Lattice calculation of static quark correlators at finite temperature

Lattice calculation of static quark correlators at finite temperature J. Weber in collaboration with A. Bazavov 2, N. Brambilla, M.Berwein, P. Petrezcky 3 and A. Vairo Physik Department, Technische Universität München, Garching, 2 University of Iowa, 3 Brookhaven National Lab 33rd International Symposium on Lattice Field Theory, 7/7/25 Kobe

Overview & introduction Static quark free energy Onset of thermal effects Screening mass Summary & outlook

Introduction Commonly known facts Asymptotic freedom: QCD in thermal medium at short distances vacuum-like with weak coupling Hard thermal loop approximation: QCD in thermal medium for large distances and weak coupling: Debye screening of colour charges For weak coupling: description of heavy quarkonia in NREFTs Transition between vacuum-like and screening regions Quarkonium suppression indicates temperature in heavy-ion collisions Distance of the transition between regions? Q Q free energies are observables sensitive to the regions Determine screening mass in screening region Our lattice setup HISQ 2+ flavours: N τ = 4, 6, 8,, 2, aspect ratio 4 Temperature range 5 MeV T.4 GeV, M π 6 MeV

Vacuum-like and screening regime Effective coupling α qq αqq(r, T ).8.6.4.2.8.6 [MeV] T 33 47 54 62 7 77 86 94 24 222 242 257 266 28 32 369 445 52 546 62 666 84 947 Nt = 6 αqq(r, T ).2.8.6.4 [MeV] T 54 59 68 8 92 27 22 239 267 3 327 4 488 568 674 799 Nt =.4.2.2..2.3.6. r [fm].2.3.6..2.3.6. r [fm].2.3.6 Effective coupling α qq = 3/4r 2 E(r), E(r) = {F r (T, r), V (r)} max α qq.5 for T 3 MeV indicates strongly coupled plasma Regions separated by max α qq: vacuum physics or Debye screening

Static Q Q free energy Wilson lines represent static Q, Q: ψ(τ, r) = W (, τ; r)ψ(, r),... Polyakov loop correlator gives exponentiated free energy P c =e (F Q Q F )/T = TrW (/T, ; r) TrW (, /T ; r2) N 2 c P c formally splits into singlet and octet contributions [ Tr W ( T P c =, ; r)w (, ; r2)] [ T Tr W ( +, ; r)t a W (, ; r2)t a] T T Nc 3 T F Nc 2 = exp [ F /T ] + N2 c exp [ F 8/T ] Nc 2 Nc 2

Singlet and octet free energies exp [ F /T ] and exp [ F 8/T ] undergo mixing, gauge-dependent For lattice QCD, singlet (octet) free energy from Cyclic Wilson loop - loop closed by spatial Wilson lines (gauge invariant) - path dependence leads to extra divergences 2 Coulomb gauge Wilson line correlator (aka singlet free energy correlator) - no spatial lines required - gauge dependence leaves physical interpretation questionable Both correlators agree with static energy only at leading order

Cyclic Wilson loop Cyclic Wilson loop.2 T h = 5 MeV log( L C s )/Nt -.2 -.4 P W W W 2 W 5 -.6.5.5 2 rt Logarithm of ratios over Coulomb gauge Wilson line correlator C s Different iterations of spatial HYP smearing for cyclic Wilson loops W N Singlet fraction in C s and W N decreases for larger rt (cf. P c)

Cyclic Wilson loop Perturbative predictions HTL at one loop for rt > F (T, r) = N2 c α sm D N2 c exp ( m D r) α s 2N c 2N c r F 8(T, r) = N2 c α sm D + exp ( m D r) α s 2N c 2N c r F Q Q(T, r) = N2 c α sm D α 2 exp ( 2m D r) 2N c Nc 2 s r 2 T Magnetic mass contributes at even larger distances in EQCD: F Q Q(T, r) 2F Q Q(T, r ) exp ( m A r) exp ( m M r) # + # 2 T r r [ m A < m M despite power counting ma 2m D gt, m M g 2 T ]

Single quark free energy Bare single quark free energy F Q 8 7 6 5 4 3 FQ T N t 2 86 4 2 5.5 6 6.5 7 7.5 8 β 8.5 Quark free energy from Polyakov loop: F bare Q /T = N τ log Tr W (, T )

Single quark free energy Single quark free energy F Q 6 5 4 3 2 FQ ren [MeV] Std. ren. τ = 8 τ = 6 N ref N ref - -2 2 3 T [MeV] 4 5 6 7 Combine three renormalisation schemes & extend F Q to T 7 MeV

Single quark free energy Single quark free energy F Q 6 5 4 3 2 Squares Bazavov et al. Circles Borsanyi et al. Diamonds our data 2 4 6 8 2 Combine three renormalisation schemes & extend F Q to T 7 MeV Bazavov and Petreczky, Phys. Rev. D 87 (23) 9 : consistent

Single quark free energy Single quark free energy F Q 8 6 2FQ ren T log 9 [MeV] 4 2-2 Bands Borsanyi et al. Points our data -4-6 -8-5 2 25 3 T [MeV] 35 4 Combine three renormalisation schemes & extend F Q to T 7 MeV Bazavov and Petreczky, Phys. Rev. D 87 (23) 9 : consistent Borsanyi et al., JHEP 54 (25) 38 : for T = 2 MeV set to zero

Single quark free energy Single quark free energy F Q with gradient flow 6 FQ ren [MeV] 5 4 6 8 2 Our res. 3 2 5 2 25 3 35 T [MeV] 4 F Q w. gradient flow (finite N τ ) from H.-P. Schadler (Wed. 5th, 7:5) Constant flow time in physical units, same for all temperatures

Quark-Antiquark free energy Quark-Antiquark free energy F Q Q.6.3 F Q Q T log(9) [GeV] -.3 -.6 -.9 -.2 -.5 -.8..2.3 6 575 55 525 5 475 45 425 4.4 T [MeV] 375 35 325 3 275 25 225 2.5.6 r [fm].7 Short distance & low temperature: reproduce static energy (T = )

Quark-Antiquark free energy Quark-Antiquark free energy F Q Q.4.2 F Q Q T log(9) [GeV] -.2 -.4 -.6 -.8 -..2.3 Bands Borsanyi et al. Points our data.4.5.6.7 r [fm].8 Short distance & low temperature: reproduce static energy (T = ) Borsanyi et al., JHEP 54 (25) 38 : for T = 2 MeV set to zero

Singlet free energy Singlet free energy F.6.4.2 -.2 -.4 F [GeV] T [MeV] 2 225 25 275 3 325 35 4 45 5 55 6 r [fm] -.6..2.3.4.5.6.7.8.9 Reproduce static energy (T = ): larger distances & higher T

Singlet free energy Singlet free energy F F [GeV].5 -.5 -.2.4 55 46 425 365 325 28 T [MeV] 26 24 225 25 2.6.8 r [fm] Reproduce static energy (T = ): larger distances & higher T Kaczmarek, PoS CPOD 7 (27) 43, N τ = 4, 6, M π 22MeV Our N τ = 6 and continuum results are higher: chiral or cutoff effects?

Lattice calculation Onset of thermal effects: V (r) F (T, r) 8 V(r) F(T, r) [MeV] 6 4 2 N t = 4 N t = 6 N t = 8 N t = N t = 2 T=399 T=49 T=4 T=47 C.L..3.6. r [fm].2.3 Static (T = ) and singlet free energies: v(t, r) = V (r) F (T, r) Large cutoff effects, minimum visible only for N τ Continuum extrapolation with only N τ = 8,, 2

Lattice calculation Onset of thermal effects: V (r) F (T, r) 5 V(r) F(T, r) [MeV] 5-5 4 37 33 3 27 24 -.3.6. r [fm].2.3 Static (T = ) and singlet free energies: v(t, r) = V (r) F (T, r) T independent falling slope at rt.5, minimum at rt.25

Lattice calculation Onset of thermal effects: V (r) F (T, r) V(r) F(T, r) [MeV] 2 5 5 T=47 T=488 T=6 T=84 T=222..2.3.4 rt.6.8 Static (T = ) and singlet free energies: v(t, r) = V (r) F (T, r) Estimate cutoff effects as -2 MeV from data at T 4 MeV For rt.3 almost constant, for rt >.3 rapid rise

Screening mass Screening mass At large distance & weak coupling F (T, ) = F Q Q(T, ) = 2F Q (T ) Subtract asymptotic constant to define screening functions S (T, rt ) = rt F(T, r) 2F Q(T ) T S avg(t, rt ) = (rt ) 2 F Q Q(T, r) 2F Q (T ) T HTL N2 c α s exp ( m D r), 2N c HTL N 2 c α 2 s exp ( 2m D r) Theoretically ideal: extract m D from F Q Q, but data is too noisy Singlet free energy: extra rt dependendence due to Z ren Cyclic Wilson loop extra linear divergence due to self-energy 2 Coulomb gauge correlator: gauge dependence, but cleanest probe

Screening mass Screening mass: extraction 4.5 4 3.5 T 333 MeV N t 2 8 4 3 log S rt 2.5 2.5.5 rt 2 rt, weak coupling: log S (T, r) = A M T rt HTL log (C F α s) m D T rt

Screening mass Screening mass: extraction 4.5 4.25 M T M T rt 4 3 4 3.75 3.5 plateau data derivative candidates 2-3.25 3.4 T = 334 MeV.6.8.2.4 rt -2-3 rt, weak coupling: log S (T, r) = A M HTL rt log (C T F α s) m D T rt Estimate systematical uncertainty with derivative ( M )/ (rt ) T

Screening mass Screening mass: extraction 6.5 6 5.5 5 M T 4 6 8 2 full errors 4.5 4 3.5 3 T [MeV] 2 3 4 5 6 7 8 Largest rt before signal loss, increase errors w. systematic effects Jackknife estimate of extraction method dependence w. 8 4 methods

Screening mass Screening mass: comparison with other lattice results 5.5 5 M T PoS CPOD 7 N τ = 6 m D /T 4.5 4 3.5 3 2 3 4 5 6 7 T [MeV] 8 Kaczmarek, PoS CPOD 7 (27) 43: lower due to heavier M π Borsanyi et al., JHEP 54 (25) 38: magnetic mass m M (T )

Summary Extend numerical results for static quark free energy to T 7 MeV and Q Q free energy to T 6 MeV Study the onset of screening with new observable V (r) F (T, r), field-theoretically cleaner than α qq(t, r) Extract screening mass in screening region Outlook Comparison of V (r) F (T, r) with weak coupling (pnrqcd) Need finer zero temperature lattices [a.25,.3,.35 fm] Extract spectral function from static correlators, extract imaginary part of potentialwith 3 dynamical flavours

Cyclic Wilson loop.5 T h = 5 MeV log( L C s )/Nt exp [+3rT ] -.5 -. -.5 P W W W 2 W 5.5.5 2 rt Boost w. factor exp (3rT ), log P c/c s w. error reduced by factor /2 Small rt : negative values indicate larger octet fraction than C s (cf. P c) Unsmeared Wilson loop on top of Polyakov loop correlator for rt

Scale r and renormalisation constant 8 r 4 via r via r polynomial spline 2 6 6.5 7 7.5 8 β Use T = data from Bazavov et al., Phys. Rev. D 9 (24) 9 Scale setting with r (r ) for fine (coarse) lattices

Scale r and renormalisation constant -.65 -.7 Zren -.75 -.8 via r via r polynomial spline -.85 6 6.5 7 7.5 8 β Use T = data from Bazavov et al., Phys. Rev. D 9 (24) 9 Scale setting with r (r ) for fine (coarse) lattices Renormalisation constant Z ren (β) from T = static energy

Renormalisation schemes for F Q 6 5 4 3 2 FQ ren [MeV] N t, Z ren 2, std. ren., std. ren. 8, std. ren. 6, std. ren. 2, Nτ ref = 8, Nτ ref = 8 8, Nτ ref = 8 6, Nτ ref = 8 - -2 2 3 T [MeV] 4 5 6 7 Old (standard) scheme: use Z ren (β) (rescaled with N τ ) for each N τ

Renormalisation schemes for F Q 6 5 4 3 2 FQ ren [MeV] N t, Z ren 2, std. ren., std. ren. 8, std. ren. 6, std. ren. 2, Nτ ref = 8, Nτ ref = 8 8, Nτ ref = 8 6, Nτ ref = 8 - -2 2 3 T [MeV] 4 5 6 7 Old (standard) scheme: use Z ren (β) (rescaled with N τ ) for each N τ New scheme: avoid extrapolated Z ren (β) assuming small cutoff effects: Z ren (T, N τ ) = Z ren (T, Nτ ref ) + 2 ( FQ bare (T, N τ ) FQ bare (T, Nτ ref ) )

Screening functions.5.3.2 S(rT, T) T=49 MeV Interpolation.5.3.2 S(rT, T) T=4 MeV Interpolation..8..8.5.5.3.3.2.2..2.4.6.8 rt.2.4..2.4.6.8 rt.2.4.2 Savg(rT, T).2 Savg(rT, T)...8.8.5.5.3.2 T=49 MeV Interpolation..2.3.4.5 rt.6.7.8.3.2 T=4 MeV Interpolation..2.3.4.5 rt.6.7.8

rt dependence of screening mass 5.5 5 rt.8.9.. 4.5 m/t 4 3.5 3 2.5 2 3 4 5 6 7 8 T [ MeV ]

Screening mass: comparison with weak coupling calculations 6.5 6 5.5 5 4.5 M T A =.85, µ = 2.4πT A =.85, µ = 2.πT.85.6πT A =.85, µ =.3πT A =.85, µ =.πt A =.5, µ = πt full errors 4 3.5 3 2 3 4 5 6 7 T [MeV] Braaten and Nieto, Phys. Rev. D 53 (996) 342: electric screening mass ( mel 2 = 4πα st 2 Nc 3 + N ) f + O(αs 2 ) 6 Plot A m 2 el [T, µ, g(µ)]/t w. m el at one or two-loop, α s(µ) at two-loop 8

Short distance cutoff effects Data with equal temperature and neighbouring N τ at short distance Estimate systematic uncertainties by studying rt -dependence of cutoff effect Compute difference of correlators C S C L and subtract interval average Absolute maximum of remainder is considered as systematical uncertainty and applied to all distances up to largest rt of interval Quadratically added to statistical errors Significant only for V (r) F (T, r)

Cutoff effects in V (r) F (T, r).8.6.4.2 -.2 δ[v (r) F (T, r)]/t T = 45 MeV N t =, N t = 2 -.4 -.6 -.8 δ(v S ) δ(v L ) 47 4..25.5 (v S v L ) v S v L (v S v L ) max (v S v L ).75.2 rt.225

Cutoff effects in S (T, rt ).8.6 δs (T, r) T = 45 MeV N t =, N t = 2.4.2 -.2 -.4 -.6 -.8 δ(s S ) δ(s L ) 47 4..25.5 (S S S L ) S S S L (S S S L ) max (S S S L ).75.2 rt.225

Cutoff effects in S avg (T, rt ).8.6 δs avg (T, r) T = 45 MeV N t =, N t = 2.4.2 -.2 -.4 -.6 -.8 δ(s S ) δ(s L ) 47 4..25.5 (S S S L ) S S S L (S S S L ) max (S S S L ).75.2 rt.225

Scaling behaviour: V (r) F (T, r) (V F) [MeV] T = 3 MeV (V F) [MeV] T = 33 MeV 6 6 2 2 8 8 4 4-4 χ 2 =.536 χ 2 =.25 χ 2 =.39 χ 2 =.56 rt =.3 rt =.25 rt =.2 rt =.5-4 χ 2 =.74 χ 2 =.25 χ 2 =.54 χ 2 =.6 rt =.3 rt =.25 rt =.2 rt =.5 2 2 2 8 2 6 2 2 2 2 8 2 6 2 (V F) [MeV] T = 35 MeV (V F) [MeV] T = 4 MeV 6 6 2 2 8 8 4 4-4 χ 2 =.88 χ 2 =.52 χ 2 =.99 χ 2 =.2 rt =.3 rt =.25 rt =.2 rt =.5-4 χ 2 =.962 χ 2 =.9 χ 2 =.58 χ 2 =.62 rt =.3 rt =.25 rt =.2 rt =.5 2 2 2 8 2 6 2 2 2 2 8 2 6 2

Scaling behaviour: S (T, rt ) 8 S [MeV] T = 3 MeV 8 S [MeV] T = 35 MeV 6 4 2 8 6 4 2 2 2 2 8 2 6 2 χ 2 =.3 χ 2 =.52 χ 2 =.424 χ 2 =.39 χ 2 =.7 χ 2 =.22 χ 2 =.86 χ 2 =.9 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 rt =.5 rt =.6 rt =.7 Nt 2 4 2 6 4 2 8 6 4 2 2 2 2 8 2 6 2 χ 2 =.56 χ 2 =.39 χ 2 =.25 χ 2 =.536 χ 2 =.56 χ 2 =.39 χ 2 =.25 χ 2 =.536 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 rt =.5 rt =.6 rt =.7 Nt 2 4 2 8 S [MeV] T = 4 MeV 8 S [MeV] T = 5 MeV 6 4 2 8 6 4 2 2 2 2 8 2 6 2 χ 2 =.38 χ 2 =.2 χ 2 =.67 χ 2 =.29 χ 2 =. χ 2 =.53 χ 2 =.84 χ 2 =.33 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 rt =.5 rt =.6 rt =.7 Nt 2 4 2 6 4 2 8 6 4 2 2 2 2 8 2 6 2 χ 2 =.6 χ 2 =.67 χ 2 =.69 χ 2 =.8 χ 2 =.34 χ 2 =.6 χ 2 =.64 χ 2 =.9 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 rt =.5 rt =.6 rt =.7 Nt 2 4 2

Scaling behaviour: S avg (T, rt ) 8 8 Savg [MeV] T = 3 MeV Savg [MeV] T = 35 MeV 7 7 6 5 4 χ 2 =.53 χ 2 =.47 χ 2 =.59 χ 2 =.246 χ 2 =.4 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 6 5 4 χ 2 =.9 χ 2 =.32 χ 2 =.29 χ 2 =.85 χ 2 =.25 rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 3 3 N 2 t N 2 t 2 8 2 2 2 8 2 6 2 4 2 2 8 2 2 2 8 2 6 2 4 2 Savg [MeV] T = 4 MeV Savg [MeV] T = 5 MeV 7 7 6 5 4 χ 2 =.5 χ 2 =.23 χ 2 =.97 χ 2 =.29 χ 2 =. rt =.5 rt =.2 rt =.25 rt =.3 rt =.4 6 5 4 χ 2 =.26 χ 2 =. χ 2 =.8 χ 2 =.2 χ 2 =.8 rt=.5 rt=.2 rt=.25 rt=.3 rt=.4 3 3 N 2 t N 2 t 2 2 2 2 8 2 6 2 4 2 2 2 2 2 8 2 6 2 4 2