Standard Model and Beyond · Problem 3.13

Gauge Theories

Physics — Particle Physics · isospin-pion-nucleon-scattering

Problem

The pion and nucleon transform as isospin triplets and doublets, respectively. The approximate isospin invariance of the strong interactions implies that all of the amplitudes for πiNa→πjNb\pi_i N_a \to \pi_j N_b can be expressed in terms of two amplitudes MIM_I, corresponding to total isospin I=1/2I = 1/2 and 3/23/2. Write the amplitudes for π+p→π+p\pi^+ p \to \pi^+ p, π−p→π−p\pi^- p \to \pi^- p, π−p→π0n\pi^- p \to \pi^0 n, and π0n→π0n\pi^0 n \to \pi^0 n in terms of the MIM_I. (Note that at low energy the M3/2M_{3/2} amplitude, which is dominated by the Δ\Delta resonance, strongly dominates. Such calculations can be extended to SU(3)SU(3) by means of isoscalar factors (de Swart, 1963; Patrignani, 2016).)

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