Symmetric Group and Meson Born Amplitudes

In this paper we construct Born amplitudes for a theory of nonstrange meson interactions. It is convenient to summarize our results in two parts. (i) Four-particle Born amplitude, ${A}_{4}$ Here we decide to retain two and reject two of four properties possessed by the unit-intercept Euler-$B$-function model. We retain (a) no odd daughters and (b) summability over permutations; we reject (c) the supplementary condition on the sum of trajectory functions and (d) the Plahte phase identities. The reason for rejecting (c) is simply that the physical masses do not in general fulfill such a condition; one is then forced to reject (d) because without (c) the phase identies are inconsistent. Having made these decisions, we show that retention of (a) and (b) guarantees that there be new singularities in the integrand. Reggc behavior dictates that these singularities be at the fixed points $x={e}^{\ifmmode\pm\else\textpm\fi{}\frac{i\ensuremath{\pi}}{3}}$ of a six-element ${\mathrm{S}}_{3}$ group with elements $x\ensuremath{\rightarrow}(1\ensuremath{-}x)\ensuremath{\rightarrow}\frac{x}{(x\ensuremath{-}1)}$. The general form of ${A}_{4}$ is given; it contains a function which is (1) invariant under an ${\mathrm{S}}_{3}$ group and (2) analytic everywhere except for possible simple or multiple poles at the fixed points. Some examples lead to the selection of a particularly simple form for ${A}_{4}$ as a physically interesting model for $\ensuremath{\pi}\ensuremath{\pi}$ elastic scattering. Some phenomenological consequences of this model are given including the elastic-resonance widths and the status of the soft-pion consistency condition. (ii) $N$-particle Born amplitude, ${A}_{N}$. By introducing redundant variables and rewriting ${A}_{4}$ in terms of projective-invariant anharmonic ratios, the integrand becomes invariant under a larger ${\mathrm{S}}_{4}$ group. This is convenient because it enables us to extend all factors, including those with fixed-point singularities, from ${A}_{4}$ to ${A}_{N}$. The general formula for ${A}_{N}$ is given; its integrand is ${\mathrm{S}}_{N}$-invariant. An extension of the specific physically interesting four-pion model is constructed. Evident in the multipionic amplitude is an anticommutative algebra which we interpret as the signature of underlying fermionic quarks. In our final remarks we urge that the spectrum uniquely implied by these multipionic amplitudes be analyzed and compared in detail with experimental observations of the nonstrange mesonic spectrum. In an appendix we point out that the symmetric group may be used to unify three different previous proposals. Special cases, extrapolated to unphysical mass values, of the present approach include (1) the unit-$\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$-intercept ${A}_{4}$ of Veneziano and its ${A}_{N}$ extension in the Bardakci-Chan-Goebel-Koba-Nielsen-Ruegg-Sakita-Tsun-Virasoro theory; (2) the unit-p-intercept ${A}_{4}$ of Lovelace and Shapiro and its ${A}_{N}$ extension by Neveu and Schwarz; (3) the equal-$\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$-intercept ${A}_{4}$ of Mandelstam and its ${A}_{N}$ extension by Gervais and Neveu.