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 2 v 0 + + • Charm and bottom quarks much 0.4 D , D , D* average, | y |<0.8 ALICE {EP, | |>0.9} , = 5.02 TeV v ∆ η s 2 NN {EP, | ∆ η |>0} , = 2.76 TeV v s 2 NN heavier than RHIC/LHC tempera- 0.3 PRL 111 (2013) 102301 ± , | |<0.5, = 2.76 TeV π y s tures NN 0.2 {SP, | ∆ η |>0.9} , JHEP 06 (2015) 190 v 2 v {EP, | ∆ η |>2} , PLB 719 (2013) 18 2 • Can be used to probe early time 0.1 physics 0 Syst. from data 30 − 50% Pb − Pb • Experimental results for nuclear Syst. from B feed-down − 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 suppression factor R AA and elliptic (GeV/ ) p c T s (T) TD flow ν 2 differ from simple perturba- α s pQCD LO Lattice QCD =0.4 pQCD LO α S π Ding et al. 2 tive estimates 30 Banerjee et al. Kaczmarek et al. • Both R AA and ν 2 can be calculated F i x V = - M a t r T 20 from diffusion constant D of heavy QPM (Catania) - LV D-meson (TAMU) quark in medium D-meson (Ozvenchuk) PHSD 10 T-Matrix V=U QPM (Catania) - BM Duke (Bayesian) • D can be tuned to match experi- MC@sHQ AdS/CFT mental results 0 1 1.5 2 T/T c Figures from: ALICE PRL120 (2017), X. Dong CIPANP (2018) 1 / 27
Heavy Quark in medium • Heavy quark energy doesn’t change much in collision with a thermal quark √ E k ∼ 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 dp i � ξ ( t ) ξ ( t ′ ) � = κδ ( t − t ′ ) dt = − η D p i + ξ i ( t ) , • κ : strength of stochastic interaction: property of medium • The drag coefficient η D = κ/ ( 2 MT ) • Relaxation time τ R = 1 /η D • In position space � x 2 ( t ) � = 6 Dt with D = 2 T 2 /κ Moore et.al. PRC71 (2005), Caron-Huot et.al. JHEP02 (2008) 2 / 27
Perturbation theory • Perturbation theory has poor convergence 0.6 NLO 0.5 LO truncated LO 0.4 g 4 T 3 0.3 κ 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 g • Non-perturbative methods needed Moore et.al. PRC71 (2005), Caron-Huot et.al. JHEP02 (2008) 3 / 27
Quarkonium in medium • Quarkonium characterized by energy-scales: mass M, Bohr radius a 0 , and binding Energy E • M ≫ 1 a 0 ≫ E • Environment at energy scale π T and correlation time τ E ∼ 1 /π T • Evolution of system characterized by relaxation time τ R 1 • Assume a 0 ≫ π T and π T ≫ E 1 → τ R ∼ 0 ( π T ) 3 and τ R ≫ τ E a 2 • 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 Brambilla et.al. PRD96 (2017), Brambilla et.al. PRD97 (2018) 4 / 27
Transport Coefficients from in Medium Quarkonium Dynamics 5 / 27
The singlet self-energy in pNRQCD � t Σ S ( t ) = r i r j d t ′ � gE a , i ( t , 0 ) gE a , j ( t ′ , 0 ) � = r i r j κ ij ( t ) + i ˆ γ ij ( t ) � � ˜ 2 N C 2 t 0 • In large time limit, assuming time translation invariance: � ∞ 1 gE a , i ( t , 0 ) gE a , j ( 0 , 0 ) � � κ = d t � � 6 N c 0 � ∞ γ = − i gE a , i ( t , 0 ) gE a , j ( 0 , 0 ) � � d t � � 6 N c 0 • Where κ is just the heavy quark momentum diffusion coefficient Brambilla et.al. PRD96 (2017), Brambilla et.al. PRD97 (2018) 6 / 27
In medium mass shift and width • From previous slide it follows: r 2 κ = Σ s + Σ † s = − 2 Im ( − i Σ s ) r 2 γ = − i Σ s + i Σ † s = 2 Re ( − i Σ s ) • The self-energies provide the in medium induced mass shifts δ M s and widths Γ s • For 1S Coulombic quarkonium state: δ M ( 1 S ) = 3 Γ( 1 S ) = 3 a 2 2 a 2 0 κ 0 γ • For Coulombic system: Solve a 0 from self-consistency equation with 1-loop 3 flavor running coupling: a 0 = 2 / [ MC F α s ( 1 / a 0 )] Brambilla et.al. PRD96 (2017), Brambilla et.al. PRD97 (2018) 7 / 27
Scales • Use recent lattice determination of δ M ( 1 S ) and Γ( 1 S ) (Aarts et.al. JHEP11 (2011), Kim et.al. JHEP11 (2018)) • Available lattice data for J /ψ at T = 251MeV and Υ( 1 S ) at T = 251MeV and T = 407MeV • Use masses M b = 4 . 78GeV and M c = 1 . 67GeV • Scale lattice masses to above values • The binding energies are E Υ( 1 S ) = − 0 . 1 GeV and E J /ψ = − 0 . 24 GeV 1 1 • The Bohr radiuses are a 0 = 0 . 84 GeV for J /ψ and a 0 = 1 . 5 GeV for Υ( 1 S ) • Scale hierarchies E ≪ π T ≪ 1 / a 0 satisfied 8 / 27
κ Measurement 2 3 S 1 (vector) Upsilon 1.5 Γ/ T 1 0.5 0 0 0.5 1 1.5 2 T/T c • 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 / T c ≈ 2, T c = 220 MeV • Un-quenched determination of κ from lattice data Brambilla et.al. TUM-EFT122/18 (2019), Fig from: Aarts et.al. JHEP11 (2011) 9 / 27
κ Result − 0 . 24 � κ T 3 � 4 . 2 • Un-quenched determination of κ from lattice data Brambilla et.al. TUM-EFT122/18 (2019) 10 / 27
γ Measurement 0 0 Δ m [MeV] (J/ ψ 3 S 1 ) -50 -50 Δ m [MeV] ( Υ 3 S 1 ) -100 -100 -150 -150 n=8 charmonium 3 S 1 n=4 bottomonium 3 S 1 [BR T>0] calib. [BR T=0 trunc.] [BR T>0] calib. [BR T=0 trunc.] -200 -200 140 160 185 223 251 140 160 185 223 251 333 407 T [MeV] T [MeV] • Use results from (Kim et.al. JHEP11 (2018)) • Two distinct mass shifts J /ψ and Υ( 1 S ) • Use two different temperatures T = 251MeV and T = 407MeV • Unquenched measurement Brambilla et.al. TUM-EFT122/18 (2019), Figs from: Kim et.al. JHEP11 (2018) 11 / 27
γ Results − 3 . 8 � γ T 3 � − 0 . 7 • First non-perturbative determination of γ Brambilla et.al. TUM-EFT122/18 (2019) 12 / 27
Transport Coefficients from lattice euclidean correlator 13 / 27
Directly from HQ current correlator • Operator of interest: HQ current correlator ¯ Q γ i Q • problematic observable: • Quite insensitive to D (Teaney PRD74 (2006), Petreczky EPJC62 (2008)) • Narrow transport peak around zero in spectral function • better approach needed 0.05 G vc low ( τ )/T 3 D=1/(2 π T) 0.045 0.04 0.035 0.03 0.025 τ T 0.25 0.3 0.35 0.4 0.45 0.5 fig: Petreczky EPJC62 (2008) 14 / 27
Euclidean Correlator derivation • In the M → ∞ limit we can define: � ˆ � ∞ 3 ˆ J i ( t , x ) J i ( 0 , 0 ) � � � 1 ��� 1 d t e i ω ( t − t ′ ) � M →∞ M 2 d 3 x κ = lim lim , kin 3 T χ d t d t 2 ω → 0 −∞ i = 1 J i ( x ) = ¯ • Where ˆ ψ ( x ) γ µ ψ ( x ) is the heavy quark current • The heavy quark force in static limit: M d ˆ J i � � φ † E i φ − θ † E i θ = d t Q operators, E i color-electric field • Where φ , θ are HQ and H ¯ • Now the euclidean correlator is defined as: 3 � 1 �� � � � � � d 3 x φ † gE i φ − θ † gE i θ φ † gE i φ − θ † gE i θ G E ( τ ) = − lim ( τ, x ) ( 0 , 0 ) 3 T χ M →∞ i = 1 Following Caron-Huot et.al. JHEP04 (2009) 15 / 27
Euclidean correlator • After simplifying the propagators of φ and θ in M → ∞ : 3 G E ( τ ) = − 1 � Re Tr [ U ( β, τ ) gE i ( τ, 0 ) U ( τ, 0 ) gE i ( 0 , 0 )] � � 3 � Re Tr [ U ( β, 0 )] � i = 1 • Related to the Diffusion coefficient by: N f = 0, T = 3 T c 4.0 2 ) O(g � � β � ∞ cosh 2 − τ ω 4 ) d ω O(g 3.0 G E ( τ ) = π ρ ( ω ) sinh βω 0 2 2 ρ E / ω T 2.0 2 T κ = lim ω ρ ( ω ) ω → 0 1.0 • In general inversion problem is ill-defined 0.0 0.0 1.0 2.0 3.0 4.0 5.0 ω / T • No ω → 0 transport-peak like with direct HQ-current measurement Following Caron-Huot et.al. JHEP04 (2009), fig from: Burnier et.al. JHEP08 (2010) 16 / 27
Multilevel algorithm N t periodic periodic x t N tsl • Algorithm for quenched simulations • At large N t observables like Polyakov loop can have poor signal • Idea: Divide the lattice to temporal slices of size N tsl • update each sublattice independently keeping boundaries fixed • Average over different boundary configurations → Allows reaching better statistics with less configurations Lüscher et.al. JHEP09 (2001) 17 / 27
Euclidean � EE � correlator periodic periodic − − x t τ 3 G E ( τ ) = − 1 � Re Tr [ U ( β, τ ) gE i ( τ, 0 ) U ( τ, 0 ) gE i ( 0 , 0 )] � � 3 � Re Tr [ U ( β, 0 )] � i = 1 • Renormalization: Z E = 1 + g 2 0 × 0 . 137718569 . . . + O ( g 4 0 ) (Christensen et.al. PLB02 (2016)) 18 / 27
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