Geistes-, Natur-, Sozial- und Technikwissenschaften gemeinsam unter einem Dach. Olaf Kaczmarek. University of Bielefeld

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1 Geistes-, Natur-, Sozial- und Technikwissenschaften gemeinsam unter einem Dach (some selected) Recent Developments in Lattice Studies for Quarkonia Olaf Kaczmarek University of Bielefeld 5th International Workshop on Heavy Quark Production in Heavy-Ion Collisions Utrecht

2 Motivation - Quarkonium in Heavy Ion Collisions Heavy Ion Collision QGP Expansion+Cooling Hadronization J/ J/?? c J/ c J/ D D Charmonium+Bottmonium is produced (mainly) in the early stage of the collision Depending on the Dissociation Temperature - remain as bound states in the whole evolution - release their constituents in the plasma [Kaczmarek, Zantow] [S.Bass] ψ S(J/ ) In In, SPS Pb Pb, SPS Au Au, RHIC, y < 0.35 Au Au, RHIC, y =[1.2,2.2] Charmonium yields at SPS/RHIC ε (GeV/fm 3) First estimates on Dissociation Temperatures from detailed knowledge of Heavy Quark Free Energies and Potential Models χ c J/ψ Ψ F(r,T) [MeV] T=0 0.81T c 1.02T c 1.50T c r [fm] 4.01T c

3 Motivation - Transport Coefficients Transport Coefficients are important ingredients into hydro models for the evolution of the system. R AA AAR (a) 0-10% central Armesto et al. (I) van Hees et al. (II) 3/(2 T) 12/(2 T) Moore & Teaney (III) Usually determined by matching to 0.8 experiment (see right plot) here: Heavy Quark Diffusion Constant D also: Electrical conductivity σ (light quarks) v2 HF 2 HF v s NN = 200 GeV (b) minimum bias 0 R AA 0 v 2, p > 2 GeV/c T HF e R, e AA v 2 Need to be determined from QCD using first principle lattice calculations! [GeV/c] [PHENIX Collaboration, Adare et al.,arxiv: & PRL98(2007)172301] p T

4 Motivation PHENIX/STAR results for the low-mass dilepton rates pp-data well understood by hadronic cocktail large enhancement in Au+Au between MeV indications for thermal effects!? Need to understand the contribution from QGP spectral functions from lattice QCD [PHENIX PRC81, (2010)] Dileptonrate directly related to vector spectral function: [STAR preliminary, arxiv: ] dw dωd 3 p = 5α2 54π 3 1 ω 2 (e ω/t 1) ρ V(ω, ~p, T)

5 Vector correlation functions at high temperature G(τ,~p, T ) = Z 0 dω ρ(ω,~p, T )K(τ, ω,t) 2π cosh ω(τ 1 K(τ, ω,t)= sinh ω 2T 2T Lattice observables: G μν (τ,~x) = hj μ (τ,~x)j ν(0,~0)i q Γ H Γ H (0,0) q (τ,x) J μ (τ,~x) = 2κZ V ψ(τ,~x)γμ ψ(τ,~x) local, non-conserved current, needs to be renormalized G μν (τ,~p) = X ~x G μν (τ,~x)e i~p~x ~p =0 only used here How to extract spectral properties from correlation functions?

6 Spectral functions at high temperature Free theory (massless case): free non-interacting vector spectral function (infinite temperature): ρ free 00 (ω) = 2πT 2 ωδ(ω) ρ free ii (ω) = 2πT 2 ωδ(ω)+ 3 2π ω2 tanh(ω/4t ) δ-functions exactly cancel in ρ V (ω)=-ρ 00 (ω)+ρ ii (ω)

7 Spectral functions at high temperature Free theory (massless case): free non-interacting vector spectral function (infinite temperature): ρ free 00 (ω) = 2πT 2 ωδ(ω) ρ free ii (ω) = 2πT 2 ωδ(ω)+ 3 2π ω2 tanh(ω/4t ) δ-functions exactly cancel in ρ V (ω)=-ρ 00 (ω)+ρ ii (ω) With interactions (but without bound states): while ρ 00 is protected, the δ-funtion in ρ ii gets smeared: Ansatz: ρ 00 (ω) = 2πχ q ωδ(ω) ρ ii (ω) = 2χ q c BW ωγ/2 ω 2 +(Γ/2) π (1 + κ) ω2 tanh(ω/4t ) Ansatz with 3-4 parameters: works fine for light quarks at 1.5 Tc (χ q ),c BW, Γ, κ κ = α s π at leading order [ Thermal dilepton rate and electrical conductivity, H.T.-Ding, OK et al., PRD83 (2011) ]

8 Light Quarks - Vector Correlation Function continuum extrapolation Use our Ansatz for the spectral function ρ 00 (ω) = 2πχ q ωδ(ω) and fit to the continuum extrapolated values T 2 G V ( T)/[ q G V free ( T)] x x x x48 Extrapolation fit [0.2:0.5] [ Thermal dilepton rate and electrical conductivity, H.T.-Ding, OK et al., PRD83 (2011) ] ρ ii (ω) = 2χ q c BW ωγ/2 ω 2 +(Γ/2) π (1 + κ) ω2 tanh(ω/4t ) G V (τ, T) Ḡ 00 G free V (τ, T) & G (2) V 2C BW χ q Γ = 1.098(27) Γ T = 2.235(75) κ = (27) T 2 G V ( T)/( q G V free ( T)) 2Tc BW / =1.098 /2T= T/T c = T T

9 Light Quarks - Spectral function and electrical conductivity Use our Ansatz for the spectral function ρ 00 (ω) = 2πχ q ωδ(ω) [ Thermal dilepton rate and electrical conductivity, H.T.-Ding, OK et al., PRD83 (2011) ] ρ ii (ω) = 2χ q c BW ωγ/2 ω 2 +(Γ/2) π (1 + κ) ω2 tanh(ω/4t ) Θ(ω 0, ω ) ii ( T T/T c =1.5 0 /T=0, /T=0 0 /T=1.5, /T=0.5 HTL free Hard thermal loop (HTL) [E.Braaten, R.D.Pisarski, NP B337 (1990) 569] /T Analysis of the systematic errors using truncation of the large ω contribution Θ(ω 0, ω )= σ T = C em 6 ³1+e (ω2 0 ω2 )/ω ω 1 lim ω 0 ρ ii (ω,~p =0,T) ωt electrical conductivity 1/3 1 C em σ T 1

10 Light Quarks - Dilepton rates and electrical conductivity Dileptonrate directly related to vector spectral function: [ Thermal dilepton rate and electrical conductivity, H.T.-Ding, OK et al., PRD83 (2011) ] 1e-05 1e-06 1e-07 1e-08 1e-09 dw dωd 3 p = 5α2 54π 3 1 ω 2 (e ω/t 1) ρ V(ω, ~p, T) dn l + l -/d d 3 p BW+continuum: 0 /T=0, /T=0 0 /T=1.5, /T=0.5 HTL Born T/T c =1.5 p=0 1e-10 1e-11 1e-12 Hard thermal loop (HTL) [E.Braaten, R.D.Pisarski, NP B337 (1990) 569] /T

11 Preliminary results at 1.1 T c PRACE-Project: Thermal Dilepton Rates and Electrical Conductivity in the QGP (JUGENE Bluegene/P in Jülich) [M.Müller et al., preliminary 2012] study of T-dependence of dilepton rates and electrical conductivity fixed aspect ratio to allow continuum limit at finite momentum: ~p T =2π~ k N τ N σ

12 Preliminary results at 1.1 T c [M.Müller et al., preliminary 2012] Use our Ansatz for the spectral function ρ 00 (ω) = 2πχ q ωδ(ω) ρ ii (ω) = 2χ q c BW ωγ/2 ω 2 +(Γ/2) π (1 + κ) ω2 tanh(ω/4t ) 6 5 ii ( ) / T 1.1 T c T=1.45Tc T=1.09Tc HTL Born Analysis of the systematic errors T c in progress 3 2 electrical conductivity σ T = C em 6 lim ω 0 ρ ii (ω,~p =0,T) ωt 1 /T

13 Preliminary results at 1.1 T c Dileptonrate directly related to vector spectral function: dw dωd 3 p = 5α2 54π 3 1 ω 2 (e ω/t 1) ρ V(ω, ~p, T) [M.Müller et al., preliminary 2012] 1e-05 1e-06 1e-07 dw/d d 3 p 1.1 T c 1.5 T c T=1.09T c T=1.46T c HTL Born Rate 1e-08 1e-09 1e-10 1e-11 /T 1e

14 Non-zero momentum T 2 free,lat. G VL ( T,p) / q G VL ( T,p=2) V L p/t = [M.Müller et al., preliminary 2012] T 2 free,lat. G VT ( T,p) / q G VT ( T,p=2) V T p/t = N = 64 N = 48 N = T continuum extrapolation N = 64 N = 48 N = T continuum extrapolation indications for non-trivial behaviour of spectral functions at small frequencies: [Hong+Teaney, PRC82 (2010)044908]

15 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 T= ω + zero-mode contribution at ω=0: ρ(ω) =2πχ 00 ωδ(ω)

16 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 TÀT c T= ω + zero-mode contribution at ω=0: + transport peak at small ω: ρ(ω) =2πχ 00 ωδ(ω) ρ(ω T ) ' 2χ 00 T M ωη ω 2 + η 2, η = T MD

17 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 T>T c TÀT c T= ω + zero-mode contribution at ω=0: + transport peak at small ω: ρ(ω) =2πχ 00 ωδ(ω) ρ(ω T )=2χ 00 T M ωη ω 2 + η 2, η = T MD

18 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 T<T c T>T c TÀT c ω + zero-mode contribution at ω=0: + transport peak at small ω: ρ(ω) =2πχ 00 ωδ(ω) ρ(ω T )=2χ 00 T M ωη ω 2 + η 2, η = T MD

19 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 T=0 T<T c ω + zero-mode contribution at ω=0: + transport peak at small ω: ρ(ω) =2πχ 00 ωδ(ω) ρ(ω T )=2χ 00 T M ωη ω 2 + η 2, η = T MD

20 Quarkonium spectral function What do we expect!? ρ(ω) ω 2 T=0 T<T c T>T c TÀT c T= ω + zero-mode contribution at ω=0: + transport peak at small ω: ρ(ω) =2πχ 00 ωδ(ω) ρ(ω T )=2χ 00 T M ωη ω 2 + η 2, η = T MD

21 Spatial Correlation Function and Screening Masses Correlation functions along the spatial direction Z G(z, T) = dxdy Z 1/T 0 dτhj(x, y, z, τ)j(0, 0, 0, 0)i are related to the meson spectral function at non-zero spatial momentum G(z, T) = Z dp z e ip zz Z 0 dω σ(ω,p z,t) ω exponential decay defines screening mass M scr : zà1/t e M scrz bound state contribution σ(ω,p z,t) δ(ω 2 p 2 z M 2 ) high-t limit (non-interacting free limit) σ(ω,p z,t) σ free (ω,p z,t) M scr = M indications for medium modifications/dissociation M scr =2 p (πt ) 2 + m c 2

22 Spatial Correlation Function and Screening Masses [ Signatures of charmonium modification in spatial correlation functions, F.Karsch, E.Laermann, S.Mukherjee, P.Petreczky, (2012) arxiv: ] 2+1 flavor QCD using p4-improved staggered action 32 3 N t with N t =6,8,12 and 32 physical m s and m l =ms/10 (m π ' 220 MeV) M PS (T)/M PS (T=0) N =6 N =8 N =12 quenched M scr =2 p (πt ) 2 + m c A PS (T)/A PS (T=0) N =8 N = T/T c T/T c M scr = M indications for medium M scr =2 p (πt ) 2 + m 2 c modifications/dissociation

23 Spatial Correlation Function and Screening Masses M PS (T)/M PS (T=0) N =8 N =12 anti-periodic boundary conditions (closed symbols) M scr =2 p (πt ) 2 + m c M scr = M T/T c periodic boundary conditions (open symbols) M scr =2m c screening masses for bound states insensitive to boundary conditions due to bosonic nature of the basic degrees of freedom the change in the behavior of the charmonium screening masses around T=1.5T c is likely due to the melting of the meson states. [ Signatures of charmonium modification in spatial correlation functions, F.Karsch, E.Laermann, S.Mukherjee, P.Petreczky (2012) arxiv: ]

24 Vector correlation and spectral function at finite temperature G(τ,~p, T ) = Z 0 dω ρ(ω,~p, T )K(τ, ω,t) 2π cosh ω(τ 1 K(τ, ω,t)= sinh ω 2T 2T Lattice observables: G μν (τ,~x) = hj μ (τ,~x)j ν(0,~0)i J μ (τ,~x) = 2κZ V ψ(τ,~x)γμ ψ(τ,~x) local, non-conserved current, needs to be renormalized G μν (τ,~p) = X ~x G μν (τ,~x)e i~p~x ~p =0 only used here q Γ H Γ H (0,0) q (τ,x)

25 Charmonium spectral function in quenched QCD Quenched SU(3) gauge configurations (separated by 500 updates) at 4 temperatures 3 Lattice size N σ N τ with N σ = 128 N τ = 16, 24, 32, 48, 96 Non-perturbatively O(a) clover improved Wilson fermions Non-perturbative renormalization constants Quark masses close to charm quark mass [H.T.Ding, OK et al., arxiv: ] cut-off dependence volume dependence close to continuum (m c a 1) Temperature dependence

26 Charmonium Correlators vs Reconstructed Correlators G(,T)/G rec (,T) PS 1.46 T c 2.20 T c 2.93 T c G(,T)/G rec (,T) V ii 1.46 T c 2.20 T c 2.93 T c [fm] [fm] Z main T-effect due to zero-mode contribution G rec (τ,t) = σ 0 (ω, 0.75T c )K(ω, τ,t) well described by small ω-part of σ T (ω,t) G μμ (τ,t) = G ii (τ,t)+g 00 (τ,t) explains the rise in the vector channel no zero-mode contribution in PS-channel (similar to discussions by Umeda, Petreczky)

27 Charmonium Correlators vs Reconstructed Correlators G(,T)/G rec (,T) PS 1.46 T c 2.20 T c 2.93 T c diff sub G(,T)/G rec (,T) V ii 1.46 T c 2.20 T c 2.93 T c diff sub [fm] [fm] Z main T-effect due to zero-mode contribution G rec (τ,t) = σ 0 (ω, 0.75T c )K(ω, τ,t) effectively removed by diff/sub correlators G μμ (τ,t) = G ii (τ,t)+g 00 (τ,t) G diff (τ,t) = G(τ,T) G(τ +1,T) G sub (τ,t) = G(τ,T) G(τ = N t /2,T) almost constant small-ω contribution similar T-dependence in PS and V channel in the large frequency part of spectral fct.

28 Charmonium Correlators vs Reconstructed Correlators (G( T) - G rec ( T))/T 3 PS 1.46 T c 2.20 T c 2.93 T c (G( T) - G rec ( T))/T 3 V ii 1.46 T c 2.20 T c 2.93 T c T T negative difference for all T indications for thermal modifications in the bound state frequency region remember: no transport contribution in this channel positive diff. due to small-ω contr. positive slope indicates modifications in the bound state frequency region remember: small-ω contribution determines transport coefficient First estimate from fit to vector channel: 2πT D

29 Charmonium Spectral function [H.T.Ding, OK et al., arxiv: ] from sophisticated Maximum Entropy Method analysis: statistical error band from Jackknife analysis no clear signal for bound states above 1.46 T c study of the interesting region closer to T c on the way!

30 Charmonium Spectral function [H.T.Ding, OK et al., arxiv: ] from sophisticated Maximum Entropy Method analysis: statistical error band from Jackknife analysis no clear signal for bound states above 1.46 T c study of the interesting region closer to T c on the way!

31 Charmonium Spectral function Transport Peak [H.T.Ding, OK et al., arxiv: ] 4 2 TD D = 0.5 T/T c π ρ ii (ω,~p =0,T) lim 3χ 00 ω 0 ωt Perturbative estimate (α s ~0.2, g~1.6): Strong coupling limit: LO: 2πTD ' πTD = 1 NLO: 2πTD ' 8.4 [Moore&Teaney, PRD71(2005)064904, [Kovtun, Son & Starinets,JHEP 0310(2004)064] Caron-Huot&Moore, PRL100(2008)052301]

32 Charmonium Spectral function momentum dependence [H.T.Ding, arxiv: ] preliminary results for the momentum dependence:

33 Charmonium Spectral function momentum dependence [H.T.Ding, arxiv: ] preliminary results for the momentum dependence:

34 Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP11(2011)103] In non-relativistic QCD the Lagrangian is expanded in terms of v= p /M with L 0 = ψ µ D τ D2 2M L = L 0 + δl ψ + χ µd τ + D2 2M χ and δl = c 1 8M 3 ψ (D 2 ) 2 ψ χ (D 2 ) 2 χ ig +c 2 ψ (D E E D) ψ + χ (D E E D) χ 8M 2 g c 3 ψ σ (D E E D) ψ + χ σ (D E E D) χ 8M 2 g c 4 ψ σ Bψ χ σ Bχ 2M which is correct up to order O(v^4) [G.T.Bodwin,E.Braaten,G.P.Lepage, PRD 51 (1995) 1125]

35 Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP11(2011)103] correlation function calculated as an initial value problem no costly matrix inversion no restriction by thermal boundary conditions N s N τ T (MeV) T/T c N cfg gauge configurations from n f =2 dynamical Wilson fermion action a s ' fm 1/a t ' 7.35 GeV a s /a τ =6 anisotropic lattice

36 Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP11(2011)103] Kernel is T-independent, contributions at ω < 2M absent no small-ω contribution no information on transport properties G(τ) = Z 2M dω 0 2π exp( ω0 τ)ρ(ω 0 ), ω 0 = ω 2M

37 Bottomonium - Lattice NRQCD [G.Aarts et al., JHEP11(2011)103] Kernel is T-independent, contributions at ω < 2M absent no small-ω contribution no information on transport properties G(τ) = Z 2M dω 0 2π exp( ω0 τ)ρ(ω 0 ), ω 0 = ω 2M

38 Lattice cut-off effects free spectral functions [G.Aarts et al., JHEP11(2011)103] gauge configurations from n f =2 dynamical Wilson fermion action a s ' fm 1/a t ' 7.35 GeV [H.T.Ding, OK et al., arxiv: ] gauge configurations from quenched action a ' 0.01 fm 1/a ' 19 GeV vc ( )/ 2 N =24 N =32 N =48 continuum 0.6 anisotropic NRQCD cut-off effects and energy resolution determined by spatial lattice spacing isotropic Wilson /T M +1.5GeV +2.9GeV +4.4GeV 0GeV 16GeV 40GeV 73GeV no continuum limit in NRQCD, a s MÀ1 only small energy region accessible continuum limit straight forward, but expensive transport properties accessible [see also F.Karsch et al., PRD68 (2003) ]

39 Heavy Quark Momentum Diffusion Constant [A.Francis,OK,M.Laine,J.Langelage, arxiv: ] Heavy Quark Effective Theory (HQET) in the large quark mass limit leads to a (pure gluonic) color-electric correlator [J.Casalderrey-Solana, D.Teaney, PRD74(2006)085012, S.Caron-Huot,M.Laine,G.D. Moore,JHEP04(2009)053] G E (τ) 1 3 3X i=1 D h ie Re Tr U( 1 T ; τ) ge i(τ, 0) U(τ;0)gE i (0, 0) E DRe Tr[U( 1T ;0)] 2T ρ E (ω) κ =lim ω 0 ω Heavy quark (momentum) diffusion:, D = 2T 2 κ

40 Heavy Quark Momentum Diffusion Constant [A.Francis,OK,M.Laine,J.Langelage, arxiv: ] due to the gluonic nature of the operator, signal is extremely noisy multilevel combined with link-integration techniques used to improve the signal tree-level improvement (right figure) to reduce discretization effects [similar studies by H.B.Meyer, New J.Phys.13 (2011) and D.Banerjee,S.Datta,R.Gavai,P.Majumdar,PRD85(2012)014510]

41 Heavy Quark Momentum Diffusion Constant [A.Francis,OK,M.Laine,J.Langelage, arxiv: ] D ( )/T Model spectral function: transport contribution + NLO [Y.Burnier et al. JHEP 1008 (2010) 094)] n ρ model (ω) max ρ NLO (ω), ωκ o 2T Z dω G model (τ) 0 π ρ model(ω) cosh 1 2 τt ω T sinh ω 2T Still large uncertainties but very promising thermodynamic+continuum limit needed more constraints on the spectral function other operators and observables from EFT?

42 Conclusions Charmonium: [F.Karsch, E.Laermann, S.Mukherjee, P.Petreczky, arxiv: ]: the change in the behavior of the charmonium screening masses around T=1.5T c is likely due to the melting of the meson states. [H.T.Ding, OK et al., arxiv: ]: Detailed knowledge of the vector correlation function at various T in quenched QCD continuum extrapolation of correlation function still needed! Results so far depend on MEM analysis Ansätze more difficult due to m q dependence Heavy quark diffusion constant: 2πDT 2 No signs for bound states at and above 1.46 T c

43 Conclusions Effective Field Theory on the Lattice: [FASTSUM Collaboration, G.Aarts et al., JHEP11(2011)103]: NRQCD study of bottomonium: survival of ground state till 2 T c melting of excited states above T c qualitative agreement with CMS results [A.Francis,OK,M.Laine,J.Langelage, arxiv: and Banerjee et al., PRD85(2012)014510]: HQET operator for Heavy Quark Momentum Diffusion constant first results promising Need to understand systematic uncertainties in both approaches in more detail