Existence Proof by a Fixed-Point Theorem for Solutions of the Low Equation

Schauder's fixed-point theorem may be used to show that certain crossing-symmetric $S$-matrix equations have solutions. The method is illustrated in the case of the one-meson Low equation. It is proved that a sufficient condition for the existence of a solution is that the coupling constant be less than a certain bound which depends on the cutoff and the crossing matrix. The proof works for an arbitrary $n\times n$ crossing matrix with weak conditions on the cutoff function [for instance, $v\left(k\right)=O\left(k^{-2-\epsilon }\right)$, $k\to \infty$]. The allowed range of coupling constants is such as to rule out resonant scattering. A related circumstance is that for the solution in question the baryon is elementary in the sense that it corresponds to a Castillejo-Dalitz-Dyson pole of an appropriate $D$ function. The technique of applying Schauder's theorem differs from that of Atkinson's similar work in that the dispersion relations are approached directly without the aid of the $\frac{N}{D}$ method. Hence the problem of $D$-function ghosts is avoided, and complete crossing symmetry is ensured.