Lattice QCD thermodynamics Kalman Szabo Bergische Universitat, Wuppertal Wuppertal-Budapest collaboration
Outline Equation of State [1309.5258] Fluctuations [1305.5161] T > 0 with Wilson-type fermions [1205.0440]
Equation of state As of 2010 height and position are different:
Crossover [Wuppertal-Budapest,WB,’06] volume dependence of chiral susceptibility:
T c � ∼ 190 MeV [BNL-Bielefeld-RIKEN-Columbia,’06] T c = = ? ∼ 150 MeV [WB,’06] 1 ∆ l,s f K scale 0.8 [WB’06’09’10] T c = 147(2)(3)MeV 0.6 asqtad: N τ =8 N τ =12 0.4 HISQ/tree: N τ =6 N τ =8 [hotQCD ’12] T c = 154(9)MeV 0.2 N τ =12 N τ =8, m l =0.037m s stout cont. 0 T [MeV] 120 140 160 180 200
T c [hotQCD ’12] 220 T c [MeV] 210 200 N t a [fm] 4 0.30 190 6 0.20 180 8 0.15 170 160 2 1/N τ 150 0 0.02 0.04 0.06 0.08 0.1 N t = 4 results are unreliable
T c : scaling regime quark number susceptibility [WB,’11] a [fm] N t 6 0.20 8 0.15 10 0.12 12 0.10 16 0.075 a 2 -scaling for N t � 10 or a � 0 . 12 fm
Equation of state As of 2012 only the height is different:
Equation of state A couple of improvements made: [WB,arXiv:1309.5258]
Equation of state Finer lattices: N t = 6 , 8 , 10 → 6 , 8 , 10 , 12 , 16 -2 10 -2 -2 -2 12 8 6 3.2 tree level correction 3 no correction 2.8 2.6 4 p/T 2.4 � free � p cont p corr = p LAT · p LAT 2.2 2 1.8 1.6 0.005 0.01 0.015 0.02 0.025 0.03 2 1/N t tree-level improvement is as good as Naik,p4, but much cheaper not the slope matters, but the beginning of the a 2 -scaling-regime
Equation of state Thermodynamics is more expensive N 4 system size increase → t N 2 autocorrelation → t Hybrid-Monte Carlo stepsize → N t fermion inversion → N t � O � T − � O � T =0 remove UV divergence of N 4 N 8 → t t N − 4 volume contributes to statistics → t Total cost ∼ N 12 t
Equation of state Extreme fine lattice spacings downto a ≈ 0 . 02 fm • T > 0 is easier than T = 0 (smaller box needed, faster inverter, no tunneling problem) → use T / 2 to cancel the divergence: � O � T − � O � T / 2 • L = 1 fm is easier, than L = 6 fm → use Wilson-flow based step-scaling to determine running coupling and mass scale from w 0 based step scaling 4.6 scale from w 0 along LCP 4.4 fit to f K scale fit to w 0 scale 4.2 β 4 3.8 3.6 3.4 0.05 0.1 0.15 0.2 a[fm]
Equation of state Trace anomaly is unchanged 6 hotQCD HISQ N t =6 8 10 12 5 s95p-v1 4 HRG 4 ( ε -3p)/T 3 2 2 stout continuum WB 2010 1 4 stout crosscheck 0 200 300 400 500 T[MeV] extended the estimation of systematic error (896 methods)
Equation of state Independent check from a different action -2 -2 -2 -2 -2 -2 -2 -2 6 8 10 12 16 12 10 8 6 6 5 5 4 4 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.005 0.01 0.015 0.02 2 2 1/N t 1/N t tree level correction no correction 4-stout smearing with charm quark has different β LAT ǫ − 3 p ∼ d β LAT d log a · ( � O � T − � O � T =0 ) T 4
Equation of state As of 2013 discrepancy remains ... 6 hotQCD HISQ N t =6 8 10 12 5 s95p-v1 HRG 4 4 ( ε -3p)/T 3 2 2 stout continuum WB 2010 1 4 stout crosscheck 0 200 300 400 500 T[MeV] WB is “full-result”: continuum with physical mass
Equation of state Zero point of the pressure p ( T ∗ ) � T � T p ( T ) − p ( T ∗ ) dT ′ dp ( T ′ ) dT ′ ǫ − 3 p = = T 4 T 4 T ′ 5 dT ′ T ∗ T ∗ ∗ Previously: set p ( T ∗ ) to “0” or to HRG at T ∗ = 100 MeV Now: integrate in mass instead of T, divergence is only N 2 t � m = ∞ p ( T ) dm ′ dp ( m ′ ) = T 4 dm ′ m phys 0.8 N t =12 N t =16 light 0.6 strange � ln(ma) 0.4 4 � lnZ 0.2 -N t 0 -0.2 0 1 2 3 4 5 6 log(m q /m phys )
Equation of state Pressure 6 SB lattice continuum limit 5 4 4 p/T 3 2 HTL NNLO 1 HRG 0 200 300 400 500 T[MeV] at low T agreement with the HRG model (result, not an input!)
Fluctuations
Fluctuations B,Q,S fluctuations ≡ u,d,s susceptibilities ∂ i ∂ j ∂ k 1 � � χ BQS = log Z ijk VT 3 ∂ ( µ B / T ) i ∂ ( µ Q / T ) j ∂ ( µ S / T ) k χ 1 = � Tr M − 1 ∂ µ M � χ 2 = � Tr . . . 2 � − � Tr . . . � 2 + � ∂ µ Tr . . . � Expensive! higher orders bring volume penalty (disconnected) err( χ 4 ) ∼ VT 3 1 1 √ √ √ err( χ 1 ) ∼ err( χ 2 ) ∼ N · VT 3 N N traces are calculated with noisy estimators (multigrid, TSA, GPU)
Fluctuations HRG at low T , rapid rise, upto 90%SB, s “melts” at higher T
Uses of fluctuations 1 dominant degrees of freedom 2 thermodynamics at finite µ B ( T c and EoS) 3 compare with experiment → freezout parameters
Low temperature diagnose the breakdown of hadronic description, eg. uncorrelated hadron gas: � � p ( µ B , µ S ) = p mes , S cosh( S µ S ) + p bar , S cosh( µ B − S µ S ) S =0 ... 3 S use fluct-combinations, which are zero
High temperature How does strangeness behave? 1 bound with other partons (eg. sg,sq) 2 do not couple with other partons (quasi-quark) C BS = − 3 � δ B δ S � � δ S 2 �
High temperature How does strangeness behave? 1 bound with other partons (eg. sg,sq) 2 do not couple with other partons (quasi-quark) C BS = − 3 � δ B δ S � � δ S 2 � 6 SB lattice continuum limit 5 4 4 p/T 3 2 HTL NNLO 1 HRG 0 200 300 400 500 T[MeV]
Uses of fluctuations 1 dominant degrees of freedom 2 thermodynamics at finite µ B ( T c and EoS) � O � µ B = � O � + µ 2 � � � O χ B 2 � − � O �� χ B 2 � + � 2 O ′ χ B B 2 · 1 + O ′′ � + . . . 3 compare with experiment → freezout parameters
The transition line from change of chiral-condensate with µ B (only LO) is there an endpoint along this line? higher orders, sign problem
The freezout line extract T , µ B , µ S , µ Q , V of a heavy-ion collision from abundancies Can a hadronic approach be applied at such high temperatures? → Freezout parameters from the lattice
Event-by-event What fluctuates in a heavy-ion collision, if you start with a fixed number of conserved charges (Z=82, A=207)? Consider particles coming only from a small part of the whole system, defined by imposing kinematical constraints. Charges from subvolumes will fluctuate from one event to the other.
Experiment vs. lattice ? � fluct. � exp = � fluct. � latt , T ,µ B ,µ Q ,µ S 1 use 4 fluctuations to determine 4 freezout T , µ B , µ Q , µ S 2 do they originate from thermal & chemical equilibrium ? 3 how is the freezout curve (latt+exp) related to QCD transition line (latt)? is the freezout curve close enough to the QCD endpoint ?
Freezeout parameters [BNL-Bielefeld ’12] How to get freezout T , µ B , µ Q , µ S ? in a lead-lead collision (Z=82, A=207): � Q � = 82 � S � = 0 , 207 � B � can be used to obtain µ Q = µ Q ( T , µ B ) and µ S = µ S ( T , µ B ) choose two fluctuations ( 1. thermometer, 2. baryometer ) and find T and µ B , where � . . . � exp = � . . . � latt , T ,µ B to cancel Vol of the subsystem work with fluctuation ratios
Thermometer experimentally charge is the cleanest � δ Q 3 � / � Q � T f � 157 MeV
Baryometer � Q � / � δ Q 2 � ∼ µ B
Freezout vs. transition line [BNL-Bielefeld,’13] ab-initio determinations of freezout T , µ : STAR electric charge + lattice PHENIX electric charge + lattice STAR proton number + lattice baryon number good agreement with QCD transition line
Wilson/overlap at T > 0
Problems with staggered quarks rooting is not necessarily universal → probably correct finite µ at finite “ a ” is not well-defined (det M ) 1 / 4 = j k λ 1 / 4 � , where j k = { +1 , − 1 , + i , − i } k k chiral limit at finite “ a ” is not well-defined → confusion about n f = 2 order of the transition correlators are just awful ( − 1) t
Check staggered rooting is not necessarily universal → check it by Wilson/overlap check only meaningful in continuum limit for fully renormalized nite quantities for the check only, physical quark masses not essential (heavier quarks also universal)
Wilson setup fixed β approach: change N t with β fixed three pion-masses, strange mass is physical four lattice spacings ( N t = 8 , 12 , 16 , 20 at T c )
Wilson comparison renormalized chiral condensate m R � ¯ � ψ R ψ R � T , 0 = 2 m 2 PCAC Z 2 � P ( x ) P (0) � T , 0 A x agreement between Wilson and staggered also for Polyakov-loop and strange number fluctuation
Overlap setup 2HEX smeared links Zolotarev-approximation project low lying modes by Krylov-Schur fixed topology by determinant suppression w 0 for scale setting m π = 350 MeV N t = 6 , 8 , 10 , 12 N s / N t = 2
Overlap comparison T [MeV] 120 140 160 180 200 220 240 260 0.01 staggered 0 6 × 12 3 8 × 16 3 -0.01 10 × 20 3 12 × 24 3 -0.02 ψψ R /m 4 π -0.03 -0.04 – m R -0.05 -0.06 Preliminary -0.07 -0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 Tw 0 agreement between overlap and staggered
Summary Equation of State → full result Fluctuations → ab-initio freeze-out parameters Wilson/overlap at T > 0 → staggered is OK
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