Hydroization and Equilibration in Heavy Ion Collisions

Hydroization and Equilibration in Heavy Ion
Collisions
¨
Andreas Schafer
(Regensburg)
The problem
Different approaches
Our work on AdS/CFT
1/λ and 1/N corrections to holographic calculations
Conclusions
1 / 40
A key question in heavy ion physics:
Does one reach thermal equilibrium fast enough to really probe
the quark gluon plasma ?
Observable: Elliptic flow vn ∼ cos(nφ)
2 / 40
How large are the systematic errors, e.g. on the extracted η/s ?
Gale et al. 1209.6330
0.2
v1
v2
v3
v4
v5
〈vn 〉
2 1/2
0.15
0.1
RHIC 200GeV, 30-40%
filled: STAR prelim.
open: PHENIX
η/s = 0.12
0.2
0.05
〈vn2〉1/2
0
0.2
v1
v2
v3
v4
v5
2 1/2
0.15
〈vn 〉
v2
v3
v4
v5
0.15
0.1
RHIC 200GeV, 30-40%
filled: STAR prelim.
open: PHENIX
ATLAS 10-20%, EP
η/s =0.2
0.1
0.05
η/s(T)
0
0
0.05
0.5
1
pT [GeV]
1.5
2
0
0
0.5
1
pT [GeV]
1.5
2
3 / 40
How is thermalization at all possible ?
Thermalization requires massive entropy production at early
stages (< 1 fm/c)
QCD is basically time-reversal invariant. Entropy is only
produced by information loss, typically by incomplete
measurements ⇒ Coarse-graining.
Q1: How can entropy be produced at all before any
measurement takes place ?
Q2: How can it be produced so fast ?
A fundamental problem relevant for many fields of physics, e.g.
reheating after inflation, quantum computers, ...
Heavy ion physics could offer the best chances to understand
this !
4 / 40
Different stages of entropy production in a HIC
cYM & AdS/CFT
hydrodynamics
τ hydro
pQCD+CGC
τdeco
1/Q s
τ therm
τ hadro
t
0.5−2 fm/c ?!
AdS/QCD
About 50% of entropy has to be produced in the thermalization
phase.
entropy production = information loss
⇒ requires measurement; otherwise only an entangled state
5 / 40
The full QCD problem is not tractable. Therefore, one studies
various limits and approximations, hoping that in combination
they will provide a consistent picture
One can solve the classical Yang-Mills equations
numerically:
T. Kunihiro et al. Phys. Rev. D 82 (2010) 114015 [arXiv:1008.1156 [hep-ph]
J. Berges et al. Phys. Rev. D 77 (2008) 034504 [arXiv:0712.3514 [hep-ph]]
One can treat the weak-coupling limit
J. Berges et al. Phys. Rev. D 89 (2014) 114007 [arXiv:1311.3005 [hep-ph]]
α1/2
S
Occupancy nHard
1
α-1
S
α
=
α1/3
S
BD
SU(2) Lattice
Tu SS
rb (e
ul la
en st
ce ic
-2
/3
ex sca
,β
po tte
=
ne rin
0,
nt g)
γ=
s:
1
/3
(pla
sm
a in
sta
bil
itie
s)
α1/7
S
δS
KM (p
ξ0=6
ξ0=4
lasm
a inst
abilitie
Higher anisotropy
Momentum space anisotropy: ΛL/ΛT
BM
s)
ξ0=2
BGLMV (const. anisotropy)
1
Smaller occupancy
ξ0=1
n0=1/4 n0=1
One can solve the strong coupling limit and AdS/CFT – this
talk
6 / 40
The big picture
phase
Q1:mechanism
Q2: τtherm
∆S
1.) pQCD
mixing
0.3 fm/c
20-40 %
2.) cYM
Husimi trafo
2 fm/c
40-50%
2.) AdS/CFT
Bekenstein
0.1 ⇒ 2 fm/c
40-50 %
3.) hydrodynamics
viscosity
1-2 fm/c
10-20 %
4.) hadron gas
fragmentation
1 fm/c
10 %
Interpretation:
“hydroization” time <<1 fm/c, equilibration time 2 fm/c
7 / 40
The basic idea of cYM and AdS/CFT is the same: not
retrievable information can be regarded as lost information
cYM: Coarse graining in non-linear mechanics
p
p
p
∆p
∆x
x
x
x
Husimi-Transformation & Wehrl entropy: No measurement can
beat the uncertainty principle.
8 / 40
BH formation and annihilation is probably adiabatic, e.g. BH
formation at LHC ?!?
T=0 S=0
TH = 0 S BH = 0
TH = 0 S BH = 0
Polchinski et al.: Unmeasurable information is lost information,
Hawking radiation is not “really” a thermal but an entangled
state
the time for thermalization is the time for BH formation.
Bekenstein entropy SB = kB A/(4`2P )
9 / 40
The ideas behind AdS/CFT nice review: Ramallo 1310.4319
renormalization flow of a SU(N) vertex function on ever coarser
lattices
V (x, a) → V (x, 2a) → V (x, 4a) → ...
u
∂
g(u)
∂ log u
=
a, 2a, 4a
=
β(u)
J|UV
=
Φ|∂
UV
u
∂
z
IR
10 / 40
geometric interpretation of new coordinate called z
ds2 = Ω2 (z) [dt 2 − dx i dx i − dz 2 ]
The properties of the renormalization flow is only simple for
conformal theories.
z → λz
L
Ω(z) =
→ λ−1 Ω(z)
z
L2
ds2 =
[dt 2 − dx i dx i − dz 2 ]
z2
SU(N), N = 4 is conformal
8
quantum corrections ∼ `LPl
=
string theory formulation AdS5 ×
π4
2N 2
S5;
AdS − metric
are small for large N.
√
RS4 /(α0 )2 = 4π λN.
11 / 40
planar amplitude
string amplitude
Thus the crucial questions are:
is N = 3 already large ?
answer: dedicated lattice calculations
Is QCD an approximately conformal theory ?
answer: literature, e.g., Braun, Korchemsky and Muller,
¨
hep-ph/0306057
conformal perturbation theory e.g. hep-ph/0605237
NLO→NNLO for GPDs
The answer to both questions is: YES
12 / 40
Pressure
1
0.9
4
p / T , normalized to the SB limit
0.8
0.7
0.6
0.5
SU(3)
SU(4)
SU(5)
SU(6)
SU(8)
improved holographic QCD model
0.4
0.3
0.2
0.1
0
0.5
1
1.5
2
T / Tc
2.5
3
3.5
SU(N) pure gauge theory in 1+3 dimensions
M. Panero, 0907.3719
13 / 40
Energy density
Entropy density
0.4
0.6
0.55
0.35
0.5
0.3
0.45
s/[T (N -1)]
0.35
2
2
ε/[T (N -1)]
0.4
0.25
0.3
2
3
0.2
0.15
SU(2)
SU(3)
SU(4)
SU(5)
SU(6)
holographic model, for α = 3 / 2
0.1
0.05
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
T / Tc
5
5.5
6
6.5
7
0.25
SU(2)
SU(3)
SU(4)
SU(5)
SU(6)
holographic model, for α = 3 / 2
0.2
0.15
0.1
0.05
7.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
T / Tc
5
5.5
6
6.5
7
7.5
energy and entropy in 1+2 dimensions
M. Caselle et. al, 1111.0580
14 / 40
5
m / √σ
4
3
SU(∞)
SU(2)
SU(3)
SU(4)
SU(5)
SU(6)
SU(7)
SU(17)
2
1
0
^
Fπ
^
fρ
ρ
a0
a1
b1
π∗
ρ∗
a*0
a*1
b*1
T = 0 meson spectrum and decay constants
G. Bali et. al, 1304.4437
15 / 40
Equilibration times from AdS/CFT
Idea: Probe black brane formation with a string or membrane.
PRL: 1012.4753 PRD: 1103.2683
fire ball
probing string
falling shell
event horizon
16 / 40
The change in geodetic length is sensitive to equal time
correlators of high dimension gluonic operators.
hO(tshell , x)O(tshell , 0)ishell
≈ e−∆δL(tshell ,x)
hO(tshell , x)O(tshell , 0)iAdS
∆L
0.020
0.015
0.010
0.005
4
6
8
10
x0
17 / 40
We solved analytically and numerically different cases:
AdS3 ∼ CFT (1 + 1), AdS4 ∼ CFT (1 + 2), AdS5 ∼ CFT (1 + 3)
and analyzed how the length of the geodesic/the area of the
surface approaches its thermal value, as a function of ` and t0 .
18 / 40
-0.3
0.
~
-0.1
-0.2
~
-0.2
~
~
-0.2
-0.3
1.
2.
t0
0.
∆L - ∆L thermal
-0.1
~
~
-0.1
0.
∆L - ∆L thermal
0.
∆L - ∆L thermal
0.
-0.3
1.
2.
t0
0.
1.
2.
t0
δ L˜ − δ L˜thermal (L˜ ≡ L/`) for d = 2, 3, 4 (left,right, middle) and
` = 1, 2, 3, 4 (top to bottom curve).
19 / 40
0.
0.
~
-0.02
-0.03
~
-0.06
∆V - ∆V thermal
~
~
~
~
-0.03
0.
∆A - ∆Athermal
∆A - ∆Athermal
0.
-0.04
1.
2.
t0
-0.06
0.
1.
2.
t0
0.
1.
2.
t0
˜ − δV
˜thermal
δ A˜ − δ A˜thermal (A˜ ≡ A/πR 2 and δ V
˜ ≡ V /(4 πR 3 /3) as a function of t0 for radii R = 0.5, 1, 1.5, 2
(V
(top to bottom) in d = 3 (left panel) and d = 4 (middle and right
panel) CFT
20 / 40
Observations
Thermalization is approached as fast as compatible with
causality.
Note: speed of light (gluons), not speed of sound (density
fluctuations)
For heavy ion collisions this implies
τ ∼ 1/(2Qs ) ∼ 0.1fm/c
Clear contradiction to YM result ?
Short distances thermalize first, top-down rather than
bottom-up thermalization
Unavoidable in the AdS dual theory. A fundamental
difference between strong and weak coupling ???
21 / 40
One has to include realistic fluctuations, requiring a 1+4
dimensional numerical code
(Chesler & van der Schee & Yaffe & AS)
0.7
0.6
DΕΕ
0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
z HfmL
B. Schenke, Tribedy, Venugopalan 1206.6805
B.M. & A.S. 1111.3347
energy density at 0.2 fm/c
22 / 40
PRL 1307.1487; JHEP 1307.7086
We included fluctuations in one spatial dimension of a scalar
field on a 1+2 dimensional boundary and calculated in AdS4
using perturbative expansions.
2 2
2
2
φ(t, x) = 1 + e−µ x e−(t−ν) /σ
We compared with
free streaming
second order viscous hydrodynamics
3
1
4
(x, y ) ⇒ “local temperature” = 8πG
πT
3
N
shear and bulk viscosity η =
1
16πGN
4
3 πT
2
and
ζ=0
23 / 40
⇒ [Hubeny and Rangamani,
shear
stress
1006.3675]
Γ0 (−1/3)
γ
3
relaxation time τΠ = 4πT 1 + 3 + 3Γ(−1/3)
“We” (i.e. primarily A. Bernamonti and F. Galli) solve
D = −ut ∂t + ux ∂x
∂ hα u βi
M αβ =
θ
θ = ∂ρ u ρ
1
σ αβ = P αρ P βσ ∂hρ uσi − Pρσ θ
2
Dσ αβ = 2(Dθ)M αβi + 2θDM αβ
h
1 αβ i
h
αβ
Παβ
=
ητ
Dσ
+
σ θ + ...
Π
(2)
2
shear tensor
From the stress tensor Π one can read of px − py
24 / 40
px-py
¶
0.012
0.00001
0.010
t= 0.10Μ
0.008
t= 0.16Μ
t= 0.22Μ
-300
-200
-100
0.006
-0.00001
0.004
-0.00002
100
200
300
x
t= 0.04Μ
t= 0.07Μ
-300
-200
-100
100
200
300
x
t= 0.10Μ
solid: AdS; dashed: free streaming; dotted: 2nd order hydro
Conclusion: hydroization is reached long before equilibration
25 / 40
px-py
5. ´ 10-6
¶
0.012
0.010
t= 0.10Μ
-600
-400
200
-200
400
600
x
-6
-5. ´ 10
t= 0.16Μ
0.008
-0.00001
t= 0.22Μ
0.006
-0.000015
0.004
-600 -400 -200
-0.00002
200
400
600
x
t= 0.04Μ
t= 0.07Μ
t= 0.10Μ
-0.000025
solid: AdS; dashed: free streaming; dotted: 2nd order hydro
26 / 40
px-py
¶
0.010
0.0098
0.00001
0.0022
0.009
-120
0.008
-90
-400
0.0018
0.006
40
80
120
t= 0.04Μ
t= 0.12Μ
0.004
-200
-400
-200
x
400
-0.00001
t= 0.04Μ
-0.00002
t= 0.20Μ
0.002
200
t= 0.08Μ
-0.00003
200
400
x
t= 0.12Μ
-0.00004
solid: AdS; dashed: free streaming; dotted: 2nd order hydro
27 / 40
to become more realistic one needs heavy numerics van der
Schee, Romatschke, Pratt, Hogg 1307.2539 and 1301.2635
AdS/CFT analytic: short time expansion ⇒ starting
condition in local rest frame
T µν = diag(−, PT , PT , −1.5PT )
AdS/CFT numerical solution of Einstein equations,
enforcing convergence
ds2 = −Adτ 2 + Σ2 e−B−C dξ 2 + eB dρ2 + eC dθ2
+2drdτ + 2Fdρdτ
6
X
bi (τ, ρ)r −i
B(r , τ, ρ) → B0 (r , τ, ρ) +
1 + σ 7 r −7
i=0
change at some time to a hydro code
hadronization by MC code describing particle scattering
28 / 40
The general scheme
i) early time expansion ii) numerical AdS iii) hydro iv) kinetic
theory
29 / 40
Time evolution of energy density at center of fireball for different
values of σ , τinit and τhydro .
Central Energy Density
Pb+Pb @ √s = 2.76 TeV
analytic τ<<1
start AdS/CFT code
AdS/CFT
start hydro code
hydro
start cascade code
no hydro matching
4
ε(ρ=0) [GeV ]
1
0.1
0.01
0.1
τ [fm/c]
1
10
30 / 40
PL /PT at center of fireball for different values of τhydro
Pressure Anisotropy
Pb+Pb @ √s = 2.76 TeV
10
no hydro matching
1
0.5
PL/PT(ρ=0)
0
-0.5
analytic τ<<1
start AdS/CFT code
AdS/CFT
start hydro code
hydro
start cascade code
perfect isotropy
-1
-1.5
-2
-2.5
-3
0.1
1
τ [fm/c]
10
31 / 40
Comparison with ALICE data
Effect of choosing τhydro
Pb+Pb @ √s=2.76 TeV
2000
Multiplicity
1800
ALICE
dN/dY
1900
1700
1600
0.52
0.515
ALICE
<pT>[GeV]
0.525
0.51
0.505
0.3
Pion Mean Transverse Momentum
0.35
0.4
0.45
0.5
0.55
τ hydro [fm/c]
0.6
0.65
0.7
32 / 40
Comparison with ALICE data
Light Particle Spectra
Pb+Pb @ √s = 2.76 TeV
+
1000
π ,π
-
AdS+hydro+cascade
hydro @ 1 fm/c + cascade
dN/dY/dpT
100
10
1
+
-
K , K x 0.1
0.1
0.01
0.001
0
1
2
3
4
pT[GeV/c]
33 / 40
Comments:
i.) The instability visible in PL /PT is physically irrelevant
One has to average over the scale of the wave length BM & AS
tbp, hopefully :-)
harmonic oscillator in AdS
Agon, Guijosa and Pedraza 1402.5961
34 / 40
Is this all to good to be true ?
D. Steineder, S. A. Stricker and A. Vuorinen, 1304.3404 claim
2
huge corrections γ = ζ(3)λ−3/2 , λ = NgYM
Z
h
i
√
1
1
1
d 10 x −g R10 − (∂φ)2 −
(F5 )2
2κ10
2
4 · 5!
Z
4
6
√
L
d 10 x −g e−3/2 φ C + T (F5 )
+ γ
2κ10
SIIB =
quasinormal modes for the transverse electric field
eigenmodes with complex frequency, poles of thermal photon
propagator
For λ → ∞ one gets analytically ωn (q = 0) = 2πTn(±1 − i).
35 / 40
0
0
-1
1
2
Re w`
4
3
5
6
7
Ê
Ê
ÊÊ
l=1000
Im w`
-2
Ï
‡
‡
‡‡‡
‡
l=2000
Ú
Ï
Ï
-3
ÏÏ
l=3000
Ï
Ú
Ú
-4
l=•
Ú
Ú
l=10000
Ù
l=5000
The four lowest q = 0 quasinormal poles on the complex ω
plane for different values of λ
36 / 40
AS, A. Vuorinen, S. Waeber, L. Yaffe, tbp
this might be due to an ill-defined expansion
Only one of the higher order corrections
37 / 40
comparison to the situation for retarded Greens function and
shear viscosity
observable
O(γ 0 )
O(γ 1 )-correction
partial O(γ 2 )-corr.
R
iGxy,xy
(ω,0)8π 2
Nc2 r03 w
1
195γ
695γ 2
η( π8 Nc2 T 3 )−1
1
145γ
−8171γ 2
ω
ˆ2
2
−2i
(1.34 · 105
+0.43 · 105 i)γ
(−4.18 · 1010
+3.07 · 1010 i)γ 2
38 / 40
Conclusions
Understanding entropy production, hydroization and
thermalization in HICs is a problem of fundamental
importance.
Thermalization for strong coupling as described by
AdS/CFT suggests τhydroization ≈ 0.1fm/c and
τequilibration ≈ 1fm/c.
One needs 1+4 dim numerical AdS studies, including
realistic initial fluctuations.
Photon production does not provide a counter argument
The initial magnetic field could have significant impact, see
˝ talk
Gergely Endrodi
39 / 40
Many Thanks to
V. Balasubramanian, A. Bernamonti, J. de Boer, L. Castagnini,
P. Chesler, N. Copland, B. Craps, L. Franti, F. Galli, U. Gursoy,
¨
M. P. Heller, E. Keski-Vakkuri, N. Kilbertus, S. Mages, B. Muller,
¨
M. Panero, A. Rabenstein, M. Shigemori, W. Staessens, W. van
der Schee, A. Vuorinen, S. Waeber, L. Yaffe
40 / 40

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