1. 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
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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]. 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. 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
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5. 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
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6. 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
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7. 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
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8. - 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
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9. 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].
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10. 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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11. 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.
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12. 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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13. 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
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14. 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 GeV/c.
P0 =1.73 ГэВ/с
Fig.7
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15. 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− .
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16. ‘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.
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17. 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.
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18. 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.
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19. 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) :
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20. 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.
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21. 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.
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22. 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 , 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: ,1972. (in russian).
10. A.P.Jerusalimov et al. Study of the Reaction np → npπ+ π− at Intermediate Energies.
11. A.B. Kaydalov and A.F. Nilov. YaF, v.41(3),pp , 1985 ;
YaF, v.52(6), pp , 1990.
12. NN and ND interactions - a compilation. UCRL NN, august 1970.
13. V.Barashenkov and B.Kostenko. JINR Comm , 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.
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23. Thank You
for
attention !