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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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 2
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 3
Conjectured QCD phase diagram for physical Pion
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 4
∼15
5M
eV
Conjectured QCD phase diagram for physical Pion
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 4
∼15
5M
eV
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 5
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 6
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 7
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 .
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 8
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 9
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 10
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 11
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 12
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 13
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 14
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 15
−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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 16
−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].
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 17
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 18
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 19
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}
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 20
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 21
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).
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 22
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 23
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 24
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 25
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 26
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 27
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].
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 28
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].
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 28
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].
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 28
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 29
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
}
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 30
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
}
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 30
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
}
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 30
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 31
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 32
QCD in 2nd RW plane
Under Z(2) transformation :
{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
QCD in 2nd RW plane
Under Z(2) transformation :
{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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 34
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 35
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).
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 36
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.
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 37
mπ = 140 MeV
mπ = 140 MeV
mπ = 140 MeV
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 38
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].
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 39
Nσ = 24
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
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 40
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??
Anirban Lahiri (Bielefeld University) for HotQCD QCD phase diagram 41
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 42
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 42
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