XQCD 2011 @ San Carlos, MXJuly 18-20, 2011

Phase Structure of QCD from the LatticeKazuyuki KanayaUniversity of Tsukubakanaya@ccs.tsukuba.ac.jp

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Accelerator Center5

K. Fukushima and T. HatsudaRep. Prog. Phys. 74, 014001 (’11)

To which extent are these pictures confirmed on the lattice?

“Columbia plot”

QCD PHASE STRUCTURE

at physical mq’s (“physical point”)at µ ≈ 0

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✤ µ = 0★ O(4) scaling / Scaling tests on the lattice★ Tc -- Resolution of the disagreement / Remaining issues

✤ µ ≠ 0★ Difficulties and tricks★ Physics / phase structure at µ ≈ 0 ★ Around the heavy quark limit

✓Descriptions of works presented at Lattice 2011 are mostly based on my rough memo. Apologize if they are incorrect!

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µ = 0

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PROSPECTED PHASE STRUCTURE AT µ = 0

tricrit.point

Physicalpoint? ??

Lattice studies with staggered-type quarks

=> Physical point locates in the crossover region

GL effective models + lattice results

Z(3) Potts

Z(3) Potts + ext. field

effective ϕ6 theorymud ∝ (mstri – ms)5/2

SU(2)×SU(2)×U(1) σ model

SU(3)×SU(3)×U(1) σ model

In fixing details of the plot,

critical scaling based on universality argument plays an essential role.

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QCD transition around the chiral limit

Effective 3d σ model with the same flavor-chiral symmetry of massless QCD (continuum) Pisarski-Wilczek, PRD(’84); Wilczek, IJMP(‘92); Rajagopal-Wilczek NPB(‘93)

NF ≥ 3: 1st order

NF = 2: depends on the magnitude of the anomaly

when anomaly strong: the σ model ≈ O(4) Heisenberg model

=> 2nd order with established critical properties

when anomaly negligible around Tc => fluctuation-induced (weakly)1st order

<= chiral violating coupling (external mag. field)<= chiral symmetric coupling (reduced temperature)

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In case anomaly negligible around Tc,NF = 2: 1st order chiral trans. with Ising crit. end pointNF = 3: smaller 1st order region

<= anomaly was a source of the M3 term

O(4) scaling is a powerful guide here.

To discriminate the pictures, non-perturbative test on the lattice needed.

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Flavor-chiral sym. on the latticeNo-go theorem (Nielesen-Ninomiya): one flavor lattice fermion cannot be local and chiral simultaneously.

Chiral symmetry cannot be simply realized on the lattice.=> several options for the quark action with different flavor-chiral properties:

Wilson-type / staggered-type / domain-wall / overlap / ...

Wilson-type quarks: violate the chiral symmetry at a > 0.

Pros: ✓ Describes a single flavor. => Flavor symmetry exact.

✓ Continuum limit exists. <= The chiral sym. is restored in the cont. limit.

Cons: ❖ Explicit violation of the chiral sym. at a > 0.

❖ Light quarks expensive.

Many studies are being made.

Lattice chiral quarks: domain-wall / overlap

Still quite expensive to simulate.Real applications to T > 0 have just started! => HotQCD, JLQCD

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So far, most large-scale simulations at finite T and µ have been made with

Staggered-type quarksPros: ✓ Relatively cheap to simulate.

✓ A modified chiral sym. preserved: U(1) [=O(2)] taste-chiral sym.

Cons: ❖ 4 copies of identical fermions (“tastes”) in the cont. lim. for each flavor.

=> “4th root trick” to remove unwanted 3 : detM => [detM]1/4

⇓❖ Non-local => Universality arguments fragile.

? continuum limit?　　<= Empirically OK if the continuum limit is taken first.

? chiral scaling on finite lattices? ???

❖ Taste violation problem at a>0 => errors in flavor identifications.

(e.g.) many π’s in the taste space, one is light due to the taste-chiral sym.

Lightest π (pNG π) usually treated as “physical”.Other π’s do contribute in dynamical effects.

=> lattice artifacts.

mq is effectively much heavier.

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Improved staggered quarksVarious actions proposed to milden lattice artifacts including the taste violation:

asqtad / p4 / HYP / stout / HISQ / ...The extent of improvement differs depending on the action.

Recently, it was noted that a good control of the taste-violation is essential to obtain physical results with staggered-type quarks.

Orginos et al, hep-lat/9909087

mπ/mρ=0.55

HISQ

Bazavov-Petreczky (HotQCD)arXiv:1012.1257

The magnitude of the taste violationHISQ < stout < asqtad < p4

heavy “π” masses (at T~170MeV with mπpNG ~135 MeV) Nt~8 ~ 400-600 asqtad < p4 ~ 300-500 stout ~ 200-400 HISQNt~12 ~ 200-350 stout

asqtadunimproved

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O(4) scaling tests on the latticeWilson-type quarks (NF=2)

Iwasaki et al. (QCDPAX)PRL78(’97)‣ Iwasaki gauge + Wilson‣ Nt=4, mπ ~ 600-900 MeV

AliKhan et al. (CP-PACS) PRD63(’01)‣ Iwasaki gauge + Clover‣ Nt=4, mπ ~ 600-1000 MeV

Bornyakov et al. (QCDSF)PRD82(’10)‣ plaquette gauge + Clover‣ Nt = 8,10,12, mπ ≈ 420-1300 MeV

Proper renormalization needed to recover the chiral symmetry in the continuum limit.

M ~ via axial W.I Bochicchio et al.(’85)

QCD data vs. O(4) scaling function and exponents

➡ Consistent with the O(4) scaling, though quarks are heavy.

No indication of 1st order chiral transition.QCD data well described by the O(4) scaling function with O(4) exponents.

O(4) scaling fit for Tc

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Nógrádi (Budapest-Wuppertal) @ Lat11‣ Symanzik/tree + 6-level-stout-clover NF=2+1‣ fixed scale approach at 6 ß values

Comparison with staggered (stout, Nt=8-12) at mπ≈545 MeV, mK≈612 MeV susceptibilities, renormalized Polyakov loop => well consistent with each other.

Continuum thermodynamics feasible with improved Wilson.

No O(4) scaling tests yet.

Burger (tmfT) @ Lat11, 1102.4530‣ Symanzik/tree + maximally twisted Wilson NF=2‣ Nt=8–12 mπ≈320-480 MeV

O(4) fit for mπ≈320-480 MeV works well => Tc = 160–270 MeV.

Difficult to discriminate between O(4) and 1st order yet.

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O(4) scaling tests on the latticeStaggered-type quarks

★ O(4) vs. O(2)The symmetry of 4-taste staggered quark action is the O(2) taste-chiral symmetry.

This is so also with the 4th root trick detM ⇒ [detM]1/4, therefore,

Sym. of the system in the chiral limit = O(2) for any NF.

=> When the chiral transition is 2nd order on the lattice, we expect O(2) scaling not O(4). O(4) may be realized when (1) continuum extrapolation, and then (2) chiral extrapolation.

In practice, O(2) ≈ O(4) numerically.

Caveat: Universality may be inapplicable on finite lattices due to the non-locality.

Ejiri et al. (BNL-Bielefeld)PRD80(’09)‣ p4, NF=2+1‣ Nt = 4, mud/ms ≈ 1/80-1/20

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Improved staggered quarks (NF=2+1)Ejiri et al. (BNL-Bi) PRD80(’09) (Nt = 4); Lat10 (Nt = 8)

‣ p4, Nt=4, ms≈physical, ml/ms= 1/80 – 1/20 (mπpNG ≈ 75 – 150 MeV)

➡ Consistent with O(2)

O(2)

It turned out from intensive studies of T>0 QCD with staggered-type quarks around ’09-’11,

a good control of taste violation essential to extract physical conclusions with staggered-type quarks.

HotQCD @ QM11, Lat11

‣ HISQ‣ ms ≈ physical, ml/ms = 1/27 – 1/20

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Improved staggered quarks (NF=2+1)

➡ The phys. point dominated by the O(N) scaling ➡ 2nd order chiral transition for NF=2➡ Tricritical point locates lower than ms

phys

With staggered-type quarks, we should have taken the cont. limit prior to the chiral scaling studies.

If these properties remain also after taking the cont. limit, ...

Consistent with small mc for NF=3:

mπc ≤ 45 MeV HISQ Nt=6 Ding et al. @ Lat11

mc/mudphys ≤ 0.12 stout Nt=6 Endrodi et al. @ Lat07

Consistent with broken U(1)A at T≈Tc and above:HISQ, DW NF=2+1 HotQCD @ Lat11Overlap NF=2 Cossu (JLQCD) @ Lat11

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Tc at the physical point

Crossover => uncertainty in the value of Tc depending on the definition.

There was a big discrepancy about the value of Tc beyond the uncertainty (’06-’09). => major discrepancy resolved in ’10-’11. <= importance of controlling the taste violation (see KK @ Lat10)

Tc <= susceptibility peak (chiral / quark# / Polyakov loop / ...) etc. + scale determined by T=0 simulations

Tc at the phys. pt. in the cont. limit: available so far with stag. quarks only.

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Latest values of Tc with improved staggered quarks (NF=2+1)at the physical point and in the continuum limit

Tc = 157(4)(3)(1) MeVpreliminary

Tc =

Stout Borányi et al. (Wuppertal-Budapest) JHEP 09 (‘10)

HISQ + asqtad HotQCD @ QM 11, Lat 11

MeV

Major discrepancy in Tc removed. <= good control of the taste violation important

Remaining disagreements: shape of χ, EOS, operator dependence in Tc, etc.

with χchiral

(similar with Polyakov suscept.)

Nt=8 Nt=8[-----------------------------------------------> identified with “inflection point”

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Tc under external magnetic field

Motivated by the chiral magnetic effect, phase structure under uniform background magnetic field is studied.

T. Ishikawa (HotQCD) @ Lat11‣ DW NF=2 => Ls ≥ 94 needed to study low modes

which are relevant to the CME.

Endrõdi (Wuppertal-Budapest) @ Lat11‣ stout NF=2+1 => Tc lower with B.‣ Nt=8–12 mπ≈135 MeV Remains crossover up to eB≥1 GeV2.

D’Elia’s result recovered at large mq.

D’Elia-Mukherjee-Sanfilippo, 1005.5365‣ KS NF=2 => Tc higher, trans. sharper‣ Nt=4 mπpNG≈195-480 MeV with B.

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µ ≠ 0

µ

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Difficulties of the lattice at µ ≠ 0

✤ LQCD at µ≠0

✤ Complex phase problem (sign problem)=> Exponential cancellation due to the phase fluctuation of detM

✤ Techniques for small µ/T✦ Taylor expansion✦ Multi-parameter reweighting✦ Canonical ensemble✦ Analytic continuation from imaginary µ✦ Complex Langevin✦ Combination of them

[ + density of state method / cumulant expansion / ... ]

✦ => µ/T ≤ O(1) accessible

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Physics at µ/T ≤ O(1)

Ejiri et al. (WHOT-QCD)PRD 82 (’10)

Clover, NF=2, mPS/mV=0.65, n≤2

Allton et al. (Bielefeld-Swansea)PRD 71 (’05)

p4, NF=2, mq/T=0.4, n≤6

thru measurement of quark propagators

etc.

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Ejiri et al. (WHOT-QCD)PRD 82 (’10)

Clover, NF=2, mPS/mV=0.65, n ≤ 2

Allton et al. (Bielefeld-Swansea)PRD 71 (’05)

p4, NF=2, mq/T=0.4, n ≤ 6

etc.

quark number susceptibility

isospin susceptibility

charge fluctuation

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Generalized/non-linear susceptibilities: S.Gupta@CPOD2009, Karsch-Redlich, PLB695(’11)

Comparison with experiment on the chemical freeze-out line

V-independent combinations :

variance

skewness

kurtosis

S.Gupta et al., Science 332, 2011(’11) X. Luo for STAR, arXiv:1106.2926

Event-by-event proton distribution by

Mukherjee @ QM2011, arXiv:1107.0765

Mumbai: NF=2 KS, Nt=4,6, MπpNG≈230MeV

BNL-Bi: NF=2+1 p4, Nt=8, MπpNG≈160MeV

Lattice artifacts yet to be studied:* cutoff effects / taste violation* scale setting error / heavy Mπ* finite V effects* extrapolation to the freeze-out pt. etc.

See Karsch-Redlich, arXiv:1107.1412 for caveats.

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Phase structure at small µ/T

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Transition/crossover curve at µ≈0 Taylor expansion

Endröli et al., JHEP1104stout, NF=2+1, Nt=6-10, Physical pt., continuum limit.

reflection point:

BB

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Transition/crossover curve at µ≈0 O(4) scaling around the physical point at µ = 0.

O(4) scaling at small µLeading contribution from µ is µ2 and chiral symmetric.

Ejiri et al. (WHOT-QCD) PoS(Lat10)181

clover, NF=2 O(4) fitc ≈ 0.02--0.05 (preliminary)

Kaczmarek et al. (BNL-Bi) PRD 83, 014504 (’11)

p4, NF=2+1 O(2) fit

<= chiral violating terms<= chiral symmetric terms

c => curvature of the crossover line in the (ß, µ) plane

=> κ = 0.0066(5) => κ ~ 0.008

=> We can extract κ by an O(4) scaling fit too.

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Imaginary µNo sign problem => can simulate at µ2<0

A recent example:

Nagata-A.NakamuraPRD83(’11)

clover, NF=2, Mps/Mv=0.8

Analytic continuation: fit µ2<0 => extrapolate to µ2>0

Cea et al., PRD80(’09)

KS, NF=2, isospin µKim et al., PoS(Lat05)

Potts model

Model studies: extrapolation works to some extent.

But many precise data points are required.

µI2 = – µ2

I come back to imag. µ later.κ = (d1/d0)2 ≤ 0.006

=> The sign and the magnitude of the curvatures roughly consistent: κ ≈ 0.005 -- 0.009 in contrast to κ ≈ 0.02 for the freeze-out.

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Critical point

Levkova @ Lat11

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Radius of convergence of the Taylor series

<= if the singularity locates on the real axis e.g., if cn>0 after some finite n.

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Radius of convergence of the Taylor series

<= if the singularity locates on the real axis e.g., if cn>0 after some finite n.

=> lower bound for the distance to the crit. pt.

But,

✦ only quite short series available (at most 8, but models often require more)✦ various estimators for the radius (different at small n)✦ various quantities for the series (χ bettter than p? Karsch et al. PLB698(’11))✦ alternative estimators using, e.g., Padé series

etc.In addition to these ambiguities:✦ control of lattice artifacts /taste violation✦ continuum extrapolation✦ physical point extrapolation

=> Syst. errors not well controlled yet.34

Canonical ensemble approach

Ejiri, PRD 78 (’08)p4, NF=2, 163x4, mq/T=0.41st order for µq/T >≈ 2.4

Li et al. (χQCD) arXiv:1103.3045Clover, NF=3, 63x4, mπ~700-800MeV

=> 1st order region directly accessible by lattice simulations.Issues:

• Coarse lattices • Physical point extrapolation• Thermodynamic limit keeping ρ = N/V fixed (control of finite V effects)

Many trials since decades.

=> Syst. errors not well controlled yet.35

A histogram method(a density of state method)

★ µ = 0 around the heavy quark limitEffective potential for the plaquette distribution

standard plaquette gauge action + Wilson quark action

Reweighting in ß => Combine data at different ß to obtain accurate Veff in a wide range of P.

Reweighting in κ~1/mq [with the hopping parameter expansion]

=> phase structure from the shape of Veff

H. Saito et al. (WHOT-QCD), arXiv:1106.0974

NF=2

NF=2+1

0.54 0.545 0.55 0.555

P0

250

500

Histogram

5.685.685

5.695.6925

5.7

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★ µ ≠ 0Reweighting factor in κ and µ with the hopping parameter expansion

Veff for P and the Polyakov loop Ω=> phase structure at µ≠0 from the shape of Veff Lines of dVeff/dP=0 and dVeff/dΩR=0

typical double well typical single well

Talk by EjiriPoster by H. Saito

ß = 5.69, K = µ = 0

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.544 0.545 0.546 0.547 0.548 0.549 0.55 0.551 0.552-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P

!R

0.12 0.16 0.2 0.24 -0.04 0

0.04-4

0

4

8

0.12 0.16 0.2 0.24 -0.04 0

0.04 0.12 0.16 0.2 0.24 -0.04

0 0.04

0.12 0.16 0.2 0.24 0.28-0.04 0

0.04 0.08

-4

0

4

8

0.12 0.16 0.2 0.24 0.28-0.04 0

0.04 0.08

0.12 0.16 0.2 0.24 0.28-0.04 0

0.04 0.08

!/T

"cp(!)0.6

* κcp at small µ

κcp for isospin chem. pot. (leading order hopping param. expansion)

0.547

0.548

0.549

0.55

0.551

0.552

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

-30-20-10

0 10 20 30 40

V0(P, R)

P

R

10 5 0 -5

-10 -15 -20

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★ Application to the light quark region

Veff for P and gluon energy quark energy

<= For reweighting with dynamical quarks, |detM| is important, in place of the Ω for the heavy quark case.

Simulation with |detM(µ)| => reweight to detM(µ)

Sign problem in the phase average<= cumulant expansion + Gaussian approx. with confirming

Talks by Ejiri and Nakagawa

0.442 0.444 0.446 0.448 0.45 0.452 0.454 0.456 0.458 0.46P 2

4 6

8 10

12 14

16 18

F

-2.5-2

-1.5-1

-0.5 0

0.5 1

1.5

-lnw (β,µ)

0.442 0.444

0.446 0.448

0.45 0.452

0.454 0.456

0.458 0.46

P 2 4

6 8

10 12

14 16

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F

12

12.5

13

13.5

14

14.5

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<θ2>c

0.442 0.444 0.446 0.448 0.45 0.452 0.454 0.456 0.458 0.46P 2

4 6

8 10

12 14

16 18

F

4 4.5

5 5.5

6 6.5

7 7.5

8 8.5

-lnW (β,µ)

Veff with |detM(µ)| + reweighting by the phase average => full Veff

The method looks feasible. Investigation under way.

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Imaginary µ

★ Similar to the case of Tc(µ)/Tc(0)de Forcrand-Philipsen, JHEP0701; 0811KS Nt=4

de Forcrand et al.@Lat11KS Nt=4

must turn back to have the crit. point at mqphys.

Or, more complicated phase structure at finite µ?

Caveat: unimproved staggered quarks on Nt=4 have large lattice artifacts.

µ

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Summary

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K. Fukushima and T. HatsudaRep. Prog. Phys. 74, 014001 (’11)

To which extent are these pictures confirmed on the lattice?

“Columbia plot”

QCD PHASE STRUCTURE

at mq’s ≈ physical valuesat µ ≈ 0

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K. Fukushima and T. HatsudaRep. Prog. Phys. 74, 014001 (’11)

“Columbia plot”

QCD PHASE STRUCTURE

at mq’s ≈ physical valuesat µ ≈ 0

µ/T ≤ O(1)O(4) probable42

Qualitative understanding on the phase structure has much progressed.

Tuning to the physical point / extrapolation to the cont. limit require quantitative precisions.

Currently, the staggered-type quarks are the closest to the goal. Here,

A good control of taste violation essential to extract physical conclusions with staggered-type quarks. Studies of EOS, fluctuations, etc. with improved staggered quarks underway.

But, because the fundamental uneasiness with staggered-type quarks,

Need to confirm the results with theoretically sound lattice quarks (Wilson-Clover, DW, Overlap).

=> Scaling studies, EOS, .... underway (many developments reported @ Lat10-11)

Besides, many new ideas, steady/new progresses, ... appeared. => The next reviewer will be busy too!.

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thank you!

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