Transport Coefficients of Quark Gluon Plasma - Viljami Leino Péter Petreczky, Indico
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Transport Coefficients
of
Quark Gluon Plasma
Viljami Leino
Nora Brambilla, Péter Petreczky, Antonio Vairo
Technische Universität München, Brookhaven National Laboratory
RU-A: Fundamental Particles and Forces
Research Day
20.10.2020
Heavy Ion Collisions and Phase Diagram • Understanding the strongly coupled Quark Gluon Plasma (QGP) generated at particle accelerators • Ultra-relativistic collisions of hea- vy nuclei at particle accelerators produce hot and dence fireball • A strongly coupled medium Figure from: T. Nayak (2020) 1 / 15
Transport Coefficients
• The QGP can be described in terms of transport coefficients such as:
• Shear viscosity • Electric conductivity
• Diffusion coefficient • Transverse momentum distribution q̂
• Usually real time quantities, need special attention on lattice:
• Only Euclidean observables can be defined on lattice
→ need to invert spectral function
• Transport peaks make measurements difficult,
especially at higher quark masses the peaks become sharp
• Related to observables measured experimentally such as RAA and ν2
2 / 15
Diffusion Coefficient
• We don’t have RAA and ν2 direct-
2 πTD s
T)
α s(
Lattice QCD LO
D
C pQCD LO α S=0.4
ly in our theories, instead different Ding et al.
Banerjee et al.
pQ
30
(hydrodynamical) models depend Kaczmarek et al.
on spatial diffusion coefficient Dx T-Matrix V=
F
20
- LV
• Observed ν2 is larger than expec- (Cata
nia)
D-
QPM
m
es
D-
on
PHSD
ted from kinetic models but in 10
m
(T
=U ia) - B
M
es
trix V
AM
T-Ma (Catan
on
QPM
U)
ayesian)
(O
Duke (B
zv
good agreement with hydrodyna-
en
c
hu
MC@sHQ AdS/CFT
k)
0
mic models 1 1.5 2
T/T c
• Indicates the medium has fluidlike properties
• RAA tells how heavy quarks see the nuclear medium
Figure from: X. Dong CIPANP (2018) 3 / 15
Heavy Quark in medium
• Heavy quark energy doesn’t change much in collision with a
thermal quark √
Ek ∼ T , p ∼ MT T
• HQ momentum is changed by random kicks from the medium
→ Brownian motion; Follows Langevin dynamics
dpi κ
=− pi + ξi (t) , hξ(t)ξ(t 0 )i = κδ(t − t 0 )
dt 2MT
• Heavy quark momentum diffusion coefficient κ related to many
interesting phenomena
Such as: Spatial diffusion coefficient Ds = 2T 2 /κ,
Drag coefficient ηD = κ/(2MT ),
−1
Heavy quark relaxation time τQ = ηD
Moore and Teaney PRC71 (2005), Caron-Huot and Moore JHEP02 (2008) 4 / 15
Quarkonium in medium
• Quarkonium in QGP (environment energy scale πT )
1
M πT E , τR τE ∼ 1/π
a0
• HQ mass M, Bohr radius a0 , binding energy E , correlation time τE
• Quarkonium in fireball can be described by Limbland equation
dρ X
n m† 1 m† n
= −i[H, ρ] + hnm Li ρLi − {Li Li , ρ}
dt n,m
2
• All terms depend on two free parameters κ and γ
• κ turns out to be the heavy quark diffusion coefficient and related to
thermal width Γ(1S) = 3a02 κ
• γ is correction to the heavy quark-antiquark potential and related to
mass shift δM(1S) = 2a02 γ/3
• Unquenched lattice measurements of Γ(1S) and δM(1S) available
Brambilla et.al.PRD96 (2017), Brambilla et.al.PRD97 (2018) 5 / 15
Importance of non-perturbative κ determination
g 4 Cf Nc
κ 2T mE
3
= ln +ξ + C
T 18π mE T
ξ ' −0.64718 , C NLO ' 2.3302
κ ∈ {κL (T),κC (T),κU (T)}, γ = -1.75, Tf = 250 MeV
1.0
Υ(1s)
Υ(2s)
0.8
Υ(3s)
• Huge perturbative variation 0.6
RAA
⇒ needs non-perturbative measu-
0.4
rements
• Also large scale dependence 0.2
trough mE = g (µ)T 0.0
• κ has considerable effect on RAA 0 100 200 300 400
Npart
Moore and Teaney PRC71 (2005), Caron-Huot and Moore JHEP02 (2008), RAA fig by P. Vander Griend 6 / 15Heavy quark diffusion from lattice
• Traditional approach using current correlators has transport peak
• HQEFT inspired Euclidean correlator free of transport peaks
3
X hRe Tr [U(1/T , τ )Ei (τ, 0)U(τ, 0)Ei (0, 0)]i
GE (τ ) = −
3 hRe Tr U(1/T , 0)i
i =1
• To get momentum diffusion coefficient κ, a spectral function ρ(ω)
needs to be reversed:
ω 1
∞
cosh τT −
Z
dω T 2
GE (τ ) = ρ(ω, T )K (ω, τ T ) , K (ω, τ T ) = ω
0 π sinh 2T
2T ρ(ω)
κ = limω→0
ω
• Measure using the multilevel algorithm in pure gauge (quenched)
• Compared to earlier studies we measure extremely wide range of
temperatures
• Create model ρ(ω) by matching to perturbation theory at high T
• Invert the spectral function equation by varying the model ρ
7 / 15Lattice correlator
105
483×24 T =1.1Tc
483×24 T =1.5Tc
104 483×24 T =3Tc
483×24 T =6Tc
483×24 T =10Tc
103
483×24 T =10000Tc
ZE G E
102
101
100
0.0 0.1 0.2 0.3 0.4 0.5
τT
• Normalize lattice data with the LO Perturbative result:
cos2 (πτ T )
1
GEnorm = π 2 T 4 +
sin4 (πτ T ) 3 sin2 (πτ T )
• Perform tree-level improvement by matching lattice and continuum
perturbation theories
Caron-Huot et.al.JHEP04 (2009), Francis et.al.PoSLattice (2011) 8 / 15When do thermal effects start
GE (Nt , β) GE (2Nt , β)
R2 (Nt ) = norm .
GE (Nt ) GEnorm (2Nt )
• On small physical separation every T shares a scaling
(apart from finite size effects)
• Thermal effect nonexistent for τ < 0.10, then grow
9 / 15Continuum limit and finite size effects
• Use 3 largest lattices for continuum limit
• Systematic include tadpole and extrapolations with and without
Nt = 12 point or a4 term.
• Finite size effects are in control, we can go without extrapolation.
10 / 15High Temperature
0.68
Continuum limit / 1.2
0.66 ρNLO
step LO
0.64
GE/GEnorm
0.62
0.60
0.58
0.56
0.20 0.25 0.30 0.35 0.40 0.45
τT
• NLO spectral function works only at very high temperatures
• Different ansatze have different ω ∼ T behavior
• Good matching between perturbation theory and lattice
11 / 15κ extraction
1.10 1.01
ρNLO
step
ρNLO
line (0.01,2.2) 1.00
1.05
ZEGE/(CNGENLO+)
ZEGE/(CNGENLO+)
0.99
1.00
0.98
0.95 0.97
0.96
0.90
0.95 ρNLO
step
0.85 ρNLO
line (0.01,2.2)
0.94
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
τT τT
• Take continuum limit of the lattice data
• Normalize with different models for spectral function
• Extract κ as all values that normalize to unity in
0.19 ≥ τ T ≥ 0.45
12 / 15Lattice results for Ds
30 NLO Brambilla 2019
Banerjee 2012 ALICE 2018
Francis 2015 Our result
Ding 2012
20
2πDsT
10
0
1.0 1.5 2.0 2.5 3.0
T /Tc
• On low temperature close to Tc , agreement with other results,
including ALICE
Brambilla et al 2020: Accepted to PRD, hep-lat/2007.10078 13 / 15Lattice results for κ
5 fit
NLO
4 Our result
κ/T 3
3
2
1
0
100 101 102 103 104
T /Tc
• Unprecedented temperature range: κNLO g 4 CF Nc
2T mE
= ln +ξ+C .
T = 1.1 − 104 Tc T3 18π mE T
• Can fit temperature dependence C = 3.81(1.33)
Brambilla et al 2020: Accepted to PRD, hep-lat/2007.10078 14 / 15Conclusions and Future prospects
• We have measured κ in wide range of temperatures and fitted the
temperature dependence
• Observe κ/T 3 decreasing when T increases, similar to perturbation
theory
• Future prospects:
• Measure γ
• Implement gradient flow to go un-quenched
• 1/M corrections
• Other operators?
15 / 15Thank You
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