The Low Energy $\pi\,\pi$ Amplitude to One and Two Loops

Abstract

The low-energy $\pi\pi$ amplitude is computed explicitly to two-loop accuracy in the chiral expansion. It depends only on six independent (combinations of) low-energy constants which are not fixed by chiral symmetry. Four of these constants are determined {\it via} sum rules which are evaluated using $\pi\pi$ scattering data at higher energies. Dependence of the low-energy phase shifts and of the threshold parameters on the remaining two constants (called $\alpha$ and $\beta$) are discussed and compared to the existing data from $K_{l4}$ experiments. Using generalised $\chi$PT, the constants $\alpha$ and $\beta$ are related to fundamental QCD parameters such as the quark condensate $\langle 0|\bar{q}q|0\rangle$ and the quark mass ratio $m_s/\widehat{m}$. It is shown that forthcoming accurate low-energy $\pi\pi$ data can be used to provide, for the first time, experimental evidence in favour of or against the existence of a large quark-antiquark condensate in the QCD vacuum.Comment: 61 pages, LaTeX, 10 figures in separate tarred, compressed and uuencoded Postscript fil