Hadronic correlation and spectral functions in the QGP from Quenched Lattice QCD. Olaf Kaczmarek

Transcription

1 Geistes-, Natur-, Sozial- und Technikwissenschaften gemeinsam unter einem Dach Hadronic correlation and spectral functions in the QGP from Quenched Lattice QCD Olaf Kaczmarek in collaboration with: HengTong Ding (BNL), Anthony Francis (Bielefeld), Frithjof Karsch (BNL+Bielefeld), Edwin Laermann (Bielefeld) and Wolfgang Söldner (Regensburg) [Phys. Rev. D83(2011)034504, arxiv: arxiv: , arxiv: ] Seminar Instytut Fizyki UJ, Krakow 1

2 Motivation PHENIX results for the 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 Dileptonrate directly related to vector spectral function: [PHENIX PRC81, (2010)] dw dωd 3 p = 5α2 54π 3 1 ω 2 (e ω/t 1) ρ V(ω, ~p, T)

3 Charmonium 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] 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 χ c J/ψ from detailed knowledge of Ψ Heavy Quark Free 0 Energies and Potential Models F(r,T) [MeV] T=0 0.81T c 1.02T c 1.50T c r [fm] 4.01T c

4 Outline Introduction Vector correlation function on the lattice (light quarks) Temporal correlators vs. free correlators Volume and cut-off dependence Thermal moments of the spectral function Continuum extrapolation Electrical conductivity and (low mass) dilepton rates at T 1.5 T c Using an Ansatz for the spectral function in the continuum Charmonium spectral functions Heavy quark diffusion constant and dissociation in the medium Conclusions & Outlook

5 Vector correlation functions at high temperature G(τ,~p, T ) = Z d cosh ω(τ 1 2T 0 dω 2π ρ(ω,~p, T )K(τ, ω,t) cosh K(τ, ω,t)= sinh ω 2T Lattice observables: G μν( (τ,, ~x) ) = hj (τ, ~x)j ν(0,~0)i μ, ) ν (, 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 only ~p =0 used here q Γ H Γ H (0,0) (τ,x) q

6 Vector correlation functions at high temperature G V (τ,~p) = G 00 (τ,~p)+g ii (τ,~p) J 0 is a conserved ed current G 00 is τ independent at p=0 quark number susceptibility χ q 1.2 G 00 (τ, ~p =0) -G 3 00 ( T)/T χ q T + O(a 2 ) T c x x x x T ratios of correlators free of renormalization ambiguities: R(τ) G V (τ) G 00 (τ) ; R(τ) G V (τ) G 00 (τ)g free V (τ)

7 Spectral functions at high temperature Free theory: free non-interacting vector spectral function (infinite temperature): ρ free (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)

8 Spectral functions at high temperature Free theory: free non-interacting vector spectral function (infinite temperature): ρ free (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: while ρ 00 is protected, the δ-funtion in ρ ii gets smeared: Ansatz: κ = α s π at leading order ρ 00 (ω) = 2πχ q ωδ(ω) ρ ii (ω) = ωγ/2 2χ q c BW ω + (Γ/2) 2 2π (1 + κ) ω2 tanh(ω/4t ) Ansatz with 3-4 parameters: (χ q ),c BW, Γ, κ

9 Electrical Conductivity Electrical Conductivity slope of spectral function at ω=0 0 (Kubo formula) σ = C em lim ρ ii(ω,~p =0,T) T 6 ω 0 ωtt n X f C em = e 2 Q 2 = 5/9 e2 for n f =2 f 6/9 e 2 for n f = 3 f=1 Using our Ansatz for ρ ii (ω): σ T = 2 3 χ q T 2 T Γ c BW C em previous studies using staggered fermions (need to distinguish ρ even and ρ odd ): S.Gupta, PLB 597 (2004) 57: N τ =8-14, N σ 44 G.Aarts et al., PRL 99 (2007) : N τ =16,24, N σ =64

10 Vector correlation function on large & fine lattices Quenched SU(3) gauge configurations at T/T c =1.5 (separated by 500 updates) 3 Lattice size N σ N τ with N σ = N τ = 16, 24, 32, 48 Non-perturbatively O(a) clover improved Wilson fermions Non-perturbative renormalization constants Quark masses close to the chiral limit, κ ' κ c m MS /T[μ=2GeV] 0.1 Volume dependence N τ N σ β c SW κ Z V 1/a[GeV] a[fm] #conf cut-off dependence & continuum extrapolation close to continuum

11 Vector Correlation Function volume & cut-off dependence. 135 G ( T)/G free 1.35 V V ( T) x16 only small volume dependence x x x x24(i) x24(ii) x x T Vector correlation function normalized by free (non-interacting) correlator analytic in the continuum (for massless quarks): µ 2 T T V (τ) =6T 2 π(1 2τT ) 1 + cos2 (2πτT ) sin 3 +2 cos(2πτt ) (2πτT) sin 2 (2πτT) G free

12 Vector Correlation Function volume & cut-off dependence. 135 G ( T)/G free ( T) 1.35 V V x x x x48 only small volume dependence fix lattice volume V=128 3 vary N t at fixed T=1.5T c 1.2 cut-off effects are more severe G V (τ,~p) = G 00 (τ,~p)+g ii (τ,~p) T large N t needed continuum extrapolation incomplete cancelation between G 00 (τt) and BW-contribution to G ii (τt)? information on transport properties?

13 Vector Correlation Function continuum extrapolation. 1.8 G ii ( T)/[G 00 G V free ( T)] T=0.500, lat T=0.375, lat T=0.250, lat cont cont cont 1/N τt G ii (τ, T ) Ḡ 00 G free V (τ,t) (11) (9) (7) normalization with G 2 00 χ q /T defines renormalization invariant quantity extrapolation in (at) 2 =(1/N τ ) 2 under control for τ T 0.2 extrapolation at o is independent of the chosen normalization at o extrapolation at other values use spline interpolation at fixed cut-off well defined continuum extrapolated values for this ratio

14 Vector Correlation Function curvature at τt = 1/2. Taylor expansion around the mid-point τt = 1/2 : Thermal moments of (2n) the spectral function G V (τt ) = G (0) V G (n) V = 1 n! X n=0 G (2n) V G (0) V dg V (τt ) d(τt ) n µ 1 2 τt = 1 τt =1/2 n! Z 0 dω 2π ³ ω T n ρ V (ω) sinh(ω/2t ) Ratio of midpoint subtracted correlator: (τt ) = = G V (τt ) G V (1/2) G free V (τt ) G free V (1/2) G (2) V G (2),free V Ã! 1+(R (4,2) V R (4,2) V,free ) µ 1 2 τt T 2 V ( T)/ q 128x16 128x24 128x32 128x48 cont fit [0.2:0.5] T

15 Vector Correlation Function curvature at τt = 1/2. Quadratic fit to (τt) with varying lower fit range [(τt) min, 1/2] : G (0),free V = 2T 3 G (0) V (2) f 2 G V V = 28π 5 T 3 G (2) V V = 124π4 T 3 21 V G (2),free G (4),free G (0),free = ± G (2),free = ± Ratio of midpoint subtracted correlator: (τt ) = G V (τt ) G V (1/2) G free V (τt ) G free V (1/2) T 2 V ( T)/ q 128x16 128x24 128x32 128x48 cont fit [0.2:0.5] = G (2) V G (2),free V Ã! 1+(R (4,2) V R (4,2) V,free ) µ 1 2 τt normalized curvature close to that of the free vector correlator 1.15 T

16 Vector Correlation Function continuum extrapolation Use our Ansatz for the spectral function 1.45 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 ) and fit to the continuum extrapolated values free T G V ( T)/[ q G V ( T)] x x x x48 Extrapolation fit [0.2:0.5] G V (τ, T) Ḡ 00 G free V (τ, T) & G (2) V C 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

17 Spectral function and electrical conductivity 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 ) shaded area by varying Γ/T within its errorband ii ( T 2C BW χ q Γ = 1.098(27) Γ T = 2.235(75) κ = (27) BW+continuum free /T electrical conductivity σ T =(0.37 ± 0.01)C em

18 Spectral function and electrical conductivity 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 ) Θ(ω 0, ω ) ii ( T Analysis of the systematic errors using truncation of the large ω contribution (ω2 ω 2 Θ(ω ω 1 0, ω ) = ³1 + e 0 ω )/ω /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) electrical conductivity /T 1/3 1 σ C em T 1

19 Dilepton rates and electrical conductivity 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) 1e-05 1e-06 dn l + l -/d d 3 p T/T c =1.5 p=0 1e-07 BW+continuum: 0 /T=0, /T=0 0 /T=1.5, /T=0.5 1e-08 HTL Born 1e-09 1e-10 1e-11 Hard thermal loop (HTL) E.Braaten, R.D.Pisarski, NP B337 (1990) 569 /T 1e

20 Spectral Functions Maximum Entropy Method How to obtain continuous spectral function σ(ω,t) from discrete (and small) number of correlators? G(τ, T ) = Z dωk(τ, ω, T )σ(ω, T ) 1 K(τ, ω,t) = cosh ω(τ 2T sinh ω 2T 0 Best method on the market: Maximum Entropy Method (MEM) based on Bayesian theorem [Asakawa et al. 01] most probable spectral function properly renormalized correlators as input non-perturbative renormalization constants for vector [Lüscher et al. 1997] TI perturbative renormalization constants for pseudo-scalar prior knowledge needed as input default model m(ω) result should be independent of default model usually not the case

21 Maximum Entropy Method Maximize the conditional probability P[σ GH] exp(αs - L) G is the input correlator and H the additional prior information L is the standard chi 2 S is the Shannon-Jaynes entropy: S = Z 0 σ(ω) m(ω) σ(ω) log( σ(ω) m(ω) ) prior knowledge through default model m(ω) α is the relative weight of S and L, integrated t out at the end Prior information necessary and important Should have the correct large ω behavior and funtional form

22 Free spectral functions lattice vs. continuum Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05] σ H = N c 8π 2 Θ(ω2 4m 2 ) ω 2 tanh( ω 4T ) r 1 ( 2m h 2 ω )2 a H +( 2m i ω )2 b H T 2 + N c ωδ(ω) 3 2 f H ( )

23 Free spectral functions lattice vs. continuum Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05] σ H = N c 8π 2 Θ(ω2 4m 2 ) ω 2 tanh( ω 4T ) r 1 ( 2m h 2 ω )2 a H +( 2m i ω )2 b H Lattice cut-off effects: T 2 + N c ωδ(ω) 3 2 f H ( ) ω max =2log(7+ma) 0.18 V ( )/ Cont x x x48 1/a=19 GeV /a=13 GeV /a=6.5 GeV 0.02 /T

24 Free spectral functions lattice vs. continuum Free (non-interacting) spectral function [Karsch et al. 03, Aarts et al. 05] σ H = N c 8π 2 Θ(ω2 4m 2 ) ω 2 tanh( ω 4T ) r 1 ( 2m h 2 ω )2 a H +( 2m i ω )2 b H T 2 with interactions: + N c ωδ(ω) 1 η 3 2 f H ( ) δ(ω) πω2 + η 2 zero mode contribution at ω ' 0 [Umeda 07] [Petreczky+Teaney 06 Aarts et al. 05]

25 Maximum Entropy Method (MEM) Results so far based on correlator level Use results of our fit as default model in MEM analysis MEM analysis of N τ =48 data τ (i) fitted BW and continuum parameters (ii) fitted BW and lattice free spectral function 5 4 fit vs. MEM fit MEM:BW+continuum MEM:BW+lattice /T

26 Conclusions and Outlook - Light Quarks Conclusions: Detailed knowledge of the vector correlation function at T=1.5T c in quenched QCD continuum extrapolation of correlation function and thermal moments G V (τ T) well reproduced by Breit-Wigner plus continuum Ansatz for σ V (ω) Electrical conductivity: 1/3 1 C em σ T 1 Dilepton rate approaches leading order Born rate for ω/t 4 enhancement at small ω/t Outlook: include HTL result for σ V (ω) at large ω/t in the Ansatz vector correlation function at other temperatures and non-zero momentum

27 Charmonium 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] 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 χ c J/ψ from detailed knowledge of Ψ Heavy Quark Free 0 Energies and Potential Models F(r,T) [MeV] T=0 0.81T c 1.02T c 1.50T c r [fm] 4.01T c

28 Lattice set-up Quenched SU(3) gauge g 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 cut-off dependence volume dependence close to continuum (m c a 1) Temperature dependence

29 Charmonium Correlators Temporal Correlators 1e+07 G ( )/T 3 1e+06 vector G ( )/T 3 vector 1e+06 pesudo-scalar pesudo-scalar scalar scalar axial-vector axial-vector T=0.75 T c x96 T=1.5 T c x [fm] [fm] non-degenerate states still at 1.50 T c (almost) close to free correlators at (very) small separations largest distance 0.25 fm due to compact temporal direction only small distance regime ( fm) relevant for thermal effects and bound state effects

30 Charmonium Correlators vs Free Correlators T G( )/G free ( ) c PS 1.46 T c ,2020 T c 2.93 T c [fm] T G( )/G free ( ) c V 1.46 T ii c ,2020 T c 2.93 T c [fm] small temperature effects visible in both channels relevant distance regime compressed to smaller distances largest distance 1/(2T) due to compact temporal direction only small distance regime relevant for thermal effects and bound state effects Problem: quark mass dependence of free correlation function!

31 Charmonium Correlators cut-off dependence 1.15 G(,1.49T c )/G rec (,1.49T c ) V ii PS 1 [fm] G rec (τ,t) = Z σ 0 (ω, 0.75T c )K(ω, τ,t) only small cut-off/volume dependence in vector and pseudo-scalar channel G μμ (τ,t) = G ii (τ,t)+g 00 (τ,t) at 1.5 T c

32 Charmonium Correlators vs Free Correlators G(,T)/G rec (,T) PS 1.46 T c 220T 2.20 c 2.93 T c 1.25 G(,T)/G V ii 1.46 T rec (,T) c 2.20 T c T 2.93 c [fm] [fm] Z G rec (τ,t) = σ 0 (ω, 0.75T c )K(ω, τ,t) G μμ (τ,t) = G ii (τ,t)+g 00 (τ,t) main T-effect due to zero-mode contribution well described by small ω-part of σ T (ω,t) explains the rise in the vector channel no zero-mode contribution in PS-channel (similar to discussions by Umeda, Petreczky)

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

34 Charmonium Correlators vs Free Correlators 0.1 (G( T) - G rec ( T))/T T c PS 2.20 T c 0 293T 2.93 c 0.5 (G( T) - G rec ( T))/T T c V ii 2.20 T c T 2.93 c T T negative difference for all T positive diff. due to small-ω contr. indications for thermal modifications positive slope indicates modifications in the bound state t frequency region in the bound state t frequency region remember: no transport contribution remember: small-ω contribution in this channel determines transport coefficient First estimate from fit to vector channel: 2πT D

35 Charmonium Spectral function below T c ( )/ 2 V 0.73 T c DM 1 spf 1 DM 2 spf 2 DM 3 spf [GeV] [GeV] N σ =128 and N τ =96 (a GeV) cut-off effects well separated Pronounced ground state peak close to J/ψ mass no zero-mode/transport contribution observed below T c in any channel structure at small and intermediate ω independent of default model

36 Charmonium Spectral function above T c 0.25 ( )/ T c 2.20 T c 2.93 T c 0.2 DM 1 spf 1 DM 2 spf DM 3 spf 3 DM spf [GeV] [GeV] [GeV] various default models used to understand systematic uncertainties small default model dependence only in small-ω region no strong default model dependence in the intermediate ω region no zero-mode contribution in pseudo-scalar scalar channel observed no pronounced peak bound states melted? / threshold enhancement?

37 Charmonium Spectral function above T c 0.25 ( )/ 2 V 1.46 T c DM spf 1 DM 2 spf DM 3 spf 3 DM 4 spf V 2.20 T c V 2.93 T c [GeV] [GeV] [GeV] small-ω region accessible hope to extract transport properties small default model dependence only in small-ω region no strong default model dependence in the intermediate ω region unique transport contribution in vector channel observed no pronounced peak bound states melted? / threshold enhancement?

38 Charmonium Spectral function Transport Peak 0.35 ( )/( T) 1.46 Tc 2.20 Tc Vii DM 1 spf 1 DM 2 spf 2 DM 3 spf 3 DM 4 spf 2.93 Tc 0.05 /T /T /T vary the large-ω behavior of the default models transport peak is a unique feature for all DMs increases with increasing temperature at least lower and upper limits for transport coefficients should be possible

39 Charmonium Spectral function Transport Peak 0.5 ( )/( T) V 1.46 T c V 2.20 T c V 2.93 T c DM 1 spf 1 DM 2 spf 2 DM 3 spf /T /T /T vary the low-ω behavior of the default models transport peak is a unique feature for all DMs increases with increasing temperature at least lower and upper limits for transport coefficients should be possible

40 Charmonium Spectral function our current best guess: statistical error band from Jackknife analysis no clear signal for bound states t above 1.46 T c systematic uncertainties still need to be analyzed in more detail!

41 Charmonium Spectral function our current best guess: statistical error band from Jackknife analysis no clear signal for bound states t above 1.46 T c systematic uncertainties still need to be analyzed in more detail!

42 Charmonium Spectral function Transport Peak TD T/T c D = 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]

43 Heavy Quark Diffusion constant TD Charm Light PHENIX Francis et al. NLO pqcd estimate from HQ momentum diffusion Francis, OK, Laine, Langelage, arxiv: (see talk by Mikko Laine) estimate t from charmonium correlation fct. Ding, Francis, OK, Karsch, Satz, Soeldner, arxiv: AdS/CFT T/T c estimate from light quark correlation fct. Ding, Francis, OK, Karsch, Laermann, Soeldner, PRD83 (2011) OK, Francis, arxiv: (see also Aarts et al. PRL 99 (2007) ) * LOPT similar calculation on HQ momentum diffusion by Banerjee, Datta, Gavai, Majumdar, arxiv: (see also Meyer, New J.Phys. 13 (2011) ) DT PHENIX T/T c There is improvement in extracting transport coefficients from LQCD! still more improvement needed to reduce systematic uncertainties and to move from estimates to real numbers!

44 Conclusions and Outlook Charm Quarks Conclusions: 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 Outlook: understand systematic uncertainties in more detail need more sophisticated t Ansätze for the spectral function vector correlation function at non-zero momentum