Constituent-counting rule in photoproduction of hyperon resonances

We analyze the CLAS data on the photoproduction of hyperon resonances, as well as the available data for the ground state $\mathrm{\Lambda }$ and ${\mathrm{\Sigma }}^0$ of the CLAS and SLAC-E84 collaborations, by considering a constituent-counting rule suggested by perturbative QCD. The counting rule emerges as a scaling behavior of cross sections in hard exclusive reactions with large scattering angles, and it enables us to determine the number of elementary constituents inside hadrons. Therefore, it could be used as a new method for identifying internal constituents of exotic hadron candidates. From the analyses of the $\gamma p\to K^{+}\mathrm{\Lambda }$ and $K^{+}{\mathrm{\Sigma }}^0$ reactions, we find that the number of elementary constituents is consistent with $n_{\gamma }=1$, $n_p=3$, $n_{K^{+}}=2$, and $n_{\mathrm{\Lambda }}=n_{{\mathrm{\Sigma }}^0}=3$. Then, an analysis is made for the photoproductions of the hyperon resonances $\mathrm{\Lambda }(1405)$, $\mathrm{\Sigma }(1385{)}^0$, and $\mathrm{\Lambda }(1520)$, where $\mathrm{\Lambda }(1405)$ is considered to be a $\overline{K}N$ molecule, and hence its constituent number could be 5. However, we find that the current data are not enough to conclude the numbers of their constituents. It is necessary to investigate the higher-energy region at $\sqrt{s}>2.8\mathrm{GeV}$ experimentally beyond the energy of the available CLAS data to count the number of constituents clearly. We also mention that our results indicate energy dependence in the constituent number, especially for $\mathrm{\Lambda }(1405)$. If an excited hyperon is a mixture of three-quark and five-quark states, the energy dependence of the scaling behavior could be valuable for finding its composition and mixture.

Figure 1

Hard exclusive reaction $\gamma +b\to c+d$.

Figure 2

A typical hard-gluon exchange process in the exclusive reaction $\gamma +b\to c+d$ at high energies.

Figure 3

Experimental data of the $\gamma p\to K^{+}\mathrm{\Lambda }$ cross section $d\sigma /dt$ [7, 26] multiplied by $s^{n-2}/f({\theta }_{\mathrm{cm}})$ ($n=10.0$) with the function $f({\theta }_{\mathrm{cm}})$ in Eq. (6). The scaling factor $n$ is fixed at 10.0 by fitting in the region $\sqrt{s}\ge 2.6\mathrm{GeV}$ as shown in Table 1. The solid line indicates a fit to the data.

Figure 4

Experimental data of the $\gamma p\to K^{+}{\mathrm{\Sigma }}^0$ cross section $d\sigma /dt$ [7, 27] multiplied by $s^{n-2}/f({\theta }_{\mathrm{cm}})$ ($n=9.2$) with the function $f({\theta }_{\mathrm{cm}})$ in Eq. (6). The scaling factor $n$ is fixed at 9.2 by fitting in the region $\sqrt{s}\ge 2.6\mathrm{GeV}$ as shown in Table 1. The solid line is a fit to the data.

Figure 5

Behavior of the scaling factor $n$ by changing the minimal value of the energy $\sqrt{s_{\min }}$ for fitting.

Figure 6

Experimental data of the $\gamma p\to K^{+}\mathrm{\Lambda }(1405)$ cross section $d\sigma /dt$ [24] multiplied by $s^{n-2}/f({\theta }_{\mathrm{cm}})$ ($n=10.6$) with the function $f({\theta }_{\mathrm{cm}})$ in Eq. (6). The scaling factor $n$ is fixed at 10.6 by fitting in the region $\sqrt{s}\ge 2.5\mathrm{GeV}$ as shown in Table 2. The solid line is a fit to the data. For a better visualization, we slightly shift the data horizontally.

Figure 7

Experimental data of the $\gamma p\to K^{+}\mathrm{\Sigma }(1385{)}^0$ cross section $d\sigma /dt$ [24] multiplied by $s^{n-2}/f({\theta }_{\mathrm{cm}})$ ($n=11.4$) with the function $f({\theta }_{\mathrm{cm}})$ in Eq. (6). The scaling factor $n$ is fixed at 11.4 by fitting in the region $\sqrt{s}\ge 2.5\mathrm{GeV}$ as shown in Table 2. The solid line is a fit to the data. For a better visualization, we slightly shift the data horizontally.

Figure 8

Experimental data of the $\gamma p\to K^{+}\mathrm{\Lambda }(1520)$ cross section $d\sigma /dt$ [24] multiplied by $s^{n-2}/f({\theta }_{\mathrm{cm}})$ ($n=9.8$) with the function $f({\theta }_{\mathrm{cm}})$ in Eq. (6). The scaling factor $n$ is fixed at 9.8 by fitting in the region $\sqrt{s}\ge 2.5\mathrm{GeV}$ as shown in Table 2. The solid line is a fit to the data. For a better visualization, we slightly shift the data horizontally.

Figure 9

Behavior of the scaling factor $n$ by changing the minimal value of the energy $\sqrt{s_{\min }}$ for fitting.