Contribution of the $\rho$-Exchange Diagram to Photoproduction of $\eta$ Mesons on Protons*

G. M. Radutskii, V. A. Serdyutskii, A. N. Tabachenko

(Presented by the scientific seminar of the high-energy sector of the Research Institute of Nuclear Physics, Electronics, and Automation)

Using $\rho$-photon analogy, we calculated the angular distribution of $\eta$ mesons produced by $\gamma$ quanta with energies of 1 and 1.5 GeV on protons, taking into account the structure of the proton. The results obtained do not contradict the existing experimental data.

Introduction

At present there is very little information on the behavior of the cross section for photoproduction of $\eta$ mesons. Experiments [1–5], carried out at $E_\gamma$ from the reaction threshold $\gamma p \to \eta p$ up to 1100 MeV and at angles of about $90^\circ$ in the c.m.s., indicate the existence of an $\eta N$ resonance.

At higher energies, the role of peripheral diagrams may possibly increase. The most important of these will be the diagram with $\rho$-meson exchange. The contribution of the $\varphi$-exchange diagram will be suppressed because of the smallness of the interaction constant of the $\varphi$ meson with nucleons. As for the $\omega$-exchange diagram, its contribution will be smaller than that of the $\rho$-exchange diagram, since, first, $g_{\rho\eta\gamma}/g_{\omega\eta\gamma}=3$, according to $SU(6)$ symmetry, and, second, the contributions of the $\omega pp$ and $\rho pp$ vertices are related as the isoscalar and isovector magnetic moments of the nucleon. The diagram with one-photon exchange gives a contribution roughly $10^4$ times smaller than the $\rho$-exchange one.

In the present work, to calculate the contribution of the $\rho$-exchange diagram, a model known as “$\rho$-photon analogy” [6] is used. The basic idea of the model is that the meson is regarded as a “heavy” isovector photon when the expression for the matrix element is written down. For the present reaction this statement reduces to the equality

\[ \langle P|J_\mu^\rho(0)|P\rangle = \frac{f_{\rho NN}}{e} \langle P|J_\mu^\gamma(V)(0)|P\rangle \]

in the region of transferred-momentum values under consideration. In addition, the existence of an effective interaction constant $f_{\rho\eta\gamma}$ is assumed; its magnitude is chosen in accordance with the requirements of $SU(6)$ symmetry [7].

* This work is a report by the authors at the International Conference on Electromagnetic Interactions at Low and Intermediate Energies, Dubna, 1967.

Calculation of the differential cross section

The invariant amplitude of the reaction under consideration has the form:

\[ \frac{f_{\rho\eta\gamma} f_{\rho NN}}{e m_\rho}\, \varepsilon_{\mu\nu\lambda\sigma} e_\mu q_\lambda \kappa_\sigma\, \frac{\langle p_2 | J_\nu^V(0) | p_1\rangle}{(q-\kappa)^2 - m_\rho^2}, \]

where \(e_\mu\) is the photon polarization vector, and \(q,\ \kappa,\ p_2,\ p_1\) are the 4-momenta of the \(\eta\)-meson, photon, final and initial protons, respectively. The vertex \(\rho \to \eta+\gamma\) was written in the form:

\[ \frac{f_{\rho\eta\gamma}}{m_\rho}\, \varepsilon_{\mu\nu\lambda\sigma} e_\mu(\gamma) e_\nu(\rho) p_\lambda \kappa_\sigma . \]

For the matrix element of the isovector part of the electromagnetic current, the known expression was taken:

\[ \langle p_2 | J_\nu^V(0) | p_1\rangle = \bar e u(p_2) \left\{ F_1^V(t)\gamma_\nu + i\sigma_{\nu\mu}(\kappa-q)_\mu F_2^V(t) \right\} u(p_1). \]

Here \(F_1^V\) and \(F_2^V\) are the isovector electromagnetic form factors of the nucleon, and \(t=(q-\kappa)^2\) is the square of the transferred momentum.

Hence the following formula is obtained for the differential cross section of photoproduction of \(\eta\)-mesons on protons in the c.m. system:

\[ \frac{d\sigma}{dt} = -\frac{\pi}{4} \left(\frac{1}{4\pi\omega W}\right)^2 \frac{f_{\rho\eta\gamma}^{\,2} f_{\rho NN}^{\,2}}{(t-m_\rho^2)^2} \left\{ \frac{ G_E^{V\,2}(t)-\dfrac{t}{4M^2}G_M^{V\,2}(t) }{ 1-\dfrac{t}{4M^2} } \times \right. \]

\[ \left. \times \left[ W^2 t^2 + t\left((W^2-M^2)^2-W^2-M^2\right) + M^2 \right] + G_M^{V\,2}(t)\, \frac{t(t-1)^2}{2} \right\}. \]

Here \(W\) is the total energy, \(\omega\) is the photon energy, with \(\hbar=c=m_\eta=1\),

\[ G_E^V(t)=F_1^V(t)+\frac{t}{2M}F_2^V(t), \]

\[ G_M^V(t)=F_1^V(t)+2M F_2^V(t). \]

\(M\) is the proton mass.

Fig. 1. Dependence of the differential cross section of the process
\(\gamma+p\to\eta+p\) on the emission angle of the \(\eta\)-meson in the c.m. system at energy \(E_\gamma=1\ \text{GeV}\) (a) and \(E_\gamma=1.5\ \text{GeV}\) (b).

Results

The differential cross section for photoproduction of \(\eta\)-mesons was calculated for \(\gamma\)-quantum energies \(E_\gamma=1\ \text{GeV}\) and \(1.5\ \text{GeV}\). The proton form factors were taken from Ref. [8]. The following values of the coupling constants were chosen:

\[ \frac{f_{\rho\eta\gamma}^{\,2}}{4\pi}=0.013; \qquad \frac{f_{\rho NN}^{\,2}}{4\pi}=2. \]

The dependence of the differential cross sections on the angle at the selected photon energies is presented in Fig. 1. The total cross sections are equal to \(2.3\cdot10^{-30}\ \text{cm}^2\) and \(2.6\cdot10^{-30}\ \text{cm}^2\) at \(E_\gamma=1\ \text{GeV}\) and \(1.5\ \text{GeV}\), respectively.

The experimental data currently available are insufficient to speak of any agreement of the proposed model with experiment. However, they do not contradict the results obtained.

Note

During the International Conference on Electromagnetic Interactions at Low and Medium Energies in Dubna (February 7–15, 1967), a series of experiments on the photoproduction of \(\eta\)-mesons on protons in the energy region under consideration was carried out. An analysis of the data, performed in [9, 10] with the aid of a resonance model, showed that a good description of the reaction cross section is achieved when contributions from the nucleon pole, the vector-meson pole, and the \(S_{11}(1570)\) and \(D_{13}(1512)\) resonances are taken into account. In addition, the importance of the contribution of the \(\rho\)-exchange diagram to the photoproduction of \(\eta\)-mesons on protons was pointed out in [11] on the basis of the vector-dominance model of electromagnetic interactions.

References

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