The QCD equation of state at high temperatures

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1 The QCD equation of state at high temperatures Alexei Bazavov (in collaboration with P. Petreczky, J. Weber et al.) Michigan State University Feb 1, 2017 A. Bazavov (MSU) GHP2017 Feb 1, / 16

2 Introduction Earlier results on the equation of state Lattice QCD setup Trace anomaly Results Conclusion A. Bazavov (MSU) GHP2017 Feb 1, / 16

3 QCD phase diagram 1 Collins, Perry (1975), Cabbibo, Parisi (1975) A. Bazavov (MSU) GHP2017 Feb 1, / 16

4 QCD phase diagram Study response of the system to change of external parameters, i.e. temperature and baryon density, asymptotic freedom suggests a weakly interacting phase 1 1 Collins, Perry (1975), Cabbibo, Parisi (1975) A. Bazavov (MSU) GHP2017 Feb 1, / 16

5 QCD phase diagram Study response of the system to change of external parameters, i.e. temperature and baryon density, asymptotic freedom suggests a weakly interacting phase 1 Experimental program: RHIC, LHC, FAIR, NICA 1 Collins, Perry (1975), Cabbibo, Parisi (1975) A. Bazavov (MSU) GHP2017 Feb 1, / 16

6 QCD phase diagram Study response of the system to change of external parameters, i.e. temperature and baryon density, asymptotic freedom suggests a weakly interacting phase 1 Experimental program: RHIC, LHC, FAIR, NICA High-temperature phase: deconfinement, restoration of chiral symmetry 1 Collins, Perry (1975), Cabbibo, Parisi (1975) A. Bazavov (MSU) GHP2017 Feb 1, / 16

7 QCD phase diagram Study response of the system to change of external parameters, i.e. temperature and baryon density, asymptotic freedom suggests a weakly interacting phase 1 Experimental program: RHIC, LHC, FAIR, NICA High-temperature phase: deconfinement, restoration of chiral symmetry QCD equation of state at zero baryon density has been recently calculated up to T = 400 MeV 1 Collins, Perry (1975), Cabbibo, Parisi (1975) A. Bazavov (MSU) GHP2017 Feb 1, / 16

8 Earlier results on the equation of state First perturbative EoS calculation 2 (left) First lattice pure gauge SU(2) EoS calculation 3 (right) 2 Kapusta (1979) 3 Engels et al. (1981) A. Bazavov (MSU) GHP2017 Feb 1, / 16

9 Recent results up to T = 400 MeV stout HISQ (ε-3p)/t 4 p/t 4 s/4t 4 T [MeV] Comparison of the continuum results with HISQ 4 and stout 5 for the trace anomaly, pressure and entropy density About 2σ deviations in the integrated quantities at the highest temperature 4 Bazavov et al. [HotQCD] (2014) 5 Borsanyi et al. [WB] (2014) A. Bazavov (MSU) GHP2017 Feb 1, / 16

10 Approach to the perturbative limit 3 2 ε/p-3 HISQ stout O(g 6 ) EQCD 3-loop HTL 1 T [MeV] The ratio of the trace anomaly and the pressure Θ µµ p = ɛ p 3 compared with perturbative calculations in the Hard Thermal Loop (HTL) 6 and Electrostatic QCD (EQCD) 7 schemes 6 Haque et al. (2014) 7 Laine and Schroder (2006) A. Bazavov (MSU) GHP2017 Feb 1, / 16

11 Approach to the perturbative limit 4 (ε-3p)/t 4 HISQ O(g 6 ) EQCD 3-loop HTL p/p ideal HISQ O(g 6 ) EQCD 3-loop HTL T [MeV] T [MeV] The trace anomaly (left) and pressure (right) compared with HTL and EQCD calculations The black line is the HTL calculation with the renormalization scale µ = 2πT A. Bazavov (MSU) GHP2017 Feb 1, / 16

12 Approach to the perturbative limit 4 (ε-3p)/t 4 HISQ O(g 6 ) EQCD 3-loop HTL p/p ideal HISQ O(g 6 ) EQCD 3-loop HTL T [MeV] T [MeV] The trace anomaly (left) and pressure (right) compared with HTL and EQCD calculations The black line is the HTL calculation with the renormalization scale µ = 2πT Need to extend the lattice equation of state to higher temperature - THIS TALK A. Bazavov (MSU) GHP2017 Feb 1, / 16

13 Lattice QCD Switch from Minkowski to Euclidean space imaginary time formalism A. Bazavov (MSU) GHP2017 Feb 1, / 16

14 Lattice QCD Switch from Minkowski to Euclidean space imaginary time formalism Define the theory on discrete space-time lattice N 3 s N τ A. Bazavov (MSU) GHP2017 Feb 1, / 16

15 Lattice QCD Switch from Minkowski to Euclidean space imaginary time formalism Define the theory on discrete space-time lattice N 3 s N τ This is a gauge-invariant regularization scheme with the momentum cut-off π/a, a lattice spacing A. Bazavov (MSU) GHP2017 Feb 1, / 16

16 Lattice QCD Temperature is set as T = 1/(aN τ ) Switch from Minkowski to Euclidean space imaginary time formalism Define the theory on discrete space-time lattice N 3 s N τ This is a gauge-invariant regularization scheme with the momentum cut-off π/a, a lattice spacing A. Bazavov (MSU) GHP2017 Feb 1, / 16

17 Lattice QCD Temperature is set as T = 1/(aN τ ) Switch from Minkowski to Euclidean space imaginary time formalism Define the theory on discrete space-time lattice N 3 s N τ This is a gauge-invariant regularization scheme with the momentum cut-off π/a, a lattice spacing Fix N τ, dial the lattice spacing to cover a temperature range A. Bazavov (MSU) GHP2017 Feb 1, / 16

18 Lattice QCD Temperature is set as T = 1/(aN τ ) Switch from Minkowski to Euclidean space imaginary time formalism Define the theory on discrete space-time lattice N 3 s N τ This is a gauge-invariant regularization scheme with the momentum cut-off π/a, a lattice spacing Fix N τ, dial the lattice spacing to cover a temperature range The continuum limit is reached as 1/N τ 0 A. Bazavov (MSU) GHP2017 Feb 1, / 16

19 Static quark potential and setting the scale r 1 V(r) β=6.740 β=6.880 β=6.950 β=7.030 β=7.150 β=7.280 β=7.373 β=7.596 β=7.825 r/r Fit the static quark potential to the form: V (r) = C + B r + σr Define an interpolating quantity, r 1 : r 2 dv dr = 1 r1 A. Bazavov (MSU) GHP2017 Feb 1, / 16

20 Static quark potential and setting the scale r 1 V(r) β=6.740 β=6.880 β=6.950 β=7.030 β=7.150 β=7.280 β=7.373 β=7.596 β=7.825 r/r Fit the static quark potential to the form: V (r) = C + B r + σr Define an interpolating quantity, r 1 : r 2 dv dr = 1 r1 The physical value r 1 = (14)(8)(4) fm Measuring r 1 /a allows one to define the lattice spacing Other choices of scale setting are, of course, possible, e.g. f K, m Ω, w 0, etc. A. Bazavov (MSU) GHP2017 Feb 1, / 16

21 Setting the scale spline interpolation data global fit R β spline interpolation global fit 2-loop a/r 1 /f(β) β β a = c 0f (β) + c 2 (10/β)f 3 (β) r d 2 (10/β)f 2 (β) ( d(r1 /a) R β = a dβ da = r 1 a dβ ( ) b1 /(2b 10b0 0 2), f (β) = exp( β/(20b 0 ) β ) 1, R 2 loop β = 20b b 1 /β A. Bazavov (MSU) GHP2017 Feb 1, / 16

22 Trace anomaly The partition function Z = DUD ψdψ exp{ S}, S = S g + S f A. Bazavov (MSU) GHP2017 Feb 1, / 16

23 Trace anomaly The partition function Z = DUD ψdψ exp{ S}, S = S g + S f The trace anomaly Θ µµ ε 3p = T V d ln Z d ln a p T 4 p 0 T 4 0 = T T 0 dt ε 3p T 5 A. Bazavov (MSU) GHP2017 Feb 1, / 16

24 Trace anomaly The partition function Z = DUD ψdψ exp{ S}, S = S g + S f The trace anomaly Θ µµ ε 3p = T V d ln Z d ln a p T 4 p 0 T 4 0 = T T 0 dt ε 3p T 5 Requires subtraction of UV divergences (subtract divergent vacuum contribution evaluated at the same values of the gauge coupling): ε 3p T 4 = R β [ S G 0 S G T ] R β R m [2m l ( ll 0 ll T ) + m s ( ss 0 ss T )] R β (β) = a dβ da, R m(β) = 1 m dm dβ, β = 10 g 2 A. Bazavov (MSU) GHP2017 Feb 1, / 16

25 HISQ data sets We use the Highly Improved Staggered Quarks 8 action for two degenerate light quarks and physical-mass strange quark and the tree-level Symanzik-improved gauge action Previous data set: m l = m s /20 N τ = 6, 8, 10, 12 β = 5.9,..., Follana et al. [HPQCD] (2007) A. Bazavov (MSU) GHP2017 Feb 1, / 16

26 HISQ data sets We use the Highly Improved Staggered Quarks 8 action for two degenerate light quarks and physical-mass strange quark and the tree-level Symanzik-improved gauge action Previous data set: New data set: m l = m s /20 N τ = 6, 8, 10, 12 β = 5.9,..., m l = m s /5 N τ = 8, 10, 12 β = 8, 8.2, Follana et al. [HPQCD] (2007) A. Bazavov (MSU) GHP2017 Feb 1, / 16

27 Results: trace anomaly 5 4 N τ =8 N τ =10 N τ =12 (e-3p)/t T [MeV] The trace anomaly with HISQ m l = m s /20 and m l = m s /5 at T > 400MeV A. Bazavov (MSU) GHP2017 Feb 1, / 16

28 Results: trace anomaly 5 4 N τ =8 N τ =10 N τ =12 p4, N τ =6 (e-3p)/t T [MeV] The trace anomaly with HISQ m l = m s /20 and m l = m s /5 at T > 400MeV A. Bazavov (MSU) GHP2017 Feb 1, / 16

29 Results: pressure p/t N τ =6 N τ =8 N τ =10 1 p4, N τ =6 p4, N τ =8 stout cont T [MeV] Pressure with HISQ at N τ = 6, 8 and 10. Continuum stout result and p4 at N τ = 6 and 8 are shown for comparison. The cutoff effects with HISQ are consistent with those of free theory. A. Bazavov (MSU) GHP2017 Feb 1, / 16

30 Conclusion Previous result by the HotQCD collaboration for the 2+1 QCD equation of state at zero baryon chemical potential is being extended to higher temperatures At temperatures above 400 MeV we use ensembles with m l = m s /5 Quark mass (in)dependence at high temperatures needs to be quantified More statistics is required for N τ = 12 ensembles to do the continuum extrapolation The continuum limit at high temperature may be somewhat above the stout result (as earlier results also indicate) A. Bazavov (MSU) GHP2017 Feb 1, / 16