Solitons in an effective theory of CP violation

Abstract

We study an effective field theory describing CP-violation in a scalar meson sector. We write the simplest interaction that we can imagine, $$\mathcal{L}\sim {ϵ}_{i_1\cdots i_5}{ϵ}^{{\mu }_1\cdots {\mu }_4}{\varphi }_{i_1}{\partial }_{{\mu }_1}{\varphi }_{i_2}{\partial }_{{\mu }_2}{\varphi }_{i_3}{\partial }_{{\mu }_3}{\varphi }_{i_4}{\partial }_{{\mu }_4}{\varphi }_{i_5}$$ which involves 5 scalar fields. The theory describes CP-violation only when it contains scalar fields representing mesons such as the $K_0^{\ast }$, 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 $KK\to \pi \pi \pi$. 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.

Comments: 15 pages, 1 figure