Instantons and chiral Lagrangian

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