Heavy Quark Diffusion from the Lattice Viljami Leino Technische - - PowerPoint PPT Presentation

Heavy Quark Diffusion from the Lattice

Viljami Leino

Technische Universität München, t30f

In collaboration with: Nora Brambilla, Saumen Datta, Miguel Escobedo, Peter Petreczky, Antonio Vairo, and Peter Vander Griend From Euclidean spectral densities to real-time physics 15.03.2019 CERN

Outline

• Introduction
• Transport coefficients from in medium quarkonium dynamics
• Lattice measurement
• Conclusions

0 / 27

Introduction

• Charm and bottom quarks much

heavier than RHIC/LHC tempera- tures

• Can be used to probe early time

physics

• Experimental results for nuclear

suppression factor RAA and elliptic flow ν2 differ from simple perturba- tive estimates

• Both RAA and ν2 can be calculated

from diffusion constant D of heavy quark in medium

• D can be tuned to match experi-

mental results

) c (GeV/

T

p 2 4 6 8 10 12 14 16 18 20 22 24

2

v 0.1 − 0.1 0.2 0.3 0.4

|<0.8 y average, |

+

, D*

+

, D D = 5.02 TeV

NN

s , |>0.9} η ∆ {EP, |

2

v = 2.76 TeV

NN

s , |>0} η ∆ {EP, |

2

v PRL 111 (2013) 102301 = 2.76 TeV

NN

s |<0.5, y , |

±

π , JHEP 06 (2015) 190 |>0.9} η ∆ {SP, |

2

v , PLB 719 (2013) 18 |>2} η ∆ {EP, |

2

v

• Syst. from data
• Syst. from B feed-down

ALICE Pb − 50% Pb − 30

c

T/T

1 1.5 2 s

TD π 2

10 20 30

D-meson (TAMU) D-meson (Ozvenchuk) =0.4

S

α pQCD LO (T)

s

α pQCD LO T

• M

a t r i x V = F T-Matrix V=U PHSD QPM (Catania) - LV QPM (Catania) - BM MC@sHQ AdS/CFT Duke (Bayesian)

Lattice QCD

et al. Ding et al. Banerjee et al. Kaczmarek

1 / 27

Figures from: ALICE PRL120 (2017), X. Dong CIPANP (2018)

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
• Successive collisions with medium are uncorrelated

→ Brownian motion

• The physics can be simulated with Langevin dynamics

dpi dt = −ηDpi + ξi(t) , ξ(t)ξ(t′) = κδ(t − t′)

• κ: strength of stochastic interaction: property of medium
• The drag coefficient ηD = κ/(2MT)
• Relaxation time τR = 1/ηD
• In position space x2(t) = 6Dt with D = 2T 2/κ

2 / 27

Moore et.al.PRC71 (2005), Caron-Huot et.al.JHEP02 (2008)

Perturbation theory

• Perturbation theory has poor convergence

0.0 0.5 1.0 1.5 2.0 2.5

g

0.0 0.1 0.2 0.3 0.4 0.5 0.6

κ g 4T 3

NLO LO truncated LO

• Non-perturbative methods needed

3 / 27

Moore et.al.PRC71 (2005), Caron-Huot et.al.JHEP02 (2008)

Quarkonium in medium

• Quarkonium characterized by energy-scales:

mass M, Bohr radius a0, and binding Energy E

• M ≫ 1

a0 ≫ E

• Environment at energy scale πT and correlation time τE ∼ 1/πT
• Evolution of system characterized by relaxation time τR
• Assume

1 a0 ≫ πT and πT ≫ E

→ τR ∼

1 a2

0(πT)3 and τR ≫ τE

• With these assumptions quarkonium can be described by Limbland

equation

• The Limbland model depends only on two transport coefficients: κ

and γ

• κ turns out to be the heavy quark diffusion coefficient
• γ is correction to the heavy quark-antiquark potential

4 / 27

Brambilla et.al.PRD96 (2017), Brambilla et.al.PRD97 (2018)

Transport Coefficients from in Medium Quarkonium Dynamics

5 / 27

The singlet self-energy in pNRQCD

ΣS(t) = rirj 2NC t

t0

dt′ gE a,i(t, 0)gE a,j(t′, 0) = rirj 2

• ˜

κij(t) + iˆ γij(t)

• In large time limit, assuming time translation invariance:

κ = 1 6Nc ∞ dt

• gE a,i(t, 0)gE a,j(0, 0)
• γ = −i

6Nc ∞ dt

• gE a,i(t, 0)gE a,j(0, 0)
• Where κ is just the heavy quark momentum diffusion coefficient

6 / 27

Brambilla et.al.PRD96 (2017), Brambilla et.al.PRD97 (2018)

In medium mass shift and width

• From previous slide it follows:

r2κ = Σs + Σ†

s = −2Im(−iΣs)

r2γ = −iΣs + iΣ†

s = 2Re(−iΣs)

• The self-energies provide the in medium induced mass shifts δMs

and widths Γs

• For 1S Coulombic quarkonium state:

Γ(1S) = 3a2

0κ

δM(1S) = 3 2a2

0γ

• For Coulombic system: Solve a0 from self-consistency equation with

1-loop 3 flavor running coupling: a0 = 2/ [MCFαs(1/a0)]

7 / 27

Brambilla et.al.PRD96 (2017), Brambilla et.al.PRD97 (2018)

Scales

• Use recent lattice determination of δM(1S) and Γ(1S)

(Aarts et.al.JHEP11 (2011), Kim et.al.JHEP11 (2018))

• Available lattice data for J/ψ at T = 251MeV and Υ(1S) at

T = 251MeV and T = 407MeV

• Use masses Mb = 4.78GeV and Mc = 1.67GeV
• Scale lattice masses to above values
• The binding energies are EΥ(1S) = −0.1GeV and

EJ/ψ = −0.24GeV

• The Bohr radiuses are

1 a0 = 0.84GeV for J/ψ and 1 a0 = 1.5GeV for

Υ(1S)

• Scale hierarchies E ≪ πT ≪ 1/a0 satisfied

8 / 27

κ Measurement

0.5 1 1.5 2

T/Tc

0.5 1 1.5 2

Γ/T

3S1(vector)

Upsilon

• The thermal width in (Kim et.al.JHEP11 (2018)) preliminary and taken as

lower bound.

• For upper bound use slightly older result (Aarts et.al.JHEP11 (2011))
• T/Tc ≈ 2, Tc = 220MeV
• Un-quenched determination of κ from lattice data

9 / 27

Brambilla et.al.TUM-EFT122/18 (2019), Fig from: Aarts et.al.JHEP11 (2011)

κ Result

−0.24 κ T 3 4.2

• Un-quenched determination of κ from lattice data

10 / 27

Brambilla et.al.TUM-EFT122/18 (2019)

γ Measurement

• 200
• 150
• 100
• 50

140 160 185 223 251 n=8 charmonium 3S1 [BR T>0] calib. [BR T=0 trunc.] Δm [MeV] (J/ψ 3S1) T [MeV]

• 200
• 150
• 100
• 50

140 160 185 223 251 333 407 n=4 bottomonium 3S1 [BR T>0] calib. [BR T=0 trunc.] Δm [MeV] (Υ 3S1) T [MeV]

• Use results from (Kim et.al.JHEP11 (2018))
• Two distinct mass shifts J/ψ and Υ(1S)
• Use two different temperatures T = 251MeV and T = 407MeV
• Unquenched measurement

11 / 27

Brambilla et.al.TUM-EFT122/18 (2019), Figs from: Kim et.al.JHEP11 (2018)

γ Results

−3.8 γ T 3 −0.7

• First non-perturbative determination of γ

12 / 27

Brambilla et.al.TUM-EFT122/18 (2019)

Transport Coefficients from lattice euclidean correlator

13 / 27

Directly from HQ current correlator

• Operator of interest: HQ current correlator ¯

QγiQ

• problematic observable:
• Quite insensitive to D (Teaney PRD74 (2006), Petreczky EPJC62 (2008))
• Narrow transport peak around zero in spectral function
• better approach needed

0.025 0.03 0.035 0.04 0.045 0.05 0.25 0.3 0.35 0.4 0.45 0.5 τ T Gvc

low (τ)/T3

D=1/(2 π T)

14 / 27

fig: Petreczky EPJC62 (2008)

Euclidean Correlator derivation

• In the M → ∞ limit we can define:

κ = 1 3Tχ

3

• i=1

lim

ω→0

• lim

M→∞ M2 kin

∞

−∞

dt eiω(t−t′)

• d3x

1 2 ˆ J i(t, x) dt , ˆ J i(0, 0) dt

• Where ˆ

J i(x) = ¯ ψ(x)γµψ(x) is the heavy quark current

• The heavy quark force in static limit:

M d ˆ J i dt =

• φ†E iφ − θ†E iθ
• Where φ , θ are HQ and H¯

Q operators, E i color-electric field

• Now the euclidean correlator is defined as:

GE(τ) = − 1 3Tχ

3

• i=1

lim

M→∞

• d3x
• φ†gEiφ − θ†gEiθ
• (τ, x)
• φ†gEiφ − θ†gEiθ
• (0, 0)
• 15 / 27

Following Caron-Huot et.al.JHEP04 (2009)

Euclidean correlator

• After simplifying the propagators of φ and θ in M → ∞:

GE(τ) = −1 3

3

• i=1

Re Tr [U(β, τ)gEi(τ, 0)U(τ, 0)gEi(0, 0)] Re Tr [U(β, 0)]

• Related to the Diffusion coefficient by:

GE(τ) = ∞ dω π ρ(ω) cosh

• β

2 − τ

• ω

sinh βω

2

κ = lim

ω→0

2T ω ρ(ω)

• In general inversion problem is ill-defined

0.0 1.0 2.0 3.0 4.0 5.0 ω / T 0.0 1.0 2.0 3.0 4.0 ρE / ωT

2

O(g

2)

O(g

4)

Nf = 0, T = 3 Tc

• No ω → 0 transport-peak like with direct HQ-current measurement

16 / 27

Following Caron-Huot et.al.JHEP04 (2009), fig from: Burnier et.al.JHEP08 (2010)

Multilevel algorithm

x t periodic periodic Nt Ntsl

• Algorithm for quenched simulations
• At large Nt observables like Polyakov loop can have poor signal
• Idea: Divide the lattice to temporal slices of size Ntsl
• update each sublattice independently keeping boundaries fixed
• Average over different boundary configurations

→ Allows reaching better statistics with less configurations

17 / 27

Lüscher et.al.JHEP09 (2001)

Euclidean EE correlator

− − τ x t periodic periodic GE(τ) = −1 3

3

• i=1

Re Tr [U(β, τ)gEi(τ, 0)U(τ, 0)gEi(0, 0)] Re Tr [U(β, 0)]

• Renormalization: ZE = 1 + g2

0 × 0.137718569 . . . + O(g4 0 )

(Christensen et.al.PLB02 (2016))

18 / 27

Lattice parameter

• Currently only one spatial lattice size 483, Temporal extent 24
• For multilevel 4 sublattices with 2000 updates
• Temperatures between 1.5 − 15Tc
• Set β for each T with (Francis et.al.PRD91 (2015))
• Previous similar studies (Meyer NJP13 (2011), Ding et.al.JPG38 (2011), Banarjee

et.al.PRD85 (2012), Francis et.al.PRD92 (2015))

T/Tc 1.5 3 6 10 15 β 7.192 7.774 8.367 8.808 9.159 Confs 1956 576 525 1068 52

• All data is very preliminary

19 / 27

Lattice correlator

0.0 0.1 0.2 0.3 0.4 0.5

τT

100 101 102 103 104 105

ZEGE

GE, pert 483×24 T =1.5Tc 483×24 T =3Tc 483×24 T =6Tc 483×24 T =10Tc 483×24 T =15Tc 0.0 0.1 0.2 0.3 0.4 0.5

τT

1.0 1.5 2.0 2.5 3.0 3.5 4.0

ZEGE/GE, pert

483×24 T =1.5Tc 483×24 T =3Tc 483×24 T =6Tc 483×24 T =10Tc 483×24 T =15Tc

GE, pert = π2T 4 cos2(πτT) sin4(πτT) + 1 3 sin2(πτT)

• 20 / 27

Tree level improvement

GE, LOlat = 1 3a4 π

−π

d3q (2π)3 e ¯

qNτ (1−τT) + e ¯ qNτ τT

e ¯

qNτ − 1

˜ q2 sinh ¯ q , ¯ q = 2arsinh ˜ q 2

• ,

˜ q2 =

3

• i=1

4 sin2 qi 2

• GE, pert(¯

τ ¯ T) = GE, LOlat(τT) 0.1 0.2 0.3 0.4

τT

1.0 1.5 2.0 2.5 3.0 3.5 4.0

ZEGE, imp/GE, pert

483×24 T =1.5Tc 483×24 T =3Tc 483×24 T =6Tc 483×24 T =10Tc 483×24 T =15Tc 21 / 27

Francis et.al.PoSLattice (2011)

Spectral function

GE(τ) = ∞ dω π ρ(ω)

ω T cosh

• τT − 1

2

• sinh ω

2T

κ = lim

ω→0

2T ω ρ(ω) , γ = − 1 3Nc ∞ dω 2π ρ(ω) ω

• Assume simple behavior on IR (ω ≪ T):

ρIR(ω) = κω 2T

• Perturbative behavior in UV (ω ≫ T):

ρLO

UV(ω) = g2(µω)CFω3

6π , ρNLO

UV

from (Burnier et.al.JHEP08 (2010)) µω = max(ω, πT)

• r from EQCD
• Use 5-loop running for the coupling

22 / 27

Perturbative ρ Behavior

10−1 100 101 102 103

ω/T

10−2 10−1 100 101 102 103 104 105

ρE/(ωT 2)

ρLO ρNLO ρNLO , log ρNLO , T = 0 0.0 0.1 0.2 0.3 0.4 0.5

τT

1.0 1.5 2.0 2.5 3.0 3.5 4.0

GE/Gnorm

NLO (NLO norm) NLO (NLO T=0 norm) NLO (LO norm, µ = 50T) NLO T=0 (NLO T=0 norm) NLO T=0 (LO norm, µ = 50T) LO (LO norm) 483 × 24, T =1.5Tc

0.0 0.1 0.2 0.3 0.4 0.5

τT

1.0 1.2 1.4 1.6 1.8 2.0 2.2

GE/Gnorm

NLO (NLO norm) NLO (NLO T=0 norm) NLO (LO norm, µ = 50T) NLO T=0 (NLO T=0 norm) NLO T=0 (LO norm, µ = 50T) LO (LO norm) 483 × 24, T =10.0Tc

0.0 0.1 0.2 0.3 0.4 0.5

τT

1.0 1.5 2.0 2.5 3.0 3.5

GE/Gnorm

NLO (NLO norm) NLO (NLO T=0 norm) NLO (LO norm, µ = 50T) NLO T=0 (NLO T=0 norm) NLO T=0 (LO norm, µ = 50T) LO (LO norm) 483 × 24, T =15.0Tc

23 / 27

Spectral function fit ansatz

• Usual ansatzes in literature:
• Interpolate between IR and UV behavior
• aωΘ(ω − Λ) + bω3 (Banerjee et.al.PRD85 (2011))
• max(aω, bω3) (Francis et.al.PRDD92 (2015))
• ctanh ωβ

2 Θ(ω − Λ) + bω3 (Banerjee et.al.PRD85 (2011))

• [1 + cn sin(πny)][κω/2T + bω3] with y = x/(1 + x) and

x = ln(1 + ω/[πT]) (Francis et.al.PRD92 (2015))

• max(aω, bω3) (Francis et.al.PRD92 (2015))
• For this talk we use the sine type fit

24 / 27

Preliminary fits

0.0 0.1 0.2 0.3 0.4 0.5 τT 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ZEGE, imp/GE, pert

T=1.5Tc, κ/T 3 =2.7(2.9) T=3Tc, κ/T 3 =2.4(2.7) T=6Tc, κ/T 3 =2.3(2.5) T=10Tc, κ/T 3 =2.3(2.5) T=15Tc, κ/T 3 =2.3(2.6)

LO

0.0 0.1 0.2 0.3 0.4 0.5 τT 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ZEGE, imp/GE, pert

T=1.5Tc, κ/T 3 =3.6(3.6) T=3Tc, κ/T 3 =6.4(6.1) T=6Tc, κ/T 3 =5.3(5.8) T=10Tc, κ/T 3 =3.9(3.7) T=15Tc, κ/T 3 =3.9(5.5)

NLO

10−1 100 101 102 103 104

ω/T

10−2 100 102 104 106 108

ρE/(ωT 2)

fit ρLO ρNLO ρNLO , T = 0

25 / 27

Comparison to existing results

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

T/Tc

2 4 6 8 10

κ T 3

NLO LO Banarjee 2011 Francis 2015 Brambilla 2019 This talk

• The analysis still needs to be refined
• Need more statistics / Volumes / continuum limit

26 / 27

Conclusions and Future prospects

• High T quenched lattice data seems to agree with NLO prediction
• Results still preliminary
• Next: see if one can improve lower T fit from knowledge from high

T

• κ and γ can be measured un-quenched from lattice determinations
• f Υ(1S) and J/ψ
• More accurate lattice data would allow constrain the values even

more

• Is it possible to extract κ un-quenched using the lattice euclidean

field correlator?

• Gradient flow?
• Is it possible to measure γ from lattice?

27 / 27