Aspects of relativistic heavy-ion collisions

Georg Wolschin∗

Institut fur Theoretische Physik der Universitat Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany, EU

The rapid thermalization of quarks and gluons in the initial stages of relativistic heavy-ion col-lisions is treated using analytic solutions of a nonlinear diffusion equation with schematic initialconditions, and for gluons with boundary conditions at the singularity. On a similarly short timescale of t ≤ 1 fm/c, the stopping of baryons is accounted for through a QCD-inspired approach basedon the parton distribution functions of valence quarks, and gluons. Charged-hadron production isconsidered phenomenologically using a linear relativistic diffusion model with two fragmentationsources, and a central gluonic source that rises with ln3(sNN ). The limiting-fragmentation conjec-ture that agrees with data at energies reached at the Relativistic Heavy Ion Collider (RHIC) is foundto be consistent with Large Hadron Collider (LHC) data for Pb-Pb at

√sNN = 2.76 and 5.02 TeV.

Quarkonia are used as hard probes for the properties of the quark-gluon plasma (QGP) through acomparison of theoretical predictions with recent CMS, ALICE and LHCb data for Pb-Pb and p-Pbcollisions.

PACS numbers: 25.75.-q,25.75.Dw,25.75.C

I. INTRODUCTION

This article covers aspects of relativistic heavy-ion col-lisions in the energy regions reached at the RelativisticHeavy Ion Collider (RHIC) and the Large Hadron Col-lider (LHC). Starting with the thermalization of partonsin the initial stages, stopping of the incoming baryonsis discussed, followed by charged-hadon production andthe modification of quarkonia in the hot quark-gluonplasma (QGP), with an emphasis on bottomonia thatprovide clear signals of QGP formation. The overallapproach is phenomenological and in close comparisonwith data. In some cases such as stopping and quarko-nia, nonequilibrium-statistical considerations are mergedwith quantum chromodynamics (QCD). The work con-tains both new developments and re-views of our previ-ously published work in a new context.

The fast local thermalization of partons in the initialstages of relativistic heavy-ion collisions is a sufficientcondition to apply hydrodynamic descriptions [1] of thesubsequent collective expansion and cooling of the hotfireball that is created in the collision. Typical local equi-libration times for gluons are about 0.1 fm/c [2], withinitial central temperatures of the order of 500–600 MeVreached in a Pb-Pb collision at

√sNN = 5.02 TeV at

the Large Hadron Collider [3]. Thermalization times forquarks are typically by an order of magnitude larger [4]due to the smaller color factor and the different statis-tical properties (Pauli’s principle). Whereas the ther-mal Bose–Einstein/Fermi–Dirac distributions and alsothe initial distribution for quarks are known, plausibleassumptions are being made for the primordial gluon dis-tribution before the onset of the collision.

Studies such as Ref. [5] have found that local thermal-ization may not be a necessary condition for the appli-

∗ g.wolschin@thphys.uni-heidelberg.de

cability of hydrodynamics to relativistic heavy-ion colli-sions. This would imply that hydrodynamics could bea valid approach away from local equilibrium [6] – butstill, it remains important to investigate how and onwhich timescale local thermalization is achieved. Ob-viously, an investigation of the local thermalization ofquarks and gluons makes sense only in a weakly coupleddescription based on an effective kinetic theory that re-lies on the Boltzmann equation. It is recognized that astrongly coupled paradigm built on the anti-de Sitter-conformal field theory (AdS/CFT) correspondence [7]discovered in the investigation of D-branes in string the-ory [8] may be relevant at RHIC and LHC energy scalessuch that partons would not be the relevant degrees offreedom any more. In this work, however, I rather as-sume quarks and gluons to be well-defined and long-livedexcitations in QCD at temperatures close to the criticalvalue.

What is not known precisely is the development withtime towards local equilibrium. An easy way to approxi-mate the time evolution is given by the linear relaxation-time approximation (RTA), which provides a simple an-alytic solution of the problem by enforcing a linear tran-sition from the initial to the local equilibrium state, butdoes not properly account for the known nonlinearity ofthe system. A more ambitious approach for bosons is tonumerically solve gluon transport equations that includethe effect of Bose statistics, as has been done in Ref. [9]and subsequent works. Initially, only elastic scatteringwas considered with the possibility of gluon condensateformation due to particle-number conservation in over-populated systems, but it was recognized that inelastic,number-changing radiative processes cannot be neglected[10], and hinder the formation of a Bose condensate.

Whereas also other numerical calculations relying ona quantum Boltzmann collision term to account for theinitial local equilibration are available such as Ref. [11], itis of interest to have an exactly solvable analytic model

2

to better understand the physics of the fast equilibra-tion. A corresponding nonlinear boson diffusion equa-tion (NBDE) has been presented in Ref. [4] and solvedfor a simplified case that did, however, not yet considerthe singularity at the chemical potential µ < 0. Thenonlinear partial differential equation preserves the es-sential features of Bose–Einstein statistics that are con-tained in the collision term. In particular, the thermalequilibrium distribution emerges as a stationary solutionand hence, the equation appears suitable to model thethermalization of gluons in relativistic collisions. It isused in this work to generate new exact solutions forthe time-dependent gluon equilibration problem that in-clude boundary conditions at the singularity. Regard-ing fermionic thermalization, the corresponding nonlin-ear fermion diffusion equation is easier to solve [4, 12] be-cause no singularity appears, and the analytic solutionswill be reviewed.

(1.9) The relevant bulk properties of relativistic heavy-ion collisions mostly arise from charged-hadron produc-tion. In many of the available macroscopic and micro-scopic models, hadronization occurs from the fireball atthe phase boundary between the QGP and the hadronicphase. However, it has been proposed [13, 14] thatthe production of charged hadrons from the fragmen-tation sources at larger rapidity values is also relevantin the overall distributions and should be treated sep-arately from the fireball source. At midrapidity, thesecontributions are relatively small, but become relevantmore forward or backward. Corresponding new results ofthe phenomenological three-source relativistic diffusionmodel (RDM) are compared with recent LHC pseudora-pidity data on charged-hadron production in 5.02 TeVPb-Pb, which are also shown to be consistent with thelimiting-fragmentation hypothesis that had been foundto agree with the hadron production data at RHIC ener-gies [15–17].

Regarding more direct evidence for transient QGP for-mation in relativistic heavy-ion collisions, jets may pro-vide the most direct manifestation for quarks and gluonsin the system. In particular, the suppression of away-side jets in the hot QGP had already been predicted byBjorken [18], and confirmed experimentally by the STARcollaboration [19]. Meanwhile, jet suppression at LHCenergies has been investigated in detail (e. g. Ref. [20]),and it has been shown how strong final-state interactionscause high-pT jets to lose energy to the plasma.

In this work, however, I consider quarkonia as anotherindicator for the properties of the quark-gluon plasmasuch as its initial central temperature Ti. Quarkonia arebound states of heavy quark-antiquark pairs that can beformed in initial hard partonic collisions. In the originalprediction for a suppression of the J/ψ yields in the pres-ence of a QGP, only the medium-effect on the real partof the quark-antiquark potential was considered [21]. Ithas later been realized that due to the presence of the hotmedium, the potential has an imaginary part [22]. Opti-cal potentials had also been used in the theory of nuclear

reactions to account for channels that are not treated ex-plicitly. In case of quarkonia, the imaginary part causestheir dissociation, in addition to melting of the quarkoniastates because the real potential is screened. It is pos-sible to treat quarkonia dissociation by thermal gluonsseparately from the imaginary part [23].

In case of charmonium at LHC energies, statistical re-combination of charm and anticharm quarks turns out tobe important, but it is not possible to separate the pro-cess from the dissociation in the QGP. One may there-fore concentrate on the heavier bottomonium system,where recombination is much less pronounced. Throughdetailed investigations of the transverse-momentum-and centrality-dependent suppression of the spin-tripletΥ(1S, 2S) states in Pb-Pb collisions at LHC energies andcomparisons with CMS data, we can deduce QGP prop-erties such as the initial central temperature Ti and studyits dependence on the center-of-mass energy.

In asymmetric collisions such as p-Pb, the situation isquite different from symmetric systems, because most ofthe system remains cold due to the much smaller over-lap. Cold nuclear matter (CNM) effects therefore providea certain understanding of the measured quarkonia mod-ifications, but a complete agreement with the availabledata remains impossible – unless one also considers thehot QGP zone that is still produced, even though it isinitially considerably less extended as compared to sym-metric systems. Indeed the bottomonia dissociation inthe hot QGP provides a clue for the interpretation of thedata in asymmetric collisions as well.

The paper follows the above-mentioned series of top-ics: In Section 2, the nonlinear diffusion equation forgluons and quarks is solved explicitly to account for thefast local thermalization at the beginning of relativisticheavy-ion collisions. Stopping is considered in Section 3,hadron production and limiting fragmentation in Section4, bottomonia modification in the medium in Section 5.The conclusions are drawn in Section 6.

II. FAST THERMALIZATION OF GLUONSAND QUARKS: AN ANALYTIC NONLINEAR

MODEL

For a given initial nonequilibrium gluon distributionat t = 0, solutions of the nonlinear boson diffusion equa-tion describe the time-dependent equilibration towardsthe thermal distribution with the local temperature T .In Ref. [4] such solutions were calculated with the freeGreen’s function. Whereas this accounts for local ther-malization in the ultraviolet (UV) with the correspondingequilibration time τeq, in the infrared (IR) the popula-tions decrease due to diffusion into the negative-energyregion. To avoid such an unphysical behaviour, one hasto consider the boundary condition at the singularity|p| = p = ε = µ with the chemical potential µ < 0,and the corresponding bounded Green’s function in thesolution of the NBDE. With this Green’s function, gluon

3

populations indeed attain the Bose–Einstein limit alsoin the infrared for nonequilibrium initial conditions thatinclude the singularity.

The nonlinear model and the solution of the combinedinitial- and boundary-value problem are first briefly re-viewed. Subsequently, the thermalization problem issolved for a schematic initial gluon distribution that char-acterizes the relativistic collision at t = 0. Adding theboundary condition at the singularity, n(ε = µ < 0, t)→∞, the time-dependent partition function that includesinitial and boundary conditions is obtained using ana-lytic expressions for both, the bound Green’s function,and the function that contains an integral over the ini-tial conditions. The resulting occupation-number distri-bution function n(ε, t) is calculated, and it is shown toapproach the equilibrium distribution both in the UVand in the IR.

Before we proceed to the nonlinear model, it is usefulto consider a linear time-dependent transition from theinitial distribution

ni(ε) = θ(1− ε/Qs) θ(ε) (1)

to the thermal distribution

neq(ε) =1

e(ε−µ)/T − 1(2)

with the chemical potential µ < 0 in a finite boson systemin the relaxation-time approximation (RTA), ∂ nrel/∂t =(neq − nrel)/τeq, with solution

nrel(ε, t) = ni(ε) e−t/τeq + neq(ε)(1− e−t/τeq) . (3)

Time-dependent RTA results for gluons are shown inFig. 1 for t = 0.02, 0.08, 0.15, 0.3, and 0.6 fm/c. The ther-mal distribution with initial central temperature T = 513MeV as inferred in Ref. [3] for central Pb-Pb collisions at√sNN = 5.02 TeV is approached linearly, the disconti-

nuities at ε = Qs persist.For a more realistic account of thermalization, one

needs to consider the inherent nonlinearity of the sys-tem. I had derived a nonlinear partial differential equa-tion for the single-particle occupation probability distri-butions n ≡ nth(ε, t) from the bosonic/fermionic Boltz-mann collision term in Ref. [4]. The transport coefficientsin this nonlinear boson diffusion equation (NBDE) de-pend on energy, time, and the second moment of the in-teraction. They incorporate the complicated many-bodyphysics. The drift term v(ε, t) is responsible for dissi-pative effects, the diffusion term D(ε, t) for diffusion ofparticles in the energy space. For the simplified case ofenergy-independent transport coefficients, the nonlineardiffusion equation for the occupation-number distribu-tion n±(ε, t) becomes

∂n±

∂t= −v ∂

∂ε

[n (1± n)

]+D

∂2n

∂ε2(4)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

ε (GeV)

nrel(ε,t)

Figure 1. Local thermalization of gluons in the linearrelaxation-time approximation (RTA) for µ < 0. Startingfrom schematic initial conditions Eq. (13) in the cold systemat t = 0 (box distribution with cut at ε = Qs = 1 GeV),a Bose–Einstein equilibrium distribution with temperatureT ' 513 MeV (solid curve) is approached. Time-dependentsingle-particle occupation-number distribution functions areshown at t = 0.02, 0.08, 0.15, 0.3, and 0.6 fm/c (decreasingdash lenghts). The lower dotted curve is Boltzmann’s distri-bution.

where the + sign represents bosons, and the − signfermions. A stationary solution is given by the ther-mal distribution Eq. (2) for bosons and, correspondingly,the Fermi–Dirac distribution for fermions (quarks). Inspite of its simple structure, the nonlinear diffusion equa-tion with constant transport coefficients preserves the es-sential features of Bose–Einstein/Fermi–Dirac statisticswhich are contained in the quantum Boltzmann equa-tion.

The transport equation can be solved exactly for agiven initial condition n±i (ε) using the nonlinear trans-formation outlined in Ref. [4]. For gluons, the nonlin-ear boson diffusion equation (NBDE) is more difficult tosolve analytically due to the singularity at the chemicalpotential ε = µ < 0, and the need to consider the bound-ary conditions at the singularity. For fermions, there isno corresponding singularity, such that the exact solutionof the nonlinear problem can be obtained with the freeGreen’s function as performed in Refs. [4, 12]. For gluons,the bounded solution n+(ε, t) [24] is briefly reviewed. Itcan be written as

n+(ε, t) = −Dv

∂

∂εlnZ+(ε, t)− 1

2= −D

v

1

Z+

∂Z+

∂ε− 1

2(5)

with the time-dependent partition function Z+(ε, t) ≡Z(ε, t) obeying a linear diffusion equation

∂

∂tZ(ε, t) = D

∂2

∂ε2Z(ε, t) . (6)

4

In the absence of boundary conditions, the free partitionfunction becomes

Zfree(ε, t) = a(t)

∫ +∞

−∞Gfree(ε, x, t)F (x) dx , (7)

The energy-independent prefactor a(t) in the partitionfunction drops out when taking the logarithmic deriva-tive in the calculation of the occupation-number distribu-tion. The function F (x) with the integral over the initialconditions covers the full energy region −∞ < x < ∞.Due to the occurence of the singularity, however, oneeventually will have to consider boundary conditions atε = µ < 0.

For a solution without boundary conditions as inRefs. [4], Green’s function Gfree(ε, x, t) of Eq. (6) is a sin-gle Gaussian

Gfree(ε, x, t) = exp(− (ε− x)2

4Dt

), (8)

but it becomes more involved once boundary conditionsare considered. The function F (x) depends on the initialoccupation-number distribution ni according to

F (x) = exp[− 1

2D

(vx+ 2v

∫ x

0

ni(y) dy)]. (9)

As discussed in Ref. [25], the definite integral can bereplaced w.l.o.g. by the indefinite integral Ai(x) over theinitial distribution with ∂xAi(x) = ni(y), resulting in

F (x) = exp[− 1

2D

(vx+ 2vAi(x)

)]. (10)

For any given initial distribution ni, one can now com-pute the partition function and the overall solution forthe occupation-number distribution function Eq. (5) an-alytically. The solution technique has been developed inRefs. [25, 26] for the case of a cold bosonic atom gas thatundergoes evaporative cooling. Here, and in Ref. [24] fordifferent initial conditions, the approach is carried overto equilibrating gluons at relativistic energies. To solvethe problem exactly, the chemical potential is treated asa fixed parameter. With limε↓µ n(ε, t) = ∞ ∀ t, one ob-tains Z(µ, t) = 0, and the energy range is restricted toε ≥ µ. This requires a new Green’s function that equalszero at ε = µ ∀ t. It can be written as

G(ε, x, t) = Gfree(ε− µ, x, t)−Gfree(ε− µ,−x, t) , (11)

and the partition function with this boundary conditionbecomes

Z(ε, t) =

∫ ∞0

G(ε, x, t)F (x+ µ) dx . (12)

The function F remains unaltered with respect toEq. (10), except for a shift of its argument by the chemicalpotential. With a given initial nonequilibrium distribu-tion ni, the NBDE can now be solved including bound-ary conditions at the singularity. The solution is givenby Eq. (5).

A. Thermalization of Gluons

For massless gluons at the onset of a relativistichadronic collision, an initial-momentum distributionni(|p|) ≡ ni(p) = ni (ε) has been proposed by Mueller[27] based on Ref. [28]. It accounts, in particular, forthe situation at the start of a relativistic heavy-ion col-lision [9]. It amounts to assuming that all gluons up toa limiting momentum Qs are freed on a short time scaleτ0 ∼ Q−1

s , whereas all gluons beyond Qs are not freed.Thus the initial gluon-mode occupation in a volume V istaken to be a constant up to Qs, as in Eq. (1) that was al-ready used before in the relaxation-time approximation.Typical gluon saturation momenta for a longitudinal mo-mentum fraction carried by the gluon x ' 0.01 turn outto be of the order Qs ' 1 GeV [29], which is chosen forthe present model investigation.

Results for the gluon thermalization from ni(ε) toneq(ε) according to Eq. (4) have been calculated in Ref. [4]for the free case, without considering boundary condi-tions at the singularity. As a consequence, diffusion intothe negative-energy region occured, depleting the occu-pation in the infrared such that the asymptotic distribu-tion differed from Bose–Einstein.

As a remedy, one has to extend the energy scale inEq. (1) to µ ≤ ε < ∞, and include the boundary condi-tions at the singularity ε = µ < 0. This will cause thetime-dependent solutions of the NBDE to properly ap-proach the thermal Bose–Einstein distribution over thefull energy scale as t→∞. The initial condition is thusmodified to include a δ-function singularity at ε = µ < 0according to

ni(ε) = θ(1− ε/Qs) θ(ε− µ) + δ(ε− µ) . (13)

The δ-function singularity in the initial conditions of theNBDE has an analogous role as the singularity that canbe added to the Boltzmann equation in order to act as aseed condensate [30] since the time evolution of the so-lutions without singularity does not lead to condensateformation. In Ref. [24], another choice of the initial con-ditions had been explored, with a thermal distribution inthe negative-energy region that also has a singularity atε = µ. Although the results differ in detail, the overallrepresentation of thermalization is similar to the presentresults.

The time-dependent partition function with the aboveinitial condition can now be calculated using the boundGreen’s function Eq. (11), and the function F (x) fromEq. (10). The latter contains an indefinite integral overthe initial condition Eq. (13) that can be carried out toobtain (with x → x + µ in the argument of F (x) asrequired by the boundary conditions)

F (x) = exp

[−v(x+ µ)

2D

]× exp[−(v/D) θ(x)((µ−Qs)θ(µ−Qs)+ (Qs − x− µ) θ(x+ µ−Qs) + x+ 1)] . (14)

5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

50

100

150

200

250

x (GeV)

F(x)

Figure 2. The function F (x) of Eq. (10) (solid curve) withx → x + µ as required by the boundary conditions, and asingularity at the origin. F (x) contains the integral over aninitial nonequilibrium gluon distribution ni(x) according toEq. (14). The parameters are given in the text.

The function F (x) is plotted in Fig. 2. Due to the singu-larity in its argument, F (x) has a discontinuity at x = 0.It is continuous, but not differentiable at x = Qs − µ.Both properties are essential to account for the equili-bration near the singularity, and in the UV region. TheGreen’s function of Eq. (11) that includes the IR bound-ary condition can explicitly be written as

G(ε, x, t) = exp

[−(ε− µ− x)2

4Dt

]−exp

[−(ε− µ+ x)2

4Dt

].

(15)With F (x) and G(ε, x, t), the partition function Z(ε, t) ofEq. (12) and its derivative ∂Z/∂ε can now be calculated,as well as the occupation-number distribution n(ε, t) fromEq. (5). The full calculation may in principle be carriedout analytically. In the case of initial conditions that areappropriate for evaporative cooling of atomic Bose gasesat very low energy, we have performed such an exact cal-culation including the boundary conditions at the singu-larity in Ref. [25]. Here I compute the partition functionand its derivative using the NIntegrate and Derivativeroutines of Mathematica.

B. Discussion of the Solutions for Gluons

The bosonic equilibration time τeq is taken as τeq =4D/(9v2) ' 0.1 fm/c. This expression has been deter-mined in Ref. [4] for a θ-function initial distribution inthe UV. The approach to equilibrium provided by thesolutions of the NBDE for the gluon distribution func-tions is shown in Fig. 3 at t = 6 × 10−5, 6 × 10−4,6 × 10−3, 4 × 10−2, 0.12, and 0.36 fm/c, with decreas-ing dash lenghts. The steep cutoff in the UV at ε = Qs

is smeared out at short times – this was the case al-ready in the free solution without boundary conditions[4]. The diffusion coefficient is D = 1.17 GeV2c/fm, the

drift coefficient v = −2.28 GeVc/fm. Correspondingly,the equilibrium temperature in this model calculation isT = −D/v ' 513 MeV = 0.513 GeV, as expected for theinitial central temperature in a Pb-Pb collision at theLHC energy of

√sNN = 5 TeV [3]. Solutions for related,

but different initial conditions (a thermal distribution inthe energy region µ ≤ ε ≤ 0) have been discussed inRef. [24].

The assumption of a constant negative chemical po-tential µ < 0 used in this work is, of course, an idealiza-tion that facilitates analytical solutions of the nonlinearproblem. Here, the value of µ is calculated from particle-number conservation

Ni = Nf =

∫ ∞0

neq(ε) g(ε) dε (16)

with the density of states g(ε). For constant density ofstates and the above parameter values (T = 513 MeV),the result is µ = −0.08 GeV. For the more realistic den-sity of states g(ε) ∝ ε2 that is valid for a zero-mass rela-tivistic system of gluons, the particle number is then alsoapproximately conserved.

In general, particle-number conservation is strictly ful-filled e. g. for atomic Bose gases at much lower energy,but not for gluons in high-energy collisions. Driven byparticle-number conservation, cold bosonic atoms canmove into the condensed phase, thus diminishing thenumber of particles in the thermal cloud. The chemi-cal potential in the equilibrium solution of the NBDEthen becomes time dependent, as has been discussed inRef. [25], albeit without a full quantum treatment of thecondensed phase. It would become zero only in the limitof an inifinite number of particles in the condensed phase.Instead, it approaches a small but finite negative valuefor a finite number of particles.

In case of relativistic heavy-ion collisions, however, glu-ons can be created and destroyed. It is therefore unlikelythat a condensed phase is actually formed, as had beenproposed in model investigations where only soft elastic,number-conserving gluon collisions were considered [9].Gluon condensate formation in relativistic collisions isessentially prevented by number-changing inelastic pro-cesses that correspond to splitting and merging of gluons,although a transient condensate formation is still beingdebated [10].

Hence, since inelastic collisions cannot be neglected,the gluon equilibrium distribution is expected to havea nearly vanishing, but still slightly negative, chemi-cal potential which should be approached by the time-dependent solutions of the NBDE. It would therefore beof interest to repeat the present calculation for a time-dependent chemical potential, with µ(t)→ 0 for t→∞,as was done in Ref. [25] for the case of cold atoms. Thisrequires, however, numerical work that goes substantiallybeyond the present analytic approach.

6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

ε (GeV)

n(ε,t)

Figure 3. Local thermalization of gluons as represented by time-dependent solutions of the nonlinear boson diffusion equation(NBDE) for µ < 0. Starting from schematic initial conditions Eq. (13) in the cold system at t = 0 (box distribution with cut atε = Qs = 1 GeV), a Bose–Einstein equilibrium distribution with temperature T = 513 MeV (lower solid curve) is approached.Time-dependent single-particle occupation-number distribution functions are shown at t = 6× 10−4, 6× 10−3, 0.04, 0.12, and0.36 fm/c (decreasing dash lenghts). The lower dotted curve is Boltzmann’s distribution.

C. Discussion of the Solutions for Quarks

The local thermalization of valence quarks towardsthe Fermi–Dirac equilibrium distribution is easier tocalculate in the nonlinear model because no singular-ity occurs for fermions. Hence, the free solutions canbe used as was already discussed in Ref. [4] and inmore detail in Ref. [12]. Hadron production in theearly thermalization phase was implicitly considered, be-cause the negative-energy region corresponds to particle-antiparticle production. A typical result for the fermionictime-dependent occupation-number distribution as func-tion of the transverse energy taken from Ref. [31] is shownin Fig. 4. Local thermalization for quarks occurs moreslowly as for gluons due to Pauli’s principle, and alsobecause of the larger color factor for gluons.

The bounded solutions of the NBDE and the corre-sponding nonlinear fermion diffusion equation are, in par-ticular, tailored to local thermalization processes that oc-cur in relativistic heavy-ion collisions at energies reachedat RHIC and LHC. In the present example, they areapplied to the local equilibration of quarks and gluonsin central Pb-Pb collisions at a center-of-mass energyof 5 TeV per nucleon pair, leading to rapid thermaliza-tion with a local temperature of T ' 500 MeV. Sincethe thermalization occurs very fast – before anisotropicexpansion fully sets in –, the analytic solution of theproblem in 1+1 dimensions appears permissible. Thehot system will subsequently expand anisotropically andcool rapidly, as is often modeled successfully by relativis-tic hydrodynamics [1], until hadronization is reached atT ' 160 MeV. Further refinements of the thermaliza-tion model such as time-dependent transport coefficientsare conceivable, but are unlikely to allow for analyticsolutions. A microscopic calculation of the transport co-efficients with an investigation of their dependencies onenergy and time would be very valuable. Extensions of

-1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

T (GeV )

n-(εT,t)

Figure 4. Occupation-number distribution n−(εT, t) of arelativistic fermion system (valence quarks) as function ofthe transverse energy εT including antiparticle production(εT < 0). It is evaluated analytically from Eq. (4) at differ-ent times for the initial distribution n−

i (εT) = θ (1 − εT/µ).The parameters are µ = 1 GeV, T = −D/v = 500 MeV,τ−eq = 4D/v2 = 0.9 fm/c. The distribution n−(εT, t) is dis-played at t = 0.003, 0.015, 0.06, 0.15, and 0.6 fm/c (orderedby decreasing dash length towards the solid equilibrium distri-bution). The upper dot-dashed curve is Boltzmann’s distribu-tion. The negative-energy region corresponds to antiparticleproduction. From Ref. [31].

the NBDE itself to higher dimensions in order to accountfor possible anisotropies should also be investigated.

III. STOPPING: NET-PROTONDISTRIBUTIONS

The incoming baryons with energies available at SPS,RHIC or LHC are being stopped on an equally short time

7

Figure 5. Schematic representation of the three-source modelfor relativistic heavy-ion collisions at RHIC and LHC ener-gies in the center-of-mass system: Following the collision andslowing down (stopping) of the two Lorentz-contracted slabs(blue), the fireball region (center, yellow) expands anisotropi-cally in longitudinal and transverse direction. At midrapidity,it represents the main source of particle production. The twofragmentation sources (red) contribute to particle production,albeit mostly in the forward and backward rapidity regions.In stopping, these are the only sources. From Ref. [32].

scale as the local equilibration occurs: In the course ofthe collision shown schematically in Fig. 5, the system isbeing slowed down, essentially through collisions of theincoming valence quarks with soft gluons in the respec-tive other nucleus. Various models to account for thisprocess and its energy dependence have been developed,in particular in Refs. [29, 33] and related works which arerelying on the appropriate parton distribution functions(PDFs). They yield agreement with the available net-proton (proton minus antiproton) stopping data. Differ-ent from the nonequilibrium-statistical approach to ini-tial thermalization, such models do not consider a timedependence. However, by using the rapidity distributioncalculated from the PDFs, and the initial valence-quarkdistribution, one can, in addition, account for the timedevelopment from the initial to the final distribution withan appropriate fluctuation–dissipation relation.

The fragmentation peaks in stopping occur mainly dueto the interaction of valence quarks with soft gluons inthe respective other nucleus. For net protons (protonsminus antiprotons), their positions in rapidity space y =0.5 ln[(E||+p||)/(E||−p||)] with mp = mp can be obtainedfrom Ref. [29], and references therein, as

dNp−p

dy=

C

(2π)2

∫d2pT

p2T

x1qv(x1, pT)fg(x2, pT) . (17)

This expression accounts for the peak in the forward re-gion, and there is a corresponding one with y → −y forthe backward peak. The longitudinal momentum frac-tion of the valence quark v that experiences stopping is

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 50

20

40

60

80

100

120

y

dN/dy

Figure 6. Rapidity distributions of net protons in central Pb-Pb collisions at SPS energies of

√sNN = 17.3 GeV compared

with NA49 data [34]. Red curves are the initial distributionsbroadened by the Fermi momentum, the final distribution isfrom a QCD-inspired model [29] with the saturation-scale ex-ponent λ = 0.2 and Q2

0 = 0.09 GeV2 (see text). It agrees withthe data, and corresponds to the distribution at the interac-tion time t = τint in a time-dependent formulation. The sixintermediate solid curves at t/τint = 0.01, 0.02, 0.05, 0.1, 0.2and 0.5 account for the time dependence in a nonequilibrium-statistical relativistic theory. From Hoelck and Wolschin [35].

x1 = pT/√s exp(y), the one for the soft gluon g in the

target is x2 = pT/√s exp(−y). The distribution function

of the valence quarks is qv(x1, pT), the one of the gluons isfg(x2, pT). The latter represents the Fourier transform ofthe forward dipole scattering amplitude N(x2, rT) for aquark dipole of transverse size rT [33]. To account for thecorrect normalization in net-proton or net-baryon distri-butions, the normalization constant C is adjusted suchthat the integral of Eq. (17) agrees with the total numberof participant protons or baryons.

Fragmentation peaks are then found to occur at y =±ypeak in rapidity space. At suffiently high energy –in particular, at LHC energies – these positions becomesensitive to the gluon saturation scale

Q2s = A1/3Q2

0x−λ . (18)

Here, A is the mass number, Q0 the momentum scale,x < 1 the momentum fraction carried by the gluon, andλ the saturation-scale exponent.

Rapidity distributions dNp−p/dy at SPS and RHICenergies had been calculated within our QCD-inspiredmodel of Ref. [29] for several values of the gluon satura-tion scale and found to agree with net-proton data fromSPS and RHIC. We are presently incorporating theseresults into a time-dependent nonequilibrium-statisticalrelativistic theory [35]. A typical outcome of our ap-proach for central Pb-Pb at SPS energies of

√sNN = 17.3

GeV is shown in Figure 6 where it is compared with NA49data [34] in rapidity space. The initial distributions areshown as red curves, with the initial broadening due tothe Fermi motion. The final distribution agrees with theone from our QCD-inspired model [29] with a saturation-scale exponent λ = 0.2 and Q2

0 = 0.09 GeV2, translatinginto a gluon saturation momentum of Qs ' 0.55 GeV/cat x = 10−4. At the interaction time t = τint, the time-

8

dependent distribution agrees with the data. The six in-termediate solid curves at t/τint = 0.01, 0.02, 0.05, 0.1, 0.2and 0.5 show the time dependence in our nonequilibrium-statistical relativistic theory [35].

A larger gluon saturation momentum Qs was foundto produce more stopping, as does a larger mass num-ber A [29]. In the context of an investigation of particleproduction, the agreement between the calculated stop-ping distributions and the data will be taken as evidencefor the importance of fragmentation contributions also incharged-hadron production.

In Ref. [36], Mehtar-Tani and I found that the frag-mentation peak positions in stopping depend in a largec.m. energy range 6.3 GeV ≤ √sNN ≤ 200 GeV linearlyon the beam rapidity ybeam and the saturation-scale ex-ponent λ according to

ypeak =1

1 + λ(ybeam − lnA1/6) + const . (19)

At the current LHC energy of 5.02 TeV Pb-Pb corre-sponding to ybeam = ± ln(

√sNN/mp) = ± 8.586 and

with a gluon saturation-scale exponent λ ∼ 0.2 one there-fore expects ypeak ' ± 6. Due to the lack of a suitableforward spectrometer at LHC, the rapidity region of thepeaks will thus not be accessible for identified protonsin the coming years at LHC energies. Nevertheless, thepartonic processes that mediate stopping also contributeto hadron production at LHC energies and hence, oneexpects fragmentation events in particle production. Ex-perience with stopping versus hadron production at SPSand RHIC energies [13, 14] has shown that the fragmen-tation peaks in particle production occur consistently atsomewhat smaller absolute rapidities than the ones instopping.

In net-baryon (proton) distributions charged baryonsproduced from the gluonic source cancel out because par-ticles and antiparticles are generated in equal amounts.In charged-hadron production, however, this is not thecase. Instead, three sources contribute provided theenergy is sufficiently high,

√sNN > 20 GeV. It was

found in Ref. [14] that the dependence of their parti-cle content on c.m. energy differs: The fragmentationsources contain Nqg

ch ∝ ln(sNN/s0) charged hadrons,whereas the midrapidity-centered source that arises es-sentially from the interaction of low-x gluons containsNgg

ch ∝ ln3(sNN/s0) charged hadrons. Due to the strong

∝ ln3 dependence, it becomes more important than thefragmentation sources at LHC energies [37].

With the three sources, the total rapidity distributionfor produced charged hadrons becomes

dN totch (y, t = τint)

dy= Nqg,1

ch R1(y, τint)

+Ngq,2ch R2(y, τint) +Ngg

chRgg(y, τint) . (20)

The fragmentation distributions R1,2(y, t) and gluonicdistributions Rgg(y, t) can be calculated in a time-dependent phenomenological model such as the relativis-tic diffusion model [13], or in microscopic theories. The

strong interaction ceases to act at the interaction time(≡ freezeout-time) t = τint and theoretical distributionsmay be compared to data in a χ2-optimization.

A transparent phenomenological model to calculateand predict the distribution functions of produced par-ticles is the relativistic diffusion model [13, 38]. In theRDM, the initial distribution functions are evolved upto τint/τy with the rapidity relaxation time τy using theanalytical moments equations. The mean values 〈y1,2〉of the fragmentation distributions that are related toτint/τy can be determined from the data. The detailsof the model calculations are given in the correspondingRefs. [13, 14, 37].

IV. CHARGED-HADRON PRODUCTION

A. Transverse-momentum distributions

As discussed frequently in experimental papers [39]and also in Ref. [14], the transverse-momentum distribu-tions of produced charged hadrons in relativistic heavy-ion collisions show an exponential behaviour in the ther-mal regime as accounted for by the Maxwell–Juttner dis-tribution [40] (c ≡ 1)

f(pT) =1

4πm2TK2(m/T )exp

[−γ(pT)m

T

](21)

with the modified Bessel function of the second kindK2(m/T ), the Lorentz-factor

γ(pT) =√

1 + (pT/m)2, (22)

freeze-out temperature T and hadron mass m. BeyondpT ' 4 GeV/c, however, a transition to a power-law(straight lines in a log-log plot) occurs. It is attributedmostly to the recombination of soft partons, and frag-mentation of hard partons. In addition to detailed theo-retical approaches, this transition can be modelled phe-nomenologically using distribution functions of the QCD-inspired form (see references in Wilk and Wong [41]) usedby Hagedorn [42] for high-energy pp and pp collisions

Ed3σ

dp3= C (1 + pT/p0)−n (23)

with a normalization constant C and parameters p0, n.This expression describes the transition from exponentialfor pT → 0 as in the Juttner distribution Eq. (21) (withp0 = nT and pT → mT), to power-law behaviour (∝(pT/p0)−n for pT →∞) .

Using Eq. (23), Fig. 7 shows pT-distributions of pro-duced charged hadrons at four centralities in 5.02 TeVPb-Pb compared with ALICE data [43] (peripheral spec-tra are scaled for better visibility; statistical and system-atic error bars are smaller than the symbol size). Thedata are well represented through many orders of mag-nitude with a power index n = 8.2 and p0 = 3 GeV/c

9

0.1 1 10

1x10-141x10-121x10-101x10-81x10-61x10-4

0.01

1

100

1x104

Import

Figure 7. Transverse momentum distributions of producedcharged hadrons in

√sNN = 5.02 TeV Pb-Pb collisions calcu-

lated from Hagedorn’s formula (see text) compared with AL-ICE data [43] for 0–5%, 20-30% (×10−3), 50–60% (×10−6)and 70–80% (×10−8) centralities (top to bottom). The dot-dashed curve has power index n = 6.2, the dashed curven = 10.2, solid curves are for n = 8.2. Error bars are smallerthan the symbol size.

(Fig. 7), but at high pT deviations occur which are at-tributed to hard processes that require a pQCD treat-ment. This corresponds to the occurence of a minimumin the nuclear modification factor for produced chargedhadrons as function of pT found already at

√sNN = 2.76

TeV [44], and confirmed at 5.02 TeV [43].There is presently no theoretical derivation for the

value of the power index n that is needed to reproducethe experimental pT-distributions in relativistic heavy-ion collisions. It is therefore not obvious from the presentanalysis which fraction of low-pT particles could be due tononequilibrium processes that differ from thermal emis-sion out of a single expanding fireball. In particular, onecan not distinguish particles emitted from the fireballand those arising from the fragmentation sources at lowpT. Hence, the analysis of transverse momentum distri-butions in terms of Eq. (23) is presently only suitable todistinguish high-pT hard events from the bulk of (thermaland nonequilibrium) charged-hadron emission.

B. Pseudorapidity distributions

The distinction of particles emitted from the fireballand those from the fragmentation sources is more trans-parent in rapidity or pseudorapidity distributions of pro-duced charged hadrons. The existence of the fragmenta-tion sources is evident from the measurements of stoppingin heavy-ion collisions as discussed in Section 3. Pseudo-

rapidity distributions dNch/dη with η = − ln [tan(θ/2)]depend only on the scattering angle θ. Particle identifi-cation is not needed here and hence, they are much easierto obtain at large η-values (small scattering angles) com-pared to rapidity distributions.

The pseudorapidity distributions dNch/dη for pro-duced charged hadrons in relativistic heavy-ion collisionsemerge from a superposition of the fragmentation sourcesand a midrapidity source. According to the discussion inRef. [37] and at the end of Section 3, the particle con-tent of the low-x gluon source rises rapidly accordingto Ngg ∝ ln3(sNN/s0). To convert rapidity distribu-tions dNch/dy to pseudorapidity distributions dNch/dη,the corresponding Jacobian is required

dN

dη=

dN

dy

dy

dη= J(η,m/pT)

dN

dy, (24)

J(η,m/pT) = cosh(η)·[1 + (m/pT)2 + sinh2(η)]−1/2 .(25)

The hadron mass is m and the transverse momentumpT. Since the transformation depends on the squaredratio (m/pT)2 of mass and transverse momentum of theproduced particles, its effect increases with the mass ofthe particles and is most pronounced at small transversemomenta. In Ref. [45], we have determined the JacobianJ0 at η = y = 0 in central 2.76 TeV Pb-Pb collisions foridentified π−,K−, and antiprotons from the experimen-tal values dN

dη |exp and dNdy |exp as J0 = 0.856. We can solve

Eq. (25) for pT ≡ 〈peffT 〉 to obtain

〈peffT 〉 =

〈m〉 J0√1− J2

0

. (26)

The mean mass 〈m〉 can be calculated from the abun-dancies of pions, kaons, and antiprotons. Using J0, theJacobian can be written independently from the valuesof 〈m〉 and 〈peff

T 〉 as

J (η, J0) =cosh(η)√

1 +1−J2

0

J20

+ sinh2(η). (27)

The result for central 2.76 TeV Pb-Pb col-lisions was found in Ref. [45] to be J(η) =cosh(η)[1.365 + sinh2(η)]−1/2. The Jacobian affectsthe pseudorapidity distributions mostly near midrapid-ity, where it generates a dip as is obvious from Figure 8:A prediction in the RDM with linear drift from Ref. [14]is compared with ALICE data for central Pb-Pb at 5.02TeV [46] (dashed upper curve), and a χ2-optimizationwithin the five parameter RDM is carried out, uppersolid curve. It differs only slightly from the prediction.

The nonequilibrium evolution of all three partial dis-tribution functions Rk(y, t) (k = 1, 2, gg) towards the

10

thermodynamic equilibrium (Maxwell–Juttner) distribu-tion for t→∞

Ed3N

dp3

∣∣∣eq∝ E exp

(−E/T

)= mT cosh (y) exp

(−mT cosh(y)/T

)(28)

is accounted for in the Relativistic Diffusion Model[13, 14, 38, 47] through solutions of the Fokker-Planckequation

∂∂tRk(y, t) =

− ∂∂y

[Jk(y, t)Rk(y, t)

]+ ∂2

∂y2

[Dk(y, t)Rk(y, t)

]. (29)

The drift functions Jk(y, t) and diffusion functionsDk(y, t) depend on rapidity and time. However, if the dif-fusion coefficients are taken as constantsDk, and the driftfunctions assumed to be linearly dependent on the rapid-ity variable y, the FPE acquires the Ornstein-Uhlenbeckform [48] which can be solved analytically in rapidityspace [38]. In this case it is easy to show that fort→∞ all three subdistributions approach a single Gaus-sian in y space which is centered at midrapidity y = 0for symmetric systems, or at the appropriate equilibriumvalue y = yeq for asymmetric systems [13, 14, 37]. Forstopping, only the two fragmentation distributions con-tribute, approaching the thermal equilibrium distribu-tion for t → ∞, as discussed in the previous section. Ifthe system reaches a stationary distribution that differsfrom the thermal one as discussed in Section 3 for thecase of stopping calculated in a QCD-based model, theunderlying fluctuation-dissipation relation becomes morecomplicated [35].

To derive the above FPE (29) in the context of rela-tivistic heavy-ion collisions, one can make use of a theoryfor a special class of non-markovian processes in space-time discussed in Refs. [49, 50], which are equivalent torelativistic Markov processes in phase space (RMPP).These processes give rise to a generalised FPE which issuitable for describing relativistic diffusive particle dy-namics. Our Eq. (29) is conceptually a special case ofsuch a RMPP formalism for charged-hadron productionin rapidity space. An application of RMPP to baryonstopping will be shown in Ref. [35].

The t → ∞ limit of the FPE solution for constantdiffusion and linear drift is found to deviate slightlyfrom the Maxwell–Juttner distribution. The discrepan-cies are small and become visible only for sufficientlylarge times. To ensure that the asymptotic solution yieldsthe Maxwell–Juttner distribution Eq. (28), a RDM withthe sinh-drift is required

Jk(y, t) = −Ak sinh(y) , (30)

as was discussed in Refs. [51, 52]. The correspondingfluctuation-dissipation relation (FDR) that connects driftand diffusion becomes [52]

Ak = mTDk/T . (31)

If the asymptotic distribution is not Maxwell–Juttner,but – as in the case of stopping, where it may be providedby a QCD-based distribution from a calculation as per-formed in Ref. [29] –, a different form of the fluctuation-dissipation relation will result, as will be discussed inRef. [35].

The strength of the drift force in the fragmentationsources k = 1, 2 depends on the distance in y-space fromthe beam rapidity, which enters through the initial condi-tions. With Eq. (28), the rapidity distribution at thermalequilibrium can then be derived [53] as

dNeq

dy= C

(m2

TT +2mTT

2

cosh y+

2T 3

cosh2 y

)

× exp

(−mT cosh y

T

), (32)

where C is proportional to the overall number of pro-duced charged hadrons N tot

ch , or – in case of stopping –to the number of net baryons (protons) in the where C isproportional to the overall number of produced chargedhadrons N tot

ch , or – in case of stopping – to the numberof net baryons (protons) in the respective centrality bin.Since the actual distribution functions remain far fromthermal equilibrium, the total particle number is evalu-ated based on the nonequilibrium solutions of the FPE,which are adjusted to the data in χ2-optimizations.

In particular, one can determine the drift amplitudesAk from the position of the fragmentation peaks as in-ferred from the data, and then calculate theoretical diffu-sion coefficients as Dk = AkT/mT. Since the fireball andboth fragmentation sources also expand collectively, theactual distribution functions will, however, be broaderthan what is obtained from Eq. (31). To account for thisbroadening through collective expansion, we use diffusioncoefficients (or widths of the partial distributions) thatare adapted to the data in both stopping and particleproduction. From the integral of the overall distributionfunction the total particle number can be obtained. Incase of the FPE with sinh-drift, the FPE must be solvednumerically as described in Ref. [52].

Results for central collisions of symmetric systems inthe three-source RDM with linear drift are summarizedin Figure 8. The dependence of the pseudorapidity distri-butions on c.m. energy in central Au-Au collisions at 19.6GeV, 130 GeV, and 200 GeV RHIC energies as well as incentral Pb-Pb at 2.76 TeV and 5.02 TeV LHC energiesare displayed. In addition to RDM calculations with pa-rameters for the lower energies from [13] compared withdata from Refs. [16, 44, 54], a prediction for 5.02 TeVPb-Pb (dashed upper curve) from Ref. [14] is comparedwith recent ALICE data at 0-5 % centrality [46]. A fit tothe data within the five-parameter RDM is also displayed(solid curve).

Only the fragmentation sources contribute at the low-est RHIC energy of 19.6 GeV that is shown here – whichis comparable to the highest SPS energy of 17.3 GeV –

11

-10 -5 0 5 10

η

0

500

1000

1500

2000

dN/dη

Figure 8. The RDM pseudorapidity distribution functions forcharged hadrons in central Au-Au (RHIC) and Pb-Pb (LHC)collisions at c.m. energies of 19.6 GeV, 130 GeV, 200 GeV,2.76 TeV, and 5.02 TeV shown here are optimized in χ2-fitswith respect to the PHOBOS [16, 54] (bottom) and ALICE[46, 56] (top) data, with parameters from Refs. [13, 14]. Theupper dashed distribution function at 5.02 TeV is a predictionfrom Ref. [14] within the relativistic diffusion model. The 5.02TeV midrapidity data point is from Ref. [57], the 5.02 TeVdata from Ref. [46]. The 19.6 GeV distribution has only twosources (dot-dashed), the other ones have three.

(see dot-dashed curves), but at higher energies the glu-onic source rapidly rises and becomes the largest sourceof particle production at an energy of ∼ 2 TeV, which isbetween energies reached at RHIC and LHC.

I have investigated the dependence of the particle con-tent of the three sources on center-of-mass energy perparticle pair

√sNN in Ref. [37]. The gluonic source is ab-

sent for√sNN . 20 GeV, see the 19.6 GeV Au-Au PHO-

BOS result in Figure 8. At this relatively low energy,charged-hadron production arises only from the fragmen-tation sources which overlap in rapidity space and hence,appear like a single gaussian (“thermal”) source. Thetotal charged-hadron production has been found experi-mentally to depend linearly on ln(sNN/s0), see for exam-ple central Pb-Pb NA50 data at 8.7 GeV and 17.3 GeV[55] and low-energy Au-Au PHOBOS results [16].

In my RDM-analysis with three sources published inRef. [37], it has turned out that the dependence of thefragmentation sources Nqg

ch ∝ ln(sNN/s0) indeed contin-ues at higher energies, see Figure 9. In addition to whatwas shown in Figure 4 of Ref. [37] with an extrapolationto 5.02 TeV, now my results based on an analysis of newALICE 5.02 TeV data [57] are included in this figure. Thegluonic source is confirmed to have a strong energy de-pendence Ngg

ch ∝ ln3(sNN/s0). As discussed in Ref. [37],the rise of the cross section in the central distribution isdriven by the growth of the gluon density at small x andtheoretical arguments [59] suggest a ln2s asymptotic be-haviour that satisfies the Froissart bound [60]. Becausethe beam rapidity is ∝ ln(sNN ), the integrated yield from

0.01 0.1 1 1010

100

1000

10000

Figure 9. The total charged-hadron production in centralAu-Au and Pb-Pb collision in the energy region 19.6 GeV to5.02 TeV is following a power law (solid upper line), whereasthe particle content in the fragmentation sources is Nqg ∝ln (sNN/s0), dash-dotted curve. The particle content in themid-rapidity source obeys Ngg ∝ ln3 (sNN/s0), dashed curve.The energy dependence of the measured mid-rapidity yieldsis shown as a dotted line, with PHOBOS data [54] at RHICenergies, and ALICE data [57, 58] at 2.76 and 5.02 TeV. Thevertical line indicates 5.02 TeV.

the gluonic source then becomes proportional to ln3s, inagreement with the phenomenological analysis, and thenew 5.02 TeV data. It was mentioned in Ref. [37] thatthere exist also further experimental confirmations of thisresult at RHIC energies based on STAR data for dijetproduction, see [61] and references therein.

The 5.02 TeV Pb-Pb data confirm that the sum ofproduced charged hadrons integrated over η is close to apower law N tot

ch ∝ (sNN/s0)0.23 with s0 = 1 TeV2 asshown in Figure 9 for central Au-Au and Pb-Pb col-lisions, upper line. At RHIC energies Busza noticedthat the integrated charged-particle multiplicities scaleas ln2(sNN/s0) [62], but the energy dependence up tothe maximum LHC energy of 5.02 TeV has turned to beeven stronger due to the high gluon density. In Ref. [37] itwas shown that the midrapidity yields for central Au-Auand Pb-Pb collisions are

dN totch

dη

∣∣∣∣η'0

= 1.15 · 103(sNN/s0)0.165 (33)

with s0 = 1 TeV2 (dotted line, data points from Phobos[54] and ALICE [57, 58]).

More detailed aspects of the interplay between frag-mentation sources and gluonic source appear wheninvestigating the centrality dependence of charged-hadron pseudorapidity distributions, as has been donein Refs. [63, 64] for the asymmetric systems 200 GeV d-Au and 5.02 TeV p-Pb, and in Refs. [13, 45] for 2.76 TeVPb-Pb.

12

C. Limiting Fragmentation at RHIC and LHCenergies

Using the RDM with both, linear and sinh-drift, wehave investigated whether the limiting fragmentationconjecture is fulfilled at energies reached at RHIC andLHC [32]. The significance of the fragmentation re-gion in relativistic heavy-ion collisions had been real-ized when data on Au-Au collisions in the energy range√sNN =19.6 GeV to 200 GeV became available at RHIC

[15–17]. The pseudorapidity distributions of producedcharged particles for a given centrality bin scale withenergy according to the limiting fragmentation (LF), orextended longitudinal scaling, hypothesis: Over a largerange of pseudorapidities η ′ = η− ybeam in the fragmen-tation region with the beam rapidity ybeam, the charged-particle pseudorapidity distribution is found to be energyindependent.

The phenomenon was first shown to be present in ppdata, in a range from 53 up to 900 GeV [65], followinga prediction for hadron-hadron and electron-proton col-lisions in Ref. [66]. With increasing collision energy thefragmentation region grows in pseudorapidity space. Itcan cover more than half of the pseudorapidity rangeover which particle production occurs. Especially in rel-ativistic heavy-ion collisions, the approach to a universallimiting curve is a remarkable feature of the particle pro-duction process.

As discussed in Ref. [32] and references cited therein,it is an interesting question whether limiting fragmen-tation will persist at the much higher incident energiesthat are available at the LHC, namely,

√sNN = 2.76 and

5.02 TeV in Pb-Pb collisions. At these energies, experi-mental results in the fragmentation region are not avail-able due to the lack of a dedicated forward spectrome-ter. If one wants to account for the collision dynamicsmore completely, however, this region is most interest-ing. We have therefore investigated in Ref. [32] to whatextent limiting fragmentation can be expected to occurin heavy-ion collisions at LHC energies. The result fromthat investigation is summarized in Figure 10: Limiting-fragmentation scaling can be expected to hold at both,RHIC and LHC energies. This conclusion agrees withmicroscopic numerical models such as AMPT [67], butit disagrees with expectations from simple parametriza-tions of the rapidity distributions such as the differenceof two Gaussians. It also disagrees with predictions fromthe thermal model, which does not explicitly treat thefragmentation sources but refers only to particles pro-duced from the hot fireball. In contrast, in our approachthe fragmentation sources play an essential role. Onlyfuture upgrades of the detectors would make it possibleto actually test the limiting-fragmentation conjecture ex-perimentally at LHC energies.

−15 −10 −5 0 5η− ybeam

0

500

1000

1500

2000

2500

dNch/dη

−1 0 10

200

Figure 10. Comparison of the three-source RDM-distributions with linear and sinh-drift, PHOBOS data [54],and ALICE data [46, 56] with emphasis on limiting fragmen-tation. From bottom to top: central Au-Au at

√sNN =19.6,

62.4, 130 and 200 GeV (RHIC), Pb-Pb at√sNN = 2.76 and

5.02 TeV. The difference between the model with sinh-drift(solid curves) and the one with linear drift (dot-dashed anddashed curves) is small, but visible in the fragmentation re-gion. The zoom into this region shows that the RDM withsinh-drift is consistent with limiting fragmentation at RHICand LHC energies. From Kellers and Wolschin [32], wheredetails and parameters are given.

V. QUARKONIA AND THE QGP

Among the hard probes in relativistic heavy-ion colli-sions, the modification of quarkonia yields in the presenceof the quark-gluon plasma has an outstanding role. Char-monia (J/ψ) suppression due to the screening of the realpart of the Cornell-type quark-antiquark potential in thehot medium had initially been suggested by Matsui andSatz in 1986 as a QGP signature [21]. It was realizedlater that the potential in the medium is an optical one,with the imaginary part [22] causing dissociation of thequarkonia states in the hot medium and thus, quarkoniasuppression when compared to the production rates frompp collisions at the same center-of-mass energy scaledwith the number of binary collisions. In addition to theassociated collisional damping widths of the quarkoniastates, thermal gluons can dissociate these states, andthe gluon-induced dissociation widths [23] can be treatedseparately from the damping [68]. An important roleis played by the reduction of feed-down in the heavy-ion case as compared to pp, because feed-down from thehigher to the lower states is hindered if the higher statesare screened away, or depopulated. If the medium con-tains a large number of heavy quarks as is the case forcharm quarks in Pb-Pb collisions at LHC energies, sta-tistical recombination cannot be neglected at sufficientlylow transverse momentum, and there is an interplay ofdissociation and recombination. There is meanwhile alarge number of publications and reviews about charmo-

13

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30

R PbP

b[ Υ(1

S)]

pT(GeV/c

)

5.02 TeV RQGPPbPb

5.02 TeV RPbPb

CMS (2018)

Figure 11. Transverse-momentum dependence of the sup-pression factor RAA(Υ(1S)) for the spin-triplet ground statein minimum-bias Pb-Pb collisions at

√sNN = 5.02 TeV.

The (upper) dashed curve shows the suppression in the hotmedium, the (lower) solid curve the suppression including re-duced feed-down, which is important for the ground state,but not for excited states. The theoretical prediction is fromRef. [3], data are from CMS [69].

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400

R PbP

b[ Υ(1

S)]

〈Npart〉

5.02 TeV RQGPPbPb

5.02 TeV RPbPb

CMS (2018)

Figure 12. Centrality-dependent suppression factorRAA(Υ(1S)) in Pb-Pb collisions at

√sNN = 5.02 TeV (solid

line, from Ref. [3]) together with data from CMS (dots,|y| < 2.4, Ref. [69]), as function of the number of participants〈Npart〉 (averaged over centrality bins). The suppression inthe QGP-phase is the dashed curve, the solid curve includesreduced feed-down. Theoretical prediction from Ref. [3].

nia physics in relativistic heavy-ion collisions available[39].

Bottom quarks are about three times heavier thancharm quarks, have a correspondingly smaller productioncross section, and hence, are less abundant even at LHCenergies. Statistical recombination is therefore less im-portant, and expansions in terms of (1/m) are more pre-cise. Consequently, bottomonia provide a cleaner probeof the QGP properties such as the initial central tem-perature, and have been investigated in detail both the-oretically and experimentally. The bottom quarks areproduced on a very short time scale of 0.02 fm/c in theinitial stages of the collision, before the QGP of lightquarks and gluons is actually being produced. The for-

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

R PbP

b[ Υ(2

S)]

〈Npart〉

5.02 TeV RQGPPbPb

5.02 TeV RPbPb

CMS (2018)

Figure 13. Suppression factor for the first excited spin-tripletstate RAA(Υ(2S)) in Pb-Pb collisions at

√sNN = 5.02 TeV

(solid line) together with data from CMS. The suppressionfactor RAA(Υ(2S)) in the QGP-phase (dashed) accounts formost of the calculated total suppression (solid) for the Υ(2S).Theoretical predictions are from Hoelck, Nendzig, Wolschin[3], CMS data are from Ref. [69].

mation time of bottomonia states is larger, in the rangeτF ' 0.3 − 0.6 fm/c. It is less precisely known, may dif-fer for the individual states such as Υ(1S, 2S, 3S) andχb(1P, 2P, 3P ), and could depend on the temperature ofthe emerging QGP, which would further enlarge it [70].Since the spin-triplet Υ(1S) state is particularly stablewith a binding energy of ' 1.1 GeV, it has a sizeableprobability to survive as a color-neutral state in the col-ored hot quark-gluon medium of light quarks and gluonsthat is created in a central Pb-Pb collision at LHC ener-gies, even at initial medium temperatures of the order of400 MeV or above.

There exists a considerable literature on the dissocia-tion of quarkonia, in particular of the Υ meson [71–73], inthe hot quark-gluon medium; see Ref. [74] and referencestherein for a review. In minimum-bias Pb-Pb-collisionsat LHC energies of

√sNN = 5.02 TeV in the midrapid-

ity range, the strongly bound Υ(1S)-state is found tobe suppressed down to about 38 % as compared to theexpectation from scaled pp collisions at the same energy.The Υ(1P ) state has a smaller binding energy and is evenmore suppressed, down to 12 % [69].

Various theoretical approaches such as Refs. [75–78],and their recent updates to higher energy, are availablethat allow for an interpretation of the data. In the nextsection, results of our model are reported that aims to ac-count for the previous Pb-Pb results at 2.76 TeV and topredict results for the higher energy of 5.02 TeV, whichare then compared to the data that have meanwhile be-come available.

A. Υ(1S, 2S) suppression in Pb-Pb at LHC energies

In Refs. [3, 68, 79] we have devised a model that ac-counts for the screening of the real part of the potential,the gluon-induced dissociation of the various bottomo-

14

nium states in the hot medium (gluodissociation), andthe damping of the quark-antiquark binding due to thepresence of the medium which generates an imaginarypart of the temperature-dependent potential. Screeningis less important for the strongly bound Υ(1S) groundstate, but it is relevant for the bb excited states, andalso for all cc bound states.

Due to screening and depopulation of the excited statesin the hot medium, the subsequent feed-down cascade to-wards the Υ(1S) ground state differs considerably fromwhat is known based on pp collisions. The LHCb col-laboration has measured a feed-down fraction of Υ(1S)originating from χb(1P ) decays in pp collisions at

√s = 7

TeV of 20.7 % [80], and the total feed-down from excitedstates to the ground state is estimated to be around 40 %[81] at LHC energies. If feed-down was completely absentbecause of screening and depopulation of excited statesin the hot medium, a modification factor RAA(Υ(1S)) '0.6 would thus result, whereas the measured modifica-tion factor of the Υ(1S) state in minimum-bias Pb-Pbcollisions at 2.76 TeV is RAA(Υ(1S)) = 0.453 ±0.014(stat)±0.046 (syst) [82], and at 5.02 TeV RAA(Υ(1S)) =0.378 ± 0.013 (stat)±0.035 (syst) [69]. Hence, thereclearly exist in-medium suppression mechanisms for thestrongly bound Υ(1S) state which we aim to account forin detail, together with the suppression of the excitedstates, and the reduced feed-down.

In our model calculation [3], we thus determine the re-spective contributions from in-medium suppression, andfrom reduced feed-down for the Υ(1S) ground state,and the Υ(2S) first excited state in Pb-Pb collisions atboth LHC energies, 2.76 TeV and 5.02 TeV. The pT-dependence and the role of the relativistic Doppler ef-fect on the measured transverse-momentum spectra areconsidered. For the Υ(2S) state, the QGP effects areexpected to be much more important than reduced feed-down. We compare in Ref. [3] with centrality-dependentCMS data [71, 82] for the Υ(1S) and Υ(2S) states in2.76 TeV Pb-Pb collisions, and predict the pT- andcentrality-dependent suppression at the higher LHC en-ergy of

√sNN = 5.02 TeV. The predictions at 5.02 TeV

are compared with recent CMS data in Fig. 11 for thetransverse-momentum dependence, Fig. 12 for the cen-trality dependence of Υ(1S), and Fig. 13 for the central-ity dependence of Υ(2S).

For symmetric systems such as Au-Au at RHIC orPb-Pb at LHC, we do not include an explicit treatmentof CNM effects such as shadowing in the present study.These are, however, important in asymmetric collisionssuch as p-Pb where most of the system remains cold dur-ing the interaction time, and we have considered them inour corresponding calculations [83]. Statistical recombi-nation of the heavy quarks following bottomonia dissoci-ation is disregarded: Although this is certainly a relevantprocess in the J/ψ case, the cross section for Υ produc-tion is significantly smaller.

The anisotropic expansion of the hot fireball is ac-counted for using hydrodynamics for a perfect fluid that

Figure 14. The ellipticity v2(pT) of the Υ(1S) momentumdistribution in 5− 60% Pb-Pb collisions at

√sNN = 5.02TeV

as calculated from anisotropic escape is consistent with zero,green symbols. (From Fritsch, BSc thesis HD 2020, unpub-lished. Reproduced with permission.) The data (red) arefrom ALICE [86].

includes transverse expansion. Such a simplified nonvis-cous treatment [3, 68] of the bulk evolution appears tobe tolerable because conclusions on the relative impor-tance of the in-medium suppression versus reduced feed-down are not expected to depend much on the details ofthe background model. When calculating the in-mediumdissociation, we consider the relativistic Doppler effectthat arises due to the relative velocity of the bottomo-nia with respect to the expanding medium. It leads tomore suppression at high pT, and to an overall rather flatdependence of RAA on pT.

Our predictions for the pT-dependent Υ-suppressionin 5.02 TeV Pb-Pb collisions are shown together withrecent CMS data [69] in Fig. 11; see the caption for de-tails. The in-medium modification factor (dashed) firstrises at small transverse momentum, because escape fromthe hot zone becomes more likely with increasing pT, butthen falls off when the increase in effective temperaturebecomes more pronounced. For the Υ(1S) state, a sub-stantial fraction of the suppression, in particular at lowpT, is due to reduced feed-down, solid curve. The cor-responding centrality-dependent suppression (integratedover pT) is shown in Fig. 12 to be in agreement with thedata [69] for the Υ(1S) state. Related ALICE data atmore forward rapidities 2.5 < y < 4 are roughly consis-tent within the error bars [84]. The suppression of theΥ(2S) state in Fig. 13 is mostly in-medium, with only asmall contribution due to reduced feed-down. The pre-diction shows less suppression than the data in periph-eral collisions. We have shown in Ref. [85] that the ex-tra suppression of the loosely bound Υ(2S) state is mostlikely not due to the strong electromagnetic fields in moreperipheral collisions. Hence, the origin of this effect ispresently unknown.

It is of considerable interest to determine if the bot-tomonia distributions in more peripheral collisions be-

15

Figure 15. Overlap (red) of the thickness functions in thetransverse plane for lead (green) and proton (blue). (FromDinh, MSc thesis HD 2019, unpublished. Reproduced withpermission.)

come anisotropic, as has been found for produced par-ticles in general. The quadrupole part of the momen-tum anisotropy is due to the almond-shaped spatialanisotropy of the overlap region, which translates to mo-mentum space. It is more pronounced for lighter mesonssuch as pions, and can be quantified by the ellipticityv2 of the momentum distribution in a Fourier decompo-sition of the experimentally determined, event-averagedparticle distribution [1]

d〈N〉dφ

=〈N〉2π

1 + 2

∞∑n=1

〈vn〉 cos[n(φ− 〈ψn〉)]

(34)

with the azimuthal angle φ, the mean flow angle 〈ψn〉,and 〈N〉 the mean number of particles of interest perevent (charged hadrons or identified particles of a spe-cific species). The flow planes are, however, not exper-imentally known, and hence, the anisotropic flow coeffi-cients are obtained using azimuthal angular correlationsbetween the observed particles. The experimentally re-ported anisotropic flow coefficients from two-particle cor-relations can then be obtained as the root-mean-squarevalues, vn{2} ≡ vn ≡

√〈v2n〉, and the flow coefficients are

being measured not in individual events, but in centralityclasses.

Whereas for charged hadrons the flow coefficients havebeen measured very precisely [1] with v2-values up to20− 30% for pions, kaons, and antiprotons and maximanear pT ' 3 GeV/c, this is more difficult for quarko-nia due to the much smaller production rates. For thecharmonium ground state, the ellipticity coefficient atforward rapidity (2.5 < y < 4) in the centrality class5− 60% is v2 ' 3− 8%, depending on pT [86], implyingthat J/ψ shows elliptic flow, albeit on a smaller scale dueto the larger mass. It may, therefore, appear possiblethat even bottomonium exhibits flow, following the massordering of lighter particles resulting from a collective ex-pansion of the medium [90]. Indeed, the large statisticalerror bars on the presently available bottomonium datafor v2 [86] do not yet exclude such a possibility, althoughit is quite doubtful whether this meson – which is aboutthree times heavier than charmonium – flows with theexpanding hot medium. Instead, the Υ(1S) is expectedto essentially maintain its trajectory in the hot QGP, un-

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10-6 10-5 10-4 10-3 10-2 10-1 100

Ri(x,µ

2)

at µ =

10

GeV

x

gluonvalence down

valence up

antishadowingshadowing

EMC

Figure 16. Modification of the nuclear PDFs EPPS16 [87] forgluons, up- and down-quarks as function of the momentumfraction x at 10−6 < x ≤ 1: Shadowing at x ≤ 0.02, antishad-owing at 0.02 < x < 0.3. (From Dinh, MSc thesis HD 2019,unpublished. Reproduced with permission.)

less it is dissociated. Still, its momentum distribution inmore peripheral collisions may exhibit a finite v2 due tothe anisotropic escape from the fireball, because the pathlength from Υ formation to escape in the transverse planedepends on the azimuthal angle. This mechanism had al-ready been suggested for J/ψ by Wang and Yuan at SPSand RHIC energies [91], and has been used in Ref. [92] forthe bottomonium states in non-central 2.76 TeV Pb-Pb.There, the maximum of v2(Υ(1S)) including feed-downcontributions is found to be below 1% in the 40 − 50%centrality class. We have performed a corresponding cal-culation within our model in the 5− 60% centrality classand compare the pT dependence with the available AL-ICE data [86] , see Figure 14. The anisotropy is verysmall, compatible with zero. For more definite conclu-sions, one has to wait for a reduction of the experimentaluncertainties in run 3.

−6 −4 −20

24

68

10 −6−4

−20

24

60

100

200

300

400

500

x1 (fm) x2 (fm)

T(b

;x1 ,

x2 )( M

eV)

central

min. bias

Figure 17. Initial temperature profiles of the hot QGP gen-erated in p-Pb collisions at

√sNN = 8.16 TeV as functions of

the transverse coordinates (x1, x2) at two centralities: Centralcollisions with Ncoll ' 15.6, left, and minimum-bias collisionswith Ncoll ' 7, right. From Ref. [83].

16

0

0.2

0.4

0.6

0.8

1

1.2

1.4−5.0 < ycms < −2.5

R pPb[ Υ(1

S)]

CNMCNM + QGPLHCb (2018)

1.5 < ycms < 4.0

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

−5.0 < ycms < −2.5

R pPb[ Υ(2

S)]

p⊥(GeV/c

) 0 5 10 15 20

1.5 < ycms < 4.0

p⊥(GeV/c

)

Figure 18. Calculated transverse-momentum-dependent nuclear modification factors Rp-Pb for the Υ(1S, 2S) spin-triplet groundand first excited state in p-Pb collisions at

√sNN = 8.16 TeV with LHCb data [88] in the backward (Pb-going, left) and forward

(p-going, right) region, for minimum-bias centrality. Results for CNM effects that include shadowing, energy loss, and reducedfeed-down (dashed curves, blue) are shown together with calculations that incorporate also QGP effects (solid curves, red). Theerror bands result from the uncertainties of the parton distribution functions that enter the model. Calculations from Ref. [83].

B. Υ(1S, 2S) modification in p-Pb at√sNN = 8.16

TeV

Regarding bottomonia in asymmetric collisions, p-Pbat√sNN = 8.16 TeV has been investigated experimen-

tally by the LHCb [88] and ALICE [93] collaborations,and cold nuclear matter predictions had been publishedby a group of theorists [94]. Clearly, CNM effects aremuch more relevant than in symmetric systems, becausethe bulk of the hadronic matter remains cold during theinteraction, see Fig. 15. The most relevant CNM effect isthe modification of the parton distribution functions inthe nuclear medium, which have been studied by manyauthors. A typical result for the PDF modifications withshadowing at small values of Bjorken-x, and antishadow-ing at intermediate x-values as obtained with EPPS16[87] is shown in Fig. 16. Shadowing causes a reductionof the Υ(nS) yields in p-Pb as compared to scaled pp,whereas antishadowing results in an enhancement. Shad-owing is somewhat more pronounced if one, in addition,considers coherent energy-loss mechanisms in the coldmedium. Still, these are not sufficient to interpret theavailable data in terms of CNM effects, as becomes obvi-ous from direct comparisons, in particular, for the Υ(2S)state.

There is, however, a spatially small hot zone (fire-ball) with an initial central temperature that is compa-

rable to the one in a symmetric system (Fig. 17), andduring its expansion and cooling, it contributes to bot-tomonia dissociation in regions where the temperatureremains above the critical value. We have investigatedthe respective cold-matter and hot-medium effects on Υ-dissociation in 8.16 TeV p-Pb collisions in Ref. [83]. Rep-resentative results from this work are shown in Fig. 18for the transverse-momentum dependence, and Fig. 19 forthe rapidity dependence. The plots show CNM (blue, up-per bands) and CNM plus QGP (red, lower bands) effectson the Υ(1S) and Υ(2S) yields in 8.16 TeV p-Pb colli-sions at the LHC. The transverse-momentum dependencein the backward direction (top) shows enhancement dueto antishadowing when only the CNM effects are consid-ered, whereas the data for Υ(1S) are clearly suppressedat pT < 10 GeV/c and for Υ(2S) at all measured pT val-ues. This discrepancy is cured through the considerationof the momentum-dependent dissociation in the QGP, asshown in our cold-matter plus hot-medium calculation(red).

The forward/backward asymmetric shape of thenuclear modification factors as functions of rapidity(Fig. 19) arises from the different cold-matter effects inthe forward and backward regions (in particular, shadow-ing/antishadowing of the parton distribution functions,but also energy loss in the relatively cold medium). Theadditional suppression due to the dissociation in the hot

17

0

0.2

0.4

0.6

0.8

1

1.2

1.4p⊥ < 25 GeV/c

R pPb[ Υ(1

S)]

CNMCNM + QGPLHCb (2018)ALICE (2018)

0

0.2

0.4

0.6

0.8

1

−4 −2 0 2 4

p⊥ < 25 GeV/c

R pPb[ Υ(2

S)]

ycms

Figure 19. Calculated rapidity-dependent nuclear modifica-tion factors Rp-Pb for the Υ(1S) (top) and Υ(2S) state (bot-tom) in p-Pb collisions at

√sNN = 8.16 TeV with preliminary

ALICE data [89] (triangles) and LHCb data [88] (circles).Results for CNM effects that include shadowing, energy loss,and reduced feed-down (dashed curves, blue) are shown to-gether with calculations that incorporate also QGP effects(solid curves, red). The error bands result from the uncer-tainties of the parton distribution functions that enter thecalculations. From Dinh, Hoelck, Wolschin [83].

fireball is again shown in the lower (red) curves, which arein better agreement with the data for the Υ(1S) groundstate not only in the backward, but also in the forwarddirection. The substantial role of the hot-medium ef-fects is even more pronounced for the Υ(2S) first excitedstate, where the CNM calculation shows enhancement inthe backward region, whereas the full calculation within-medium dissociation displays a suppression down toabout 70% – in agreement with the LHCb data [88] andthe ALICE data point [93].

There have been attempts to explain the discrepancybetween CNM calculations and data for the Υ(nS)suppression in p-Pb in terms of interactions withcomoving hadrons, in particular, pions [95]. We havenot included this process in our calculations – initiallyon the grounds that interactions of the bottomoniastates with comovers were found to be unimportantat LHC energies in the work of Ko et al. [96] aboutΥ absorption in hadronic matter. Probably one even-

tually has to consider both, comover interactions plussuppression in the hot QGP zone in order to fully un-derstand the Υ modification data in asymmetric systems.

VI. CONCLUSIONS

This article presents aspects of relativistic heavy-ioncollisions with an emphasis on energies reached at theRelativistic Heavy Ion Collider RHIC and the LargeHadron Collider LHC. It does not attempt to be a reviewof the field, which is available in recent textbooks andin the proceedings of Quark Matter conferences such asRef. [39]. Instead, a specific phenomenological viewpointwith experimental data as a guiding principle is taken,but also QCD-based and nonequilibrium-statistical ar-guments are considered. The rapid local equilibrationof gluons and quarks in the initial stages of a relativis-tic heavy-ion collision is being modeled through exactanalytical solutions of a nonlinear diffusion equation.On a similar time scale, stopping is accounted for ina QCD-inspired model, which is also incorporated in anonequilibrium-statistical approach to compute the timeevolution from the initial to the measured distributionfunctions of net protons. The production of chargedmesons such as pions, kaons, and antiprotons is discussedin a phenomenological three-source relativistic diffusionmodel that emphasises the importance of the fragmenta-tion distributions in addition to the usual fireball source.These are also shown to be essential in the phenomenonof limiting fragmentation, which had been confirmed ex-perimentally at RHIC energies, and turns out to be con-sistent with the present LHC energies. The investiga-tion of the dissociation of quarkonia in the QGP providesinsights into the QGP properties, including an indirectmeasurement of its initial central temperature before theanisotropic expansion sets in. In asymmetric systemssuch as p-Pb at the current maximum LHC energy, theinterplay of cold-matter and hot-medium effects has beenstudied, achieving a detailed understanding of the avail-able data from the Large Hadron Collider. The moreprecise measurements from the forthcoming run 3 at theLHC are expected to provide deeper insights.

ACKNOWLEDGMENTS

I am grateful to members of UHD’s Multiparticle Dy-namics Group for discussions and collaborations, whichresulted in common publications that are quoted in thereferences. Special thanks go to the Heidelberg studentsViet Hung Dinh (now Orsay), Johannes Holck, BenjaminKellers, Niklas Rasch, Philipp Schulz, and Alessandro Si-mon.

18

[1] U. Heinz and R. Snellings, Collective flow and viscosity inrelativistic heavy-ion collisions, Annu. Rev. Nucl. Part.Sci. 63, 123 (2013).

[2] W. Florkowski, E. Maksymiuk, and R. Ryblewski, Cou-pled kinetic equations for fermions and bosons in therelaxation-time approximation, Phys. Rev. C 97, 024915(2018).

[3] J. Hoelck, F. Nendzig, and G. Wolschin, In-medium Υsuppression and feed-down in UU and PbPb collisions,Phys. Rev. C 95, 024905 (2017).

[4] G. Wolschin, Equilibration in finite Bose systems, Phys-ica A 499, 1 (2018).

[5] M. P. Heller, A. Kurkela, M. Spalinski, and V. Svensson,Hydrodynamization in kinetic theory: Transient modesand the gradient expansion, Phys. Rev. D 97, 091503(R)(2018).

[6] P. Romatschke, Do nuclear collisions create a locally equi-librated quark-gluon plasma?, Eur. Phys. J. C 77, 21(2017).

[7] D. T. Son and A. O. Starinets, Viscosity, black holes, andquantum field theory, Annu. Rev. Nucl. Part. Sci. 57, 95(2007).

[8] J. M. Maldacena, The large N limit of superconformalfield theories and supergravity, Adv. Theor. Math. Phys.2, 231 (1998).

[9] J.-P. Blaizot, F. Gelis, J. Liao, L. McLerran, andR. Venugopalan, Bose–Einstein condensation and ther-malization of the quark gluon plasma, Nucl. Phys. A 873,68 (2012).

[10] J.-P. Blaizot, J. Liao, and Y. Mehtar-Tani, The thermal-ization of soft modes in non-expanding isotropic quarkgluon plasmas, Nucl. Phys. A 961, 37 (2017).

[11] Z. Xu, K. Zhou, P. Zhuang, and C. Greiner, Thermal-ization of gluons with Bose-Einstein condensation, Phys.Rev. Lett. 114, 182301 (2015).

[12] T. Bartsch and G. Wolschin, Equilibration in fermionicsystems, Annals Phys. 400, 21 (2019).

[13] G. Wolschin, Particle production sources at LHC ener-gies, J. Phys. G: Nuclear and Particle Physics 40, 45104(2013).

[14] G. Wolschin, Beyond the thermal model in relativisticheavy-ion collisions, Phys. Rev. C 94, 024911 (2016).

[15] I. G. Bearden et al. (BRAHMS Collaboration), Pseudo-rapidity distributions of charged particles from Au+Aucollisions at the maximum RHIC energy, Phys. Rev. Lett.88, 202301 (2002).

[16] B. B. Back et al. (PHOBOS Collaboration), The sig-nificance of the fragmentation region in ultrarelativisticheavy ion collisions, Phys. Rev. Lett. 91, 052303 (2003).

[17] J. Adams et al. (STAR Collaboration), Multiplicity andpseudorapidity distributions of charged particles andphotons at forward pseudorapidity in Au + Au collisionsat√sNN = 62.4 GeV, Phys. Rev. C 73, 034906 (2006).

[18] J. D. Bjorken, Energy loss of energetic partons inquark-gluon plasma: Possible extinction of high pT jetsin hadron-hadron collisions., Fermilab-Pub-82/59-THY(1982).

[19] J. Adams et al. (STAR Collaboration), Evidence fromd+Au measurements for final-state suppression of high-pthadrons in Au+Au collisions at RHIC, Phys. Rev. Lett.91, 072304 (2004).

[20] S. Chatrchyan et al. (CMS Collaboration), Observa-tion and studies of jet quenching in PbPb collisions at√sNN = 2.76 TeV, Phys. Rev. Lett. 84, 024906 (2011).

[21] T. Matsui and H. Satz, J/ψ Suppression by Quark-GluonPlasma Formation, Phys. Lett. B 178, 416 (1986).

[22] M. Laine, O. Philipsen, M. Tassler, and P. Romatschke,Real-time static potential in hot QCD, JHEP 2007, 54.

[23] F. Brezinski and G. Wolschin, Gluodissociation andscreening of Υ states in PbPb collisions at

√sNN =

2.76 TeV, Phys. Lett. B 707, 534 (2012).[24] G. Wolschin, Local thermalization of gluons in a nonlin-

ear model, Nonlin. Phenom. Complex Syst. 23, 72 (2020).[25] N. Rasch and G. Wolschin, Solving a nonlinear analytical

model for bosonic equilibration, Physics Open 2, 100013(2020).

[26] G. Wolschin, Time-dependent entropy of a cooling Bosegas, EPL 129, 40006 (2020).

[27] A. H. Mueller, The Boltzmann equation for gluons atearly times after a heavy ion collision, Phys. Lett. B 475,220 (2000).

[28] L. McLerran and R. Venugopalan, Green’s functions inthe color field of a large nucleus, Phys. Rev. D 50, 2225(1994).

[29] Y. Mehtar-Tani and G. Wolschin, Baryon stopping as anew probe of geometric scaling, Phys. Rev. Lett. 102,182301 (2009).

[30] D. V. Semikoz and I. I. Tkachev, Kinetics of Bose con-densation, Phys. Rev. Lett. 74, 3093 (1995).

[31] G. Wolschin, Local equilibration of fermions and bosons,Results Phys. 13, 102197 (2019).

[32] B. Kellers and G. Wolschin, Limiting fragmentation atLHC energies, Prog. Theor. Exp. Phys. 2019, 053D03(2019).

[33] Y. Mehtar-Tani and G. Wolschin, Baryon stopping andsaturation physics in relativistic collisions, Phys. Rev. C80, 054905 (2009).

[34] H. Appelshauser et al. (NA49 Collaboration), Baryonstopping and charged particle distributions in centralPb+Pb collisions at 158 GeV per nucleon, Phys. Rev.Lett. 82, 2471 (1999).

[35] J. Hoelck and G. Wolschin, Baryon stopping as a rel-ativistic Markov process in phase space, in preparation(2020).

[36] Y. Mehtar-Tani and G. Wolschin, Stopping in centralPb+Pb collisions at SPS energies and beyond, EPL 94,62003 (2011).

[37] G. Wolschin, Ultraviolet energy dependence of particleproduction sources in relativistic heavy-ion collisions,Phys. Rev. C 91, 014905 (2015).

[38] G. Wolschin, Relativistic diffusion model, Eur. Phys. J.A 5, 85 (1999).

[39] B.-W. Zhang, E. Wang, and F. Liu, (eds.) Quark Matter2019 - the XXVIIIth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions, Nucl. Phys. A, inpress (2020).

[40] F. Juttner, Das Maxwellsche Gesetz derGeschwindigkeitsverteilung in der Relativitatstheo-rie, Annalen Phys. 339, 856 (1911).

[41] C. Y. Wong and G. Wilk, Tsallis fits to pt spectra andmultiple hard scattering in pp collisions at the LHC,Phys. Rev. D 87, 114007 (2013).

19

[42] R. Hagedorn, Multiplicities, pt distributions and theexpected hadron → quark-gluon phase transition, Riv.Nuovo Cimento 6, 1 (1983).

[43] S. Acharya et al. (ALICE Collaboration), Transversemomentum spectra and nuclear modification factors ofcharged particles in pp, p-Pb and Pb-Pb collisions at theLHC, JHEP 2018, 013.

[44] B. Abelev et al. (ALICE Collaboration), Centrality de-pendence of charged particle production at large trans-verse momentum in Pb-Pb collisions at

√sNN = 2.76

TeV, Phys. Lett. B 720, 52 (2013).[45] D. Rohrscheid and G. Wolschin, Centrality dependence

of charged-hadron pseudorapidity distributions in PbPbcollisions at LHC energies in the RDM, Phys. Rev. C 86,024902 (2012).

[46] J. Adam et al. (ALICE Collaboration), Centrality de-pendence of the pseudorapidity density distribution forcharged particles in Pb-Pb collisions at

√sNN = 5.02

TeV, Phys. Lett. B 772, 567 (2017).[47] M. Biyajima, M. Ide, T. Mizoguchi, and N. Suzuki, Scal-

ing behavior of (Nch)−1dNch/dη at√sNN = 130 GeV

by PHOBOS collaboration and its implication: A possi-ble explanation by the Ornstein-Uhlenbeck process, Prog.Theor. Phys. 108, 559 (2002).

[48] G. E. Uhlenbeck and L. S. Ornstein, On the theory ofthe Brownian motion, Phys. Rev. 36, 823 (1930).

[49] S. I. Denisov, W. Horsthemke, and P. Hanggi, General-ized Fokker-Planck equation: Derivation and exact solu-tions, Eur. Phys. J. B 68, 567 (2009).

[50] J. Dunkel, P. Hanggi, and S. Weber, Time parame-ters and Lorentz transformations of relativistic stochasticprocesses, Phys. Rev. E 79, 010101(R) (2009).

[51] A. Lavagno, Anomalous diffusion in nonequilibrium rel-ativistic heavy ion rapidity spectra, Physica A 305, 238(2002).

[52] A. Simon and G. Wolschin, Examining nonextensivestatistics in relativistic heavy-ion collisions, Phys. Rev.C 97, 044913 (2018).

[53] F. Forndran and G. Wolschin, Relativistic diffusionmodel with nonlinear drift, Eur. Phys. J. A 53, 37 (2017).

[54] B. Alver et al. (PHOBOS Collaboration), Phobos resultson charged particle multiplicity and pseudorapidity dis-tributions in Au+Au, Cu+Cu, d+Au, and p+p collisionsat ultra-relativistic energies, Phys. Rev. C 83, 024913(2011).

[55] F. Prino et al. (NA50 Collaboration), Charged particlemultiplicity in Pb-Pb collisions from the NA50 experi-ment, J. Phys. Conf. Series 5, 008 (2005).

[56] E. Abbas et al. (ALICE Collaboration), Centrality de-pendence of the pseudorapidity density distribution forcharged particles in Pb-Pb collisions at

√sNN = 2.76

TeV, Phys. Lett. B 726, 610 (2013).[57] J. Adam et al. (ALICE Collaboration), Centrality de-

pendence of the charged-particle multiplicity density atmidrapidity in Pb-Pb collisions at

√sNN = 5.02 TeV,

Phys. Rev. Lett. 116, 222302 (2016).[58] K. Aamodt et al. (ALICE Collaboration), Centrality de-

pendence of the charged-particle multiplicity density atmid-rapidity in Pb-Pb collisions at

√sNN = 2.76 TeV,

Phys. Rev. Lett. 106, 032301 (2011).[59] M.-F. Cheung and C. B. Chiu, Gluon-gluon elastic scat-

tering amplitude in classical color field of colliding pro-tons, arXiv:1111.6945 (2011).

[60] M. Froissart, Asymptotic behavior and subtractions inthe mandelstam representation, Phys. Rev. 123, 1053(1961).

[61] T. A. Trainor and D. J. Prindle, Charge-multiplicitydependence of single-particle transverse-rapidity yt andpseudorapidity η densities and 2d angular correlationsfrom 200 GeV p-p collisions, Phys. Rev. D 93, 014031(2016).

[62] W. Busza, Trends in multiparticle production and some’predictions’ for pp and PbPb collisions at LHC, J. Phys.G: Nucl. Part. Phys. 35, 044040 (2008).

[63] G. Wolschin, M. Biyajima, T. Mizoguchi, and N. Suzuki,Local thermalization in the d + Au system, Phys. Lett.B 633, 38 (2006).

[64] P. Schulz and G. Wolschin, Diffusion-model analysis ofpPb and PbPb collisions at LHC energies, Mod. Phys.Lett. A 17, 1850098 (2018).

[65] G. J. Alner et al. (UA5 Collaboration), Scaling of pseu-dorapidity distributions at c.m. energies up to 0.9 TeV,Z. Physik C 33, 1 (1986).

[66] J. Benecke, T. T. Chou, C. N. Yang, and E. Yen, Hypoth-esis of limiting fragmentation in high-energy collisions,Phys. Rev. 188, 2159 (1969).

[67] Z. W. Lin, C. M. Ko, B. A. Li, B. Zhang, and S. Pal,A multi-phase transport model for relativistic heavy ioncollisions, Phys. Rev. C 72, 064901 (2005).

[68] F. Nendzig and G. Wolschin, Bottomium suppression inPbPb collisions at LHC energies, J. Phys. G: Nuclear andParticle Physics 41, 095003 (2014).

[69] A. M. Sirunyan et al. (CMS Collaboration), Measure-ment of nuclear modification factors of Υ(1S), Υ(2S),and Υ(3S) mesons in PbPb collisions at

√sNN = 5.02

TeV, Phys. Lett. B 790, 270 (2019).[70] T. Song, C. M. Ko, and J. Xu, Quarkonium formation

time in relativistic heavy-ion collisions, Phys. Rev. C 91,044909 (2015).

[71] S. Chatrchyan et al. (CMS Collaboration), Observationof sequential Upsilon suppression in PbPb collisions,Phys. Rev. Lett. 109, 222301 (2012).

[72] B. Abelev et al. (ALICE Collaboration), Suppression ofΥ(1S) at forward rapidity in Pb-Pb collisions at

√sNN =

2.76 TeV, Phys. Lett. B 738, 361 (2014).[73] L. Adamczyk et al. (STAR Collaboration), Suppression

of Υ production in d+Au and Au+Au collisions at√sNN = 200 GeV, Phys. Lett. B 735, 127 (2014).

[74] A. Andronic, F. Arleo, R. Arnaldi, A. Beraudo, et al.,Heavy-flavour and quarkonium production in the LHCera: from proton-proton to heavy-ion collisions, Eur.Phys. J. C 76, 107 (2016).

[75] A. Emerick, X. Zhao, and R. Rapp, Bottomonia in thequark-gluon plasma and their production at RHIC andLHC, Eur. Phys. J. 48, 72 (2012).

[76] M. Strickland and D. Bazow, Thermal bottomonium sup-pression at RHIC and LHC, Nucl. Phys. A 879, 25(2012).

[77] Y. Liu, B. Chen, N. Xu, and P. Zhuang, Υ production asa probe for early state dynamics in high energy nuclearcollisions at RHIC, Phys. Lett. B 697, 32 (2011).

[78] T. Song, K. C. Han, and C. M. Ko, Bottomonia sup-pression in heavy-ion collisions, Phys. Rev. C 85, 014902(2012).

[79] F. Nendzig and G. Wolschin, Υ suppression in PbPb col-lisions at energies available at the CERN Large HadronCollider, Phys. Rev. C 87, 024911 (2013).

20

[80] R. Aaij et al. (LHCb Collaboration), Measurement of thefraction of Υ(1S) originating from χb(1P ) decays in ppcollisions at

√s = 7 TeV, JHEP 2012, 031.

[81] G. Manca (LHCb Collaboration), private communication(2018).

[82] V. Khachatryan et al. (CMS Collaboration), Suppressionof Υ(1S),Υ(2S), and Υ(3S) quarkonium states in PbPbcollisions at

√sNN = 2.76 TeV, Phys. Lett. B 770, 357

(2017).[83] V. H. Dinh, J. Hoelck, and G. Wolschin, Hot-medium

effects on Υ yields in pPb collisions at√sNN = 8.16

TeV, Phys. Rev. C 100, 024906 (2019).[84] S. Acharya et al. (ALICE Collaboration), Υ suppression

at forward rapidity in Pb-Pb collisions at√sNN = 5.02

TeV, Phys. Lett. B 790, 89 (2019).[85] J. Hoelck and G. Wolschin, Electromagnetic field effects

on Υ-meson dissociation in PbPb collisions at LHC en-ergies, Eur. Phys. J. A 53, 241 (2017).

[86] S. Acharya et al. (ALICE Collaboration), Measurementof Υ(1S) elliptic flow at forward rapidity in Pb-Pb colli-sions at

√sNN = 5.02 TeV, Phys. Rev. Lett. 123, 192301

(2019).[87] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Sal-

gado, EPPS16: Nuclear parton distributions with LHCdata, EPJC 77, 163 (2017).

[88] R. Aaij et al. (LHCb Collaboration), Study of Υ produc-tion in pPb collisions at

√sNN = 8.16 TeV., JHEP 2018,

194.[89] E. Scomparin et al. (ALICE Collaboration), Inclusive

Υ production in p-Pb collisions at√sNN = 8.16 TeV,

ALICE-PUBLIC-2018-008 (2018).[90] K. Reygers, A. Schmah, A. Berdnikova, and X. Sun,

Blast-wave description of Upsilon elliptic flow at LHC,arXiv:1910.14618v1 (2019).

[91] X.-N. Wang and F. Yuan, Azimuthal asymmetry of J/ψsuppression in non-central heavy-ion collisions, Phys.Lett. B 540, 62 (2002).

[92] P. P. Bhaduri, N. Borghini, A. Jaiswal, and M. Strick-land, Anisotropic escape mechanism and elliptic flow ofbottomonia, Phys. Rev. C 100, 051901(R) (2019).

[93] S. Acharya et al. (ALICE Collaboration), Inclusive Υproduction in p-Pb collisions at

√sNN = 8.16 TeV, Phys.

Lett. B, submitted (2019).[94] J. L. Albacete et al., Predictions for cold nuclear matter

effects in p+Pb collisions at√sNN = 8.16 TeV, Nucl.

Phys. A 972, 18 (2018).[95] E. G. Ferreiro and J. P. Lansberg, Is bottomonium sup-

pression in proton-nucleus and nucleus-nucleus collisionsat LHC energies due to the same effects?, JHEP 2018,094.

[96] Z. Lin and C. M. Ko, Υ absorption in hadronic matter,Phys. Lett. B 503, 104 (2001).