Critical Temperature and Equation of state from N f = 2 twisted mass lattice QCD

Critical Temperature and Equation of state from N f = 2 twisted mass lattice QCD Florian Burger Humboldt University Berlin for the tmft Collaboration: E. M. Ilgenfritz, M. Müller-Preussker, M. Kirchner (HU Berlin), M. P. Lombardo (INFN Frascati), C. Urbach (Uni Bonn), O. Philipsen, C. Pinke, L. Zeidlewicz (Uni Frankfurt) INFN Frascati November, 26 212 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 1 / 36

1 Introduction I 2 Signals for T c 3 Chiral Limit and Universality Studies 4 Introduction II 5 Thermodynamic Equation of State 6 Conclusions Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 2 / 36

Outline 1 Introduction I 2 Signals for T c 3 Chiral Limit and Universality Studies 4 Introduction II 5 Thermodynamic Equation of State 6 Conclusions Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 3 / 36

Motivation Explore finite T phase transition/crossover for N f = 2 QCD at vanishing chemical potential Order of transition in the chiral limit not known yet for N f = 2 Argued to be in universality class of O(4) 3-dim spin modell [R. D. Pisarski, F. Wilczek, 84] However 1 st order not ruled out. Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 4 / 36

QCD Phase Diagram µ B = The QCD phase diagram at vanishing chemical potential: m tric s O(4)? N f = 2 Z(2) Crossover 1 st pure gauge phys. pt. N f = 2 + 1 m s N f = 3 Nf = 1 1 st Z(2) m ud N f = 2: 2 nd order O(4) in the chiral and continuum limit limit? Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 5 / 36

Path Integral at Finite Temperature Starting point: Grand canonical partition function in continuum (with k B = 1) t i τ Euclidean Path Integral Z(V,T,µ) = Tr (e ) (Ĥ µˆn q)/t = DψD ψda e (Sg[A]+S f [ψ, ψ,a]) with gluon and quark action: S g [A] = S f [ψ, ψ,a] = 1/T 1/T dτ dτ V V d 3 x 1 4 F a µνf a µν d 3 x ψ ( /D + m q µγ ) ψ Have to impose: periodic BC for A, anti-periodic BC for ψ Lattice discretization : 1/T N τ a Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 6 / 36

Twisted Mass Lattice Regularization N f = 2 lattice QCD with Wilson fermions at maximal twist. S f [U,ψ,ψ] = x χ(x) ( 1 κh[u] + 2iκaµγ 5 τ 3) χ(x) Advantage: when κ tuned to critical value κ c : Automatic O(a) improvement [R. Frezzotti, G.C. Rossi, 24] Disadvantage: explicit flavor symmetry breaking (mostly small, but has to be checked) Tree level improved gauge sector: S g [U] = β (c [1 1 3 ReTr(U P)] + c 1 [1 1 ) 3 ReTr (U R)] P R Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 7 / 36

Phase Space Phase Diagram: κ confinement? Doubler region thermal transition/crossover surface deconfinement µ confinement κ c (β, T = ) β β cusp β qu Aoki phase bulk transition quenched limit [Phys.Rev.D8:9452, 29] Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 8 / 36

Current and Future Simulations Z A B C D N τ = 6 T c 8 1 12 28 32 4 48 7 m π [MeV] Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 9 / 36

Studied Observables Polyakov loop susceptibility (no convincing signal).9.85.8.75.7.65.6 χ Re(L).55 3.85 3.9 3.95 4. 4.5 β 4.1 4.15 4.2 Disconnected chiral susceptibility: σ ψψ = V /T ( ( ψψ) 2 ψψ 2) Renormalized Polyakov loop and chiral condensate Equation of State ( second part) Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 11 / 36

Susceptibility of ψψ m π 28 MeV: 1.5 1.3 1.1.9.7.5.3.1 3.8 σ 2 ψψ Z12 Gaussian fit 3.83 3.86 3.89 3.92 β m π 4 MeV:.5.4.3.2.1 σ 2 ψψ 3.84 3.9 B12 Gaussian fit 3.96 β 4.2 m π 32 MeV: A12 Gaussian fit Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 12 / 36.8.7.6.5.4.3.2 3.82 σ 2 ψψ 3.88 m π 48 MeV:.4.35.3.25.2.15.1.5 3.88 σ 2 ψψ 3.94 β 3.94 C12 Gaussian fit 4. β 4.6 4.

Comparison with DIK-Collaboration (G. Schierholz et. al) from σ ψψ :.75.7.65.6.55.5.45.4.35.3.2 r T c.4 DIK: arxiv:91.2392 DIK: arxiv:112.4461 our data (tmft).6.8 1 1.2 1.4 1.6 1.8 r m π Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 13 / 36

Renormalized Re(L), ψψ ψψ R = ψψ (T,µ) ψψ (,µ)+ ψψ (,) ψψ (,) 1.8 N τ = 12 N τ = 1 <ψ ψ> R.6.4 Re(L) R = Re(L)exp (V (r )/2T).2 18 2 22 24 26 28 T [MeV] <Re(L)> R 2 1.5 1.5 N τ = 12 N τ = 1 18 22 26 3 34 38 42 T [MeV] Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 14 / 36

Renormalized Re(L) Fit to Re(L) R = Re(L)exp (V (r )/2T): 1.8 1.6 1.4 1.2 1.8.6.4.2 2 Re(L) R Fit B12 24 28 32 T [MeV] 36 Ensemble A12 B12 C12 D8 β c from σ 2 ψψ : 3.89(3) 3.93(2) 3.97(3) 3.78(1) T c[mev] from σ 2 ψψ : 22(7) 217(5) 229(5) 251(1) T c[mev] from Re(L) R : - 249(5) 258(5) 266(11) Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 15 / 36

O(4) Magnetic EoS Near critical point of 2 nd order transition: universality Expect data to collaps on universal curve ψψ h 1/δ c f (d z) + a τ τh + b 1 h +... }{{} scaling violating terms τ = β β chiral, h = 2 a µ, z = τ/h 1/( βδ) (J. Engels, T. Mendes: [hep-lat/991128]) Connection to spin model: ψψ M µ H β T Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 17 / 36

O(4) Magnetic EoS ψψ /h 1/δ.13.12.11.1.9.8.7.6.5 A12 B12 C12.4.5 1 1.5 2 2.5 3 3.5 z Result: β chiral 3.76(2) T c (m π = ) 166 MeV 4 ψψ /h 1/δ - (scaling corr.).35.3.25.2.15 Pion Mass: C12: 48 MeV B12: 4 MeV A12: 32 MeV.1.5 1 1.5 A12 corr. B12 corr. C12 corr. 2 z 2.5 3 3.5 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 18 / 36

Chiral Limit Scenarios χ = M h = h1/δ 1 ( 1 δ f (z) f (z)z ) h 1/δ 1 F(z) 1 χ h 1/δ 1 τ = 1 F (z) =! z B h 1/( βδ) 26 24 T c (m π ) = T c () + Am 2/(βδ) π z = τ/h 1/( βδ) β c (h) = β chiral + B h 1/( βδ) 22 T c (MeV) 2 18 1 st order Z(2) m π,c = 18 MeV 16 14 m π,c = MeV O(4) 1 2 3 4 5 m π (MeV) O(4): T c () = 152(26) MeV Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 19 / 36

Heavy Ion Collisions Quark Gluon Plasma studied experimentally at RHIC and LHC Description includes relativistic hydrodynamics ǫ = (ǫ + p)( u) ṅ B = n B ( u) u µ = µ p ǫ + p plus :ǫ(p) ǫ(p) accessible on the lattice (at n B ) currently at N f = 2, should be extended by going to N f = 2 + 1 + 1 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 21 / 36

Motivation Differences in the results for the thermodynamic equation of state from different staggered simulations resolved now? Other discretizations of QCD worthwhile to study systematics and universality Useful to study onset of mass thresholds in thermodynamics Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 22 / 36

Trace Anomaly I T 4 = ǫ 3p T 4 = T = N 4 τ ( a dβ da + κ c β χh[u]χ sub d lnz d lna sub T 4 V ) ( c 3 ReTrU P sub + c 1 ( 2aµ κ c β + 2κ c 3 ReTrU R sub (aµ) β ) χiγ5 τ 3 χ sub ) Starting point for p(t) and ǫ(t) by integral method Subtracted expectation values interpolations for T = data preliminary results for m π 4 MeV and m π 7 MeV Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 24 / 36

T = Subtractions & Interpolations example: plaquette:.64 ReTr U P.6.56 β = 3.7, aµ =.9, m π 4 MeV 3 Residuals of fit -3 3.7 3.9 β 4.1 4.3.56.558.556 beta=3.7, 16^3x24 plaquette.554.552.55 1 2 3 4 5 6 7 8 9 No traj. Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 25 / 36

β-function, Scale Setting II [M. Cheng et al.: Phys.Rev. D77:14511, 28] ` dβ rχ `a da = a «d( rχ 1 a ) dβ ` rχ a (β) = 1+n R(β) 2 d (a 2L (β)+d 1 R(β) 2 ) R(β) = a 2L(β) a 2L (3.9) r =.42(15) fm 14 12 1 8 r χ /a r /a Interpolation -.1 -.2 -.3 -.4 -.5 a dβ da 2-loop 6 -.6 -.7 4 3.65 3.8 3.9 4.5 4.2 β 4.35 4.6 -.8 -.9 3.65 3.8 3.9 4.5 4.2 β 4.35 4.6 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 26 / 36

Trace anomaly - Lattice Artifacts m π 4 MeV: 12 1 8 6 4 2-2 1 ǫ 3p T 4 3 N τ = 12 N τ = 1 N τ = 8 N τ = 6 N τ = 4 5 7 T [MeV] 9 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 27 / 36

Trace Anomaly, Tree Level Corrections Observe large lattice artifacts in I T 4 Leading lattice corrections for slide) p L p SB Twisted mass fermions (next [P. Hegde et al. Eur.Phys.J. C55 (28)] [O. Philipsen, L. Zeidlewicz (21)] Corrected by division by psb L /pc SB [S. Borsanyi: JHEP, 111:77, 21] m π 7 MeV: 8 7 6 5 4 3 2 1 ǫ 3p T 4 uncorrected corrected T = 282 MeV.5.1.15.2.25.3 1/Nτ 2 8 7 6 5 4 3 2 1 ǫ 3p T 4 uncorrected corrected T = 318 MeV.5.1.15.2.25.3 1/Nτ 2 8 7 6 5 4 3 2 1 ǫ 3p T 4 uncorrected corrected T = 392 MeV.5.1.15.2.25.3 1/Nτ 2 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 28 / 36

Tree Level Corrections II, SB (Free) Limit Lattice: Continuum: Correction: pf L(µ) T 4 = 3 [,2π) 3 G(k) = (am) + 2r µ g(µ/t) = 36 7π 4 d 3 N k 1 t 1 (2π) 3 N t n= sin 2 ( akµ 2 pf C(µ/T) T 4 = 2 7 8 µ/t dx x lndet ( G(k) 2 + (aµ) 2) ) + ı µ π 2 9 g(µ/t) γ µ sin(ak µ ) x 2 (µ/t) 2 ln(1 + e x ) N τ 4 6 8 1 12 p L SB /pc SB 2.586 1.634 1.265 1.134 1.84 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 29 / 36.

Tree Level Corrections III uncorrected m π 4 MeV: 12 1 8 6 4 2-2 1 ǫ 3p T 4 3 m π 7 MeV: 12 1 8 6 4 2 2 ǫ 3p T 4 4 N τ = 12 N τ = 1 N τ = 8 N τ = 6 N τ = 4 5 7 T [MeV] N τ = 1 N τ = 8 N τ = 6 N τ = 4 6 8 T [MeV] 9 1 corrected N τ = 12 N τ = 1 N τ = 8 N τ = 6 N τ = 4 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 3 / 36 12 1 8 6 4 2-2 12 1 8 6 4 2 2 ǫ 3p T 4 ǫ 3p T 4 3 4 5 T [MeV] N τ = 1 N τ = 8 N τ = 6 N τ = 4 6 8 T [MeV] 7 1

Interpolation of I/T 4, T integration (preliminary results) I T 4 = T ( p ) T T 4 p T 4 p T 4 = T dτ ǫ 3p T τ 5 LCP Using interpolation for corrected I/T 4 : m π 7MeV : 1 8 6 4 2-2 1 ǫ 3p T 4 3 I T 4 = exp ( h 1 t h 2 t 2) Interpolation N τ = 1 N τ = 8 N τ = 6 5 7 T [MeV] ( ) h + f {tanh f 1 t+f 2 } 1+g 1 t+g 2 t 2 [S. Borsanyi: JHEP, 111:77, 21] 9 m π 4MeV : Interpolation N τ = 12 N τ = 1 N τ = 8 N τ = 6 N τ = 4 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 31 / 36 1 8 6 4 2-2 1 ǫ 3p T 4 3 5 7 T [MeV] 9

Lines of Constant Physics, constant m π m π 7 MeV: presently fulfilled up to 1 % 9 85 8 75 7 65 6 55 mps [MeV] D8 5 m π 4 MeV: 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 β 48 46 44 42 4 38 36 34 mps [MeV] ETMC tmft B mass 3.7 3.8 3.9 4.5 4.2 β Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 32 / 36

Pressure (preliminary results) m π 7 MeV : 16 12 8 4 1 1.5 2 2.5 3 D mass (ǫ 3p)/T 4 ǫ/t 4 3p/T 4 3p SB T Tc 2 3 4 5 6 7 8 T [MeV] m π 4 MeV : 16 12 8 4 1 1.5 2 2.5 B mass (ǫ 3p)/T 4 ǫ/t 4 3p/T 4 3p SB T Tc 2 3 4 5 6 T [MeV] Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 33 / 36

Conclusions & Outlook Conclusions: - T c for pion masses in the range 27-7 MeV - Chiral limit so far inconclusive - O(4) scaling compatible up to scaling violation terms - Thermodynamic Equation of State presented for two pion masses - Have to further increase statistics at T > and T = Outlook: - N f = 2 + 1 + 1 Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 35 / 36

Thank you Florian Burger (HU Berlin) Thermodynamics with twisted mass fermions October 212 36 / 36