Solitons in an effective theory of CP violation

We study an effective field theory describing CP-violation in a scalar meson sector. We write the simplest interaction that we can imagine, $${cal L}sim epsilon_{i_1cdots i_5}epsilon^{mu_1cdotsmu_4}phi_{i_1}partial_{mu_1}phi_{i_2}partial_{mu_2}phi_{i_3}partial_{mu_3}phi_{i_4}partial_{mu_4}phi_{i_5}$$ which involves 5 scalar fields. The theory describes CP-violation only when it contains scalar fields representing mesons such as the $K^*_0$, sigma, $f_0$ or $a_0$. If the fields represent pseudo-scalar mesons, such as B, K and $pi$ mesons then the Lagrangian describes anomalous processes such as $KKto pipipi$. We speculate that the field theory contains long lived excitations corresponding to $Q$-ball type domain walls expanding through space-time. In an 1+1 dimensional, analogous, field theory we find an exact, analytic solution corresponding to such solitons. The solitons have a U(1) charge $Q$, which can be arbitrarily high, but oddly, the energy behaves as $Q^{2/3}$ for large charge, thus the configurations are stable under disintegration into elementary charged particles of mass $m$ with $Q=1$. We also find analytic complex instanton solutions which have finite, positive Euclidean action.