a}_{n}{(\ensuremath{\Delta}{m}_{0})}^{n}$. It is shown that the resulting scattering matrix is unitary and that for $n\ensuremath{\le}4$ all of the coefficients in this new expansion are explicitly finite. It is not yet known whether the proposed method yields finite coefficients for all $n$. Finally, it is noted that these techniques can be applied to a number of other nonrenormalizable theories.