The low energy pi pi amplitude to one and two loops

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 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-14 experiments. Using generalized chi PT, the constants cu and beta are related to fundamental QCD parameters such as the quark condensate [0\<(q)over bar q\ 0] and the quark mass ratio m(s)/(m) over cap. 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.