Open Physics Journal




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ISSN: 1874-8430 ― Volume 5, 2019

Emergence and Violation of Geometrical Scaling in pp Collisions§



Michal Praszalowicz*
M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Krakow, Poland

Abstract

We argue that geometrical scaling (GS) proposed originally in the context of Deep Inelastic Scattering (DIS) at HERA works also in pp collisions at the LHC energies and in NA61/SHINE experiment. We show that in DIS GS is working up to relatively large Bjorken x ~ 0.1 . As a consequence negative pion multiplicity pT distributions at NA61/SHINE energies exhibit GS in mid rapidity region. For y ≠ 0 clear sign of scaling violations can be seen when one of the colliding partons has Bjorken x ≥ 0.1 . Finally, we argue that in the case of identified particles GS scaling is still present but the scaling variable is a function of transverse mass rather than pT.

PACS number(s): 13.85.Ni,12.38.Lg.

Keywords:: Gluon saturation, low x physics, geometrical scaling.


Article Information


Identifiers and Pagination:

Year: 2014
Volume: 1
Issue: Suppl 1: M5
First Page: 43
Last Page: 47
Publisher Id: PHY-1-43
DOI: 10.2174/1874843001401010043

Article History:

Received Date: 25/11/2013
Revision Received Date: 29/1/2014
Acceptance Date: 30/1/2014
Electronic publication date: 31/12/2014
Collection year: 2014

© Michal Praszalowicz; Licensee Bentham Open.

open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.


* Address correspondence to this author at the Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Krakow, Poland; Tel: +48 12 663 56 77; Fax: +48 12 633 40 79; E-mail: michal@if.uj.edu.pl§ Presented at the Low x Workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel.






1. INTRODUCTION

This talk based on refs. [1(a) McLerran L, Praszalowicz M. Saturation and scaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV Acta Phys Pol B 2010; 41: 1917-26.; (b) Saturation and dcaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV – Addendum Acta Phys Pol B 2011; 42: 99-103.-6Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.] (where also an extensive list of references can be found) follows closely an earlier report of ref. [7Praszalowicz M. Geometrical scaling in high energy collisions and its breaking Acta Phys Pol B Proceeding 2013; 6: 809-16.]. We shall discuss the scaling law, called geometrical scaling (GS), which has been introduced in the context of DIS [8Stasto AM, Golec-Biernat KJ, Kwiecinski J. Geometric scaling for the total gamma*p cross section in the low x region Phys Rev Lett 2001; 86: 596-9.]. It has been also shown that GS is exhibited by the pT spectra at the LHC [1(a) McLerran L, Praszalowicz M. Saturation and scaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV Acta Phys Pol B 2010; 41: 1917-26.; (b) Saturation and dcaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV – Addendum Acta Phys Pol B 2011; 42: 99-103., 3(a) Praszalowicz M. Geometrical scaling in hadronic collisions Acta Phys Pol B 2011; 42: 1557-66.; Geometrical Scaling in in High Eneregy Hadronic Collisions [arXiv:1205.4538 [hep-ph]]] and that an onset of GS can be seen in heavy ion collisions at RHIC energies [3(a) Praszalowicz M. Geometrical scaling in hadronic collisions Acta Phys Pol B 2011; 42: 1557-66.; Geometrical Scaling in in High Eneregy Hadronic Collisions [arXiv:1205.4538 [hep-ph]]]. At low Bjorken x < xmax gluonic cloud in the proton is characterized by an intermediate energy scale Qs (x), called saturation scale [9(a) Gribov LV, Levin EM, Ryskin MG. Semihard processes in QCD Phys Rept 1983; 100: 1-150.; (b) Mueller AH, Qiu JW. Gluon recombinantion and shadowung at Small Values of x Nucl Phys 1986; 268: 427-52.; (c) Mueller AH. Parton saturation at small xand in large nuclei Nucl Phys 1999; B558: 285-303., 10(a) Golec-Biernat KJ, Wüsthoff M. Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction Phys Rev D 1998; 59: 014017.; (b) Saturation in diffractive deep inelastic scattering Phys Rev D 1999; 60: 114023.]. Qs (x) is defined as the border line between dense and dilute; gluonic systems (for review see e.g. refs. [11Mueller AH. Parton Saturation An Overview [arXiv:hep-ph/0111244], 12McLerran L. Matter at very high energy density: 3 Lectures in Zakopane Acta Phys Pol B 2010; 41: 2799-6.]). In the present paper we study the consequences of the very existence of Qs (x); the details of saturation phenomenon are here not of primary importance.

Here we shall focus of four different pieces of data which exhibit both emergence and violation of geometrical scaling. In Sect. 2 we briefly describe the method used to assess the existence of GS. Secondly, in Sect. 3 we describe our recent analysis [4(a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.] of combined HERA data [13(a) Adloff C, Andreev V, Andrieu B, et al. [H1 Collaboration] Deep-inelastic inclusive ep scattering at low x and a determination of alpha_s Eur Phys J C 2001; 21: 33-61.; (b) Chekanov S, Derrick M, Krakauer D, et al. [ZEUS Collaboration] Measurement of the neutral current cross section and F_2 structure functionfor deep inelastic e+ p scattering at HERA Eur Phys J C 2001; 21: 443-71.] where it has been shown that GS in DIS works surprisingly well up to relatively large xmax~0.1 (see also [14Caola F, Forte S, Rojo J. HERA data and DGLAP evolution: Theory and phenomenology Nucl Phys A 2011; 854: 32-44.]). Next, in Sect. 4, on the example of the CMS pT spectra in central rapidity [15(a) Khachatryan V, Sirunyan AM, Tumasyan A, et al. [CMS Collaboration] Transverse momentum and pseudorapidity distributions of charged hadrons in pp collisions at sqrt(s) = 09 and 2 6 TeV J High Energy Phys 2010; 1002: 041.; Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at sqrt(s) = 7 TeV Phys Rev Lett 2010; 105: 022002.; Charged particle multiplicities in pp interactions at sqrt(s) = 09, 236, and 7 TeV J High Energy Phys 2011; 1101: 079.], we show that GS is also present in hadronic collisions. For particles produced at non-zero rapidities, one (larger) Bjorken x=x1 may be outside of the domain of GS, i.e.x1>xmax, and violation of GS should appear. In Sect. 5 we present analysis of the pp data from NA61/SHINE experiment at CERN [16(a) Abgrall N, Aduszkiewicz A, Antićić T, et al. [NA61/SHINE Collaboration]. Report from the NA61/SHINE experiment at the CERN SPS CERN-SPSC-2012-029 SPSC-SR-107 ; (b) Aduszkiewicz A. Ph.D. Thesis, University of Warsaw 2013.; (c) Pulawski S. Talk at 9th Polish workshop on Relativistic Heavy-Ion Collisions Kraków and private communication 2012 November; ] and show that GS is indeed violated once rapidity is increased. Finally in Sect. 6 we analyze identified particles spectra where the particle mass provides another energy scale which may lead to the violation of GS, or at least to some sort of its modification [6Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.]. We conclude in Sect. 7.

2. ANALYZING DATA WITH METHOD OF RATIOS

Geometrical scaling hypothesis means that some observable σ depending in principle on two independent kinematical variables, like x and Q2, depends in fact only on a given combination of them, denoted in the following as τ:


Here function F in Eq. (1) is a dimensionless universal function of scaling variable τ:


and


is the saturation scale. Here Q0 and x0 are free parameters which, however, are not of importance in the present analysis, and exponent λ is a dynamical quantity of the order of λ ~ 0.3. Throughout this paper we shall test the hypothesis whether different pieces of data can be described by formula (1) with constantλ, and what is the kinematical range where GS is working satisfactorily.

As a consequence of Eq. (1) observables σ(xi,Q2) for different xi's should fall on a universal curve, if evaluated in terms of scaling variable τ. This means that ratios


should be equal to unity independently of τ. Here for some xref we pick up all xi < xref which have at least two overlapping points in Q2.

For λ≠0 points of the same Q2 but different x's correspond in general to different τ's. Therefore one has to interpolate σ(xref,τ(xref,Q2;λ)) to Q2k,ref such that τ(xref,Q2k,ref;λ)=τk. This procedure is described in detail in ref. [4(a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.].

By adjusting λ one can make Rxi,xref(λ;τk)→ 1 for all τk in a given interval. In order to find an optimal value λmin which minimizes deviations of ratios (4) from unity we form the chi-square measure


where the sum over k extends over all points of given xi that have overlap with xref, and Nxi,xref is a number of such points.

3. GEOMETRICAL SCALING IN DIS AT HERA

In the case of DIS the relevant scaling observable is γ* p cross section and variable x is simply Bjorken x. In Fig. (1) we present 3-d plot of λmin(x,xref) which has been found by minimizing (5).

Fig.(1)

Three dimensional plot of λmin(x, xref) obtained by minimization of Eq. (5).



Qualitatively, GS is given by the independence of λmin on Bjorken x and by the requirement that the respective value of χ2x,xrefmin) is small (for more detailed discussion see ref. [4(a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.]). One can see from Fig. (1) that the stability corner of λmin extends up to xref~0.1, which is well above the original expectations. In ref. [4(a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.] we have shown that:


4. GEOMETRICAL SCALING OF CENTRAL RAPIDITY pT SPECTRA AT THE LHC

In hadronic collisions at c.m. energy W=s particles are produced in the scattering process of two patrons (mainly gluons) carrying Bjorken x's


For central rapidities x=x1 ~ x2 In this case charged particles multiplicity spectra exhibit GS [1(a) McLerran L, Praszalowicz M. Saturation and scaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV Acta Phys Pol B 2010; 41: 1917-26.; (b) Saturation and dcaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV – Addendum Acta Phys Pol B 2011; 42: 99-103.]


where F is a universal dimensionless function of the scaling variable (2). Therefore the method of ratios can be applied to the multiplicity distributions at different energies (Wi taking over the role of xi in Eq. (4))1. For Wref we take the highest LHC energy of 7 TeV. Hence one can form two ratios RWref,Wi with W1 = 2.36 and W2 = 0.9 TeV. These ratios are plotted in Fig. (2) for the CMS single non-diffractive spectra for λ = 0 and for λ = 0.27, which minimizes (5) in this case. We see that original ratios plotted in terms of pT range from 1.5 to 7, whereas plotted in terms of they are well concentrated around unity. The optimal exponent λ is, however, smaller than in the case of DIS. Why this so, remains to be understood.

Fig.(2)



5. VIOLATION OF GEOMETRICAL SCALING IN FORWARD RAPIDITY REGION

For y > 0 two Bjorken x's can be quite different: x1 > x2. Therefore by increasing y one can eventually reach x1 > xmax and GS violation should be seen. For that purpose we shall use pp data from NA61/SHINE experiment at CERN [16(a) Abgrall N, Aduszkiewicz A, Antićić T, et al. [NA61/SHINE Collaboration]. Report from the NA61/SHINE experiment at the CERN SPS CERN-SPSC-2012-029 SPSC-SR-107 ; (b) Aduszkiewicz A. Ph.D. Thesis, University of Warsaw 2013.; (c) Pulawski S. Talk at 9th Polish workshop on Relativistic Heavy-Ion Collisions Kraków and private communication 2012 November; ] at different rapidities y = 0.1-3.5 and at five scattering energies W1,...,5 = 17.28, 12.36, 8.77, 7.75 , and 6.28 GeV.

In Fig. (3) we plot ratios R1i = RW1,Wi (4) for π- spectra in central rapidity for λ = 0 and 0.27. For y = 0.1 the GS region extends down to the smallest energy because xmax is as large as 0.08. However, the quality of GS is the worst for the lowest energy W5. By increasing y some points fall outside the GS window because x1 ≥ xmax, and finally for y ≥ 1.7 no GS is present in NA61/SHINE data. This is illustrated nicely in Fig. (4).

Fig.(3)



Fig.(4)



6. GEOMETRICAL SCALING FOR IDENTIFIED PARTICLES

In ref. [6Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.] we have proposed that in the case of identified particles another scaling variable should be used in which pT is replaced by mT=mTm=mT2+mT2m(mTscaling), i.e.


This choice is purely phenomenological for the following reasons. Firstly, the gluon cloud is in principle not sensitive to the mass of the particle it finally is fragmenting to, so in principle one should take pT as an argument of the saturation scale. In this case the proper scaling variable would be


10 

However this choice (mT=pT – scaling) does not really differ numerically from the one given by Eq. (9).

To this end let us see how scaling properties of GS are affected by going from scaling variable τpT = τ (2) to mT (9) and what would be the difference in scaling properties if we had chosen pT as an argument in the saturation scale leading to scaling variable mTpT (10). This is illustrated in Fig. (5) where we show analysis [6Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.] of recent ALICE data on identified particles [17(a) Aamodt K, Abel N, Abeysekara U, et al. [ALICE Collaboration] Production of pions, kaons and protons in pp collisions at sqrt(s)=900 GeV with ALICE at the LHC Eur Phys J C 2011; 71: 1655.; (b) Velasquez AO. [ALICE Collaboration] Production of pions, kaons and protons at high p_T in sqrt(s_{NN})=2.76 TeV Pb-Pb collisions Nucl Phys A 2013; 904-5. 763c-766c(ALICE preliminary)]. In Fig. (5a-c) full symbols refer to the pT – scaling (2) and open symbols to mT – scaling or mTpT – scaling. One can see very small difference between open symbols indicating that scaling variables mT (9) and mTpT (10) exhibit GS of the same quality. On the contrary pT – scaling in variable τpT(2) is visibly worse.

Fig.(5)



Finally in Fig. (5d), on the example of protons, we compare mT – scaling (open symbols) and mT – scaling (full symbols) in variable


11 

for λ =0. 27. One can see that no GS has been achieved in the latter case. Qualitatively the same behavior can be observed for other values of λ.

7. CONCLUSIONS

In ref. [4(a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.] we have shown that GS in DIS works well up to rather large Bjorken x's with exponent λ = 0.32-0.34. In pp collisions at the LHC energies in central rapidity GS is seen in the charged particle multiplicity spectra, however, λ = 0.27 in this case [1(a) McLerran L, Praszalowicz M. Saturation and scaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV Acta Phys Pol B 2010; 41: 1917-26.; (b) Saturation and dcaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV – Addendum Acta Phys Pol B 2011; 42: 99-103.]. By changing rapidity one can force one of the Bjorken x's of colliding patrons to exceed xmax and GS violation is expected. Such behavior is indeed observed in the NA61/SHINE pp data [5Praszalowicz M. Violation of geometrical scaling in pp collisions at NA61/SHINE Phys Rev D 2013; 87: 071502(R).]. Finally we have shown that for identified particles scaling variable τ of Eq. (2) should be replaced by mT defined in Eq. (9) and the scaling exponent λ ≈0.3 [6Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.].

CONFLICT OF INTEREST

The authors confirm that this article content has no conflicts of interest.

NOTES

For pp collisions we define ratios RWref,Wi as an inverse of (4).

ACKNOWLEDGEMENTS

Many thanks are due to the organizers of this successful series of conferences. This work was supported by the Polish NCN grant 2011/01/B/ST2/00492.

REFERENCES

[1] (a) McLerran L, Praszalowicz M. Saturation and scaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV Acta Phys Pol B 2010; 41: 1917-26.; (b) Saturation and dcaling of multiplicity, mean pT and pT distributions from 200 GeV < sqrt(s) < 7 TeV – Addendum Acta Phys Pol B 2011; 42: 99-103.
[2] Praszalowicz M. Improved geometrical scaling at the LHC Phys Rev Lett 2011; 106: 142002.
[3] (a) Praszalowicz M. Geometrical scaling in hadronic collisions Acta Phys Pol B 2011; 42: 1557-66.; Geometrical Scaling in in High Eneregy Hadronic Collisions [arXiv:1205.4538 [hep-ph]]
[4] (a) Praszalowicz M, Stebel T. Quantitative study of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1303: 090.; (b)Quantitative study of different forms of geometrical scaling in deep inelastic scattering at HERA J High Energy Phys 2013; 1304: 169.
[5] Praszalowicz M. Violation of geometrical scaling in pp collisions at NA61/SHINE Phys Rev D 2013; 87: 071502(R).
[6] Praszalowicz M. Geometrical scaling for indentified particles Phys Lett B 2013; 727: 461-7.
[7] Praszalowicz M. Geometrical scaling in high energy collisions and its breaking Acta Phys Pol B Proceeding 2013; 6: 809-16.
[8] Stasto AM, Golec-Biernat KJ, Kwiecinski J. Geometric scaling for the total gamma*p cross section in the low x region Phys Rev Lett 2001; 86: 596-9.
[9] (a) Gribov LV, Levin EM, Ryskin MG. Semihard processes in QCD Phys Rept 1983; 100: 1-150.; (b) Mueller AH, Qiu JW. Gluon recombinantion and shadowung at Small Values of x Nucl Phys 1986; 268: 427-52.; (c) Mueller AH. Parton saturation at small xand in large nuclei Nucl Phys 1999; B558: 285-303.
[10] (a) Golec-Biernat KJ, Wüsthoff M. Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction Phys Rev D 1998; 59: 014017.; (b) Saturation in diffractive deep inelastic scattering Phys Rev D 1999; 60: 114023.
[11] Mueller AH. Parton Saturation An Overview [arXiv:hep-ph/0111244]
[12] McLerran L. Matter at very high energy density: 3 Lectures in Zakopane Acta Phys Pol B 2010; 41: 2799-6.
[13] (a) Adloff C, Andreev V, Andrieu B, et al. [H1 Collaboration] Deep-inelastic inclusive ep scattering at low x and a determination of alpha_s Eur Phys J C 2001; 21: 33-61.; (b) Chekanov S, Derrick M, Krakauer D, et al. [ZEUS Collaboration] Measurement of the neutral current cross section and F_2 structure functionfor deep inelastic e+ p scattering at HERA Eur Phys J C 2001; 21: 443-71.
[14] Caola F, Forte S, Rojo J. HERA data and DGLAP evolution: Theory and phenomenology Nucl Phys A 2011; 854: 32-44.
[15] (a) Khachatryan V, Sirunyan AM, Tumasyan A, et al. [CMS Collaboration] Transverse momentum and pseudorapidity distributions of charged hadrons in pp collisions at sqrt(s) = 09 and 2 6 TeV J High Energy Phys 2010; 1002: 041.; Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at sqrt(s) = 7 TeV Phys Rev Lett 2010; 105: 022002.; Charged particle multiplicities in pp interactions at sqrt(s) = 09, 236, and 7 TeV J High Energy Phys 2011; 1101: 079.
[16] (a) Abgrall N, Aduszkiewicz A, Antićić T, et al. [NA61/SHINE Collaboration]. Report from the NA61/SHINE experiment at the CERN SPS CERN-SPSC-2012-029 SPSC-SR-107 ; (b) Aduszkiewicz A. Ph.D. Thesis, University of Warsaw 2013.; (c) Pulawski S. Talk at 9th Polish workshop on Relativistic Heavy-Ion Collisions Kraków and private communication 2012 November;
[17] (a) Aamodt K, Abel N, Abeysekara U, et al. [ALICE Collaboration] Production of pions, kaons and protons in pp collisions at sqrt(s)=900 GeV with ALICE at the LHC Eur Phys J C 2011; 71: 1655.; (b) Velasquez AO. [ALICE Collaboration] Production of pions, kaons and protons at high p_T in sqrt(s_{NN})=2.76 TeV Pb-Pb collisions Nucl Phys A 2013; 904-5. 763c-766c(ALICE preliminary)
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