A master formula for chiral symmetry breaking

We derive a master formula for chiral SU(2) × SU(2) breaking, based on the Veltman-Bell equations and the Peierls-Dyson relation. Our approach does not rely on the use of the soft pion limit or an expansion around the chiral limit, and yields exact results for on-shell pions. Threshold theorems for πN → πN, γN → πN, πN → ππN, γN → γπN, γN → ππN and πN → πγN are recovered, and corrections to them are given. The reactions π → eνγ, π → eνe⁺e^-, γπ → γπ and γγ → ππ are also discussed. A general formula for ππ scattering and a new one loop effective action are obtained. The new effective action reproduces the KSFR relation, and yields specific estimates for the pion polarisabilities. A detailed comparison with baryon-free chiral perturbation theory to one loop is made. An extension of our effective action to two loops is outlined.