Equation of state from N f = twisted mass lattice QCD Florian Burger Humboldt University Berlin for the tmft Collaboration: E. M. Ilgenfritz, M. Kirchner, M. Müller-Preussker (HU Berlin), M. P. Lombardo (INFN Frascati), C. Urbach (Uni Bonn), O. Philipsen, C. Pinke (Uni Frankfurt) Lattice 1 June, 5 1 Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 1 /
1 Introduction T c and chiral limit 3 Thermodynamic Equation of State Outlook Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 /
Outline 1 Introduction T c and chiral limit 3 Thermodynamic Equation of State Outlook Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 3 /
Motivation Explore finite T phase transition/crossover for N f = QCD at vanishing chemical potential Order of transition in the chiral limit not known yet for N f = Differences in the results for the thermodynamic equation of state from different staggered simulations Other discretizations of QCD worthwhile to study systematics and universality Useful to study onset of mass thresholds in thermodynamics Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 /
Lattice Setup N f = lattice QCD with Wilson fermions at maximal twist. S f [U,ψ,ψ] = x χ(x) ( 1 κh[u]+iκaµγ 5 τ 3) χ(x) 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 κ tuned to critical value κ c : Automatic O(a) improvement β-scans along κ c (β) with fixed m π :.175.17.165.16.155 κc ETMC Pade.15.15 g 6.35.15 3.65 3.83.9.5..35 β = 6/g Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 5 /
Outline 1 Introduction T c and chiral limit 3 Thermodynamic Equation of State Outlook Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 6 /
Suszeptibility of ψψ σ ψψ = V/T ( ( ψψ) ψψ ) m π 3 MeV:.6e-.e- 3.e- σ ψψ A1 Gaussian fit m π MeV:.8e-.e-.e- 1.6e- 1.e- 8.e-5.e-5 σ ψψ B1 Gaussian fit.8e-.e- 1.6e- 1.e- 3.8 3.88 m π 8 MeV:.e-.e- 1.6e- 1.e- 8.e-5.e-5 σ ψψ β 3.9 C1 Gaussian fit. 3.8 3.9 3.96. 3.88 3.9..6 β β Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 7 /
Tc(m π ) Comparison with DIK-Collaboration (G. Schierholz et. al) from σ ψψ :.75.7.65.6.55.5.5 r T c DIK: arxiv:91.39 DIK: arxiv:11.61 our data (tmft) T c (m π ) = T c ()+Am /(βδ) π O(): T c () = 15(6) MeV 6..35.3...6.8 1 1. 1. 1.6 1.8 r m π T c (MeV) 18 1 st order Z() m π,c = MeV 16 m π,c = MeV O() 1 1 3 5 m π (MeV) Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 8 /
Renormalized Re(L), ψψ ψψ R = ψψ (T,µ) ψψ (,µ)+ ψψ (,) ψψ (,) 1.8 N τ = 1 N τ = 1 <ψ ψ> R.6. Re(L) R = Re(L)exp(V(r )/T). 18 6 8 T [MeV] <Re(L)> R 1.5 1.5 N τ = 1 N τ = 1 18 6 3 3 38 T [MeV] Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 9 /
Outline 1 Introduction T c and chiral limit 3 Thermodynamic Equation of State Outlook Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 1 /
Trace Anomaly I = ǫ 3p = d lnz d lna sub = T V ( da)( a dβ c ReTrU P +c sub 1 ReTrU R sub + κ c β χh[u]χ sub ( aµ κ c β +κ c ) (aµ) χiγ5 τ 3 χ β sub) Starting point for p(t) and ǫ(t) by integral method subtracted expectation values interpolations for T = data preliminary results for m π MeV and m π 7 MeV Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 11 /
T = interpolations, β-function example: plaquette interpolation: ReTr UP.6.6.56 3 Residuals of fit -3 3.7 3.9.1.3 β ( a dβ da ( rχ a ) = ( r χ a ) ( ) d( rχ 1 a ) dβ ) (β) = 1+n R(β) d (a L (β)+d 1 R(β) ) R(β) = a L(β) a L (3.9) [M. Cheng et al.: Phys.Rev. D77:1511, 8] -.1 -. -.3 -. -.5 -.6 -.7 -.8 -.9 3.65 a dβ da 3.8 3.9.5. β.35 -loop.6 Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 1 /
Lines of Constant Physics, constant m π m π 7 MeV: presently fulfilled up to 1 % 9 D8 85 mps [MeV] 8 75 7 65 6 55 5 m π MeV: 3.6 3.7 3.8 3.9.1..3..5 β mps [MeV] B1 38 36 3.8 3.9 β.5. Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 13 /
Trace anomaly, tree level corrections Observe large lattice artifacts in I Leading lattice corrections for p L p SB twisted mass fermions [P. Hegde et al. Eur.Phys.J. C55 (8)] [O. Philipsen, L. Zeidlewicz (1)] Corrected by division by p L p SB [S. Borsanyi: JHEP, 111:77, 1] m π 7 MeV: 8 7 6 5 3 1 ǫ 3p uncorrected corrected T = 8 MeV.5.1.15..5.3 1/Nτ 8 7 6 5 3 1 ǫ 3p uncorrected corrected T = 318 MeV.5.1.15..5.3 1/Nτ 8 7 6 5 3 1 ǫ 3p uncorrected corrected T = 39 MeV.5.1.15..5.3 1/Nτ Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 1 /
Tree level corrections II uncorrected m π MeV: 1 1 8 6 ǫ 3p 3 m π 7 MeV: 8 7 6 5 3 1 ǫ 3p 3 N τ = 1 N τ = 1 N τ = 8 N τ = 6 N τ = 5 6 T [MeV] N τ = 1 N τ = 8 N τ = 6 5 6 T [MeV] 7 7 8 8 corrected N τ = 1 N τ = 1 N τ = 8 N τ = 6 N τ = Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 15 / 1 8 6 6 5 3 1 ǫ 3p ǫ 3p 3 5 6 T [MeV] N τ = 1 N τ = 8 N τ = 6 N τ = 7 8-1 3 5 6 7 8 9 1 T [MeV]
Interpolation of I/, T integration (preliminary results) p p = T dτ ǫ 3p T τ 5 LCP Using interpolation for uncorrected I/ : I = exp ( h 1 t h t ) m π 7MeV : 8 6 ǫ 3p 3 ( h + f {tanhf 1 t+f } 1+g 1 t+g t ), t = 3MeV [S. Borsanyi: JHEP, 111:77, 1] Interpolation N τ = 1 N τ = 8 N τ = 6 5 6 T [MeV] 7 8 m π MeV : Interpolation N τ = 1 N τ = 1 N τ = 8 N τ = 6 N τ = Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 16 / 1 1 8 6 ǫ 3p 3 5 6 T [MeV] 7 8
Pressure (preliminary results) m π 7MeV : 5 3 p N τ = 8 p SB 1 3 preliminary m π MeV : 5 6 T [MeV] 7 8 5 3 p m π MeV p SB 1 3 T [MeV] preliminary 5 Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 17 /
Outline 1 Introduction T c and chiral limit 3 Thermodynamic Equation of State Outlook Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 18 /
Conclusions & Outlook Conclusions: - T c for pion masses in the range 3-5 MeV - Thermodynamic Equation of State presented for two pion masses - Improvement on T = interpolations and LOC on the way Outlook: - N f = +1+1 Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 19 /
Thank you Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 /
Phase Space Phase Diagram: κ confinement? Doubler region thermal transition/crossover surface deconfinement µ confinement β cusp β qu Aoki phase bulk transition quenched limit κ c (β, T = ) β [Phys.Rev.D8:95, 9] Florian Burger (HU Berlin) EoS with twisted mass fermions June 1 1 /