Minijet Initial Conditions For Non-Equilibrium Parton Evolution at RHIC and LHC Fred Cooper, Emil Mottola and Gouranga C. Nayak T-8, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract An important ingredient for the non-equilibrium evolution of partons at RHIC and LHC is to 3 0 have some physically reasonable initial conditions for the single particle phase space distribution 0 2 functions for the partons. We consider several plausible parametrizations of initial conditions for n a the single particle distribution function f (x,p) and fix the parameters by matching f(x,p)pµdσ J i µ R 4 totheinvariantmomentumspacesemi-hardpartondistributionsobtainedusingperturbativeQCD 2 (pQCD),aswellasfittinglowordermomentsofthedistributionfunction. Weconsiderparametriza- 2 v 1 tions of fi(x,p) with both boost invariant and boost non-invariant assumptions. We determine the 9 3 initial number density, energy density and the corresponding(effective) temperature of the minijet 0 1 plasma at RHIC and LHC energies. For a boost non-invariant minijet phase-space distribution 2 0 function we obtain 30(140) /fm3 as the initial number density, 50(520) GeV/fm3 as the / ∼ ∼ h p initial energy density and 520(930) MeV as the corresponding initial effective temperature at ∼ - p RHIC(LHC). e h : v PACS numbers: PACS: 12.38.Mh; 12.38.Bx; 13.87.Ce;25.75.-q i X r a Typeset by REVTEX 1 In order to study thermalization of quark-gluon plasma at RHIC and LHC in any kinetic approach, oneneedssuitable initialconditionsforthesingleparticlephase-spacedistribution of partons. The hard and semi hard parton production can be calculated by using perturba- tiveQCD(pQCD)[1]. Thesoftpartonproductioncannotbecalculatedinakineticapproach within perturbative QCD. One approximate strategy for looking at soft parton production is to assume thecreation ofa classical chromofield [2, 3, 4]. The first problem one facesin spec- ifying initial conditions for the phase space single particle distribution function f(x,p,t ) of 0 partons is that from pQCD one has information only about the momentum space distribu- tion of minijets. The pQCD calculation (using scattering matrix element squared and the structure functions) is done in an approach which calculates probabilities between initial and final states so that any information about space-time evolution is lost. Thus we need to supplement pQCD with assumptions about the dependence of f(x,p) on x. InthispaperourstrategywillbetousepQCDtodeterminethesingleparticledistribution functionforthepartonsinmomentum space, andthenbytaking simple parametricformsfor f(x,p) constrain the parametrization by fitting the single particle distribution as well as the average transverse momentum. This process can be refined by adding more parameters and fitting further moments of the jet distribution function. Ofcourse, onehopes that thedetails of the parametrization of the initial conditions do not significantly effect the thermalization process. The lowest order pQCD inclusive (2 2) minijet cross section per nucleon in A-A → collision is given by: 2πp σjet = Z dpTdy1dy2 sˆT x1 fi/A(x1,p2T) x2 fj/A(x2,p2T) σˆij→kl(sˆ,tˆ,uˆ), (1) X ijkl where σˆij→kl is the elementary pQCD parton cross sections for the process ij kl. As → gluons are the dominant part of the total minijet production, we will only consider the processes gq(q¯) gq(q¯) and gg gg in this paper. The partonic level cross sections for → → these processes are given by: α2 1 4 σˆgq→gq = sˆs(sˆ2 +uˆ2)[tˆ2 − 9sˆuˆ], (2) 9α2 uˆtˆ uˆsˆ sˆtˆ σˆgg→gg = 2sˆs[3− sˆ2 − tˆ2 − uˆ2]. (3) The rapidities y , y and the momentum fractions x , x are related by, 1 2 1 2 x = p (ey1 +ey2)/√s, x = p (e−y1 +e−y2)/√s, (4) 1 T 2 T 2 with the kinematic relations: sˆ y y sˆ= x x s, and tˆ= [1 tanh( 1 − 2)]. (5) 1 2 −2 − 2 To compute the minijet cross section using Eq. (1) we need to specify the minimum trans- verse momentum cut-off p above which the incoherent parton picture is applicable. The 0 momentum cut-off is 1 GeV at RHIC and 2 GeV at LHC, which are obtained from ∼ ∼ the argument that there is saturation of the gluon structure function at low x [5]. For our quantitative calculation we use p = 1 GeV at RHIC and 2 GeV at LHC. The nuclear 0 modified parton distribution functions which includes the shadowing effects are given by f (x,Q2) = R (x,Q2)f (x,Q2) where A and N stand for nucleus and free nucleon re- i/A A i/N spectively. In this paper we use the GRV98 set of parton distributions for the free nucleon distribution function f (x,Q2) [6] (in momentum space). The GRV98 analysis uses the i/N low x HERA data on deep inelatstic scatterings along with data from other hard scattering processes at fixed Q. The Q2 evolution of the parton distribution function is performed by using perturbative QCDevolution equations. WeusetheEKS98 numerical parametrizations for the ratio function R (x,Q2) [7]. The EKS98 parametrization uses NMC and E665 struc- A ture functions from deep inelastic lepton-nucleus collisions and E772 Drell-Yan data from proton-nucleus collisions at fixed Q. The Q2 evolution of the structure function is studied by using the DGLAP evolution equation. We multiply the above minijet cross section by a standard K factor (K=2), to account for the higher order contributions. Forcentralcollisions, theminijetcrosssection(Eq.(1))canberelatedtothetotalnumber of partons (Njet) by dNjet 2πp dydpT = K T(0) Z dy2 sˆT Xijklx1 fi/A(x1,p2T) x2 fj/A(x2,p2T) σˆij→kl(sˆ,tˆ,uˆ), (6) where T(0) = 9A2/8πR2 fm is the nuclear geometrical factor for head-on AA collisions (for A a nucleus with a sharp surface). Here R = 1.1A1/3 is the nuclear radius. Similarly the A transverse energy distribution of the minijets are given by: dEjet 2πp2 dydTpT = K T(0) Z dy2 sˆT Xijklx1 fi/A(x1,p2T) x2 fj/A(x2,p2T) σˆij→kl(sˆ,tˆ,uˆ). (7) Our numerical results for the p distribution of the minijets computed from eq. (6) are T plotted in Fig. 1. 3 5 10 1/2 LHC (Pb−Pb, s =5.5 TeV) 1/2 RHIC (Au−Au, s =200 GeV) 4 10 3 10 ) 1 − V e G 2 10 ( T p d / N d 1 10 0 10 −1 10 2 3 4 5 6 7 8 9 10 p (GeV) T FIG. 1: Transverse momentum distribution of the initial minijets at RHIC and LHC The rapidity distributions of the minijet production at RHIC and LHC are obtained by integrating over p for p > p are shown in Fig. 2. It can be seen that the minijet T T 0 distribution is not flat over the whole rapidity range. Hence it is not a particularly good approximation to assume boost invariance [8] and restrict oneself to the mid rapidity region. One important question that the transport equations should answer is how different the final rapidity distribution of the hadrons (or other signatures) from the rapidity distribution of the minijets at the pre-equilibrium stage are [9]. Of course this needs to be supplemented by further dynamical evolution near the hadronic phase transition. For this reason it is not appropriate to take into account just the flat minijet rapidity distribution seen in the mid rapidity region when evolving the minijet plasma. Partons formed outside the mid 4 2500 1/2 LHC (Pb−Pb, s =5.5 TeV, p = 2 GeV) 0 1/2 RHIC (Au−Au, s =200 GeV, p =1 GeV) 0 2000 1500 y d / N d 1000 500 0 −8 −6 −4 −2 0 2 4 6 8 y FIG. 2: Rapidity distribution of the initial minijets at RHIC and LHC rapidity region do propagate in the pre-equilibrium stage with their rescatterings and hence one must consider all the partons in determining the initial condition of the QGP in the pre-equilibrium stage. Thus we require additional information about how these partons are distributed in coor- dinate space. A simple ansatz based on scaling ([8]) allows extracting energy density from the transverse energy E distribution via: T 1 dE T ǫ = (8) πR2τ dy A 0 This formula is correct if partons are uniformly distributed in coordinate space and the mo- mentum rapidity y is equal to the coordinate rapidity η. For an 1+1 dimensional expanding system coordinate rapidity η is defined via: t = τ coshη and z = τ sinhη. 5 Traditionally the above formula is widely applied to obtain energy density from E dis- T tribution of minijets. Although the above formula might be applicable in a situation where the minijet distribution is flat in rapidity, it is precisely the scale breaking effects that we hope to understand from our transport approach. As it can be seen from Fig. 2 the initial minijet distribution function is not flat in the entire rapidity range and hence the above formula is not applicable in the pre-equilibrium region. Also it is not clear that partons are uniformly distributed at the initial time and just dividing by a volume (see eq. (8)) to obtain the energy density from the total energy is a rather crude estimate of initial condi- tions. To get a better estimate for the initial conditions one should take into account the correlation between coordinate rapidity η and momentum rapidity y. We will assume this correlation has a particular form, and then look at the sensitivity of the global quantities to the parametrization. For an expanding system the minijet number distribution eq. (6) can be related to the phase-space distribution function via [10]: d3Njet = g f(x,p) pµdσ . (9) πdydp2 CZ µ T where g = 16 is the product of spin and color degrees of freedom, C dσµ = πR2τdη(coshη,0,0,sinhη), (10) A and pµ = (p coshy,p cosφ,p sinφ,p sinhy), (11) T T T T for an 1+1 dimension expanding system. Using the above relations we get at the initial time τ (=1/p ): 0 0 dNjet = g πR2τ dη p cosh(η y) f(p ,η,y,τ ), (12) πdydp2 C A 0Z T − T 0 T where we assume a transverse isotropy at the early stage. In the following we will obtain the initial phase-space distribution function of the gluon minijets by using both boost invariant and boost non-invariant schemes. A. Boost Invariant Initial Distribution Function From Minijets First let us assume that a boost invariant description [8] is appropriate. Then the gluon distribution function depends only on the boost invariant parameters τ, p and ξ = η y. T − 6 We will parametrize the η y correlation as Gaussian: − −(η−y)2 f(pT,η y,τ0) = f(pT)e σ2(pT), (13) − where f(p ) will be determined from eq. (12) by using eq. (6) which is obtained by using T pQCD. Using the above equation in eq. (12) we get: dNjet ∞ −(η−y)2 dp = gC2π2RA2τ0 p2Tf(pT)Z dyZ−∞dηcosh(η−y)e σ2(pT) (14) T which gives: dNjet = g 2π2√πR2τ p2f(p )σ(p )eσ2(pT)/4 dy (15) dp C A 0 T T T Z T where the maximum allowed value of the momentum rapidity at RHIC and LHC depends on p and √s. Integrating over y we get from the above equation: T dNjet e−σ2(pT)/4 f(p ) = dpT , (16) T g 2π2√πR2τ p22ln(√s/2p + s/4p2 1) σ(pT) C A 0 T T q T − which gives a boost invariant initial distribution function: dNjet − ξ2 f(p ,ξ,τ ) = dpT e σ2(pT) . (17) T 0 g 2π2√πR2τ p22ln(√s/2p + s/4p2 1)σ(pT)eσ2(pT)/4 C A 0 T T q T − Using this boost invariant initial phase-space distribution function of the gluon the initial minijet number density is given by: n(τ ) = g dΓpµu f(p ,η y,τ ) = g d2p p dξcoshξf(p ,ξ,τ) 0 CZ µ T − 0 CZ T T Z T dNjet = 2π dp dpT , (18) Z T 2π2R2τ 2ln(√s/2p + s/4p2 1) A 0 T q T − where dΓ = d3p. Similarly the initial energy density is given by: p0 ǫ(τ ) = g dΓ(pµu )2f(p ,η y,τ ) = g d2p p2 dξcosh2ξf(p ,ξ,τ) 0 CZ µ T − 0 C Z T T Z T dNjet = 2π dp p dpT (e−σ2(pT)/4 +e3σ2(pT)/4). (19) Z T T 4π2R2τ 2ln(√s/2p + s/4p2 1) A 0 T q T − B. Boost Non-Invariant Initial Distribution Function From Minijets In the above we used a boost invariant distribution function f(τ,ξ,p ) as the initial T phase-space distribution function. However this is not consistent with the actual rapidity 7 distribution function obtained from pQCD. To relax the boost invariant assumption, we will introduce a function f(p ,y) instead of f(p ) in eq. (13). The form of the boost T T non-invariant distribution we take is: −(η−y)2 f(pT,η,y,τ0) = f(pT,y)e σ2(pT), (20) where f(p ,y) is obtained from dNjet. Note that f(p ) in eq. (13) is obtained from dNjet T dydpT T dpTdy after integrating over y. However, in the above form (eq. (20)), the y dependence is more general which for small σ2 is close to that obtained by using pQCD. Using the above form in eq. (12) we get: dNjet ∞ −(η−y)2 dydp = gC2π2RA2τ0 p2Tf(pT,y)Z−∞dηcosh(η −y)e σ2(pT) (21) T which gives: dNjet = g 2π2√πR2τ p2f(p ,y)σ(p )eσ2(pT)/4. (22) dydp C A 0 T T T T From the above equation we get: dNjet f(p ,y) = dydpT (23) T gC2π2√πRA2τ0p2Tσ(pT)eσ2(pT)/4 which gives a boost non-invariant initial phase-space gluon distribution function: dNjet −(η−y)2 f(p ,η,y,τ ) = dydpT e σ2(pT) . (24) T 0 gC2π2√πRA2τ0p2T σ(pT)eσ2(pT)/4 Usingtheaboveboostnon-invariantgluondistributionfunctiontheinitialminijetnumber density is given by: n(τ ,η) = g dΓpµu f(p ,η,y,τ ) = g d2p p dycosh(η y)f(p ,η,y,τ ) 0 C Z µ T 0 C Z T T Z − T 0 dNjet −(η−y)2 = 2π dp dy dydpT e σ2(pT) cosh(η y), (25) Z T Z 2π2√πRA2τ0σ(pT)eσ2(pT)/4 − and the initial minijet energy density given by: ǫ(τ ,η) = g dΓ(pµu )2f(p ,η,y,τ ) = g d2p p2 dycosh2(η y)f(p ,η,y,τ ) 0 CZ µ T 0 CZ T T Z − T 0 dNjet −(η−y)2 = 2π dp p dy dydpT e σ2(pT) cosh2(η y).(26) Z T T Z 2π2√πRA2τ0σ(pT)eσ2(pT)/4 − 8 C. Initial Energy Density and Number Density at RHIC and LHC Using the above expressions for the minijet initial phase-space distribution functions we cannowpredict theinitialenergydensity andnumber density oftheminijet plasma atRHIC and LHC. As temperature can not be defined in a non-equilibrium situation we define an effective temperature by: 15 T = [ ǫ]1/4, (27) eff 8π2 with the non-equilibrium energy density obtained above. Before we calculate the initial conditions we would like to find out the values of the unknown parameterσ2 appearingintheaboveequation. Wewillfixthisunknown parameter by equating the first p moment of the distribution function with the pQCD predicted E T T distribution as given by eq. (7). We have for the boost invariant case: dp dETjet = 2g π2R2τ dp p3f(p ) = dp pTddNpjTet (2,8) Z T dpT C A 0Z T T T Z T√π2ln(√s/2pT +qs/4p2T −1)σ(pT)eσ2(pT)/4 and for the boost non-invariant case: dp dy dETjet = 2g π2R2τ dp dy p3f(p ,y) = dp dy pTddyNdjpeTt .(29) Z T Z dydpT C A 0Z T Z T T Z T Z √πσ(pT)eσ2(pT)/4 We determine the value of σ2 by using the pQCD values of dETjet and dETjet from eq. dydpT dpT (7) in the left hand side of the above equations. Assuming σ2 independent of p we T find from the above equations: σ2 = 0.0034(0.0015) at RHIC(LHC) for boost invari- ant case and σ2 = 0.28(0.28) at RHIC(LHC) for boost non-invariant case. Note that for boost non-invariant case σ2 is same at RHIC and LHC as the normalization factor √πσ(p )eσ2(pT)/4 in eq. (29) is energy independent whereas the corresponding normalization T factor√π2ln(√s/2p + s/4p2 1)σ(p )eσ2(pT)/4 ineq. (28)dependsonenergy. Using the T q T − T above values for the boost invariant case in eq. (19) we find ǫ =31 GeV/fm3 with T = 0 eff 460 MeV at RHIC and ǫ =287 GeV/fm3 with T = 800 MeV at LHC. The number density 0 eff for the boost invariant case obtained from eq. (18) is n =20/fm3 at RHIC and 85/fm3 at 0 LHC. Using the values of the σ2 for boost non-invariant case we find from eq. (26): ǫ =51 0 GeV/fm3 with T =521 MeV at RHIC and ǫ =517 GeV/fm3 with T =932 MeV at LHC eff 0 eff for η=0. The number density for the boost non-invariant case obtained from eq. (25) is n =30/fm3 at RHIC and 138/fm3 at LHC for η=0. It can be noted that the energy density 0 9 for the boost invariant case is less than that of the boost non-invariant case. This is because we have assumed a boost invariance in the entire rapidity range. However, the actual pQCD minijet rapidity distribution is not flat in the entire rapidity range as can be seen from Fig. 2. Therefore, in order to maintain the boost invariance in the entire rapidity range the den- sity has to be smaller. However, in boost non-invariant case the rapidity distribution of the minijet is minimaly altered and the pQCD form of dN is used throughout the calculation. dpTdy Let us mention briefly other approaches for obtaining initial condition at RHIC and LHC. Commonly used way is to obtain energy density from eq. (8) by using pQCD estimate of the minijet E distribution [1, 11, 12, 13]. This formula assumes that the initial partons are T uniformlydistributedinthecoordinatespacesothatonedividesthevolumetoobtainenergy density. In [14] a boost invariant Boltzmann form f(x,p,t ) = e−pTcosh(η−y)/T was used for 0 the initial distribution function with η y (boost invariant quantity) correlation taken into − account. In the parton cascade model [15] momentum space structure function along with coordinate distribution of the nucleons inside the nucleus was used directly in the transport equation which is different from the minijet calculation by using pQCD. In the approach of this paper we have obtained a phase-space initial non-equilibrium distribution function f(x,p) for the boost non-invariant case with minimal altering the p and y distribution of T the minijet partons obtained by using pQCD calculations. To conclude, we have obtained a physically reasonable initial phase-space distribution function of the partons formed at the very early stage of the heavy-ion collisions at RHIC and LHC which can be used to study equilibration of the quark-gluon plasma [13, 14, 16]. To obtain such phase-space distribution function we have considered several plausible parametrizations of f (x,p) and have fixed the parameters by matching f(x,p)pµdσ to i µ R the invariant momentum space semi-hard parton distributions obtained using QCD (pQCD) as well as fitting low order moments of the distribution. Once the parameters of f(x,p) are found, we then have determined the initial number density, energy density and the corresponding (effective) temperature of the minijet plasma at RHIC and LHC energies. For a boost non-invariant minijet phase-space distribution function we obtain 30(140) ∼ /fm3 as the initial number density, 50(520) GeV/fm3 as the initial energy density and ∼ ∼ 520(930)MeVasthecorresponding initialeffective temperatureatRHIC(LHC).Ourminijet initial conditions are obtained above p =1(2) GeV at RHIC(LHC). As partons below these T momentum cut-off values might be described by the formation of a classical chromofield, 10