Angular Dependence of η Photoproduction in Photon-Induced Reaction

Abstract

Photoproduction of $\eta$ mesons from nucleons can provide valuable information about the excitation spectrum of the nucleons. The angular dependence of $\eta$ photoproduction in the photon-induced reaction is investigated in the multisource thermal model. The results are compared with experimental data from the $\eta \to \mathrm{3}{\pi }^{\mathrm{0}}\to \mathrm{6}\gamma$ decay mode. They are in good agreement with the experimental data. It is shown that the movement factor increases linearly with the photon beam energies. And the deformation and translation of emission sources are visually given in the formalism.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3 .

1. Introduction

The excitation spectrum of nucleons is important to understanding the nonperturbative behavior of the fundamental theory of strong interactions, Quantum Chromodynamics (QCD) [1–4]. The photon-induced meson production off nucleons is mainly used to achieve more information from the excitation spectrum of nucleons. It is very important for missing resonances that the $\eta$ meson production in photon-induced and hadron-induced reactions on free (and quasi-free) nucleons and on nuclei [5–8]. The advantage of photon-induced reactions is that the electromagnetic couplings can provide valuable information related to the details of the model wave functions. Because the electromagnetic excitations are isospin dependent, we need perform meson-production reactions off the neutron.

Recently, the photoproduction of $\eta$ mesons from quasi-free protons and neutrons was measured in $\eta \to \mathrm{3}{\pi }^{\mathrm{0}}\to \mathrm{6}\gamma$ decay mode by the CBELSA/TAPS detector at the electron accelerator ELSA in Bonn [9]. At different incident photon energies, the experiments are performed by the incident photon beam on a liquid deuterium target. A great number of $\eta$ mesons are produced in the photon-induced reaction. The experimental data are regarded as a multiparticle system. And, their angular distributions represent an obvious regularity at different incident photon energies. In order to explain the abundant experimental results, some statistical methods are proposed and developed [10–16]. In this work, we will extend a multi-source thermal model to the statistical investigation of the angular distributions in the photon-induced reaction and try to understand the $\eta$ photoproduction in the reaction. In our previous wok [17–21], the model was focused on the investigation of the particle production in intermediate-energy and high-energy collisions.

2. $\eta$ Meson Distribution in the Multi-Source Thermal Model

In the multi-source thermal model [17–21], many emission sources are expected to be formed at the final stage of the photon-induced reaction. Every source emits particles isotropically in the source rest frame. The observed $\eta$ mesons are from different emission sources. The incident beam direction is defined as an $oz$ axis and the reaction plane is defined as $yoz$ plane. In the source rest frame, the meson momentum $p_x$ , $p_y$ , and $p_z$ obeys a normal distribution. The corresponding transverse momentum $p_T=\sqrt{p_x^{\mathrm{2}}+p_y^{\mathrm{2}}}$ obeys a Rayleigh distribution: $$\begin{array}{}\text{(1)}& f_{p_T}\left(p_T\right)=\frac{p_T}{{\sigma }^{\mathrm{2}}}{e}^{-p_T^{\mathrm{2}}/\mathrm{2}{\sigma }^{\mathrm{2}}},\end{array}$$ where $\sigma$ represents a distribution width. The distribution function of the polar angle $\theta$ is $$\begin{array}{}\text{(2)}& f_{\theta }\left(\theta \right)=\frac{\mathrm{1}}{\mathrm{2}}\sin \theta \end{array}$$

Because of the interactions with other emission sources, the considered source deforms and translates along the $oz$ axis. Then, the momentum component is revised to $$\begin{array}{}\text{(3)}& p_z^{\prime }=a_z{p}_z+b_z\sigma \end{array}$$ where $a_z$ and $b_z$ represent the coefficients of the source deformation and translation along the $oz$ axis, respectively. The mathematical description of the deformable translational source is formulized simply as a linear relationship between $p_z^{\prime }$ and $p_z$ , which reflects the mean result of the source interaction. For $a_z\ne \mathrm{1}$ or $b_z\ne \mathrm{0}$ , the $p_z^{\prime }$ distribution of $\eta$ mesons is anisotropic along the $oz$ axis.

By using Monte Carlo method, $p_T$ and $p_z^{\prime }$ are given by $$\begin{array}{}\text{(4)}& p_T=\sigma \sqrt{-\mathrm{2}\ln r_{\mathrm{1}}},\text{(5)}& p_z^{\prime }=a_z\sigma \sqrt{-\mathrm{2}\ln r_{\mathrm{2}}}\cos \left(\mathrm{2}\pi r_{\mathrm{3}}\right)+b_z\sigma ,\end{array}$$ where $r_{\mathrm{1}}$ , $r_{\mathrm{2}}$ , and $r_{\mathrm{3}}$ are random numbers from 0 to 1. The polar angle $\theta$ is revised to $$\begin{array}{}\text{(6)}& {\theta }^{\prime }=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{p_T}{p_z^{\prime }}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{\sqrt{-\mathrm{2}\ln r_{\mathrm{1}}}}{a_z\sqrt{-\mathrm{2}\ln r_{\mathrm{2}}}\cos \left(\mathrm{2}\pi r_{\mathrm{3}}\right)+b_z}.\end{array}$$ We can calculate a new distribution function of the polar angle by this formula.

3. Angular Dependencies of $\eta$ Photoproduction in the Photon-Induced Reaction

Figures 1(a)–1(p) show the angular distributions of $\eta$ mesons for different bins of incident photon energy 698 MeV $\le {{\rm E}}_{\gamma }\le 1005$ MeV as a function of $\cos {\theta }_{\eta }$ . ${\theta }_{\eta }$ is the polar angle of $\eta$ meson in the beam-target cm system assuming the initial state nucleon at rest. The symbols represent the experimental data from the CBELSA/TAPS detector at the electron accelerator ELSA in Bonn [9]. The results obtained by using the multi-source thermal model are shown with the curves, which behave in the same way as the experimental data in the 16 bins of incident photon energy. By minimizing ${\chi }^{\mathrm{2}}$ per degree of freedom ( ${\chi }^{\mathrm{2}}/\mathrm{d}\mathrm{o}\mathrm{f}$ ), we determine the corresponding parameters $a_z$ and $b_z$ , which are presented in Table 1. It is found that there is an almost linear relationship between the $b_z$ and ${{\rm E}}_{\gamma }$ . As representative energies of Figure 1, we give a schematic sketch of these emission sources at the four different energies in Figure 7(a). The deformations and translations can be seen intuitively in the figure.

Table 1

Values of $a_z$ and $b_z$ taken in Figure 1 model results.

Figure 1	$E_{\gamma }$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	698	1.050	-0.170	0.118
(b)	712	1.020	-0.210	0.105
(c)	730	1.050	-0.170	0.090
(d)	750	1.040	-0.155	0.134
(e)	770	0.985	-0.105	0.182
(f)	790	0.970	-0.125	0.179
(g)	810	0.979	-0.110	0.194
(h)	830	0.960	-0.120	0.192
(i)	850	0.970	-0.110	0.200
(j)	870	0.980	-0.130	0.211
(k)	890	0.950	-0.120	0.175
(l)	910	0.970	-0.140	0.261
(m)	930	0.940	-0.110	0.179
(n)	955	0.950	-0.120	0.154
(o)	980	0.950	-0.090	0.138
(p)	1005	0.920	-0.110	0.150

[figure omitted; refer to PDF]

In Figures 2(a)–2(p) and Figures 3(a)–3(p), we present the angular distributions of $\eta$ mesons for different bins of incident photon energy 1035 MeV $\le {{\rm E}}_{\gamma }\le$ 1835 MeV as a function of $\cos {\theta }_{\eta }$ . ${\theta }_{\eta }$ is the polar angle of $\eta$ meson in the beam-target cm system assuming the initial state nucleon at rest. Same as Figure 1, the symbols represent the experimental data from the CBELSA/TAPS detector at the electron accelerator ELSA in Bonn [9]. The results obtained by using the multisource thermal model are shown with the curves, which behave in the same way as the experimental data in the 28 bins of incident photon energy. Parameters $a_z$ and $b_z$ are presented in Tables 2 and 3. As the representative energies of Figures 1 and 2, the schematic sketches of the emission sources are given at different energies in Figures 7(b) and 7(c).

Table 2

Values of $a_z$ and $b_z$ taken in Figure 2 model results.

Figure 2	$E_{\gamma }$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	1035	0.920	-0.080	0.124
(b)	1065	0.940	-0.090	0.118
(c)	1095	0.970	-0.030	0.132
(d)	1125	0.870	0.030	0.165
(e)	1155	0.850	0.050	0.170
(f)	1185	0.850	0.030	0.168
(g)	1215	0.860	0.070	0.173
(h)	1245	0.860	0.050	0.122
(i)	1270	0.850	0.110	0.121
(j)	1295	0.830	0.150	0.145
(k)	1325	0.910	0.120	0.147
(l)	1355	0.910	0.160	0.142
(m)	1385	0.890	0.110	0.115
(n)	1415	0.930	0.130	0.106
(o)	1445	0.830	0.250	0.122
(p)	1475	0.810	0.210	0.091

Table 3

Values of $a_z$ and $b_z$ taken in Figure 3 model results.

Figure 3	$E_{\gamma }$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	1505	0.915	0.260	0.243
(b)	1535	0.920	0.260	0.190
(c)	1565	0.920	0.240	0.158
(d)	1595	0.800	0.360	0.315
(e)	1625	0.890	0.310	0.179
(f)	1655	0.930	0.340	0.190
(g)	1685	0.910	0.360	0.206
(h)	1715	0.940	0.400	0.235
(i)	1745	0.960	0.460	0.085
(j)	1775	0.960	0.410	0.144
(k)	1805	1.090	0.400	0.132
(l)	1835	1.060	0.390	0.140

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

In Figures 4, 5, and 6, we show angular distributions in the $\eta$ -nucleon cm system for $\gamma p\to p\eta$ reaction for the different bins of final state energy 1488 MeV $\le W\le$ 1625 MeV, 1635 MeV $\le W\le$ 1830 MeV and 1850 MeV $\le W\le$ 2070 MeV, respectively. Same as Figure 1, the model results and experimental data are indicated by the curves and symbols, respectively. The model results can also agree with the experimental data. In the same way, the deformations and translations of these emission sources are given in Tables 4–6 and Figures 7(d)–7(f). All the parameter values taken in the above calculations are also given in Figures 8 and 9. It can be found that $a_z$ keeps almost invariable and fluctuates around 1.0 with the increasing ${{\rm E}}_{\gamma }$ . The parameter $b_z$ increases linearly with the increasing ${{\rm E}}_{\gamma }$ and their relationship can be expressed by a linearly function, $b_z=(\mathrm{0.541}\pm \mathrm{0.005})\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}{{\rm E}}_{\gamma }-(\mathrm{0.622}\pm \mathrm{0.011})$ . There are similar relationships between the parameters and different final state energies $W$ in Figure 9, where the fitting function of $b_z$ is $b_z=(\mathrm{0.808}\pm \mathrm{0.003})\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}W-(\mathrm{1.322}\pm \mathrm{0.007})$ .

Table 4

Values of $a_z$ and $b_z$ taken in Figure 4 model results.

Figure 4	$W$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	1488	0.956	-0.098	0.086
(b)	1492	0.968	-0.069	0.125
(c)	1498	0.966	-0.058	0.101
(d)	1505	0.959	-0.063	0.114
(e)	1515	0.965	-0.052	0.205
(f)	1525	0.952	-0.048	0.192
(g)	1535	0.960	-0.063	0.143
(h)	1545	0.967	-0.066	0.206
(i)	1555	0.948	-0.060	0.174
(j)	1565	0.941	-0.072	0.168
(k)	1575	0.934	-0.059	0.170
(l)	1585	0.955	-0.068	0.235
(m)	1595	0.925	-0.051	0.260
(n)	1605	0.934	-0.039	0.251
(o)	1615	0.942	-0.045	0.307
(p)	1625	0.961	-0.058	0.293

Table 5

Values of $a_z$ and $b_z$ taken in Figure 5 model results.

Figure 5	$W$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	1635	0.958	-0.042	0.305
(b)	1645	0.940	-0.028	0.350
(c)	1655	0.920	-0.021	0.263
(d)	1670	0.925	-0.009	0.187
(e)	1690	0.902	0.022	0.240
(f)	1710	0.899	0.047	0.209
(g)	1730	0.868	0.095	0.181
(h)	1750	0.843	0.102	0.194
(i)	1770	0.853	0.115	0.235
(j)	1790	0.879	0.125	0.170
(k)	1810	0.855	0.142	0.152
(l)	1830	0.867	0.159	0.138

Table 6

Values of $a_z$ and $b_z$ taken in Figure 6 model results.

Figure 6	$W$ (MeV)	$$a_z$$	$$b_z$$	$${\chi }^2/dof$$
(a)	1850	0.881	0.168	0.127
(b)	1870	0.872	0.221	0.132
(c)	1890	0.883	0.241	0.135
(d)	1910	0.856	0.252	0.153
(e)	1930	0.867	0.269	0.218
(f)	1950	0.853	0.253	0.295
(g)	1970	0.860	0.249	0.321
(h)	1990	0.875	0.261	0.184
(i)	2010	0.895	0.278	0.359
(j)	2030	0.906	0.299	0.337
(k)	2050	0.932	0.348	0.285
(l)	2070	0.945	0.337	0.304

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

[figures omitted; refer to PDF]

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

4. Discussion and Conclusions

The excitation spectrum of nucleons can especially help us to understand the strong interaction in the nonperturbative regime. Before, the hadron induced reactions is a main experimental method in the investigation. In the last two decades, the photon-induced reaction and electron scattering experiment are applied to study the electromagnetic excitation of baryons. Recently, the photoproduction of $\eta$ mesons from quasi-free protons and neutrons are measured by the CBELSA/TAPS detector. In the paper, we theoretically study the angular distribution of $\eta$ mesons for different incident photon energies ${{\rm E}}_{\gamma }$ and for different final state energies $W$ . Then, the results are compared with the experimental data in detail. The deformation coefficient $a_z$ and translation coefficient $b_z$ are extracted by the comparison. $a_z$ is almost independent of incident photon energies and final state energies. $b_z$ is linearly dependent on incident photon energies and final state energies. In particular, we visually give the deformation and translation of the emission sources by schematic sketches. From the patterns, it is intuitive and easy to better understand the motion and configuration of the emission sources.

A great number of $\eta$ mesons are produced in the photon-induced reaction. These $\eta$ mesons are regarded as a multiparticle system, which can be analyzed by the statistical method. In recent years, we develop such a model, which is called multisource thermal model. Some emission sources of final-state particles are formed in the reaction. Each emission source emits particles isotropically in the rest frame of the emission source. Due to the source interaction, the sources emit particles anisotropically. The $\eta$ mesons are emitted from these sources. In our previous work, the model can successfully describe transverse momentum spectra and pseudorapidity spectra of final-state particles produced in proton-proton ( $pp$ ) collisions, proton-nucleus ( $pA$ ) collisions, and nucleus -nucleus ( $AA$ ) collisions at intermediate energy and at high energy [17–21]. In this work, we extend the multisource thermal model to the statistical investigation of final-state particles produced in the photon-induced reaction. The model is improved to describe the angular dependence of the $\eta$ photoproduction from quasi-free protons and neutrons. The information of the source deformation and translation is obtained with different beam energies. It is helpful for us to understand the $\eta$ photoproduction.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grants No. 11247250 and No. 11575103, Shanxi Provincial Natural Science Foundation under Grant No. 201701D121005, and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) Grant No. 201802017.

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