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Veneziano Amplitude, Current Algebra, and Vertex Functions

1970; American Physical Society; Volume: 1; Issue: 6 Linguagem: Inglês

10.1103/physrevd.1.1813

1538-4500

R. Arnowitt, M. H. Friedman, Pran Nath,

Black Holes and Theoretical Physics

The possibility is examined of extending the $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ Veneziano amplitudes to the case of one or two mesons off the mass shell in a fashion consistent with chiral current algebra, partial conservation of axial-vector current, and conserved vector current. Soft-pion formulas for the pion-vector form factor and the $\ensuremath{\sigma}$-commutator vertex are deduced. The hard-meson analysis shows that if one wishes to maintain Veneziano form with two mesons off the mass shell, some of the $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ amplitudes must develop poles in one of the off-shell momenta at ${p}^{2}=0$. The soft-pion vertex function equations are seen to be a consequence of removing the leading part of this singularity at the soft-pion point ${p}^{\ensuremath{\mu}}=0$. However, the resultant amplitudes are still singular at ${p}^{2}=0$, ${p}^{\ensuremath{\mu}}\ensuremath{\ne}0$ for a continuum of $s$, $t$, and $u$. The fixed poles implied by the presence of form factors (when two mesons are off shell) are seen to occur only in those amplitudes orthogonal to the ${A}_{1}$ polarization vector and do not affect the amplitudes that make nonzero contributions for $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ scattering on the mass shell.