Probing the AdS/CFT Plasma with Heavy Quarks

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Σίβύλ Ζυγομαλάς
5 χρόνια πριν
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1 Probing the AdS/CFT Plasma with Heavy Quarks Jorge Casalderrey-Solana LBNL Work in collaboration with Derek Teaney

2 Heavy Quarks at RHIC Heavy Quarks are suppressed and participate on collective motion Model: Langevin dynamics dp dt = η Dp + ξ Fit to elliptic flow: However: RAA and v2 cannot be fitted simultaneously Goal: D = 2T 2 Compute D in AdS ξ(t)ξ(t ) = κδ(t t ) κ = 3 6 2πT Can we go beyond Langevin?

3 Momentum Distribution -T/2 +T/2 v f 0 (b) f f (b)

4 Momentum Distribution -T/2 +T/2 v f 0 (b) ρ ab (-b/2, b/2; -T) A f f (b)

5 Momentum Distribution -T/2 +T/2 v f 0 (b) ρ ab (-b/2, b/2; -T) A f f (b)

6 Momentum Distribution -T/2 +T/2 v f 0 (b) ρ ab (-b/2, b/2; -T) A f f (b)

7 Momentum Distribution -T/2 +T/2 v f 0 (b) ρ ab (-b/2, b/2; -T) A f f (b) Final distribution f f (b) = Tr [ρ(b)w c (b)] A

8 Momentum Broadening From the final distribution Transverse gradient Fluctuation of the Wilson line T F y µv µ (t 1,y 1 )Δt 1 δy 1 F y µv µ (t 2,y 2 )Δt 2 δy 2 Dressed field strength

9 Momentum Broadening From the final distribution Transverse gradient Fluctuation of the Wilson line Four possible correlators T Dressed field strength

10 t Wilson Lines in AdS/CFT t HKKKY & G z=ε v=0 x x=vt x z 0 =1 Horizon Moving string is straighten by the coordinates World sheet black hole at z 2 ws =1/γ At v=0 both horizons coincide. Coordinate singularity

11 t Wilson Lines in AdS/CFT t HKKKY & G z=ε v=0 x x=vt x c(z)=v z 2 =1/γ z 0 =1 Horizon Moving string is straighten by the coordinates World sheet black hole at z 2 ws =1/γ At v=0 both horizons coincide. Coordinate singularity

12 t Wilson Lines in AdS/CFT t HKKKY & G z=ε v=0 x x=vt x c(z)=v z 2 =1/γ z 0 =1 Horizon Moving string is straighten by the coordinates World sheet black hole at z 2 ws =1/γ At v=0 both horizons coincide. Coordinate singularity

13 Kruskal Map As in black holes, (t, r) coordinates are only defined for r>r 0 Proper definition of coordinate, two copies of (t, r) related by time reversal. In the presence of black branes the space has two boundaries (L and R) SUGRA fields on L and R boundaries are type 1 and 2 sources Maldacena Herzog, Son (02) The presence of two boundaries leads to properly defined thermal correlators (KMS relations)

14 String Solution in Global Coordinates x 3 = vt + ξ(z) v=0 smooth crossing v 0 logarithmic divergence in past horizon. Artifact! (probes moving from - ) String boundaries in L, R universes are type 1, 2 Wilson Lines. Transverse fluctuations transmit from L R

15 String Fluctuations The fluctuations live in the world-sheet S NG = R2 2πl 2 s dˆtdẑ ẑ 2 [ ( ( ŷ) )] 2 f(ẑ) f(ẑ)(ŷ ) 2 S2 T Close to the world-sheet horizon the solution behaves as Ŷ (ˆω, ẑ) = e iˆωˆt (1 ẑ) iˆω/4 kkkkkkkkkkkkkkkkkkkkkkkkkk Ŷ (ˆω, ẑ) = e iˆωˆt (1 ẑ) +iˆω/4 [ ( ) The infalling solution leads to the retarded correlator. infalling outgoing ˆω = γω R 2 G R (ˆω) = lim z 0 2πls 2 (πt ) 3 Ŷ ( ˆω, ẑ) ẑŷ (ˆω, ẑ) ẑ

16 The Boundary Action The solution is extended to the full Kruskal plane: In the world sheet black hole Positive energy modes infalling Negative energy modes outgoing Boundary action Herzog, Son (02) S T = 1 dω γ 2 2π Y 1 ( ω)y 1 (ω) (ReG R (ˆω) + i(1 + 2ˆn)ImG R (ˆω)) + Y 2 ( ω)y 2 (ω) ( ReG R (ˆω) + i(1 + 2ˆn)ImG R (ˆω)) + Y 1 ( ω)y 2 (ω)e iω/2t ( i2ˆnimg R (ˆω)) + Y 2 ( ω)e iω/2t Y 1 (ω) ( i2(1 + ˆn)ImG R (ˆω)) ˆn = 1 e ω γ/t 1 KMS- like relations (even at v >0!)

17 Performing derivatives of the boundary action κ = γ lim ˆω 0 (1 + 2ˆn)ImG R (ˆω) = γλπt 3 From the Einstein relations (v=0) Momentum Broadening (Gubser hep-ph/ ) Grows with Energy η D = κ 2MT = λπt 2 2M Direct computation of the energy lost: (Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser) Putting numbers: dp dt = η Dp where is the noise? Comparable to RHIC value! Can we go beyond Langevin?

18 String Partition Function (in progress D. Teaney, D. T. Son, JCS) The string partition function is Z = DX t 1(t 1 )DX t 2(t 2 ) boundary values Dx t 1(t, r)dx t 2(t, r)e is NG+iS B In the HKKKY & G solution, the finite flux leads to λt 2 π S B = dte X dx 1 dte X dx 2. E X = 2 Quantum correction: Small fluctuations over classical path x t (t, z) = (vt + ξ(r) + x(t, r), y(t, r)) v 1 v 2,

19 Quadratic Fluctuations Large Mass Quadratic fluctuations S NG = S CL + S 2 Same problem as in broadening. We integrate out the fluctuations for z>0 and obtain Z = C DX 1 (t 1 )DX 2 (t 2 )DY 1 (t 1 )DY 2 (t 2 )e is L e is T Divergent constant S L = 1 dω 2 γ2 γ 2π X 1 ( ω)x 1 (ω) (ReG R (ˆω) + i(1 + 2ˆn)ImG R (ˆω)) + X 2 ( ω)x 2 (ω) ( ReG R (ˆω) + i(1 + 2ˆn)ImG R (ˆω)) + X 1 ( ω)x 2 (ω)e iω/2t ( i2ˆnimg R (ˆω)) + X 2 ( ω)e iω/2t X 1 (ω) ( i2(1 + ˆn)ImG R (ˆω)) This is the partition function of the heavy quark.

20 Kinetic term and ra basis γgr (ˆω) = GR is divergent: R2 2πl 2 s 1 γω 2 (πt )3 λγ z m 2π C ( ˆω πt ) MQ Redefined GR (finite) We obtain a kinetic term Stadard procedure: introduce ra basis (Feynman, Vernon, Caldeira, Legget, C. Greiner) X a = X 1 (t) X 2 (t + iβ/2) X r = X 1(t) + X 2 (t + iβ/2) 2 Conjugate to momentum

21 Z T = Random Force { ( ) } The integration of Xa leads to Z=ZTZL ( t ( ) ) t t DY r D ξ T δ γmqÿr(t) 0 + dt G R Y r (t ) ξ T (t) γ { exp 1 ( ) } t t dtdt ξ T (t)g 1 sym ξ T (t ) 2 γ Z L = ( t ( ) ) t t DX r D ξ L δ γ( 3 MQẌr(t) 0 ) + ( dt γ 2 G R ) X r (t ) ξ L (t) γ { exp 1 ( ) } t t dtdt ξ 2γ 2 L (t)g 1 sym ξ L (t ) γ ( t ( ) ) ξ L (t)ξ L (t ) = γ 2 G sym ( t t γ ) ( ) t t ξ T (t)ξ T (t ) = G sym ( ) γ G sym (ˆω) = (1 + 2ˆn) ImG R (ˆω) ( ) Force distribution: ω 0 ( ) ( ) ξ L (t)ξ L (t ) = γ 2 γλπt 3 δ(t t ) ξ T (t)ξ T (t ) = γλπt 3 δ(t t )

22 ( ) ( ) Equation of Motion t ( ) t t γ 3 MQẌr(t) 0 + dt γ 2 G R X r (t ) = ξ L (t) γ t ( ) t t γmqÿr(t) 0 + dt G R ( ) Y r (t ) = ξ T (t) γ ( ) γgr ( γω) = iγ λπt 2 2 } small ω + γ 3/2 λt 2 ω2 fluctuations on top of X=vt ( ) Large mass low frequency approximation of GR λπt γ 3 2 M kin Ẍ r (t) + γ 3 Ẋ r (t ) = ξ L (t) 2 λπt 2 γm kin Ÿ r (t) + γẏr(t ) = ξ T (t) 2 Since these are small fluctuations: The effective mass is v dependent! M kin = M 0 Q γλt (Same as HKKKY at v=0) 2 dp dt = µp + E ˆx + ξ, γ << M 2 Q λt 2

23 Conclusions The momentum broadening κ of the heavy quark is large and depends on the velocity. Its numerical value is comparable to values extracted from RHIC data on v2 Quantum fluctuation of the string lead to the appearance of the noise distribution. The 1-2 formalism is important. We have obtained finite frequency corrections to Langevin dynamics. These may be used in phenomenology. We obtained a velocity dependent kinetic mass. The effective is reduced for fast particles. Consistency demands: γ < M 2 λt 2

24 Back up Slides

25 Computation of (Radiative Energy Loss) (Liu, Rajagopal, Wiedemann) Dipole amplitude: two parallel Wilson lines in the light cone: t Order of limits: L r 0 String action becomes imaginary for For small transverse distance: entropy scaling

26 Energy Dependence of (JC & X. N. Wang) From the unintegrated PDF Evolution leads to growth of the gluon density, In the DLA HTL provide the initial conditions for evolution. Saturation effects For an infinite conformal plasma (L>L c ) with Q 2 max =6ET. At strong coupling

27 Noise from Microscopic Theory HQ momentum relaxation time: Consider times such that microscopic force (random) charge density electric field

28 Heavy Quark Partition Function McLerran, Svetitsky (82) YM + Heavy Quark states YM states Integrating out the heavy quark Polyakov Loop

29 Changing the Contour Time

30 κ as a Retarded Correlator κ is defined as an unordered correlator: From Z HQ the only unordered correlator is Defining: In the ω 0 limit the contour dependence disappears :

31 Force Correlators from Wilson Lines Integrating the Heavy Quark propagator: Which is obtained from small fluctuations of the Wilson line E(t 1,y 1 )Δt 1 δy 1 E(t 2,y 2 )Δt 2 δy 2

32 Force Correlators from Wilson Lines Integrating the Heavy Quark propagator: Which is obtained from small fluctuations of the Wilson line Since in κ there is no time order:

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