Dynamical model of $\varphi$ meson photoproduction on the nucleon and ${}^4\mathrm{He}$

We investigate $\varphi$ meson photoproduction on the nucleon and ${}^4\mathrm{He}$ targets within a dynamical model approach based on a Hamiltonian which describes the production mechanisms by the Pomeron exchange, meson exchange, $\varphi$ radiation, and nucleon resonance excitation mechanisms. The final $\varphi N$ interactions are included and described by the gluon-exchange, direct $\varphi N$ couplings, and the box diagrams arising from the couplings with $\pi N, \rho N, K\mathrm{\Lambda }$, and $K\mathrm{\Sigma }$ channels. The parameters of the Hamiltonian are determined by the experimental data of $\gamma p\to \varphi p$ from the CLAS Collaboration. The resulting Hamiltonian is then used to predict the coherent $\varphi$-meson production on the ${}^4\mathrm{He}$ targets by using the distorted-wave impulse approximation. For the proton target, the final $\varphi N$ rescattering effects, as required by the unitarity condition, are found to be very weak, which supports the earlier calculations in the literature. For the ${}^4\mathrm{He}$ targets, the predicted differential cross sections are in good agreement with the data obtained by the LEPS Collaboration. The role of each mechanism in this reaction is discussed and predictions for a wide range of scattering angles are presented, which can be tested in future experiments.

Figure 1

Total amplitude of the $\gamma N\to \varphi N$ reaction: $B$ is the production amplitude, $t$ is the $\varphi N$ scattering amplitude, and $N^{*}$ is the nucleon resonance contributions.

Figure 2

Diagrams for the $\varphi N\to \varphi N$ interactions. (a) Gluon-exchange within QCD, (b), (c) mechanisms by the direct $\varphi N$ coupling, and (d)–(f) box diagrams arising from the $\varphi \to K\overline{K}$ and $\varphi \to \pi \rho$ decays.

Figure 3

Born terms of the $\gamma N\to \varphi N$ reaction.

Figure 4

Total cross sections for photoproduction of ${\rho }^0, \varphi , J/\psi$, and $\mathrm{{\rm Y}}\left(1s\right)$ on proton targets. The data are from Ref. [46].

Figure 5

$N^{*}$ contributions to the total cross section of $\gamma p\to \varphi p$.

Figure 6

Differential cross sections $d\sigma /dcos\theta$ of the $\gamma p\to \varphi p$ reaction without FSI as functions of $cos\theta$ for energies $\sqrt{s}=W=(1.985–2.815)$ GeV. The blue dotted lines are from the calculations including Pomeron exchange only. The green dot-dashed lines show the sum of the Pomeron and meson exchange mechanisms. The black solid lines and the red dashed lines represent the full contribution with and without $N^{*}$, respectively, which are almost identical at $\sqrt{s}\ge 2.57$ GeV. The experimental data are from the CLAS Collaboration [53].

Figure 7

Total cross section as a function of the photon energy in the laboratory frame $E_{\gamma }^{\mathrm{lab}}$. The blue dotted line stands for the Pomeron exchange only. The green dot-dashed and the magenta dot-dashed-dashed lines indicate the meson exchanges and $N^{*}$ contributions, respectively. The black solid and the red dashed lines represent the full contributions with and without $N^{*}$, respectively. The experimental data are from Refs. [56] (open squares), [57] (open circles), [58] (open triangles), and [59] (stars).

Figure 8

Differential cross sections of $\gamma p\to \varphi p$ without FSI at lower c.m. energies as functions of $t^{\prime }=\left|t\right|-{\left|t\right|}_{\min }$. The blue dotted lines are from the calculations including Pomeron exchange only. The black solid lines and the red dashed lines represent the full contribution with and without $N^{*}$, respectively. The experimental data are from Ref. [53] (gray filled circles) and Ref. [60] (green filled squares).

Figure 9

Total cross sections of $\gamma p\to \varphi p$ at low energies $E_{\gamma }$. (a) The red dashed and the blue dotted lines stand for the impulse and the FSI terms, respectively. The solid black line indicates the full contribution. The experimental data are from Refs. [56] (open squares) and [57] (open circles). (b) The individual contributions of the FSI terms.

Figure 10

Differential cross sections $d\sigma /dcos\theta$ of the $\gamma p\to \varphi p$ reaction as functions of $cos\theta$ for different energies $\sqrt{s}=(1.985–2.815)$ GeV with and without the FSI effects. The red dashed and the blue dotted lines stand for the impulse and the FSI terms, respectively. The solid black lines indicate the full contributions. The experimental data are from the CLAS Collaboration [53].

Figure 11

Differential cross sections $d\sigma /dt$ at $t=t_{\min }$ as a function of $E_{\gamma }$ for the reaction of $\gamma {}^4\mathrm{He}\to \varphi {}^4\mathrm{He}$. The experimental data are from the LEPS Collaboration [14].

Figure 12

Differential cross sections $d\sigma /dt$ of $\gamma {}^4\mathrm{He}\to \varphi {}^4\mathrm{He}$ as functions of $t^{\prime }\equiv \left|t\right|-{\left|t\right|}_{\min }$ at $1.685<E_{\gamma }<2.385$ GeV. The dotted lines and the solid lines stand for the Pomeron and full contributions, respectively. The experimental data are from Ref. [14].

Figure 13

Same as Fig. 12 but for a wider range of $t^{\prime }=\left|t\right|-{\left|t\right|}_{\min }$. The dotted lines and the solid lines stand for the Pomeron and full contributions, respectively. The dot-dashed lines are from the $N^{*}$ contributions. The experimental data are from Ref. [14].

Figure 14

Same as Fig. 13 but for a wider range of $t^{\prime }=\left|t\right|-{\left|t\right|}_{\min }$. The dotted lines are the results with impulse approximation and FSI contributions are shown by the dashed lines. Their sums are given by the solid lines. The experimental data are from Ref. [14].