1. Introduction
Considering that the nucleon is the most popular hadron and the cornerstone of the observable world, the study of the nucleon and its resonance is an important topic in hadron physics. The nucleon resonance is composed of three quarks, the sum of which mass is much smaller than the total mass of a nucleon. Hence, spectroscopy of nucleon resonance is very important in understanding how quarks combine to form a hadron. Although many efforts have been made to understand the internal structure of nucleon resonances in the past, many puzzles in this research area still exist, such as the well-known ‘missing resonance’ problem [1, 2]. The nucleon resonance is usually studied in the meson production off a nucleon, especially a proton target. The pion-induced production process is an important route to nucleon resonance in history, and the high-precision pion beams will be available in many large facilities, such as J-PARC and COMPASS [3, 4].
In the conventional constituent quark model, the nucleon resonance is composed of the three u /d quarks. In the early days, most of the nucleon resonances were observed in the pion-nucleon scattering [1]. In such a process, only the u /d quarks are involved, which leads to large production possibility of the nucleon resonances. However, to carry out further study of the internal structure of the nucleon resonances, the pion-induced strange production, where a strange quark pair is created, has special advantages. In such a process, the nucleon resonances can be further distinguished compared with the pion-nucleon scattering [5, 6].
In recent years, the strange component in nucleon resonances has attracted much interest in the community. It is suggested that a considerable hidden strange pentaquark component exists in the nucleon, which is important in understanding the properties of a nucleon [7]. Such a picture was also applied to understand the issues in the nucleon resonances, such as the reverse mass order problem for the Λ(1405) and N (1535) [8]. It was further extended to the hidden-charm sector to predict the hidden-charm pentaquarks [9 –11], which was confirmed by the LHCb Collaboration [12, 13]. Such experimental confirmations of the hidden-charm pentaquark give us more confidence about the hidden strange components in the nucleon resonances [14]. The pion-induced production of the hidden-charm/bottom pentaquarks were also predicted in the literature [15, 16].
To detect the hidden strange components in the nucleon resonance, a natural choice is the strange meson production process. In such a process, the strange quarks in the final state are from the hidden strange components in the nucleon resonance. In [17 –23], the Λ(1520) and Σ(1385) photoproduction with a kaon meson was studied and the results indicate the states N (2100) and N (1875) may be candidates of the hidden strange molecular states [24]. Hence, the strange meson production will be more important in the future study of nucleon resonance.
In the literature, many strange meson photoproductions have been studied, such as with the final states ${K}^{* }{\rm{\Lambda }}$ [25 –29], $K{{\rm{\Sigma }}}^{* }$ [22, 30], K Λ (1520) [17, 19, 23, 31, 32], and Ξ KK [33]. Some of the pion- or kaon-induced strange production can also be found in the literature [34 –40] . However, the study of the process ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ is scarce, which we focus on in this work. Moreover, considering that the total spin of ${K}^{* }{{\rm{\Sigma }}}^{* }$ is different from the total spin of $K{{\rm{\Sigma }}}^{* }$ , the two scattering processes of ${\pi }^{-}p\to K{{\rm{\Sigma }}}^{* }$ and ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ may have their own advantages when studying strange baryons with different spin quantum numbers. For example, for nucleons with a spin-parity of 1/2−, it seems more advantageous to study them through the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction process. Fortunately, some experimental data on the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction exists [41 –45]. Based on those data, we conduct a preliminary analysis of this reaction in this work, which may provide some valuable references for the experimental and theoretical research into the hidden strange pentaquark.
This paper is organized as follows. After an introduction, the formalism for the calculation is presented. The numerical result and discussion are given in section 3 . Finally, this paper ends with a brief conclusion.
2. Formalism
The basic tree level Feynman diagrams of the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction are illustrated in figure 1 . These include s -channel nucleon exchanges, t -channel $K/{K}^{* }$ meson exchange, and the u channel with an intermediate Σ baryon.
Figure 1. Feynman diagrams for the |
2.1. Lagrangians and amplitudes
To gauge the pion-induced production of ${K}^{* }{{\rm{\Sigma }}}^{* }$ , the relevant Lagrangian densities are required, which have been used in [21, 28, 30, 37, 46]. Wherein, the Lagrangians corresponding to the s channel are written as $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{{K}^{* }N{{\rm{\Sigma }}}^{* }} & = & -{\rm{i}}\displaystyle \frac{{f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(1)}}{{m}_{{K}^{* }}}{\overline{K}}_{\mu \nu }^{* }{\bar{{\rm{\Sigma }}}}^{* \mu }\cdot \tau {\gamma }^{\nu }{\gamma }_{5}N\\ & & -\displaystyle \frac{{f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(2)}}{{m}_{{K}^{* }}}{\overline{K}}_{\mu \nu }^{* }{\bar{{\rm{\Sigma }}}}^{* \mu }\cdot \tau {\gamma }_{5}{\partial }^{\nu }N\\ & & -\displaystyle \frac{{f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(3)}}{{m}_{{K}^{* }}}{\partial }^{\nu }{\overline{K}}_{\mu \nu }^{* }{\bar{{\rm{\Sigma }}}}^{* \mu }\cdot \tau {\gamma }_{5}N+{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray}$ $ \begin{eqnarray}{{ \mathcal L }}_{\pi {NN}}=-{\rm{i}}{g}_{\pi {NN}}\bar{N}{\gamma }_{5}\vec{\tau }\cdot \vec{\pi }N,\end{eqnarray}$ for the t channel: $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{{K}^{* }K\pi } & = & {g}_{{K}^{* }K\pi }[\bar{K}({\partial }^{\mu }\vec{\tau }\cdot \vec{\pi })-({\partial }^{\mu }\bar{K})\vec{\tau }\cdot \vec{\pi }]{K}_{\mu }^{* }\\ & & +{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray}$ $ \begin{eqnarray}{{ \mathcal L }}_{\pi {K}^{* }{K}^{* }}={g}_{\pi {K}^{* }{K}^{* }}{\varepsilon }^{\mu \nu \alpha \beta }{\partial }_{\mu }{\bar{K}}_{\nu }^{* }\vec{\tau }\cdot \vec{\pi }{\partial }_{\alpha }{K}_{\beta }^{* },\end{eqnarray}$ $ \begin{eqnarray}{{ \mathcal L }}_{{KN}{{\rm{\Sigma }}}^{* }}=\displaystyle \frac{{f}_{{KN}{{\rm{\Sigma }}}^{* }}}{{m}_{K}}\bar{N}{{\rm{\Sigma }}}^{* \mu }\cdot \vec{\tau }({\partial }_{\mu }K)+{\rm{H}}.{\rm{c}}.,\end{eqnarray}$ for the u channel: $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{{K}^{* }N{\rm{\Sigma }}} & = & -{g}_{{K}^{* }N{\rm{\Sigma }}}\bar{N}{\rm{\Sigma }}\left({\rlap{/}{K}}^{* }-\displaystyle \frac{{\kappa }_{{K}^{* }N{\rm{\Sigma }}}}{2{m}_{N}}{\sigma }_{\mu \nu }{\partial }^{\nu }{K}^{* \mu }\right)\\ & & +{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray}$ $ \begin{eqnarray}{{ \mathcal L }}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}=\displaystyle \frac{{g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}}{{m}_{\pi }}{\bar{{\rm{\Sigma }}}}^{* \mu }({\partial }_{\mu }\vec{\tau }\cdot \vec{\pi }){\rm{\Sigma }}+{\rm{H}}.{\rm{c}}.,\end{eqnarray}$ where ${K}_{\mu \nu }^{* }={\partial }_{\mu }{K}_{\nu }^{* }-{\partial }_{\nu }{K}_{\mu }^{* }$ . The $N,{K}^{* },{{\rm{\Sigma }}}^{* },\pi ,K,$ and Σ are the nucleon, ${K}^{* }(890)$ , ${{\rm{\Sigma }}}^{* }(1385)$ , π, K meson, and Σ baryon field, respectively. The $\vec{\tau }$ is the Pauli matrix.
For the πNN interaction vertex, we adopt ${g}_{\pi {NN}}^{2}$ /4π as 12.96 [47]. In addition, we take the following coupling constant values [28, 37, 46, 48], i.e. ${g}_{{K}^{* }N{\rm{\Sigma }}}=-2.46$ , ${\kappa }_{{K}^{* }N{\rm{\Sigma }}}=-0.47,{g}_{\pi {K}^{* }{K}^{* }}=7.45\,{\mathrm{GeV}}^{-1}$ . In addition, the coupling ${f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(1)}$ can be obtained from SU(3) flavor symmetry relations, which gives ${f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(1)}=-2.6$ [49]. In this work, we take ${f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(2)}={f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(3)}=0$ because of a lack of relevant information [49].
Moreover, the coupling constants ${g}_{{K}^{* }K\pi },$ ${g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}$ can be determined from the decay widths of the ${{\rm{\Gamma }}}_{{K}^{* }\to K\pi }$ and ${{\rm{\Gamma }}}_{{{\rm{\Sigma }}}^{* }\to \pi {\rm{\Sigma }}}$ , respectively, as $ \begin{eqnarray}{{\rm{\Gamma }}}_{{K}^{* }\to K\pi }=\displaystyle \frac{{g}_{{K}^{* }K\pi }^{2}}{2\pi }\displaystyle \frac{| {\vec{p}}_{\pi }^{\,{\rm{c}}.{\rm{m}}.}{| }^{3}}{{m}_{{K}^{* }}^{2}},\end{eqnarray}$ $ \begin{eqnarray}{{\rm{\Gamma }}}_{{{\rm{\Sigma }}}^{* }\to \pi {\rm{\Sigma }}}=\displaystyle \frac{{f}_{{\rm{I}}}{g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}^{2}}{12\pi }\displaystyle \frac{| {\vec{p}}_{{\rm{\Sigma }}}^{\,{\rm{c}}.{\rm{m}}.}{| }^{3}({E}_{{\rm{\Sigma }}}+{m}_{{\rm{\Sigma }}})}{{m}_{\pi }^{2}{M}_{{{\rm{\Sigma }}}^{* }}},\end{eqnarray}$ with
$ \begin{eqnarray*}\begin{array}{rcl}{E}_{{\rm{\Sigma }}} & = & \displaystyle \frac{{M}_{{{\rm{\Sigma }}}^{* }}^{2}+{m}_{{\rm{\Sigma }}}^{2}-{m}_{\pi }^{2}}{2{M}_{{{\rm{\Sigma }}}^{* }}},\\ | {\vec{p}}_{{\rm{\Sigma }}}^{\,{\rm{c}}.{\rm{m}}.}{| }^{3} & = & \sqrt{{E}_{{\rm{\Sigma }}}^{2}-{m}_{{\rm{\Sigma }}}^{2}},\\ | {\vec{p}}_{\pi }^{\,{\rm{c}}.{\rm{m}}.}{| }^{3} & = & \displaystyle \frac{\sqrt{\left[{m}_{{K}^{* }}^{2}-{\left({m}_{K}+{m}_{\pi }\right)}^{2}\right]\left[{m}_{{K}^{* }}^{2}-{\left({m}_{K}-{m}_{\pi }\right)}^{2}\right]}}{2{m}_{{K}^{* }}},\end{array}\end{eqnarray*}$
where the isospin factor${f}_{{\rm{I}}}$ for ${{\rm{\Sigma }}}^{* }\to \pi {\rm{\Sigma }}$ is 2. By taking the experimental data of the relevant particles in the PDG book [1], the coupling constants of ${g}_{{K}^{* }K\pi }$ and ${g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}$ can be calculated. With mass ${m}_{{K}^{* }}=891.7\,\mathrm{MeV}$ and ${M}_{{{\rm{\Sigma }}}^{* }}=1383.7\,\mathrm{MeV}$ , the total decay widths can be obtained as ${{\rm{\Gamma }}}_{{K}^{* }}=50.8\,\mathrm{MeV}$ and ${{\rm{\Gamma }}}_{{{\rm{\Sigma }}}^{* }}=36.0\,\mathrm{MeV}$ . And with the branching ratio ( ${Br}[{K}^{* }\,\to K\pi ]=1.00$ , Br [ ${{\rm{\Sigma }}}^{* }\to \pi {\rm{\Sigma }}$ ] = 0.117), we obtain ${g}_{{K}^{* }K\pi }=3.26$ and ${g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}=0.68$ .
where the isospin factor
To consider the size of the hadron, for the s and u channels with intermediate baryons, the form factors are adopted in our calculation as follows [21, 28, 46, 47, 50], $ \begin{eqnarray}{{ \mathcal F }}_{s/u}({q}_{\mathrm{ex}})=\displaystyle \frac{{{\rm{\Lambda }}}_{s/u}^{4}}{{{\rm{\Lambda }}}_{s/u}^{4}+{\left({q}_{\mathrm{ex}}^{2}-{m}_{\mathrm{ex}}^{2}\right)}^{2}}\,,\end{eqnarray}$ where q ex and m ex are the four-momentum and mass of the exchanged hadron, respectively. To reduce the free parameters, one takes ${{\rm{\Lambda }}}_{s}={{\rm{\Lambda }}}_{u}.$ For the t -channel $K/{K}^{* }$ meson exchange [47], the form factor ${{ \mathcal F }}_{t}({q}_{V})$ consisting of ${{ \mathcal F }}_{{K}^{* }V\pi }\,=({{\rm{\Lambda }}}_{t}^{2}-{m}_{V}^{2})/({{\rm{\Lambda }}}_{t}^{2}-{q}_{V}^{2})$ and ${{ \mathcal F }}_{{VBB}}=({{\rm{\Lambda }}}_{t}^{2}-{m}_{V}^{2})/({{\rm{\Lambda }}}_{t}^{2}-{q}_{V}^{2})$ are adopted with q V and m V being the four-momentum and mass of the exchanged meson, respectively. The values of the cutoffs Λ will be determined by fitting the experimental data.
According to the above Lagrangians, the scattering amplitude of the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reads as $ \begin{eqnarray}-{\rm{i}}{{ \mathcal M }}_{i}={\epsilon }_{\mu }({k}_{2}){\bar{u}}_{\nu }({p}_{2}){{ \mathcal A }}_{i}^{\mu \nu }u({p}_{1}),\end{eqnarray}$ where ${\epsilon }_{\mu }$ is the polarization vector of the ${K}^{* }$ meson, and ${\bar{u}}_{\nu }$ and u are dimensionless Rarita–Schwinger and Dirac spinors, respectively.
The reduced amplitudes ${{ \mathcal A }}_{i}^{\mu \nu }$ for s, t, and u channel contributions read as $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{s,(N)}^{\mu \nu } & = & {\rm{i}}\sqrt{2}{g}_{\pi {NN}}\displaystyle \frac{{f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(1)}}{{m}_{{K}^{* }}}{{ \mathcal F }}_{s}({q}_{\mathrm{ex}}){\gamma }_{\sigma }{\gamma }^{5}\\ & & ({k}_{2}^{\nu }{g}^{\mu \sigma }-{k}_{2}^{\sigma }{g}^{\mu \nu })\displaystyle \frac{({\rlap{/}{q}}_{N}+{m}_{N})}{s-{m}_{N}^{2}}{\gamma }_{5},\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{t,(K)}^{\mu \nu } & = & \sqrt{2}{g}_{{K}^{* }K\pi }\displaystyle \frac{{f}_{{KN}{{\rm{\Sigma }}}^{* }}}{{m}_{K}}{{ \mathcal F }}_{t}({q}_{V})\displaystyle \frac{1}{t-{m}_{K}^{2}}\\ & & {\left({k}_{1}-{k}_{2}\right)}^{\nu }[{k}_{1}^{\mu }+{\left({k}_{1}-{k}_{2}\right)}^{\mu }],\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{t,({K}^{* })}^{\mu \nu } & = & \sqrt{2}{g}_{\pi {K}^{* }{K}^{* }}\displaystyle \frac{{f}_{{K}^{* }N{{\rm{\Sigma }}}^{* }}^{(1)}}{{m}_{{K}^{* }}}{{ \mathcal F }}_{t}({q}_{V}){\varepsilon }^{\sigma \mu \alpha \beta }\displaystyle \frac{{{ \mathcal P }}_{\beta \xi }}{t-{m}_{{K}^{* }}^{2}}\\ & & {\gamma }_{\eta }{\gamma }_{5}[{\left({k}_{1}-{k}_{2}\right)}^{\nu }{g}^{\xi \eta }-{\left({k}_{1}-{k}_{2}\right)}^{\eta }{g}^{\xi \nu }]\\ & & {k}_{2\sigma }{\left({k}_{1}-{k}_{2}\right)}_{\alpha },\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{u,({\rm{\Sigma }})}^{\mu \nu } & = & -{\rm{i}}\sqrt{2}{g}_{{K}^{* }N{\rm{\Sigma }}}\displaystyle \frac{{g}_{{{\rm{\Sigma }}}^{* }\pi {\rm{\Sigma }}}}{{m}_{\pi }}{{ \mathcal F }}_{u}({q}_{\mathrm{ex}}){k}_{1}^{\nu }\displaystyle \frac{({\rlap{/}{q}}_{{\rm{\Sigma }}}+{m}_{{\rm{\Sigma }}})}{u-{m}_{{\rm{\Sigma }}}^{2}}\\ & & \left({\gamma }^{\mu }+\displaystyle \frac{{\kappa }_{{K}^{* }N{\rm{\Sigma }}}}{4{m}_{N}}({\gamma }^{\mu }{\rlap{/}{k}}_{2}-{\rlap{/}{k}}_{2}{\gamma }^{\mu })\right),\end{array}\end{eqnarray}$ with $ \begin{eqnarray}{{ \mathcal P }}^{\beta \xi }={\rm{i}}\left({g}^{\beta \xi }+{q}_{{K}^{* }}^{\beta }{q}_{{K}^{* }}^{\xi }/{m}_{{K}^{* }}^{2}\right),\end{eqnarray}$ where $s={({k}_{1}+{p}_{1})}^{2}$ , $t={({k}_{1}-{k}_{2})}^{2}$ , and $u={({p}_{2}-{k}_{1})}^{2}$ are the Mandelstam variables.
2.2. Reggeized treatment
The Reggeized treatment is often used to analyze hadron production in the middle and high energy regions [21, 26, 28, 47, 51]. For the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ , the energy range of the relevant experimental data is mainly concentrated in the middle energy zone, thus we will study this reaction channel in the Feynman model and the Regge model, respectively. Usually, it can be introduced by replacing the t -channel Feynman propagator in the Feynman amplitudes in equations (13 ) and (14 ) by the Regge propagator as [21, 26, 28, 47, 51], $ \begin{eqnarray}\displaystyle \frac{1}{t-{m}_{K}^{2}}\to {\left(\displaystyle \frac{s}{{s}_{\mathrm{scale}}}\right)}^{{\alpha }_{K}(t)}\displaystyle \frac{\pi {\alpha }_{K}^{{\prime} }}{{\rm{\Gamma }}[1+{\alpha }_{K}(t)]\sin [\pi {\alpha }_{K}(t)]},\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{1}{t-{m}_{{K}^{* }}^{2}}\to {\left(\displaystyle \frac{s}{{s}_{\mathrm{scale}}}\right)}^{{\alpha }_{{K}^{* }}(t)-1}\displaystyle \frac{\pi {\alpha }_{{K}^{* }}^{{\prime} }}{{\rm{\Gamma }}[{\alpha }_{{K}^{* }}(t)]\sin [\pi {\alpha }_{{K}^{* }}(t)]}.\end{eqnarray}$ The scale factor s scale is fixed at 1 GeV. Moreover, the meson Regge trajectories ${\alpha }_{K}(t)$ and ${\alpha }_{{K}^{* }}(t)$ are given by [26], $ \begin{eqnarray}{\alpha }_{K}(t)=0.70(t-{m}_{K}^{2}),{\alpha }_{{K}^{* }}(t)=1+0.85(t-{m}_{{K}^{* }}^{2}).\end{eqnarray}$
With the Regge model, no additional parameters are added. The only free parameter is the cutoff Λ, and its value needs to be discussed later in conjunction with the experimental data.
3. Numerical results
After the above preparation, one can calculate the differential cross-section of the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction and combine the experimental data [41 –45] for correlation analysis. The differential cross-section in the center of mass (c.m.) frame is written as follows $ \begin{eqnarray}\displaystyle \frac{{\rm{d}}\sigma }{\mathrm{dcos}\theta }=\displaystyle \frac{1}{32\pi s}\displaystyle \frac{\left|{\vec{k}}_{2}^{{\rm{c}}.{\rm{m}}.}\right|}{\left|{\vec{k}}_{1}^{{\rm{c}}.{\rm{m}}.}\right|}\left(\displaystyle \frac{1}{2}\displaystyle \sum _{\lambda }{\left|{ \mathcal M }\right|}^{2}\right),\end{eqnarray}$ where $s={({k}_{1}+{p}_{1})}^{2}$ , and θ denotes the angle of the outgoing ${K}^{* }$ meson relative to the π beam direction in the c.m. frame. Here, ${\vec{k}}_{1}^{{\rm{c}}.{\rm{m}}.}$ and ${\vec{k}}_{2}^{{\rm{c}}.{\rm{m}}.}$ are the three-momenta of the initial π beam and final ${K}^{* }$ , respectively.
3.1. Fitting procedure
With the help of the MINUIT code in the CERNLIB, the experimental data [41 –45] of the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction will be fitted in two schemes: the Regge model and the Feynman model. As shown in figures 2 –3, there are a total of 18 experimental data points for the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ , including six total cross-section data and 12 data of the t distribution. Hence, one minimizes χ 2 per degree of freedom (d.o.f.) for the 18 data points by fitting three parameters, which include the coupling constant ${f}_{{KN}{{\rm{\Sigma }}}^{* }}$ and the cutoffs Λt and Λu . As mentioned above, to minimize the free parameters, one sets Λs = Λu . The fitted values of the free parameters in two schemes are listed in table 1 .
Figure 2. The t distribution for the reaction |
Figure 3. The total cross-section for the |
One notices that the values of the parameters fitted by the two models are close, except that the difference in the Λt value is relatively large. Moreover, the best-fitted χ 2 for the two schemes are 2.38 and 1.54 for the Feynman and the Regge models, respectively, which may mean that the Regge model is more consistent with the experimental data than the Feynman model. The best-fitted results given by the two schemes are presented in figures 2 and 3 .
3.2. t distribution for reaction ${\pi }^{-}p \rightarrow {K}^{* }{{\rm{\Sigma }}}^{* }$
The t distribution for the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ at P lab = 3.95 GeV/c in two models is presented in figure 2 . It is found that although both models can explain the experimental data well, the fitting of the data by the Regge model is significantly better than the fitting of the Feynman model to the experiment. In addition, we found that the contributions of the s and u channels are small in the results given by the two models, and the cross-section is basically derived from the contribution of the t channel. In the Feynman model, the contribution of the t -channel K exchange is dominant, and the contribution from the t -channel ${K}^{* }$ exchange is very small. However, in the Regge model, the contributions of the t -channel K and ${K}^{* }$ exchanges are almost equivalent, and they each give their own contribution at different t .
3.3. Cross section of reaction ${\pi }^{-}p \rightarrow {K}^{* }{{\rm{\Sigma }}}^{* }$
The total cross sections of the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ obtained in two schemes are illustrated in figure 3, which show that the contribution from the s channel is small, but the u channel plays a very important role, especially at higher energy. In both models, the contribution of the t channel is mainly concentrated at low energy, and the relative contributions of K and ${K}^{* }$ exchanges are similar to that in figure 2 . The contribution of the t channel exhibits the same pattern as the results in figure 2 . That is, in the Feynman model, the contribution of the K exchange is much larger than the contribution of the ${K}^{* }$ exchange, while in the Regge model, the contributions of the K exchange and the ${K}^{* }$ exchange are comparable. Moreover, it is found that the total cross-section we obtained only reached the lower limit of the experimental data points at P lab = 3.1–4.2 GeV/c, which may mean that there is also a contribution from s channel resonance in this region. When we consider the error bars of the fitting parameters, the total cross-section obtained will fit better with the data at ${P}_{\mathrm{lab}}$ = 3.1–4.2 GeV/c, but it still does not reach the highest point of the data. Since the data points of this energy region are very old at present, and there are few studies on the resonance state that can decay to ${K}^{* }{{\rm{\Sigma }}}^{* }$ , it is difficult to clarify the contribution of the resonance state of this energy region. Therefore, the measurement of experimental data with higher precision in this energy region and the theoretical prediction of the relevant resonance state are greatly needed.
The predictions of the differential cross-section of the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ in two schemes at different beam momenta are presented in figure 4, which shows that the discrepancy of the differential cross sections in the two models is small at low beam momenta but gradually increases in the higher beam momenta, especially in the forward angle region. Moreover, one notices that the contribution from the t channel is dominant at forward angles, while the contribution from the u channel becomes more and more important at backward angles with the increase in the energy. As in the previous case, the contribution of the s channel is small and negligible.
Figure 4. The differential cross-section |
4. Summary and discussion
Within an effective Lagrangian approach, a three-parameter fitting is applied to the experimental data of the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ with the Feynman and Regge models, which leads to best-fitted values of χ 2 for the two schemes that are 2.38 and 1.54, respectively. For the t distribution for the reaction ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ at P lab = 3.95 GeV/c, the fitting of the Regge model to the experimental data is better than the Feynman model. But the quality of the fitting of the two models to the total cross-section data is similar.
From the fitting results, the contribution from the t channel is very large, the contribution of the s channel is small and negligible, and the contribution of the u channel is mainly concentrated in the high energy. The only difference is that the contributions of the t -channel K exchange and the ${K}^{* }$ exchange are comparable in the Regge model, while in the Feynman model, the contribution of the K exchange is greater than that of the ${K}^{* }$ exchange. In addition, the fitting results for the total cross-section seem to indicate a contribution from the s -channel resonance state at a center of mass energy of 2.6–2.97 GeV. Therefore, the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction may be an effective channel for finding the hidden strange pentaquark. More theoretical predictions about the resonance state that can decay to ${K}^{* }{{\rm{\Sigma }}}^{* }$ are greatly needed for our subsequent research.
Finally, we also give the prediction of the differential cross-section of the ${\pi }^{-}p\to {K}^{* }{{\rm{\Sigma }}}^{* }$ reaction at different beam momentum. The results show that the difference in the shape of the cross-section given by the two models increases with the increase in energy, especially in the forward angle region. This situation will help to clarify the role of the Regge propagator, and can be verified by future experiments.
The J-PARC, COMPASS, and JLAB are capable of producing high-intensity pion meson beams, which are expected to yield high-quality data for the ${K}^{* }{{\rm{\Sigma }}}^{* }$ production. Our theoretical results will provide valuable information for such experiments. Combined with the corresponding photoproduction of ${K}^{* }{{\rm{\Sigma }}}^{* }$ at JLAB 12 GeV upgraded, it is of great significance for a deeper understanding of the relevant reaction mechanism and the search for the hidden strange pentaquark.