Instantons and chiral Lagrangian

We derive the chiral Lagrangian including instanton corrections in a consistent way. CP violation associated with the θ term in QCD is discussed in terms of this chiral Lagrangian. A particular emphasis is given to a phenomenological determination of mᵤ and θ. It is shown that, contrary to the previous conclusion that mᵤ=0.56md and θ=0, all three distinct cases (i) 0.56≥mᵤmd>0, θ=0, (ii) mᵤ=0, (iii) 0.56≥mᵤmd>0, θ=π are phenomenologically acceptable depending on the strength of the instanton corrections. In cases (i) and (iii), the ratio mᵤmd can be much smaller than the usually quoted value 0.56. As a result, the models of spontaneous CP violation which predict |θ|10^-9 can be phenomenologically viable. Also in the axion models, because of the possibility of mᵤmd being arbitrarily small, the cosmological upper bound on the axion decay constant can be relaxed.