Investigation of electroproduction of $\phi$ mesons off protons

We investigate $\phi$-meson electroproduction off the proton target, i.e., $\gamma^* p \to \phi p$, by employing a tree-level effective Lagrangian approach in the kinematical ranges of $Q^2$ = (0$-$4) $\mathrm{GeV}^2$, $W$ = (2$-$5) GeV, and $|t| \leq 2\,\mathrm{GeV}^2$. In addition to the universally accepted Pomeron exchange, we consider various meson exchanges in the $t$ channel with the Regge method. Direct $\phi$-meson radiations in the $s$- and $u$-channels are also taken into account. We find that the $Q^2$ dependence of the transverse ($\sigma_{\mathrm{T}}$) and longitudinal ($\sigma_{\mathrm{L}}$) cross sections are governed by Pomeron and $(a_0,f_0)$ scalar meson exchanges, respectively. Meanwhile, the contributions of $(\pi,\eta)$ pseudoscalar- and $f_1(1285)$ axial-vector-meson exchanges are much more suppressed. The results of the interference cross sections ($\sigma_{\mathrm{LT}}, \sigma_{\mathrm{LT}}$) and the spin-density matrix elements indicate that $s$-channel helicity conservation holds at $Q^2$ = (1$-$4) $\mathrm{GeV}^2$. The result of the parity asymmetry yield $P \simeq 0.95$ at $W$ = 2.5 GeV, meaning that natural-parity exchange dominates the reaction process. Our numerical results are in fair agreement with the experimental data and thus the use of our effective Reggeized model is justified over the considered kinematical ranges of $Q^2$, $W$, and $t$.