Nucleon elastic scattering in quark-diquark representation with springy Pomeron
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Nucleon elastic scattering in quark-diquark representation withspringy Pomeron
V.M. Grichine a Lebedev Physical Institute, Moscow, RussiaReceived: date / Revised version: date
Abstract.
A model for elastic scattering of nucleons (and anti-nucleons) based on the quark-diquark repre-sentation of the nucleon with springy Pomeron, providing increased real part of the scattering amplitude,is developed. The model predictions are compared with experimental data for the differential elastic cross-sections of nucleons in the energy range from few GeV up to 7 TeV using available databases.
PACS.
The quark-diquark (qQ-) model for description of the proton-proton elastic scattering proposed in [1] was recently up-dated and compared with experimental data in [2]. Fig. 1shows the differential cross-section of proton-proton elas-tic scattering corresponding to the total energy in the cen-ter of mass system, √ s =7 TeV. The curve corresponds tothe qQ-model [2] with the standard Pomeron parametriza-tion: exp (cid:26) ˜ α (cid:20) ln ss o − iπ (cid:21)(cid:27) , (1)where the Pomeron trajectory slope ˜ α =0.15 GeV − andthe term − iπ/ s o =1 GeV ; we use units: ¯ h = c = 1).It is seen that the curve overestimates the dip value, andthe reason is that the scattering amplitude real part valueis not big enough.The picture here is similar to the elastic scattering ofhadrons on nuclei. Fig. 2 shows the differential hadronelastic scattering cross-sections of protons with the en-ergy 1 GeV on lead versus the polar scattering angle [4].The pure diffuse diffraction model proposed in [4] doesnot describe the minimum regions of the differential cross-section. If we add, however, the Coulomb amplitude, whichincreases the scattering amplitude real part, the modelbecomes to be more close to the experimental data. TheCoulomb amplitude can increase the scattering amplitudereal part for heavy nuclei with high atomic number only.In the case of nucleons one should modify directly thehadronic amplitude. Therefore one can assume, that if weincrease the imaginary part of the Pomeron parametriza- a e-mail: [email protected] tion (1), i.e. apply:exp (cid:26) ˜ α (cid:20) ln ss o − α p iπ (cid:21)(cid:27) , (2)with the empirical parameter, | α p | >
1, which can benamed as the Pomeron elasticity, we can improve the de-scription of the hadron-hadron elastic scattering. This mod-ification is proposed in the current paper together with anextension of the qQ-model to the case of the elastic scat-tering of different nucleons (and even hadrons).We discuss below the main features of the extendedqQ-model suitable for numerical calculations of elasticscattering of different nucleons and provide comparisonswith the experimental data for the pp , np , and ¯ pp elas-tic differential cross-section in the energy range from fewGeV up to 7 TeV in the center of mass system. The nucleon-nucleon differential elastic cross-section, dσ el /dt ,can be expressed in terms of the scattering amplitude F ( s, t ): dσ el dt = πp | F ( s, t ) | , (3)where p is the nucleon momentum in the center of masssystem, and t is the four-momentum transfer squared.We consider here an extension of the model [2] describ-ing the elastic scattering of two different nucleons. Thefirst nucleon consists of quark (index 1), diquark (2), andthe second one consists of quark (3) and diquark (4). Themodel [2] limits the consideration of the scattering am-plitude by contributions from one- and two-Pomeron ex-changes between quark-quark (1-3), diquark-diquark (2-4) V.M. Grichine: Nucleon elastic scattering in quark-diquark representation with springy Pomeron ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 ) / d t ( m b / G e V σ d -5 -4 -3 -2 -1 Qq-model, Gauss FFTOTEM data at low |t|TOTEM data at high |t| =7 TeV vs. |t|s/dt at σ p-p elastic d Fig. 1.
The proton-proton differential elastic cross-section ver-sus | t | at √ s =7 TeV. The curve is the prediction of quark-diquark model [2]. The open and closed circles are the LHCTOTEM experimental data from [3]. (degree) θ ( m b / s r ) Ω d e l σ d nucl+coulombnuclcoulombexperiment Differential elastic cross-section of 1 GeV protons on Pb
Fig. 2.
The differential hadron elastic scattering cross-sectionsof protons with the energy 1 GeV on lead versus the polarscattering angle [4]. The curves show the pure nuclear (dash-dash) and with the Coulomb correction (solid) models. Thedot-dot line corresponds to the pure electromagnetic Coulombscattering. The open points are experimental data. and two quark-diquark (1-4, 2-3). In this approximation F ( s, t ) can be expressed as: F ( s, t ) = F ( s, t ) − F ( s, t ) − F ( s, t ) , (4)where F ( s, t ) is the scattering amplitude with one-Pomeronexchange, while F ( s, t ) corresponds to two-Pomeron ex-changes between the nucleon constituents, quark and di-quark, and F ( s, t ) corresponds to two-Pomeron exchangesbetween the quark (or diquark) of one nucleon and thequark and the diquark of another nucleon at the same time. The amplitude F ( s, t ) reads: F ( s, t ) = ipσ tot ( s )4 π [ f + f + f + f ] , (5) f = B exp[( ξ + β λ + δ η ) t ] ,f = B exp[( ξ + β λ + γ η ) t ] ,f = B exp[( ξ + α λ + δ η ) t ] ,f = B exp[( ξ + α λ + γ η ) t ] , where σ tot ( s ) is the total nucleon-nucleon cross-section.The coefficients B jk , parametrize the quark-quark, diquark-diquark and quark-diquark cross-sections: σ = B σ tot ( s ) , σ = B σ tot ( s ) ,σ = B σ tot ( s ) , σ = B σ tot ( s ) . The model assumes the quark-diquark cross-section, σ = σ = √ σ σ . The coefficients α = γ = 1 /
3, and β = δ = 2 / λ and η are equalto the the first and second nucleon radius squared dividedby four, respectively.The coefficients ξ jk , ( j, k = 1 ,
2) are derived takinginto account the Gauss distribution of quark and diquarkin nucleon together with the Pomeron parametrization dis-cussed above in relation (2). They read: ξ jk = r j + r k
16 + ˜ α (cid:20) ln ss o − α p iπ (cid:21) . (6)Here r j , r k are the quark or diquark radii. The quarkand diquark radii r ( r ) and r ( r ) were found by thefitting of experimental data to be 0 .
173 and 0 .
316 of thecorresponding nucleon radius, respectively.The amplitudes F ( s, t ) and F ( s, t ) are: F ( s, t ) = ipσ tot ( s )16 π [ f , + f , ] , (7) f , = B B ξ + ξ + λ + η exp (cid:8)(cid:2) ξ + α λ + γ η −− ( ξ + αλ + γη ) ξ + ξ + λ + η (cid:21) t (cid:27) ,f , = B B ξ + ξ + λ + η exp (cid:8)(cid:2) ξ + α λ + δ η −− ( ξ + αλ + δη ) ξ + ξ + λ + η (cid:21) t (cid:27) , and: F ( s, t ) = ipσ tot ( s )32 π [ f , + f , + f , + f , ] , (8) f , = B B ξ + ξ + η exp (cid:8)(cid:2) ξ + β λ + γ η −− ( ξ + γη ) ξ + ξ + η (cid:21) t (cid:27) , .M. Grichine: Nucleon elastic scattering in quark-diquark representation with springy Pomeron 3 (GeV) s10 p α average of qQ-model s vs. p α Fig. 3.
The dependence of α p versus the √ s . f , = B B ξ + ξ + η exp (cid:8)(cid:2) ξ + α λ + γ η −− ( ξ + γη ) ξ + ξ + η (cid:21) t (cid:27) ,f , = B B ξ + ξ + λ exp (cid:8)(cid:2) ξ + α λ + δ η −− ( ξ + αλ ) ξ + ξ + λ (cid:21) t (cid:27) ,f , = B B ξ + ξ + λ exp (cid:8)(cid:2) ξ + α λ + β η −− ( ξ + αλ ) ξ + ξ + λ (cid:21) t (cid:27) , respectively.To simplify the numerical calculations, one can assumethat all nucleons involved to the elastic scattering havethe same radius and the quark-quark and diquark-diquarkcross-sections do not depend on the what quarks ( u or d ,as well as ¯ u or ¯ d ) interact. Then the quark-quark cross-section, σ , the nucleon radius and the Pomeron elastic-ity, α p , are the free parameters defining (together with σ tot ( s )) the s -dependence of the dσ el /dt . The diquark-diquark cross-section, σ , and the parameter B are de-rived from the optical theorem, as it was discussed in [2].The s-dependencies of the nucleon radius and the quark-quark cross-section are essentially the same as it was shownin [2]. The averaged s-dependence of the Pomeron elastic-ity, α p is shown in fig. 3. Fig. 4, 5, and 6 show the proton-proton differential elasticcross section at √ s =7 TeV versus | t | and the antiproton-proton differential elastic cross-sections at, √ s =1960 GeV ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 ) / d t ( m b / G e V σ d -5 -4 -3 -2 -1
10 ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 ) / d t ( m b / G e V σ d -5 -4 -3 -2 -1 Qq-model, Gauss FFTOTEM data at low |t|TOTEM data at high |t| =7 TeV vs. |t|s/dt at σ p-p elastic d Fig. 4.
The proton-proton differential elastic cross-section ver-sus | t | at √ s =7 TeV. The curve is the prediction of quark-diquark model with springy Pomeron. The open and closedcircles are the LHC TOTEM experimental data from [3]. ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 3 ) / d t ( m b / G e V σ d -6 -5 -4 -3 -2 -1
10 ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 3 ) / d t ( m b / G e V σ d -6 -5 -4 -3 -2 -1 Qq-model, Gauss FF=1960 GeV datas =1960 GeV vs. |t|s/dt at σ -p elastic dp Fig. 5.
The antiproton-proton differential elastic cross-sectionversus | t | at √ s =1960 GeV. The curve is the prediction of ourmodel. The open circles are the experimental data [5]. and √ s =546 GeV, respectively. The curves are the pre-dictions of our model. We see that the proposed qQ -modelwith springy Pomeron describes reasonably the differentialelastic cross sections of the antiproton-proton and proton-proton scattering in the TeV-region of energy.Fig. 7 and 8 show the neutron-proton differential elas-tic cross sections at the neutron momentum in the labo-ratory system 100 and 9 GeV/c, respectively. Fig. 10 theproton-proton differential elastic cross-section versus | t | atthe proton momentum in the laboratory system 3 GeV/c.One can see the satisfactory agreement of the qQ-modelwith experimental data in the GeV-range of energy. V.M. Grichine: Nucleon elastic scattering in quark-diquark representation with springy Pomeron ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 3 ) / d t ( m b / G e V σ d -6 -5 -4 -3 -2 -1
10 ) momentum transfer, |t| (GeV0 0.5 1 1.5 2 2.5 3 ) / d t ( m b / G e V σ d -6 -5 -4 -3 -2 -1 Qq-model, Gauss FF=546 GeV datas =546 GeV vs. |t|s/dt at σ -p elastic dp Fig. 6.
The antiproton-proton differential elastic cross-sectionversus | t | at √ s =546 GeV. The curve is the prediction of ourmodel. The open circles are the experimental data [6]. ) |t| (GeV0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) ( m b / G e V d t e l σ d -4 -3 -2 -1 qQ-modelexperiment Fig. 7.
The neutron-proton differential elastic cross-sectionversus | t | at the neutron momentum in the laboratory system100 GeV/c. The curve is the prediction of our model. The opencircles are the experimental data [7]. We have considered the qQ -model of the nucleon-nucleonelastic scattering with springy Pomeron and compared itwith experimental data. It was obtained reasonable de-scription of the differential cross section of elastic pp ¯ pp and np scattering in a wide range of energies from fewGeV to 7 TeV in the center of mass system. The dip po-sition and its value of the dσ el /dt are in the satisfactoryagreement with experimental data.The qQ-model, in the framework of relations (5)-(8),can be easily generalized to the case of elastic scattering ofmesons on nucleons or mesons on mesons. Fig. 10 shows ) |t| (GeV1 2 3 4 5 6 7 8 ) ( m b / G e V d t e l σ d -5 -4 -3 -2 -1 qQ-modelexperiment Fig. 8.
The neutron-proton differential elastic cross-sectionversus | t | at the neutron momentum in the laboratory system9 GeV/c. The curve is the prediction of our model. The opencircles are the experimental data [8]. ) |t| (GeV0 0.2 0.4 0.6 0.8 1 1.2 1.4 ) ( m b / G e V d t e l σ d -1 Differential elastic cross-section of 3 GeV/c protons on hydrogen qQ-modelexperiment
Fig. 9.
The proton-proton differential elastic cross-section ver-sus | t | at the proton momentum in the laboratory system3 GeV/c. The curve is the prediction of our model. The opencircles are the experimental data [9]. the π + − p differential elastic cross-section versus | t | atthe π + momentum in the laboratory system 4.122 GeV/c,as an example of the qQ-model application to the caseof meson-nucleon elastic scattering. The meson-nucleonelastic scattering in terms of the qQ-model with springyPomeron will be reported in next paper. Acknowledgment
The author is thankful to S. Bertolucci, S. Giani and M.Mangano for stimulating discussions and support. The .M. Grichine: Nucleon elastic scattering in quark-diquark representation with springy Pomeron 5 ) |t| (GeV2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 ) ( m b / G e V d t e l σ d -4 -3 -2 on hydrogen + π Differential elastic cross-section of 4.122 GeV/c qQ-modelexperiment
Fig. 10.
The π + − p differential elastic cross-section versus | t | at the π + momentum in the laboratory system 4.122 GeV/c.The curve is the prediction of our model. The open circles arethe experimental data [10]. meetings and e-mail communication with N. Starkov andN. Zotov were powerful for clarification of the qQ-modeldetails. The work was also partly supported by the CERN-RAS Program of Fundamental Research at LHC. References
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