Theoretical study on pp --> p n pi+ reaction at medium energies
aa r X i v : . [ nu c l - t h ] F e b October 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs
International Journal of Modern Physics Ec (cid:13)
World Scientific Publishing Company
THEORETICAL STUDY ON pp → pnπ + REACTION AT MEDIUMENERGIES
ZHEN OUYANG
Institute of Modern Physics, CAS, Lanzhou 730000, ChinaGraduate University of Chinese Academy of Sciences, Beijing 100049, ChinaTheoretical Physics Center for Science Facilities, CAS, Beijing 100049, [email protected]
JU-JUN XIE
Institute of High Energy Physics, CAS, Beijing 100049, ChinaGraduate University of Chinese Academy of Sciences, Beijing 100049, ChinaTheoretical Physics Center for Science Facilities, CAS, Beijing 100049, [email protected]
BING-SONG ZOU
Institute of High Energy Physics, CAS, Beijing 100049, ChinaTheoretical Physics Center for Science Facilities, CAS, Beijing 100049, [email protected]
HU-SHAN XU
Institute of Modern Physics, CAS, Lanzhou 730000, ChinaTheoretical Physics Center for Science Facilities, CAS, Beijing 100049, [email protected]
Received (received date)Revised (revised date)The pp → pnπ + reaction is a channel with the largest total cross section for pp collisionin COSY/CSR energy region. In this work, we investigate individual contributions fromvarious N ∗ and ∆ ∗ resonances with mass up to about 2 GeV for the pp → pnπ + reaction.We extend a resonance model, which can reproduce the observed total cross section quitewell, to give theoretical predictions of various differential cross sections for the presentreaction at T p = 2 .
88 GeV. It could serve as a reference for identifying new physics inthe future experiments at HIRFL-CSR.
1. Introduction
The study of excited nucleon states is very important for understanding the inter-nal structure of nucleon and the strong interaction in the nonperturbative QCDdomain 1. In the early years, our investigation on the N ∗ and ∆ ∗ baryon spec-troscopy was mainly based on πN experiments, which made observations unsatis-factory 2 ,
3. An outstanding problem is that, in many of its forms, the quark model ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou predicts a large amount of “missing” N ∗ and ∆ ∗ states around 2 GeV/ c , whichhave not to date been observed 3 , ,
5. Therefore, it is of necessity to search for these“missing” N ∗ and ∆ ∗ states from other production processes. Moreover, even forthose well-established resonance states, properties like mass, width and branchingratios still suffer large experimental uncertainties 6, which also need further studiesin more other production processes. Here we propose to look for “missing” N ∗ and∆ ∗ resonances in pp → pnπ + reaction. At COSY/J¨ulich, Experiments for studying N ∗ and ∆ ∗ resonances through pp collisions are being carried out, but there is alack of a good 4 π detector for complete measurement of diverse differential crosssections. At present, a heavy ion cooler-storage ring HIRFL-CSR—an acceleratorsystem of the same beam energy region with maximum incoming-proton kinetic en-ergy up to 2 .
88 GeV 7, has already been completed at Lanzhou. With its scheduled4 π hadron detector 7, it will have a special advantage for studying excited nucleonstates through pp collisions.Recently, BES collaboration has produced quite a few novel findings on N ∗ reso-nances by using various N ∗ production processes from J/ψ or ψ ′ decays 8 , , , , J/ψ → pπ − ¯ n + c.c. decay by BES collaboration showed twonew, clear N ∗ peaks in the pπ invariant mass spectrum around 1360 MeV/ c and2065 MeV/ c , respectively 9. Of them the former one was identified as the firstdirect observation of the N ∗ (1440) peak in the πN invariant mass spectrum, whichwas confirmed by the CELSIUS-WASA Collaboration in their nπ + invariant massspectrum of pp → pnπ + reaction 13. For the latter one, it is very likely to bea long-sought missing N ∗ peak around 2 GeV/ c . However, similar searches forit in ψ ′ decays are inconclusive 11 ,
12. Therefore, it is of necessity to look for thenew N ∗ resonance in other reaction processes, such as the pp → pnπ + reaction.Furthermore, in Ref. 14, the authors found that the ∆ ∗ ++ (1620) resonance givesan overwhelmingly large contribution in the pp → nK + Σ + reaction by t-channel ρ + exchange. If so, it is also expected to make a significant contribution in the pp → pnπ + reaction, as can be checked in the present work. In Ref. 15, we havestudied the pp → pnπ + reaction for beam energies below 1.3 GeV. Here we extendthe study of this reaction to higher energies and investigate individual contributionsfrom various N ∗ and ∆ ∗ resonances with mass up to 2 GeV/ c for this reaction. Weextend a resonance model, which can describe the experimental data of the totalcross section for beam energies ranging from 0.8 GeV to 3.0 GeV quite well, togive theoretical prediction of various differential cross sections for the pp → pnπ + reaction at T p = 2 .
88 GeV. It can be used for the subsequent comparison withthe experimental results at COSY and HIRFL-CSR. Meanwhile, it could serve as areference for the construction of the scheduled 4 π hadron detector at HIRFL-CSR.
2. Formalism and ingredients
We study the pp → pnπ + reaction within an effective Lagrangian approach. Inour model,all the mesons,baryons and resonances are treated as fundamental fields.ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Theoretical study on pp → pnπ + reaction at medium energies p pπ + , ρ + ( π , ρ ) n ( p ) p ( n ) π + ∆ ∗ ++ (∆ ∗ + ) p n π + π , η, ρ , σ N ∗ + ( a ) ( b ) p p ( c ) ( d ) Fig. 1. Feynman diagrams for pp → pnπ + reaction. All the basic Feynman diagrams involved in our calculation for this reaction aredepicted in Fig. 1. In view of overall system invariant mass about 3 GeV for T p =2 .
88 GeV, we have checked contributions from all the well-established N ∗ and∆ ∗ resonances (overall status 3 or 4 stars) below 2 GeV/ c , but only present theresults of the relatively significant ones in next section. Meanwhile, we investigatethe contribution from N ∗ (2065) resonance for the present reaction. Explicitly, welist in Table 1 all the N ∗ , ∆ ∗ resonances and the meson exchanges considered inour present calculation.For N ∗ (2065), according to results in Ref. 9, its spin-parity is limited to be 1 / + and 3 / + , and it is more likely that both are needed. In the quark model there arepredictions for the existence of N ∗ resonances with spin-parity 1 / + and 3 / + between 2.0 and 2.1 GeV/ c , ,
5. Since the spin-parity of the new resonance(s)ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou was not well determined, we assume that this peak consists of exactly those reso-nances with J p = 1 / + , / + predicted in Ref. 5, which are N ∗ (1975)( J p = 1 / + ), N ∗ (2030)( J p = 3 / + ) and N ∗ (2065)( J p = 1 / + ). Among them N ∗ (2065) hasmuch stronger coupling to πN than the other two, which is in accord with BESresults. As in Ref. 16, here we also treat the observed N ∗ (2065) peak as an effective N ∗ (2065)( J p = 1 / + ) resonance which represents all contributions of the three res-onances N ∗ (1975)( J p = 1 / + ), N ∗ (2030)( J p = 3 / + ) and N ∗ (2065)( J p = 1 / + ).In so doing, the coupling constant g πNN ∗ (2065) / π is scaled by a factor of 1.122.See Ref. 16 for details of this effective treatment. Of course, we have used the BESobserved values for the mass and width of N ∗ (2065) to determine its relevant cou-pling constant, see Table 1. In view of scanty information for its decay branchingfractions, here we regard N π as the dominant decay mode of N ∗ (2065), whereasthe N ∗ (2065) peak in invariant mass M pπ − spectrum is so strong and highly sig-nificant. So, in our calculations we have taken an artificial branching ratio up to 1for N π decay mode.The effective Lagrangian densities involved for describing the meson-
N N ver-tices are: L πNN = − f πNN m π u N γ γ µ ~τ · ∂ µ ~ψ π u N , (1) L ηNN = − ig ηNN u N γ ψ η u N , (2) L σNN = g σNN u N ψ σ u N , (3) L ρNN = − g ρNN u N ( γ µ + κ m N σ µν ∂ ν ) ~τ · ~ψ µρ u N . (4)At each vertex a relevant off-shell form factor is used. In our computation, wetake the same form factors as used in the well-known Bonn potential model 17: F NNM ( k M ) = ( Λ M − m M Λ M − k M ) n (5)with n=1 for π , η and σ mesons and n=2 for ρ meson. k M , m M and Λ M arethe 4-momenta, mass and cut-off parameter for the exchanged meson ( M ), respec-tively. The coupling constants and the cut-off parameters are taken as the follow-ing ones 14 , , , , g πNN / π = 14 . g ηNN / π = 0 .
4, Λ π = Λ η = 1 . g σNN / π = 5 .
69, Λ σ = 2 . g ρNN / π = 0 .
9, Λ ρ = 1 .
85 GeV, and κ = 6 .
1. Notethat the constant g πNN is related to f πNN of Eq.(1) as g πNN = ( f πNN /m π )2 m N N ∗ and ∆ ∗ resonances. In Ref. 21,a Lorentz covariant orbital-spin scheme for N ∗ N M couplings has been describedin detail, which can be easily extended to describe all the couplings that appearctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs
Theoretical study on pp → pnπ + reaction at medium energies in the Feynman diagrams depicted in Fig. 1. By using that scheme, we can easilyctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou obtain the effective couplings: L πN ∆(1232) = g ∆(1232) Nπ u N ∂ µ ψ π u ∆(1232) µ + h.c. , (6) L πNN ∗ (1440) = g N ∗ (1440) Nπ u N γ γ µ ∂ µ ψ π u N ∗ (1440) + h.c. , (7) L σNN ∗ (1440) = g N ∗ (1440) Nσ u N ψ σ u N ∗ (1440) + h.c. , (8) L πNN ∗ (1520) = g N ∗ (1520) Nπ u N γ γ µ p µπ p νπ ψ π u N ∗ (1520) ν + h.c. , (9) L ρNN ∗ (1520) = g N ∗ (1520) Nρ u N ψ µρ u N ∗ (1520) µ + h.c. , (10) L πNN ∗ (1535) = g N ∗ (1535) Nπ u N ψ π u N ∗ (1535) + h.c. , (11) L ηNN ∗ (1535) = g N ∗ (1535) Nη u N ψ η u N ∗ (1535) + h.c. , (12) L ρNN ∗ (1535) = g N ∗ (1535) Nρ u N γ ( γ µ − q µ γ ν q ν q ) ψ µρ u N ∗ (1535) + h.c. , (13) L πN ∆ ∗ (1600) = g ∆ ∗ (1600) Nπ u N ∂ µ ψ π u ∆ ∗ (1600) µ + h.c. , (14) L πN ∆ ∗ (1620) = g ∆ ∗ (1620) Nπ u N ψ π u ∆ ∗ (1620) + h.c. , (15) L ρN ∆ ∗ (1620) = g ∆ ∗ (1620) Nρ u N γ ( γ µ − q µ γ ν q ν q ) ψ µρ u ∆ ∗ (1620) + h.c. , (16) L πNN ∗ (1650) = g N ∗ (1650) Nπ u N ψ π u N ∗ (1650) + h.c. , (17) L ηNN ∗ (1650) = g N ∗ (1650) Nη u N ψ η u N ∗ (1650) + h.c. , (18) L ρNN ∗ (1650) = g N ∗ (1650) Nρ u N γ ( γ µ − q µ γ ν q ν q ) ψ µρ u N ∗ (1650) + h.c. , (19) L πNN ∗ (1675) = g N ∗ (1675) Nπ u N p µπ p νπ ψ π u N ∗ (1675) µν + h.c. , (20) L πNN ∗ (1680) = g N ∗ (1680) Nπ u N γ γ µ p µπ p νπ p λπ ψ π u N ∗ (1680) νλ + h.c. , (21) L πNN ∗ (1700) = g N ∗ (1700) Nπ u N γ γ µ p µπ p νπ ψ π u N ∗ (1700) ν + h.c. , (22) L ρNN ∗ (1700) = g N ∗ (1700) Nρ u N ψ µρ u N ∗ (1700) µ + h.c. , (23) L πN ∆ ∗ (1700) = g ∆ ∗ (1700) Nπ u N γ γ µ p µπ p νπ ψ π u ∆ ∗ (1700) ν + h.c. , (24) L ρN ∆ ∗ (1700) = g ∆ ∗ (1700) Nρ u N ψ µρ u ∆ ∗ (1700) µ + h.c. , (25) L πNN ∗ (1710) = g N ∗ (1710) Nπ u N γ γ µ ∂ µ ψ π u N ∗ (1710) + h.c. , (26) L ηNN ∗ (1710) = g N ∗ (1710) Nη u N γ γ µ ∂ µ ψ η u N ∗ (1710) + h.c. , (27) L σNN ∗ (1710) = g N ∗ (1710) Nσ u N ψ σ u N ∗ (1710) + h.c. , (28) L πNN ∗ (1720) = g N ∗ (1720) Nπ u N ∂ µ ψ π u N ∗ (1720) µ + h.c. , (29) L ηNN ∗ (1720) = g N ∗ (1720) Nη u N ∂ µ ψ η u N ∗ (1720) µ + h.c. , (30) L πN ∆ ∗ (1905) = g ∆ ∗ (1905) Nπ u N γ γ µ p µπ p νπ p λπ ψ π u ∆ ∗ (1905) νλ + h.c. , (31) L πN ∆ ∗ (1910) = g ∆ ∗ (1910) Nπ u N γ γ µ ∂ µ ψ π u ∆ ∗ (1910) + h.c. , (32) L ρN ∆ ∗ (1910) = g ∆ ∗ (1910) Nρ u N ( p Nµ − k ρµ − ( m N − k ρ ) q µ q ) ψ µρ u ∆ ∗ (1910) + h.c. , (33) L πN ∆ ∗ (1920) = g ∆ ∗ (1920) Nπ u N ∂ µ ψ π u ∆ ∗ (1920) µ + h.c. , (34) L πN ∆ ∗ (1930) = g ∆ ∗ (1930) Nπ u N p µπ p νπ ψ π u ∆ ∗ (1930) µν + h.c. , (35) L πNN ∗ (2065) = g N ∗ (2065) Nπ u N γ γ µ ∂ µ ψ π u N ∗ (2065) + h.c. . (36)ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Theoretical study on pp → pnπ + reaction at medium energies For the relevant vertices involving N ∗ and ∆ ∗ resonances, the off-shell formfactors are adopted as follows: F M ( k M ) = ( Λ ∗ M − m M Λ ∗ M − k M ) n (37)where n=1 for all the resonances except for n=2 for ∆(1232). All the couplingconstants and cut-off parameters used in the present paper are listed in Table 1.In addition, we also introduce form factors for the off-shell baryon resonances as inRefs. 22 , , F R ( q ) = Λ Λ + ( q − m R ) , (38)with Λ= 0.8 GeV.The propagators can be written as G ( q ) = q + m R q − m R + im R Γ R (39)for the spin- resonances, G µν ( q ) = − P µν ( q ) q − m R + im R Γ R (40)with P µν ( q ) = − ( q + m R )[ g µν − γ µ γ ν − m R ( γ µ q ν − γ ν q µ ) − m R q µ q ν ] , (41)for the spin- resonances, and G µναβ ( q ) = − P µναβ ( q ) q − m R + im R Γ R (42)with P µναβ ( q ) = − ( q + m R )[ 12 (˜ g µα ˜ g νβ + ˜ g µβ ˜ g να ) −
15 ˜ g µν ˜ g αβ (43)+ 110 (˜ γ µ ˜ γ α ˜ g νβ + ˜ γ ν ˜ γ β ˜ g µα + ˜ γ µ ˜ γ β ˜ g να + ˜ γ ν ˜ γ α ˜ g µβ )] , (44)˜ g µν ( q ) = − g µν + q µ q ν m R , ˜ γ µ = − γ µ + qq µ m R . (45)for the spin- resonances.After the effective Lagrangians, coupling constants and propagators fixed, theamplitudes for various diagrams can be written down straightforwardly by followingthe Feynman rules. And the total amplitude is just their simple sum. Here we givectober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou
Table 1. Relevant parameters of N ∗ and ∆ ∗ included in our calculations. The widths and branchingratios are taken from PDG and the cut-off parameters are from Refs. , , , . Here the g / π for N ∗ (2065) Nπ vertex has already been scaled by a factor of 1.122. Resonance Width/GeV Decay mode Branching ratio g / π Cut-off/GeV∆(1232) 0.118
N π N ∗ (1440) 0.3 N π
N σ N ∗ (1520) 0.115 N π
N ρ N ∗ (1535) 0.15 N π
N η
N ρ ∗ (1600) 0.35 N π ∗ (1620) 0.145 N π
N ρ N ∗ (1650) 0.165 N π
N η
N ρ N ∗ (1675) 0.15 N π N ∗ (1680) 0.13 N π N ∗ (1700) 0.1 N π
N ρ ∗ (1700) 0.3 N π
N ρ N ∗ (1710) 0.1 N π
N η
N σ N ∗ (1720) 0.2 N π
N η ∗ (1905) 0.33 N π ∗ (1910) 0.25 N π
N ρ ∗ (1920) 0.2 N π ∗ (1930) 0.36 N π N ∗ (2065) 0.165 N π ∼ N ∗ (1440)( π exchange), as anexample, M ( N ∗ (1440) , π ) = √ f πNN m π g N ∗ Nπ ¯ u n ( p n , s n ) γ p π G N ∗ (1440) ( q ) γ k π u ( p , s ) ik π − m π ¯ u ( p , s ) γ k π u ( p , s ) + (exchange term with p ↔ p ) , (46)ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Theoretical study on pp → pnπ + reaction at medium energies where u n ( p n , s n ), u ( p , s ), u ( p , s ), u ( p , s ) denote the spin wave functions ofthe outgoing neutron, proton in the final state and two initial protons, respectively. p π and k π are the 4-momenta of the outgoing and the exchanged pion mesons. p and p represent the 4-momenta of the two initial protons. The coupling constantappearing herein can be determined from the experimentally observed partial decaywidth of N ∗ (1440) resonance as follows,Γ N ∗ (1440) → Nπ = 3 g N ∗ (1440) Nπ p cmN π [ m π ( E N − m N ) m N ∗ (1440) + 2( p cmN ) ] , (47)with p cmN = vuut ( m N ∗ (1440) − ( m N + m π ) )( m N ∗ (1440) − ( m N − m π ) )4 m N ∗ (1440) , (48) E N = q ( p cmN ) + m N . (49)All the other coupling constants can be obtained similarly. See Refs. 14 ,
15 fordetails.Then the calculation of the cross section σ ( pp → pnπ + ) is straightforward, dσ ( pp → pnπ + ) = 14 m p F X s i ,s f |M| m p d p E d p π E π m n d p n E n δ ( p + p − p − p π − p n )(50)with the flux factor F = (2 π ) q ( p · p ) − m p . (51)The factors 1/4 and P s i ,s f emerge for the simple reason that the polarization ofinitial and final particles is not considered.
3. Numerical results and discussion
With the formalism and ingredients discussed in the former section, we com-puted the total cross section versus the kinetic energy of the proton beam (T P )for the pp → pnπ + reaction by using a Monte Carlo multi-particle phase spaceintegration program. The results for T P ranging from 0.8 to 3.0 GeV are shown inFig. 2 along with experimental data 26 for comparison.As one can see from Fig. 2, the experimental data of total cross section are repro-duced reasonably well by our theoretical calculations over the entire energy range.Note that we have considered the interference terms between the direct amplitudes(diagram a,c in Fig. 1) and the corresponding exchange amplitudes (diagram b,din Fig. 1) in our calculations. However,the interference terms between differentresonance-excitation processes and between various meson-exchange diagrams areignored. We also show contributions of various components which are large and notctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou -4 -3 -2 -1 ( m b ) T P (GeV) -4 -3 -2 -1 ( m b ) T P (GeV) Fig. 2. Total cross section and contributions from various N ∗ (left) and ∆ ∗ (right) resonancesas a function of T P for the pp → pnπ + reaction with the solid line as the simple incoherentsum of all contributions, compared with data . Left: the dashed, dotted, dot-dashed, dot-dot-dashed, short-dashed and short-dotted lines represent individual contributions from N ∗ (1440), N ∗ (1520), N ∗ (1650), N ∗ (1675), N ∗ (1680), and N ∗ (2065), respectively. Right: the dashed, dot-ted, dot-dashed, dot-dot-dashed, and short-dashed lines represent individual contributions from∆(1232), ∆ ∗ (1600), ∆ ∗ (1620), ∆ ∗ (1700), and ∆ ∗ (1905), respectively. negligible there, N ∗ contributions in Fig. 2 (left) and ∆ ∗ contributions in Fig. 2(right), respectively. Individual contributions from N ∗ (1440), N ∗ (1520), N ∗ (1650), N ∗ (1675), N ∗ (1680), and N ∗ (2065) are presented in Fig. 2 (left) by dashed, dot-ted, dot-dashed, dot-dot-dashed, short-dashed and short-dotted lines, respectively.And contributions from ∆(1232), ∆ ∗ (1600), ∆ ∗ (1620), ∆ ∗ (1700), and ∆ ∗ (1905)are shown in Fig. 2 (right) by dashed, dotted, dot-dashed, dot-dot-dashed, andshort-dashed lines, respectively. One can find that contributions from ∆(1232) and N ∗ (1440) are still dominant in present energy region and the contribution from N ∗ (1680) becomes significant for kinetic energy above 2.0 GeV. We also give ourpredictions of invariant mass spectra and Dalitz plot in Fig. 3 and the momentumand angular distributions of the final charged particles in Fig. 4 for pp → pnπ + reaction at T P = 2 .
88 GeV.The pp → pnπ + reaction is a channel with the largest total cross section forpp collision in the present energy region. Since the kinetic energy T P of the protonbeam at HIRFL-CSR can reach 2 .
88 GeV with luminosity above 10 cm − s − π hadron detector at HIRFL-CSR will be particularly competent.As mentioned above, the spin-parity of the N ∗ (2065) peak was not well deter-ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Theoretical study on pp → pnπ + reaction at medium energies pp → pnπ + reaction atT P = 2 .
88 GeV. The dashed curves stand for pure phase space distribution while the solid curvesinclude the interaction amplitudes. mined by the BES collaboration, therefore we have used an effective treatment forthe N ∗ (2065) resonance(s) as the authors did in Ref. 16. This effective descriptionwould indeed be a very good approximation for the total cross section, but generallyspeaking, it might reproduce the differential cross sections not so well. However,due to the dominance of N ∗ (2065) (its magnitude is much stronger than the othertwo) among the three resonances predicted in Ref. 5, hence even for the descriptionof various differential cross sections, it would be acceptable. Of course, this issuestill waits for an exact answer from future experimental results at HIRFL-CSR.To sum up, in this paper we investigate individual contributions from diverse N ∗ and ∆ ∗ resonances up to 2 GeV/ c for the pp → pnπ + reaction. We extenda resonance model, which can describe the observed total cross section for beamctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs Zhen Ouyang, Ju-Jun Xie, Hu-Shan Xu, Bing-Song Zou
Fig. 4. The momentum and angular distributions of the final proton and charged pion for pp → pnπ + reaction at T P = 2 .
88 GeV, compared with pure phase space distributions (dashed curves). energies ranging from 0.8 GeV to 3.0 GeV quite well, to give theoretical predictionof various differential cross sections for this reaction at T p = 2 .
88 GeV. It can beused for identifying new physics in the future experiments at HIRFL-CSR. It couldalso serve as a reference for the construction of the scheduled 4 π hadron detectorat HIRFL-CSR, which is quite possible to offer more physical information and tohelp us understanding the relevant physics better. Acknowledgements:
We thank C. Zheng and B.C. Liu for useful discussions.This work is partly supported by the National Natural Science Foundation of Chinaunder grants Nos. 10435080, 10521003, 10635080, and by the Chinese Academy ofSciences under project No. KJCX2-SW-N18, KJCX3-SYW-N2,CXTD-J2005-1.ctober 27, 2018 7:31 WSPC/INSTRUCTION FILE ws-ijmpe-zoubs
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