Lower bounds to the I = 0, I = 2 ππ s-wave scattering lenghts

Abstract

Roy's exact partial wave equations allow us to find bounds for any linear combination of the isospin I = 0 and s-wave scattering lengths, with positive coefficients. The bound is a function of the quantity a_D = a₂⁽⁰⁾ + a₂⁽²⁾, where a₂^(I) are the D-wave sacttering lengths. Thus, we can draw on the (a₀⁽⁰⁾, a₀⁽²⁾) plane an allowed domain whose boundary is fairly close to the phenomenological region. For a value of a_D = 1.7×10⁻³, we find the following particular bounds: a₀⁽⁰⁾⩾−0.49, a₀⁽²⁾⩾−0.29.