OPER-model for π–mesons production NN-interactions in 1. Introduction A.P. Jerusalimov, JINR-LHEP, Dubna Outlook 1. Introduction 2. Reaction np → npπ+ π− at P0 > 3 GeV/c 3. Reaction pbar p → pbar p π+ π− at P0 = 7.23 GeV/c 4. Reaction np → npπ+ π− at P0 < 3 GeV/c 5. OPER model and other reactions 6. Conclusion References Orsay April 3-4, 2013

1. Introduction: Various modifications of the one pion exchange models (OPE) are used to describe the experimental data of the inelastic NN-, NbarN- and πN-interactions. At that parameters of these models are different for various processes and even for concrete reactions at various energies. Various models differ also in respect of the reggeization of π-meson: at times an exchange by elementary π-meson [1] is used at other times - by reggeized π-meson [2]. The models of Regge pole exchange [3,4] are based on the method of complex momenta and consider an exchange in t-channel by a virtual state R that has quantum numbers of particle (resonances) with variable spin and is on some trajectory αR(t) named Regge trajectory. The most developed and detailed model of reggeized π -meson exchange is the model suggested in ITEP [5]. The advantages of this model are: ● small number of free parameters (3 in our case), ● wide region of the described energies (2 ÷ 200 GeV), ● calculated values are automatically normalized to the reaction cross-section. Orsay April 3-4, 2013

Amplitude of binary and quasi-binary processes a + b → c + d [3] where gRac(t), gRbd(t) – vertex functions αR(t) - Regge trajectory - signature factor with signature σ =(-1)l for interger l (bosons) σ =(-1)l±½ for interger l (fermions) Orsay April 3-4, 2013

Reaction NN→ NNππ [5] S=(Q1+Q2)2 S1=(q1+k1)2 S2=(q2+k2)2 t1=(Q1-q1)2 t =(Q1-q1-k1)2 =(Q2-q2-k2)2 at small|t| → η(t)~1/(t-m2π) κ2i= k2i┴+m2π-c(t-m2π)(C=0.08) απ(t)- Regge trajectory Orsay April 3-4, 2013

2. Reaction np → npπ+π− at P0=5.20 GeV/c Interference between diagrams (a,b and c) is negligible at P0> 3.0 GeV/c [6] This «hanged» diagrams (d and e) are important only at P0 > 10 GeV/с Fig.1 Regge trajectory of π-meson: απ(t)= α'π(t-m2π) with α'π=0.7 The data of elastic πN→πN were taken from PWA [7] and factor c in κ2i was taken in the form Orsay April 3-4, 2013

We have studied the reaction np → npπ+ π− under condition of 4π geometry at P0=1.25 ÷5.20 GeV/c. The use of some specific cuts[8] permits to select the kinematic region of the reaction np → npπ+ π− in which the contribution of the diagrams (1a) consists up to 95 % at P0 > 3 GeV/c. Fig.2 shows some distributions for the reaction np → npπ+ π− for this region at P0 = 5.20 GeV/c (red curves – calculations using OPER model) Fig.2 Orsay April 3-4, 2013

the matrix element for which is written in the form like in [9] : But the diagrams shown in Fig.1 are insufficient to describe totally the characteristics of the reaction np → npπ+ π−. It is necessary to take into account the diagrams of the following type: Fig. 3 the matrix element for which is written in the form like in [9] : where TπN→ππN - off mass shell amplitudes of inelastic πN→ππN scattering Orsay April 3-4, 2013

- formfactor An important detail in determination of value κ2 in formfactor F13 The reaction πN→ππN is in fact the sum of separate 2-particles channels (see Appendix): πN → N*(Δ*) → Δ π , → N*(Δ*) → N ρ , → N*(Δ*) → N ε , → N*(Δ*) → N*1440 π , Therefore there are 4 formfactors: F13Δ for πN → Δ π with κ2= k2π┴+m2π-c(t-m2π) and F13ρ for πN → Nρ with κ2= k2ρ┴+m2ρ-c(t-m2ρ) and F13ε for πN → Nε with κ2= k2ε┴+m2ε-c(t-m2ε) and F13N for πN → N*π with κ2= k2π┴+m2π-c(t-m2π) and This choice of the formfactor provides the explanation for the absence of the clear signal of ρ-meson production in the effective masses of ππ – combinations from NN→NNππ reactions due to the suppression by a considerably larger value of κ2 in formfactor F13 Orsay April 3-4, 2013

np→ppπ–. Therefore it is necessary to take into account such processes It was shown in [9] that the processes of diffractive production of N*1440 - and N*1680-resonances make also sizeable contribution into the reaction np→ppπ–. Therefore it is necessary to take into account such processes for the reaction n p →n pπ+π− that is described by the diagrams similar to diagrams in Fig.3 with the replacement of π-meson exchange by the exchange of vacuum pole (pomeron) The matrix element for the diagrams of pomeron exchange is written in following form: where gPN(t)=gPN(0)exp(-R2N | t |) – vertex function, αP(t)=αP(0)+ α‘P(t) – Regge trajectory of pomeron. Values gPN(0), R2N , αP(0) and α‘P were taken from [3]. Orsay April 3-4, 2013

in Fig.1 and Fig.3 and diagram of pomeron exchange : It permits to get a good description of the experimental characteristics of the reaction np → npπ+ π− at P0=5.20 GeV/c [10] (Fig. 4) taking into account diagrams shown in Fig.1 and Fig.3 and diagram of pomeron exchange : Fig. 4 Orsay April 3-4, 2013

3. Reaction pbar p → pbar p π+ π− at P0 = 7.23 GeV/c Using OPER model we try to describe the experimental distributions from the reaction pbar p → pbar p π+ π− at P0 = 7.23 GeV/c [8] Fig.5 It is observed a good agreement between experimental data and theory. Orsay April 3-4, 2013

4. Reaction np → npπ+ π− at P0 < 3 GeV/c The study of effective mass spectra of np – combinations at P0=1.73 and 2.23 GeV/c shows the clear peack close the threshold (Mnp= mn+mp) that can not be described within the framework of OPER-model с using the diagrams 1a, 1b, 1c, 3a, 3b, 3c и 3d. P0=2,23 ГэВ/с P0=1,73 ГэВ/с Fig.6 Orsay April 3-4, 2013

suggested in ITEP [11] was used to describe these features. The model of Regge poles with baryon exchange and nonlinear trajectories, suggested in ITEP [11] was used to describe these features. The following diagrams of one baryon exchange (OBE) were taken into account within the framework of this model: Fig. 7 Orsay April 3-4, 2013

reaction np → npπ+ π− at P0 =1.73 and 2.23 GeV/c. The vertex function of elastic np → np scattering was calculated using the data from [12]. The vertex functions of ΔN → np, NN → ΔN и ΔN → ΔN scattering were calculated corresponding to [13]. In result one can get the good description of the experimental distribution from the reaction np → npπ+ π− at P0 =1.73 and 2.23 GeV/c. P0 =1.73 ГэВ/с Fig.7 Orsay April 3-4, 2013

5. OPER model and other reactions The following reactions were simulated for HADES experiment: ● pp → pp π+ π− at Tkin= 3.5 GeV (for A.Belyaev) ● np → np π+ π− at Tkin= 1.25 GeV (for A.Kurilkin) The bump in π+π– effective masses at ~ 300 MeV/c2 was shown in the report presented by A.Kurilkin. The preliminary calculations using ‘hanged’ diagrams 1d and 1e has given only qualitative description of this mass spectrum. The similar bump one can see in Fig.7b at P0=1.73 GeV/c (Tkin= 1.0 GeV) obtained under condition of 4π geometry, but statistics is small. May be it is necessary to take into account the interference of ‘usual’ and ‘hanged’ diagrams? But at first we test the contribution of the ‘hanged’ diagrams with pomeron exchange into the reaction np → np π+ π− . Orsay April 3-4, 2013

‘Hanged’ diagrams for the reaction np → npπ+π– including π-meson and pomeron (P) exchanges are shown in figure below : Squared matrix element of the reaction np → npπ+π– was written in the form: neglecting the interference of the diagrams for the present. Orsay April 3-4, 2013

is written in the following form: The matrix element of the ‘hanged’ diagrams being the result of π-meson exchange is written in the following form: where u(Qi)γ5u(Qj) - vertex functions, Fπ - formfactors in the form taken from [14], Tππ - off shell amplitude of elastic –scattering , G - the constant of strong interaction (G2/4π = 14.6), t1 = (Q0 - Q1)2, t2 = (QT - Q2)2, SNππ: (Q1 + q1 + q2)2 and (Q2 + q1 + q2)2, Sππ = (q1 + q2)2. The corresponding matrix element for the pomeron (P) exchange is written in the form: where gP(t) - vertex functions [3], FP - formfactors with parameters taken from [3], T00 - the S-wave (I=0, L=0) amplitude of elastic ππ –scattering. Orsay April 3-4, 2013

The results of the calculations for the reactions np → npπ+π– at P0 = 1.73 GeV/c are shown in figure below. Solid line - the result of taking into account ‘hanged’ diagrams, dashed line - without "hanged" diagrams, dash-dotted line - the contribution of ‘hanged’ diagrams. One can see that taking into account the ‘hanged’ diagrams permits to get the noticeably better description of the π+π– masses close to 300 MeV/c2. Orsay April 3-4, 2013

Within the framework of OPER-model we study the reactions with dielectron production such as np → npe+e− at Tkin=1.25 GeV using πN → N e+e− vertex function (calculated by G.Lykasov) : Orsay April 3-4, 2013

The other reactions of np interactions are now under calculation by means of OPER model: − np → ppπ− π0 − np → pp π+ π− π− − np → pp π+ π− π− π0 (including η0 and ω0 production) − np → np π+ π+ π− π− Similar reactions of p p − , pbap p − and π N − interactions also can be described by OPER model. Next step of the development of OPER model is to use more appropriate approximation of the signature factor η(t), not the simple pole like 1/(t-m2π). The use of other Regge trajectories such as ρ-meson and strange particle (K-mesons and Λ-hyperon) can expand the region of application of the model. But it will be another model, not only ONE PION EXCHANGE. Orsay April 3-4, 2013

OPER – model permits to describe another N(barN)-N reactions Conclusion OPER – model permits to describe another N(barN)-N reactions with the production of some π-mesons. The further development of OPER – model can be very promising to describe the production of e+e− -pairs in hadronic interactions. OPER – model can be used as an effective tool to simulate various reactions of hadronic interactions. The experimental data are successfully described by the further development of OPER – model. However at lower energies ( P0 < 3 GeV/c) it is necessary to take into account another mechanism of the reactions (such as OBE). OPER – model is the peripheral model suggested to describe the reactions at enough high momenta. Therefore it can be more successful used for upgraded HADES set-up at SIS100. Orsay April 3-4, 2013

References: 1. G. Wolf. PR182, 1969, p.1538. 2. E.L. Berger. PRL21, 1968, p.701. 3. Yu.P.Nikitin and I.L.Rozental. Nuclear Physics of High Energies. Atomizdat, Moscow,1980. (in russian) 4. P.D.B Collins. An Introduction to Regge Theory and High Energy Physics. Cambridge University Press, 1977. 5. L.Ponomarev. Part. and Nucl., v.7(1), pp. 186-248, 1976, JINR, Dubna (in russian). 6. A.P.Jerusalimov et al. JINR Rapid Comm., v.35(2) pp.21-26, 1989, JINR, Dubna. (in russian). 7. R.A.Arndt et al. IJMP A18(3), 2003, p.449. 8. G.W. van Apeldoorn et al. NP B156, 1979, p.111. 9. K.G.Boreskov et al. Yad.Fiz.15:557-565,1972. (in russian). 10. A.P.Jerusalimov et al. Study of the Reaction np → npπ+ π− at Intermediate Energies. http://arxiv.org/pdf/1102.1574.pdf 11. A.B. Kaydalov and A.F. Nilov. YaF, v.41(3),pp. 768-776, 1985 ; YaF, v.52(6), pp. 1683-1696, 1990. 12. NN and ND interactions - a compilation. UCRL-20000 NN, august 1970. 13. V.Barashenkov and B.Kostenko. JINR Comm. 4-84-761, 1984, JINR, Dubna. (in russian). 14. A.P.Jerusalimov et al. Analysis of the Reaction np → npπ+ π− from the Point of View of OPER- Model. http://arxiv.org/pdf/1203.3330.pdf Orsay April 3-4, 2013

Thank You for attention !