Bjorken scaling violation in deep inelastic electron-proton scattering (DIS) is related to the rise of hadronic cross sections by using the additive quark model. Of special interest is the connection between saturation in the low-x behavior of the DIS structure functions (SF) and possible slow-down of the pp cross section rise due to saturation effects. We also identify saturation effects in the DIS SF with phase transition that can be described by the Van der Waals equation of state

Celiberto F. G.

Fiore R.

Jenkovszky L.

English

Archivio Istituzionale della Ricerca - Università degli Studi di Pavia

Collective Phenomena in pp and ep ScatteringFrancesco Giovanni Celiberto1,2,b), Roberto Fiore1,c) and László Jenkovszky3,a)1Dipartimento di Fisica, Università della Calabria, Arcavacata di Rende, 87036 Cosenza, Italy2Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Cosenza, Arcavacata di Rende, 87036 Cosenza, Italy3Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, 03680 Ukrainea)Corresponding author: jenk@bitp.kiev.uab)francescogiovanni.celiberto@fis.unical.itc)roberto.fiore@fis.unical.itAbstract. Bjorken scaling violation in deep inelastic electron-proton scattering (DIS) is related to the rise of hadronic cross sectionsby using the additive quark model. Of special interest is the connection between saturation in the low-x behavior of the DIS structurefunctions (SF) and possible slow-down of the pp cross section rise due to saturation effects. We also identify saturation effects inthe DIS SF with phase transition that can be described by the Van der Waals equation of state.INTRODUCTIONAlthough deep inelastic lepton-hadron scattering (DIS) is thought mainly as a “microscope” identifying individualpoint-like constituents in the hadron, collective effects may come into play with decreasing x (Bjorken variable) andincreasing photon virtuality Q2. In this paper we present two examples of such phenomena. The first example, given inthe second Section, is a connection between expected saturation in the low-x, high-Q2 DIS structure function (SF) andthe high-energy behavior of the hadronic total cross section. The other one, given in the third Section, is a statisticalphysics interpretation of the DIS SF.BJORKEN SCALING VIOLATION AND RISING CROSS SECTIONSThe main idea of this section is that the rise of hadron-hadron cross sections at high energies is related to the amountof Bjorken scaling violation at low x.By definition, the virtual photon-proton total cross section is related to the relevant DIS SF byF2(x,Q2) =Q2(1 − x)4πα(1 + 4m2x2Q2)σγ∗pt , (1)s = W2 = Q2(1 − x)/x + m2.Real photon-proton total cross section is known [1] to be about two orders of magnitude below that of proton-proton cross section.In [2] the Q2−dependent structure function was related to on-mass-shall hadronic cross sections by means ofdispersion relations and duality in Q2,F1(s,Q2) =∫ ∞Q20Φ(s,Q′2)(Q′2 + Q2)2dQ′2 , 2xF1 = F2, (2)where the asymptotic part of Φ was saturated by ρ and ω vector meson masses. With a simple phenomenologicalmodel of the SF, e.g.FAs.2 (x,Q2) = 0.2(1 − 0.05 lnQ2Q20lnxx0), (3)arXiv:1612.00797v2  [hep-ph]  5 Dec 2016after integration in Q2, with lower integration limits corresponding to vector meson masses, and the relation σρN ≈σωN ≈ σπN, one gets a reasonable expression for the rising meson-baryon total cross sectionσπN(s) = 24(1 + 0.1 lnsQ20)mb , Q20 = 3 GeV2 . (4)In the additive quark model, the hadron-hadron total cross section can be written as a product of the cross sectionsof the constituents, σqq [2], e.g.σ(s)tpp = σqq[nV + nS (s)]2 , (5)where nV is the number of valence quarks and nS (s) is that of sea quarks, their number increasing with energy.We suggest that the increasing number of sea quarks is related to the Bjorken scaling-violating contribution tothe deep inelastic lepton-hadron structure function (DIS SF), namely to the momentum fraction of the relevant quarksgiven by the integral over the DIS structure function F2(x,Q2).The number of quarks in a reaction can be calculated from the SF by means of sum rules. The momentum fractioncarried by a parton is∫ 10 dxF2(x,Q2), and hence the number of partons in the reaction is its inverse. Focusing on theglobal and asymptotic behavior of both the structure functions (at low x) and of the total cross sections (large s)and possible signs of the interrelated saturation effects, here we disregard details related to the flavor content of thehadrons.A simple model for the DIS structure function results in the following expression for the total cross section,compatible with the dataσ(s)tpp = σqqn2V [1 + 0.016 ln(s/Q20)] , (6)where σqq is a free parameter; Q20 was fitted to the DIS data, and nV = 3.Of special interest is the expected correlation between the onset of saturation in DIS (see, e.g. [3–5] and thepossible slow-down of the rise of hadronic cross sections to follow).DIS AS A VAN DER WAALS SYSTEM OF VIRTUAL PARTONSIt was suggested in [6, 7] that saturation in DIS, predicted by QCD and observed experimentally, corresponds to thecondensation of the partonic gas to a fluid. The interior of the nucleon exited in DIS or DVCS undergoes a phasetransition from a parton gas (high- and intermediate x, x ∼ 0.05) to a partonic fluid. The division line is locatedroughly at those values of x and Q2 where Bjorken scaling is violated, as shown in Figure 1, left icon.Formally, saturation may be treated in the context of the BFKL equation [8]∂N(x, k2T )∂ ln(1/x)= αsKBFKL ⊗ N(x, k2T ) , (7)where KBFKL is the BFKL integral kernel (splitting function). Since the number of partons increases with energy, atcertain “saturation” value of x and Q2, an inverse process, namely recombination of pairs of partons comes into play,and the BFKL equation is replaced by∂N(x, k2T )∂ ln(1/x)= αsKBFKL ⊗ N(x, k2T ) − αs[KBFKL ⊗ N(x, k2T )]2 , (8)based on the simple idea that the number of recombinations is roughly proportional to the number of quark partonpairs, N2. Saturation sets in when the second, quadratic term overshoots the first, linear one, thus tempering theincrease of the number of produced partons and securing unitarity.The transition between the partonic gas, via a mixed foggy phase, to the partonic liquid (see Figure 1) mayquantified by the Van der Waals equation of state (EoS)(P + N2a/V2)(V − Nb) = NT , (9)where a and b are parameters depending on the system, N is the number of particles and V is the volume of the“container”, V(s) = πR3(s), R(s) ∼ ln s being the nucleon radius. For point-like particles (perfect partonic gas)a = b = 0 and the VdW equation reduced to pV = NT , and, since N/V ∼ T 3, we get in this approximation p ∼ T 4.QCD evolution withrecombination2 1.5 0.5 QCD 0.001 • FSC ... ,, ... ,, .... ,· · ,,,,,.,,. ," .,. ..... ........ ,.,, ... , ................... � ••• .? ... ,,,,... -O.D1 0.1 X GAS /'"\ / \ j--.. \ /i \. \ ·11// \ \ !( \ \ ' I\ � ' ' \ 'l ' ..\\ xFIGURE 1. Left icon: Gaseous, foggy and liquid states in various kinematic regions of DIS (schematic). Right icon: Same as onthe left, but with the data at Q2 = 15 GeV2 also shown.Alternatively, the Van der Waals equation may be written as(P + a/V2)(V − b) = RT , (10)or, equivalently,P =RTV − b− a/V2 . (11)The parameter b, responsible for the finite dimension of the constituent, in our case is ∼ 1/Q, while the term a/V2 isrelated to the long-range forces between the constituents. From this equation one finds for critical values, V = Vc, P =Pc, T = Tc in terms of the parameters a and b:Vc = 3b, Pc = a/(27b2), Tc = 8a/(27Rb) . (12)The number of particles may be calculated asN(s) =∫ 10dxF2(x,Q2) , (13)where F2(x,Q2) is the nucleon SF, measured in DIS.CONCLUSIONSCollective properties of the nuclear matter are studied mostly in heavy-ion collisions. In this paper we presentedexamples of collective effects in DIS. As energy is pumped into a nucleon by a virtual photon, the number of partonspopulating it increases, reaching critical densities, thus causing phase transitions. The difference between the case ofhadronic (heavy nuclei) collisions and DIS is that in the latter case the nucleon (nucleus) is heated by a highly virtualphoton (heavy vector meson), while in heavy-ion (nucleon) collisions we deal with on-shall (real) objects. However,the produced hot and dense matter is of the same nature.ACKNOWLEDGMENTSWe thank the organizers, for their hospitality and for the inspiring atmosphere at the conference. L.J. acknowledgesthe financial support during the conference.REFERENCES[1] K.A. Olive et al. (Particle Physics Booklet), Chin. Phys. C, 38 090001 (2014).[2] L.L. Jenkovszky and B.V. Struminsky, Yadernaya Fizika (English translation: Sov. J. Nucl.Phys.), 38 (1983), 1568; see also: http://inspirehep.net/record/1456006/files/mpiatlhc2015.pdf;http://indico.ictp.it/event/a14280/ , p.171.[3] F.G. Celiberto, L. Jenkovszky, and V. Myronenko, EPJ Web Conf. 125, 04012 (2016)doi:10.1051/epjconf/201612504012 [arXiv:1608.04646 [hep-ph]].[4] F.G. Celiberto, L. Jenkovszky, and V. Myronenko, Low-x structure functions and rising total cross sections,to be published in the Proceedings of the “Low-x” Workshop held in Gyöngyös, Hungary, 2016.[5] H. Abramowicz et al. [H1 and ZEUS Collaborations], Eur. Phys. J. C 75 (2015) no.12, 580doi:10.1140/epjc/s10052-015-3710-4 [arXiv:1506.06042 [hep-ex]]; I. Abt, A. M. Cooper-Sarkar, B. Fos-ter, V. Myronenko, K. Wichmann and M. Wing, Study of HERA ep Data at Low Q2 and Low xB j and theNeed for Higher-Twist Corrections to Standard pQCD Evolution, arXiv:1604.02299 [hep-ph].[6] L.L. Jenkovszky, Andrea Nagy, S.M. Troshin, Jolán Turóci, and N.E. Tyurin, Int’l J. Mod. Phys. A25, 5667-5682 (2010).[7] L.L. Jenkovszky, A.O. Muskeev, and S.N. Yezhov, Physics of Atomic Nuclei 75, 721-724 (2012).[8] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45, 199-205 (1977; I.I. Balitsky, L.N. Lipatov, Sov.J. Nucl. Phys., 28, 822-830 (1978).

Collective phenomena in pp and ep scattering

https://iris.unipv.it/bitstream/11571/1358792/3/010_ArXiv_proc_Diffraction_2016_1612.00797.pdf

Collective phenomena in pp and ep scattering

Abstract

Similar works

Full text

Available Versions

Archivio Istituzionale della Ricerca - Università degli Studi di Pavia