What do we learn from the $\rho-\pi$ puzzle

Abstract

The experimental observation indicates that the branching ratio of $\psi'\to \rho\pi$ is very small while the $\rho-\pi$ channel is a main one in $J/\psi$ decays. To understand the puzzle, various interpretations have been proposed. Meanwhile according to the hadronic helicity selection rule the decay mode $J/\psi\to\rho\pi$ should be suppressed, but definitely, numerical computation is needed to determine how it is suppressed. We calculate the branching ratios corresponding to subprocesses $J/\psi\to ggg\to\rho\pi$ and $\to ggg\to\pi\pi$ in the framework of QCD. The results show that the branching ratios are proportional to $({m_u+m_d\over M_{J/\psi}})^2$ for $\rho\pi$ mode and $({m_u-m_d\over M_{J/\psi}})^2$ for $\pi\pi$ mode which is an isospin-violation channel. If only the OZI process is considered, the theoretical prediction on the ratio of $J/\psi\to \rho\pi$ is smaller than data, but not too drastically small. Meanwhile, a possible interpretation for the $\rho\pi$ puzzle is proposed that the suppression is due to interference between OZI and electromagnetic (EM) contributions. Thus based on this observation, we suggest that if the amplitudes of the strong OZI process via an s-channel three-gluon intermediate state and electromagnetic one via an s-channel virtual photon intermediate state have the same order of magnitude, and constructively interfere for $J/\psi\to\rho\pi$, but destructively interfere for $\psi'\to\rho\pi$, thus simultaneously the $\rho\pi$ puzzle disappears and the sizable width of $J/\psi\to\rho\i$ is understandable. However, so far, we cannot derive the phase difference from an underlying principle of QCD yet. Alternative interpretations are also discussed in the text.