Probing the AdS/CFT Plasma with Heavy Quarks

Jorge Casalderrey-SolanaLBNL

Work in collaboration with Derek Teaney

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =3$ 6

2#T(0.34)

dp

dt= $&Dp + $ (0.35)

"$(t)$(t!)# = '((t$ t!) (0.36)

3

Heavy Quarks at RHICHeavy Quarks are suppressed and participate on collective motion

Model: Langevin dynamics

Fit to elliptic flow:

However: RAA and v2 cannot be fitted simultaneously

Goal:

Compute D in AdS

Can we go beyond Langevin?

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =3$ 6

2#T(0.34)

dp

dt= $&Dp + $ (0.35)

"$(t)$(t!)# = '((t$ t!) (0.36)

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

3

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

ρ ab

(-b/

2, b

/2; -T)A

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

ρ ab

(-b/

2, b

/2; -T)A

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

ρ ab

(-b/

2, b

/2; -T)A

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

!!GR(

!!") = "i!

!$#T 2

2" + !3/2

!$T

2"2 (0.30)

!MkinYr(t) +

!$#T 2

2!Yr(t

!) = &T (t) (0.31)

!3MkinXr(t) +

!$#T 2

2!3Xr(t

!) = &L(t) (0.32)

ff (b) = #Tr [((b)Wc(b)]$A (0.33)

2

Final distribution

ρ ab

(-b/

2, b

/2; -T)A

Momentum BroadeningFrom the final distribution

Transverse gradient ⇒ Fluctuation of the Wilson line

Dressed field strength

Fyµvµ(t1,y1)Δt1δy1

Fyµvµ(t2,y2)Δt2δy2

T

Momentum BroadeningFrom the final distribution

Transverse gradient ⇒ Fluctuation of the Wilson line

Dressed field strength

Four possible correlators

T

Wilson Lines in AdS/CFT

x

t

x

t

z0=1

v=0 x=vt

Horizon

z=ε

Moving string is “straighten” by the coordinates

“World sheet black hole” at z2ws=1/γ

At v=0 both horizons coincide. Coordinate singularity

HKKKY & G

Wilson Lines in AdS/CFT

x

t

x

t

z0=1

v=0 x=vt

Horizon

z=ε

c(z)=v z2=1/γ

Moving string is “straighten” by the coordinates

“World sheet black hole” at z2ws=1/γ

At v=0 both horizons coincide. Coordinate singularity

HKKKY & G

Wilson Lines in AdS/CFT

x

t

x

t

z0=1

v=0 x=vt

Horizon

z=ε

c(z)=v z2=1/γ

Moving string is “straighten” by the coordinates

“World sheet black hole” at z2ws=1/γ

At v=0 both horizons coincide. Coordinate singularity

HKKKY & G

Kruskal MapAs in black holes, (t, r) –coordinates are only defined for r>r0

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

Herzog, Son (02)

The presence of two boundaries leads to properly defined thermal correlators (KMS relations)

Maldacena

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

3

String Solution in Global Coordinates

v=0 smooth crossingv≠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

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

Y (), z) = ei!t (1$ z)#i!/4 (0.40)

Y (), z) = ei!t (1$ z)#i!/4 (0.41)

Y $(), z) = ei!t (1$ z)+i!/4 (0.42)

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

Y (), z) = ei!t (1$ z)#i!/4 (0.40)

Y (), z) = ei!t (1$ z)#i!/4 (0.41)

Y $(), z) = ei!t (1$ z)+i!/4 (0.42)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

Y $(), z) = ei!t (1$ z)+i!/4 (0.43)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

Y (), z) = ei!t (1$ z)#i!/4 (0.40)

Y (), z) = ei!t (1$ z)#i!/4 (0.41)

Y $(), z) = ei!t (1$ z)+i!/4 (0.42)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

Y $(), z) = ei!t (1$ z)+i!/4 (0.43)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

ST2 (0.44)

3

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

Y (), z) = ei!t (1$ z)#i!/4 (0.40)

Y (), z) = ei!t (1$ z)#i!/4 (0.41)

Y $(), z) = ei!t (1$ z)+i!/4 (0.42)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

Y $(), z) = ei!t (1$ z)+i!/4 (0.43)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

ST2 (0.44)

) =!

!) (0.45)

3

String Fluctuations

The fluctuations “live” in the world-sheet

Close to the world-sheet horizon the solution behaves as

infalling

outgoing

The infalling solution leads to the retarded correlator.

GR = !R2!T 2

!"!limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

Z = C

#DX1(t1)DX2(t2)DY1(t1)DY2(t2)e

iSLeiST (0.8)

ST = !1

2

"'

#d#

2!(0.9)

Y1(!#)Y1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.10)

+ Y2(!#)Y2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.11)

+ Y1(!#)Y2(#)ei!/2T (!i2nImGR(#)) (0.12)

+ Y2(!#)e$i!/2T Y1(#) (!i2(1 + n)ImGR(#)) (0.13)

SL = !1

2'2"'

#d#

2!(0.14)

X1(!#)X1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.15)

+ X2(!#)X2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.16)

+ X1(!#)X2(#)ei!/2T (!i2nImGR(#)) (0.17)

+ X2(!#)e$i!/2T X1(#) (!i2(1 + n)ImGR(#)) (0.18)

1

The Boundary ActionThe solution is extended to the full Kruskal plane:

In the world sheet black holePositive energy modes ⇒ infalling

Negative energy modes ⇒ outgoingHerzog, Son (02)

KMS- like relations (even at v >0 !)

!3MkinXr(t) +

!"#T 2

2!3Xr(t

!) = $L(t) (0.32)

ff (b) = "Tr [%(b)Wc(b)]#A (0.33)

D =2T 2

&=

3$ 6

2#T(0.34)

dp

dt= $'Dp + $ (0.35)

"$(t)$(t!)# = &((t$ t!) (0.36)

x3 = vt + $(z) (0.37)

SNG = $ R2

2#l2s

!dtdz

z2

"1$ 1

2

#( ˙y)2

f(z)$ f(z)(y!)2

$%(0.38)

GR()) = $ limz"0

R2

2#l2s

(#T )3

zY ($), z)*zY (), z) (0.39)

Y (), z) = ei!t (1$ z)#i!/4 (0.40)

Y (), z) = ei!t (1$ z)#i!/4 (0.41)

Y $(), z) = ei!t (1$ z)+i!/4 (0.42)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

Y $(), z) = ei!t (1$ z)+i!/4 (0.43)

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

ST2 (0.44)

) =!

!) (0.45)

n =1

e!%

"/T $ 1(0.46)

3

Boundary action

! =!

"! (0.45)

n =1

e!!

"/T " 1(0.46)

# = "!" lim!"0

(1 + 2n)ImGR(!) =!

"$%T 3 (0.47)

4

! =!

"! (0.45)

n =1

e!!

"/T " 1(0.46)

# = "!" lim!"0

(1 + 2n)ImGR(!) =!

"$%T 3 (0.47)

&D =#

2MT=

!$%T 2

2M(0.48)

4

! =!

"! (0.45)

n =1

e!!

"/T " 1(0.46)

# = "!" lim!"0

(1 + 2n)ImGR(!) =!

"$%T 3 (0.47)

&D =#

2MT=

!$%T 2

2M(0.48)

dp

dt= "&Dp (0.49)

4

Momentum BroadeningPerforming derivatives of the boundary action

Grows with EnergyFrom the Einstein relations (v=0)

(Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser)

Direct computation of the energy lost:

Putting numbers:

Comparable to RHIC value!

Can we go beyond Langevin?

where is the noise?

(Gubser hep-ph/0612146)

String Partition Function

The string partition function is

(in progress D. Teaney, D. T. Son, JCS)

boundary values

In the HKKKY & G solution, the finite flux leads to

I. MOVING STRING

We repeat the analysis for the drag string solution which moves in the X direction. Inthis case, the action is

S = SNG + SB (1.1)

where SNG is the Nambu-Goto action and SB is

SB =

!dtEXdX1 !

!dtEXdX2 . (1.2)

The electric field sets the velocity of the probe

EX =

"!T 2"

2

v"1! v2

, (1.3)

which cancels the boundary value of the dragg string solution.Defining, as before, the “ra” basis

Xa =X1 + X2 (t + i#/2)

2, (1.4)

Xr = X1 !X2 (t + i#/2) . (1.5)

after integrating the bulk coordinates as in the previous case,

iSeff = (1.6)

!1

2

!dtdt!

"Xr(t)$

2iGR

#t! t!"

$

$Xa(t

!) + Yr(t)iGR

#t! t!"

$

$Ya(t

!)

%

!1

2

!dtdt!

"Xa(t)$

2iGA

#t! t!"

$

$Xr(t

!) + Ya(t)iGA

#t! t!"

$

$Yr(t

!)

%

!1

2

!dtdt!

"Xr(t)$

2Gsym

#t! t!"

$

$Xr(t

!) + Yr(t)Gsym

#t! t!"

$

$Yr(t

!)

%,

where the position of the string is endpoint is at Xi = (Xi,Yi) and, as before,

Gsym

#t! t!"

$

$="

$

!d%

2"e"i!(t"t!) (1 + 2n) Im GR(%) , (1.7)

with % = %"

$ and n = 1/(exp{%/T}! 1)Identifying, as before, the divergent part of the retarded correlator as the mass, we obtain

iSeff = !i

!dtXr(t)$

3M0QXa(t)! i

!dtYr(t)$M0

QYa(t) (1.8)

!!

dtdt!"Xr(t)$

2iGR

#t! t!"

$

$Xa(t

!) + Yr(t)iGR

#t! t!"

$

$Ya(t

!)

%

!1

2

!dtdt!

"Xr(t)$

2Gsym

#t! t!"

$

$Xr(t

!) + Yr(t)Gsym

#t! t!"

$

$Yr(t

!)

%,

1

I. MOVING STRING

We repeat the analysis for the drag string solution which moves in the X direction. Inthis case, the action is

S = SNG + SB (1.1)

where SNG is the Nambu-Goto action and SB is

SB =

!dtEXdX1 !

!dtEXdX2 . (1.2)

The electric field sets the velocity of the probe

EX =

"!T 2"

2

v"1! v2

, (1.3)

which cancels the boundary value of the dragg string solution.Defining, as before, the “ra” basis

Xa =X1 + X2 (t + i#/2)

2, (1.4)

Xr = X1 !X2 (t + i#/2) . (1.5)

after integrating the bulk coordinates as in the previous case,

iSeff = (1.6)

!1

2

!dtdt!

"Xr(t)$

2iGR

#t! t!"

$

$Xa(t

!) + Yr(t)iGR

#t! t!"

$

$Ya(t

!)

%

!1

2

!dtdt!

"Xa(t)$

2iGA

#t! t!"

$

$Xr(t

!) + Ya(t)iGA

#t! t!"

$

$Yr(t

!)

%

!1

2

!dtdt!

"Xr(t)$

2Gsym

#t! t!"

$

$Xr(t

!) + Yr(t)Gsym

#t! t!"

$

$Yr(t

!)

%,

where the position of the string is endpoint is at Xi = (Xi,Yi) and, as before,

Gsym

#t! t!"

$

$="

$

!d%

2"e"i!(t"t!) (1 + 2n) Im GR(%) , (1.7)

with % = %"

$ and n = 1/(exp{%/T}! 1)Identifying, as before, the divergent part of the retarded correlator as the mass, we obtain

iSeff = !i

!dtXr(t)$

3M0QXa(t)! i

!dtYr(t)$M0

QYa(t) (1.8)

!!

dtdt!"Xr(t)$

2iGR

#t! t!"

$

$Xa(t

!) + Yr(t)iGR

#t! t!"

$

$Ya(t

!)

%

!1

2

!dtdt!

"Xr(t)$

2Gsym

#t! t!"

$

$Xr(t

!) + Yr(t)Gsym

#t! t!"

$

$Yr(t

!)

%,

1

Quantum correction: Small fluctuations over classical path

GR = !R2!T 2

!"! limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

1

GR = !R2!T 2

!"! limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

1

GR = !R2!T 2

!"! limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

Z = C

#DX1(t1)DX2(t2)DY1(t1)DY2(t2)e

iSLeiST (0.8)

1

GR = !R2!T 2

!"!limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

Z = C

#DX1(t1)DX2(t2)DY1(t1)DY2(t2)e

iSLeiST (0.8)

ST = !1

2

"'

#d#

2!(0.9)

Y1(!#)Y1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.10)

+ Y2(!#)Y2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.11)

+ Y1(!#)Y2(#)ei!/2T (!i2nImGR(#)) (0.12)

+ Y2(!#)e$i!/2T Y1(#) (!i2(1 + n)ImGR(#)) (0.13)

SL = !1

2'2"'

#d#

2!(0.14)

X1(!#)X1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.15)

+ X2(!#)X2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.16)

+ X1(!#)X2(#)ei!/2T (!i2nImGR(#)) (0.17)

+ X2(!#)e$i!/2T X1(#) (!i2(1 + n)ImGR(#)) (0.18)

1

GR = !R2!T 2

!"!limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

Z = C

#DX1(t1)DX2(t2)DY1(t1)DY2(t2)e

iSLeiST (0.8)

ST = !1

2

"'

#d#

2!(0.9)

Y1(!#)Y1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.10)

+ Y2(!#)Y2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.11)

+ Y1(!#)Y2(#)ei!/2T (!i2nImGR(#)) (0.12)

+ Y2(!#)e$i!/2T Y1(#) (!i2(1 + n)ImGR(#)) (0.13)

SL = !1

2'2"'

#d#

2!(0.14)

X1(!#)X1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.15)

+ X2(!#)X2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.16)

+ X1(!#)X2(#)ei!/2T (!i2nImGR(#)) (0.17)

+ X2(!#)e$i!/2T X1(#) (!i2(1 + n)ImGR(#)) (0.18)

SNG = SCL + S2 (0.19)

1

Quadratic Fluctuations

Large Mass ⇒ Quadratic fluctuations

Same problem as in broadening. We integrate out the fluctuations for z>0 and obtain

This is the partition function of the heavy quark.

Divergent constant

GR = !R2!T 2

!"!limu"0

1

u1/2Y # (!#, u) $uY (#, u) (0.1)

% = limu"0

!2T

#ImGR (#) =

"&!T 3 (0.2)

Y i (u; #)# Y (u; #) (0.3)

Y o (u; #)# Y # (u; #) (0.4)

' <

!M"&T

"2

(0.5)

1) v # 1

2) M # $

Z =

#DXt

1(t1)DXt2(t2)

#Dxt

1(t, r)Dxt2(t, r)e

iSNG+iSB (0.6)

xt(t, z) = (vt + ((r) + x(t, r),y(t, r)) (0.7)

Z = C

#DX1(t1)DX2(t2)DY1(t1)DY2(t2)e

iSLeiST (0.8)

ST = !1

2

"'

#d#

2!(0.9)

Y1(!#)Y1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.10)

+ Y2(!#)Y2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.11)

+ Y1(!#)Y2(#)ei!/2T (!i2nImGR(#)) (0.12)

+ Y2(!#)e$i!/2T Y1(#) (!i2(1 + n)ImGR(#)) (0.13)

SL = !1

2'2"'

#d#

2!(0.14)

X1(!#)X1(#) (ReGR(#) + i(1 + 2n)ImGR(#)) (0.15)

+ X2(!#)X2(#) (!ReGR(#) + i(1 + 2n)ImGR(#)) (0.16)

+ X1(!#)X2(#)ei!/2T (!i2nImGR(#)) (0.17)

+ X2(!#)e$i!/2T X1(#) (!i2(1 + n)ImGR(#)) (0.18)

SNG = SCL + S2 (0.19)

"'GR(#) = ! R2

2!l2s

1

zm'#2 !

$&'

(!T )3

2!C

!#

!T

"(0.20)

1

Kinetic term and “ra” basisGR is divergent:

MQ (finite)Redefined GR

We obtain a kinetic term

(Feynman, Vernon, Caldeira, Legget, C. Greiner)

Stadard procedure: introduce “ra” basis

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

2

Conjugate to momentum

where we have redefine the correlators to include only the finite part.Introducing, as before, the longitudinal and trasverse foces, !L and !T and splitting the

partition function into a longitudinal and transverse piece

Z = ZLZT , (1.9)

with

ZL =

!DXrD !L "

"#3M0

QXr(t) +

! t

dt!#2GR

"t! t!"

#

#Xr(t

!)! !L(t)

#

exp

$! 1

2#2

!dtdt!!L(t)G"1

sym

"t! t!"

#

#!L(t!)

%(1.10)

ZT =

!DYrD !T "

"#M0

QYr(t) +

! t

dt!GR

"t! t!"

#

#Yr(t

!)! !T (t)

#

exp

$!1

2

!dtdt!!T (t)G"1

sym

"t! t!"

#

#!T (t!)

%(1.11)

where!

dt!Gsym

"t! t!"

#

#G"1

sym

"t! ! t!!"

#

#= "(t! t!!) (1.12)

In the low frequency limit,

GR

"t"

#

#="

#

!d$

2%e"i!tGR($) =

!d$

2%e"i!t

&!i#

&

2T$ + #3/2

"'T

2$2

',(1.13)

Gsym

"t"

#

#="

#&"(t) , (1.14)

and the equations of motion are

#3MkinXa(t) +&

2T#3Xa ! !L = 0 (1.15)

#MkinYa(t) +&

2T#Ya ! !T = 0 (1.16)

with

Mkin = M0Q !

"#'T

2(1.17)

Note that the condition that (M0Q !Mkin)/M0

Q << 1 implies

# <<M2

Q

'T 2(1.18)

By recaling that Xa and Ya are small fluctuations on the classical path we find

dP

dt= !µP +

&

2T

v"1! v2

x + ! , (1.19)

2

where we have redefine the correlators to include only the finite part.Introducing, as before, the longitudinal and trasverse foces, !L and !T and splitting the

partition function into a longitudinal and transverse piece

Z = ZLZT , (1.9)

with

ZL =

!DXrD !L "

"#3M0

QXr(t) +

! t

dt!#2GR

"t! t!"

#

#Xr(t

!)! !L(t)

#

exp

$! 1

2#2

!dtdt!!L(t)G"1

sym

"t! t!"

#

#!L(t!)

%(1.10)

ZT =

!DYrD !T "

"#M0

QYr(t) +

! t

dt!GR

"t! t!"

#

#Yr(t

!)! !T (t)

#

exp

$!1

2

!dtdt!!T (t)G"1

sym

"t! t!"

#

#!T (t!)

%(1.11)

where!

dt!Gsym

"t! t!"

#

#G"1

sym

"t! ! t!!"

#

#= "(t! t!!) (1.12)

In the low frequency limit,

GR

"t"

#

#="

#

!d$

2%e"i!tGR($) =

!d$

2%e"i!t

&!i#

&

2T$ + #3/2

"'T

2$2

',(1.13)

Gsym

"t"

#

#="

#&"(t) , (1.14)

and the equations of motion are

#3MkinXa(t) +&

2T#3Xa ! !L = 0 (1.15)

#MkinYa(t) +&

2T#Ya ! !T = 0 (1.16)

with

Mkin = M0Q !

"#'T

2(1.17)

Note that the condition that (M0Q !Mkin)/M0

Q << 1 implies

# <<M2

Q

'T 2(1.18)

By recaling that Xa and Ya are small fluctuations on the classical path we find

dP

dt= !µP +

&

2T

v"1! v2

x + ! , (1.19)

2

Random Force The integration of Xa leads to Z=ZTZL

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

2

Force distribution:

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

2

⇒ω→0

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

!!GR(

!!") = "i!

!$#T 2

2" + !3/2

!$T

2"2 (0.30)

2

Equation of Motion

} small fluctuations on top of

X=vt

Large mass ⇒ low frequency approximation of GR

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

!!GR(

!!") = "i!

!$#T 2

2" + !3/2

!$T

2"2 (0.30)

!MkinYr(t) +

!$#T 2

2!Yr(t

!) = &T (t) (0.31)

!3MkinXr(t) +

!$#T 2

2!3Xr(t

!) = &L(t) (0.32)

2

SNG = SCL + S2 (0.19)

!!GR(") = " R2

2#l2s

1

zm!"2 "

!$!

(#T )3

2#C

""

#T

#(0.20)

Xr =X1(t) + X2(t + i%/2)

2(0.21)

Xa = X1(t)"X2(t + i%/2) (0.22)

Gsym(") = " (1 + 2n) ImGR(") (0.23)

#&L(t)&L(t!)$ = !2Gsym

"t" t!!

!

#(0.24)

#&T (t)&T (t!)$ = Gsym

"t" t!!

!

#(0.25)

#&L(t)&L(t!)$ = !2!

!$#T 3'(t" t!) (0.26)

#&T (t)&T (t!)$ =!

!$#T 3'(t" t!) (0.27)

!3M0QXr(t) +

$ t

dt!!2GR

"t" t!!

!

#Xr(t

!) = &L(t) (0.28)

!M0QYr(t) +

$ t

dt!GR

"t" t!!

!

#Yr(t

!) = &T (t) (0.29)

!!GR(

!!") = "i!

!$#T 2

2" + !3/2

!$T

2"2 (0.30)

!MkinYr(t) +

!$#T 2

2!Yr(t

!) = &T (t) (0.31)

!3MkinXr(t) +

!$#T 2

2!3Xr(t

!) = &L(t) (0.32)

2

where P is the momentum and ! = (!L, !T). Using Eq. (1.3) we find

dP

dt= !µP + E x + ! , (1.20)

After integrating the coordinates fluctuation are related to the noise term

Xr(t) =1

"3M0Q

!dt!G(t! t!)!L(t!) , (1.21)

Yr(t) =1

"M0Q

!dt!G(t! t!)!T (t!) , (1.22)

where the Fourier transformed of G(t) is

G(#) =1

!#2 + GR(!"

")"

"M0Q

(1.23)

For large M0Q the retarded correlator can be neglected and the propagartor is

G(t) = t$(t) . (1.24)

II. PHOTOM BREMSSTRAHLUNG

The induced photon radiation from the quark can be computed from the classical radia-tion field

dN

d3k=

1

#

dE!

d3k= ! #

4%2

!dt1dt2

"jµ(t1)jµ(t2)e

#i!(t1#t2)+ik(x(t1)#x(t2))#

, (2.1)

where the path x(t) and the current jµ(t) are

x(t) = (vt + Xr(t)) x + Yr(t) , (2.2)

jµ(t) = "tot (1, x(t)) , (2.3)

with "tot = 1/$

1! x(t)2

Since the position x(t) is a linear functional of the noise !, after defining

!!(t) = !(t)! ik

"M

!dt!GA(t!; t1, t2)Gsym

%t! ! t"

"

&, (2.4)

GA(t!; t1, t2) = G(t1 ! t!)! G(t2 ! t!) (2.5)

the average can be simplified to

"jµ(t1)jµ(t2)e

#i!(t1#t2)+ik(x(t1)#x(t2))#

= #jµ(t1)jµ(t2)$! e#i(!#vkx)(t1#t2) (2.6)

exp

'!

(k2

x

"4M2Q

+k2$

"2M2Q

)!dt!dt!!GA(t!; t1, t2)Gsym

%t! ! t!!"

"

&GA(t!!; t1, t2)

*

3

where we have redefine the correlators to include only the finite part.Introducing, as before, the longitudinal and trasverse foces, !L and !T and splitting the

partition function into a longitudinal and transverse piece

Z = ZLZT , (1.9)

with

ZL =

!DXrD !L "

"#3M0

QXr(t) +

! t

dt!#2GR

"t! t!"

#

#Xr(t

!)! !L(t)

#

exp

$! 1

2#2

!dtdt!!L(t)G"1

sym

"t! t!"

#

#!L(t!)

%(1.10)

ZT =

!DYrD !T "

"#M0

QYr(t) +

! t

dt!GR

"t! t!"

#

#Yr(t

!)! !T (t)

#

exp

$!1

2

!dtdt!!T (t)G"1

sym

"t! t!"

#

#!T (t!)

%(1.11)

where!

dt!Gsym

"t! t!"

#

#G"1

sym

"t! ! t!!"

#

#= "(t! t!!) (1.12)

In the low frequency limit,

GR

"t"

#

#="

#

!d$

2%e"i!tGR($) =

!d$

2%e"i!t

&!i#

&

2T$ + #3/2

"'T

2$2

',(1.13)

Gsym

"t"

#

#="

#&"(t) , (1.14)

and the equations of motion are

#3MkinXa(t) +&

2T#3Xa ! !L = 0 (1.15)

#MkinYa(t) +&

2T#Ya ! !T = 0 (1.16)

with

Mkin = M0Q !

"#'T

2(1.17)

Note that the condition that (M0Q !Mkin)/M0

Q << 1 implies

# <<M2

Q

'T 2(1.18)

By recaling that Xa and Ya are small fluctuations on the classical path we find

dP

dt= !µP +

&

2T

v"1! v2

x + ! , (1.19)

2

where we have redefine the correlators to include only the finite part.Introducing, as before, the longitudinal and trasverse foces, !L and !T and splitting the

partition function into a longitudinal and transverse piece

Z = ZLZT , (1.9)

with

ZL =

!DXrD !L "

"#3M0

QXr(t) +

! t

dt!#2GR

"t! t!"

#

#Xr(t

!)! !L(t)

#

exp

$! 1

2#2

!dtdt!!L(t)G"1

sym

"t! t!"

#

#!L(t!)

%(1.10)

ZT =

!DYrD !T "

"#M0

QYr(t) +

! t

dt!GR

"t! t!"

#

#Yr(t

!)! !T (t)

#

exp

$!1

2

!dtdt!!T (t)G"1

sym

"t! t!"

#

#!T (t!)

%(1.11)

where!

dt!Gsym

"t! t!"

#

#G"1

sym

"t! ! t!!"

#

#= "(t! t!!) (1.12)

In the low frequency limit,

GR

"t"

#

#="

#

!d$

2%e"i!tGR($) =

!d$

2%e"i!t

&!i#

&

2T$ + #3/2

"'T

2$2

',(1.13)

Gsym

"t"

#

#="

#&"(t) , (1.14)

and the equations of motion are

#3MkinXa(t) +&

2T#3Xa ! !L = 0 (1.15)

#MkinYa(t) +&

2T#Ya ! !T = 0 (1.16)

with

Mkin = M0Q !

"#'T

2(1.17)

Note that the condition that (M0Q !Mkin)/M0

Q << 1 implies

# <<M2

Q

'T 2(1.18)

By recaling that Xa and Ya are small fluctuations on the classical path we find

dP

dt= !µP +

&

2T

v"1! v2

x + ! , (1.19)

2

Since these are small fluctuations:

The effective mass is v dependent!

(Same as HKKKY at v=0)

ConclusionsThe 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

We have obtained finite frequency corrections to Langevin dynamics. These may be used in phenomenology.

Quantum fluctuation of the string lead to the appearance of the noise distribution. The 1-2 formalism is important.

We obtained a velocity dependent kinetic mass. The effective is reduced for fast particles. Consistency demands:

! =!

"! (0.45)

n =1

e!!

"/T " 1(0.46)

# = "!" lim!"0

(1 + 2n)ImGR(!) =!

"$%T 3 (0.47)

&D =#

2MT=

!$%T 2

2M(0.48)

dp

dt= "&Dp (0.49)

" <M2

$T 2(0.50)

4

Back up Slides

Computation of (Radiative Energy Loss)(Liu, Rajagopal, Wiedemann)

t

L

r0

Dipole amplitude: two parallel Wilson lines in the light cone:

For small transverse distance:

Order of limits:

String action becomes imaginary for

entropy scaling

Energy Dependence of (JC & X. N. Wang)

From the unintegrated PDF

Evolution leads to growth of the gluon density,

In the DLA

Saturation effects ⇒

For an infinite conformal plasma (L>Lc) with Q2max=6ET.At strong coupling

HTL provide the initial conditions for evolution.

Noise from Microscopic Theory

HQ momentum relaxation time:

Consider times such that

⇒

microscopic force (random)

charge density electric field

Heavy Quark Partition FunctionMcLerran, Svetitsky (82)

Polyakov Loop

YM + Heavy Quark states YM states

Integrating out the heavy quark

Changing the Contour Time

κ as a Retarded Correlator

κ is defined as an unordered correlator:

From ZHQ the only unordered correlator is

Defining:

In the ω→0 limit the contour dependence disappears :

Force Correlators from Wilson Lines

Which is obtained from small fluctuations of the Wilson line

Integrating the Heavy Quark propagator:

E(t1,y1)Δt1δy1

E(t2,y2)Δt2δy2

Force Correlators from Wilson Lines

Which is obtained from small fluctuations of the Wilson line

Integrating the Heavy Quark propagator:

Since in κ there is no time order: