Joint description of the $e^{+}{e}^{-}$ annihilation into both four-pion channels

The $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ reaction cross section as a function of the incident energy is calculated using a model that is an extension of our recently published model of the $e^{+}{e}^{-}$ annihilation into four charged pions. The latter considered the intermediate states with the $\pi$, $\rho$, and $a_1$ mesons and fixed the mixing angle of the $a_1\rho \pi$ Lagrangian and other parameters by fitting the cross section data. Here we supplement the original intermediate states with those containing $\omega (782)$ and $h_1(1170)$, but keep unchanged the values of those parameters that enter both charged and mixed channel calculations. The inclusion of $\omega$ is vital for obtaining a good fit to the cross section data, while the intermediate states with $h_1$ further improve it. Finally, we merge our models of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ and $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ reactions and obtain a simultaneous good fit.

Figure 1

Two Feynman diagrams of a pure-$a_1$ model of $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$. Two others can be obtained by exchanging ${\pi }^0$’s.

Figure 2

The generic Feynman diagram describing the $\omega$ contribution to $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$. The other five diagrams are obtained by obvious modifications.

Figure 3

One of the two Feynman diagrams with the contact $\omega 3\pi$ term. Another is obtained by exchanging ${\pi }^0$’s.

Figure 4

Comparison of the $a_1+\omega +h_1$ model with the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ data for two approaches to $\omega$.