Anirban Lahiri (Bielefeld University) for HotQCD collaboration · 2018. 11. 22. · Anirban Lahiri...

Chiral phase transition of (2+1)-flavor QCD

Anirban Lahiri (Bielefeld University)for

HotQCD collaboration

Anirban Lahiri (Bielefeld University) for HotQCD Chiral phase transition of (2+1)-flavor QCD 1

Overview

1 Introduction

2 Scaling analysis : basic definitions and some insights

3 Estimation of T 0c from generic scaling arguments

4 Summary and Outlook


Overview

1 Introduction





Introduction?

In QCD, transition for physical quark masses and vanishing or smallbaryon density is a crossover.

Tpc for physical mass QCD has been determined with very goodaccuracy from various observables : Tpc(µB = 0) = 156.5± 1.5 MeV⇒ “QCD transition at zero and non-zero baryon densities”by PatrickSteinbrecher on 16 May 2018, 11:30 for recent HotQCD results.

Important question : How do singular terms of the chiral phasetransition at vanishing quark mass affect observables at physicalvalues of the quark mass? ⇒ How far the physical regime is fromchiral limit w.r.t. scaling window?

Key question : What is the chiral transition temperature, T 0c ?


Introduction?

In this talk, we are trying to address the followings :

Key question : What is the chiral transition temperature, T 0c ?

Possibly another question : What is the nature of the thermal phasetransition in the chiral limit?

Two possible scenarios : [O. Philipsen and C. Pinke. Phys. Rev. D93, 114507, 2016.]

mud

ms

PureGauge

1st

1st

crossover

N f=

3phys.point

Z(2)

Z(2)

Nf = 2

mtrics

N f=

1

O(4)?U(2)L ⊗ U(2)R/U(2)V ?

chiral limit Nf = 2

mud

ms

PureGauge

1st

1st

crossover

N f=

3phys.point

Z(2)

Z(2)

Nf = 2

N f=

1

chiral limit Nf = 2

We are trying to pin down the correct one.


Overview

1 Introduction





Basic quantities

In terms of temperature T and symmetry breaking field H = ml/ms thescaling variables are defined as :

t =1

t0

T − T 0cT 0c

and h =1

h0

mlms

=1

h0H

Scaling variable :

z =t

h1βδ

= z0

(T − T 0cT 0c

)(1

H1/βδ

); z0 =

h1βδ

0

t0

Chiral condensate : 〈ψ̄ψ〉f =T

V

∂ lnZ

∂mf

Chiral susceptibility : χfgm =∂

∂mg〈ψ̄ψ〉f


Scaling relations

Renormalization group invariant (RGI) definition of order parameter :

M =msf4K

((〈ψ̄ψ〉u + 〈ψ̄ψ〉d

)− mu +md

ms〈ψ̄ψ〉s

)RGI definition of order parameter susceptibility :

χM =T

Vms

(∂

∂mu+

∂

∂md

)M

Close to chiral limit, singular part behaves as :

M = h1/δfG(z)

χM =1

h0h1/δ−1fχ(z)

fG(z) and fχ(z) are universal scaling functions which have beenprecisely determined from various spin models.


Scaling functions : Some intriguing factsConventional approach to determineT 0c : determine pseudo-criticaltemperature Tpc(H) from the peaklocation of χM (T,H) and

Tpc(H) = T0c

(1 +

zpz0H1/βδ

)Our approach in this work :determine temperature T−60%(H) at60% of peak height of χM (T,H) and

T−60%(H) = T0c

(1 +

z−60%z0

H1/βδ

)0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-2 -1 0 1 2 3

z=t/h1/βδ

fχ(z)

O(4)

O(2)

60% of peak

z=0

1/δ

zp

Advantage : z−60% ' 0 ⇒ reducesinfluence of regular terms, simplifies

scaling analysis.

Dependence on quark mass (H)reduced by two orders of magnitude

⇒zp z

−60%

O(2) 1.56 -0.009

O(4) 1.37 -0.01


Scaling functions : Some intriguing facts

Other approach used for consistency check : Crossing points of

HχMM

=fχ(z)

fG(z)+ regular terms.

is unique for H → 0.Ratio has a curvature in z ⇒ Non-linear part of scaling function isimportant.

0.00

0.20

0.40

0.60

0.80

1.00

-2 -1 0 1 2 3

z=t/h1/βδ

fχ(z)/fG(z) O(4)

O(2)

1/δz=0

0.00

0.20

0.40

0.60

0.80

1.00

-2 -1 0 1 2 3

z=t/h1/βδ

fχ(z)/fG(z) O(4)

O(2)

1/δz=0

-0.6 -0.3 0.0 0.3 0.6

0.12

0.16

0.20

0.24

0.28

0.32

0.36

z

-0.6 -0.3 0.0 0.3 0.6

0.12

0.16

0.20

0.24

0.28

0.32

0.36

z

1/δ

O(2) 0.209

O(4) 0.207

Difference between O(4) andO(2) is negligible in the zrange we are working.


Scaling functions : Some intriguing facts

fχ(z)

fG(z)=

0 , z → −∞

1/δ , z = 01 , z → +∞

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

f χ/f

G

(T-T0c)/T

0c

1/δ

O(2)

z0=1

H=1/20

H=1/40

H=1/80

Ratio is expected to havea constant value at thecrossing point, z = 0,i.e. in chiral limit at T 0c .

Uniqueness of thecrossing point getspoiled in presence ofregular terms.

Around T 0c :slope ∝ H−1/βδ andcurvature ∝ H−2/βδ.

Determine temperature Tδ(H) which satisfies :

HχM (Tδ, H)

M (Tδ, H)=

1

δ⇒ T 0c = lim

H→0Tδ(H)

.Anirban Lahiri (Bielefeld University) for HotQCD Chiral phase transition of (2+1)-flavor QCD 11

Disclaimer

Here we will mainly show results for Nτ = 8 only.

Same analyses also carried out for Nτ = 6.

Same analyses for Nτ = 12 are still at preliminary stage.

Our goal is to find out the chiral transition temperature, T 0c , incontinuum limit.

More lattices are still being generated for low masses and the analysesare not settled yet for those parameter sets.

We are working on the systematic error estimation for variousanalyses and most of the results shown here are just indicative.


Overview

1 Introduction





No evidence for 1st order transition

Volume dependence ofχM is studied forH = 1/80 whichcorresponds tomπ = 80 MeVin continuum.

χmaxM is not proportionalto volume.

150

200

250

300

350

400

450

135 140 145 150 155 160 165 170χ

M

T [MeV]

HotQC

D preli

minary

Nτ=8

ml/ms=1/80

Ns=32

Ns=40

Ns=56

Ns=∞

Possibility for 1st order phase transition can beruled out at mπ = 80 MeV for Nτ = 8.We only focus on O(2), O(4) and Z(2).Similar results are also obtained for Nτ = 6 and 12.Nτ = 12 : 42

3 × 12 and 603 × 12 lattices have beenused for ml/ms = 1/40.


Estimation of T 0c from HχM/M

Behavior of the ratio is like Binder cumulant at critical point.[F. Karsch and E. Laermann. Phys. Rev. D50, 6954, 1994.]

For H = 1/27and 1/20, regularcontributionsmay be largerthan lowest threemasses.

Crossing pointsfor three lowestmasses gives anestimateT 0c ∼ 144 MeVfor Nτ = 8.

0.0

0.2

0.4

0.6

0.8

1.0

130 140 150 160 170 180

H⋅χ

M/M

T [MeV]

HotQC

D prelim

inary

Nτ=8

ml/ms=1/160

1/80

1/40

1/27

1/20

140 144 148

0.12

0.16

0.20

0.24

0.28

0.32

0.36


Fitting χfKM and MfK with rational functions

■ ■ ■■ ■■

■ ■ ■ ■ ■ ■★ ★

★★

★ ★ ★★

★▲ ▲▲ ▲

▲▲▲ ▲ ▲ ▲●

●●●

● ●

●

●

◆

◆

◆◆

◆

◆

◆

◆

■ H=1/20★ H=1/27▲ H=1/40● H=1/80◆ H=1/160

130 140 150 160 170 180

100

200

300

400

500

600

700

T [MeV]

χM

Nτ=8

■■ ■■

■■

■

■■■

■■

★★

★

★

★

★

★

★★

▲▲▲▲▲

▲▲

▲

▲▲

●●●●

●

●

●●

◆◆◆

◆

◆

◆◆ ◆

■ H=1/20★ H=1/27

▲ H=1/40

● H=1/80

◆ H=1/160

130 140 150 160 170 180

5

10

15

20

T [MeV]

M

Nτ=8

χM for different H are fittedwith rational functions :

1 peak location : Tpc(H).2 peak height : χmaxM (H).3 T−60%, defined by :

χM(T−60%

)=

60

100χmaxM

M at Tpc and T−60% are

calculated for different H byfitting with rational functions.


Sanity checks for T 0cT−60% for three lowestmasses indicatesT 0c ∼ 142 MeV.

141.0

141.5

142.0

142.5

143.0

1/16

01/

801/

40

55 80 110

T-60% [MeV]

ml/ms

mπ [MeV]

HotQCD preliminary

Nτ=8

Joint scaling fit of M andχM to MEOS with O(2)exponents also givesT 0c ∼ 145 MeV forNτ = 8.

0.10

0.15

0.20

0.25

0.30

0.35

0.40

140 141 142 143 144 145 146 147 148

H⋅χM/M

T [MeV]

HotQC

D prelim

inary

T0c

1/δNτ=8

H=1/40

H=1/80

H=1/160

Simultaneous fit (with regular term) ofH · χM/M to the three lowest massesgives T 0c ∼ 144 MeV for Nτ = 8.�� Current estimate of T 0c for Nτ = 8 is 144(2) MeV.


Towards continuum T 0c

Current estimate of T 0c is 147(2) MeVfor Nτ = 6 and 138(3) MeV forNτ = 12.

T−60% is approximately independent of Hestablishes it as a good estimator of T 0c .

■ ■■ ■

■ ■ ■■

■★★

★★

★ ★★

★★

▲

▲▲

▲ ▲▲

▲

●

●

● ●

●

●

●

■ H=1/20★ H=1/27▲ H=1/40● H=1/80

130 140 150 160 170 180

100

200

300

400

T [MeV]

χM Nτ=12

125

130

135

140

145

150

155

160

165

1/162

1/122

1/82

1/62

HotQ

CD pr

elimi

nary

Tc0 [

Me

V]

1/Nτ

2

from peak, ml/ms= 1/271/401/80

1/160from 60% of peak, ml/ms= 1/40

Calculations at different ml/ms suggestthat cut-off effects are only weakly(not) dependent on quark mass.

Experience from ml/ms = 1/27calculations suggest that extrapolationto continuum limit based on Nτ = 8and 12 is safe.�� We find T 0c = 138(5) MeV in continuum.


Order of chiral transition : work in progress....

M

χM= H

fG(z)

fχ(z)

For small H the dataseems to be linear.

Lines are NOT fittedcurves ratherexpectations forO(2) and O(4).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1/16

01/

801/

601/

401/

271/

20

55 80 90 110 140 160

M/χ

M

ml/ms

mπ [MeV]

HotQC

D preli

minary

solid : Nτ=6

open : Nτ=8

half-filled : Nτ=12

@χmaxM

@0.6⋅χmaxM

Z(2) transition, at some finite Hc, will results into a strong Vdependence and sudden drop in the ratio ⇒ 1st order transition isunlikely for mπ > 55 MeV.

Additional low H measurements : slope can be directlydetermined as fit parameter.


Order of chiral transition : work in progress....

0.00

0.02

0.04

0.06

0.08

0.10

1/16

01/

801/

601/

401/

271/

20

55 80 90 110 140 160

M/χM

HotQ

CD pr

elimi

nary

solid : Nτ=6

open : Nτ=8

half-filled : Nτ=12

@χmaxM

1/16

01/

801/

601/

401/

271/

200.00

0.05

0.10

0.15

0.20

0.25

0.3055 80 90 110 140 160

M/χM

ml/ms

mπ [MeV]

HotQ

CD pr

elimi

nary

colored : O(N)

black solid : Z(2)@Hc=1/120

black dashed : Z(2)@Hc=1/240

@0.6⋅χmaxM

Z(2) lines are schematic.

If M is not exactly order parameter then the Z(2) lineswill have a curvature.

Mixing becomes weak as Hc becomes small.

Data seems to favor O(N) compared to Z(2).


Overview

1 Introduction





Summary and Outlook

Summary :1 Measurements are according to the expectations even without being

specific about the universality class.2 Proposed method can distinguish between Z(2) and O(N).3 Scaling fits with O(2) exponents worked reasonably.4 Current estimate of T 0c , in continuum, is 138(5) MeV.

Outlook :1 Finer lattices are being generated for H = 1/40 and 1/80.2 Calculations at more masses have been planned to have better control

on the functional dependence of various quantities.3 Additional low H measurements will help us to be confident about the

scaling behavior.


Summary and Outlook

Summary :1 Measurements are according to the expectations even without being

specific about the universality class.2 Proposed method can distinguish between Z(2) and O(N).3 Scaling fits with O(2) exponents worked reasonably.4 Current estimate of T 0c , in continuum, is 138(5) MeV.

Outlook :1 Finer lattices are being generated for H = 1/40 and 1/80.2 Calculations at more masses have been planned to have better control

on the functional dependence of various quantities.3 Additional low H measurements will help us to be confident about the

scaling behavior.


IntroductionScaling analysis : basic definitions and some insightsEstimation of Tc0 from generic scaling argumentsSummary and Outlook

Anirban Lahiri (Bielefeld University) for HotQCD collaboration · 2018. 11. 22. · Anirban Lahiri...

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