Kämpfer, Burkhard

Pavlenko, O. P.

English

Qucosa – Hemholtz-Zentrum Dresden-Rossendorf

Probing early parton kinetics by photons, dileptons and charm* B. ~ ä m ~ f e r " ~  and O.P. PavlenkoC "Research Center Rossendorf, PF  5101 19, 01314 Dresden, Germany bInst. Theor. Phys. (KAI e.V.), Technical University Dresden, Germany 'Inst. Theor. Phys. Kiev, Ukraine Equilibration processes in pre-equilibrium parton matter are considered. We investigate chemical quark equilibration, partial thermalization and overall thermalization, and their influence on electromagnetic (photons, dileptons) and charmed probes. 1 1. Introduction The future heavy-ion colliders RHIC and LHC will achieve energy regions where many semihard-scattered, secondary partons are produced, The estimated energy densities'are so large that the dense parton matter is expected to stay for some time in a d~econfined state. Whether the partonic interactions are efficient enough to bring the system in a quark-gluon plasma state is still matter of debate. Also the identification of the transient plasma, whence created, is a difficult task, despite of many proposed signals. Nowadays, parton cascade models [I-31 are under development, which are aimed to follow the parton evolution in the Course of a nuclear collision on grounds of improved perturbative QCD. Since such cumbersome codes need some numeric efforts, it seems to be desirable to model various evolution aspects by more transparent analytical simulations. It is the aim of the present contribution to describe equilibration processes in pre- equilibrium parton matter in a sufficiently tracktable way to get some insight in their importance for plasma probes. In the last decade most probes are calculated for a fdly locally equilibrated quark-gluon ~ l a s m a  [4]. Since the initial parton distributisn in two separated nuclei differs totally from such a plasma state, there must be a pre-equilibrium stage before the plasma era, and its relevante for the ~ robes  has been till rlecently a miracle. Here we go a step into past arid consider the ~re-ecpilibrium prior to the plasma. This does not resolve the entire history from the very beginning of a nuclear collision, but might demonstrate how important the pre-equilibrium parton matter is. We consider three scenarios: - (i) Due to the relations of total Cross sections G„ > a„ > C„, a dense and hot, tkies- malized gluon plasma should be created first, according ts Ref. [5j at proper time 7 'supported by BMFT under grant 06 DR 107; invited talk at Quark Matter '93 (Boxlänge), to be ~ublished in Nud. Phys. A, Proceedin- ~f 10th International Conference on Ultrardativistic Nucleus-Nucleus Collisions 0.5 fm/c. The quark admixture is rather low, according to Refs. [I, 3, 5, 61 30% sf its chemical equilibrium value. Relying on a simple transport model sf Boltzmann type we calculate within lowest order a, processes the quark chemical equilibration process. As application we consider Photon production in such an initially gluon-enriched plasma. - (ii) In Iine with the above quoted Cross sections the glue system thermalizes earlier than quarks. Assuming primordial secondary-parton prolduction in minijets, the component might possess some time another distribution than gluons till ecamplete tber- malization. Since the minijet picture predicts a power-law high pl-tail, such tails might be favorable for producing large invariant mass probes such as charm Pairs. - (iii) At very early times all partons are expected to be off-equilibrium. Within a sim- plified kinetic model we follow the evolution towards thermal equilibrium. As usefull frame-independent probe we consider dileptons. 2. Chemical equilibration of quarks/photons 2.1. Quark cooking The boost invariant Boltzmann equation describes the evolution of parton species a = g, q, q with energy E, in tcrms of their distribution functions fa due to collisions (C = collision term) as function of the proper time T = d m  and rapidity E (6 = 17 - y, q/y = space-timelparton rapidity). Phase space integration gives the evolution equation of densities ( n  = J drf ,  dI' zz g ( 2 ~ ) - ~ d ~ * )  Using, according to the above outlined evolution scenario (i), thermal distributisns f„ in Maxwell Boltzmann approximation, one gets where n = n, + n, + nq and X = n/n, stand for total paston density and gluon weight, and ,L? = 419. In case of lowest order gg + qq reactions the reactivity R becomes 5 W mZ Z(m, T ) ,  Z = d z Z 2 ~ ( Z )  I { ~ ( ~ ) ,  24m2 with regularizing thermal quark mass m2 = T22ra,/3. The evolution of the relative quark weight proceeds as dispayed in Fig. 1. In accord with other estimates [3, 6, 81 one observes a rather slow cooking of quarks. The longitudinal exgansion hinders oviously the quark chemical equilibration. Processes gg -t ggg [6] prevent the fast gluon dilution Figure 1. Time evolution of the relative quark Figure2. Thephoton yieldper four- weight for different initial values. Full and dashed volume in units of the equilibrium lines use xo = 0.8, no = 10 fm-3, T. = 500 MeV, yield (X„ = 0.4) for different photon 70 = 0.5 fm/c, a, = 0.58, while the dotted line de- energies (full/dashed/dotted lines - picts an example with a three times smaller value E = 31211 GeV). In both yields of n07-O~ö2; dashed line: without expansion. For the same total parton density is as- details see Ref. [7]. sumed. (mimicked here by switching off the expansion), and higher order prscesses gg --+ ng, n 2 4 further damp the gluon dilution [9]. But the dashed line in Fig. 1, which is without expansion, indicates anyway a slow equilibration. The inelastic processes also change the temperature evolution, which proceeds for 2 + 2 reactions according to eq. (1) as T = T ~ ( T ~ / T ) ~ / ~ .  2.2. Photon yield The photon yield according to Ref. [10] is for q?j 4 gy and qg 4 qy processes (CE = Euler's constant) with an infrared cut-off kC = T247ra,/3. AS seen in Fig. 2 the yield is strongly suppressed for the gluon enriched plasma, i.e., at X 4 1. There are indieations from the HIJING simulations 161 that in early stages the parton density is diminished, compared to full equilibrium density at given temperature. ThPs wouM cause an additional suppression. The parton cascade [I] does not show such a pronounced parton suppression; this might be due to different low-rnomentum cut-sdf's. The time integrated yield is displayed in Fig. 3 in relative uwits. As seen in the lower part the main affect on the yield csmes from the initial teniperature. ompased with t h s e  large variations the suppression of the photon yield due t s  reduced quark csncentsation is a minor effect as long as the initial quark contributilsn is not smaller than 3 chemical equilibrium value. A sufilciently fast chernical equilibration aEects the slopes, Pigure 3. Time integrated photon yield = dNq /7cRZSn:P as function of d v d 3 p l E  energy. Lower part : equilibrium yields for different initial times (in fmJc) and temperatures (in MeV), rO/TO. Upper part: different evolutions of the quark weight (fullJdashed [long-dashed] lines - X = x„/xo = 0.8 and slow [fast] equilibra- tion) . Figure 4. Number of charmed pairs diVcc/dM2d~ (Ncc = N c c ~ ( ~ a , n o ~ o ) 2 )  as function of invariant mass. Full (dashed) line: pairs from power-law (thermal) quarks, dotted line: yield from thermal gluons. The evolution is terminated when reaching an energy density corresponding to Tf = 200 MeV. 7-0 = l / p l  is used. which might be used to measure the degree of quark equilibration. The absolute yields might serve as thermometer. As Ref. [3] we conclude that a higher initial temperature in the hat glue scenario more than overcompensates the lack of quarks. Not~e however that the photon energy is frame dependent. A substantial transverse expansion wsuld enhance the large-E yield [ll]. For a comparison of the parton yield with later hadronic sources see Refs. [3, 10, 111. 3. Partial  thermalization/charm Minijets are known to obey a power-law pl distribution [E]. Due to the correlation of momentum space and configuration space, the very early parton distribution in a localized voluma element might be parametrized as with p*, as regularizin low-momnturn cut-0% and N as nsrmalization. As shown in Refs. [13, 141 an ini tiaP distribution fo evolves, according to the relaxation time approximation to eq. (11, as The corresponding charm pair production rate per space-time unit reads (V„. = relative velocity of fusing partons). For a boost invariant distribution in the rapidity interval dY the yield, according to lowest order gg -+ cC and qp + cc processes, for pairs with invariant mass M and transverse momentum q l  can be cast in the form In case of b = qq one has The Jl term describes fusion processes of partons stemming from the initial distribution fo, while J2,3 incorporate processes of initial distribution partons with already equilibrated partons (described by F) or among the equilibrated ones. The corresponding expressions for J2,3 are rather involved and not displayed here. In Fig. 4 the rate for b = q4 is displayed for > T O , ~ ,  i.e., an idealized situation where the minijet structure of the quark distribution persists some time before relaxing. This rate is compared with the thermal rate 115, 161 where H(x) = Iio (x)x2(x2 + 8) + ICl (x)4s(x2 + 4). For b = gg one has and b = qq follows analsgously with X,@ = X g g ( l  - zO)gg ~ ~ 9 ~ ~ ) .  Here we kave normalized the parton energy density and particle number in such ay that nth = n,i. - n, eth = e,.l. = e an$ n, = sn ,  nqs = ( 1  - x)n. This enables us ts cornpare the minijet rate (I with the thermal rate (14), both ones having the same n, e, see dxhed IPne in Fig. 4. As expected the minijet quark-fusion rate exceeds also partidly the thermal gluon rate (dotted line). This points to an enhanced charm productisn by a not yet eqaiilibrated quark distribution irnmersed in an dready thermalized um bath, accordi glue scenario 151. However, since the ratio of q ~ /  g rates is proportisnd to suppmxkm of the qua& weight 1 X, aa discussed in the previous s suppresses the gij source: sco - 0. instead of s, = 0.4 ac in FP by a fax;tos 37. Thereif~re~ iif the qua& admixture is Indeed so small, charrn probes very on diskribution f„ while the greviously discuissed phstons probe .fgfqaq. 4.1. IParton thermalization Assaiming that an initial distribution of partons scattered into a small volume dement at midrapidity evolves according to a kinetic theory of Boltzmann type eq. (I), then the parton distribution of secondaries evolves according to eq. (8) with [13, 141 In this form one employs a relaxation time approximation of the collision integral. Al1 details of the Cross sections are accumulated in the time dependent relaxation time T„.(T). Inelasitic processes are included too, since the particle number is not conserved. The tem- perature parameter T is determined by the evolution of the energy density e = 1 dI'Ef, cf. [13, 171. Since we are going to consider dileptons with invariant mass M N 2 - 3 GeV the cut-off in the minijet motivated initial distribution (7) and possible contributions of softer partons prevents a straightforward use of fo(7). In this situation we resort to an initial distribution fo which is motivated by the Schwinger mechanism of parton pair production with Ii as parameter of average transverse parton mornentum < p~ >= and N as normalization. 4.2. Dilepton yield Within such a framework the dilepton yield is similar to eq. (9). For lowest order cr„ a qij -t ET processes it becomes - (1312 = \l% - 2 M ~ ~ f i c h ( ? - b ) + p : ~ ,  s h b  = L ( M ~ s h g  P.L~ - pl l shh) ,  PI - $M2[M~ ch(7 - 61) $ q~]-', R is the transverse radius of the radiating region) and has been analyzed in Ref. [14]. According to estimates in Refs. [14, 18, 191 the yield at M > 2 GeV is expected t s  exceed the Drell Yan background (which ultimately dominates at higher M) at  RHIC and LHC energies. The time evolution of the yield is displayed in Fig. 5. Accosding to eqs. (8,16,l ) t he yield contains terms f f oc fo fo(a), and foF(b), and FF(c ) .  As seen in Fi . 5 fsr large rdaxation times (taken for simplicity as constant) the fo fo term dominates, hile fsr very short rdaxation times ~~~1 5 q, also the thermalized partons cc FF gives a substantial csntribution. Adding all contributions we find that of the total yield is prsduced within 1/3 sf the lifetime of secondary partons. There- fsre, if the parton ssurce dorninates the total yield (including also the hadron sources), the dilleptons seexn to probe mainly tbe isitial diatributisn of secandaries and carry little Figure ferent scaled 5. Time contributions dilepton yield evolution of dif- (see text) to the dN,i X i  = & w d g : d ~  1 o3 12M2(h~)4 / (R2a2) (in fm2 GeV2) for pairs with M = 2 GeV, q~ = 0.2 GeV (full/dashed/dotted lines: rTel = 10, 1, 0.13 fm/c). ro = 0.13 fm/c, I< = 1.2 GeV2. rf is determined by the freeze- out energy density corresponding to 200 MeV. Figure 6. The dilepton spectrum eq. (18) Nii -* (f .~. ,  in GeV-") as func- tion of transverse momentum at ML = 2.6 GeV. The charged hadron rapidity density is d N J d y  = 103, and R = 7 fm, Ti = lS0 MeV, I< = 1.2 GeV2, r0 = 0.13 (f.s.) and 1 (QGP) fm/c. ~f is determined by the freeze-out en- ergy density corresponding ts Tf. QGP shows the ideal scaling of a longitudinally expanding massless quark-gluon plasma. information from the later equilibrium (say quark-gluon plasma) era, unless the latter one lasts very long (e.g., in a long-living mixed phase). Another interesting outcome of our investigations [14, 171 is that once a thermalized state is achieved the lateron evolution does not affect much the invariant mass speetrum of dileptons. E.g., in the HIJING code rather early the partons, due to trivial kinematics, pass through a state whicli looks to some extent thermal 161. I.e., the dilepton spectrum is determined mainly by this state and the prior stages. In case of rTel > 70,f the dominating fofo term results in As seen in Pig. 6 (curve labeled by f. s.) this shows the so-cakd ML scdinng: at const the yield is fairly independent of We find that this also Pmolds for the case sf  a relaxi distribution f (T) ( [Z?O]. This generalizes the statement of Ref. [21] where both the Drdl Yan centribution and the yidd frsm a tnransversalPy expanding uon pfiasnntz are found to clisplay appsoximate ML s c a h  . Any scale destroys the scaling, e.g., formfactors in the hadron gas cause a change of the yield by more than a factor 30 when going from q~ = 0 to 2 GeV at A l l  = 2.6 GeV. Recent parton cascade resullts [22] display also strong violations of the ML scaling, however, it needs to be clarified whether this is a consequence (or artifact) of low-pl cut-offs and parton formfactors due to higher twist effects implemented in the code. 5. Parton masses The above presented probes rely partially on undressed parton masses. To regularize certain Cross sections we employ thermal masses M*(T) oc g(T)T. Such thermal masses might also be extracted from available lattice QCD data. SU(3) pure gauge theory data for a 1 6 ~  lattice [23] can be fitted by a family of thermodynamically self consistent equations of state [24] for interacting quasiparticles Obvious choices are (i) dp/dM* = 0, which interrelates M* and p* 224, 251, or (ii) p* = B = const, or (iii) M* = M = const. Such fits [26] yield (i) M* cc T for T/T, = 1.4 - 2.2, s r  (ii) M* = 0.5375g(T)T7 g2 = 16r2/(11 ~ o ~ ( T / T , ) ~ ) ,  B = 0 for T/T, = 1.1 - 2.2, or (iii) P* ec T, M = 0. Besides the convenient parametrizations of the equation of state [27], whicli'deviates strongly from the earlier popular bag model, the Ansätze (i, ii) point to substantial effective parton masses m N 0(3)Tc in the available temperature range up to 2.2 T, and indicate a divergent behaviour at T,. Similar results are found in SU(2) gauge theory lattice data too [28]. These finite screening masses of Course affect the various primordial plasma probes, as demanstrated in Ref. [25] for charm production. Screeoing masses in the off-equilibrium plasma represent a yet unsolved problem. Such masses might also be reponsible for destroying the ML scaling, which is conjectured to appear in any massless parton system. 6. Summary In Summary we present here analytical approaches to specific equilibration processes in evolving palrtsn matter. As analog work 161 our presentation is aimed to analyze qual- itatively some aspects which are thrsughoutly implemented in complex partan cascade codes. As pointed out by M. Gyulassy [29j the transition region of semihard (i.e., per- turbatively tracktable) &CD prsceses and softer ones represents a difficult region which needs more investigations with respect of reIiable predictions of future RHIC and LHC resul t s. REFERENCES 1. K. Geiger, B. Müller, Nucl. Phys. B369 (1992) 600, K. Geiger, Phys. Rev. D46 (1992) 4965, 4986, D47 (1993) 133, NUC-MINN-93/5-T 2. X.N. Wang, M. Gyulassy, Phys. Rev. D45 (1992) 844 3. E.V. Shuryak, L. Xiong, Phys. Rev. Lett. 70 (1993) 2241 4. Proc. Quark Matter '91, Nucl. Phys. A544 (1992) lc,  (Eds.) T.C. Awes et al. 5. E.V. Shuryak, Phys. Rev. Lett. 68 (1992) 3270 6. T.  Birb, E. van Dorn, B. Müller, M. Thoma, X.N. Wang, Duke-TH-93-46 7. 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Probing early parton kinetics by photons, dileptons and charm

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Probing early parton kinetics by photons, dileptons and charm

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