Khuri–Treiman equations for \(\pi \pi \) scattering

The Khuri–Treiman formalism models the partial-wave expansion of a scattering amplitude as a sum of three individual truncated series, capturing the low-energy dynamics of the direct and cross channels. We cast this formalism into dispersive equations to study \(\pi \pi \) scattering, and compare their expressions and numerical output to the Roy and GKPY equations. We prove that the Khuri–Treiman equations and Roy equations coincide when both are truncated to include only S- and P-waves. When higher partial waves are included, we find an excellent agreement between the Khuri–Treiman and the GKPY results. This lends credence to the notion that the Khuri–Treiman formalism is a reliable low-energy tool for studying hadronic reaction amplitudes.