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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, $${\cal L}\sim ε_{i_1\cdots i_5}ε^{μ_1\cdotsμ_4}ϕ_{i_1}\partial_{μ_1}ϕ_{i_2}\partial_{μ_2}ϕ_{i_3}\partial_{μ_3}ϕ_{i_4}\partial_{μ_4}ϕ_{i_5}$$ 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 $π$ mesons then the Lagrangian describes anomalous processes such as $KK\to πππ$. 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.