Hot and Dense QCD on the lattice Frithjof Karsch, BNL Introduction: - - PowerPoint PPT Presentation
Hot and Dense QCD on the lattice Frithjof Karsch, BNL Introduction: T , gT , g 2 T ,... screening and the running coupling Bulk thermodynamics T c and the equation of state in (2+1)-flavor QCD with an almost realistic quark mass spectrum
Frithjof Karsch, BNL Introduction: T , gT , g2T ,... screening and the running coupling Bulk thermodynamics Tc and the equation of state in (2+1)-flavor QCD with an almost realistic quark mass spectrum Thermodynamics at non-zero baryon number density hadronic fluctuations isentropic equation of state Conclusions
F . Karsch, apeNEXT, Florence 2007 – p.1/32
MeV 170 ~ few times nuclear matter density
color superconductor
T
hadron gas quark-gluon plasma deconfined,
χ-symmetric
confined,
χ
F . Karsch, apeNEXT, Florence 2007 – p.2/32
MeV 170 ~ few times nuclear matter density
color superconductor
T
hadron gas quark-gluon plasma deconfined,
χ-symmetric
confined,
χ
continuous/rapid (crossover) transition continuous transition for small chemical potential and small quark masses
F . Karsch, apeNEXT, Florence 2007 – p.2/32
MeV 170 ~ few times nuclear matter density
color superconductor
T
hadron gas quark-gluon plasma deconfined,
χ-symmetric
confined,
χ
continuous/rapid (crossover) transition continuous transition for small chemical potential and small quark masses chiral critical point 2nd order phase transition; Ising universality class
location of CCP uncertain: volume and quark mass dependence
F . Karsch, apeNEXT, Florence 2007 – p.2/32
MeV 170 ~ few times nuclear matter density
color superconductor
T
hadron gas quark-gluon plasma deconfined,
χ-symmetric
confined,
χ
continuous/rapid (crossover) transition continuous transition for small chemical potential and small quark masses chiral critical point 2nd order phase transition; Ising universality class
location of CCP uncertain: volume and quark mass dependence improving accuracy on Tc , ǫc, ǫ(p) and the phase boundary is mandatory to make contact to HIC phenomenology
F . Karsch, apeNEXT, Florence 2007 – p.2/32
Perturbation theory provides a hierachy of length scales T ≫ gT ≫ g2T ...⇒ guiding principle for effective theories, resummation, dimensional reduction... Early lattice results show that g2(T ) > 1 even at T ∼ 5Tc
...one has to conclude that the temperature dependent running coupling has to be large, g2(T ) ≃ 2 even at T ≃ 5Tc the Debye screening mass is large close to Tc the spatial string tension does not vanish above Tc √σs = 0 ⇒ the QGP is ”non-perturbative” up to very high T
F . Karsch, apeNEXT, Florence 2007 – p.3/32
pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510
singlet free energy
500 1000 0.5 1 1.5 2 2.5 3 r [fm] F1 [MeV] 0.76Tc 0.81Tc 0.90Tc 0.96Tc 1.00Tc 1.02Tc 1.07Tc 1.23Tc 1.50Tc 1.98Tc 4.01Tc
T ≃ Tc : screening for r> ∼0.5fm F1(r, T ) ∼ α(T ) r e−µ(T )r +const. F1(r, T ) follows linear rise of V¯
qq(r, T = 0) = −4α(r, T = 0)
3r + σr for T < ∼1.5Tc, r< ∼0.3 fm
F . Karsch, apeNEXT, Florence 2007 – p.4/32
pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling:
αeff = 3r2 4 dF1(r, T ) dr
(in Coulomb gauge)
0.1 0.2 0.3 0.4 0.5 0.6 0.01 0.1 r [fm] αqq(r,T) T/Tc T=0 1.05 1.10 1.20 1.30 1.50 1.60 3.00 6.00 9.00 12.0
. Karsch, apeNEXT, Florence 2007 – p.5/32
pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling:
αeff = 3r2 4 dF1(r, T ) dr
(in Coulomb gauge)
0.1 0.2 0.3 0.4 0.5 0.6 0.01 0.1 r [fm] αqq(r,T) T/Tc T=0 1.05 1.10 1.20 1.30 1.50 1.60 3.00 6.00 9.00 12.0
short distance physics ⇔ vacuum physics
α ≡ π/16
α(r) from (T = 0), 3-loop
(S. Necco, R. Sommer,
Coulomb term (string model) T-dependence starts in non-perturbative regime for T <
∼3 Tc
. Karsch, apeNEXT, Florence 2007 – p.5/32
pure gauge: O.Kaczmarek, FK, P . Petreczky, F. Zantow, PRD70 (2005) 074505 2-flavor QCD: O.Kaczmarek, F. Zantow, Phys. Rev. D71 (2005) 114510 singlet free energy defines a running coupling:
αeff = 3r2 4 dF1(r, T ) dr
(in Coulomb gauge)
0.1 0.2 0.3 0.4 0.5 0.6 0.01 0.1 r [fm] αqq(r,T) T/Tc T=0 1.05 1.10 1.20 1.30 1.50 1.60 3.00 6.00 9.00 12.0
short distance physics ⇔ vacuum physics
α ≡ π/16
α(r) from (T = 0), 3-loop
(S. Necco, R. Sommer,
Coulomb term (string model) T-dependence starts in non-perturbative regime for T <
∼3 Tc
confinement
αeff ∼ σr2
F . Karsch, apeNEXT, Florence 2007 – p.5/32
leading order perturbation theory: mD = g(T )T
6 Tc < T < ∼10Tc: non-perturbative effects are well represented by an ”A-factor”: mD ≡ Ag(T )T, A ≃ 1.5 perturbative limit is reached very slowly (logarithms at work!!)
1 2 3 4 1 1.5 2 2.5 3 3.5 4 T/Tc mD/T mD/T=Ag(T) A=1.42(2) Nf=0 Nf=2 10
−3
10
−2
10
−1
x
0.0 1.0 2.0 3.0 4.0
m/gT
0.5 1 10 10
10
10
100
βG=144 32
3
βG=72 24
3
βG=40 24
3
βG=24 24
3
βG=12 24
3
T/Λ MS SU(3)
O.Kaczmarek,F .Zantow, PRD 71 (2005) 114510 K.Kajantie et al, PRL 79 (1997) 3130 g(T) ≃ 1.5 ⇔ α(T) ≃ 0.18
F . Karsch, apeNEXT, Florence 2007 – p.6/32
Non-perturbative, vanishes in high-T perturbation theory: √σs = − lim
Rx,Ry→∞ ln W (Rx, Ry)
RxRy √σs g2(T )T = cMfM(g(T )) , cM = 0.553(1)
cM : 3-d SU(3), LGT gM ≡ g2fM : dim. red. pert. th.
0.4 0.6 0.8 1 1.2 1 2 4
T/σ1/2 T/Tc
Nf=0 Nf=2+1
4-d SU(3) and QCD
RBC-Bielefeld, preliminary
dimensional reduction works for T>
∼2Tc
c = 0.566(13) [SU(3)] c = 0.594(39) [QCD]
g2(T) ≃ 2 ⇔ α(T) ≃ 0.16
F . Karsch, apeNEXT, Florence 2007 – p.7/32
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
T/Tc
ε/T4 εSB/T4 3p/T4 3 flavor, Nτ=4, p4 staggered mπ=770 MeV strong deviations from ε = 3p 15% dev.
160 180 200 220 240 260 280 500 1000 1500 2000 2500 3000 3500
mPS [MeV] Tc [MeV]
nf=2, p4 nf=3, p4 nf=2, std
QCD EoS strong deviations from ideal gas behavior (ǫ = 3p) for Tc ≤ T ∼ 3Tc and even at high T transition temperature Tc = (173 ± 8 ± sys) MeV weak quark mass and flavor dependence
FK, E. Laermann, A. Peikert, Nucl. Phys. B605 (2001) 579
improved staggered fermions but still on rather coarse lattices: Nτ = 4, i.e. a−1 ≃ 0.8 GeV with moderately light quarks
F . Karsch, apeNEXT, Florence 2007 – p.8/32
Goal: QCD thermodynamics with realistic quark masses and controlled extrapolation to the continuum limit Tc, EoS, µq > 0, ... use an improved staggered fermion action that removes O(a2) errors in bulk thermodynamic quantities and reduces flavor symmetry breaking inherent to the staggered formulation RBC-Bielefeld choice: p4-action + 3-link smearing (p4fat3)
MILC: Naik-action + (3,5,7)-link smearing (asqtad); Wuppertal: standard staggered + exponentiated 3-link smearing (stout)
F . Karsch, apeNEXT, Florence 2007 – p.9/32
Goal: QCD thermodynamics with realistic quark masses and controlled extrapolation to the continuum limit Tc, EoS, µq > 0, ... use an improved staggered fermion action that removes O(a2) errors in bulk thermodynamic quantities and reduces flavor symmetry breaking inherent to the staggered formulation RBC-Bielefeld choice: p4-action + 3-link smearing (p4fat3)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2 4 6 8 10 12 14 16 18 20 Nτ standard action p4 action naik action µ/T=0.0, m/T=0.0 µ/T=1.0, m/T=0.0 µ/T=0.0, m/T=1.0 µ/T=1.0, m/T=1.0
p4-action: smooth high-T behavior for bulk thermodynamics on lattice with temporal extent Nτ
p(Nτ)/T 4 = pcont
SB /T 4 + O(N −4 τ
)
p4&Naik: similarly small cut-off dependence
number susceptibilities
F . Karsch, apeNEXT, Florence 2007 – p.9/32
US/RBRC QCDOC 20.000.000.000.000 ops/sec BI – apeNEXT 5.000.000.000.000 ops/sec
. Karsch, apeNEXT, Florence 2007 – p.10/32
Goal: QCD thermodynamics with realistic quark masses and controlled extrapolation to the continuum limit Tc, EoS, µq > 0, ... use an improved staggered fermion action that removes O(a2) errors in bulk thermodynamic quantities and reduces flavor symmetry breaking inherent to the staggered formulation RBC-Bielefeld choice: p4-action + 3-link smearing (p4fat3) use the newly developed RHMC algorithm to remove ’step-size errors’ in the numerical simulation perform simulations with (3-4) different light quark masses corresponding to 150 MeV< ∼mπ< ∼500 MeV at 2 different values of the lattice cut-off controlled by the spatial lattice size Nτ = 4, 6 to perform the chiral and continuum extrapolation previous results with p4-action: 2-flavor QCD: Nτ = 4, mπ ≃ 770 MeV
F . Karsch, apeNEXT, Florence 2007 – p.11/32
crossover rather than phase transition: need to determine location of the transition from various susceptibilities: (disconnected part of the) light and strange quark chiral susceptibility; Polyakov loop and quark number susceptibility,... thermodynamic limit: need to control finite volume effects; continuum limit: need to analyze cut-off dependence in T > 0 and T = 0 calculations; large statistics; several ten thousand trajectories find little volume dependence of location of transition point
find weak cut-off dependence
F . Karsch, apeNEXT, Florence 2007 – p.12/32
10 20 30 40 50 60 70 80 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χl / T 2 ml = 0.05 ms 83x4 163x4 10 20 30 40 50 60 70 80 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χl / T 2 ml = 0.10 ms 83x4 163x4
10 20 30 40 50 60 70 80 3.40 3.42 3.44 3.46 3.48 3.50 3.52 3.54 3.56 3.58 β χl / T 2 ml = 0.4 ms ml = 0.2 ms ml = 0.1 ms
163 × 6
weak volume dependence peak location consistent with that of Polyakov loop suscep- tibility and maximum of quar- tic fluctuation of quark number density
F . Karsch, apeNEXT, Florence 2007 – p.13/32
10 20 30 40 50 60 70 80 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χl / T 2 ml = 0.05 ms 83x4 163x4
F . Karsch, apeNEXT, Florence 2007 – p.14/32
10 20 30 40 50 60 70 80 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χl / T 2 ml = 0.05 ms 83x4 163x4
2.5% error band ⇔ 5 MeV
data sample for smallest quark mass
β
3.2900 38960 3.3000 40570 3.3050 32950 3.3100 42300 3.3200 39050
200,000/10 trajectories enter Ferrenberg-Swendsen sample
F . Karsch, apeNEXT, Florence 2007 – p.14/32
10 20 30 40 50 60 70 80 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χl / T 2 ml = 0.05 ms 83x4 163x4 1 2 3 4 5 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 β χL / T 2 mq = 0.05 ms , 83×4 163×4
2.5% error band ⇔ 5 MeV
data sample for smallest quark mass
β
3.2900 38960 3.3000 40570 3.3050 32950 3.3100 42300 3.3200 39050
F . Karsch, apeNEXT, Florence 2007 – p.14/32
0.000 0.005 0.010 0.015 0.020 0.025 0.5 0.5 0.5
ml/ms
βL - βl βl - βs
163x6 ml/ms
βL - βl βl - βs
163x4 ml/ms
βL - βl βl - βs
83x4
2.5% (Nτ = 4) or 4% (Nτ = 6) error band ⇔ 5 or 8 MeV
differences of pseudo-critical couplings locating peaks in light (βl), strange (βs) and Polyakov loop (βL) susceptibilities
differences in the location of pseudo-critical couplings are taken into account as systematic error
F . Karsch, apeNEXT, Florence 2007 – p.15/32
use r0 or string tension to set the scale for Tc = 1/Nτa(βc) V (r) = −α r + σr , r2 dV (r) dr |r=r0 = 1.65
0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 1 2 3
T=4.34Tc, β=3.94, ms=0.01625 T=3.01Tc, β=3.76, ms=0.02 T=2.51Tc, β=3.68, ms=0.026 T=2.12Tc, β=3.61, ms=0.0325 T=1.70Tc, β=3.518, ms=0.0382 T=1.28Tc, β=3.41, ms=0.052 T=1.20Tc, β=3.382, ms=0.052 T=1.10Tc, β=3.351, ms=0.0591
V(r/r0)r0 rimp/r0
a
no significant cut-off dependence when cut-off varies by a factor 4 i.e. from the transition region
we use r0 = 0.469(7) fm determined from quarkonium spectroscopy
F . Karsch, apeNEXT, Florence 2007 – p.16/32
extrapolation to chiral and continuum limit (r0Tc)Nτ = (r0Tc)cont. + b (mP Sr0)d + c/N 2
τ
(d=1.08 (O(4), 2nd ord.), d=2 (1st ord.))
0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40
mps r0
Tc r0
Nτ=4 (squares, triangles) 6 (circles) nf=2+1 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40
mps r0
Tc/√σ
Nτ=4 (squares, triangles) 6 (circles) nf=2+1
⇒ r0Tc = 0.456(7)+3
−1
, Tc/√σ = 0.408(7)+3
−1 at phys. point
⇒ Tc = 192(7)(4) MeV (1st error: stat. error on βc and r0; 2nd error: N −2
τ
extrapolation)
F . Karsch, apeNEXT, Florence 2007 – p.17/32
staggered fermions Nτ = 4, 6
0.35 0.40 0.45 0.50 0.55 0.60 0.0 0.5 1.0 1.5 2.0 2.5 3.0
mps r0
Tc r0
stag.,p4fat3: Nτ=4,6 continuum extrapolation
RBC-Bielefeld (p4fat3 (p4))
F . Karsch, apeNEXT, Florence 2007 – p.18/32
staggered fermions Nτ = 4, 6
0.35 0.40 0.45 0.50 0.55 0.60 0.0 0.5 1.0 1.5 2.0 2.5 3.0
mps r0
Tc r0
stag.,p4fat3: Nτ=4,6 stag.,asqtad: Nτ=4,6 stag.,stout: Nτ=4,6
RBC-Bielefeld (p4fat3 (p4)) vs. MILC (asqtad (Naik)) and Wuppertal (stout (stand. staggered)) asqtad results for Nτ = 4 and 6 agree with p4 results within statistical errors; (C.Bernard et al., PR D71, 034504 (2005)) results obtained with stout action for Nτ = 4 and 6 are about 15% lower;
βc from Nτ = 8, 10 covers (151 − 176) MeV;
(Y. Aoki et al., hep-lat/0609068) asqtad data for Tcr1 rescaled with
r0/r1 = 1.4795
asqtad: continuum extrapolation: quoted Tc from mq/ms ≤ 1 and fit in mπ/mρ yields
Tc = 169(12)(4) MeV
using mq/ms ≤ 0.4 and fit in mπr0 yields
Tc = 173(13)(4) MeV
F . Karsch, apeNEXT, Florence 2007 – p.18/32
staggered fermions Nτ = 4, 6 and Wilson fermions Nτ = 6 − 10
0.35 0.40 0.45 0.50 0.55 0.60 0.0 0.5 1.0 1.5 2.0 2.5 3.0
mps r0
Tc r0
stag.,p4fat3: Nτ=4,6 stag.,asqtad: Nτ=4,6 stag.,stout: Nτ=4,6 Clover, Ejiri et al Nτ=4,6 Clover, Bornyakov et al, Nτ=6,8,10
RBC-Bielefeld (p4fat3 (p4)) vs. MILC (asqtad (Naik)) and Wuppertal (stout (stand. staggered))
Tc from Wilson/Clover fermions so far only for mpsr0 > 1.5;
consistent with staggered results Wilson for Nτ ≥ 6 show no significant cout-off effects (V.G. Bornyakov et al., hep-lat/0509122) scale setting uncertainties: staggered: r0 = 0.469(7) fm (MILC + heavy quark spec. ) Clover: r0 = 0.516(21) fm (CP-PACS+JLQCD, light quark spec.)
F . Karsch, apeNEXT, Florence 2007 – p.19/32
150 160 170 180 190 200 210 0.00 0.02 0.04 0.06 0.08 0.10
1/Nτ
2
Tc [MeV]
p4fat3 stout, χm χs P
RBC-Bielefeld (p4fat3 (p4)) vs. Wuppertal (stout (stand. staggered)) results for Nτ = 4, 6 differ by 15% but show similar cut-off dependence stout results for different observables no longer consistent with each other for Nτ = 8, 10
r0 = 0.469 fm
F . Karsch, apeNEXT, Florence 2007 – p.20/32
The pressure
p T 4 ˛ ˛ ˛
β β0
= N4
τ
Z β
β0
dβ′ » 1 N3
σNt
(Sg0 − SgT ) − „ 2( ¯ ψψl0 − ¯ ψψlT ) + ˆ ms ˆ ml ( ¯ ψψs0 − ¯ ψψsT ) « „∂ ˆ ml ∂β′ «
ˆ ms/ ˆ ml
− ˆ ml ` ¯ ψψs0 − ¯ ψψsT ´ „∂ ˆ ms/ ˆ ml ∂β′ «
ˆ ml
#
The interaction measure for Nf = 2 + 1
ǫ − 3p T 4 = T d dT “ p T 4 ” = „ adβ da «
LCP
∂p/T 4 ∂β = „ ǫ − 3p T 4 «
gluon
+ „ǫ − 3p T 4 «
f ermion
+ „ ǫ − 3p T 4 «
ˆ ms/ ˆ ml
F . Karsch, apeNEXT, Florence 2007 – p.21/32
2 4 6 8 10 100 200 300 400 500 600 700 0.4 0.6 0.8 1.0 1.2 1.4 1.6
T [MeV] Tr0 (ε-3p)/T4
p4: Nτ=4 6 asqtad: Nτ=4 6
Using an RG-inspired 2-loop β-function underestimates (ǫ − 3p)/T 4 in the transition region and stretches the temperature interval in the low temperature regime artifically, i.e. makes the tran- sition region look broader than it is.
RBC-Bielefeld, preliminary differences in the transition region partly arise from differences in the β-functions used in the crossover region
Note: T-scale is not dependent on Tc deter- mination asqtad data:
F . Karsch, apeNEXT, Florence 2007 – p.22/32
5 10 15 20 100 200 300 400 500 600
T [MeV] s/T3
sSB/T3
p4: Nτ=4 6 asqtad, Nτ=4 6 1 2 3 4 5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
T/Tc p/T4
pSB/T4 RBC-BI preliminary
nf=2+1, LCP, Nτ=4 6 nf=3, mq/T=0.4, Nτ=4
RBC-Bielefeld vs. MILC: the RBC-Bi energy/entropy density on Nτ = 4 lattices rises more steeply; direct consequence of the use of a non-perturbative β-function directly deduced from calculated r0/a values
Note: T-scale does not depend on Tc determination!! band marks T = (192 ± 11) MeV
RBC-Bielefeld, preliminary
pressure increased slightly with smaller quark mass
F . Karsch, apeNEXT, Florence 2007 – p.23/32
LGT: Tc ≃ 190 MeV T = Tc: ǫc/T 4
c ≃ 6 ⇒ ǫc ≃ 1 GeV/fm3
T ≥ 1.5Tc: ǫ/T 4 ≃ (13 − 14) T = 1.5Tc: ǫ ≃ 11 GeV/fm3 T = 2.0Tc: ǫ ≃ 35 GeV/fm3 ⇓
J/ψ suppression RHIC RAu ≃ 7 fm; τ0 ≃ 1 fm ET ≃ 1 GeV dN/dy ≃ 1000 ⇓ ǫBj ≃ 7 GeV/fm3 maybe: τ0 ≃ 0.5 fm ⇓ ǫBj ≃ 14 GeV/fm3
F . Karsch, apeNEXT, Florence 2007 – p.24/32
LGT: Tc ≃ 190 MeV T = Tc: ǫc/T 4
c ≃ 6 ⇒ ǫc ≃ 1 GeV/fm3
T ≥ 1.5Tc: ǫ/T 4 ≃ (13 − 14) T = 1.5Tc: ǫ ≃ 11 GeV/fm3 T = 2.0Tc: ǫ ≃ 35 GeV/fm3 ⇓ χc, ψ′ suppression at RHIC direct J/ψ suppression unlikely ⇓ S(J/ψ) ≃ 0.6 + 0.4S(χc) (assume S(χc) ≃ S(ψ′)) RHIC RAu ≃ 7 fm; τ0 ≃ 1 fm ET ≃ 1 GeV dN/dy ≃ 1000 ⇓ ǫBj ≃ 7 GeV/fm3 maybe: τ0 ≃ 0.5 fm ⇓ ǫBj ≃ 14 GeV/fm3
F . Karsch, apeNEXT, Florence 2007 – p.24/32
Z(V , T , µ) =
ψ e−SE(V ,T ,µ) =
f e−SG(V ,T )
F . Karsch, apeNEXT, Florence 2007 – p.25/32
Z(V , T , µ) =
ψ e−SE(V ,T ,µ) =
f e−SG(V ,T )
ways to circumvent this problem:
reweighting: works well on small lattices; requires exact evaluation of detM
Taylor expansion around µ = 0: works well for small µ;
R.V. Gavai, S. Gupta, Phys. Rev. D68 (2003) 034506 imaginary chemical potential: works well for small µ; requires analytic continuation
. Lombardo, Phys. Rev. D64 (2003) 014505
F . Karsch, apeNEXT, Florence 2007 – p.25/32
Thermodynamics: (NB: continuum ˆ m ≡ mq lattice ˆ m ≡ mqa, implicit T-dependence) pressure p T 4 ≡ 1 V T 3 ln Z(T, µq) =
∞
cn(T, ˆ m) µq T n energy density from ”interaction measure” ǫ − 3p T 4 =
∞
c′
n(T, ˆ
m) µq T n , c′
n(T, ˆ
m) ≡ T dcn(T, ˆ m) dT entropy density s T 3 ≡ ǫ + p − µqnq T 4 =
∞
m) + c′
n(T, ˆ
m) µq T n
F . Karsch, apeNEXT, Florence 2007 – p.26/32
Taylor expansion of pressure up to O
` (µq/T)6´ p T 4 = 1 V T 3 ln Z =
∞
X
n=0
cn(T) „µq T «n ≃ c0 + c2 „µq T «2 + c4 „µq T «4 + c6 „ µq T «6
quark number density
nq T 3 = 2c2 µq T + 4c4 „µq T «3 + 6c6 „µq T «5
quark number susceptibility
χq T 2 = 2c2 + 12c4 „µq T «2 + 30c6 „ µq T «4
an estimator for the radius of convergence
„ µq T «
crit
= lim
n→∞
˛ ˛ ˛ ˛ c2n c2n+2 ˛ ˛ ˛ ˛
1/2
cn > 0 for all n;
singularity for real µ
F . Karsch, apeNEXT, Florence 2007 – p.27/32
Taylor expansion of pressure up to O
` (µq/T)6´ p T 4 = 1 V T 3 ln Z =
∞
X
n=0
cn(T) „µq T «n ≃ c0 + c2 „µq T «2 + c4 „µq T «4 + c6 „ µq T «6
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8 1 SB limit SB (Nτ=4)
c2
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1 0.15 0.2 0.25 SB limit SB (Nτ=4)
c4
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1
c6
F . Karsch, apeNEXT, Florence 2007 – p.27/32
Taylor expansion of pressure up to O
` (µq/T)6´ p T 4 = 1 V T 3 ln Z =
∞
X
n=0
cn(T) „µq T «n ≃ c0 + c2 „µq T «2 + c4 „µq T «4 + c6 „ µq T «6
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8 1 SB limit SB (Nτ=4)
c2
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1 0.15 0.2 0.25 SB limit SB (Nτ=4)
c4
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1
c6
cn > 0 for all n and T < ∼ 0.95 Tc ⇔
singularity for real µ (positive µ2)
F . Karsch, apeNEXT, Florence 2007 – p.27/32
Taylor expansion of pressure up to O
` (µq/T)6´ p T 4 = 1 V T 3 ln Z =
∞
X
n=0
cn(T) „µq T «n ≃ c0 + c2 „µq T «2 + c4 „µq T «4 + c6 „ µq T «6
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8 1 SB limit SB (Nτ=4)
c2
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1 0.15 0.2 0.25 SB limit SB (Nτ=4)
c4
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.05 0.1
c6
irregular sign of cn for T >
∼ Tc ⇔
singularity in complex plane
F . Karsch, apeNEXT, Florence 2007 – p.27/32
high-T, ideal gas limit
p T 4 ˛ ˛ ˛
∞
= nf „ 7π2 60 + 1 2 “µq T ”2 + 1 4π2 “µq T ”4«
1 2 3 4 5 100 200 300 400 500 600 T [MeV] p/T4
pSB/T4
3 flavour 2+1 flavour 2 flavour pure gauge
µq = 0, 163 × 4 lattice
improved staggered fermions;
nf = 2, mπ ≃ 770 MeV
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8
µq/T=1.0
∆p/T
4
µq/T=0.8 µq/T=0.6 µq/T=0.4 µq/T=0.2
C.R. Allton et al. (Bielefeld-Swansea), PRD68 (2003) 014507 contribution from µq/T > 0 Taylor expansion, O((µ/T)4) RHIC: µq/T<
∼0.1
F . Karsch, apeNEXT, Florence 2007 – p.28/32
1 2 3 4 5 100 200 300 400 500 600 T [MeV] p/T4
pSB/T4
3 flavour 2+1 flavour 2 flavour pure gauge
µq = 0, 163 × 4 lattice
improved staggered fermions;
nf = 2, mπ ≃ 770 MeV
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8
µq/T=1.0
∆p/T
4
µq/T=0.8 µq/T=0.6 µq/T=0.4 µq/T=0.2
C.R. Allton et al. (Bielefeld-Swansea), PRD68 (2003) 014507 contribution from µq/T > 0 NEW: Taylor expansion, O((µ/T)6) PRD71 (2005) 054508
0.8 1 1.2 1.4 1.6 1.8 2
T/T0
0.2 0.4 0.6 0.8
µq/T=1.0
∆p/T
4
µq/T=0.8 µq/T=0.6 µq/T=0.4 µq/T=0.2
pattern for µq = 0 and µq > 0 similar; quite large contribution in hadronic phase;
O((µ/T)6) correction small for µq/T< ∼1
RHIC: µq/T<
∼0.1
F . Karsch, apeNEXT, Florence 2007 – p.28/32
dense matter created in a HI-collision expands and cools at fixed entropy and baryon number ⇒ lines of constant S/NB in the QCD phase diagram for example: isentropic expansion, ”mixed phase model”:
V.D. Toneev, J. Cleymans, E.G. Nikonov,
F . Karsch, apeNEXT, Florence 2007 – p.29/32
dense matter created in a HI-collision expands and cools at fixed entropy and baryon number ⇒ lines of constant S/NB in the QCD phase diagram high T: ideal gas S NB = 3
32π2 45nf + 7π2 15 +
µq
T
2
µq T + 1 π2
µq
T
3 S/NB = constant ⇔ µq/T constant low T: nucleon + pion gas T → 0: µq/T ∼ c/T
F . Karsch, apeNEXT, Florence 2007 – p.29/32
150 200 250 300 100 200 300 400 500 600 700 800 900 µB [MeV] T [MeV] RHIC SPS AGS (low-E RHIC)
µB/T=0.24 µB/T=1.63 µB/T=2.47
S/NB=30 45 300 freeze-out 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
p/ε T/T0
S/Nq=10 15 100 p4fat3, Nq=0 asqtad, Nq=0
p/ǫ vs. ǫ shows almost no dependence on S/NB softest point: p/ǫ ≃ 0.075 phenomenological EoS for T0 < ∼ T < ∼ 2T0 p ǫ = 1 3
1.2 1 + 0.5 ǫ fm3/GeV
. Karsch, apeNEXT, Florence 2007 – p.30/32
150 200 250 300 100 200 300 400 500 600 700 800 900 µB [MeV] T [MeV] RHIC SPS AGS (low-E RHIC)
µB/T=0.24 µB/T=1.63 µB/T=2.47
S/NB=30 45 300 freeze-out 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
p/ε T/T0
S/Nq=10 15 100 p4fat3, Nq=0 asqtad, Nq=0
p/ǫ vs. ǫ shows almost no dependence on S/NB softest point: p/ǫ ≃ 0.075 phenomenological EoS for T0 < ∼ T < ∼ 2T0 p ǫ = 1 3
1.2 1 + 0.5 ǫ fm3/GeV
so far analyzed only for mπ ≃ 770 MeV awaits confirmation in (2+1)-flavor QCD with light quarks
F . Karsch, apeNEXT, Florence 2007 – p.30/32
non-perturbative QGP the QGP is non-perturbative up to high temperatures; the running of αs reflects ”remnants of confinement” bulk thermodynamics the transition between a HG and the QGP is signaled by a rapid change in the energy density; calculations with different O(a2) improved staggered fermions yield a consistent description of the high temperature phase; the transition temperature at the physical point of (2+1)-flavor QCD our calculation of Tc yields Tc = 192(7)(4)MeV
F . Karsch, apeNEXT, Florence 2007 – p.31/32
Taylor expansion
⇒ estimates for radius of convergence ρ2n = s˛ ˛ ˛ ˛ c2n c2n+2 ˛ ˛ ˛ ˛
2 4 6 8 10 12 0.5 1 1.5 2 2.5
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
SB(ρ2) SB(ρ0)
0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
ρ2, ρ4 res. gas
T < T0: ρn ≃ 1.0 for all n ⇒ µcrit
B
≃ 500 MeV
. Karsch, apeNEXT, Florence 2007 – p.32/32
Taylor expansion
⇒ estimates for radius of convergence ρ2n = s˛ ˛ ˛ ˛ c2n c2n+2 ˛ ˛ ˛ ˛
2 4 6 8 10 12 0.5 1 1.5 2 2.5
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
SB(ρ2) SB(ρ0)
0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
ρ2, ρ4 res. gas
T < T0: ρn ≃ 1.0 for all n ⇒ µcrit
B
≃ 500 MeV
HOWEVER still consistent with resonance gas!!! HRG analytic, LGT consistent with HRG ⇒ infinite radius of convergence not yet ruled out
. Karsch, apeNEXT, Florence 2007 – p.32/32
Taylor expansion
⇒ estimates for radius of convergence ρ2n = s˛ ˛ ˛ ˛ c2n c2n+2 ˛ ˛ ˛ ˛
2 4 6 8 10 12 0.5 1 1.5 2 2.5
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
SB(ρ2) SB(ρ0)
0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4
ρ0 ρ2 ρ4 Tc(µq)
T/T0 µq/T0
ρ2, ρ4 res. gas
T < T0: ρn ≃ 1.0 for all n ⇒ µcrit
B
≃ 500 MeV
HOWEVER still consistent with resonance gas!!! HRG analytic, LGT consistent with HRG ⇒ infinite radius of convergence not yet ruled out
JHEP 0404 (2004) 050
F . Karsch, apeNEXT, Florence 2007 – p.32/32