CHIRAL PERTURBATION THEORY AND THE 1/Nc-EXPANSION

1 Expansion in the quark masses and 1/Nc It is generally accepted that QCD possesses an exact U(3)R×U(3)L flavour symmetry in the limitmu=md=ms= 0,Nc→∞. We furthermore assume that (a) the axial part of this symmetry is spontaneously broken and (b) that, in the vicinity of this point, the low energy properties of the theory are governed by the Goldstone bosons associated with this symmetry breakdown. These assumptions provide the basis for the effective theory, where the nine Goldstone degrees of freedom are collected in a matrix U(x) ∈ U(3). Apart from the Wess-Zumino term, the effective Lagrangian represents the most general expression formed with the field U and the source fields for the quark currents that is consistent with chiral symmetry. Furthermore, in order to study the consequences of the U(1)A-anomaly, we include a source field for the winding number density ω = 1/(16π) trc GμνG̃ μν . We denote this field by θ(x); in the QCD-Lagrangian it enters in the form LQCD = −θ ω + ... The terms in the effective Lagrangian are ordered by introducing a counting parameter δ, Leff = L1 + Lδ + Lδ2 + . . . A coherent expansion emerges if powers of momenta, quark masses and 1/Nc are counted according to 2