# Phenomenological analysis of the double pion production in nucleon-nucleon collisions up to 2.2 GeV

###### Abstract

With an effective Lagrangian approach, we analyze several channels by including various resonances with mass up to 1.72 GeV. For the channels with the pion pair of isospin zero, we confirm the dominance of in the near threshold region. At higher energies and for channels with the final pion pair of isospin one, we find large contributions from , double-, , and . There are also sizeable contributions from , , and nucleon pole at energies close to the threshold. We well reproduce the total cross sections up to beam energies of 2.2 GeV except for the channel at energies around 1.1 GeV and our results agree with the existing data of differential cross sections of , and which are measured at CELSIUS and COSY.

###### pacs:

13.75.-n, 13.75.Cs, 14.20.Gk, 25.75.Dw## I Introduction

Double pion production in both pion- and photo-induced reactions has been an intriguing field to study baryon spectrum and given insight to the properties of strong interaction reviewlee (); schneider (). These reactions close to threshold are also an interesting area to test chiral symmetry and have been extensively explored experimentally twopiphoto () and theoretically twopicpt (). Recently the double pion production in the electro-production off protons has advanced a important step twopielectro (). All essential contributions are identified from the data and the major isobar channels are well determined. On the other hand, as another fascinating platform for studying resonances properties, double pion production in nucleon-nucleon collisions has been accurately measured at the facilities of CELSIUS and COSY in the past few years, and the comprehensive data of various differential cross sections are obtained up to beam energies 1.3 GeV celsius (); dataIJMPA (); skorodko (); cosy (); data2000 (); otherdata (). However, on the theoretical side, the study of this reaction is scarce. The state-of-art one is still the Valencia model calculation Alvarez () of more than ten years ago after some much earlier calculations of one pion exchange (OPE) model Ferrari () of more than 45 years ago. Thus a more comprehensive analysis matching the modern data is very necessary.

The early OPE model, which mainly focused on the old data at beam energies of 2.0 GeV and 2.85 GeV olddata (), included two types of diagrams with the final two pions produced from a single and two baryon line(s), respectively. It used the amplitudes of and extracted from limited data and the off-shell corrections were considered under several assumptions. It did not account for the explicit production mechanisms of double pion and other exchanged meson besides -meson. The Valencia model is characteristic by the the dominance of in the near threshold region in the isospin allowed channels while the double- and rise up at higher energies and in channels where is forbidden by the isospin conservation. Recently, the experimental data celsius (); cosy () confirm the predicted behavior close to threshold and this makes and good places to study whose structure is still controversial. Current data seem to show a weaker than that listed in Particle Data Group pdg2008 () and support the explanation of as the monopole excitation of the nucleon. Contrarily, the case is much more complicated at higher energies. The dominance of the double- mechanism in the Valencia model results in that the total cross section of is about a factor of four larger than that of , while the new exclusive and the old bubble-chamber data are consistent to conclude an approximate equal value of these two channels. The isospin decomposition unambiguously reveal that more isospin 3/2 resonances besides is required to explain the data celsius (); skorodko (), and this is also the reason that the Valencia model including simply the and achieved merely a rough agreement in most channels. Indeed, at higher energies the contribution from higher lying resonances, especially those having large double pion decay channels, should become relevant. The recent detailed measurements performed by CELSIUS and COSY make the further exploration of these problems possible.

In the present work, we try to incorporate the resonances with mass up to 1.72 GeV in an effective Lagrangian model with the motivation to give a reasonable explanation to the six isospin channels of simultaneously and get better understanding of dynamics for this kind of reactions. Our paper is organized as follows. In Sect. II, we present the formalism and ingredients in our computation. The numerical results and discussion are demonstrated in Sect. III and a brief summary is given in Sect. IV.

## Ii Formalism and Ingredients

We consider nearly all possible Feynman diagrams as depicted in Fig. 1, and exchanged diagrams are also included. We use the commonly used interaction Lagrangians for , , , and couplings,

(1) |

(2) |

(3) |

(4) |

(5) |

At each vertex a relevant off-shell form factor is used. In our caculation, we take the same form factors as that used in the Bonn potential model bonn (),

(6) |

with n=1 for - and -meson and n=2 for -meson. , and are the 4-momentum, mass and cut-off parameters for the exchanged meson, respectively. The coupling constants and the cutoff parameters are taken as bonn (); tsushima (); singlepi (): , , , , = = 1.0 GeV, = 1.3 GeV, = 1.6 GeV, and = 6.1. We use from the quark model reviewlee (); Alvarez (). The mass and width of -meson are adopted as 550 MeV and 500 MeV, respectively.

We include all and resonances with spin-parity , , and mass up to 1.72 GeV listed in Particle Data Group (PDG) tables pdg2008 (). The resonances with further higher masses are expected to give negligible contributions in the energy region considered here and their two pion branching ratios have large uncertainties, so we do not include them at present. The effective Lagrangians for the relevant resonance couplings are zoucoupling (); ouyang (),

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

For the Resonance-Nucleon-Meson vertices, form factors with the following form are used:

(30) |

with n=1 for resonances and n=2 for resonances. We employ = = = = 1.0 for all resonances except = 0.8 for . We also use Blatt-Weisskopf barrier factors in the -- vertices zouform (),

(31) |

Here is the hadron ¡°scale¡± parameter GeV/c, where R is the radius of the centrifugal barrier in the unit of fm and is tuned to be 1.5 fm to fit the data. and is defined as,

(32) |

(33) |

with being the invariant energy squared of particle. Because the mass of -meson is near the two- threshold, the following Lagrangians and form factor are employed for the -- vertex reviewlee (); schneider (); juelich (),

(34) |

(35) |

(36) |

where is the relative momentum of the emitted -mesons. We use GeV and GeV to normalize this form factor to unity when - and - meson are all on-shell. The decay width of and yield = and = .

Resonance | Pole Position | BW Width | Decay Mode | Decay Ratio | |
---|---|---|---|---|---|

(1210, 100) | 118 | 1.0 | 19.54 | ||

(1365, 190) | 300 | 0.65 | 0.51 | ||

0.075 | 3.20 | ||||

0.135 | 4.30 | ||||

(1510, 110) | 115 | 0.6 | 1.73 | ||

0.09 | 1.32 | ||||

0.2 | 0.01 | ||||

(1510, 170) | 150 | 0.45 | 0.037 | ||

0.525 | 0.34 | ||||

0.02 | 0.15 | ||||

(1600, 300) | 350 | 0.175 | 1.09 | ||

0.55 | 59.9 | ||||

0.225 | 289.1 | ||||

(1600, 118) | 145 | 0.25 | 0.06 | ||

0.14 | 0.37 | ||||

0.45 | 83.7 | ||||

(1655, 165) | 165 | 0.775 | 0.06 | ||

0.065 | 0.026 | ||||

0.08 | 0.011 | ||||

0.04 | 0.063 | ||||

(1660, 135) | 150 | 0.4 | 2.16 | ||

0.55 | 3077.5 | ||||

(1675, 120) | 130 | 0.675 | 5.53 | ||

0.125 | 4.45 | ||||

0.09 | 0.32 | ||||

0.1 | 9.39 |

Resonance | Pole Position | BW Width | Decay Mode | Decay Ratio | |
---|---|---|---|---|---|

(1680, 100) | 100 | 0.1 | 0.075 | ||

0.07 | 0.043 | ||||

0.04 | 0.003 | ||||

(1650, 200) | 300 | 0.15 | 1.02 | ||

0.125 | 0.69 | ||||

0.45 | 0.072 | ||||

(1720, 230) | 100 | 0.15 | 0.012 | ||

0.062 | 0.042 | ||||

0.25 | 0.085 | ||||

0.15 | 36.1 | ||||

0.275 | 0.12 | ||||

(1675, 195) | 200 | 0.15 | 0.12 | ||

0.04 | 0.28 | ||||

0.775 | 190.7 |

The form factor for the resonance, , is taken as,

(37) |

with = 1.0 GeV. The same type of form factors are also applied to the nucleon pole with = 0.8 GeV. The propagators of the exchanged meson, nucleon pole and resonance can be written as bonn (); tsushima (),

(38) |

(39) |

(40) |

(41) |

(42) |

(43) |

(44) |

Here is the total width of the corresponding resonance, and and is defined as,

(45) |

(46) | |||||

(47) |

(48) |

Because constant width is used in the Breit-Wigner (BW) formula, we adopt the pole positions of various resonances for parameters appearing in the propagators.

The coupling constants appearing in relevant resonances are determined by the empirical partial decay width of the resonances taken from PDG pdg2008 (), and then we adjust the values of cut-off in form factors to fit the data. The relations between the branching ratios of the adopted resonances and the corresponding coupling constants squared can be calculated straightforwardly with above Lagrangians, and most of them can be found in the appendix of Ref. tsushima (). The detailed calculations of and from the decay are given in Ref. xierho (). The values of coupling constants used in our computation are compiled in the Table 1, together with the properties of the resonances and the central value of branch ratios. It should be noted that we adopt a nearly half of the decay width of in PDG as the recent data favored cosy (); celsius (); roperdecay ().

Then the invariant amplitudes can be obtained straightforwardly by applying the Feynman rules to Fig. 1. As to the different isospin channels, isospin coefficients are considered. We do not include the interference terms among different diagrams because their relative phases are not known, and the Valencia model seems to show that such terms are very small.

## Iii Numerical RESULTS AND DISCUSSION

As a starting point, in Fig. 2 we demonstrate our calculated total cross sections of six isospin channels compared with the existing data olddata (); cosy (); celsius (); data2000 (); otherdata (). Our numerical results give an overall good reproduction to all six channels. The pre-emission diagrams (see (2), (4), (6) in Fig. 1) tend to be negligibly small, consistent with the Valencia model, so we do not include them in our concrete computation. In Fig. 2 we do not show the following negligible contributions: double-, , , , , , , , , , , double-, , double-, , , double-, , , and . These terms are minor either because of their small branching ratios of double pion channel such as , and , or belonging to higher partial waves such as and , or lying beyond the considered energies such as , , and . It should be mentioned that -meson exchange is much smaller than -meson exchange in the available diagrams except for nucleon poles but we still include the -meson exchange in the calculation for the completeness of our model.

Our results underestimate the data in the close-to-threshold region where the final state interactions (FSI) should be relevant. We do not consider the initial state interaction (ISI) either, because at present we do not have an unambiguous method at hand to simultaneously include the FSI and ISI in our model. The ISI usually has a weak energy dependence, so adjusting cut-off parameters in the form factors may partly account for it effectively ouyang (). We would give some qualitative observations of FSI.

Next we shall first address the channel because it has negligible contribution to be more clean. Then we shall discuss other channels and explore the different situation at each channel. In the following we assume the same definitions of various differential cross sections as graphically illustrated in the experimental articles celsius (); skorodko (). The and are the invariant mass spectra, and the angular distributions are all defined in the overall center of mass system. The is the scattering angle of , and is the opening angle between and particles. The (or corresponding to defined in Ref. celsius (); skorodko ()) is the scattering angle of in the rest frame of and with respect to the beam axis (or the sum of momenta of and ). The values of vertical axis are all arbitrarily normalized.

### iii.1 The channel of

In this channel, we find that the term is dominant below 1000 MeV while the and double-nucleon-pole terms are also important. The term is not included in the Valencia model Alvarez (). Our model seems to overestimate the COSY-TOF upper-limit by a factor of around four. The terms in two models are consistent with each other because we use the same coupling constant of from quark model but our double- term contributes smaller as we use a smaller cut-off parameter in form factor. Between 1000 MeV and 1700 MeV, the contribution of the double- term is the most important one, and the invariant mass distribution at 1100 MeV do show a clear peak as can be seen in Fig. 3. We also find that the and terms are crucial to get the right shape of differential cross sections at 1100 MeV. Though the data is of poor statistics, the invariant mass spectrum does not show obvious low-mass peak and this is realized by the inclusion of the and in our model. The has also a significant improvement compared to the double- alone. These distributions should be very useful to constrain the poorly known coupling constant of . The particular enhancement compared to our model without FSI in the invariant mass spectrum is probably an indication of strong FSI.

The contribution from term has a steep rise and begins to take over as the largest one for above 1700 MeV. Besides, at large energies contributions from the and become significant. So in these energy region of , it is a good place to explore the properties of these resonances. We would like to point out that these behaviors together with the dominance of and close to threshold alleviate the isospin problem of the and channels mentioned at the beginning of our article, because the isospin coefficients of these terms in are bigger than that in channel and this is contrary to the case of double-. As a result, we get an improvement on the description of all isospin channels.

It should be addressed that it is very useful to pin down the cut-off values in form factors of the relevant and contributions using the data of at first, and then it makes much easier for us to determined the contributions in other channels. The new value of total cross section of measured at CELSIUS celsius () is in line with previous data and this gives our some confidence on the extracted parameters. Further accurate measurements of the channel should be very helpful for the improvement of the model.

### iii.2 The channel of

Below 1000 MeV the term is the largest while the term is the second. Of these two terms the -meson exchange gives much bigger contribution than the -meson exchange as depicted in Fig. 4 and this shows the importance of isoscalar excitation of . The Double- term is negligible at this low energies as well as the term. Contributions from the nucleon pole and terms are visible below 700 MeV. The proton and pion angular distributions in the center of mass system at 650 and 680 MeV are trivially isotropic and the model does agree with the measured data celsius (). So we do not show them here. The differential cross sections at 750, 775, 800 and 895 MeV are given in Fig. 5, Fig. 6, Fig. 7 and Fig. 8, respectively. Our model calculations reproduce the published data well and are also compatible to the very preliminary data of CELSIUS at 895 MeV skorodko () as shown in Fig. 8. The role of is evident in invariant mass spectrums and the data is fitted better than that including alone. Most obviously, the anisotropic shape of is well fitted after including the while term is symmetric, so together with and is used to determined the ratio of partial decay widths of and celsius (). The results are strongly energy dependent and give a smaller decay width of than that listed in PDG. This is believed to support the breathing mode of ropertheory (). The FSI is evident in invariant mass spectrum, but seems to be much weaker than in the case of channel.

In the Valencia model the double- is dominant above 1300 MeV. However, because we use smaller cut-off parameter for the form factor in order to fit both and channels, our model shows that begins to take over above 1100 MeV, and double- and are also important and comparable. We give the differential cross sections at 1100MeV and 1360 MeV in Fig. 9 and Fig. 10 which can be tested by the measured data of CELSIUS skorodko (). The prominent features are the double hump structure in and the upward bend in which arise from the . The Valencia model give very similar results because and are sensitive to the appearance of . These seem to somewhat incompatible to the preliminary data dataIJMPA (); skorodko () which show the phase space behavior in these two spectrums. The same phenomena also happen in the channel of at high energies, and we will discuss them altogether later.

### iii.3 The channel of

The term dominates below 1000 MeV and the nucleon pole term also gives significant contribution below 800 MeV. The and double- contributions are comparable in this energy region. So it should be cautious to use this channel to extract the ratio of the partial decay widths for the decay of . Indeed, the extracted ratios from are about one third of those from at the same nominal mass under the assumption that dominates in this energy range celsius (); skorodko (). The significant double- and nucleon pole contributions might account for this discrepancy and should be reasonably incorporated in the fit.

Above 1100 MeV, the double- term dominates; the and are also important and give similar contributions. Other contributions are much smaller. The most striking feature in this energy region is that a level-off behavior happens in the total cross section between 1000 and 1200 MeV, while other channels rise smoothly when increasing the incident energy. Our model fails to describe this behavior and also overestimates the high energy data. It is possible that this shape is caused by the interference of different diagrams which are not included in our model, but this would require a peculiar energy dependence of as shown by the isospin decomposition celsius (); skorodko (). Another possible explanation is that there maybe exist a steep rise of some kind of contribution when other contributions are saturated in this energy region. This happens in the channel of where a weak level-off at 1600 MeV is caused by the steep rise of . However, this is not the case for the channel where gives much smaller contribution due to the isospin factor. So this problem is left for further clarification.

In Fig. 11, Fig. 12 and Fig. 13, we show the differential cross sections of at beam energies of 775, 895 and 1000 MeV, respectively. The data at 775 MeV are well reproduced and is overwhelmingly dominant. Some of the angular distributions are sensitive to the presence of the contribution, and hence can be used to determine the partial decay ratios of , although this is somewhat complicated by the double- and nucleon pole contributions as we pointed out earlier. The contribution of and double- terms become much clearer at 895 and 1000 MeV, though slight discrepancy in invariant mass spectrums at 895 MeV exists between our model and the measured data, which remind us that it needs a further improvement in the crossover region. The phase space shapes of and begin to appear ever since 895 MeV and up to high energies of this channel as shown in Fig. 14, Fig. 15 and Fig. 16 which are the differential cross sections of at 1100, 1200 and 1300 MeV. However, because the influence of the does not decrease much or disappear at these energies, our model gives a double hump structure in and forward peak in which are contradictory to the CELSIUS data. In order to explain the data, it is required that the shows up at low energies but immediately saturated at about 1000 MeV. That is a peculiar energy dependence behavior which does not supported by our model. Except for and , other spectrums at high energies are well fitted by our model both in and . So we would rather conclude that something happens in the system which needs a more thorough investigation as the next step. The - rescattering is found to be negligible at these energies piscatter ().

The effect of FSI is not obvious at low energies compared to our calculated curve but enhancement seems to happen at high energies. It is possible that this is related to the behavior of system.

### iii.4 The channels of , and

The does not present in the reaction, so the double- term is the most important one in a wide energy range. The and terms have significant contribution below 800 MeV and also have some contribution at higher energies together with the and terms. The agreement with the data is very good and the FSI may influence the near-threshold region since our model slightly underestimates this part.

The channel of is another reaction where the does not contribute. Since the charged meson exchange is allowed in this channel, the term is very important and is of the same order as the double- term in the whole energy region. The contributions from the nucleon pole and terms are also quite significant near the threshold. Our results reproduce the new bubble chamber data measured by KEK data2000 () very well, but underestimate the old data olddata () by about a factor of 5. Since the double- contribution has been well determined by the channel of , we think that the main ambiguity comes from the term. If future experiments confirm the old data, then the isovector mesons like - and -meson should play more important role in the excitation of . On the other hand, the new data of KEK support the isoscalar excitation of which is favored by our model.

The channel is interesting because it can shed light on the low mass enhancement in , known as the ABC effect of double pion production in nuclear fusion reactions abceffect (). Below 900 MeV, the is found to be dominant while the double- and terms also give some contribution. The nucleon pole and terms are also important close to threshold. Above 1000 MeV, the double- term is the most important one and the gives sizable contribution at high energies. The total contribution gives a reasonable description to the new KEK data while the underestimation of the data close to threshold may be due to the omission of the FSI. Similar to the channel, our model does not favor the old bubble chamber data which need large isovector excitation of . Very recently the ABC effect is experimentally established in at beam energies of 1.03 and 1.35 GeV, and has been interpreted as an s-channel double- resonance abceffect (). According to the observation of our model, the emerges at these energies so it is necessary to take a further look at the mechanism of ABC effect in reaction. As a matter of fact, it has been demonstrated that at beam momentum of 1.46 GeV (corresponding to beam energies of 800 MeV) where the is expected to be dominant, the deuteron momentum spectra can be reasonably explained by the interference of the and Alvarez (); Alvarezabc ().

### iii.5 Final State Interaction

As discussed above, the effect of FSI is anticipated to influence the results close to threshold where the s-wave is expected to be dominant. Usually the Jost function is used to account for the FSI enhancement factor,

(49) |

where is the relative momentum of subsystem in the final state. The corresponding scattering length and effective range are:

(50) |

with = -7.82fm and = 2.79fm for interaction, = 5.42fm and = 1.76fm for isoscalar interaction, and = -18.45fm and = 2.83fm for interaction. At higher energies, the high partial waves become important and above approximate treatment would deteriorate rapidly. Fortunately, the effect of FSI should significantly decrease. So we may just ignore it above 1.4 GeV. To investigate the influence of the FSI to the energy dependence of cross sections, we assume the Jost function for the FSI and normalize this factor to the unity at the beam energy of 1.4GeV. For the final states with the pair, we assume it is mainly in the isoscalar state.

Though above prescription is quite rough, the agreement with the data are considerably improved. In Fig. 17, we demonstrate the total cross section below 1.4GeV. The effect of FSI can be seen in some of the differential cross sections especially spectrums. In Fig. 18 we take channel as a typical example. The FSI gives a sharp peak in the spectrums which agree with the data. However, the data in other channels do not favor this sharp peak and this reflects the drawbacks of our formalism. The FSI slightly improves the fit of the and spectrums but increase the slope of . FSI has very small influence on other angular distributions. The situations for other channels are similar.

## Iv Summary

In this article, we present a simultaneous analysis of varous isospin channels of double pion production in nucleon-nucleon collisions up to 2.2 GeV within an effective Lagrangian approach. We study the contributions of various resonances with mass up to 1.72 GeV and demonstrate that , , , and nucleon pole constitute the main ingredients to reasonably explain the measured data while the contribution of other resonances are negligible. We suggest that it is necessary to consider other influence such as the double- and nucleon pole contributions when one studies the properties of in the channels of and . Our model well describe the measured differential cross sections except some spectra which are left as an open problem. Compared with the Valencia model, the main differences are: (1) Among 3 major ingredients, double-, and terms, considered in the Valencia model, our model increases significantly the relative contribution from the term by reducing the relative branching ratio of and assuming a smaller cut-off parameter for the coupling; (2) In addition, our model introduces significant contributions from at energies near threshold and from and at energies above 1.5 GeV. Though the model should be improved to reasonably incorporated the ISI and FSI, the conclusions reached from our model should be helpful to the future experiments to be performed at COSY and HIRFL-CSR as well as further theoretical study on related problems. Our results also give hints to the ABC effect in the and He reactions which need to be further explored.

###### Acknowledgements.

Useful discussions with E. Oset, J. J. Xie and Z. Ouyang are gratefully acknowledged. We also thank T. Skorodko, E. Dorochkevitch, H. Clement and B. Höistad for providing the data files. Special thanks to C. Wilkin for careful reading of the manuscript and stimulating comments. This work was supported by the National Natural Science Foundation of China (Nos. 10635080, 10875133, 10821063, 10925526 0701180GJ0).## References

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