Quarkonium Free Energy on the lattice and in effective field theories

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1 Quarkonium Free Energy on the lattice and in effective field theories J. H. Weber 1,2 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 Multi-Scale Problems Using Effective Field Theories, INT-18-1b, Seattle, 05/21/2018 PRD (2016); arxiv: / 27

2 Introduction In-medium quarkonia from heavy-ion collisions Quarkonium as thermometer of QGP T. Matsui, M. Satz, PL B (1986) Oversimplified picture real-time processes are important. First-principle calculation are feasible in an EFT framework 2 / 27

3 Introduction Hierarchies of scales for in-medium quarkonia Non-relativistic EFTs with non-relativistic hierarchy of scales M Mv p 1 r Mv 2 E Integrate out heavy scales NRQCD and pnrqcd The thermal medium introduces the thermal scales T gt g 2 T Suitable for dimensionally reduced thermal EFTs 4-dimensional with one compact direction effectively 3-dimensional Many different hierarchies between NR and thermal scales are possible p T, p gt,... 3 / 27

4 Introduction EFTs for in-medium quarkonia Thermal hierarchies are manifest for asymptotically high temperatures T g(t ) 0 where g = 4πα s Phenomenologically interesting (HIC): T < 1 GeV: α s 0.4, g 2 Is the weak-coupling approach appropriate for phenomenology? Are the postulated hierarchies actually realized and distinguishable? We test the different hierarchies and regimes using realistic lattice QCD simulations. We consider heavy quarks in the static limit. We aim at establishing whether the EFT descriptions for quarkonium are suitable for (experimentally) relevant temperatures. 4 / 27

5 Overview Quarkonium Free Energy on the lattice and in effective field theories Overview & Introduction Correlators of Polyakov loops and Q Q free energy F Q Q on the lattice Deconfinement and onset of color screening: entropy S Q Comparison to weak-coupling EFTs Summary 5 / 27

6 Overview f bare Q What is new about the lattices of the TUMQCD study? 9 bare f N Q τ = log P T 4 Continuum limit with 8 realistic quark masses Two different volumes and two quark masses: Controlled finite volume and light quark mass dependence TUMQCD collaboration, arxiv: β 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 action, errors: O(α sa 2, a 4 ); lattice artefacts are reduced. Ensembles: m π 160 MeV; a 0.04 fm & m π 320 MeV; a fm 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. A. Bazavov et al., PRD (2012), PRD (2014) [HotQCD] A. Bazavov et al., PRD (2016), PRD (2018), arxiv: [TUMQCD] r 1 scale for β > 8.4 from non-perturbative β function PRD (2014) 6 / 27

7 Free energies of static quark states 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 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 Q Q (r,t ) = 1 9 e F S /T e F A/T = 1 9 C S(r, T ) C A(r, T ). Singlet & octet free energies are gauge dependent. S. Nadkarni, PRD 33, 34 (1986) C P is related to the gauge-invariant free energies f s,o of pnrqcd C P (r, T ) = e F Q Q (r,t ) = 1 9 e f S /T e fo/t + O(g 6 ) for rt 1. N. Brambilla et al., PRD 82 (2010) 7 / 27

8 Free energies of static quark states 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 = F b Q + 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 ) + F b Q(β ref, N ref τ ) F b Q(β, N τ ) } S. Gupta et al., PRD (2008) 8 / 27

9 Free energies of static quark states 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 = F b Q + 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) 8 / 27

10 Free energies of static quark states Static quark-antiquark correlators periodic boundaries τ N x x Q Q r N τ On the lattice static quarks are temporal Wilson lines W = N τ τ/a=1 U 0(τ, x). 9 / 27

11 Polyakov loop correlators Polyakov loop correlator and Q Q free energy _ For very small rt F QQ (r,t)-t ln 9 [GeV] For large rt and T [MeV] and T 500 MeV T 200 MeV F Q Q T ln 9-1 we can explicitly is very close to the static energy V S of the vacuum. The relevant scale hierarchy is α s/r T Continuum limit! r [fm] make connection to the asymptotic rt behavior F Q Q 2F Q. Free energy of a Q Q pair, F Q Q, is also called color-averaged potential: C P (r, T ) = P (0)P (r) ren T = e F Q Q (r,t ) T = 1 F S (r,t ) 9 e T + 8 F A (r,t ) 9 e T. 10 / 27

12 Polyakov loop correlators Static meson correlator and singlet free energy in Coulomb gauge For rt 0.3 F S (r,t) [GeV] For large rt and and T 2.2 GeV 0 F S T 200 MeV is very close to the -1 we can explicitly static energy V S of the vacuum. -2 make connection The relevant scale hierarchy is α s/r T Continuum limit! T [MeV] r [fm] to the asymptotic rt behavior F S 2F Q. The singlet free energy is related to the gauge-fixed static meson correlator at τ = 1/T in Coulomb gauge 3 ren CS ren (r, T ) = 1 W a(0)w a (r) = e F S (r,t )/T. 3 a=1 T 11 / 27

13 Polyakov loop correlators Effective coupling: vacuum-like and screening regimes _ α QQ (r,t) T [MeV] Continuum limit! r [fm] The effective coupling α Q Q(r, T ) is a suitable proxy for the force between the Q Q pair and for the QCD coupling α s running with 1/r. α Q Q (r, T ) = r 2 V S (r) C F r We generalize α Q Q with the singlet free energy F S instead of V S (r). 12 / 27

14 Polyakov loop correlators Effective coupling: vacuum-like and screening regimes Running coupling (Coulomb-like) 0 _ α QQ (r,t) T [MeV] Continuum limit! r [fm] Vacuum-like regime Screening regime max(α Q Q ) rt 0.2 rt 0.3 r maxt 0.4 Running coupling (string tension-like) Strongly-coupled QGP max(α Q Q) for 300 MeV Weakly-coupled QGP Effective 3d theory (running of α Q Q stalls) r max defined through max(α Q Q), which is proxy for the maximal force. Weak-coupling approaches may work for T 300 MeV (α Q Q 0.5). 12 / 27

15 Polyakov loop correlators Effective coupling: vacuum-like and screening regimes T [MeV] r 2 sub _ TF QQ (r,t) α s/r T Continuum limit! 1/16 1/12 1/10 1/8 1/6 1/4 rt T α s/r α QQ _ (r,t) T [MeV] α s/r T Finest lattices N τ = 16, 12 r [fm] F Q Q has for rt 1 two distinct regimes α s/r T and T α s/r. Singlet-dominance Singlet-octet cancellation Screening regime rt 0.05,..., 0.15 rt 0.3 rt 0.3 We define a running coupling in terms of F Q Q in the regime αs/r T. 12 / 27

16 A single static quark: the Polyakov loop periodic boundaries τ N x x Q N τ / 27

17 Deconfinement and onset of screening T χ from chiral observables vs T S from the peak of the entropy F Q Q(r ) = 2F Q The entropy S Q = df Q dt of an isolated quark is independent of 2 the volume and of 1.5 the renormalization 1 scheme. 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 ) Nτ 2 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] 14 / 27

18 Deconfinement and onset of screening T χ from chiral observables vs T S from the peak of the entropy 5 S 4.5 Q Reminder: an τ = 1/T Discretization errors: F Q Q (r ) = 2F Q The entropy S Q = df Q dt of an isolated quark is independent of 2 the volume and of 1.5 the renormalization 1 scheme. T χ (N τ ) local fit global 1/N τ 4 fit global 1/N τ 2 fit HRG N τ =8 T [MeV] T c (N τ ) = T c + O( 1 ) Nτ 2 Bazavov et al. [TUMQCD] PRD (2016) ds Q dt > 0 for T < Tc Hadron resonance gas (HRG) is limited to only below T 125 MeV. 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. ds Q dt Large number of extra states or strong thermal modification of (low-lying) states are needed already substantially below T c. 14 / 27

19 Deconfinement and onset of screening T χ from chiral observables vs T S from the peak of the entropy F Q Q (r ) = 2F Q The entropy S Q = df Q dt 10 of an isolated quark 4 is independent of 3 the volume and of 2 the renormalization 1 scheme. 0 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 = P. Petreczky, K. Petrov PRD (2004) O. Kaczmarek, F. Zantow, hep-lat/ (2005) Bazavov et al. [TUMQCD] PRD (2016) ds Q dt < 0 for T > Tc < 0 for T > Tc: the static quark interacts with the medium only T 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. 14 / 27

20 Onset of weak coupling Onset of weak coupling in the entropy S Q (T) LO NLO, µ=(1-4)π T NNLO lattice Continuum limit! T [MeV] Free energy at leading order F Q = C F α s m D 2 + O(g 4 ) m D gt S Q g 3. known to NNLO: M. Berwein, et al., PRD (2016) Poor convergence of expansion in g NLO still missing NLO in α s. Continuum results and NNLO agree for T 10 T c. Late onset of weak-coupling behavior: static Matsubara mode is dominant. A. Bazavov et al. [TUMQCD] PRD (2016) 15 / 27

21 pnrqcd and the vacuum-like regime The vacuum-like regime The vacuum-like regime is defined in terms of rt 1. For r 1/T multipole expansion is appropriate the appropriate EFT is pnrqcd. The vacuum-like regime has two sub-regimes: α s/r T and α s/r T For α s/r T weak-coupling calculations are available up to O(g 7 ). M. Berwein, et al., PRD (2016), PRD (2017) For α s/r T weak-coupling calculations are not available. Medium effects are exponentially suppressed as e (Vo Vs )/T e αs /rt. Brambilla et al., PRD (2008) 16 / 27

22 pnrqcd and the vacuum-like regime Static energy and singlet free energy on the lattice 60 V S (r) -F S (r,t) [MeV] 50 T [MeV] r [fm] [V S (r)-f S (r,t)]/t T [MeV] rt V S (T = 0) F S (T > 0) up to O(αs 3 ) M. Berwein et al., PRD (2017) Cancellations in V S F S smoother for r/a < 3, no renormalization. For rt 0.1 & T > 300 MeV: V S F S 0.02T, mild N τ dependence. Only mild T dependence up to rt 0.3. For rt 0.3 strong medium effects set in rapidly. 17 / 27

23 pnrqcd and the vacuum-like regime Static energy and singlet free energy at weak coupling [V S (r)-f S (r,t)]/t Lattice(HISQ) O(g 5 )+const T=500 MeV rt Band: NNNLO Resummation scale µ = (1 4)πT Input QCD scale Λ QCD = 320 MeV Bazavov et al., PRD (2014) Weak-coupling result for hierarchy α s/r T vanishes for r 0 as V S (T = 0) F S (T > 0) αs 2 rt M. Berwein et al., PRD (2017) Partial compensations of non-static gluons/quarks by static gluons. Constant term αs 3 T in F S from matching of pnrqcd and NRQCD If α s/r T thermal effects exponentially suppressed. Brambilla et al., PRD (2008) 18 / 27

24 pnrqcd and the vacuum-like regime Polyakov loop correlator in pnrqcd pnrqcd: C P is given in terms of gauge-invariant color-singlet and color-octet free energies up to O(g 6 (rt ) 4 ) as N. Brambilla et al., PRD 82 (2010) C P (r, T ) = e F Q Q (r,t ) = 1 N 2 c e fs /T + N2 c 1 e fo/t. Nc 2 The decomposition of C P into gauge-invariant singlet and octet is defined assuming weak coupling realized for which temperatures? For rt 0 C P is expressed in terms of potentials V s and V o 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 N 2 c e Vs /T + N2 c 1 L Nc 2 A e Vo/T + O(g 6 (rt ) 0 ). The non-trivial temperature dependence of C P is mainly due to the interplay and cancellations between color-singlet and color-octet. 19 / 27

25 pnrqcd and the vacuum-like regime Color octet contribution in the Polyakov loop correlator repulsive tail F O (r,t) [GeV] N τ = 8 C P (r, T ) = e F Q Q (r,t ) = T [MeV] r [fm] N 2 c e Vs /T + N2 c 1 L Nc 2 A e Vo/T + O(g 6 (rt ) 0 ). Use lattice quantities as proxies (static energy ( V S for singlet potential V s) to define an octet free energy e F O /T = 9 8 e F Q Q (r,t ) ) 1 9 e V S /T F O decreases rapidly for higher T : the color-octet contribution becomes large, the regime α s/r T is restricted to shorter distances. 20 / 27

26 pnrqcd and the vacuum-like regime Test of pnrqcd for the Polyakov loop correlator r 2 sub _ TF QQ (r,t) _ F QQ, data rec V S only N τ = 12 T=172 MeV rt r 2 sub _ TF QQ (r,t) T=666 MeV F QQ _, data rec V S only rec V S, V O rec V S, V O, δv o N τ = rt C P (r, T ) = e F Q Q (r,t ) = 1 e Vs /T + N2 c 1 L Nc 2 Nc 2 A e Vo/T + O(g 6 (rt ) 0 ). Low T = 172 MeV: color-singlet, i.e. V S, is enough for reconstructing C P (no sensitivity to color-octet). Data are in the regime α s/r T. High T = 666 MeV: cancellation between color-singlet and -octet leads to 1/r 2 behavior in F Q Q. Data are in the regime α s/r T. Casimir scaling violation 8V o+v s = 3 α3 s [ π2 3] B. Kniehl et al., PLB 607 (2005) r 4 21 / 27

27 pnrqcd and the vacuum-like regime Direct comparison for the Polyakov loop correlator r 2 sub _ TF QQ (r,t) Band: NNNLO T [MeV] rt Resummation scale µ = (1 4)πT Input QCD scale Λ QCD = 320 MeV Bazavov et al., PRD (2014) C P (r, T ) = e F Q Q (r,t ) = 1 e Vs /T + N2 c 1 L Nc 2 Nc 2 A e Vo/T + O(g 6 (rt ) 0 ). A recent calculation of C P at NNNLO using pnrqcd in the regime α s/r T up to order g 7. M. Berwein et al., arxiv: Both results agree, although the uncertainty of the weak-coupling result is large even at T 10 T c. For rt 0.2 the hierarchy α s/r T eventually breaks down. 22 / 27

28 EQCD and the electric screening regime The screening regime The screening regime is defined in terms of r 1/m D. Hierarchy is automatically built into dimensionally-reduced EFT. The appropriate EFT is EQCD. The screening regime has two sub-regimes: r 1/m D and r 1/m D In the electric screening regime, r 1/m D, chromo-electric fields are important. Weak-coupling calculations are available up to O(g 5 ). S. Nadkarni, PRD 33 (1986) M. Laine et al., JHEP (2007) M. Berwein, et al., PRD (2017) In the asymptotic screening regime, r 1/m D, Chromo-magnetic fields are dominant. Non-perturbative methods are required. M. Laine, M. Vepsalainen, JHEP (2009) 23 / 27

29 EQCD and the electric screening regime Weak-coupling prediction for F S in the screening regime 10 0 sub T [MeV] -rf S (r,t)/cf [a.u.] Leading order 260 FS sub LO = C F α s e rm D r Using the NLO Debye mass E. Braaten, A. Nieto PRD (1996) Continuum limit! 10-3 rt Hashed bands: LO Solid bands: NLO Resummation scale µ = (1 4)πT Input QCD scale Λ QCD = 320 MeV Bazavov et al., PRD (2014) NLO singlet free energy (two-gluon exchange is deferred to NNLO) F sub S ( NLO = F sub LO 1+αsN crt [2 ln(2x) γ E +e 2x E ) 1(x)], x = 2rm D S Correction due to field renormalization: ( δfs sub = FS sub LO 1 rm D ) 2 δz 1 M. Berwein, et al., PRD (2017) F S in the electric screening regime is controlled by the parameter m D. 24 / 27

30 EQCD and the electric screening regime Weak-coupling prediction for F Q Q in the screening regime T [MeV] -9r 2 sub _ TF QQ (r,t) [a.u.] Leading order Dash-dotted line: LO 5814 Solid bands: NLO F sub Q Q LO = N2 c [ -2 Resummation scale ] 2 α s e rm D µ = (1 4)πT N c rt 10-3 Input QCD scale at scale µ = 4πT Using the NLO Debye mass 10-4 E. Braaten, A. Nieto 10-5 PRD (1996) T [MeV] rt Continuum estimate Λ QCD = 320 MeV Bazavov et al., PRD (2014) Strong signal-to-noise problem calculation requires larger volumes. Use data for N τ = 4 with (estimated) correction for cutoff effects. We compare to the full O(g 5 ) result. M. Berwein, et al., PRD (2017) Previous EFT calculations had been missing important pieces. 25 / 27

31 EQCD and the electric screening regime 6 5 m S /T Asymptotic screening regime N τ _ m QQ /T 7 6 N τ PT A m D NLO / T A=2.00 A= T [GeV] 3 2 A 2m D NLO / T A=1.25 Non-pert T [GeV] Severe signal-to-noise problem no continuum limit. Cutoff effects are mild for N τ 8, but require estimates of asymptotic behavior. Asymptotic screening mass factor larger than m D for F S Asymptotic screening mass only slightly larger than 2m D for F Q Q Good agreement with results from direct EQCD simulations. A. Hart, et al., NPB (2000) 26 / 27

32 Summary Summary We study color screening and deconfinement using the renormalized Polyakov loop correlator and related observables. We extract the continuum limit of static quark correlators in N f = 2+1 QCD up to T 2 GeV and down to r 0.01 fm. Static quark correlators are vacuum-like up to rt 0.3 and are well-described by pnrqcd for T > 300 MeV. In C P we find numerical evidence for the distinction between the regimes of singlet dominance, α s/r T, and singlet-octet cancellaton, α s/r T. For singlet dominance we can define an effective coupling. Static quark correlators have an electric screening regime up to 0.3 rt 0.6 and are well-described by EQCD for T > 300 MeV. The perturbative Debye mass controls this regime. We identify in the entropy S Q = df Q crossover behavior at T Tc dt and extract T S = MeV from the entropy, in agreement with T χ = 160(6) MeV (chiral susceptibilities, O(2) scaling fits, m l m s = 1 ). 20 S Q becomes weakly coupled only at very high temperatures, T 10T c. 27 / 27