Mixing Effects forφ,ω, andρ0Mesons

A method based on the properties of the propagator which was developed for treating the time-dependent behavior of almost degenerate unstable particles is used to discuss mixing effects in resonance phenomena. General equations for the location of the poles and the mixing of states in the production amplitudes are presented. The method is applied to the mixing of $\phi$, $\omega$, and ${\rho }^0$ mesons. Mixing coefficients and poles are determined explicitly in terms of the elements of the matrix representing the square of the mass. In particular an expression is given for the complex amplitude, $s$, of $\phi$, $\omega$ mixing but it is indicated that $\mathrm{Im}s\ll \mathrm{Re}s$ so that the usual approximation of treating $s$ as real is probably good enough for most purposes. The $2\pi$ and $3\pi$ production amplitudes due to ${\rho }^0$ and $\omega$ production are shown to depend strongly on the production mechanism, as already noted by Bernstein and Feinberg. For example it is found that production of the $2\pi$ model will have its maximum at the $\omega$ pole in a process dominated by $\rho$ exchange. It is also shown that ${\rho }^0$ and $\omega$ masses could be taken to be nearly equal if the apparent splitting of the masses is caused by destructive interference. It is found that, in general, $\mathrm{PT}$ invariance implies that the mass matrix is symmetric.