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A MASTER FORMULA FOR CHIRAL SYMMETRY BREAKING

We derive a master formula for chiral $SU(2)\times SU(2)$ 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 $πN\rightarrow πN$, $γN\rightarrow πN$, $πN\rightarrowππN$, $γN\rightarrow γπN$, $γN\rightarrow ππN$ and $πN\rightarrow πγN$ are recovered, and corrections to them are given. The reactions $π\rightarrow eνγ$, $π\rightarrow eνe^+e^-$, $γπ\rightarrow γπ$ and $γγ\rightarrow ππ$ 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.