We show that the p_T spectra measured in pp collisions at the LHC exhibit
geometrical scaling introduced earlier in the context of deep inelastic
scattering. We also argue that the onset of geometrical scaling can be seen in
nucleus-nucleus collisions at lower RHIC energies.Comment: 10 pages, 5 figures, based on a talk presented at Cracow Epiphany
  Conference "On the First Year of the LHC", Krakow, January 10-12, 2011. Small
  corrections in v

Praszalowicz, Michal

Acta Physica Polonica B

English

arXiv

M. Praszalowicz

Crossref

Praszałowicz, Michał

Jagiellonian Univeristy Repository

Vol. 42 (2011) ACTA PHYSICA POLONICA B No 7GEOMETRICAL SCALINGIN HADRONIC COLLISIONS∗Michal PraszalowiczThe Marian Smoluchowski Institute of Physics, Jagiellonian UniversityReymonta 4, 30-059 Krakow, Polandmichal@if.uj.edu.pl(Received April 29, 2011)We show that the pT spectra measured in pp collisions at the LHCexhibit geometrical scaling introduced earlier in the context of deep inelasticscattering. We also argue that the onset of geometrical scaling can be seenin nucleus–nucleus collisions at lower RHIC energies.DOI:10.5506/APhysPolB.42.1557PACS numbers: 13.85.–t, 11.80.Fv1. IntroductionWith the start of the LHC we have been confronted with a wealth of dataon multiparticle production at high energies both in pp [1,2,3] and in heavyion collisions [4]. One of the remarkable results is that total multiplicity ofcharged particles produced in central rapidity in pp collisions is rising like apower of sdNchdη∼ sλ̃ with λ̃ ' 0.23 . (1)The power law behavior (1) is expected in saturation models [5, 6, 7, 8, 9,10]. In this paper we show that power like behavior of total multiplicityfollows naturally if pT spectra of charged particles exhibit geometrical scaling[11, 12]. In Sec. 2 we remind basic properties of geometrical scaling whichwas introduced in the context of small x deep inelastic scattering (DIS). InSec. 3 we introduce geometrical scaling in pp collisions. In Sec. 4 we brieflydiscuss a possibility of geometrical scaling in heavy ion collisions. Finally,we summarize and give conclusions in Sec. 5.∗ Based on a talk presented at the Cracow Epiphany Conference on the First Year ofthe LHC, Cracow, Poland, January 10–12, 2011.(1557)1558 M. Praszalowicz2. Geometrical scaling in DISIn a successful description of small x DIS proposed in seminal papers byGolec-Biernat and Wüsthoff (GBW model) [13], a cross-section for virtualphoton–proton scattering in DIS readsσγ∗p =∫dr2∣∣ψ(r,Q2)∣∣2 σdP (r2Q2s (x)). (2)Here ψ is the wave function describing dissociation of a virtual photon intoa qq̄ dipole and σdP is a dipole-proton cross-section. The main assumptionof the GBW model is that σdP which in principle is a function of two inde-pendent variables: dipole size r and dipole-proton energy W (or Bjorken x),depends in practice only on a certain combination of these two variables,namely on the product r2Q2s (x), whereQ2s (x) = Q20(x0x)λ(3)is called a saturation scale. Bjorken x is defined asx =Q2Q2 +W 2. (4)Here Q0 ∼ 1 GeV and x0 ∼ 10−3 are free parameters whose precise valuescan be extracted by fitting (2) to the HERA data. Power λ is known to beof the order of λ ∼ 0.2÷ 0.3.For transverse photons (neglecting quark masses)∣∣ψT (r,Q2)∣∣2 = 1∫0dz[z2 + (1− z)2]Q2K21(Qr), (5)whereQ2 = z(1− z)Q2 (6)and K1 is a modified Bessel function. From Eq. (5) it follows that∣∣ψT (r,Q2)∣∣2 = Q2 ∣∣∣ψ̃T(rQ)∣∣∣2 , (7)where we have explicitly factored out Q2. Defining new variable u = Q2r2,new function φ(u) = u|ψ̃T(u)|2 and scaling variable ττ =Q2Q2s (x)(8)Geometrical Scaling in Hadronic Collisions 1559we arrive atσγ∗p =∫duuφ(u)σdP (uτ) . (9)It follows that σγ∗p is a function of scaling variable τ , rather than a functionof two variables Q2 and x. This phenomenon is known as geometrical scaling(GS) [14]. GS has been observed in DIS data for x < 0.01.In the Golec-Biernat–Wüsthoff modelσdP(r2Q2s (x))= σ0(1− exp(−r2Q2s (x))), (10)where σ0 is dimensional constant; σ0 ' 23 mb.In practice, Q2s may also have some residual dependence on Q2 if DGLAPevolution in (2) is taken into account [15]. Indeed, it can be shown that inthe GBW model — up to logarithmic corrections — DIS structure functionis proportional to Q2sσγ∗p(x,Q2)∼ σ0Q2s (x)Q2and F2(x,Q2)∼ σ0Q2s (x) . (11)At first sight Eq. (11) may look contradictory, since left-hand sides dependnon-trivially on Q2 and the right-hand sides do not. In practice, exactcalculation of the integral in (2) renders some mild Q2 dependence of theright-hand sides. Moreover, DGLAP evolution introduces Q2 dependence ofQ2s (x). Therefore, the effective saturation scale can be conveniently param-eterized asQ2s,eff(x,Q2)= Q20(x0x)λeff(Q2). (12)1 0 1 0 00 . 1 00 . 1 50 . 2 00 . 2 50 . 3 00 . 3 50 . 4 00 . 4 50 . 5 0 H 1 Z e u s 0 . 1 3  +  0 . 1  ( Q  2 / 1 0 ) 0 . 3 5  effQ  2  [ G e V 2 / c 2 ]Fig. 1. Dependence of λeff on Q2 from HERA (HERA data points [16] afterRef. [17]).1560 M. PraszalowiczExponent λeff(Q2) has been extracted from the HERA data [16]. This isshown in Fig. 1 (after Ref. [17]) together with an eyeballing fit [12]λeff(Q) = 0.13 + 0.1(Q210)0.35. (13)3. Geometrical scaling in pp collisionsIn pp collisions we do not have a viable model of low and medium pT par-ticle production at high energies. Nevertheless one often uses a kT factorizedform of a cross-section describing production of a pT gluon at rapidity y [18]Edσd3p=3π21p2T∫dk2Tαs(kT)ϕ1(x1, k2T)ϕ2(x2, (k − p)2T), (14)wherex1,2 =pT√se±y (15)are Bjorken xs of colliding partons. Here ϕs are unintegrated gluon densities.Introducing “regular” gluon distributionxG(x,Q2)=Q2∫dk2Tϕ(x, k2T)(16)one obtains for p2T > Q2sEdσd3p=3π2αs(Qs)p2T{ϕ1(x1, p2T)x2G(x2, p2T)+ ϕ2(x2, p2T)x1G(x1, p2T)}.(17)There have been recently more involved model calculations of particle mul-tiplicity based on Eq. (14) [19,20,21].Kharzeev an Levin proposed a simple Ansatz for unintegrated gluondistribution [22]ϕ(x, p2T)=3σ0π2αs (Q2s ) 1 for p2T < Q2s ,Q2s/p2T for Q2s < p2T .(18)Hence, up to the logarithmic corrections due to the running coupling con-stant, we arrive at geometrical scaling for the multiplicity distributiondNchdηd2pT=1σinelEdσd3p=1Q20F (τ) , (19)Geometrical Scaling in Hadronic Collisions 1561where Q0 ∼ 1 GeV and σinel is the inelastic cross-section. Although wehave used a very simple Ansatz (18) for unintegrated gluon distribution ϕ,it satisfies the generic property that xG(x,Q2s ) ∼ Q2s which is enough forGS to hold.F (τ) is a universal function of the scaling variableτ =p2TQ2s, (20)where in view of (3) and (15)Q2s = Q20(pTW)−λ, (21)where W =√s × 10−3. Here factor 10−3 corresponds to the (arbitrary atthis moment) choice of x0.The power like growth of the multiplicity can be easily understood as aconsequence of geometrical scaling. IndeeddNchdy=∫dp2TQ20F (τ) . (22)Simple change of variables gives [11]dp2TQ20=22 + λ(WQ0) 2λ2+λτ−λ2+λdτ . (23)The integral over dτ is convergent and universal, i.e. it does not depend onenergy. It follows from Eq. (23) that the effective power of the multiplicitygrowth isλ̃ =2λ2 + λ< λ (24)rather than λ. For λ = 0.27 we have that λ̃ = 0.238.In Refs. [11] it was shown that CMS charged particle pT spectra [2] atmid rapidity |η| < 2.4 plotted as functions of scaling variable τ fall on oneuniversal curve (19). This is depicted in Fig. 2, where we plot pT spectrafor three LHC energies as functions of p2T (left panel) and as functions ofscaling variable τ for λ = 0.27 (right panel).In order to examine the quality of geometrical scaling in pp collisions weplot in Fig. 3 ratios of spectra measured at 7 TeV to spectra at 0.9 and 2.36TeV in function of pT (left panel) and√τ (right panel). We see that originalratios plotted in terms of pT range from 1.5 to 7, whereas plotted in termsof√τ they are well concentrated around unity. This is further illustrated in1562 M. Praszalowicz0 5 1 0 1 5 2 0 2 5 3 0 3 51 0- 51 0 - 41 0 - 31 0 - 21 0 - 11 0 01 0 1dNch/d 2 p Tdyp 2TC M S : 7  T e V 2 . 3 6  T e V 0 . 9  T e V0 5 1 0 1 5 2 0 2 5 3 0 3 51 0- 51 0 - 41 0 - 31 0 - 21 0 - 11 0 01 0 1 C M S 7  T e V 2 . 3 6  T e V 0 . 9  T e V  =  0 . 2 7dNch/d 2 p TdyFig. 2. Charged particle multiplicity at mid rapidity |η| < 2.4 as measured byCMS [2], plotted as functions of p2T (left) and scaling variable τ (right) for λ = 0.27.0 1 2 3 4012345678 R 7 / 0 . 9 R 7 / 2 . 3 6Rp T  [ G e v / c ]0 1 2 3 4 5012345678 R 7 / 0 . 9 R 7 / 2 . 3 6     =  0 . 2 7R 1/2Fig. 3. Ratios of CMS pT spectra [2] at 0.7 TeV to 0.9 (blue circles) and 2.36 TeV(red triangles) plotted as functions of pT (left) and scaling variable√τ (right) forλ = 0.27.the left panel of Fig. 4 which presents the enlarged view of the right panel ofFig. 3. With this accuracy we see a small systematic increase of the ratios(apart from the first 4, 5 points which correspond to particles of very lowpT where a different production mechanism may dominate) which suggestssome weak dependence of exponent λ on pT. The value of λ = 0.27 hasbeen obtained by minimizing deviations of ratios R7/0.9 and R7/2.36 from 1for central pT points (i.e. rejecting first 5 and last 4 points).Geometrical Scaling in Hadronic Collisions 15630 1 2 3 4 50 . 51 . 01 . 5 R 7 / 0 . 9 R 7 / 2 . 3 6     =  0 . 2 7R 1/20 1 2 3 4 50 . 51 . 01 . 5 R 7 / 0 . 9 R 7 / 2 . 3 6     =   D I S ( 2 p T )R 1/2Fig. 4. Enlarged plot of the right panel of Fig. 3 for λ = 0.27 (left) and for λ =λeff(2pT).Residual dependence of exponent λ on pT is in agreement with small xdependence of the DIS structure function as measured in HERA [16]. InRef. [12] we have argued that this dependence can be well approximatedby use of the effective exponent λeff of Eq. (13) with argument Q = 2pT.This is demonstrated in the right panel of Fig. 4, where we used λeff(2pT)to calculate the ratios R7/0.9 and R7/2.36. An impressive improvement ofgeometrical scaling (i.e. of the equalities R7/0.9 ' 1 and R7/2.36 ' 1) can beindeed seen.4. Onset of geometrical scaling in heavy ion collisionsHeavy ions provide much reacher information on the characteristics ofparticle production at high energies. Indeed, one can study not only energydependence but also atomic number A-dependence, rapidity dependence (atRHIC much larger rapidity range has been covered than so far at the LHC)and finally centrality dependence. The production of quark-gluon plasmaand its ability to “remember” the initial conditions of the saturated glu-onic matter are here of primary interest. Unfortunately RHIC energies arepresumably too low for geometrical scaling to work. Nevertheless we showbelow, that approximate GS can be seen in the RHIC data. To this end wechoose the PHOBOS pT distributions measured in gold–gold and copper–copper collisions at 62.4 and 200 GeV per nucleon [23,24].Here a new scaling law is particularly interesting. Namely the satura-tion scale in nucleus–nucleus collisions scales with A as [25] (for review seeRef. [26])1564 M. PraszalowiczQ2A s = A1/3Q2s (25)which implies that the relevant scaling variable readsτA =p2TA1/3Q2s=1A1/3p2TQ20(pTW)−λ. (26)In Fig. 5 we plot multiplicity distribution for central Au–Au and Cu–Cucollisions in function of p2T and τA. In this case a slightly higher value of theexponent λ is used, namely λ = 0.3. We see again that the rescaled spectraseem to fall on one curve, although the alinement of Au and Cu spectrais not perfect for small and medium values of τA. Nevertheless a tendencytowards geometrical scaling is clearly seen. Similar conclusions can be drawnfor more peripheral collisions. A detailed study of the onset of geometricalscaling in heavy ion collisions will be presented elsewhere.0 5 10 1510-410-310-210-1110102 p 2T  [ G e V  2 / c 2 ]P H O B O S  0  −  6  % A u   2 0 0  G e V A u  6 2 . 4  G e V C u  2 0 0  G e V C u  6 2 . 4  G e V     =  0multiplicity0 5 10 1510-410-310-210-1110102 P H O B O S  0  −  6  % A u   2 0 0  G e V A u  6 2 . 4  G e V C u  2 0 0  G e V C u  6 2 . 4  G e V     =  0 . 3 0multiplicity AFig. 5. Multiplicity distribution in heavy ion collisions for Au–Au and Cu–Cu attwo RHIC energies 200 and 62.4 GeV [23, 24] plotted in terms of p2T (left panel)and scaling variable τA (right panel).5. ConclusionsIn this paper we have demonstrated that geometrical scaling originallypostulated in deep inelastic scattering [14] is also exhibited by the pT spectrain hadronic collisions [11, 12]. To this end recent CMS data [2] have beenanalyzed and shown to scale with scaling variable τ defined in Eqs. (20),(21). A simplified model of Gribov, Levin and Ryskin [18] has been used tomotivate the appearance of GS in hadronic collisions. This model can be apriori used to study the shape of the universal scaling function F (τ) whichdeserves a separate study.Geometrical Scaling in Hadronic Collisions 1565A notable difference between DIS and hadronic collisions is that in DISwe deal with totally inclusive cross-section, whereas in pp both hadronizationand final state interactions play essential role. Nevertheless, the imprintof the saturation scale Qs is visible in the spectra, which means that theinformation on the initial fireball survives until final hadrons are formed.It has been shown that the quality of geometrical scaling is improved ifthe exponent λ becomes pT-dependent [12] in accordance withQ-dependenceof λeff(Q = 2pT) obtained from DIS. This is a remarkable feature that sup-ports the picture in which medium pT particles are produced from saturatedgluonic matter irrespectively of the scattering states.If so, geometrical scaling should be also present in heavy ion collisions.The detailed studies will be certainly carried out at the LHC. Here we haveanalyzed PHOBOS data [23, 24] for two RHIC energies and for two differ-ent nuclei: gold and copper, and the onset of geometrical scaling has beenclearly seen. Interestingly, we have found that the exponent λ that governsgeometrical scaling in heavy ion collisions is higher than the one in pp. Thisis in striking agreement with the fact that multiplicity growth with energyobserved by ALICE [4] is faster in heavy ions than in pp. Question arisesto what extent the hydrodynamical evolution of the quark-gluon plasma isgoing to wash out geometrical scaling that is present in the initial state. Fur-ther studies should also concentrate on centrality and rapidity dependenceof GS.The author wants to thank Larry McLerran for a number of stimulatingdiscussions that triggered this work and Andrzej Białas for discussion andencouragement. Special thanks are due to Barbara Wosiek for the guidancethrough the wealth of heavy ion RHIC data. Part of this work has beencompleted during a short visit at CERN TH Department.REFERENCES[1] K. Aamodt et al. [ALICE Collaboration], Eur. Phys. J. C65, 111 (2010);Eur. Phys. J. C68, 89 (2010); Eur. Phys. J. 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Geometrical scaling in hadronic collisions

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