Bose Symmetry Interference Effects of $4\pi$ Final States

We carefully analyze the relative branching ratios of $4\pi$ final states ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, ${\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ and ${\pi }^0{\pi }^0{\pi }^0{\pi }^0$, from various resonances of $J^{PC}=0^{++}$, $0^{-+}$, $2^{++}$. We find that the Bose symmetry interference effects would make their ratios to obviously differ from the naive counting values without considering these effects. The results should be applied to estimate correctly various $4\pi$ decay branching ratios of relevant resonances.