anirban lahiri (bielefeld university) for hotqcd …...qcd phase diagram anirban lahiri (bielefeld...

QCD phase diagram

Anirban Lahiri (Bielefeld University)for

HotQCD collaboration

Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 1

Overview

1 QCD at non-vanishing real µB : Taylor expansion approachIntroductionBasic definitions and some notationsResults

Pseudo-critical temperature Tpc at µB = 0 in continuumCurvature of the crossover line in continuumFluctuations along the crossover line Tpc(µB)

2 QCD with imaginary chemical potentialMotivationResults

Nature of RW endpoint for physical massNature of RW endpoint towards chiral limitInterplay between RW and chiral transitions towards chiral limit

3 Summary


Overview





3 Summary


Conjectured QCD phase diagram for physical Pion


∼15

5M

eV

Overview





3 Summary


Critical behavior and O(4) scaling : Basic quantities

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

t =1

t0

T − T 0c

T 0c

and h =1

h0

ml

ms=

1

h0H

Scaling variable :

z =t

h1βδ

= z0

(T − T 0

c

T 0c

)(1

H1/βδ

); z0 =

h1βδ

0

t0

Chiral condensate : 〈ψ̄ψ〉f =T

V

∂ lnZ

∂mf

Chiral susceptibility : χfgm =∂

∂mg〈ψ̄ψ〉f


Critical behavior and O(4) scaling : Basic quantities

Renormalization group invariant (RGI) definition of order parameter :

M = ms

((〈ψ̄ψ〉u + 〈ψ̄ψ〉d

)− mu +md

ms〈ψ̄ψ〉s

)≡ Σsub

RGI definition of order parameter susceptibility :

χM =T

Vms

(∂

∂mu+

∂

∂md

)M ≡ χsub

χdisc is the disconnected part of χsub.


Different estimators for pseudo-critical temperature

For finite chemical potential :

t =1

t0

(T − T 0

c

T 0c

+ κB

(µBT

)2)

In scaling regime :∂

∂T∼ ∂2

∂µ2B

Here we have used 3 different estimators for Tpc :1 Peak of chiral susceptibility : χM .

2 Inflection point of chiral condensate :∂

∂TM .

3 Minimum of∂2

∂µ2B

M .

At finite mass, different estimators of pseudo-critical temperature,in principle will give different results.

In chiral limit, all pseudo-critical temperatures should mergeto the chiral critical temperature, T 0

c .


Scaling functionsMass scaling of different estimators of Tpc :

T (zx, H) = T 0c

(1 +

zxz0H1/βδ

)x = t, p

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-3 -2 -1 0 1 2 3

z=z0 (T/T0c-1)H

-1/βδ

O(4)zt

zp

f'

G(z)

fχ(z)

In chiral limit :

M = h1/δfG(z)

χM =1

h0h1/δ−1fχ(z)

∂M

∂T=

1

t0T 0c

h1/δ−1/βδf ′G(z)

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


Taylor expansion in chemical potentials : Notations

Simplest case : µQ = µS = 0.

subtracted condensate :

Σsub

f4K

=

∞∑

n=0

cΣn

n!µ̂nB with cΣ

n =∂Σsub/f

4K

∂µ̂nB

∣∣∣∣µ=0

disconnected susceptibility :

χdisc

f4K

=

∞∑

n=0

cχnn!µ̂nB with cχn =

∂χdisc/f4K

∂µ̂nB

∣∣∣∣µ=0

same notation for strangeness neutral system : nS = 0,nQnB

= 0.4.


Overview





3 Summary


The subtracted chiral susceptibility

0

50

100

150

200

250

135 145 155 165 175

χsub/f4K

T [MeV]

HotQCD preliminary

Nτ = 161286

Peaks shift to lower T towards continuum.


Subtracted chiral condensate

0

5

10

15

20

25

135 145 155 165 175

Σsub/fk4

T [MeV]

ms/ml=27, Nτ=16

1286

Inflection points have been calculated from the fitted curves.


T derivative of subtracted chiral condensate

0

20

40

60

80

100

120

135 145 155 165 175

-T dc0Σ/dT

T [MeV]

ms/ml=27, Nτ=12

68

Inflection points also shift to lower T towards continuum.


−T ∂

∂T

Σsub

f4K

µB derivative of subtracted chiral condensate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

135 145 155 165 175

µQ=µS=0

-c2Σ/2

T [MeV]

ms/ml=27, Nτ=1286

Conjectured∂

∂T∼ ∂2

∂µ2B

seems to work.


−1

2

∂2

∂µ2B

Σsub

f4K

Tpc : continuum extrapolation

150

152

154

156

158

160

162

164

166

continuum

Nτ =

16

Nτ =

12

Nτ =

8

Nτ =

6

HotQCD preliminary

Tc(µB = 0) [MeV]

1/N2τ

χdisc

χsub

Σsub

∂2µ̂BΣsub

∂2µ̂Bχdisc

(156.5 ± 1.5) MeV

Patrick Steinbrecher (for the HotQCD collaboration),arXiv:1807.05607[hep-lat].


Tpc : comparison

140

145

150

155

160

165

170

Σsub

χdisc

χsub

∂µB

2 Σsub

∂µB

2 χ

disc

Σsub , Bonati2015

χtot , Bazavov

2012

Σsub , Borsanyi2010

T0 [MeV]

�� All estimators of Tpc agree with each other in continuum.


Overview





3 Summary


Curvature of the crossover line

Tpc(µB)

Tpc(0)= 1− κ2

(µB

Tpc(0)

)2

− κ4

(µB

Tpc(0)

)4

+O(

µBTpc(0)

)6

Following the pseudo-critical estimators in µB − T plane defines thecrossover line.

Employ Taylor expansion :

d

dT

χdisc (T, µB)

f4K

= (· · · )µ2B + (· · · )µ4

B + · · · = 0

Coefficients have to be zero order by order.

Demanding that coefficient of O(µ2B

)vanishes,

κχ2 =1

2T 2pc(0)

Tpc(0)∂cχ2∂T

∣∣∣{Tpc(0),0}

− 2 cχ2 |{Tpc(0),0}

∂2cχ0∂T 2

∣∣∣{Tpc(0),0}


Curvature of crossover line

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Σsub

χdisc

Σsub , B

ellwied

2015

κ2

κ4

nS=0, nQ/nB=0.4

κ4 is consistent with zero.

κ2 is the dominant contribution in curvature of crossover line.


The QCD crossover line

135

140

145

150

155

160

165

170

0 50 100 150 200 250 300 350 400

Tc [MeV]

µB [MeV]

HotQCD preliminary

nS = 0, nQnB

= 0.4

crossover line: O(µ4B)constant: ε

sfreeze-out: STAR

ALICE

STAR : Phys. Rev. C 96 044904 (2017). ALICE : Nucl. Phys. A 931 103 (2014).


Overview





3 Summary


Fluctuations along the QCD crossover line Tpc(µB)

Baryon-number fluctuations :

σ2B

V f3K

=1

V f3K

∂2 lnZ

∂µ̂2B

=

∞∑

n=0

cBnn!µ̂nB with cBn =

1

V f3K

∂ lnZ

∂µ̂(n+2)B

∣∣∣∣∣µB=0

σ2B diverges at the critical end point.

study increase along the crossover line :

σ2B (Tpc(µB), µB)

σ2B (Tpc(0), 0)

−1 = λ2

(µB

Tpc(0)

)2

+λ4

(µB

Tpc(0)

)4

+O(

µBTpc(0)

)6


Baryon-number fluctuations along Tpc(µB)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300

σ2B(Tc(µB), µB)/σ2B(T0, 0)− 1

µB [MeV]

HotQCD preliminary

nS = 0, nQnB

= 0.4O(µ4B)O(µ2B)HRG

Baryon number fluctuation increases smoothlyonly up to 30% for µB ≤ 250 MeV.


Scalar susceptibility along Tpc(µB)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 50 100 150 200 250 300

nS=0, nQ/nB=0.4

χdisc(Tc(µB),µB)/χdisc(T0,0) - 1

µB [MeV]

HotQCD preliminary

O(µB4)

O(µB2)

No increase in chiral susceptibility along QCD crossover line.

CEP is unlikely to be found for µB ≤ 250 MeV.


Overview





3 Summary


Nature of chiral transition for µB = 0

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

O. Philipsen and C. Pinke. Phys. Rev. D93, 114507, 2016.

Nf = 3 : No direct evidence of 1st order transition down to mπ = 80MeV. Scaling argument pushes it further to mπ = 50 MeV.A. Bazavov et. al. Phys. Rev. D95, 074505 (2017).

Nf = 2 + 1 : No evidence of 1st order transition down to mπ = 80MeV. Preliminary analyses do not give a hint for Z(2) transition downto mπ = 55 MeV. H.-T. Ding et. al. arXiv : 1807.05727 [hep-lat].


Columbia plot in RW plane

2nd 3d Ising

mu,d ms

∞1st tr.

1st tr.

∞

Nf =1

Nf = 2

0

−(π3

)2

(µT

)2

tric

tric

1st

Z(2)Z(2)

1st

phys. point

Nf = 3 crossover

1st order region isexpected to belargest at RWplane.

Critical mass inRW plane, iffound, can put abound for thesame in µB = 0plane.


Ising endpoint of a first order line

Effective Ising Hamiltonianwhich defines the universalcritical behavior of the system

Heff(t, h) = t E +h M

Energy-like

Magnetization-like

φ

)1(

)1(

)2()2(

K. Nagata and A. Nakamura; EPJ Web Conf. 20 03006 (2012).

Under Z(2) transformation :

{E → EM→ −M

}


Overview





3 Summary


QCD in 2nd RW plane


{ReL→ ReL⇒ Energy-like

ImL→ −ImL⇒ Magnetization-like

}

T < TRW

−0.05 0.00 0.05δ(ImL)

−0.02

0.00

0.02

δ(ReL

)

T > TRW

−0.05 0.00 0.05δ(ImL)

−0.02

0.00

0.02

δ(ReL

)

−0.05 0.00 0.05

−0.05

0.00

0.05

δ(ψ̄ψ

)

−0.05 0.00 0.05−0.05

0.00

0.05

0.10

δ(ψ̄ψ

)

ψ̄ψ also behaves as energy-like observable.


QCD in 2nd RW plane


{ReL→ ReL⇒ Energy-like

ImL→ −ImL⇒ Magnetization-like

}

T < TRW

−0.05 0.00 0.05δ(ImL)

−0.02

0.00

0.02

δ(ReL

)

T > TRW

−0.05 0.00 0.05δ(ImL)

−0.02

0.00

0.02

δ(ReL

)

−0.05 0.00 0.05

−0.05

0.00

0.05

δ(ψ̄ψ

)

−0.05 0.00 0.05−0.05

0.00

0.05

0.10

δ(ψ̄ψ

)

ψ̄ψ also behaves as energy-like observable.Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 33

Finite size scaling : order parameterLattice size is N3

σ ×Nτ with Nτ = 4 in the following.Free energy density :

ftot = b−dfs(bytut, b

yhuh, b−1Nσ

)+ fns

with ut ∼ ctt+O(t2) and ut ∼ chh+O(ht).

Order parameter : M ∼ N−β/νσ fM

(z0(T/Tc − 1)N

1/νσ

)

0

0.01

0.02

0.03

0.04

0.05

175 180 185 190 195 200 205 210 215

lines: Z(2) scaling fit

Tc = 201.1(3) MeV

<|Im

L|>

T [MeV]

Nσ=24

16

12

0

0.01

0.02

0.03

0.04

0.05

175 180 185 190 195 200 205 210 215

0

0.05

0.1

0.15

0.2

0.25

-6 -4 -2 0 2 4 6

line: Z(2) scaling curve

Tc = 201.1(3) MeV

<|Im

L|>

Nσβ

/ν

z=z0 Nσ1/ν

(T-Tc)/Tc

Nσ=24

16

12

8

0

0.05

0.1

0.15

0.2

0.25

-6 -4 -2 0 2 4 6

Jishnu Goswami, LATTICE 2018.

Finite size scaling with Z(2) exponents seems to work.


mπ = 140 MeV mπ = 140 MeV

Finite size scaling : order parameter susceptibility

Free energy density :

ftot = b−dfs(bytut, b

yhuh, b−1Nσ

)+ fns

with ut ∼ ctt+O(t2) and ut ∼ chh+O(ht).

OP susceptibility : χH ∼ Nγ/νσ fχ

(z0(T/Tc − 1)N

1/νσ

)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

175 180 185 190 195 200 205 210

line: Z(2) scaling curve

Tc = 202.6(4) MeV

χh

T [MeV]

Nσ=24

16

12

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

175 180 185 190 195 200 205 210

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

-6 -4 -2 0 2 4

line: Z(2) scaling curveTc = 202.6(4) MeV

χh N

σ−γ/

ν

z=z0 Nσ1/ν

(T-Tc)/Tc

Nσ=24

16

12

8

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

-6 -4 -2 0 2 4

Jishnu Goswami, LATTICE 2018.

Finite size scaling with Z(2) exponents seems to work.


mπ = 140 MeV

mπ = 140 MeV

Finite size scaling : Binder cumulant

Binder cumulant : B4 ∼ fB(z0(T/Tc − 1)N

1/νσ

)+O

(1/Nd

σ

)

180 190 200 210T [MeV]

1

2

3

B4

Tc=202.4(3)MeV

Nσ=8Nσ=12Nσ=16Nσ=24

−7.5 −5.0 −2.5 0.0 2.5 5.0z = z0tN

1/νσ

1

2

3

B4−

(1/N

3 σ)B

ns

Tc=202.4(3)MeV

Z(2) lineNσ=8Nσ=12Nσ=16Nσ=24

RW endpoint seems to be of 2nd order and belongs to Z(2)universality class.

In agreement with calculations with stout improved staggeredfermions. Bonati et. al. ; Phys. Rev. D93 074504 (2016).


mπ = 140 MeV

mπ = 140 MeV

Finite size scaling : Chiral observable

175 180 185 190 195 200 205 210 215T [MeV]

5

10

15

20

25

30

∆ls

Nσ=12

Nσ=16

Nσ=24

175 180 185 190 195 200 205 210 215T [MeV]

0

100

200

300

400

−Td

∆ls

dT

Nσ=12

Nσ=16

Nσ=24

180 190 200 210T [MeV]

50

100

150

m2 sχdisc

ψ̄ψ/f

4 K

Ns = 12Ns = 16Ns = 24

∆ls = Σsub/f4K , in 1-flavor

normalization.

Inflection point of ∆ls is veryclose to TRW.

Behavior of χdisc is under activeinvestigation.


mπ = 140 MeV

mπ = 140 MeV

mπ = 140 MeV

Overview





3 Summary


Quark mass dependence of RW transition

180 190 200 210T [MeV]

0.5

1.0

χ|ImL|

ml = ms/27ml = ms/40ml = ms/160ml = ms/320

ms/ml mπ (MeV)

27 140

40 110

160 55

320 40

TRW decreases towards chiral limit.

No unusual qualitative change in the order parameter susceptibilitywith decreasing quark mass.

Order of the RW endpoint seems to be unchanged??

Consistent with calculations with stout improvedstaggered fermions. Bonati et. al. ; arXiv:1807.02106[hep-lat].


Nσ = 24

Overview





3 Summary


Chiral observables

175 180 185 190 195 200 205 210 215T [MeV]

5

10

15

20

25

30

∆ls

ml = ms/27

ml = ms/40

ml = ms/160

175 180 185 190 195 200 205 210 215T [MeV]

50100150200250300350400

−Td∆

ls

dT

ml = ms/27

ml = ms/40

ml = ms/160

180 190 200 210T [MeV]

0

100

200

300

400

m2 sχ

disc

ψ̄ψ/f

4 K

ml = ms/27ml = ms/40ml = ms/160ml = ms/320

Goldstone contribution in χdisc

below Tpc is prominent.

Finite volume effects in χdisc arestill being investigated.

Do chiral and RW phasetransitions coincide in chirallimit??


Nσ = 24

Nσ = 24

Nσ = 24

Summary

Taylor expansion for real µB :Precise calculation of Tpc(µB = 0) = 156.5± 1.5 MeV.For strangeness neutral system : κ2 = 0.0123± 0.003 andκ4 = 0.000131± 0.0041.There is no signal of CEP for µB ≤ 250 MeV.

QCD with imaginary µB :RW endpoint for physical mass is found to be 2nd order belonging toZ(2) universality class.Order of the RW endpoint seems to be unchanged with decreasingquark mass.Chiral and RW transitions may coincide in chiral limit.


anirban lahiri (bielefeld university) for hotqcd …...qcd phase diagram anirban lahiri (bielefeld...

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