Static quark correlators on the 2+1 flavor TUMQCD lattices

Transcription

1 Static quark correlators on the 2+1 flavor TUMQCD lattices J. H. Weber 1 in collaboration with A. Bazavov 2, N. Brambilla 1, P. Petreczky 3 and A. Vairo 1 (TUMQCD collaboration) 1 Technische Universität München 2 Michigan State University 3 Brookhaven National Lab Quarkonium Realtime Dynamics Workshop 2017, ITP Heidelberg, 09/22/2017 PRD (2016); TUM-EFT 81/16; arxiv: (2017) 1 / 33

2 Overview Static quark correlators on the 2+1 flavor TUMQCD lattices Overview & Introduction Static quark correlators with τ = N τ Free energies and weak coupling Vacuum-like regime Electric screening regime Static quark correlators with τ < N τ Method of moments Static energy at T > 0 from lattice Summary 2 / 33

3 Overview f bare Q = log P T Two different volumes and two quark masses: Controlled finite volume and quark mass dependence What is new about the TUMQCD lattices? f Q bare N τ TUMQCD collaboration, TUM-EFT 81/ β Continuum limit with realistic quark masses N τ a(β) = 1/T (N τ, β) T [135, 2325] MeV with at least four N τ N τ =4 16: ens. each, 5.9 β 9.67, a = fm. HISQ/Tree action, errors: O(α sa 2, a 4 ); lattice artefacts are reduced. N σ N τ Ensembles: = 4, m l = ms mπ = 161 MeV; β 7.825, a 0.04 fm 20 most from A. Bazavov et al., PRD (2012), PRD (2014) [HotQCD] All N τ < 16, m l = ms 5 : 3 5 ensembles each, TU each, 7.03 β 8.4, a = fm; T = 0 lattices available. r 1 scale for β > 8.4 from non-perturbative β function PRD (2014) 3 / 33

4 Overview S (r, T, τ) = 1 3 Why static quarks? Motivation: direct spectral analysis of static quark correlators is easier than of heavy meson correlators Added benefit: static correlators have been already generated for Polyakov loop & Wilson line correlators, (smeared) Wilson loops CS ren (r, T, τ) = 1 Tr [Wτ (0)W τ (r)] ren τ=n τ 3 T C ren S (r, T, N τ ) = e F S (r,t )/T W ren C ren P (r, T ) = 1 9 Tr WNτ (0)Tr W N τ (r) ren T = e F Q Q (r,t )/T Tr [Wτ (0)W r(τ)w τ (r, 0)W r (0)] ren T ren. unresolved! Idea: static Q Q correlator potential model spectral function Spectral decomposition of static correlators Real and imaginary parts of potential from correlator moments 4 / 33

5 Free energies Polyakov loops and free energies of static quark states The Polyakov loop L is the gauge-invariant expectation value of the traced propagator of a static quark (P) and related to its free energy: L(T ) = P T = Tr S Q (x, x) T = e F b Q /T. L needs renormalization. A. M. Polyakov, PL 72B (1978); L. McLerran, B. Svetitsky, PRD 24 (1981) The Polyakov loop correlator is related to singlet & octet free energies C P (r, T ) = e F b Q Q (r,t ) = 1 9 e F b S /T e F b A /T = 1 9 C S(r, T ) C A(r, T ). S. Nadkarni, PRD 33, 34 (1986) Meaning of gauge-dependent singlet & octet free energies is unclear. C P is also related to the gauge-invariant potentials V S,A of pnrqcd C P (r, T ) = e F b Q Q (r,t ) = 1 b 9 e V S /T Lb A e V b A /T + O(g 6 ) for rt 1. N. Brambilla et al., PRD 82 (2010) 5 / 33

6 Free energies Renormalization of free energies Singlet free energy and potential appear to be related for rm D 1: [ ] e F S (r, T ) = C F α r m D s + m r D + O(g 4 ) = V S (r) + O(g 3 ). N. Brambilla et al., PRD 82 (2010) F S and V S share the same renormalization 2C Q, which depends on T only through the lattice spacing: V S = V b S + 2C Q F S = F b S + 2C Q. Use V S at T = 0: fix r 1 scale & determine 2C Q using static energy. A. Bazavov et al., PRD (2012), PRD (2014) [HotQCD] Cluster decomposition theorem: F Q Q = F S = 2F Q for r 1/T. renormalize as F Q Q = F b Q Q + 2C Q and F Q = FQ b + C Q. PRD (2016) Beyond C Q (β) from T = 0 lattices use direct renormalization of F Q Infer unknown C Q (β) from known C Q (β ref ) using different N τ, Nτ ref C Q (β)= { C Q (β ref ) + FQ(β b ref, Nτ ref ) FQ(β, b N τ ) } S. Gupta et al., PRD (2008) 6 / 33

7 Free energies Renormalization of free energies Singlet free energy and potential appear to be related for rm D 1: [ ] e F S (r, T ) = C F α r m D s + m r D + O(g 4 ) = V S (r) + O(g 3 ). N. Brambilla et al., PRD 82 (2010) F S and V S share the same renormalization 2C Q, which depends on T only through the lattice spacing: V S = V b S + 2C Q F S = F b S + 2C Q. Use V S at T = 0: fix r 1 scale & determine 2C Q using static energy. A. Bazavov et al., PRD (2012), PRD (2014) [HotQCD] Cluster decomposition theorem: F Q Q = F S = 2F Q for r 1/T. renormalize as F Q Q = F b Q Q + 2C Q and F Q = FQ b + C Q. PRD (2016) Beyond C Q (β) from T = 0 lattices use direct renormalization of F Q Infer unknown C Q (β) from known C Q (β ref ) using different N τ, N ref C Q (β)= { C Q (β ref ) + F b Q(β ref, N ref τ ) F b Q(β, N τ ) + Nτ,N ref τ τ } PRD (2016) 6 / 33

8 Correlators with τ = N τ periodic boundaries τ N x x Q Q r N τ / 33

9 Polyakov loop correlators Polyakov loop correlator and Q Q free energy 0.5 F QQ _ (r,t)-t log 9 [GeV] T [MeV] Preliminary continuum! r [fm] Free energy of a Q Q pair, F Q Q, is also called color-averaged potential: CP ren (r, T ) = P (0)P (r) ren T = e F Q Q (r,t ) T = 1 F 9 e S (r,t ) T + 8 F 9 e A (r,t ) T. F Q Q T log 9 is close to the T =0 static energy V S for very small rt. 8 / 33

10 Polyakov loop correlators Singlet free energy in Coulomb gauge 0 F S (r,t) [GeV] T [MeV] Preliminary continuum! r [fm] ren Singlet free energy: C ren S (r, T ) = W a(0)w a (r) a=1 T = e F S (r,t )/T Wilson line correlator requires explicit gauge fixing (Coulomb gauge) F S is numerically close to the T =0 static energy V S for rt / 33

11 Polyakov loop correlators Effective coupling: vacuum-like and screening regimes α QQ _ (r,t) T [MeV] Preliminary! r [fm] N τ = Effective coupling α Q Q(r, T ) is a proxy for the force between Q and Q. α Q Q α Q Q (r, T ) = r 2 E(r,T ) C F r, E = {F S (r, T ), V S (r)} clearly distinguishes different regimes at small and large r. 10 / 33

12 Polyakov loop correlators Effective coupling: vacuum-like and screening regimes Vacuum-like α QQ _ (r,t) T [MeV] Preliminary! r [fm] N τ = String tension Strongly-coupled QGP max(α Q Q ) for 300 MeV Color screening Weakly-coupled QGP Vacuum-like regime Screening regime max(α Q Q) rt 0.2 rt 0.3 r maxt 0.4 r max defined through max(α Q), Q which is proxy for the maximal force. For T 300 MeV: max(α Q)(T Q ) 0.5 strongly-coupled QGP. 10 / 33

13 Deconfinement and the onset of color screening T χ from chiral observables vs T S from the peak of the entropy F Q Q(r ) = 2F Q Static quark entropy S Q = df Q dt independent of volume and renormalization scheme (continuum) S Q T χ (N τ ) local fit global 1/N τ 4 fit global 1/N τ 2 fit HRG N τ =8 T [MeV] Reminder: an τ = 1/T Discretization errors: T c(n τ ) = T c + O(1/N 2 τ ) Bazavov et al. [TUMQCD] PRD (2016) The entropy peaks at T S = MeV in the continuum limit. T S (N τ ) T χ(n τ ) for any N τ Bazavov et al., PRD (2016) [TUMQCD], suggests a tight link between chiral symmetry and deconfinement. e.g. as in glueball-sigma mixing scenarios, Y. Hatta, K. Fukushima PRD (2004). N.b. T χ defined via O(2) scaling of χ m,l (O(4): MeV lower T χ ) A. Bazavov et al., PRD (2012) [HotQCD] 11 / 33

14 Deconfinement and the onset of color screening T χ from chiral observables vs T S from the peak of the entropy F Q Q (r ) = 2F Q Static quark entropy S Q = df Q dt independent of volume and renormalization scheme (continuum) S Q T χ (N τ ) local fit global 1/N τ 4 fit global 1/N τ 2 fit HRG N τ =8 T [MeV] Reminder: an τ = 1/T Discretization errors: T c(n τ ) = T c + O(1/N 2 τ ) ds Q dt > 0 for T < Tc Bazavov et al. [TUMQCD] PRD (2016) Hadron resonance gas (HRG) is limited to only below T 125 MeV. ds Q dt static HRG results from: A. Bazavov, P. Petreczky, PRD 87, (2013) > 0 for T < Tc: the number of bound states of bound states including a static quark increases faster than HRG predictions. Large number of additional states or strong thermal modification of (low-lying) states are needed already substantially below T c. 11 / 33

15 Deconfinement and the onset of color screening T χ from chiral observables vs T S from the peak of the entropy F Q Q (r ) = 2F Q Static quark entropy S Q = df Q dt independent of volume and renormalization scheme (continuum) ds Q dt S Q T/T c N f =2+1, m π =161 MeV N f =3, m π =440 MeV N f =2, m π =800 MeV N f = O. Kaczmarek, F. Zantow, hep-lat/ (2005) P. Petreczky, K. Petrov PRD (2004) ds Q dt < 0 for T > Tc Bazavov et al. [TUMQCD] PRD (2016) < 0 for T > Tc: the static quark interacts with the medium T only inside its Debye screening radius, r 1/m D 0. Deconfinement and onset of screening are clearly defined via S Q (T S ) = 0 in the QCD crossover scenario. MPL A31 no.35, (2016) The peak is broader and lower for smaller m sea or larger N f. 11 / 33

16 pnrqcd and the vacuum-like regime Color octet contribution in the Polyakov loop correlator (I) F O (r,t) [GeV] N τ = 8 Preliminary! T [MeV] r [fm] pnrqcd: C P is given in terms of potentials V S and V A at T =0 and of the adjoint Polyakov loop L A at T >0 N. Brambilla et al., PRD 82 (2010) C P (r, T ) = e F Q Q (r,t ) = 1 9 e V S /T L A e V A/T + O(g 6 ) for rt 1. Gauge-invariant decomposition of C P into color singlet and octet is defined assuming weak coupling test if it works for lattice as well. ( Color octet contribution: define e F O /T 9 8 e F Q Q (r,t ) ) 1 9 e V S /T As F O rapidly decreases for higher T, the octet becomes important. 12 / 33

17 pnrqcd and the vacuum-like regime Color octet contribution in the Polyakov loop correlator (II) r 2 sub _ TF QQ (r,t) _ F QQ, data rec V S, exc. T=172 MeV N τ = 12 Preliminary! rt r 2 sub _ TF QQ (r,t) F QQ _, data rec V S, cor. N τ = 12 Preliminary! rt Low T = 172 MeV: color singlet V S is enough for reconstructing C P (no sensitivity to color octet due to large statistical errors). High T = 666 MeV: cancellation between color singlet and octet leads to 1/r 2 behavior in F Q Q. We use pnrqcd, i.e. 1 9 e V S /T L A e V A/T. We include Casimir scaling violation: 8V A + V S = 3 α3 s [ π2 3] + r 4 O(α4 s ). F Q Q for rt 0.4 can be understood in terms of vacuum physics only. 13 / 33

18 pnrqcd and the vacuum-like regime Static energy and singlet free energy (I) - discretization effects 0.1 [V S (r)-f S (r,t)]/t T [MeV] Preliminary! N τ = rt V S (r)-f S (r,t) [MeV] T [MeV] Preliminary! N τ = 10 r [fm] log scale! pnrqcd: V S (T =0) F S (T >0) up to O(g 6 ) M. Berwein et al., arxiv: Smooth r dependence due to cancellations in V S F S for r/a < 3. Strong N τ dependence for rt > 0.15, but N τ 12 continuum limit. V S F S 0.02T for rt 0.1 & T > 300 MeV, mild N τ dependence. T independent for small r, then sudden onset of medium effects. 14 / 33

19 pnrqcd and the vacuum-like regime Static energy and singlet free energy (II) weak coupling [V S (r) -F S (r,t)]/t r 2 d[v S (r) -F S (r,t)]/dr Preliminary! T 400 MeV Cont. N τ =12 N τ =10 full O(g 5 ) Preliminary! T 400 MeV rt N τ =12 N τ =10 O(g 5 ), µ= 2 / r O(g 5 ), µ= 1 / r O(g 5 ), µ= 1 / 2r pnrqcd: V S (T =0) F S (T >0) up to O(α 3 s ) M. Berwein et al., arxiv: r independent term at α 3 s allowed, can explain difference at small r? Constant term removed in r derivative eff. coupling α Q Q [V S F S ]. pnrqcd prediction works for V S F S in the range rt If V A V S 1/T which corresponds to rt < α s(1/r) thermal effects even exponentially suppressed Brambilla et al., PRD (2008) 15 / 33

20 pnrqcd and the vacuum-like regime Q Q force in the vacuum-like regime (I) α Q Q[V S ] α Q Q[F S ] + known O(g 4, g 5 ) + unknown O(g 6 ) T = 0 force resp. α Q Q for rt 0.15 from F S at finite temperature? Non-perturbative improvement scheme for r/a 3 is mandatory! V lat (r)/v cont (r) LW, all LW, on-axis Wilson, all Wilson, on-axis Free theory r/a [V S (r,β)-v S (r,β ref )]/V S (r,β ref ) β ref avg. HISQ/Tree, β = VS lat (r I ) for r/a > 6 on fine lattices, i.e. β ref > β, as estimate of VS cont (r) 16 / 33 r/a

21 pnrqcd and the vacuum-like regime Non-perturbative improvement of the static energy and free energies K Preliminary! r a = 3 fit average one β ref (a/r 1 ) 2 r b 2 =3a rf S sub (r,t) Preliminary! N τ = 12, β = cornell unimproved tree-level imp. non-pert. imp Correction factor K( r, a, a βref ) = V S (r,a) V S (r,β ref ) β ref avg. K( r, a) V S (r,β ref ) a Retuning of renormalization constant C Q needed at 1 level! Correction factors for fine lattices: continuum extrapolation of K Apply to renormalized F S works out of the box! For F Q Q via pnrqcd decomposition: C P = 1 9 e V S /T L A e V A/T r I /a 17 / 33

22 pnrqcd and the vacuum-like regime Q Q force in the vacuum-like regime (II) α qq N τ=6 N τ=8 N τ=10 N τ=12 N τ=16 T=0 β = 8.2 Very preliminary! 0.24 α qq r [fm] r [fm] N τ = 12 Very preliminary! Within errors α Q Q [V S] α Q Q [F S] satisfied for numerical derivatives T > 0 effects, i.e. N τ -dependence, statistically insignificant at rt 0.2 Study the force resp. α Q Q via T > 0: r is five times smaller than 2014 Bazavov et al., PRD (2014) Weak-coupling analysis of the T > 0 results ongoing stay tuned! 18 / 33

23 EQCD and the electric screening regime Confronting weak-coupling predictions in the screening regime (I) 1e0 1e-1 1e-2 1e-3 1e-4 -rf S (r,t)/c F [a.u.] N τ = 4 Preliminary! T [MeV] rt Hashed bands: LO Solid bands: NLO Scale uncertainty µ = (1 4)πT due to resummation smaller for larger T Λ QCD = 320 MeV F S (r, T ) for rm D 1 at NLO in EQCD M. Laine et al., JHEP (2007) F sub S F sub S = F S 2F Q = C F α s e rm D (1 + α r s[δz 1(µ) + rt f 1(rm D )]) on the lattice is compatible with EQCD@NLO up to rt 0.6. Weakly-coupled EQCD is reasonable in electric screening regime of F S. For rt > 0.8: asymptotic screening is inherently non-perturbative. 19 / 33

24 EQCD and the electric screening regime Confronting weak-coupling predictions in the screening regime (II) 1e-1 1e-2 T [MeV] -9r 2 sub _ TF QQ (r,t) [a.u.] Correction for cutoff effects filled symbols Hashed bands: LO Solid bands: NLO 1e-3 1e-4 1e-5 N τ = 4 Preliminary! rt Scale uncertainty µ = (1 4)πT due to resummation smaller for larger T Λ QCD = 320 MeV Singlet and octet cancel at LO in F sub Q Q = α2 s 9 F Q Q(r, T ) = α2 s 9 e 2rm D r 2 + C F α sm D. e 2rm D r 2 (1 + α s[δz 1(µ) + rt f 1(rm D )]) S. Nadkarni, PRD 33 (1986) Continuum estimate of F sub Q Q agrees with EQCD@NLO up to r 1/m D. Weakly-coupled EQCD is reasonable in the electric screening regime of F Q Q, but non-perturbative (chromo-magnetic) effects are stronger. 20 / 33

25 Correlators with τ < N τ periodic boundaries τ N x x Q Q r N τ / 33

26 From lattice correlators to spectral functions The TUMQCD lattice correlators Two static correlators for τ < N τ permit access to V S (r, T ): CS ren (r, T, τ) = 1 Tr [Wτ (0)W τ (r)] ren in Coulomb gauge 3 T WS ren (r, T, τ) = 1 Tr [Wτ (0)W r(τ)w τ (r, 0)W r (0)] ren 3 T Renormalization for WS ren (r, T, τ) more simple than for τ = N τ treat self-energy divergences with C Q (fund. rep.) or link smearing. Link smearing done for spatial links only spatial HYP smearing. τ < N τ data are available on all lattices of the color screening study. New strategies: 4-dimensional HYP smearing Wilson flow (4-dimensional, continuous smearing) Working to obtain continuum limit before further analysis 22 / 33

27 From lattice correlators to spectral functions Relation to the spectral functions Each correlator related to different spectral functions σ X (r, T, τ) via + X S (r, T, τ) = dω σ X (r, T, ω)e ωτ A. Rothkopf 2009, A. Rothkopf et al.,... V S (r, T > 0) is encoded in the peak structure of σ X (r, T, ω), i.e. peak position Re V S (r, T ), peak width Im V S (r, T ) Reconstruction of full σ X (r, T, ω) is severely ill-posed problem N τ 1 correlator data vs O(1000) points for each σ X (r, T, ω) How to effectively increase N τ beyond O(10)? Anisotropic lattices hard to do in practice G. Aarts et al. [FASTSUM] Continuum limit combines different N τ, taking CL first increases density of statistically independent points with τ/n τ < 1. Sophisticated 1-dimensional Bayesian analysis: MEM, BR, etc. Multidimensional Bayesian analysis combining different r and T Do we really need the full σ X (r, T, ω) to study V S (r, T )? Maybe not / 33

28 Moments from the lattice Moments of the spectral functions from lattice correlators Define the moments m X,n (r, T, τ) of the correlators X(r, T, τ) as m X,1 = d ln X = 1 dx, m dτ X dτ X,n = [ ] d n 1 mx,1 for n > 1. dτ Relation to mean and variance in the spectral representation: m X,1 = m X,2 = + dω ωσ X (r,t,ω)e ωτ + + dω ω2 σ X (r,t,ω)e ωτ + dω σ X (r,t,ω)e ωτ dω σ X (r,t,ω)e ωτ = ω, ω 2 = σ 2 X,ω For spectra with only one large peak find peak properties directly For lattice correlators need discrete derivatives ( time step sa): m X,1 (τ as 2 ) = 1 [ ] as ln W (τ) = V eff X(τ as) X (τ as 2 ), m X,n (τ ± as 2 ) = 1 as [m X,n 1 (τ) m X,n1 (τ as)] for n > / 33

29 Moments from the lattice Moments at T = 0 and T > 0 r=0.1 r=0.2 r=0.3 r=0.4 r=0.5 r= r=0.1 r=0.2 r=0.3 r=0.4 r=0.5 r=0.6 am 1 am τt τt Very preliminary. N τ = 12, 64, β = 7.825, r [fm], n-p. improved. Open squares: T 125 MeV counted as T 0, bullets: T 400 MeV τt 0 behavior almost T 0-like similar high ω behavior! Decrease of am 1 faster for larger distances and for higher temperatures Excited state contamination contributes to τt dependence of am 2 25 / 33

30 Moments from the lattice Operator (in-)dependence of the moments Very preliminary! N τ = 12, β = T = 408 MeV Diamonds: Coulomb gauge Triangles: smeared W. Loops Squares: bare W. Loops Approximate operator independence around midpoint τ T 0.5. Wilson loops with link-smeared spatial Wilson lines much closer to Coulomb gauge result. 26 / 33

31 Moments from the lattice Cutoff effects of the moments T 200 MeV T 400 MeV Very preliminary, only tree-level improved! Bullets, Triangles: Squares: N τ = 12, 10, 8; r [fm], V eff [MeV] Cutoff effects larger are for coarser lattices, i.e. lower temperatures Effective potentials steeper for higher T and larger r 27 / 33

32 Modeling the moments HTL spectral functions as test case (I) r=0.33gev -1 r=0.66gev -1 r=1.00gev -1 r=1.33gev -1 r=1.66gev -1 r=2.00gev -1 r=2.33gev -1 T = 2.33T c T c = 270 MeV e HTL loop spectral functions for Cb gauge Y. Burnier, A. Rothkopf 2013,... Fit the HTL with more (skewed) Lorentzian Ansatz in peak region σ C (T, ω) = 1 π Γ (ω µ) 2 +Γ 2 Θ(ω η) Not appropriate beyond peak region! η is IR-cutoff parameter. Compute Lorentzian moments analytically from σ C (T, ω) 28 / 33

33 Modeling the moments HTL spectral functions as test case (II) Lorentzian moments describes HTL moments well (except first point) High energy tail is not important beyond the first few points. Input parameters are reproduced at 1% accuracy. 29 / 33

34 Modeling the moments Fitting the lattice moments m 2 /T m 1 /T T=0 uns T=0 sub T>0 uns T>0 sub T>0 F s τt T>0 uns T>0 sub τt N τ = 12, β = 7.825, rt = 0.85, only tree-level improved! 1 Identify T = 0 ground state treat rest as excited state contamination 2 Subtract T = 0 excited state contamination from T > 0 correlator 3 Determine T > 0 moments from subtracted correlator 4 Fit skipping small and large τ T regions with constrained parameters 30 / 33

35 Modeling the moments Numerical results Re V/T T=0 sub T>0 F S fit1 fit2 fit rt Im V/T N τ = 12, β = 7.825, only tree-level improved! At small distances rt 0.5 only small difference to T 0 fit1 fit2 fit3 τt For rt 0.5 between T 0 potential and singlet free energy V S (r) Re V S (r, T ) > F S (r, T ) Im V S T as function of rt is only mildy temperature dependent 31 / 33

36 Summary We study color screening and deconfinement using the renormalized Polyakov loop correlator and related observables. We identify in the entropy S Q = df Q dt We extract T S = crossover behavior at T Tc. MeV from the entropy, in agreement with T χ = 160(6) MeV (chiral susceptibilities, O(2) scaling fits, m l m s = 1 ). 20 Continuum limit of static quark correlators in N f = 2+1 QCD up to T 1.9 GeV and down to r 0.01 fm. Color-singlet correlators are vacuum-like up to rt 0.3, exhibit color-electric screening for rm D rt 0.6 compatible with weak coupling and change to asymptotic screening for rt 0.7. The Polyakov loop correlator C P has a substantial color adjoint contribution for T 200 MeV. For rt 0.4 weakly-coupled pnrqcd describes C P well in terms of T = 0 potentials and the adjoint Polyakov loop L A. C P has a color-electric screening regime rm D 1. Non-perturbative effects (i.e. the chromo-magnetic sector) are much larger. 32 / 33

37 Summary Moments are a promising approach for obtaining the T > 0 potential. Complementary to Bayesian methods more direct contact to underlying correlation function. τ < N τ correlators are still mostly work in need of progress. New data and new techniques have not been included so far. We need further refinement of the excited state subtraction. Continuum limit! Include information from EFTs for different regimes! Tests of model dependence! 33 / 33