The ππ total cross section: Its scale and massless pion limit

The question of the scale of the ππ total cross section is considered. We first give a simple intuitive argument for expecting cross sections in the massless chiral world to be finite. Assuming the universality of P-f Regge couplings, we give a simple dispersion theoretic evaluation of the asymptotic ππ total cross section, σ^ππ. We deduce an algebraic expression for σ^ππ and find that its scale is determined by 1/m_ρ{variant}² or, equivalently, 1/f{hook}_π² - just as Pagels conjectured. This relation has a smooth, finite limit as m_π tends to zero. Numerically, we obtain σ^ππ = 17 ± 4 mb for physical mass pions and 14 ± 3 mb for massless pions.