
pmid: 10011890
We derive the chiral Lagrangian including instanton corrections in a consistent way. $\mathrm{CP}$ violation associated with the $\ensuremath{\theta}$ term in QCD is discussed in terms of this chiral Lagrangian. A particular emphasis is given to a phenomenological determination of ${m}_{u}$ and $\ensuremath{\theta}$. It is shown that, contrary to the previous conclusion that ${m}_{u}=0.56{m}_{d}$ and $\ensuremath{\theta}=0$, all three distinct cases (i) $0.56\ensuremath{\ge}\frac{{m}_{u}}{{m}_{d}}g0$, $\ensuremath{\theta}=0$, (ii) ${m}_{u}=0$, (iii) $0.56\ensuremath{\ge}\frac{{m}_{u}}{{m}_{d}}g0$, $\ensuremath{\theta}=\ensuremath{\pi}$ 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 $|\ensuremath{\theta}|\ensuremath{\gg}{10}^{\ensuremath{-}9}$ 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.
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