It is argued that the expected turn-down in $x- Q^2$ of the cross sections
(structure functions $F_2(x,Q^2)$), assumed to result from the saturation of
parton densities in the nucleon, is related to a phase transition from the
(almost) ideal partonic gas, obeying Bjorken scaling, to a partonic "liquid".
This can be quantified in the framework of statistical models, percolation and
other approaches to collective phenomena of the strongly interacting matter.
Similarities and differences between the case of lepton-hadron, hadron-hadron
and nuclear collisions are discussed.Comment: 7 pages, 2 figures, talk given by L.L. Jenkovszky at the EDS'09
  Conferenc

Jenkovszky, L. L.

Troshin, S. M.

Tyurin, N. E.

English

arXiv

It is argued that the expected turn-down in $x- Q^2$ of the cross sections (structure functions $F_2(x,Q^2)$), assumed to result from the saturation of parton densities in the nucleon, is related to a phase transition from the (almost) ideal partonic gas, obeying Bjorken scaling, to a partonic "liquid". This can be quantified in the framework of statistical models, percolation and other approaches to collective phenomena of the strongly interacting matter. Similarities and differences between the case of lepton-hadron, hadron-hadron and nuclear collisions are discussed

Jenkovszky, L L

Troshin, S M

Tyurin, N E

CERN Document Server

Saturation and Critical Phenomena in DISL.L. Jenkovszky1, S.M. Troshin2, and N.E. Tyurin21BITP, Kiev, 03680 Ukraine2IHEP, Protvino, 142281 RussiaIt is argued that the expected turn-down in x− Q2 of the cross sections (structure functionsF2(x,Q2)), assumed to result from the saturation of parton densities in the nucleon, is re-lated to a phase transition from the (almost) ideal partonic gas, obeying Bjorken scaling,to a partonic “liquid”. This can be quantified in the framework of statistical models, per-colation and other approaches to collective phenomena of the strongly interacting matter.Similarities and differences between the case of lepton-hadron, hadron-hadron and nuclearcollisions are discussed.1 IntroductionBased on different observations, models and equations governing deep inelastic scattering (DIS)and related processes, a “saturation” regime is expected when certain values of low enough xand relevant Q2 are reached. According to the dipole model of DIS, this regime already hasbeen achieved and it is characterised by the “saturation radius” [1] R20 = (x/x0)λ/Q20, withQ20 = 1 GeV2, x0 = 3 · 10−4 and λ = 0.29, found from a fit to the DIS data at x < 0.01.On more general grounds, saturation could be expected also from unitarity: the rapid (power-like) increase with 1/x of the structure functions/cross sections may suggest that unitaritycorrections will tamper this rise, although formally the Froissart bound has never been provenfor off-mass-shell particles, thus unitarity does not provide any rigorous limitation for suchamplitudes [2, 3]. One more argument is physical: the rise of the structure function F2(x, Q2)reflects the increase of the parton density (parton number in the nucleon). Since this numberincreases as a power, and the nucleon radius is known to increase as ln s (or, at most ln2 s)(shrinkage of the cone), the particle number density within a nucleon increases, inevitably,reaching a critical value where the partons start to coalesce (overlap, recombine etc). Thequalitative picture of this phenomenon in the Q2−1/x plane is well known and cited in variouscontexts (see, e.g., [4]).Quantitatively, the dynamics depend on many, poorly known, details, such as the propertiesof the constituents and their interaction within the nucleon.Below we propose a novel approach to the saturation phenomenon in DIS and related pro-cesses based on the collective properties of the excited nucleon. Namely, we suggest that, belowthe saturation reggeon, the nucleon in DIS is seen as a gas of almost free partons. With theirincreasing density, the constituent gradually overlap and, starting from a certain value of x andQ2, the gas of free partons coalesce condensing in a liquid of quarks and gluons. Saturationcorresponds to the onset of the new phase.The thermodynamic properties of such an excited nucleon are characterised by its tempera-ture, pressure etc, and a relevant equation of state (EoS). The statistical treatment of partonic415log(Q 2)log(x -1)ΛQCDSaturationGeometricalScalingFigure 1: Phase diagram of DIS, from Ref. [4].distributions is by far not new, see e.g. [5, 6, 7]. What is new in our approach, is the interpreta-tion of the saturation in DIS as a manifestation of the transition from a dilute partonic gas toa liquid. The details (nature) of this (phase?) transition are not known. It can be of the first,of second order, or, moreover, be a smooth cross-over phenomenon. Our main argument is thatthe volume of the nucleon confining the partons (quarks and gluons) in the interior increasesslower, at most as ∼ ln6 s (more likely, as ∼ ln3 s), while the volume occupied by the interior,quarks and gluons, increases as a power, thus resulting in a limiting behaviour: a gas-to-liquidcross-over or a phase transition. The present contribution is a first step in understanding thiscomplex process.In a related paper, Ref. [8], the phase structure of the hadronic matter in terms of its tem-perature T and its baryochemical potential µ, was studied in the framework of the percolationtheory. The percolation mechanism was used in [9] to obtain a limiting energy dependence ofthe hadronic matter at s→∞ .2 SaturationWe define the saturation line (in the x−Q2 plane) as the turning point (line) of the derivativesBQ(x, Q2) =∂F2(x, Q2)∂(ln Q2), Bx(x, Q2) =∂F2(x, Q2)∂(ln(1/x)), (1)called BQ or Bx slopes, where F2(x, Q2) is a “reasonable” model for the structure function,i.e. one satisfying the basic theoretical requirements, yet fitting the data. For example, themodel for F2(x, Q2) of Ref. [10] interpolates between Regge behaviour at small Q2 and thesolution of the DGLAP evolution equation at asymptotically large values of Q2, practically forL L JENKOVSZKY, S M TROSHIN, AND N E TYURIN4161e-050.00010.0010.01 1101001000-4-3-2-1012xQ**2slopeFigure 2: The surface of BQ(x,Q2) calculated in Ref. [10].all values of x. The resulting two-dimensional projection of the Q−slope is shown in Fig. 2. Inour interpretation, the critical line (saturation = phase transition) occurs along the fold line onthis figure (compare with a similar figure, Fig. 2 of Ref. [11], derived from a different model).The main goal of this paper is the identification of this line (point) with the critical line(point) on the T, µ phase diagram of an excited nucleon viewed as a thermodynamical system.The thermodynamical approach to DIS may provide a new insight to this complex phenomenon.We are aware of the limited time scales in a deep inelastic scattering from the point of viewof thermalisation, a familiar problem relevant to any thermodynamical description of hadronicsystems. Let us only remind that the thermodynamic approach to high-energy scattering andmultiple production, originated by Fermi and Landau’s papers, were applied to hadrons, ratherthan heavy ions.3 Statistical Models of Parton DistributionsThe statistical model of parton distributions in DIS was considered and developed in quite anumber of papers [5, 6, 7]. In its simplest version, one assumes [5] that inside the nucleon, thevalence quarks, as well as the sea quarks and antiquarks and gluons form a noninteracting gas inequilibrium. This simple picture may be further developed in two directions: 1. Introduction ofthe effect of the finite size (and its energy dependence!) of the nucleon on statistical expressionfor the number of states for the unit energy interval; 2) Account for the Q2 evolution, thatcan be calculated either from the DGLAP equation or phenomenologically, as e.g. in Ref. [10].A likely scenario emerging [5, 6] for this high quark density nq = (nq − nq¯)/3 system is thatSATURATION AND CRITICAL PHENOMENA IN DIS417quarks form Cooper pairs and new condensates develop.The nucleon of mass M consists of a gas of massless particles (quarks, antiquarks andgluons) in equilibrium at temperature T in a spherical volume V with radius R(s) increasingwith squared c.m.s. energy s as ln s (or ln2 s). The invariant parton number density in phasespace is given by [12]dnid3pi d3ri=dnd3p d3r=gf(E)(2pi)3, (2)where g is the degeneracy (g = 16 for gluons and g = 6 for q and q¯ of a given flavour), E,p isthe parton four-momentum and f(E) =(exp[β(E−µ)]±1)−1is the Fermi or Bose distributionfunction with β ≡ T−1. Quantities in the infinite momentum frame (IMF) are labelled by thesubscript i.The invariant parton density dni/dx in the IMF is related to dn/dE and f(E) in the protonrest frame as follows [5]dnidx=gV (s)M2x(2pi)2∫ M/2xM/2dEf(E), (3)and the structure functionF2(x) = x∑qe2q[(dnidx)q+(dnidx)q¯]. (4)Without any account for the finite volume of the hadron, this SF disagrees with the data. Finitevolume effects can be incorporated, following R.S. Bhalerao from Ref. [5], and the result isdn/dE = gf(E)(V E2/2pi2 + aR2E + bR), (5)where V and R are energy dependent and a, b, in front of the surface and curvature terms, areunknown numerical coefficients. Their values are important for the final result, but they cannotbe calculated from perturbative QCD. A rather general method to calculate these importantparameters can be found in Ref. [13]. We intend to come back to this point in a subsequentpublication.4 PercolationPercolation as a model of phase transition from colourless hadrons to a quark gluon plasma(QGP) was studied in a number of papers, recently in Ref. [8]. Below we apply the argumentsof that paper to the saturation inside a nucleon, where extended (dressed) quark and gluons,rather then mesons and baryons considered in Ref. [8] percolate into a uniform new phase ofmatter. Both objects are coloured particle inside a colourless nucleon. We start with a shortintroduction to the subject.Consider N spheres of radius R0 and hence volume V0 = (4pi/3)R30 in a “box” of size V ,with V À V0.Percolation (clustering) of spheres, in three spacial directions, was studied for the case of:a) arbitrary overlap [15] and b) for those with impenetrable hard core, allowing only partialoverlap [6].L L JENKOVSZKY, S M TROSHIN, AND N E TYURIN418For the case a), the first percolation point (partial percolation), with 30% of occupation,occurs for the density n = N/V, nm ≈ 0.35/V0, the largest cluster having the density of about1.2/V0. A second percolation point occurs at nv = 1.22/V0, when 70% of space is covered byspheres, i.e. the vacuum disappears as a large-scale entity.The existence of two percolation thresholds, one for the formation of the first spanningcluster of spheres and the second one for the disappearing of a spanning vacuum “cluster”, is ageneral feature of the 3-dimensional percolation theory.b) An impenetrable spherical core, with Rc = R0/2, and the spheres can only partiallyoverlap. Here again one has two percolation thresholds, at n¯m ≈ 0.34/V0 (close to the case a)),and vacuum percolation, at n¯v = 2.0/V0, requiring a higher density compared to a).4.1 GPD H(x, t) and Impact Parameter Parton Distribution q(x, b)The number of constituents in a nucleon can be found by integrating in b (impact parameter)the general parton distribution, e.g. that of Ref. [14].q(x, b) =12pi∫ ∞0√−td√−tH(x, t)J0(b√−t).Here, contrary to q(x), q(x, b) is dimensional, with a dimension of squared mass m2, inter-preted as the transverse size of the extended parton in the hadron, m = R−1, q(x, b) =m2q˜(x, b), q˜(x, b) being the partons number density [14]. In paper [14], m was a constant;here we choose it to depend on the photon virtuality: m → m0 ln(Q + mρ) = R−1(Q), wherem is the mass of the lightest vector meson, m = mρ. Note the inequality q(x, b) ≤ 1/Sq, whereSq(Q) = ln−2(Q + mρ).Thus, the nucleon is composed of N = 2pi∫ 10 dx∫∞0 b db q˜(x, b, Q) extended partons with thetransverse area Sq(Q).4.2 A Toy EoSOne has for mesonsNM (T ) = 3pi290T 3.On the other hand, mesons percolate whenever nv = 1.22/Vh, where Vh = (4pi/3)R3h, and weuse for the probe meson radius Rh = 1.1/ ln(Q + mρ), which in the limit of a real photon,Q→ 0, matches the relevant value Rh = 0.8 fm used in Ref. [8]. Solving nh(T ) = nf yieldsTM (T, Q) ' 171 ln(Q + mρ) MeVas the limiting temperature through meson fusion (cf. Tpi ' 240MeV of [8]). In a similaranalysis of the phase diagram of hadronic matter [8], the (Q−independent) limiting temperatureTpi ≈ 240MeV was obtained.N.B. The parameters appearing below should be rescaled with account for the replacementm→ ln(Q + mρ)!The density of point-like nucleons of mass M is at T = 0nb(µ, T = 0) =23pi2(µ2 −M2)3/2,SATURATION AND CRITICAL PHENOMENA IN DIS419or, by using the van der Waals approach of Ref. [6]nB(µ, T = 0) =nb(T, µ)1 + nb(T, µ)Ve.With increasing nucleon density, the empty vacuum disappears forn¯v ' 2/Vh(Q) ≈ 0.93 ln(Q + mρ))3 fm−3,which, for real photons (Q = 0) corresponds to about 5.5 times standard nuclear density.Solving nB(T = 0, µ) = n¯v gives µv ' 1.12 ln(Q + mρ) GeV for the limiting baryochemicalpotential at T = 0.In the region of low or intermediate µ in the Tµ diagram, one can approximate the densityof point-like nucleons by the Boltzmann limitnb(µ, T ) ' 2T3pi2(MT)2K2(M/T )eµ/T ' T32(2MpiT)3/2e(µ−M)/T .As a result, one obtains a family of curves for the phase diagram T (µ) and for the EoS p(T )for various values of Q and x (cf. Figs. 3 and 4 of Ref. [8]).5 ConclusionsIn this talk a new approach to the saturation phenomena in deeply virtual processes – DIS,DVCS, VMP – diffractive and non-diffractive – is suggested. The basic idea is a physical one:Bjorken scaling implies that the nucleon in DIS and related processes is a system of weeklyinteracting partonic gas, that can be described by means of Bose-Einstein or Fermi statistics.As the density increases (with decreasing x and relevant Q2), seen as the violation of Bjorkenscaling, the system reaches a coalescence point where the gas condenses, eventually to a liquid.The thermodynamic properties of this transition can be only conjectured, and further studiesare needed to quantify this phenomenon.Of two models presented in this note, more promising seems be the first one (Sec. 3). Furtherstudies will show its viability.An alternative measure of the onset of saturation and expected change of phase can berelated to non-linear evolution equations. Saturation and a phase transition are expected whenthe non-linear contribution overshoots the linear term.The phenomenon discussed in the present note may have much in common with the colourglass condensate proposed in the context of heavy ion collisions (see, e.g. [16] and earlier ref-erences therein). Apart from similarities (condensation of quarks and gluons as x → 0) thereare apparent differences: for example, hydrodynamical flow is not expected in DIS. In any case,the dynamics of the strong interaction is the same in lepton-hadron, hadron-hadron and heavyion collisions.AcknowledgementsWe thank Mark Gorenstein, Volodymyr Magas and Francesco Paccanoni for useful discussionsand criticism.L L JENKOVSZKY, S M TROSHIN, AND N E TYURIN420References[1] K. Golec-Biernat and M. Wu¨sthoff, Phys. Rev. D 644 131 (2007).[2] V.A. Petrov, A.V. Prokudin, Proceedings of the conference Elastic and diffractive scattering, Protvino1999, p.95.[3] S.M. Troshin, N.E. Tyurin. Eur. Phys. J. 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Saturation and Critical Phenomena in DIS

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