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Chiral Symmetry and Low Energy Pion-Nucleon Scattering

In these lectures, I examine the effect of the meson factory $π$N data on the current algebra/PCAC program which describes chiral symmetry breaking in this system. After historical remarks on the current algebra/PCAC versus chiral Lagrangians approaches to chiral symmetry, and description of the need for $π$N amplitudes with virtual (off-mass-shell) pions in nuclear force models and other nuclear physics problems, I begin with kinematics and isospin aspects of the invariant amplitudes. A detailed introduction to the hadronic vector and axial-vector currents and the hypothesis of partially conserved axial-vector currents (PCAC) follows. I review and test against contemporary data the PCAC predictions of the Goldberger-Treiman relation, and the Adler consistency condition for a $π$N amplitude. Then comes a detailed description of the current algebra Ward-Takahashi identities in the chiral limit and a brief account of the on-shell current algebra Ward-Takahashi identities. The latter identities form the basis of so-called current algebra models of $π$N scattering. I then test these models against the contemporary empirical $π$N amplitudes extrapolated into the subthreshold region via dispersion relations. The scale and the t dependence of the "sigma term" is determined by the recent data.