lattice QCD
at finite temperatureand density
kazuyuki kanaya
kanaya@.
.ac.jp08/10/15

contents
 introduction tolattice QCD at T > 0 and/or µ ≠ 0 how we calculate where we need care status: how far are we ?
 watʼs new at Lattice 2008 ?
 phase diagram(just keywords)

2

lattice QCD at T > 0 / µ ≠ 0
how we calculate ••• T > 0: Matsubara formalism
euclidian path integral! of LQCD with finite euclidian-time extent.

Z = Tre−H/T =DqDq̄DU e−S ;Ns1Nt
T =Nt a vary T by varying a in terms of g at fixed Nt

a2 −b1 /(2b20 ) −1/(2b0 g2 ) + NP correctionsa = const. × (b0 g )b0 =1
16π 2!

211 − NF
3"e!
"138
, b1 =
102−
NF: asymptotic scaling(16π 2 )23‣ or by varying Nt at fixed a3

lattice QCD at T > 0 / µ ≠ 0
how we calculate ••• (2) thermodynamic quantities from derivatives of Z

β = 6/g 2

 trace anomaly (interaction measure)T d ln Z
T! − 3p = −=−V d ln aV∂ ln Z
For derivatives in V and T, e.g., p = T,∂V∂β∂ ln Zln a$∂β!∂ "#! "# $NP beta fn.

!··· "

we need separate derivatives in at and as on anisotropic lattice and Karsch coefficients,
whose NP values not easy.

 integral method in conventional fixed Nt approach
using the thermodyn. relation for large V,TTp = ln Z =VV! β∂ ln Zdβ∂ββ0p(β0 ) ≈ 04

RBC-BiNF = 2+1
improved staggered (p4)mπ ≈ 220 MeVM. Cheng et al.
PRD 77 (2008) 014511
HotQCD (≈ RBC-Bi + MILC)
NF = 2+1
improved staggered (AsqTad, p4)
mπ ≈ 220 MeV
R. Gupta, Lattice 20085

lattice QCD at T > 0 / µ ≠ 0how we calculate ••• (3)This requires T=0 (large Nt) simulations at each ß too.
• subtraction of T=0 UV divergences • determination of LCP • etc=> 70-90% of the computer time!

 T-integral method in the fixed a approach (Talk by Umeda)
using the thermodyn. relation of grand canonical system at µ=0,T∂ ! p "∂TT4" − 3p=T4==>

p=4T! T! − 3pdT5TT0Merits:• subtraction by a common T=0 simulation • obviously on a LCP •=> large reduction of the computer time.
6

65

anisotropic lattice
"=4
#=6.1

4

43(!−3p)/T
43p/T
4!/Tcontinuum S.B. limit

21

0200

400

600

800

T[MeV]
1000

T-integral method(WHOT-QCD: Umeda et al. @ Lattice 2008)7

lattice QCD at T > 0 / µ ≠ 0how we calculate ••• (4) µ ≠ 0: sign (complex phase) problem
U4 = eiaA4=⇒ U4 eiaµq (positive direction);−iaµq
Ue(negative direction)!
4"−SZ = DqDq̄DU e ; S = Sg +q̄M [U ] q

Quark kernel not γ5 -hermete at µ ≠ 0M (µ)† = γ5 M (−µ)γ5=> complex Boltzmann weight!

Dq Dq̄e−Sq (µ) = det M (µ)[det M (µ)]∗ = det M (−µ) "= det M (µ)=> large! cancellations
duetophasefluctuatons"iθe ∼ e−V while fluctuations ~ O(1), V = lattice vol.
=> MC simulation O(e+V) expensive.
8lattice QCD at T > 0 / µ ≠ 0
how we calculate ••• (4) µ≠0Methods for small µ★ Taylor expansion in µ around µ = 0<= major studies★ reweighting from µ = 0★ analytic continuation from imaginary µ★ canonical ensemble=> RHIC/LHC region OK
crit. point ??

still challenging

Intermediate-large µ ????
9

Bielefeld-Swansea
NF = 2, improved KS (p4)mqbare / T = 0.4, Nt = 4
Allton et al., PRD 71 (2005) 054508

10

lattice QCD at T > 0 / µ > 0
where we need care ••• we are not very close to the cont. limit yet.
 fixed Nt studies mostly at Nt = 4-8
lattice artifacts due to large a and small Nt
At Tc ≈ 180 MeV, we have:=> Nt ≥ 8 hopefully

 T-integral method in fixed a approachlarge Nt around Tc (Nt > 10 with usual a)=> lattice artifacts smaller there11

lattice QCD at T > 0 / µ > 0
where we need care ••• (2) lattice quarks- naïve lattice fermions: doubler problem (improved) staggered (Kogut-Susskind) quarks‣ relatively cheap => most extensively used
‣ degeneracy of 4 quarks with O(a2) mixing.
original idea= 4 flavors, but not easy to dissolve=> “fourth root trick” to drop additional 3 “tastes”

det M =⇒ [det M ]1/4 by hand- still many additional valence “quarks” => many “hadrons”- still different flavor+taste symmetry=> universality class??
- nonlocality- O(4) scaling for NF = 2 QCD not seen yet.
12lattice QCD at T > 0 / µ > 0
where we need care ••• (3) lattice quarks (2) (improved) Wilson fermions‣ more expensive => need various improvements‣ flavor symmetry & locality naturally realized
QCDPAX/CP-PACS NF = 2, µ = 0 (ʻ96-ʼ03):
O(4) scaling confirmed, phase structure
quark masses are still heavy
WHOT-QCD (ʻ06-): screening masses, µ ≠ 0• Taylor expansion method with various improvements• T-integral method for lighter quarks
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CP-PACS
NF = 2, improved Wilson
mPS/mV = 0.65-0.95, Nt = 4
AliKhan et al., PRD 63 (2000) 034502}

QCD
O(4) Heisenberg
modelh = 2 mqt = ß - ßchiral trans.
fit withO(4) critical exponents
14lattice QCD at T > 0 / µ > 0
where we need care ••• (4) lattice quarks (3) lattice chiral fermions (DW, overlap)
still expensive (O(100) times more computer time)first results of DW simulations (RBC/HotQCD)“in infancy” (C. DeTar, plenary @ Lattice 2008)15

lattice QCD at T > 0 / µ > 0
where we need care ••• (5) and more finite volume effects and FSS
 violation of chiral symmetry MEM •••please enjoy

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status
how far are we?

how was it at?

Williamsburg,VA, USA, July 14-19, 2008

 38 talks and 6 posters on T > 0 / µ > 0Started with plenaries by C. DeTar on “QCD Thermodynamics”
 S. Ejiri on “LQCD at finite density”
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38 talks

T >0µ>06 posters
C. Miao

Lattice Calculation of Hadronic ...

19 C. DeTar on “QCD Thermodynamics”
 large NF =2+1 simulations near the physical point:
HotQCD with impr.stag. at mπ ≈ 220 MeV.
 new ideas:
T-integral method for EOS (WHOT-QCD), etc.
 Tc confusion diminished:chiral suscept. problematic for Tc
Tc ~ 170-190 MeV
 ...
20

 S. Ejiri on “LQCD at finite density”
 isentropic EOS (MILC, RBC-Bi, HotQCD)
 results with Wilson-type quarks (WHOT-QCD):
had. fluctuations enhanced toward crit. pt.
 technical developments for µ > 0 simulations
 ...

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phase diagram at µ = 0
 theoretical expectations from effective models
3d Z(3) Potts!

3d O(4) scaling!
tricritical point!
2nd order line: !
mud~ (ms*-ms)5/2!
near the !
tricritical point!

3d Ising scaling!

3d Ising scaling!
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QCDPAXPRD46 (’92)

phase diagramat µ = 0 (2)3d O(4) scaling!

Im !

-0.15

0.15

00-0.15

 LQCD simulations

0.15

Re !

3d Z(3) Potts!

Wilson-type: OK
staggered-type: not seen
DW/overlap: not tested yet
Dissent argument by the Pisa group:
1st order at NF=2
(Nt=4, unimproved staggered)Wilson-type, staggered-type: look OK
DW/overlap: not tested yet

3d Ising scaling!
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phase diagram at µ = 0 (3) location of the physical point
Intensively studied only withstaggered quarks.
=> crossoverCaveats:• Staggered quarks could not
reproduce the O(4) scaling.
• How about Wilson-type ??
or DW/ovelap ???
(old unimpr.Wil.=> 1st order)24

Y. Aoki et al., nature05120 (’06)
improved staggered (stout),continuum extrap. with Nt=4-10
finite size scaling study=> Crossover at the phys. pt.
de Forcrand and Phillipsen, Lattice 2006
unimproved staggered, Nt=4, exact algorithm
25phase diagram at µ ≠ 0 what usually assumed
based on model studies + lattice staggered quark results

ms ≈ msphys
mud ≈ 0Tms ≈ msphys
mud ≈ mudphysTquark-gluon plasma
phase

µ=0
hadronphase

color supernuclearconductor?
matterµq

Assuming
crossoverat µ = 0.

quark-gluon plasmaphasecrit.pt.
hadronphase

color supernuclearconductor?
matterµq 26

Bielefeld-Swansea
NF = 2, mud/T = 0.4
improved stag. (p4), Nt = 4
Allton et al.,
PRD71, 054508 (’05)

WHOT-QCDNF = 2, mPS/mV = 0.65
improved Wilson, Nt = 4
KK, Lattice 2008
20202

χq/T

2

χI/T

µq/T=1.2µq/T=1.0µq/T=0.8µq/T=0.6µq/T=0.4µq/T=0.2µq/T=0.0

15101510500.5µq/T=1.2µq/T=1.0µq/T=0.8µq/T=0.6µq/T=0.4µq/T=0.2µq/T=0.05

1

1.5

T/T02

00.5

1

1.5

T/T0227

RBC-BiNF = 2+1, mπ ≈ 220 MeV
improved stag. (p4), Nt = 4, 6C. Miao, Lattice 2008µ=028

phase diagram at µ ≠ 0 (2) where is the critical point?

Assuming
crossoverat µ = 0.

• Lee-Yang zero (Fodor and Katz)<= critique by Ejiri (PRD73,054502(’06))• radius of convergence of Taylor expansion:
NF=2, 2+1, staggered-type quarks, Nt=4 mostlyTcompiled byC. Schmidt,Lattice 2008

quark-gluon plasmaphasehadronphasecolor supernuclearconductor?
matterµq 29

phase diagramat µ ≠ 0 (3)Naive expectationde Forcrand-Phillipsen

imaginary µ method
unimproved stag., Nt=4
JHEP01 077, Lattice 2008=> slightly negative curvature at µ=0.
The crit. surface should bend back!
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summary
LQCD: direct bridge between
1st principles of QCD <=> hadron / QGP physics

Predictions availabele.
caveats: several systematic errors not well controlled yet.

Simulations becoming constantly realistic.
Direct studies just at the physical point started.
(plenary by Y. Kuramashi, Lattice 2008)
Feed back to finite temperature and density will be stareted soon.

thank you

lattice QCD at T > 0
how we calculate ••• line of constant physics (LCP)
A physical system (with various a, i.e various g) is given by a linein the coupling parameter space:
Nf=2 QCD with improved Wilson quarks(CP-PACS Collab., WHOT-QCD Collab.)

LCP by mPS /mV at T = 0.
Different line = different world
Our world is given by LCP formPS /mV = mπ /mρ = 135/770= 6/g 2

To heat up a given physical system in fixed Nt approaches,
we have to follow the LCP for this system.
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