Double $\phi$ Production in $\bar{p}p$ Reactions Near Threshold

We employ an effective Lagrangian approach to investigate double $\phi$ production in the $\bar{p}p$ reaction near the threshold and describe a notable violation of the Okubo-Zweig-Iizuka (OZI) rule in this reaction process through hadronic degrees of freedom, using the currently available theoretical and experimental information. The ground-state nucleon and its $s$-wave resonances, i.e., $N^*(1535,1650,1895)$, are taken into account in the $t$- and $u$-channel tree-level diagrams. The pentaquark-like nucleon resonance $P_s(2071,3/2^-)$ is also considered. In the $s$-channel diagrams, scalar and tensor mesons are incorporated, specifically $f_0(2020,2100,2200)$ and $f_2(1950,2010,2150)$, respectively, along with the pseudoscalar meson $\eta(2225)$. Our calculations suggest that contributions from the $N$ and $N^*$ resonances significantly enhance the cross-section near the threshold. At the same time, those from the $f_0$ and $f_2$ mesons produce distinctive peak structures around $W=2.2$ GeV, qualitatively reproducing the JETSET experimental data. The openings of the $\bar{\Lambda}\Lambda$ and $\bar{\Sigma}\Sigma$ channels lead to the cusp structures, which can explain the nontrivial structures in the cross-section at $W\approx2M_{\Lambda}$ and $2M_{\Sigma}$ observed in the data. It turns out that contributions from the $\eta$ and $P_s$ resonances are almost negligible. Additionally, polarization observables such as the spin density matrix element (SDME) provide crucial insights into the individual hadronic processes involved in this reaction.