MD Simulations of classical sqgp

Transcription

1 MD Simulations of classical sqgp From RHIC to LHC: Achievements and Opportunities Tuesday, November 7, 2006 Kevin Dusling Ismail Zahed

2 Outline Introduction Motivation for sqgp Motivation for classical MD simulation The Model Diffusion Color Conductivity Parton Energy Loss

3 Introduction Interested in deconfined phase of QCD Interaction Strength High T (wqgp) At RHIC T=2Tc (sqgp) Datta et al. Evidence for sqgp Low viscosity Bound states High pressure puzzle

4 Binary Bound States in QGP Coulomb Potential w/ Screening: V = " 4# s(r) 3r exp("m D r) Solve NR Schrodinger: d 2 "(r) Lattice: Bound States: Petreczky et al. + # + me & -% dr 2 2 (,- $ M D ' E = 0 " 4m 3M D # s >1.68 E.V. Shuryak and I. Zahed, Phys. Rev. C 70, (2004) [arxiv:hep-ph/ ]. Phys. Rev. D 70, (2004) [arxiv:hep-ph/ ] 2 + 4m ) s (r) 3M D r " 2 = p 2 + M 2 4(3.9T) T =1.5T c m q = 3.9T m g = 3.4T M D = 2.25T Quarks: Gluons: Weak Coupling: e *r. 0 / 0 "(r) = 0 "(r) = # r 3(2.25T) " s = 2.3" s 9 4 " 4(3.4T) 3(2.25T) # = 4.5# s s Gluons have even stronger binding! 4( 1 6 T) 3(2.25T) " s = 0.24" s

5 Binary Bound States in QGP (cont.) Main Point: nothing prevents QCD coupling from running to larger values at lower momentum scales at T T c α s ~1 Stopped at screening mass scale " s (M D ) ~ 1 Gauge coupling constant not frozen to value from vacuum potential Quarks: Gluons: 2.3" s > " s >1.68 Weak Coupling: 0.24" s >1.68 E.V. Shuryak and I. Zahed, Phys. Rev. C 70, (2004) [arxiv:hep-ph/ ]. Phys. Rev. D 70, (2004) [arxiv:hep-ph/ ]

6 The Model Assumptions: M>>T, so particles move non-relativistically colored electric coulomb potential dominates Color representation large (classical color vectors) Quantum Effects: Generate thermal-like masses Cause effective coupling to increase at small T Add localization energy to interaction Why use MD? Even though crude, directly provides real time correlators MC or Lattice QCD can get at thermodynamics but very difficult to get at transport coefficients Identify strongly coupled plasma in context of traditional EM plasma " = (ze)2 a WS T # < V > T # 3 & a WS = % ( $ 4" n' 1 3 Regimes: Γ<1 (gas) Γ 10 (liquid) Γ 100 (glass) Γ>300 (solid)

7 Model Hamiltonian H = i" p i" 2 # + 2m i" i", j$ & ( '( Q a a i" Q j$ x i" % x j$ ) + V core + * + Species: # " = q,q,g Phase Space Coordinates: x ", p ", Q " Q a " D ab (#)Q b Equation of Motion: x m n { i", p j# } = $ mn $ "# $ ij Q a b { i",q j# } = f abc c Q "i a =1,...,N c (N c "1) /2 x n i" = {H, x n i" } = p i" n m " p n i" = {H, p n i" } = ge an a i" Q i" Q a i" = {H,Q a i" } = gf abc Q b c i" A i" 0 r E i" a = # $ r a i" A i" 0 = # $ r a gq j% i" ' r x i" # r i"& j% x j%

8 Units Length Unit: r min = " * V = g2 Q i # Q j, " + r + 1 $ "' & ) n % r ( n - /. Time Unit: " 0 = # p $1 = m 4%ne 2 Mass Unit: m Examples: $ ' [ KE] = & m"2 2 ) % # 0 ( $ [ strength of potential] = & g2 % " m" 2 ' $ 1 ' 2 ) = # 0 ( % & 4*n" 3 ( )

9 Comparison to sqgp Mapping: cqgp " sqgp Adjust three length scales described before All parameters of model function of T in sqgp sqgp at T= T c Quasiparticle mass: m 5T c (lattice results at T=(1.5-3)T c ) Take m=3t Casimir: " s C =1 Density: Length: Time: n " (0.244T 3 )(8 + 6N f ) " 6.3T 3 V eff = h2 2mr " C# s 2 r m " 0 = # $1 p = 4%n & s C ' 1 5.1T r min = " = h2 mc# s " ~ 1 3T " # 3 $ # 1 3T % 0 # 1 5.1T m # 3T

10 Some Technical Details Run Size: n part =4 3 =64 Periodic Boundary Conditions: V ij = +# $ n x,n y,n z ="# V ( x r i " x r j + nl r ) Cooling: p i" n = #p i" n # = 1+ 1 & T ) ( %1. + $ ' T desired * Integration 1 st order RK, Leap-frog, Verlet all give same results

11 Simulation

12 Results Diffusion Color Diffusion and Conductivity Energy Loss

13 Diffusion D(") = 1 3N N $ i=1 r v i (") # v r i (0) Gas Liquid Glass D = # $ D D(")d" 0 D " 1 3# Γ

14 Diffusion in QGP MD Simulation (sqgp) D " 1 3# " 0.1$ 0.1 '% 2 * ' ), ( & " T * ), " 0.1 ( (3T) 2 + T Perturbation Theory Heavy Quarks (M>>T) Thermal mtm (p~sqrt(m/t) 3" = 1 2M D " 6 2#T 2 ( N f M quark D = T = 2T 2 M" D # d r 3 k d r % r 3 k # d 3 p # (2$) 9 8k 0 k # (2$) 3 & 3 ( r r r p + k # ' p # ' k r )2$&( k # ' k) q r 2 ( 0 [ n f (k)(1' n f ( k # )) + M 2 n gluon b (k)(1' n b ( k #))] % ' & 0.5 $ s Experiment ( * ) D " 1.5 2#T $ 0.2 T 2 $ s = , " 1 T G. Moore and D. Teaney, arxiv:hep-ph/

15 Color Diffusion Constitutive Eq: Current Conservation: r j a Diffusion Equation for Color: D c = 1 # r $ d" j a (") r j a (0) = g2 3C 2 3C 2 0 " g2 3C 2 $ % 0 = "D c r # Q a $ t Q a + r # % r j a = 0 N % i, j # $ 0 d# Q a i (#)Q a i (o) r r j a = gq a v (linearized theory) " t Q a ( r,t) # D c $ 2 Q a ( r,t) = 0 d" Q a i (") v r i (")Q a j (o) v r j (o) r v i (#) v r i (o)

16 Color Diffusion (cont.) Color Decorrelation: Q i a (")Q i a (o) color decorrelation time >> momentum decorrelation time Q i a (" p )Q i a (o) #1 D c " g2 3C 2 $ d# Q a (#)Q a (o) r % i v i i (#) v r i (o) " g2 3 0 D c " g 2 D $ % 0 d# r v i (#) r v i (o) D c = g2 3C 2 N # % $ i, j 0 d" Q a i (") v r i (")Q a j (o) v r j (o)

17 NA Transport Theory NA Transport Eqn: % p µ a $ ( ' D µ " gq a F µ# * f = C[ f, f,,g] = #p + u + ( f eq " f ) & $p # ) Relaxation Time Approx. f ( p,x,q) = f eq ( p, x,q) + "f ( p, x,q) "f = #g $p % u % pµ u & Q a F a µ$ f eq (1+ f eq ) Color Current: J a µ = 2g " c = Ng2 C 2 6#T$ 2 % " dp dq p µ Q a f J µ a = " ab # µ$ b u % LRF F $% J µ = " µ# F # 0 " µ# = " c g µ$ + " 1 u µ u # & p 2 dp f eq (p)( 1+ f eq (p)) " c = g2 T (Ohm s Law) " 2 p = 4#$ st 2 % N c + N f ' 9 & 2 ( * ) " c = # p 2 D c $ N c + N f /2' & % # g # ) q (

18 Color Conductivity in QGP MD Simulation (sqgp) % " c = # 2 p D c $ # 2 p D D ( c ' * & D ) " c (sqgp) =1.5 C# s $ p 2 D % 4T g 2 # ' <1 D " % 1.5g 2 $ 1 < ' <10 2.5g 2 % & ' >10 D c Weak Coupling: ' 1 " ab ij = #lim $ %0 $ Im& R ab ij ($, 0 r * ) ( ) +, = " c- ij - ab Selikhov, Gyulassy: " c = # p 2 D c $ H. Heiselberg: " c = 1.7N f T ln(1/# x ) # p 2 & 3% s T ln 1 ) ( + ' % s * # x = 0.5 $ $ % & 5T % s = 0.5,, - $ 3T

19 Energy Loss Collisional energy loss extremely important for interpretation of Jet-quenching Suppression of high p hadron spectra Steps: Start with equilibrium IC taken from prior runs Add heavy particle with p init Run Simulation until particle thermalizes Measure energy loss, diffusion, etc. Repeat Preliminary results M=10m

20 Energy Loss (Binary Collisions) What to expect for de dx What role does V core play? Classical Energy Loss given by: de dx (T > ") = 2#n T(b)bdb b max (" ) $ 0 T(b) = 2z2 e 4 " 1 % $ mv 2 b 2 2 ' # + b min & b min = e2 pv For T=0 and M>>m de dx ~ E de dx e4 = 2"n mv ln $ 4 E ' & ) 2 % # (

21 Simulation Results Γ=0.8 E=15 Γ=30 E=15 1D Fokker-Planck: "f "t = " [ "p T 1( p) f ] + " 2 [ "p T 2( p) f ] 2 T 1 (p): T 1 ( p) = "E 1 E T 1 ( p) = F = " de dx " de dx # E de dx T 1 ( p) = "p "t ( T 2 ( p) = "p ) 2 "t = F T 2 (p): T 2 ( p) = D F "f "t = A " "p ( pf ) + D F " 2 f "p 2 A = " 1 E de dx

22 A=0.1 D=3. A=0.07 D=1.

23 Average Energy Loss E = " # 0 $E = E 0 % E E f ( p,t)dp

24 Energy Loss in QGP Bjorken: de dx = 1 2 C 3"# st 2 v $ 1$ v 2 -, 2v 2 % ln 1+ v (. % ' * 01 ln 4TE ( ' 2 * & 1$ v )/ & M D )

25 Conclusions / Future Work Conclusions Diffusion # D MD " 0.1 & % $ T ' ( < # D " 1 & % $ PT T ( ' Color Conductivity D c " g 2 D sqgp poor color conductor Energy Loss Higher energy loss in sqgp compared to weak coupling estimates Future Work Dipole Energy Loss Collective Excitations: sound and plasma-color waves

26 References S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, hep-lat/ P. Petreczky, F. Karsch, E. Laermann, S. Stickan, I. Wetzorke, Nucl. Phys. Proc. Suppl. 106, 513 (2002). A. Selikhov and M. Gyulassy, Phys. Lett. B 316, 316 (1993). H. Heiselberg, Phys. Rev. Lett. 72, 3013 (1994). J. D. Bjorken, Fermilab Report No. PUB-82/59-THY (unpublished). S. Mrowczynski, in Quark-Gluon Plasma, ed. R. Hwa, World Scientific, 1990