Strangeness Production Process pp→n K + Σ + within Resonance Model
aa r X i v : . [ nu c l - t h ] A p r Strangeness Production Process pp → nK + Σ + within Resonance Model Xu Cao , , Xi-Guo Lee , ∗ , and Qing-Wu Wang ,
1. Institute of Modern Physics, Chinese Academy of Sciences,P.O. Box 31, Lanzhou 730000, P.R.China2. Center of Theoretical Nuclear Physics,National Laboratory of Heavy Ion Collisions, Lanzhou 730000, P.R.China3. Graduate School, Chinese Academy of Sciences, Beijing 100049, P.R.China
Abstract
We explore production mechanism and final state interaction in the pp → nK + Σ + channel basedon the inconsistent experimental data published respectively by COSY-11 and COSY-ANKE. Thescattering parameter a > n Σ + interaction is favored by large near-threshold cross sectionwithin a nonrelativistic parametrization investigation, and a strong n Σ + interaction comparable to pp interaction is also indicated. Based on this analysis we calculate the contribution from resonance∆ ∗ (1920) through π + exchange within resonance model, and the numerical result suggests a rathersmall near-threshold total cross section, which is consistent with the COSY-ANKE data. Withan additional sub-threshold resonance ∆ ∗ (1620), the model gives a much better description to therather large near-threshold total cross section published by COSY-11. PACS numbers: 13.75.Cs; 14.20.Gk; 13.30.Eg; 13.75.Ev ∗ Email: [email protected] . INTRODUCTION Strangeness production process in nucleon-nucleon collisions has attracted considerabletheoretical interest since high precision data have been published in the past few years[1,2, 3, 4, 5, 6, 7, 8]. This field is fascinating for several reasons. First of all, strangenessproduction in nucleon-nucleon collisions is an elementary process, and their cross sectionsare an fundamental input into transport model calculations of the strangeness production inheavy ion collisions. Furthermore, strangeness production process may provide informationon the strange components of the nucleon and deepen our knowledge on the internal structureof nucleon[22]. Finally, for the short life time of hyperons, it is difficult to accumulate a largeset of scattering events and obtain accurate scattering parameters. Final state interaction(FSI) in strangeness production can supply us with assistant information on the KN andYN interaction.A series of theoretical models[9, 10, 11, 12, 13] have given excellent description to theobserved energy dependence of the total cross section for the strangeness production process pp → pK + Λ and pp → pK + Σ . Some of these models[11, 12, 14] also give a reasonableexplanation to the energy dependence of the ratio R = σ ( pp → pK + Λ) /σ ( pp → pK + Σ ).Compared to the detailed experimental and theoretical investigation in these two channels,there is only limited data on the total cross section for pp → nK + Σ + . A recent COSY-ANKE data[26] indicates a rather small near-threshold total cross section, which is muchsmaller than previous COSY-11 data[25]. For the large error bar of these experimental data,any definite conclusion cannot be surely drawn at present, and it must be checked in futureexperiments.On the other hand, as a good isospin-3/2 filter without complication caused by N ∗ con-tribution, pp → nK + Σ + is an excellent channel for the investigation of ∆ ∗ resonance[17].It is necessary to deepen our theoretical understanding to the controversial experimentaldata. In this paper, we use a resonance model and its nonrelativistic parametrization toexplore the energy dependence of the total cross section of the pp → nK + Σ + . In sectionII we introduce the Feynman diagrams and the effective Lagrangian approach within res-onance model. In section III we present a exploration to the near threshold region withina nonrelativistic parametrization and give a set of FSI parameters. In section IV, we givethe numerical results and explore their implications, especially in the near threshold region.2ection V provides a summary. II. RESONANCE MODEL AND EFFECTIVE LAGRANGIAN APPROACH
The treatment of the elementary processes
N N → N Y K is tree level at the hadroniclevel in the resonance model[9], as illustrated in Figure 1. All interference terms betweendifferent amplitudes are neglected because the relative phases of these amplitudes can notbe fixed by the scarce experimental data. Mesons exchanged are restricted to those observedin the decay channels of the adopted resonances, and kaon exchange is not included in thecalculation. Thus, all values of the coupling constants are fixed by the experimental decayratios, and the only adjustable parameters are cut-off parameters in the form factors. Inthis model, only ∆ ∗ (1920) through π + exchange contributes to the pp → nK + Σ + channelabove the production threshold. The relevant meson-nucleon-nucleon(MNN) and resonance-nucleon(hyperon)-meson effective Lagrangians for evaluating the Feynman diagrams in Fig.1 are: L πNN = − ig πNN ¯ N γ ~τ · ~πN, (1) L ∆(1920) Nπ = g ∆(1920) Nπ m π ¯ N ∆ µ ~τ · ∂ µ ~π + h.c., (2) L ∆(1920)Σ K = g ∆(1920)Σ K m K ∆ µ ~I · ~ Σ ∂ µ K + h.c., (3)where N ,∆ µ , K and ~π are the Rarita-Schwinger spin wave function of the nucleon, theresonance, the kaon and the pion respectively, ~τ the Pauli matrices, ~I the spin-3/2 matricesand g πNN / π = 14 .
4. The effective Lagrangians for the resonance N ∗ please refer to [9, 15,16, 17, 18].For the propagators of π meson and the spin-3/2 ∆ ∗ (1920) resonance, the usual Rarita-Schwinger propagators are used: G π ( k π ) = ik π − m π , (4) G µνR ( p R ) = γ · p + m R p R − m R + im R Γ R [ g µν − γ µ γ ν − m R ( γ µ p ν − γ ν p µ ) − m R p µ p ν ] , (5)3 ABLE I: Relevant ∆ ∗ (1920) parametersWidth Channel Branching ratio Adopted value g / π ∆ ∗ (1920) 200 M eV N π K with k π and p R being the four momentum of π meson and ∆ ∗ (1920) resonance respectively, m π and m R the corresponding masses. The relations between the branching ratios and thecoupling constants then can be calculated using the relevant Lagrangians. Take ∆ ∗ (1920) → N π as an example: Γ ∆(1920) Nπ = g Nπ m π ( m N + E N )( P cmN ) πM ∆(1920) , (6) P cmN = λ / ( m , m N , m π )2 M ∆(1920) , E N = q ( P cmN ) + m N , (7)here λ ( x, y, z ) = x + y + z − xy − yz − zx . Then the coupling constants can be determinedthrough the empirical branching ratios. Values related to ∆ ∗ (1920) are summarized in TableI[9, 17].The resulting πN N and ∆(1920) N π vertexes are multiplied by form factors whichdampen out high values of the exchanged momentum: F M ( ~q ) = Λ M − m M Λ M − q M , (8)with Λ M , q M and m M being the cut-off parameter, four-momentum and mass of the ex-changed meson. Λ π = 1 . GeV as the widely used value[9, 15, 17]. The form factor of the∆(1920)Σ K vertex is 1.As to pp → nK + Σ + channel, if the only contribution comes from resonance ∆ ∗ (1920)through π + exchange, the corresponding amplitude can be obtained straightforwardly byapplying the Feynman rules to Fig. 1:M = g ∆(1920)Σ K m K ¯ u ( S Σ )( p Σ ) p µK G µν ∆(1920) p ∆(1920) g ∆(1920) Nπ m π u ( S )( p ) p νπ G π ( p π )¯ u ( S n )( p n ) √ g πNN γ u ( S )( p ) exchange term with p ↔ p ) , (9)Final state interaction influences the near-threshold behavior significantly. It has beenexperimentally and theoretically verified that p Λ FSI is essential to the pp → pK + Λ process.Similarly, here only n Σ + interaction is considered with Watson-Migdal factorization[19], and nK + interaction will not be taken into account for its weakness[8]:A = M T n Σ , (10)where T n Σ is the enhancement factor describing the n Σ + final state interaction and goes tounity in the limit of no FSI, as expected by the physical picture. Similar to the p Λ FSI inthe pp → pK + Λ channel, we assume that T n Σ can be taken to be Jost function[20]: T n Σ = q + iβq − iα , (11) q = λ / ( s, m , m n )2 √ s , s = ( ε + m n + m K + m Σ ) , (12)with ε being the excess energy. The related scattering length and effective range are: a = α + βαβ , r = 2 α + β , (13)Then the total cross section can be calculated by: σ tot = m p F Z | M | δ ( p + p − p n − p K − p Σ ) m n d p n E n d p K E K m Σ d p Σ E Σ , (14)with the flux factor F = (2 π ) q ( p · p ) − m p . | M | is the square of the invariant scatteringamplitude, averaged over the initial spins and summed over the final spins. The integrationover the phase space can be performed by Monte Carlo program.For the lack of the n Σ + scattering data, a set of data are employed referring to the p Λinteraction[21] in the Ref.[17]’s calculation: α = − . M eV, β = 200 . M eV, (15)The corresponding scattering length and effective range are: a = − . f m, r = 3 . f m, (16)5ext we will concentrate on the relation between FSI and the near-threshold total crosssection, and try to find some clues on the n Σ + interaction. As a matter fact, we fix scatteringlength and effective range with a nonrelativistic parametrization, assuming no possible quasi-bound or molecular state. III. NONRELATIVISTIC PARAMETRIZATION AND FSI
It is generally agreed that the energy variation of the total cross section for the pp → pK + Λ and pp → pK + Σ is fixed mainly by the ε factor coming from phase space, modifiedby the FSI in the near-threshold region: σ tot ∝ ε × F SI, (17)If | M | depend smoothly on the excess energy, Eq.(14) can be parameterized as[21, 23]: σ tot = √ m N m K m Σ π ( m N + m K + m Σ ) / ε λ / ( s, m N , m N ) | M | κ, (18) κ = 1 + 4 β − α ( − α + p α + 2 µε ) , (19)where κ , the FSI factor, was firstly put forward by G. F¨ a ldt and C. Wilkin[28]. At suffi-ciently large ε , κ → ǫ ≃ GeV . In order to give some clues to the n Σ + interaction, the physicalmechanism of strangeness production will not be concerned, and | M | is parameterized as | M | ∝ e − const.ε . Similar parametrization[21] has give an excellent description to the en-ergy dependence of the total cross section for the pp → pK + Λ, pp → pK + Σ and the ratio R = σ ( pp → pK + Λ) /σ ( pp → pK + Σ ).If parameters in FSI factor are employed to be the same with Eq.(16), and | M | =18 . e − . ε · µb ( ε in GeV), the total cross section for pp → nK + Σ + as a function of excessenergy can be exhibited by the solid curve in Fig. 2. It gives a reasonable description tothe total cross section in high energies and underestimates the COSY-11 data, which isconsistent with the numerical calculation below(see Fig.3). This verifies the applicability ofthe parametrization Eq.(18). 6f n Σ + FSI is negligible ( κ = 1) and | M | = 21 . e − . ε · µb , the result is dashed curvein Fig. 2. It still overestimates COSY-ANKE data by a factor of about 3. As a matter offact, the pure phase space behavior is favored by so small near-threshold total cross section,as indicated by the dotted curve with κ = 1 and | M | = 4 . · µb . This means that n Σ + FSI is negligible, similar to the p Σ FSI in pp → pK + Σ , but the scattering amplitude ismore weakly dependent on the excess energy. If the near-threshold total cross section isreally as small as COSY-ANKE data[26], the nearly constant scattering amplitude is theextraordinary character for pp → nK + Σ + channel.However, in order to reproduce COSY-11 data, totally different parameters should beadopted: | M | = 20 e − . ε · µb, (20) α = 30 M eV, β = 360
M eV, (21)The result is the dash-dotted curve in Fig. 2. The corresponding scattering length a = 7 . f m > r = 1 . f m ) is contrary to the values used in the Ref.[17]’s calculation(Eq.(16)) and implies an even stronger n Σ + interaction. It is worth noting that the scatteringparameters of pp interaction are a = − . f m and r = 2 . f m . In the next section we willgive a comparison between these two sets of n Σ + FSI parameters, and show that a betterreproduce to the large near-threshold total cross section is achieved with Eq.(21).
IV. NUMERICAL RESULTS AND IMPLICATION
The contribution from resonance ∆ ∗ (1920) through π + exchange with FSI parametersin Eq.(16) is shown as the dotted curve in Fig. 3. Note that the resonance ∆ ∗ (1920)can be treated as an effective resonance which represents all contributions of six reso-nances (∆ ∗ (1900),∆ ∗ (1905),∆ ∗ (1910),∆ ∗ (1920),∆ ∗ (1930), and ∆ ∗ (1940)) and the couplingconstants relevant to ∆ ∗ (1920) are scaled multiplying by a factor[27]. The numerical result(the dashed curve in Fig. 3) reproduces the data in high energies reasonably well but un-derestimates COSY-11 data by a factor of orders, as pointed out in above section. It hasbeen suggested[17] that the inclusion of a ignored sub-threshold resonance ∆ ∗ (1620)1 / − ρ + and π + exchange are included), together with a strong n Σ + FSI in Eq.(16), wouldachieve much better agreement, as illustrated by the solid curve in Fig. 3. Unfortunately, itstill underpredicts two near-threshold data points. It is argued[17] that improvement can beachieved by inclusion of the interference term between ρ + and π + exchange. However, thiskind of interference term is very small as demonstrated in another channel pp → pK + Λ[16],and it definitely can not provide any interpretation to this relatively large discrepancy.The numerical result showed by the dashed curve in Fig. 3 agrees COSY-ANKE data,which was obtained at a little higher energy and shows a rather small total cross section innear-threshold region, as indicated by a closed star in Fig. 3.The numerical results with the FSI parameters in Eq.(21) are given in Fig.4. The contri-bution from resonance ∆ ∗ (1920) through π + exchange is not affected much by the alterationof the FSI parameters, and still gives a small near-threshold total cross section. This in-dicates that the contribution from resonance ∆ ∗ (1920) is not sensitive to the magnitude ofthe n Σ + interaction, and we would be not able to distill any valuable information about the n Σ + interaction. However, the contribution from resonance ∆ ∗ (1620) gives an even largernear-threshold cross section, and the resulting curve reproduces all previous published datareasonably, as illustrated by the solid curve in Fig. 4. This strongly suggests a rather strong n Σ + interaction if the near-threshold cross section is as large as COSY-11 data[25]. Wewould like to stress that this is incompatible to the known YN scattering data. Besides,the p Σ interaction is found to be weak in pp → pK + Σ channel, and the naive speculationis that the n Σ + interaction should be weak, too. One may further expect that the energydependence of the pp → nK + Σ + is approximately the same with that of pp → pK + Σ [29].Anyway, due to the scarcity and uncertainty of those experimental data, the conclusion thatthese data imply a highly anomaly near-threshold behavior can not be definitely drawn yet.Based on these numerical result, we may also conclude that contribution from ∆ ∗ (1620) isindispensable for a good reproduce to the large near-threshold total cross section, which hasbeen discussed deeply in the Ref.[17]. V. SUMMARY
We presented the confusion caused by the existing data for pp → nK + Σ + channel. Thenumerical result and parametrization analysis reveal a large discrepancy between two sets of8ear-threshold data, which means totally different n Σ + interaction. The theoretical resultspresented in other works[21] also indicated that the total cross section data alone cannotdistinguish different production mechanisms. In order to clarify the confusion that if therewas a highly abnormal near-threshold behavior in pp → nK + Σ + channel, it is necessary toaccumulate a data set with high accuracy and large statistics. Further measurements arebeing planned in COSY-ANKE[29]. Besides, a significant improvement is to be expectedthrough the installation of Cooling Storage Ring (CSR) in Lanzhou, China, which is de-signed for the study of heavy-ion collisions. It can provide 1 ∼ . GeV proton beam andperform accurate measurement to differential observables and invariant mass spectrum. InRef.[21] the authors proposed a scheme to study the role of the FSI by analyzing Dalitzplot distribution, and experiment performed by COSY-TOF Collaboration[8] verified theirprediction. Similar analysis may also clarify the confusion in pp → nK + Σ + channel. SoCSR ’s data may offer an opportunity to explore the reaction mechanism for the strangeproduction process in nucleon-nucleon collisions. Only then it is possible to achieve an ac-ceptable confidence level for the near-threshold behavior and the dynamical mechanism inthe pp → nK + Σ + channel. Acknowledgments
We would like to thank J. J. Xie and B. S. Zou for fruitful discussions and programcode. We also thank Colin Wilkin for a careful reading of the draft and valuable comments.This work was supported by the CAS Knowledge Innovation Project (No.KJCX3-SYW-N2,No.KJCX2-SW-N16) and Science Foundation of China (10435080, 10575123,10710172). [1] J. T. Balewski et. al.,
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FIG. 2: Total cross section for pp → nK + Σ + as a function of excess energy. The relevant pa-rameters used for the curves refer to the text. Data are taken from Ref.[24], COSY-11[25](closedsquare) and COSY-ANKE[26] (closed star). .500 1.875 2.250 2.625 3.000 3.375 3.750 4.125 4.500 4.875 5.250 5.625 6.000 C r o ss s e c t i on ( ub ) Tp(GeV)
FIG. 3: Variation of the total cross section with the kinetic energy of the proton beam (Tp). Dottedcurve: contribution from resonance ∆ ∗ (1920) through π + exchange. Dashed curve: contributionfrom resonance ∆ ∗ (1920) scaled multiplying by a factor 5. Solid curve: numerical result in theRef.[17]. Data are the same as in Figure 2. .500 1.875 2.250 2.625 3.000 3.375 3.750 4.125 4.500 4.875 5.250 5.625 6.000 C r o ss s e c t i on ( ub ) Tp(GeV)