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Transport description of viscous effects
Che-Ming KoTexas A&M University
Introduction A multi-phase transport (AMPT) model Anisotropic flow - Elliptic flow - Higher-order anisotropic flow - Rapidity dependence - System size dependence - Flavor dependence Summary
Collaborators: Lie-wen Chen (SJTU), Zie-wei Lin (Univ. Alabama), Bin Zhang (Arkansas State Univ.)
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Zhang, Gyulassy & Ko, PLB 455, 45 (1999)
Elliptic flow from parton cascade
Based on Zhang’s parton cascade (ZPC) (CPC 109, 193 (1998))using minijet partons from HIJING for Au+Au @ 200 AGeV and b=7.5fm
v2 of partons is sensitive to their scattering cross section
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1 2 1 2 1 2 1 2p f (x,p,t) dp d v v (d /d )(f 'f '-f f )
2 2
2 2 2 2
9 9d 1,
dt 2(t- ) 2 1 /s s
s
Using αs=0.5 and screening mass μ=gT≈0.6 GeV at T≈0.25 GeV, then <s>1/2≈4.2T≈1 GeV, and pQCD gives σ≈2.5 mb and a transport cross section
σ=6 mb → μ≈0.44 GeV, σt≈2.7 mb σ=10 mb → μ≈0.35 GeV, σt≈3.6 mb
t
dd (1 cos ) 1.5mb
d
Zhang’s parton cascade (ZPC)
Bin Zhang, Comp. Phys. Comm. 109, 193 (1998)
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A multiphase transport (AMPT) model
Default: Lin, Pal, Zhang, Li &Ko, PRC 61, 067901 (00); 64, 041901 (01);
72, 064901 (05); http://www-cunuke.phys.columbia.edu/OSCAR
Initial conditions: HIJING (soft strings and hard minijets) Parton evolution: ZPC Hadronization: Lund string model for default AMPT Hadronic scattering: ART
Convert hadrons from string fragmentation into quarks and antiquarks Evolve quarks and antiquarks in ZPC When partons stop interacting, combine nearest quark and antiquark to meson, and nearest three quarks to baryon (coordinate-space coalescence) Hadron flavors are determined by quarks’ invariant mass
String melting: PRC 65, 034904 (02); PRL 89, 152301 (02)
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Based on coordinate-space coalescence Need string melting and large parton scattering cross section Mass ordering of v2 at low pT as in ideal hydrodynamic model
Lin & Ko, PRC 65, 034904 (2002)Elliptic flow from AMPT
σp= 6 mb
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Coalescence model
qq3
3
N)p,x(fE)2(
pddp
36/1gg K Mg
3ni
n i q,i i i n 1 n 1 n3i=1 i
d pN =g p d f (x ,p ) f (x ,...,x ;p ,...,p )
(2 ) E
Quark distribution function
Spin-color statistical factor
e.g. 12/1gg *K
Coalescence probabilityfunction
)p,x(fq
PRL 90, 202102 (2003); PRC 68, 034904 (2003)
Number of hadrons with n quarks and/or antiquarks
54/1gg,108/1gg pp
}]/2Δ)m-(m-)p-exp{[(p×
]/2Δ)x-exp[(x=
)p-p;x-x(f=)p,p;x,(xf
2p
221
221
2x
221
212122121M
px
For baryons, Jacobi coordinates for three-body system are used.
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Momentum-space coalescence model
Including 4th order quark flow
q T 2,q T 4,q Tf (p ) 1 2v (p )cos(2 ) 2v (p )cos(4 )
Meson flow
Baryon flow
v
v
3
1+
3
1=
v
v ,
v
v
2
1+
4
1=
v
v ⇒ 2
2,q
4,q2
B2,
B4,22,q
4,q2
M2,
M4,
)v+v(2+1
v+v2= v,
)v+2(v+1
vv2+v2=v 2
4,q22,q
22,q4,q
M4,24,q
22,q
4,q2,q2,q
M2,
)vv+v+6(v+1
v3+vv6+v3+v3= v,
)vv+v+v(6+1
vv6+v3+vv6+v3=v
4,q22,q
24,q
22,q
34,q4,q
22,q
22,q4,q
B4,4,q
22,q
24,q
22,q
24,q2,q
32,q4,q2,q2,q
B2,
Kolb, Chen, Greco, & Ko, PRC 69 (2004) 051901
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22,q4,q2
2
4 2v v 1.2 v
v
Data can be described by a multiphase transport (AMPT) model
22,q4,q v v
Data
Parton cascade
Higher-order anisotropic flow
Momentum-space coalescence
Chen, Ko, & Lin, PRC 69, 031901 (04)
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Strange quarks from AMPT Chen & Ko, PRC 73, 044903 (2006)
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Phi & Omega flows
)2/p(v2)2/p(v21
)2/p(2v)p(v T2,q
T22,q
T2,qTM2,
)3/p(v3)3/p(v61
)3/p(3v)p(v T2,q
T22,q
T2,qTB2,
n)/p(vn
1T2
same for phi & Omegabut different from quark
4,4,s 4,2 2 22,s 2, 2,
vv v 0.85 0.67, 0.61
v v v
instead of 1.1 and 0.7 from dynamicalcoalescence model
4, 4,s 4, 4,s2 2 2 22, 2,s 2, 2,s
v v v v1 1 1 1 ,
v 4 2 v v 3 3 v
Chen & Ko, PRC 73, 044903 (2006)
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2, T 2, T
2,s T
1 1 v (p / 2) v (p / 3) 2 3
v (p )
4, 4,s 4, 4,s2 2 2 22, 2,s 2, 2,s
v v v v1 1 1 1 ,
v 4 2 v v 3 3 v
4,4,s 4,2 2 22,s 2, 2,
vv v 0.85 0.67, 0.61
v v v
satisfied for large phi and Omegasizes of 4.5 fm
Flows of phi & Omega with large sizes
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Pseudorapidity dependence of v1 and v2
String melting describes data near mid-rapidity (||<1.5) At large rapidity (||>3), hadronic picture works better
Chen, Greco, Ko & Koch, PLB 605, 95 (2005)
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System size dependence of elliptic flow
Chen & Ko, nucl-th/0505044
v2 is proportional to size of colliding system
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Elliptic flow in collisions of asymmetric systems
Compared to symmetric collisions Directed flow v1 is stronger Elliptic flow v2 is more sensitive to parton cross section Both directed and elliptic flows are asymmetric in forward and backward rapidities
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Quark elliptic flow from AMPT
PT dependence of charm quark v2 is different from that of light quarks. At high pT, charm quark has similar v2 as light quarks. Charm elliptic flow is also sensitive to parton cross sections
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Charm elliptic flow from AMPTZhang, Chen & Ko, PRC 72, 024906 (05)
Current light quark masses are used in AMPT. Charmed meson elliptic flow will be larger if constituent quark masses are used. To increase v2 at low pt requires going beyond parton cascade.
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10
5
0
-5
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Summary
Elliptic flow is sensitive to parton scattering cross section The AMPT model with string melting can reproduce observed large elliptic flow and mass ordering at low pT with large parton cross section (quasi bound states in QGP and/or multiparton dynamics gg↔ggg?)
Observed hadron v4≈1.2v22 is reproduced with parton v4≈v2
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v1 and v2 at midrapidity requires formation of partonic matter, while those at large rapidity is consistent with a hadronic matter AMPT predicts: - v2 scales with the size of colliding system - Stronger v1 and more sensitivity of v2 to parton cross section in asymmetric
than symmetric collisions, and both show forward-backward asymmetry in rapidity Observed large charmed meson flow requires large charm quark scattering cross section (resonance effect?)
