LLight-Cone QCD Plasma
H.J. Pirner
Institute for Theoretical Physics, University of Heidelberg, Germany
K. Reygers
Physikalisches Institut, University of Heidelberg, Germany (Dated: October 8, 2018)We deduce the maximum-entropy state of partons created in proton-proton and nucleus-nucleuscollisions. This state is characterized by a distribution function n ( x, p ⊥ ) depending on the light-cone fraction x and the transverse momentum p ⊥ of forward or backward particles, respectively.The mean transverse momentum (cid:104) p ⊥ (cid:105) determines the single parameter of the maximum entropydistribution which is constrained by the sum of all light-cone momentum fractions being unity.The total multiplicity is related to the transverse area of the colliding Lorentz-contracted hadrons.Assuming parton-hadron duality we can compare the model to data from RHIC and LHC. I. INTRODUCTION
A great challenge in strong-interaction physics hasbeen to relate high-energy scattering experiments to thestate of the early universe. Especially, ultra-relativisticheavy-ion collisions promise to create a state of hot anddense matter similar to the quark-gluon plasma [1, 2]which dominated the early universe approximately onemicrosecond after the Big Bang. The underlying hy-pothesis is that in nucleus-nucleus collisions at centralrapidities a hot fireball is created with a temperaturewhich is higher than the phase transition or cross-overtemperature T ≈
160 MeV from a hadronic gas to thequark-gluon plasma. Note, in this fireball the longitudi-nal and transverse momenta are equilibrated. Owing tothe extended size of the nuclei and the large change in en-tropy between the quark-gluon plasma and the hadronicgas, the hot matter created in the collision has been con-sidered as a macroscopic state which lives long enoughto determine the main features of the collision. This sce-nario can in principle also be related to the rapidity dis-tribution of the produced particles if a series of fireballsspread out along the rapidity axis [3–5] is constructed.In this letter we propose a different picture basedmostly on experimental information. Our goal is to de-scribe the inclusive cross section of charged particles pro-duced in proton-proton (pp) collisions dNdyd p ⊥ = dNd ln xd p ⊥ (1)= dσσ in dyd p ⊥ (2)on the basis of a statistical distribution function whichmaximizes the entropy of the produced partons given cer-tain constraints. We consider the light-cone momenta ofthe partons with energies (cid:15) and longitudinal momenta p z relative to the light-cone momentum of the incomingproton with ( E, P z , x = (cid:15) + p z E + P z = p + P + . (3) On the basis of light-cone momentum conservation, wethen determine the maximum-entropy distribution [6, 7]for a given transverse energy which we call light-coneplasma distribution. Our approach aims at a macroscopicdescription of the very soft part of the multiplicity dis-tribution in hadronic collisions using as little dynamicalinput as possible. We refer to the literature for micro-scopic calculations of rapidity distributions based on par-ton shower Monte Carlo event generators [8] or based onunintegrated gluon distributions from saturation models[9].The light-cone plasma distribution is a new state ofmatter, different from the thermal quark-gluon plasmadistribution in the early universe. It agrees rather wellwith data when we assume a gradual transition from par-tons to hadrons, i.e., when we use parton/hadron dualityto relate our model to the measured rapidity distribu-tions.In the following we consider symmetric collision part-ners and discuss the produced partons separately in theforward and backward hemisphere. We limit ourselves togluons only, i.e., to Bose statistics. It is important to em-phasize the light-cone property of the maximum-entropydistribution. The dynamics of collisions at high energiesis governed by a light-cone Hamiltonian which is boostinvariant and determines wave functions depending ontransverse momentum and light-cone fractions which la-bel the eigenstates. The density matrix resulting afterthe collision can be built up from an incoherent mixtureof such multi-parton states. Without a boost invariantformulation one cannot define a number density of par-tons since in each reference system it will be different.In the rest system of the proton the gluons will all sitin the strings holding the quarks together, whereas infast-moving proton the virtual gluons materialize as par-tons carrying a sizable momentum fraction. Light-conephysics has been very successful in determining hard ex-clusive processes in high-energy collisions [10]. We thinkthat it is also important to describe soft inclusive cross-sections via the maximum-entropy principle.The outline of the paper is as follows. In section2 we give a theoretical motivation for the light-cone a r X i v : . [ h e p - ph ] A ug plasma distribution. In section 3 we calculate the pa-rameters of the light cone distributions for pp collisionsat √ s = 200 GeV and 7000 GeV. In section 4 we con-sider nucleus-nucleus collisions as a superposition of ppcollisions where the mean transverse momentum of thepartons is broadened. Section 5 is devoted to a com-parison of the entropy of the light-cone distribution andthe conventional smeared fireball distribution. Section 6gives our conclusions. II. MOTIVATION FOR THE LIGHT CONEDISTRIBUTION
The multiplicity distribution for partons can be mo-tivated from the maximum-entropy principle. For thispurpose we have to find the phase space for a system ofpartons which move on the light cone. The entropy of thepartonic system is then proportional to the logarithm ofthe integrated phase space. This entropy has finally to bemaximized given certain constraints which take into ac-count the conservation of light-cone momentum and thelimited transverse energy produced in hadronic collisions.On the light cone, phase space includes the transversespatial coordinate b ⊥ , the transverse momentum p ⊥ , thelongitudinal light-cone momentum p + and the longitudi-nal spatial variable x − = 1 / x − x ). These variablesare handled like in conventional thermodynamics, i.e.,the phase space multiplied by the gluon degeneracy fac-tor g = 2( N c −
1) and divided by the Planck constant h = (2 π (cid:126) ) gives the number of available quantum states G : G b ⊥ ,p ⊥ ,p + ,x − = g d b ⊥ d p ⊥ dp + dx − (2 π ) (4)= g d b ⊥ d p ⊥ (2 π ) dx dρ π . (5)For high energies, Feynman scaling is a good phenomeno-logical concept, therefore we have multiplied and dividedthis expression by P + = E + P z to obtain the light-conemomentum x = p + /P + and the longitudinal light-conedistance ρ = x − P + which are canonically conjugate vari-ables. We refer to ref. [11] to show the role of ρ inthe light cone Hamiltonian of a meson built from a va-lence quark and antiquark. We further make the simpli-fying assumption that the distribution function and con-sequently the entropy are homogeneously distributed intransverse space. The integration over the b ⊥ -coordinatecan then be executed and gives the area L ⊥ .An estimate of the integral of the scaled light-cone dis-tance (cid:82) dρ π is more subtle, since it is not independent onthe rest of the variables. It has to be done separatelyfor valence and sea partons. Let us describe the fast-moving system by a Lorentz-factor γ → ∞ . Then thevalence quarks occupy a decreasing longitudinal exten-sion ∆ x − ≈ L z γ whereas the extension of the sea partonsremains fixed. Consequently, the longitudinal distance for valence quarks scaled with the increasing light-conemomentum P + of the proton is constant and yields for m = 0 .
938 GeV a factor of O (1): (cid:90) dρ val π ≈ L z γ mγ π ≈ . (6)For sea partons with x → (cid:90) dρ sea π ≈ xP + P + → ∞ . (7)We interpolate the x dependence of these two limit-ing cases for the integral over the scaled distance in thefollowing way: (cid:90) dρ π ≈ x . (8)A possible pre-factor of 1 /x cannot be determined moreaccurately and has to be absorbed into the transversearea L ⊥ . The so motivated ansatz for the phase spaceon the light cone is crucial for all further derivations. Itdeviates from the flat measure by the factor 1 /x : G x,p ⊥ = gL ⊥ d p ⊥ (2 π ) dxx . (9)Gluons are bosons; therefore, they can occupy thephase-space cells in multiples. The binomial of the com-bined number of particles and states over the numberof states gives the number of possibilities to distribute N x,p ⊥ gluons, i.e., bosons, on G x,p ⊥ quantum states:∆Γ x,p ⊥ = ( G x,p ⊥ + N x,p ⊥ − G x,p ⊥ − N x,p ⊥ ! . (10)The entropy of the system is defined by the logarithmof the phase space. In Eq. 10 we use Stirling’s formulaand set G x,p ⊥ − ≈ G x,p ⊥ for large numbers of quantumstates and particle numbers. Then one gets the entropyfrom the summation of the individual phase space ele-ments: S = (cid:88) ln(∆Γ x,p ⊥ ) (11)= (cid:88) G x,p ⊥ [(1 + n x,p ⊥ ) ln(1 + n x,p ⊥ ) − n x,p ⊥ ln n x,p ⊥ ](12)where the mean occupation number of each quantumstate is defined as n x,p ⊥ = N x,p ⊥ G x,p ⊥ . (13)In high-energy collisions the searched-for maximum en-tropy distribution has to satisfy the following two require-ments: (cid:88) G x,p ⊥ x n x,p ⊥ = 1 (14) (cid:88) G x,p ⊥ p ⊥ n x,p ⊥ = (cid:104) E ⊥ (cid:105) . (15)The first constraint means that the x fractions of allpartons emitted in the positive hemisphere add up tounity, i.e., their light-cone momenta equal the light-conemomentum of the parent proton. The second constraintdefines the total transverse energy released in the colli-sion in the positive hemisphere. These constraints areadded with Lagrange parameters 1 /λ and w to the en-tropy above. These two constraints are sufficient to de-termine the x, p ⊥ dependence of the distribution functionsince we have fixed the distribution in coordinate space.The resulting functional S + 1 /λ · (cid:104) E ⊥ (cid:105) + w · n x,p ⊥ to obtain the maximumentropy density: δ ( S + λ (cid:80) p ⊥ n x,p ⊥ + w (cid:80) xn x,p ⊥ ) δn x,p ⊥ = 0 . (16)By choosing the cell sizes small we convert the sumsinto integrals over the continuum variables x, p ⊥ and thediscrete distribution n x,p ⊥ becomes n ( x, p ⊥ ), the light-cone plasma distribution function n ( x, p ⊥ ) = 1 e p ⊥ λ + xw − . (17)The light-cone plasma distribution together withthe measure generates a distribution for the Lorentz-invariant yield of the form dNdyd p ⊥ = gL ⊥ (2 π ) (cid:104) p ⊥ (cid:16) λ + we | y | √ s (cid:17)(cid:105) − . (18)This is the connection of the maximum-entropy dis-tribution on the light cone with the semi-inclusive crosssection. The factor 1 / (2 π ) has its origin in the trans-verse phase space cell d b ⊥ d p ⊥ (cid:126) (2 π ) . The distribution is con-sistent with the two constraints of light-cone momentumconservation and total transverse energy which have thefollowing form in continuum variables. The x integrationis executed in one hemisphere in the cm system, wherethe partons released by the proton projectile or targetrespectively are to be found: gL ⊥ (cid:90) d p ⊥ (2 π ) (cid:90) dxx x n ( x, p ⊥ ) = 1 (19) gL ⊥ (cid:90) d p ⊥ (2 π ) (cid:90) dxx p ⊥ n ( x, p ⊥ ) = (cid:104) E ⊥ (cid:105) . (20)The phenomenological description of the multiplicitydistribution has three parameters L ⊥ , λ , and w . The pa-rameter λ plays the role of an effective transverse “tem-perature”. The “softness” w is related to the mean x .With increasing center-of-mass energies we expect thatthe effective transverse temperature λ and the softness w increase: The collision becomes “hotter” and the particledistributions “softer”. The effective transverse tempera-ture λ is calculated from the mean transverse momentum which is equal to the ratio of the transverse energy andmultiplicity in one hemisphere: (cid:104) p ⊥ (cid:105) = (cid:104) E ⊥ (cid:105) / ( N/
2) (21)with N/ gL ⊥ (cid:90) d p ⊥ (2 π ) (cid:90) dxx n ( x, p ⊥ ) . (22)For a given L ⊥ , λ , and w the multiplicity is uniquelydefined when the cut on the x integration is taken as x min = (cid:112) p ⊥ + m π / √ s . We describe in the next sectionhow this distribution fits the data. III. PARAMETERS OF LIGHT-CONEDISTRIBUTIONS IN PP COLLISIONS
When we want to relate the theoretical distribution ofthe gluon plasma to multiplicities of produced particleswe must apply a simplified form of parton-hadron dual-ity. We assume that all particles are pions and replacetransverse momentum by the transverse mass. The in-terpretation of the pre-factor gL ⊥ for pions has to bechanged. For pions a smaller degeneracy factor g π = 3corresponds to a larger area L ⊥ ,π = L ⊥ at freeze out.The pion multiplicity distribution then has the followingform: dNdyd p ⊥ = gL ⊥ (2 π ) (cid:104) m ⊥ (cid:16) λ + we | y | √ s (cid:17)(cid:105) − m ⊥ = (cid:113) p ⊥ + m π . (24)For not too large rapidities, its integral over transversemomentum can be expanded in powers of m π a : dNdy ≈ πgL ⊥ λ (cid:16) wλ e | y | √ s (cid:17) (cid:18) − m π aπ + 3 m π a π − ... (cid:19) (25)with a = 1 λ + w e | y | √ s . (26)Collider experiments preferentially take data around cen-tral rapidity as a function of the pseudorapidity η whichrequires only to measure the angle of each particle rel-ative to the beam axis. Since we saturate the reactionproducts by pions, charged hadrons make up 2 / dN ch dηd p ⊥ = 23 (cid:115) − m π m ⊥ cosh y dNdyd p ⊥ . (27) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Η dN ch (cid:144) d Η FIG. 1. Data points show the charged-particle pseudorapiditydistribution in p+p collisions at √ s = 200 GeV from [15]. Thecurves represent the light-cone plasma distributions (solid line K = 0 .
5, dotted line K = 1 . In [13] the √ s dependence of dN ch /dη | η =0 and (cid:104) p T (cid:105) η =0 were parameterized as dN ch /dη | η =0 = 2 . − .
307 ln s + 0 . s (28)and (cid:104) p T (cid:105) η =0 = 0 . − . s + 0 . s. (29)We use these parameterizations as experimental inputs.The theoretical constraint of light-cone momentum con-servation (cf. Eq. 14 or 19) serves as third input to deter-mine the three unknown parameters L ⊥ , λ , and w of thelight-cone plasma distribution. Unlike in e + e − -collisionsit cannot be expected that the total center-of-mass en-ergy is available for particle production. This may bedescribed by subtracting from the cm energy the energyof the leading particles E leading , cf. [14]. Therefore, wewill represent results with a K factor which reduces theeffective cm energy and is defined as K √ s = √ s − (cid:104) E leading (cid:105) . (30)A realistic consideration of the gluons which do not alonecarry the light-cone momentum of the proton would goin the same direction.Figures 1 and 2 show the comparison of the light-coneplasma distribution with the data from RHIC [15] andLHC [13]. One sees that the light-cone plasma distribu-tion for K = 0 . L ⊥ , λ , and w determined from dN ch /dη | η =0 and themean transverse momentum (cid:104) p T (cid:105) at η = 0 for these ener-gies. From the table one can see that the increase of therapidity distribution at η = 0 is of order λ (cf. Eq. 25).A further test of the light-cone plasma distribution isgiven by a measurement of the multiplicity distribution (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) (cid:45) Η dN ch (cid:144) d Η FIG. 2. Data points show the charged-particle pseudorapiditydistribution in p+p collisions at √ s = 7000 GeV from [13],the curves represent the light-cone plasma distributions (solidline: K = 0 .
5, dotted line K = 1 . √ s L ⊥ λ w dN ch /dη | η =0 (cid:104) p T | η =0 (cid:105) (TeV) (fm) (GeV) (GeV)0.20 1.34 0.183 3.44 2.20 0.390.90 1.25 0.216 5.36 3.48 0.452.76 1.28 0.252 6.81 4.56 0.507.00 1.20 0.288 8.21 5.65 0.56TABLE I. For different cm energies and K = 0 . L ⊥ , the effective transverse temperature λ , and the softness w of the light-cone distributions. Thefollowing columns show the experimental input: the multi-plicity dN ch /dη and the mean transverse momentum (cid:104) p T (cid:105) atpseudorapidity η = 0. as a function of transverse momentum for different ra-pidities. In Fig. 3 we plot two experimental transversemomentum spectra for charged hadrons (( h + + h − ) / η = 0 (upper points) [16] and for positive pions at η = 3 . √ s = 200 GeV. The full drawn curves represent thecorresponding light-cone plasma distributions at theserapidities. They fit the inclusive cross sections up to1 GeV/ c rather well. For higher momenta significantcontributions from hard scattering are expected. Theplasma distribution describes the fall-off of the cross sec-tions with transverse momentum by an effective trans-verse temperature λ eff ( y ) which depends on rapidity y : dNdyd p ⊥ = gL ⊥ (2 π ) m ⊥ /λ eff ( y )] − λ eff ( y ) = λ wλ exp | y | / √ s . (32)Due to light-cone momentum conservation the effectivetemperature decreases with increasing rapidity. There-fore, the transverse momentum spectra fall off faster at (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) p T (cid:72) GeV (cid:144) c (cid:76) E d N (cid:144) d p (cid:72) c (cid:144) G e V (cid:76) FIG. 3. Transverse momentum dependence of the inclusivecross section at √ s = 200 GeV for charged hadrons (( h + + h − ) /
2) at η = 0 (upper points) [16] and positive pions at η =3 . K = 0 . larger rapidities. It would be good to test the light-coneplasma distribution over a larger domain in y or η and p ⊥ . IV. NUCLEUS-NUCLEUS COLLISIONS
It is possible to extend the parametrization of the mul-tiplicity distributions to nucleus-nucleus collisions. Wecan use the universality of the light-cone plasma distri-bution originating from pp collisions. In A-A collisions,we multiply the underlying pp multiplicity by the num-ber of participating nucleons N part and take into accountthe increase of (cid:104) p ⊥ (cid:105) with centrality: dN AA ch dηd p ⊥ = N part (cid:115) − m π m ⊥ cosh y dN ( (cid:104) p ⊥ (cid:105) ) dyd p ⊥ . (33)In principle, colliding rows of nucleons contain vary-ing numbers of nucleons in projectile and target nucleuswhich may lead to a small shift in the total cm rapid-ity and total cm energy. But these corrections are minorkinematic corrections in nucleus-nucleus (A-A) collisions.In p-A collisions, however, the kinematics of the differ-ent row configurations is expected to be more important.In both p-A and A-A collisions the increase of the meantransverse momentum (cid:104) p ⊥ (cid:105) with the number of partici-pants is an important feature of the collision which hasto be taken into account. The initial parton distributionsin the projectile nucleus will be broadened by the inter-action with the nucleons in the target nucleus and viceversa.We parameterize the (cid:104) p T (cid:105) data from the STAR exper-iment [18] as (cid:104) p T (cid:105)| η =0 = p +( p − p )(1 − exp( − ( N part − /p )) (34) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Η dN ch (cid:144) d Η (cid:45) (cid:37) a (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Η dN ch (cid:144) d Η (cid:45) (cid:37) b (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Η dN ch (cid:144) d Η (cid:45) (cid:37) c (cid:76) FIG. 4. Charged particle pseudorapidity distributions fromPhobos [15] in Au+Au collisions at √ s NN = 200 GeV for thefollowing centrality classes: a) 45 −
50 %, b) 25 −
30 %, andc) 0 − K = 1) with p = 0 .
390 GeV /c (35) p = 0 .
516 GeV /c (36) p = 52 . (37)We will show in a separate publication how the observedbroadening can be estimated by considering the multiplescattering of partons in pp collisions using the thicknessand density [19] of the other nucleus. Here we use theobserved experimental transverse momentum broadening[18] to fit the input λ and w values, cf. Table II. Note thatfor increasing mean transverse momentum the softness w increases because of the light-cone momentum sum rule.In Table II we show for each centrality the (cid:104) p ⊥ (cid:105) of chargedhadrons [18] and the scaled central rapidity density [15].To determine the light-cone plasma parameters λ and w we fix L ⊥ = 1 .
12 fm. centrality N part (cid:104) p T (cid:105) N part / dN AA ch dη λ w (GeV) (GeV)45-50% 65 0.478 2.96 0.238 4.51625-30% 150 0.509 3.63 0.253 5.1840-3 % 361 0.516 3.81 0.263 5.650TABLE II. For √ s = 200 GeV Au+Au collisions the tablegives the centralities, the number of participants N part , meantransverse momentum (cid:104) p T (cid:105) , and dN AA ch /dη/ ( N part /
2) at η = 0together with the resulting light-cone plasma parameters for K = 1. The theoretical rapidity distributions in Figs. 4 a, b,c reproduce the variation of the measured rapidity dis-tributions rather well for K = 1. Since for fixed energythe size of the nucleon-nucleon overlap area L ⊥ is con-stant the increase of dN AA ch /dη at η = 0 is due to thenumber of participants and λ , cf. Eq. 25, 33. The cen-tral multiplicity divided by the number of participants(cf. Fig. 5) illustrates the increase originating from thehigher mean transverse momentum or effective transversetemperature λ .In Fig. 6 we give the dependence of the mean trans-verse momentum on rapidity for √ s NN = 200 GeV. Thedata points are from STAR [20] and the the curve is cal-culated for central Au-Au collisions. As shown before forpp collisions the conservation of light-cone momentummakes the effective transverse temperature decrease forlarger rapidities. V. ENTROPY OF THE LIGHT-CONE PLASMAAND THE THERMAL PLASMA
It is instructive to compare the light-cone plasma dis-tribution with a sum of thermal distributions boostedalong the z axis [3–5]. Thereby one mimics the widerapidity plateau seen in data: dN bt0 ( y, p ⊥ ) dyd p ⊥ = gL ⊥ L z (2 π ) E p e E p /T − dN bt ( y, p ⊥ ) dyd p ⊥ = (cid:90) y max − y max du dN bt ( y − u, p ⊥ ) dyd p ⊥ . (39)For a comparison of entropies the same transverse area ofthe system ( L ⊥ = 1 .
12 fm) has to be chosen and the same (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) N part (cid:72) d N c h (cid:144) d Η (cid:76) Η (cid:61) (cid:144) (cid:72) N p a r t (cid:144) (cid:76) FIG. 5. The experimental (blue points) [15] and theoretical(red points, K = 1 for all centralities) scaled central multi-plicities in Au+Au collisions at √ s NN = 200 GeV are shownas a function of the number of participants. (cid:230) (cid:230) Η (cid:88) p T (cid:92) (cid:72) G e V (cid:144) c (cid:76) FIG. 6. Mean p T as a function of η in central Au+Au col-lisions at √ s NN = 200 GeV compared to data from STAR[20]. constraints have to be included for both distributions. Asconstraints we use the mean transverse energy, the multi-plicity, and the light-cone momentum sum rule. We takeas an example the light-cone distribution for pions corre-sponding to √ s NN = 200 GeV central Au-Au collisionsand compare it with a the boosted thermal inclusive par-ticle distribution. Both distributions are divided by thenumber of participants.The thermal spectra at fixed rapidity have a ratherdifferent functional form, but we can fit the tempera-ture to reproduce the transverse energy. The interval[ − y max , y max ] can be adjusted to be in agreement with theconstraint of light-cone momentum conservation. Sincethe transverse extension of the volume is fixed, the re-maining longitudinal extension L z of the volume is thenfitted to the total multiplicity. We find the following val- (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Η d N c h (cid:144) d Η (cid:144) (cid:72) N p a r t (cid:144) (cid:76) FIG. 7. Comparison of the boosted fireball distribution(dashed line) and the light-cone distribution (solid line) satis-fying the same constraints (see text) in comparison with data[15]. ues for the boosted thermal fireball: T = 0 .
189 GeV (40) y max = 4 . L z = 3 . n x,p ⊥ = 1 e m ⊥ λ + xw − n bt x,p ⊥ = L z π (cid:90) y max − y max d u m ⊥ cosh( y − u )exp ( m ⊥ cosh( y − u ) /T ) − . (44)The impact parameter dependence is homogeneous in anarea of size L ⊥ .We obtain for the light-cone distribution and theboosted thermal distributions rather similar entropy val-ues with a larger entropy for the light-cone distribution as it should be: S lc = 142 . S bt = 141 . L z in spiteof boosting: L z = V /L ⊥ (47)Owing to the light-cone sum rule there are only twoparameters in the light-cone distribution. This smallernumber of input parameters together with the maximumentropy argument and the better agreement with thedata favor the light-cone plasma distribution. VI. CONCLUSIONS
In regard of the simplicity of the maximum-entropyansatz, the light-cone plasma distribution is very success-ful. Minimal experimental information about the meantransverse momentum and the active area of the collid-ing hadrons optimally account for rapidity and trans-verse momentum distributions in nucleon-nucleon andnucleus-nucleus collisions. Therefore, many other fea-tures of heavy-ion collisions should be reconsidered. Theentropy of fireballs distributed uniformly in rapidity isonly slightly smaller than the entropy of the light-conedistribution. Conceptually, however, the light-cone dis-tribution presented in this paper emphasizes the non-equilibrium nature of the collision process. Its parame-ters, an effective transverse temperature and (longitudi-nal) softness reflect the asymmetry of transverse and lon-gitudinal momenta of the produced particles. In nuclei,multiple scattering of partons leads to an increase of themean transverse momentum of produced particles whichcorrelates strongly with the central rapidity density. Inthe future we plan to include different distributions forquarks and gluons and study fluctuations in more detail.This may lead to slight changes of the rapidity and trans-verse distributions. Such an extended model would alsoallow to calculate specific hadronic spectra.
ACKNOWLEDGMENTS
H.J.P. would like to thank J.P. Vary whose encourage-ments were very important to push the project ahead.We are grateful to J.P. Blaizot and L. McLerran for help-ful comments. [1] D. H. Rischke, Prog. Part. Nucl. Phys. (2004) 197-296. [nucl-th/0305030].[2] J. W. Harris, B. Muller, Ann. Rev. Nucl. Part. Sci. (1996) 71-107. [hep-ph/9602235].[3] E. Schnedermann, J. Sollfrank and U. W. Heinz, Phys.Rev. C (1993) 2462 [nucl-th/9307020].[4] P. Braun-Munzinger, J. Stachel, J. P. Wessels and N. Xu,Phys. Lett. B (1995) 43 [nucl-th/9410026].[5] F. Becattini, J. Cleymans, J. Strumpfer, PoS CPOD07 (2007) 012. [arXiv:0709.2599 [hep-ph]].[6] E. T. Jaynes, Phys. Rev. (1957) 620-630.[7] E. T. Jaynes, Phys. Rev. (1957) 171-190.[8] B. Webber, Scholarpedia, 6(12):10662 (2011).[9] D. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A (2005) 609 [hep-ph/0408050].[10] G. P. Lepage and S. J. Brodsky, Phys. Rev. D (1980)2157.[11] H. J. Pirner, B. Galow and O. Schlaudt, Nucl. Phys. A (2009) 135. [12] R. G. Roberts, Cambridge, UK: Univ. Pr. (1990) , (Cam-bridge monographs on mathematical physics), p. 160[13] V. Khachatryan et al. [ CMS Collaboration ], Phys. Rev.Lett. (2010) 022002. [arXiv:1005.3299 [hep-ex]].[14] B. B. Back et al. [PHOBOS Collaboration], Phys. Rev.C (2006) 021902.[15] B. Alver et al. [PHOBOS Collaboration], Phys. Rev. C (2011) 024913 [arXiv:1011.1940 [nucl-ex]].[16] C. Albajar et al. [UA1 Collaboration], Nucl. Phys. B (1990) 261.[17] I. Arsene et al. [BRAHMS Collaboration], Phys. Rev.Lett. (2007) 252001 [hep-ex/0701041].[18] T. S. Ullrich [STAR Collaboration], Heavy Ion Phys. (2004) 143 [nucl-ex/0305018].[19] S. Domdey, D. Grunewald, B. Z. Kopeliovich,H. J. Pirner, Nucl. Phys. A825 (2009) 200-211.[arXiv:0812.2838 [hep-ph]].[20] F. Simon [STAR Collaboration], Prog. Part. Nucl. Phys.53