Renormalization of $K^+ K^- \to \pi^0\pi^+\pi^-$

Abstract

We derive the vertices of five-meson and seven-meson anomaly processes from the four dimensional expansion form of Wess-Zumino term. Using these vertices we calculate the amplitude, both the finite part and divergent part, of {$K^+ K^- \rightarrow \pi^0 \pi^+ \pi^-$} to one loop and renormalize the lagrangian. The divergent part agrees with the result derived from path integral approach. Contribution from counter terms is estimated by using the vector meson dominance model. Test of the vertex in the $t$-channel of $K^- P\rightarrow\Sigma^0\pi^0\pi^+\pi^-$ near the threshold is discussed. We find that the amplitudes arising from chiral loop and counter terms are of opposite sign and the counter term amplitude is about twice the loop amplitude.