1. Anisotropic flow, Azimuthal Balance Function, and Two-charged-particle Azimuthal Correlations in RQMD and AMPT We are very grateful to Zhixu Liu and Jiaxin Du for offering us the data We are very grateful to Zhixu Liu and Jiaxin Du for offering us the data samples of RQMD and AMPT. samples of RQMD and AMPT. 1.Introduction 2. Behaviors of these three measurements in RQMD and AMPT in RQMD and AMPT 3. Conclusions and Suggestions IOPP, Central China Normal University, Wuhan, China

2. Nov. 19, 2006 QM’06 2 1. Introduction The observed large flow at RHIC has been argued as an indicator of early thermal equilibrium. Its flavor dependence is supposed to be highly relevant to interpretation of strongly interacting QGP. ► Anisotropic collective flow is caused by initial spatially-anisotropic participant zone and interactions spatially-anisotropic participant zone and interactions in non-central nucleus-nucleus collisions, in non-central nucleus-nucleus collisions, J. Adams, et al, (STAR Coll.), Nucl. Phys. A 757, 102(2005); K. Adcox, et al, (PHENIX Coll.), Nucl. Phys. A 757, 184(2005). K. Adcox, et al, (PHENIX Coll.), Nucl. Phys. A 757, 184(2005). Miklos Gyulassy, Larry McLerran, Nucl. Phys. A 750, 30-63(2005). B. M\"uller, Annu. Rev. Nucl. and Part. Phys.,1(2006). ψ R is azimuthal angle of reaction plane defined on an event-by-event base.

3. Nov. 19, 2006 QM’06 3 1. Introduction (cont.) ► Azimuthal charge balance function P. Bozek, Phys. Lett. B 609, 247(2005). ☺ Another independent constraint on the temperature and the on the temperature and the amount of transverse flow at the amount of transverse flow at the freeze-out. freeze-out. ☺ It is expected from the thermal model that the higher temperature model that the higher temperature and the smaller elliptic flow, the and the smaller elliptic flow, the wider the azimuthal balance function. wider the azimuthal balance function. It is interesting to see how azimuthal charge balance function It is interesting to see how azimuthal charge balance function behaves in the dynamical models RQMD and AMPT. T f = 90 MeV, β=0.6 __ T f = 90 MeV, β=0.6 T f = 165 MeV β=0.5 - - T f = 165 MeV β=0.5

4. Nov. 19, 2006 QM’06 4 1. Introduction (cont.) ► Two-charge-particle azimuthal corr. ► Why we choose these three measurements ? ☻ They are all anisotropic trans. momentum dis. related. ☻ They are all anisotropic trans. momentum dis. related. ☻ The behaviors of these three measurements in RQMD and AMPT will help ☻ The behaviors of these three measurements in RQMD and AMPT will help us in understanding the underlying dynamics. us in understanding the underlying dynamics. ☻ Both two-charge-particle azimuthal corr. and azimuthal charge balance ☻ Both two-charge-particle azimuthal corr. and azimuthal charge balance func. are reaction plane irrelevant and therefore easily to measure. func. are reaction plane irrelevant and therefore easily to measure. n ab (δφ): is the number of pairs of charged a and b particles satisfying the criteria that their relative angle equals δφ. criteria that their relative angle equals δφ. n ab : is the total number of the same kind of pairs for all possible separated angles. L. Foa, Phys. Reports 22(1975). K. Adcox, et al., (PHENIX Coll.), PRL 89, 212301(2002)

5. Nov. 19, 2006 QM’06 5 1. Introduction (cont.) ► Models A MC generator of Relativistic Quantum Molecular Dynamics ( hadron ), Without hadron rescattering, v 2 = 0; Without hadron rescattering, v 2 = 0; With hadron rescattering, v 2 > 0, but smaller than the data. With hadron rescattering, v 2 > 0, but smaller than the data. H. Sorge, Phys. Rev. C52, 3291 (1995). See Feng Liu’s talk in the same session. Zi-Wei Lin, Che Ming Ko, Bao-An Li, Bin Zhang and Subrata Pal, Phys. Rev. {\bf C72}, 064901 (2005). A MC generator of A Multi-Phase Transport Model (parton + hadron) Default AMPT, v 2 > 0,but still smaller than data; Default AMPT, v 2 > 0, but still smaller than data; AMPT with string melting, where parton transportation, AMPT with string melting, where parton transportation, hadron rescattering and quark coalescence at hadronization are hadron rescattering and quark coalescence at hadronization are implemented in the model. This model reproduces v 2 (p t ) data implemented in the model. This model reproduces v 2 (p t ) data at RHIC. at RHIC. RQMD (v2.4) RQMD (v2.4) : AMPT AMPT : Au-Au collisions at √s NN =200GeV

6. Nov. 19, 2006 QM’06 6 2. Behaviors of three measurements in models ► v 2 of all, positive and negative charged particles RQMD without rescattering: no v 2 at all. no v 2 at all. RQMD with rescattering: same middle values same middle values of v c 2, v - 2, v + 2. of v c 2, v - 2, v + 2. AMPT with string melting: same big values of same big values of v c 2, v - 2, v + 2 v c 2, v - 2, v + 2 v 2 is the biggest for mid-central coll. and is smaller for central and very peripheral collisions.

7. Nov. 19, 2006 QM’06 7 Rescattering off: ☺ No small angular corr. ☺ Large angular corr. due to momentum conservation. momentum conservation. Rescattering on: ☺ Small angular corr. appear! ☺ The diff. among them are small but visible. small but visible. ☺ Large angular corr. remains. ► C cc, C ++, C – and C +- in RQMD 2. Behaviors of three measurements in models

8. Nov. 19, 2006 QM’06 8 ► Centrality dependence of C cc, C ++, C --, C +-, in RQMD Rescattering on: The small angular corr. are the strongest for the mid-central coll., small for most central coll. and even smaller for the most peripheral coll. This centrality dependence is the same as v 2. Rescattering on: The small angular corr. are the strongest for the mid-central coll., small for most central coll. and even smaller for the most peripheral coll. This centrality dependence is the same as v 2. Rescattering off: Momentum conservation leads to the observed large angular correlations, especially for the most peripheral collisions. Rescattering off: Momentum conservation leads to the observed large angular correlations, especially for the most peripheral collisions. 2. Behaviors of three measurements in models

9. Nov. 19, 2006 QM’06 9 ► Centrality dependence of C cc, C ++, C --, C +- in AMPT ► C cc, C ++, C --, C +- in AMPT with string melting Small angle corr. have the same centrality dependence as flow. In AMPT, small angular corr. becomes stronger than large angular corr. Difference among C cc, C ++, C --, and C +- are very small. In AMPT, small angular corr. becomes stronger than large angular corr. Difference among C cc, C ++, C --, and C +- are very small. 2. Behaviors of three measurements in models

10. Nov. 19, 2006 QM’06 10 ► Two-charge-particle corr. from PHENIX Coll. K. Adcox, et al, Phys. Rev. Lett. 89, 212301(2002) 2. Behaviors of three measurements in models Data show: ● A anisotropic corr. pattern is symmetric pattern is symmetric about ∆    about ∆    ● This anisotropy increase with impact increase with impact parameter and p t. parameter and p t. Models show: ► The small angle corr. is related to anisotropy is related to anisotropy of p t distribution of p t distribution ► The same symmetry structure in AMPT structure in AMPT with string melting. with string melting. ► The same centrality dependency in both dependency in both RQMD and AMPT RQMD and AMPT

11. Nov. 19, 2006 QM’06 11 2. Behaviors of three measurements in models RQMD w ► Azimuthal charge balance function in RQMD The more central coll., the narrower BF, which are diff. from centrality dependence of v 2. The more central coll., the narrower BF, which are diff. from centrality dependence of v 2. No centrality dependence. This means that the corr. caused by momentum cons. are cancelled in the measurement. No centrality dependence. This means that the corr. caused by momentum cons. are cancelled in the measurement. RQMD w/o

12. Nov. 19, 2006 QM’06 12 2. Behaviors of three measurements in models ► Azimuthal charge balance function in AMPT In more central coll., the BF is narrower In more periph. coll., the BF is wider. This is different from the expectations from the thermal model, where the smaller elliptic flow, the wider azimuthal balance function. T f = 90 MeV, β=0.6 __ T f = 90 MeV, β=0.6 T f = 165 MeV β=0.5 - - T f = 165 MeV β=0.5

13. Nov. 19, 2006 QM’06 13 3. Conclusions and suggestions The behaviors of anisotropic flow, azimuthal charge The behaviors of anisotropic flow, azimuthal charge balance function, and two-charge-particle azimuthal corr. in RQMD and AMPT are compared. ● The large angular corr. are due to the constraint of momentum conservation. momentum conservation. ● The small angular corr. mainly come from anisotropic transverse momentum distribution caused by hadron transverse momentum distribution caused by hadron level rescattering and/or parton level transportation. level rescattering and/or parton level transportation. ● The centrality dependence of the small angular corr. is the same as the centrality dependence of v 2 is the same as the centrality dependence of v 2 in both AMPT and RQMD. in both AMPT and RQMD. ● To measure the two-charge-particle azimuthal corr. at STAR detector, where TPC covers the whole at STAR detector, where TPC covers the whole azimuthal angle. azimuthal angle.

14. Nov. 19, 2006 QM’06 14 3. Conclusions and suggestions ● From the behavior of two-charge-particle azimuthal corr., we see that azimuthal charge balance function (ACBF) well cancel the contributions caused by the constraint of momentum conservation. ● ACBF changes monotonically with centrality in both RQMD and AMPT models. This is different from the centrality dependence of v 2 and what expected from thermal model. ● It is interesting to see how the B( ∆φ) behave in current heavy ion collisions. ● These three measurements are all sensitive to the anisotropic transverse momentum distributions.

15. Nov. 19, 2006 QM’06 15Appendix AMPT default