1. 1. Introduction OPER-model for multipion production Outlook
A.P. Jerusalimov,
JINR-LHEP, Dubna
Outlook
1. Introduction
2. 2π production in NN interactions
3. 3π and 4π production in NN interactions
4. n•π production for πN interactions
5. OPER model and other reactions
6. Conclusion
7. Reference
Appendix
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2. 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] and developed in JINR [6-8] .
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.
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3. 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)
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4. 2. 2π production in NN interactions
Reaction
np → npπ+π−
Diagrams: →
Cross-sections
and contributions (in %)
of diagrams
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5. Distributions for the reaction np → npπ+π−
at P0=2.23 GeV/c (Tkin=1.48 GeV, left) and at P0=1.73 GeV/c (Tkin=1.03 GeV, right)
[A.P.Jerusalimov et al., Eur.Phys.J. A51 (2015) no.7, 83 ]
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6. Total (OPER + OBE) OPER Distributions for the reaction np → npπ+π−
at P0=2.23 GeV/c (Tkin=1.48 GeV, left) and at P0=1.73 GeV/c (Tkin=1.03 GeV, right)
Total (OPER + OBE)
OPER
OBE
Valencia model (normalization by factor 2.5 for Tkin=1.0 GeV and 1.9 for Tkin=1.5 GeV )
[A.P.Jerusalimov et al., Eur.Phys.J. A51 (2015) no.7, 83 ]
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7. Fig1. Distributions for the reaction np → npπ+π−
at P0=5.20 GeV/c (Tkin=4.35 GeV, left) and at P0=3.83 GeV/c (Tkin=3.0 GeV, left)
[A.Jerusalimov. PoS BaldinISHEPPXXII (2015) 048]
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8. Reaction np → npπ+ π− at Tkin = 1.25 GeV (HADES)
[G.Agakhishiev et al., Phys.Lett. B750 (2015) ]
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9. Mass and angular spectra of the reaction
np → ppπ– π0
at P0=5.20 GeV/c (Tkin=4.35 GeV)
[A.Jerusalimov. PoS BaldinISHEPPXXII (2015) 048]
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10. Mass and angular spectra of the reaction
np → ppπ– π0 at P0=2.23 GeV/c (Tkin=1.48 GeV)
● Total
● OPER
● OBE [A.Jerusalimov. PoS BaldinISHEPPXXII (2015) 048]
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11. HADES IPN Orsay May ,

12. σ= 2.88 mb total Ntrig=4 (Χ50.0) Reaction pp → ppπ+ π−
at Tkin = 4.5 GeV (simulation for HADES)
σ= 2.88 mb
total
Ntrig=3 (Χ2.0)
Ntrig=4 (Χ50.0)
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13. → 3. 3π and 4π production in NN interactions [A.P.Jerusalimov et al.,
EPJ Web Conf. 138 (2017) 07008]
● Reaction np → ppπ+π−π− →
Cross-section of the reaction np → ppπ+π−π− vs momentum of incident beam.
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14. The distributions for the reaction np → ppπ+π−π− at P0=5.20 GeV/c.
Red line – calculations using OPER-model.
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15. ● Reaction np → ppπ+π−π− π0
Cross-section of the reaction np → ppπ+π−π−π0 vs momentum of incident beam.
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16. The distributions for the reaction np → ppπ+π−π−π0 at P0=5.20 GeV/c.
Red line – calculations using OPER-model.
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17. ? The distributions for the reaction np → ppπ+π−π−
at P0=4.42 GeV/c (left panel) and at P0=3.83 GeV/c (right panel).
Red line – calculations using OPER-model. ?
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18. GIM → 4. n•π production in πN interactions ● Reaction π–p → n π+ π–
main π–p → Δ– π+
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19. GIM [J. Dolbeau et al. NP B78, 233(1974)]
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20. π–p→nπ–π+ simulation for HADES (GIM)
Black line –
– full accept.
Red line –
– MDC accept.
Blue line –
– MDC + FD
accept.
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21. π–p→p π+π–π– simulation for HADES
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22. 5. OPER model and other reactions
● di-electron production in quasi-free n-p interactions
[A.P.Jerusalimov and G.I.Lykasov,
Int.J.Mod.Phys. A32 (2017) no.17, ]
dp→ps+ (np→e+e– + X)
np→ npη0, η0→ e+e–γ
np→ npρ0, ρ0→ e+e–
dp→ps+ (np→e+e– + X)
np→ np e+e–
np→ npπ0 , π0→ e+e–γ
np→ npπ0 π0 , π0→ e+e–γ
np→ Δ0 p , Δ0 → ne+e–
np→ Δ+n , Δ+ → pe+e–
np→ Δ0 N π , Δ0 → ne+e– , N π = nπ+ & pπ0
np→ Δ+ N π , Δ+ → pe+e– , N π = nπ0 & pπ–
dp→ns+ (pp→e+e– + X)
pp→ ppπ0 , π0→ e+e–γ
pp→ ppπ0 π0 , π0→ e+e–γ
pp→ Δ+p , Δ+ → pe+e–
pp→ Δ+ N π, Δ+ → pe+e– , N π = pπ0 & nπ+
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23. Angular distribution of the protons in FW
Mee spectrum
Angular distribution of the protons in FW
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24. ● Reaction np → npγ* → npe+e–
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25. ● Diagram NN→NNρ0 (1x1x1) ● Diagram NN→NNη0 (1x1x1)
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26. Matrix element squared for np-npη for ‘hanging’ σ-η exchange diagram
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27. ● Diagrams NN→NNη0 (1x2) ↓
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28. Conclusion ● OPER – model permits to get a good description of the
characteristics of pions production in NN and πN interactions.
For 2π production in πN interactions used used GIM (Tkin<2.5 GeV).
● It is necessary to take into account an additional mechanism of the
reaction (such as OBE) for 2π production in NN interactions
at P0 < 3 GeV/c (Tkin<2.2 GeV).
● It was shown that there are no noticeable signal of ρ-meson
production in the considered NN reactions .
● To get a better description of π+π−π0 combinations it is necessary
take into account η and ω production.
● The further development of OPER – model can be very promising
to describe the production of e+e− -pairs in hadronic interactions.
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29. Thank You for attention !
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30. Reference
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 , 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. A.P.Jerusalimov et al. Study of the Reaction np → npπ+ π− at Intermediate Energies.
8. A.P.Jerusalimov et al. Contribution of the "hanged" diagrams into the reaction np - > np п+ п- .
9. K.G.Boreskov et al. Yad.Fiz.15: ,1972. (in russian).
OBE-model
A.B. Kaydalov and A.F. Nilov. YaF, v.41(3),pp , 1985 ;YaF, v.52(6), pp , 1990.
V.Barashenkov and B.Kostenko. JINR Comm , 1984, JINR, Dubna. (in russian).
Other models
L. Alvarez-Ruso, E. Oset, E. Hernandez. NP A633, 519 (1998).
Xu Cao, Bing-Song Zou and Hu-Shan Xu. PR C81, (2010).
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31. Appendix:
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32. ► 1. Parametrization of π N → π π N reactions
Within the framework of Generalized Isobar Model (GIM)
[D.J. Herndon et al. PR D11, 3165 (1975); D.M.Manley and E.M. Saleski, PR D45, 4002 (1992) ]
πN→ππN reactions are described as quasi-two body ones:
πN → N*(Δ*) → Δ π ,
→ N*(Δ*) → N ρ ,
→ N*(Δ*) → N ε , ε - S-wave of ππ scattering with I=0
→ N*(Δ*) → N*1440 π ,
Δ → N π ,
with the consequent ρ → π π ,
decays : ε → π π ,
N*1440 → N π .
N*(1440) P D*(1600) P33
The parameters N*(1520) D D*(1620) S31
of the following N*(1675) D D*(1700) D The spin and
resonances N*(1680) F D*(1900) S isospin relations
(**** and ***) N*(1720) P D*(1905) F were taken
were taken N*(2000) F D*(1910) P account
from RPP N*(2080) D D*(1920) P33
N*(2190) G D*(1940) D33
D*(1950) F37
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33. λ=λa- λb, μ=λc- λd – helicity variables, d jλμ(θ) – rotation matrixes.
For quasi two-body reactions like a + b → c + d
πN → N*(Δ*) → Δ π ,
→ N*(Δ*) → N ρ ,
→ N*(Δ*) → N ε ,
→ N*(Δ*) → N*1440 π ,
one can write
where
λ=λa- λb, μ=λc- λd – helicity variables, d jλμ(θ) – rotation matrixes.
The polarization components of the particles c and d from the reaction a + b → c + d is suiteable
to express through the elements of the spin density matrix (for example, for particle d):
where normalization factor for Spρ=1.
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34. Example:
π + N → N*1680 → Δ + π → (N + π) + π
RJ is taken in Breight-Wigner form
Then it is easy to get the angular distribution of Δ (in CMS):
If particle d is unstable: d → α + β (d → Δ + π)
then in the rest system of the particle d (d → Δ + π)
is the normalized angular distribution of the decay products.
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35. Comparison with experimental data
The following cross-sections were calculated using GIM :
One can see a satisfactory description of cross-sections, except π+ p → n π+ π+
May be it is necessary to put into GIM S-wave of π+π+ scattering with I=2.
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36. to study a quality of the application of GIM :
Some distributions of the reaction π– p → n π+ π– were calculated at various energies
to study a quality of the application of GIM :
[J. Dolbeau et al. NP B78, 233(1974)]
It is observed a good agreement between experimental data and theory.
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37. ► 2. OPER-model for reaction np → npπ+π−
Interference between diagrams
(a,b and c) is negligible
at P0> 3.0 GeV/c [6]
The «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
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38. the matrix element for which is written
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. 2
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 [from GIM]
and formfactor
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39. 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
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40. It was shown in [9] that the processes of diffractive production of N
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 are described by the diagrams similar to
diagrams in Fig.2 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 u(Qi)5u(Qj) - vertex functions,
F - formfactors in the form taken from [1],
Tππ - o shell amplitude of elastic -scattering
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41. It was shown in [9] that the processes of diffractive production of N
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 are described by the diagrams similar to
diagrams in Fig.2 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].
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42. The study of the reaction n p →n pπ+π− at P0=1. 73 and 2
The study of the reaction n p →n pπ+π− at P0=1.73 and 2.23 GeV/c showed that it is necessary
to take into account the mechanism of one-baryon exchange (OBE).
The model of Regge poles with baryon exchange and nonlinear trajectories, suggested in ITEP [11] was used.
The following diagrams of one baryon exchange (OBE) were taken into account within the
framework of this model:
Fig. 3
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].
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43. 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
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44. 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
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45. 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.?
It is observed a good agreement between experimental data and theory.
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46. π–p→p π+π–π–π0 simulation for HADES
Black line –
– full accept.
Red line –
– MDC accept.
Blue line –
– MDC + FD
accept.
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