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

Abstract

The low-energy $ππ$ 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 $ππ$ scattering data at higher energies. Dependence of the low-energy phase shifts and of the threshold parameters on the remaining two constants (called $α$ and $β$) are discussed and compared to the existing data from $K_{l4}$ experiments. Using generalised $χ$PT, the constants $α$ and $β$ 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 $ππ$ 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.