he XXVI International Symposium on Lattice Field heory (Lattice 28), Williamsburg, Virginia, USA, 28 July 14 19. Phase diagram and EoS from a aylor expansion of the pressure Christian Schmidt Universität Bielefeld RBC-Bielefeld: M. Cheng, N.H. Christ, S.Datta, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann, R.D. Mawhinney, C. Miao, P. Petreczky, K. Petrov, CS, W. Söldner and. Umeda. See poster by C. Miao
he XXVI International Symposium on Lattice Field heory (Lattice 28), Williamsburg, Virginia, USA, 28 July 14 19. Outline Introduction aylor expansion of the pressure he radius of convergence and the QCD critical point he isentropic equation of state Summary
Introduction 3 he phase diagram of QCD Fluctuations of B, S, Q can be measured experimentally and indicate criticality Lattice at µ = RHIC, LHC LHC RHIC cooling of the fireball: 19MeV hadron gas confined, χ-broken FAIR / low RHIC quark-gluon plasma deconfined, χ-symmetric colorsuperconductor Lattice at µ > RHIC at low energies, FAIR@GSI few times nuclear matter density µ B µ S = µ Q =
aylor expansion of the pressure 4 aylor expansion in µ B,S,Q QCD is naturally formulated with quark chemical potentials µ u,d,s we start from aylor expansion of the pressure p 4 = 1 V 3 ln Z(V,, µ u, µ d, µ s ) = i,j,k c u,d,s i,j,k ( µu ) i ( µd ) j ( µs ) k use unbiased, noisy estimators to calculate see C. Miao, CS, PoS (Lattice 27) 175. line of constant physics: c u,d,s i,j,k m q = m s /1 (physical strange quark mass) measure currently up to O(µ 8 ) (N t = 4) action: improved staggered (p4fat3) O(µ 4 ) (N t = 6)
aylor expansion of the pressure 5 aylor expansion in µ B,S,Q QCD is naturally formulated with quark chemical potentials µ u,d,s we start from aylor expansion of the pressure p 4 = 1 V 3 ln Z(V,, µ u, µ d, µ s ) = i,j,k c u,d,s i,j,k ( µu ) i ( µd ) j ( µs ) k expansion coefficients n B = (p/ 4 ) (µ B / ) = 1 3 (n u + n d + n s ) n S = (p/ 4 ) (µ S / ) = n s n Q = (p/ 4 ) (µ Q / ) = 1 3 (2n u n d n s ) choice of c u,d,s i,j,k µ u µ d is equivalent to µ Q are related to B,S,Q-fluctuations µ u = 1 3 µ B + 2 3 µ Q µ d = 1 3 µ B 1 3 µ Q µ s = 1 3 µ B 1 3 µ Q µ S
aylor expansion of the pressure 6 Current statistics N τ = 4 N τ = 6 β #Conf. #Sep. #Ran. β #Conf. #Sep. #Ran. 3.24 113 2 48 3.41 8 1 4 3.28 155 3 48 3.42 888 1 4 3.29 155 3 48 3.43 85 1 4 3.3 125 3 384 3.445 924 1 4 3.315 475 6 384 3.455 672 1 35 3.32 475 6 384 3.46 6 1 2 3.335 264 6 384 3.47 571 1 15 3.351 365 3 384 3.49 45 1 15 3.41 199 6 192 3.51 67 1 1 3.46 32 6 96 3.57 54 1 5 3.61 618 1 48 3.69 35 1 5 3.76 345 1 5 work in progress!
aylor expansion of the pressure 7 Results for expansion coefficients.6.5.4 c u 2 SB.12.1.8 c u,d,s i,j,k c u 4 n f =2+1, m! =22 MeV n f =2, m! =77 MeV filled: n t =4 open: n t =6.3.2.1.25.2.15.1.5 -.5 -.1 -.15 -.2 n f =2+1, m! =22 MeV n f =2, m! =77 MeV filled: n t =4 [MeV ] [MeV ] open: n t =6 Preliminary 15 2 25 3 35 4 45 c u 6 n f =2+1, m! =22 MeV n f =2, m! =77 MeV Preliminary -.25 15 2 25 3 35 4 45 SB.6.4.2 [MeV ] Preliminary SB 15 2 25 3 35 4 45 Cut-off dependance: Small cut-off effects in the transition region (similar to p, e-3p,...) Mass dependance: c decreases with decreasing mass Fluctuations increase with decreasing mass red: RBC-Bielefeld, preliminary blue: PRD71:5458,25.
Hadronic fluctuations and the QCD critical point 8 Baryon number fluctuations 1.2 1 (µ B = ) 2c B 2 = B 2 24c B 4 = B 4 3 B 2 2 12 1 SB.8 8 n f =2+1, m! =22 MeV.6 n f =2+1, m! =22 MeV 6 n f =2, m! =77 MeV filled: nt=4.4 n f =2, m! =77 MeV 4 open: nt=6.2 [MeV ] filled: nt=4 open: nt=6 15 2 25 3 35 4 45 2 [MeV ] 15 2 25 3 35 4 45 SB 2 1.5 1.5 12 cb 4 c B 2 [MeV ] n f =2+1, m! =22 MeV = n f =2, m! =77 MeV Resonance gas B 4 3 B 2 2 B 2 filled: nt=4 open: nt=6 15 2 25 3 35 SB fluctuations increase with decreasing mass fluctuations increase over the resonance gas value red: RBC-Bielefeld, preliminary blue: PRD71:5458,25.
Hadronic fluctuations and the QCD critical point 9 Consequences for the phase diagram: the radius of convergence he radius of convergence can be estimated from the aylor coefficients of the pressure: with ρ = lim n ρ n c B n generic picture ρ 2 ρ 4 ρ 6 root of c B 6 root of c B 8 ρ n = c B n+2 for > c, ρ n for < c, ρ n is bound by the transition line he Resonance gas limit: p = G( ) + F ( ) cosh 4 µ B ( µb ρ n = 1/(n + 2)(n + 1) ) look for non-monotonic behavior in the radius of convergence
Hadronic fluctuations and the QCD critical point 1 Consequences for the phase diagram: the radius of convergence non monotonic behavior in the radius of convergence? 1.2 1.15 / c () N t = 4 n f =2+1, m " =22 MeV n f =2, m " =77 MeV ρ 2 N τ = 4 N τ = 6 1.1 m π 22 MeV m π 77 MeV Yes No 1.5 1.95 ρ 4 N τ = 4 N τ = 6.9.85 m π 22 MeV m π 77 MeV.8.75! 2! 4 µ B / c () 1 2 3 4 5 6 first hint for a critical region at small masses? higher order approximations are needed to locate the critical point
Hadronic fluctuations and the QCD critical point 11 Consequences for the phase diagram: the radius of convergence non monotonic behavior in the radius of convergence? ρ 2 N τ = 4 N τ = 6 m π 22 MeV m π 77 MeV Yes? No 1.2 1.15 1.1 1.5 1.95 / c () m π 22 MeV N t =4 N t =6 ρ 4 N τ = 4 N τ = 6.9.85 m π 22 MeV m π 77 MeV.8.75! 2! 4 µ B / c () 1 2 3 4 5 6 first hint for a critical region higher order approximations are at small masses? (only at Nt=4?) needed to locate the critical point
Hadronic fluctuations and the QCD critical point 12 Consequences for the phase diagram: the radius of convergence non monotonic behavior in the radius of convergence? ρ 2 N τ = 4 N τ = 6 m π 22 MeV m π 77 MeV Yes? No 1.2 1.15 1.1 1.5 1.95 / c () N t = 4 r 2 r 4 ρ 4 m π 22 MeV m π 77 MeV N τ = 4 N τ = 6 Yes.9.85.8.75! 2! 4 µ B / c () 1 2 3 4 5 6 first hint for a critical region higher order approximations are at small masses? (only at Nt=4?) needed to locate the critical point ρ 4 (and maybe ρ 6 ) are needed in higher precision
Hadronic fluctuations and the QCD critical point 13 Consequences for the phase diagram: the radius of convergence non monotonic behavior in the radius of convergence? ρ 2 N τ = 4 N τ = 6 m π 22 MeV m π 77 MeV Yes? No 1.2 1.15 1.1 1.5 1.95 / c () Gavai, Gupta Nt=4 Nt=6 N t = 4 Fodor, Katz Nt=4 r 2 r 4 ρ 4 m π 22 MeV m π 77 MeV N τ = 4 N τ = 6 Yes.9.85.8.75! 2! 4 µ B / c () 1 2 3 4 5 6 first hint for a critical region higher order approximations are at small masses? (only at Nt=4?) needed to locate the critical point. ρ 4 (and maybe ρ 6 ) are needed in higher precision
Hadronic fluctuations and the QCD critical point 14 Consequences for the phase diagram: the radius of convergence unexpected behavior in the radius of convergence of the pressure expansion in µ I /? 24 22 [MeV] m! =22, r 2 r4 filled: n t =4 open: n t =6 2 root of c 6 c 8 18 16 m! µ I [MeV] pion-condensation phase should show up in the ρ profile 2 4 6 8 1
Bulk thermodynamics at µ = 15 Pressure, Energy and Entropy: the th-order.4.6.8 1 1.2 1.4 1.6.4.6.8 1 1.2 1.4 1.6 16 14 r! SB / 4 2 3 r s SB / s/ 3 12 1 8 6 4 2 [MeV]!/ 4 : N " =4 6 3p/ 4 : N " =4 6 1 2 3 4 5 6 7 15 1 5 p4: N " =4 6 asqtad: N " =6 [MeV] 1 2 3 4 5 6 7 M. Cheng et al. [RBC-Bielefeld], PRD 77 (28) 14511. p/ 4 from integrating over (ɛ 3p)/ 5 systematic error from starting the integration at = 1MeV with p( ) = use HRG to estimate systematic error: [ p( )/ 4 ].265 HRG
he EoS at non zero density 16 aylor expansion of the trace anomaly ɛ 3p 4 = n= Coefficients are defined by c n c nb (, ml, m s ) ( µb ) n B (, ml, m s ) = dcb n (, m l, m s ) d perform -derivative numerically: discretization error local version is work in progress c B n (, ˆml, ˆm s ) = a dβ da dc B n (, ˆm l, ˆm s ) dβ a d ˆm l da dc B n (, ˆm l, ˆm s ) d ˆm l a d ˆm s da dc B n (, ˆm l, ˆm s ) d ˆm s aylor expansion of energy and entropy densities ɛ 4 = s 3 = n= n= ( ) ( 3c B n (, m l, m s ) + c µ B nb (, ml, m s ) ) n ( ) ( (4 n)c B n (, m l, m s ) + c µ B nb (, ml, m s ) n= ɛ B n ) n ( µb n= s B n ) n ( ) n µb
he EoS at non zero density 17 2 1.8 1.6 1.4 1.2 1.8.6.4 Coefficients of the 3 p 2! 2 3/2 s 2.2 [MeV] 15 2 25 3 35 -expansion µ B.2.1 -.1 -.2 3 p 4! 4 3/2 s 4 -.2 -.4 -.3 -.6 [MeV] [MeV] -.4 15 2 25 3 35 15 2 25 3 35.4.2 3 p 6! 6 3/2 s 6 corrections to the leading terms are small: 1% pattern of ɛ n and s n is that of c n+2
he isentropic EoS 18 Isentropic trajectories solve numerically for S(, µ B )/N B (, µ B ) = const. 6th order is small at S/N=3 (trajectories inside estimated radius of convergence) calculate pressure and energy density along isentropic trajectories pressure and energy density increase by for S/N=3. 1% 45 4 35 3 25 2 15 RHIC SPS AGS (FAIR) S/N B =3 45 3 µ B 1 2 3 4 5 6 7 8 9 5 p/ 4 N τ = 4 5 p/ 4 N τ = 6 4 4 3 µ B = S/N B =3 3 µ B = S/N B =3 2 45 3 2 45 3 1 1 [MeV] 15 2 25 3 35 4 45 [MeV] 15 2 25 3 35 4 45
he isentropic EoS 19 Isentropic trajectories solve numerically for S(, µ B )/N B (, µ B ) = const. 6th order is small at S/N=3 (trajectories inside estimated radius of convergence) calculate pressure and energy density along isentropic trajectories pressure and energy density increase by for S/N=3. 1% 45 4 35 3 25 2 15 RHIC SPS AGS (FAIR) S/N B =3 45 3 µ B 1 2 3 4 5 6 7 8 9 2 18!/ 4 16 14 12 1 8 6 N τ = 4 N τ = 6 4 2 [MeV] 15 2 25 3 35 4 45 2!/ 4 15 µ B = S/N B =3 45 3 1 µ B = S/N B =3 45 3 5 [MeV] 15 2 25 3 35 4 45
he isentropic EoS 2 Isentropic trajectories solve numerically for S(, µ B )/N B (, µ B ) = const. 6th order is small at S/N=3 (trajectories inside estimated radius of convergence) calculate pressure and energy density along isentropic trajectories pressure and energy density increase by for S/N=3..35.3.25.2.15 p/! 1% S/N B =3 45 3 45 4 35 3 25 2 15 RHIC SPS AGS (FAIR) S/N B =3 45 3 µ B 1 2 3 4 5 6 7 8 9 he EoS along isentropic trajectories is fairly independent on S/N. Leading order corrections: p ɛ = 1 3 1 3 ɛ 3p ɛ ( 1 + [ c 2 ɛ 3p ɛ 2 ɛ ] ( ) ) 2 µb.1.5![gev/fm 3 ].1 1 1 1
Summary 21 Cut-off effect for aylor expansion coefficients are small and sizable only in the transition region (similar to the interaction measure e-3p) We find non-monotonic behavior in the radius of convergence for which could be a first hint for a critical region in the, - plane. his needs to be confirmed by. N τ = 6 Isentropic trajectories show non-monotonic behavior for N τ = 4. his needs to be confirmed by. N τ = 6 Finite density correction for EoS are small, pressure and energy density increase by 1% for S/N=3 (AGS/FAIR), corrections cancel to large extent in p/ɛ. aylor expansion method will provide valuable input for HIC phenomenology. µ B N τ = 4