Nucleon resonances in the reaction near threshold
We investigate the two-body reaction within the effective Lagrangian approach and the isobar model. In addition to the “background” contributions from -channel exchange, -channel and exchanges, and -channel nucleon pole terms, the contributions from the nucleon resonances , , and are investigated. It is shown that the inclusion of these nucleon resonances contributions leads to a good description of the experimental total and differential cross sections data at low energy region. The -channel , , and resonances and the -channel exchange give the dominant contributions below GeV, while the -channel nucleon pole, -channel and exchanges give the minor contributions.
pacs:13.75.-n.; 14.20.Gk.; 13.30.Eg.
The study of the spectrum of the nucleon resonances and the resonances coupling constants from the available experimental data are two of the most important issues in hadronic physics and they are attracting much attention Klempt:2009pi (). In the classical quark models Isgur:1978xj (); Capstick:1986bm (); Loring:2001kx (), a rich spectrum of excited nucleon states is predicted. Many of these nucleon resonances could be identified in scattering. However, there are still some of them have not been so far observed.
The strangeness production reaction is a good platform to study the properties of the nucleon resonances, especially for those which have significant couplings to and channels, because the reaction is a pure isospin two-body reaction channel in meson-nucleon dynamics, and there are no contributions from the isospin baryons. Hence, lots of experimental data have been accumulated Baker:1978qm (); Knasel:1975rr (); Bertanza:1962pt (); Baldini:1988ti (), where the total and differential cross sections of reaction are measured.
In response to this wealth of data, the theoretical activity has run in parallel. In Ref. Sibirtsev:2005mv (), the two-body reaction is investigated using the partial wave amplitudes which are constructed from the available experimental data. The total and differential cross sections of reaction can be well described by using these amplitudes, which are essential for obtaining spin-flip and spin non-flip amplitudes Sotona:1988fm (). But, from these amplitudes, it is difficult to get clear properties of some nucleon resonances, because the partial wave amplitudes could have contributions from nucleon resonances with different spins, and individual contributions are more difficult to pin down. This deficiency is also shown in Ref. Ronchen:2012eg () where the reactions are studied simultaneously within an analytic, unitary, coupled-channel approach. Moreover, it is pointed out that there are ambiguities of the scattering amplitudes obtained from the partial wave analysis, when only the observables of the differential cross section and polarization are measured Anisovich:2013tij ().
The role played by the nucleon resonance , which has proved to be a controversial resonance for many years, in the production is crucial. The couples strongly to the channel pdg2014 () but a large coupling has also been deduced GarciaRecio:2003ks (); Liu:2005pm (); Liu:2006ym (); Geng:2008cv () through the analysis of BES data on decay Yang:2005ej () and COSY data on the reaction near threshold Kowina:2004kr (). In Refs. Penner:2002ma (); Penner:2002md (); Shklyar:2005xg (); JuliaDiaz:2006is (), the analyses of recent SAPHIR Tran:1998qw (); Glander:2003jw () and CLAS Nasseripour:2008aa () data also indicate a large coupling of the resonance to the channel. Furthermore, in a chiral unitary coupled channel model it is found that the resonance is dynamically generated, with its mass, width and branching ratios in fair agreement with experiment GarciaRecio:2003ks (); Kaiser:1996js (); Inoue:2001ip (); Nieves:2001wt (); Doring:2009uc (). This approach shows that the couplings of the resonance to channel could be large compared to those for . We wish to argue in this work that the resonance might play a much wide role in associated strangeness production of reaction.
In the present work, we study the two-body reaction within an effective Lagrangian approach and the isobar model, which is an important theoretical method for describing various processes in the resonances production region Tsushima:1996tv (); Tsushima:1998jz (); Shyam:1999nm (); Xie:2007qt (); Dai:2011yr (); Liu:2011sw (); Liu:2012ge (); Lu:2013jva (); Lu:2014rla (); Xie:2014kja (). In addition to the background contributions from -channel exchange, -channel and exchange, and -channel nucleon pole terms, we also investigate the contributions from nucleon resonances , , and , which have significant couplings to and channels pdg2014 ().
This article is organized as follows. In Sect. II we present the
formalism and ingredients required for the calculation. The
numerical results and discussions are given in Sect. III. A short
given in the last section.
Ii Formalism and Ingredients
In this section, we introduce the theoretical formalism and ingredients for calculating the reaction by using the effective Lagrangian approach and the isobar model. In the following equations, we use , , and , which denote the , , and , respectively.
The basic tree level Feynman diagrams for the reaction are depicted in Fig. 1. These include -channel nucleon pole and nucleon resonances process [Fig. 1 (a)], -channel exchange [Fig. 1 (b)], and -channel and exchanges [Fig. 1 (c)].
To evaluate the invariant scattering amplitudes corresponding to theose diagrams shown in Fig. 1, the effective Lagrangian densities for relevant interaction vertexes are needed. Following Refs. Janssen:1996kx (); Xie:2008ts (); Doring:2010ap (); Mart:2013ida (); Xie:2013wfa (); Xie:2013db (), the Lagrangian densities used in present work are,
for -channel nucleon pole and nucleon resonances, , , exchange terms, and
for the -channel exchange process, while
for the -channel and exchange diagrams.
For the coupling constants in the above Lagrangian densities for -channel and -channel processes, we take (obtained by ), (obtained by ), (obtained by ), and 111Which is obtained by the partial decay width of to ., which are used in Ref. Ronchen:2012eg (). The others are obtained by flavor symmetry as shown in Table 1.
Besides, the coupling constants for the -channel nucleon resonance exchange processes, are obtained from the partial decay widths,
where is the Källen function with .
With masses and partial decay widthes of the nucleon resonances, the strong coupling constants of nucleon resonances , and are obtained as listed in Table 2. Moreover, the strong coupling constants is a free parameter, which will be determined by fitting to the experimental data of the reaction.
|Resonance ()||Mass (GeV)||Width (GeV)||Decay channel||Branching ratio||Adopted value|
Due to the fact that the relevant hadrons are not point-like particles, form factors are included. In our present calculation, we adopt the following form factors,
with for -channel and , and for -channel , nucleon pole, -channel exchange, -channel hyperon pole, and exchange. The , and are the Lorentz-invariant Mandelstam variables, while , , and are the four-momenta of the intermediate particle in the , , and -channel, and , , and are the four momenta of , , and , respectively. For the cutoff parameters, they will be determined by fitting them to the experimental data.
For propagators of the spin-1/2 particle and of the spin-3/2 particle, we adopt the simple Breit-Wigner formula,,
where and stand for the mass and full decay width of the corresponding resonances. It is worth to note that for -channel nucleon pole and -channel and exchange, we take .
While for the meson propagator , we take
Then the total invariant scattering amplitude for reaction, according to these contributions shown in Fig. 1, can be written as,
Note that we have employed the phase factor for the nucleon resonances , since we can not determine the phase for the nucleon resonances within our model. Thus, these phase angles will be determined to reproduce the experimental data.
With the above effective Lagrangian densities, we can straightforwardly evaluate the following invariant scattering amplitudes, corresponding to the Feynman diagrams in Fig. 1:
where and are the polarization variables of proton and .
The unpolarized differential cross section in the center-of-mass (c.m.) frame for the reaction reads,
with is the polar scattering angle of outgoing meson, and and are the and mesons c.m. three momenta. The differential cross section depends on and also on .
As mentioned above, the model accounts for a total of three mechanisms: -channel nucleon pole and resonances terms, -channel exchange, and the -channel pole and contributions. In principle, the free parameters of the model are: i) relative phases between different contributions, ii) the cut off parameters, , and , and iii) the coupling constant .
In the next section, we will fit the parameters of the model to the low energy differential cross section data of the reaction by using the MINUIT fitting program.
Iii Results and Discussion
First, by including the contributions from -channel nucleon pole, , and , -channel exchange, and -channel and exchange terms, we perform a four-parameter  fit (Fit I) to the experimental data Bertanza:1962pt (); Knasel:1975rr (); Baker:1978qm () on differential cross sections of reaction. The experimental data base contains differential cross sections at 20 energy points from 3 different experiments, and there is a total of data points below GeV. Second, for showing the important role played by the resonance, we have also performed another best fit, where the -channel resonance has been switched off (Fit II).
The fitted parameters of Fit I and Fit II are compiled in Table 3. The resultant of Fit I is , while for the Fit II, it is , which is turn out to be larger, since we have ignored the important contributions from the resonance. From the the fitted result, , we obtain the ratio with the value of that is obtained with the partial decay width of . This value is smaller than the one obtained in Ref. Liu:2005pm () by analyzing the and experimental data. This is because in the work of Ref. Liu:2005pm (), the energy dependent width for the resonance is used, which will decrease the contribution from the propagator of resonance above the threshold and make the coupling strength of the resonance to the channel larger. 222Similar statement can be also found in Ref. Xie:2013wfa () for the case of the . In contrast with the energy dependent width as in Ref. Liu:2005pm (), in the present work we use a constant total decay width for resonance since the channel is opened. Nevertheless, the values of this ratio obtained in the previous works are widely scattered. For instance, has been determined from the latest and largest photoproduction database by using the isobar model. Similar results, were obtained in Ref. Geng:2008cv () from the decays within the chiral unitary approach and was obtained in Ref. Sarantsev:2005tg () by the partial wave analysis of kaon photonproduction. In Ref. Bruns:2010sv () the result of the -wave scattering analysis within a unitarized chiral effective Lagrangian indicates that , whereas a coupled-channels calculation predicted a value of Penner:2002md ().
|Parameters||Fit I||Fit II|
The differential distributions calculated with the Fit I best-fit parameters are shown in Fig. 2 as a function of and for various invariant mass intervals. The contributions from different mechanisms are shown separately. Thus, we split the full result into four main contributions: effective Lagrangian approach background, -channel , and . The first one corresponds to the -channel exchange, -channel nucleon pole, and -channel hyperon pole and terms. We find an overall good description of the data for the whole range of measured invariant masses below GeV, and the contributions from the above three main mechanisms are all significant to reproduce the current experimental data.
Our best result of the total cross sections of reaction as a function of the invariant mass of the system are shown in Fig. 3 compared with the experimental data Baldini:1988ti (). There, we see that we can describe the total cross section data quite well. As mentioned above, the resonance and resonance couple strongly to the channel. Indeed, the total cross section of show a strong -wave contributions close to the reaction threshold and also a little bit beyond. From Fig. 3, it is seen that the contribution from (blue dashed curve) is predominant at a very wide energy region, and the contribution from the (red solid curve) is significant from the reaction threshold till GeV, while the contributions from -channel exchange and -channel are dominant above GeV. The -channel nucleon pole and -channel contributions are too small to be shown in Fig. 3 and can be neglected.
It is worthy to note that we do not consider the contribution from the nucleon resonance , which is mostly required by the inelastic scattering data. The role of this resonance in the context of different partial-wave analyses has been discussed extensively in Refs. Ronchen:2012eg (); Ceci:2006ra (). In the analysis of the reaction, a resonance is needed Ceci:2006ra (). But, it is pointed out in Ref. Ronchen:2012eg () that in the analysis of the reaction, there is an interplay between the and and individual contributions are difficult to pin down. Moreover, the resonance appears in the three-body hadronic calculations Khemchandani:2008rk (); MartinezTorres:2008kh () and also could be a dynamically generated resonance Suzuki:2009nj (); Kamano:2010ud (). On the other hand, a Bayesian analysis of the world data on reaction DeCruz:2011xi () show that there is no significant contribution of the resonance to the reaction. Because of those doubts of this resonance, we ignore its contribution in our present calculations.
In this paper, the reaction is investigated within an effective Lagrangian approach and the isobar model. This channel is known to receive significant nonresonant contributions which complicates the extraction of information. In addition to the background contributions from the -channel nucleon pole, -channel exchange, -channel and exchanges, we also consider the contributions from -channel nucleon resonances , , and . From fits to the available experimental data for the reaction, we get the appropriate parameters which describe the total and differential cross sections well. Our results show that the inclusion of the nucleon resonances , , and can lead to a good description of the low energy experimental total and differential cross sections data of reaction. The contribution from each individual resonance to the total and differential cross sections below GeV are also presented. The contributions from those nucleon resonances and the -channel exchange are dominant, while -channel nucleon pole and the -channel and exchange give the minor contributions and can be neglected.
We would like to thank Xu Cao for useful discussions. This work is partly supported by the Ministry of Science and Technology of China (2014CB845406), the National Natural Science Foundation of China under grants: 11105126, 11375024 and 11175220. We acknowledge the one Hundred Person Project of Chinese Academy of Science (Y101020BR0). The Project is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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