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 $\pi N\rightarrow \pi N$, $\gamma N\rightarrow \pi N$, $\pi N\rightarrow\pi\pi N$, $\gamma N\rightarrow \gamma\pi N$, $\gamma N\rightarrow \pi\pi N$ and $\pi N\rightarrow \pi\gamma N$ are recovered, and corrections to them are given. The reactions $\pi\rightarrow e\nu \gamma$, $\pi\rightarrow e\nu e^+e^-$, $\gamma\pi\rightarrow \gamma\pi$ and $\gamma\gamma\rightarrow \pi\pi$ are also discussed. A general formula for $\pi\pi$ 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.