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1 DOE/ER/ INT Frame Dependence of Parton Cascade results CU-TP-794 Bin Zhang Department of Phsics, Columbia Universit, New York, NY 27 and INT, Universit of Washington, Seatle, WA Yang Pang Department of Phsics, Columbia Universit, New York, NY 27 (6th November 996) Frame dependence of parton cascade results is studied for dierent schemes of doing cascade simulations. We show that dierent schemes do not alwas agree and results ma have strong frame dependence. When the interaction range is on the order of mean free path and the collisions are done in the two parton center of mass frame, the results are not sensitive to the global frame that the collisions are ordered. Eects of dierent dierential cross sections are also discussed. I. INTRODUCTION Cascade models have been widel used to simulate Relativistic and Ultra-Relativistic Nucleus-Nucleus collisions []. The can be used to simulate sstems that are not in thermal equilibrium and the incooperate the eect of nite mean free path automaticall. Causalit violation is inherent in cascade simulations due to the geometrical interpretation of the cross section, i.e., particles p scatter when their closest distance is smaller than ( is the scattering cross section). p One problem is that the information travels across at a time which is either the xed time step or the mean free path. This ma lead to a speed of information faster than the speed of light and ma cause superluminous shock waves. Another problem is that dierent choices of doing collisions, i.e., dierent collision schemes, ma lead to dierent collision orderings and hence dierent phsical predictions. Man groups have studied the causalit problem for energies below RHIC energ [2] and attempts have been made to reduce the problem. For Ultra-relativistic heav ion collisions at RHIC energies and beond, causalit violation ma be ver serious due to the relativel small mean free path which ma be on the order or even much less than the interaction range. The superluminous signals have been studied [3] in cascade simulations of pa collisions. In this paper, we stu the eect of collision ordering on macroscopic variables b comparing the results of dierent collision schemes. A parton cascade code, GPC (Generic Parton Cascade), we developed recentl [4] has been used in this stu. First, we describe the initial conditions and dierent collision schemes. Then we compare the results from different schemes with a discussion of the eects of dierent dierential cross sections followed b the conclusions. II. INITIAL CONDITIONS AND COLLISION SCHEMES To stu frame dependence of cascade results at RHIC energies, we prepare a sstem of gluons similar to the minijet gluon sstem that is going to be produced at RHIC. 4 gluons are uniforml distributed in the 5 to 5 space time rapidit range. Initiall, the get local thermal equilibrium at temperature 5 MeV. The occup a transverse disk of radius 5 fm at t = and are formed at longitudinal proper time =:fm (i.e., the formation time for a particular particle i is t i = cosh( i ), z i = sinh( i )). We will compare results in dierent frames of ordering the collisions for the following schemes of doing cascade: (a) Two gluons collide when their closest approach distance p in the two particle center of mass frame is smaller than ( is the scattering cross section). The collision space point ischosen to be the midpoint of the two particles in the two-bo center of mass frame at their closest distance, the collisions are ordered in a global frame, either the collider lab frame or the target frame (the dier b 6 unit of rapidit boost); (b) diers from (a) in that the collision space point for a particle (not for a collision) is the position of the particle in the two-bo center of mass frame. So, in the global frame, for a particular parton parton collision, each parton will have its own time of scattering. The collisions are ordered according to the average time of the two scattering times in the global frame; (c) diers from (b) in that the collisions are ordered according to the earlier time of the two scattering times;

2 (d) two partons p collide when their closest distance is smaller than in the global frame. The collisions are ordered according to the collision time in the global frame. Dierent schemes specif dierent parton collision frame and dierent collision ordering time. The sstem propagates from one collision to the next in the global frame instead of from one time step to the next. Partons are all on mass shell. Rutherford cross section d regulated b a screening mass, = 92 S, is taken d^t 2(^t 2 ) 2 to generate the scattering angular distribution. In the above formula, t =(p p 2 ) 2 in which p and p 2 are 4 momenta of one particle before and after the collision. The total cross section for parton collision is taken as: q 2 9 :47 = 92 S 2 2 : () We set S = Numericall [4], we divide the space into cells. We store onl the next collision time, partner and next particle in the same cell in an interaction list to save memor and memor manipulation. The interaction list is onl updated locall, i.e., onl the cell where the collision happens and its neighboring cells are updated. This reduces the number ofchecks and signicantl increases the speed of the parton cascade code. We can get one 6; collisions event in around minutes. Our results have been checked b taking awa the cells to make sure the are consistent with the ordinar calculations without space divisions. At the present stage, there are onl 2 to 2 scatterings. We are working on including the radiation processes into the GPC. III. RESULTS FOR DIFFERENT SCHEMES In our case, R =5fm, dn = 4 and t :2fm taken into account the spread of rapidit around the space time rapidit. We get n 25=fm 3. The interaction p p range. To have l >, we need to have > 4:3 fm :85 GeV. This is a condition for isolated 2 bo collisions that is generall violated. Ideall we also need man particles within the Debe screening scale: N D = n( 4 3 ) : 3 For =3fm, N D We showed below even though the isolated 2 bo collision condition is barel satised, in the situation with strong inside-outside correlations like the sstem evolving from the initial conditions we specied above, a sensible scheme can still produce reasonable results. Fig., Fig. 2 and Fig. 3 give the dn, and dn distributions with cross section = 9 fm2 ( = :47 and = 3 fm ) respectivel. In plotting the target frame ordering case, we make a shift so that the central rapidit alwas has rapidit. After evolution, dn is almost the same as the initial. has a drop from initial value 385 GeV to around 335 GeV. This shows the longitudinal work [5] done b the eective pressure. The absolute value of slope of dn log curve is increasing which clearl shows the eective cooling of the sstem. The results do not depend on the frame we use to order the collisions. Scheme (b) and (c) give similar results. A close look at the average number of collisions per event shows that for scheme (a) we get 69 collisions, for (b) 68 and for (c) 62 when collision order is in the collider lab frame. In the target frame case, each gets several hundred less. The dn, and dn distribution of partons are closel related to the hadronic observables. In this section, we are going to stu parton cascade induced changes of these distributions. First, we look at the case when the mean p free path is on the order of the interaction range ; then the case when the mean free path is much smaller. Further, we stu the dependence of the results on the dierential cross section, i.e., the angular distribution of outgoing particles from a collision. Finall, a discussion on the number of total collisions vs. the number of non-causal collisions and eects of a small particle mass. The mean free path l in which n is the parton number densit and the cross section. The n 2 densit n can be estimated through: n = dn R 2 t : dn/

3 FIG.. 2 event averaged dn distribution for scheme (a) with = :47 and = 3 fm. The dot-dashed curve is the initial distribution for reference; the solid curve is for the case that collisions are ordered in the collider center of mass frame; for the dashed curve, the collisions are ordered in the target frame. 4 3 in scheme (d), we get much more collisions even with a smaller cross section. In the collider frame ordering case, there are around 22; collisions per event; in the target frame ordering case, we get 97;! The frame dependence of scheme (d) is due to the rapidit correlation introduced b taking the closest distance in the global frame. It neglects the ow velocit and increases the densit of the particles. This is also seen in the huge increase of number of collisions with smaller cross section comparing to the other schemes. Scheme (a), (b) and (c) take into account of the ow b looking at the closest distance in the 2 particle center of mass frame. This corresponds to the time in the 2 particle center of mass frame when the 2 discs meet. / FIG event averaged distribution for scheme (a) with =:47 and =3fm. The dot-dashed, solid and target frame ordering respectivel. dn/ FIG event averaged dn distribution for scheme (d) with =:47 and =5fm. The dot-dashed, solid and target frame ordering respectivel. dn/ (GeV - ) FIG event averaged dn distribution for scheme (a) with =:47 and =3fm. The dot-dashed, solid and target frame ordering respectivel. / 2 Fig. 4, Fig. 5 and Fig. 6 give the results for scheme (d). Strong frame dependence is observed in this case. From Fig. 4 and 5, we see that a boost of the sstem b 6 unit of rapidit (the original rapidit goesto 6 before shifting back to ) makes a peak which is shifted from the central along the boost b 2 unit of rapidit. We use amuch smaller = 25 fm2 mb cross section because FIG. 5. 5event averaged distribution for scheme (d) with =:47 and =5fm. The dot-dashed, solid and target frame ordering respectivel. 3

4 dn/ (GeV - ) FIG event averaged dn The dot-dashed curve is for the initial distribution; the thick solid is for =:47 and =fm collider frame ordering; the thick dashed curve is for = :47 and = fm target frame ordering; the long dashed curve isfor= :4 and = 3 fm collider frame ordering; the dotted curve is for =:4 and =3fm target frame ordering. The dot-dashed, thick dashed and dotted curves overlap; the thick solid and long dashed curves overlap FIG. 6. 5event averaged dn distribution for scheme (d) with =:47 and =5fm. The dot-dashed, solid and target frame ordering respectivel. Fig. 7, Fig. 8 and Fig. 9 give results for = fm 2. Now the interaction range is fm and the mean free path is initiall :fm. For =:47 and =fm, 25 the dn= distribution has a dip in the middle for the collider frame ordering case, while the target frame ordering case doesn't have one. The = central rapidit plateau height is almost the same as the for = :47 and = 3 fm case. This is because of the eective screening mass is used to regulate the forward scattering cross section and larger cross section or more forward collisions does not lead to more collective work. This is clearl shown when we var the dierential cross section b change the from :47 to :4 and from fm to 3 fm. As shown in Fig. 8, the collider frame ordering case has a clear drop at rapidit from 4to 2 and from 2 to 4. In the middle, there is a peak. The target frame ordering has a lower plateau. The = is frame dependent now. The dn= for =:4 and = 3 fm has a lower temperature parameter than that of =:47 and =fm. This is because more work is been done when there are more relativel large angle scatterings and we get more cooling. Scheme (b) and (c) can not get frame independent results either. / FIG event averaged The dot-dashed curve is for the initial distribution; the thick solid is for =:47 and =fm collider frame ordering; the thick dashed curve is for =:47 and =fm target frame ordering; the long dashed curve is for = :4 and =3fm collider frame ordering; the dotted curve is for =:4 and =3fm target frame ordering. 5 4 dn/ (GeV - ) 3 dn/ FIG event averaged dn The dot-dashed curve is for the initial distribution; the thick solid is for =:47 and =fm collider frame ordering; the thick dashed curve is for =:47 and =fm target frame ordering; the long dashed curve is for = :4 and =3fm collider frame ordering; the dotted curve is for =:4 and =3fm target frame ordering. 4

5 To stu the eect of dierent dierential cross sections, we compare the results for = :47 and = 3 fm and = :4 and = 9 fm. Shown in Fig. and Fig. are = and dn= distributions. dn= distribution is the same for these two cases and the are the same as the initial. From Fig., a clear drop in the height of central rapidit plateau can be seen with more large angle scatterings. This shows that more work has been done and is also the reason that we get more cooling in dn= distribution. We get 7; collisions in the collider frame and 6; 7 in the target frame. Scheme (b) gives the similar results and the are all frame independent. / Another interesting thing to look at is the number of non-causal collisions vs. the number of total collisions. A non-causal collision here is dened to be a collision that the global ordering is not consistent with the ordering for a particular particle participating in the collision. This is better seen in scheme (b). If particle a's next collision according to global ordering is with i, but the collision time for particle a with i (i.e., the time a is going to change its momentum) is later than the collision time for a with j (even though according to global ordering, a j collision happens later), then we call this collision a noncausal collision. For scheme (b) in the collider lab frame, with =:47 and =3fm,we get a total of 6; 8 collisions per event with 9 of them non-causal. With a larger cross section ( =:47 and =fm ), as we expect, the non-causal to total ratio goes up. There are total 2; collisions with ; 4 of them non-causal. Boost to the target frame, the total number of collisions for =:47 and =3fm is 6; 5 while the number of non-causal is ;. The eects of putting in a mass for the particles are also studied. There are no signicant changes in det, dn dn and and the number of total and non-causal collisions for massless, m = : GeV and m =:GeV. Fig. 2 shows the collisions rates. We see that the results are not sensitive to small masses of the particles. FIG.. 2 event averaged distribution for scheme (a). The dot-dashed curve is for the initial distribution; the dotted curve is for =:47 and =3fm the thick solid is for =:4 and =9fm collider frame ordering; the thick dashed curve is for =:4 and =9fm target frame ordering. dn coll /dt (fm - ) dn/ (GeV - ) t (fm) FIG event averaged collision rate as a function of time for scheme (b). The upper 3 solid curves are for massless, m = :GeV and m = :GeV total collision rate and the lower 3 dashed curves are for non-causal collision rate of these 3 masses. FIG.. 2 eventaveraged dn The dot-dashed curve is for the initial distribution; the dotted curve is for =:47 and =3fm the thick solid is for =:4 and =9fm collider frame ordering; the thick dashed curve is for =:4 and =9fm target frame ordering. What we describe here is the parallel ensemble method in which average is taken over n independent events. An alternative, the full ensemble method [6], is to increase the particle number b k (b dividing the particles into small pieces) and at the same time decrease the cross section b a factor of k. In the full ensemble method, the mean free path is the same as for the parallel ensemble method, but the cross section is smaller and hence isolated 2 bo scattering condition can be satised. This 5

6 makes it possible to eliminate the causalit violation in the limit of large k. But the full ensemble method will increase the computation time signicantl due to the increasing number of checks required for the higher densit sstem. We see from the above stu that for reactions with strong inside outside correlations, the full ensemble method ma not have to be used in practice to get reasonable results as long as a sensible scattering prescription is used. IV. CONCLUSIONS G. Peter, D. Behrens and C. C. Noack, Phs. Rev. C 49 (994) [3] e.g. see G. Kortmeer, W. Bauer, K. Haglin, J. Murra and S. Pratt, it Phs. Rev. C [4] Further details will be published later. [5] P. V. Ruuskannen, Phs. Lett. 47 B (984) 465; K. J. Eskola and M. Gulass, Phs. Rev. C 47 (993) 2329; B. Zhang, M. Gulass and Y. Pang, CU-TP-795. [6] see e.g., G. Welke, R. Malied, C. Gregoire, M. Prakash and E. Suraud, Phs. rev. C 4 (989) 26; Kortemeer et.al. cited above. After stuing the frame dependence of parton cascade results, we see that dierent schemes ma give quite different results. Especiall, the global frame collision and ordering scheme results have a strong frame dependence. Other 2 particle center of mass frame collision and global frame ordering schemes can give almost frame independent results even when the interaction range is on the order of the mean free path. As we expect, the fail when the interaction is much large than the mean free path. Dierent dierential cross sections lead to dierent longitudinal work and give dierent = and dn= results. Since radiation will increase the cross section and parton densit, the quantum statistics will also change effective cross sections, further stu is necessar when we appl this to more realistic situations. ACKNOWLEDGMENTS We thank M. Asakawa, S. Gavin, K. Geiger, E. Gonzalez-Ferreiro, D. Kahana, S. Kahana, Z. Lin, R. Mattiello, C. Noack, D. Rischke, J. Randrup, R. Vogt, K. Werner for useful discussions, and M. Gulass for continuing encouragement and discussions and a critical reading of the manuscript. B. Z. gratefull acknowledges partial support from the [Department of Energ] Institute for Nuclear Theor at the Universit of Washington program INT-96-3 during the completion of this work. This work was supported b the U.S. Department of Energ under Contract No. DE-FG2-93ER4764 and DE- FG-2-92ER4699. [] Y. Pang, D. E. Kahana, S. H. Kahana and T. J. Schlagel, Nucl. Phs. A 59 (995) 565c; H. Sorge, H. Stoker and W. Greiner, Nucl. Phs. A 498 (989) 567c; K. Geiger, Phs. Rev. D 46 (992) 4965, [2] e.g., T. Kodama, S. B. Duarte, K. C. Chung, R. Donangelo and R. A. M. S. Nazareth, Phs. Rev. C 29 (984) 246; 6