exploring the qcd phase diagram with imaginary chemical ......anirban lahiri 1 jishnu goswami...

Exploring the QCD Phase diagram with imaginary chemical

potential with HISQ action

Collaborators: Frithjof Karsch, Christian Schmidt, and Anirban Lahiri

�1

Jishnu Goswami

Universität Bielefeld

Plan of the talk

• Introduction

• Status on chiral phase transition with HISQ at 𝜇 =0

• Status on chiral phase transition in the RW plane.

• Results and discussions.

!2

Introduction

!3

2nd 3d Ising

mu,d ms

11st tr.

1st tr.

1Nf

= 1

Nf = 2

0

�✓⇡3

◆2

✓µT

◆2

tric

tric

1stZ(2) Z(2)

1st

phys. point

Nf = 3 crossover

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

Central Question:Nature of the chiral symmetry restoring transition at 𝜇=0 at the chiral limit??

Possible scenario of extended 3d Columbia plot

The 1st order region is expected to be largest in the RW plane(𝜇/T=(2k+1)𝜋/3). Thus critical mass in the RW plane puts a bound on the critical mass at 𝜇=0.

Does a 1st order chiral symmetry restoring transition exist at 𝜇=0 below a certain critical quark mass (mcri) ??

Chiral transition for zero chemical potential with HISQ

• HotQCD results on chiral phase transition, [ ]

!4

Nf =3 : 1st order phase transition ruled out for 230 MeV > m𝜋 >80 MeV. Bound on critical pion mass is given as, m𝜋

cr ≲50 MeV from the scaling analysis.

μ = 0

mud

ms

PureGauge

1st

1st

crossover

Nf=

3phys.point

Z(2)

Z(2)

Nf = 2

mtrics

Nf

=1

O(4)?U(2)L ¢ U(2)R/U(2)V ?

chiral limit Nf = 2

Nf =2+1 : No hint of 1st order phase transition for m𝜋 >55 MeV. chiral transition is most likely 2nd order O(N) rather than Z(2).

A. Lahiri et. al. , QM 2018, arXiv:1807.05727

Bazavov et. al. PRD 95, 074505 (2017)

m𝜋﹥1 GeV, for the ‘heavy quark mass RW transition’

Standard staggered action: m𝜋～400 MeV (N𝝉=4) Standard Wilson action: m𝜋～930 MeV(N𝝉=4) m𝜋～680 MeV (N𝝉=6) 1st order end point (of the line of 1st order RW transitions) exist already for 𝜇/T=𝜋/3 and mcri >mphy.

The results are strongly fermion discretization scheme and cut-off(N𝝉) dependent.

!5

P. de Forcrand et. al, PRL 105, 152001(2010), Owe Philipsen et. al, PRD 89, 094504(2014), Christopher Czaban et al, PRD 93, 054507 (2016)

‘small quark mass RW transition’

Studies in the RW plane Mostly with unimproved actions

(Nf = 2)

Possible scenario of RW end point—plenary by S. Mukherjee

• Stout improved staggered fermions(Nf =2+1): At the physical quark mass point(m𝜋～135 MeV) a 2nd order transition in the 3d-Ising universality class happens instead of a 1st order at the RW endpoint.

No 1st order end point (of the line of 1st order RW transitions) for m𝜋 >50 MeV.

• HISQ(Nf =2): Order of the phase transition at physical point is not clear(large cut-off effects) .

!6

C. Bonati et. al,PRD 93, 074504 (2016)

L.K.Wu, et al. PRD 97,114514(2018)

C. Bonati et. al,arXiv:1807.02106 [hep-lat]

Studies in the RW plane Very recent studies with improved actions,

Studies with HISQ in the RW plane

!7

Mq = DHISQ(μf ) + mq

Action,

Simulation details,

(MeV)

8 4 1/27 135

12 4 1/27 135

16 4 1/27,1/40,1/60

135, 110, 90

24 4 1/27,1/40 135, 110

Nσ Nτml

ms

mπ We vary 𝛽 in the range [5.850-6.038],

Nf = 2 + 1,μT

=π3

We work on the 2nd RW plane

Z(T, μ) = ∫ [𝒟U]det[Mud(μf )]1/2det[Ms(μf )]1/4exp[−SG]

Generally we generated 20k trajectory per 𝛽 value away

from 𝛽c and 80k trajectory near 𝛽c

∼ Tc ± 0.1Tccorresponds to,

Ising endpoint of a first order line

!8

Heff(t, ξ) = tℰ + hℳEffective Ising Hamiltonian which

defines the universal critical behaviour of the system

temperature likefield

energy likeoperator

magnetic field like

magnetization like operator

(order parameter)

Corresponding critical behaviour of QCD in 2nd RW plane [Z(2) transformation],

under Z(2) transformation,ℰ → ℰℳ → − ℳ

Im L → − Im LRe L → Re L

order parameter

energy like

i.e. at μ = μRW

limh!0

limV!1

hIm Li ⌘ limV!1

h|Im L|i =(0, if � < �c

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!9

Corresponding critical behaviour of QCD at 2nd RW plane [Z(2) transformation],


order parameter

energy like

i.e. at μ = μRW

limh!0

limV!1

hIm Li ⌘ limV!1

h|Im L|i =(0, if � < �c


!9



order parameter

energy like

i.e. at μ = μRW

limh!0

limV!1

hIm Li ⌘ limV!1

h|Im L|i =(0, if � < �c


�0.02 �0.01 0.00 0.01 0.02�(ReL)

�0.04

�0.02

0.00

0.02

0.04

�(Im

L)

� = 5.925T<Tc

!9



order parameter

energy like

i.e. at μ = μRW

limh!0

limV!1

hIm Li ⌘ limV!1

h|Im L|i =(0, if � < �c


�0.02 �0.01 0.00 0.01 0.02�(ReL)

�0.04

�0.02

0.00

0.02

0.04

�(Im

L)

� = 5.925T<Tc

�0.02 �0.01 0.00 0.01 0.02�(ReL)

�0.08�0.06�0.04�0.02

0.000.020.040.06

�(Im

L)

� = 5.990T>Tc

Finite size scaling and Z(2) universality class

!10

ut ∼ ctt, ut ∼ chhnear, T → Tc

t =T − Tc

Tc∼ β − βc

χh = z1Nγ/νσ fh(z0tN1/ν

σ )

χt = z2Nα/νσ ft(z0tN1/ν

σ )

f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,

Responsible for the universal critical behaviour

Susceptibility of |Im L| Susceptibility of |Re L|

< O > = ( . . . . )∂∂h

fs( . . . ) |h→o

χt = ( . . . . )∂2

∂t2fs( . . . ) |h→o

χh = ( . . . . )∂2

∂h2fs( . . . ) |h→o susceptibility of op

order parameter

specific heatuniversal functions


!10


t =T − Tc

Tc∼ β − βc


σ )


σ )




< O > = ( . . . . )∂∂h

fs( . . . ) |h→o

χt = ( . . . . )∂2

∂t2fs( . . . ) |h→o

χh = ( . . . . )∂2


order parameter


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 curveTc = 202.6(4) MeV

χ h

T [MeV]

Nσ=241612

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

175 180 185 190 195 200 205 210

mπ ∼ 135 MeV

χ h


!10


t =T − Tc

Tc∼ β − βc


σ )


σ )




< O > = ( . . . . )∂∂h

fs( . . . ) |h→o

χt = ( . . . . )∂2

∂t2fs( . . . ) |h→o

χh = ( . . . . )∂2


order parameter


175 190 200 215T [MeV]

0.075

0.080

0.085

0.090

0.095

0.100

�R

eL

Ns = 12

Ns = 16

Ns = 24

mπ ∼ 135 MeV

χ t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

175 180 185 190 195 200 205 210


χ h

T [MeV]

Nσ=241612

8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

175 180 185 190 195 200 205 210

mπ ∼ 135 MeV

χ h


!11

ut ∼ ctt, ut ∼ chh near, T → Tc

t =T − Tc

Tc∼ β − βc



are 3d Ising Exponentsα, γ, ν

“Susceptibility” and “specific heat” scale with

corresponding Z(2) finite size universal scaling functions


!11


t =T − Tc

Tc∼ β − βc





σ )

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


χ h N

σ−γ/ν

z=z0 Nσ1/ν (T-Tc)/Tc

Nσ=241612

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

mπ ∼ 135 MeV




!11


t =T − Tc

Tc∼ β − βc





σ )

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


χ h N

σ−γ/ν


Nσ=241612

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

mπ ∼ 135 MeV


σ )

�10 �5 0 5(T�Tc)

TcN 1/⌫

�

0.0450

0.0475

0.0500

0.0525

0.0550

0.0575

0.0600

0.0625

�tN

�↵/⌫

�

N�=12

N�=16

N�=24

mπ ∼ 135 MeV



!12

Binder cumulants of order parameter |Im L|

175 190 200 215

2

3

4

BIm

L

Ns = 8

Ns = 12

190 200 215T (MeV)

binder cumulant of the imaginary part of LNs = 12

Ns = 16

190 200 215

Ns = 16

Ns = 24

Binder cumulants of Im L

Ising value=1.604

mπ ∼ 135 MeV

mπ ∼ 135 MeV

0

0.01

0.02

0.03

0.04

0.05

175 180 185 190 195 200 205 210 215

lines: Z(2) scaling fitTc = 200.8(3) MeV

<|Im

L|>

T [MeV]

Nσ=241612

0

0.01

0.02

0.03

0.04

0.05

175 180 185 190 195 200 205 210 215

mπ ∼ 135 MeV

0

0.05

0.1

0.15

0.2

0.25

-6 -4 -2 0 2 4 6


<|Im

L|>

Nσβ/ν


Nσ=241612

8 0

0.05

0.1

0.15

0.2

0.25

-6 -4 -2 0 2 4 6

Universal Z(2) scaling of theorder parameter

Crossing of the Binder cumulants approaches

to the universal value in V → ∞

Quark mass dependence of RW transition

!13

No unusual change in the susceptibility of the order parameter with respect to pion mass upto, MeV mπ ∼ 90

Order of the transition seems to be unchanged ??

Susceptibility of |Im L|

Susceptibility of |Im L|

175 190 200 215T (MeV)

0.1

0.2

0.3

0.4

0.5

0.6

�|I

mL

|

ml/ms = 1/27

ml/ms = 1/40

ml/ms = 1/60

185 190 195 200 205 210T (MeV)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

�|I

mL

|

ml/ms = 1/27

ml/ms = 1/40

Nσ = 24

Nσ = 16

Chiral limit and RW transition

!14

⟨ψ̄ ψ⟩ ∼ const + m1/2

χdiscψ̄ψ ∼ m−1/2 For, T < Tc

Susceptibility of ⟨l̄l⟩

Sub. Chiral condensate

Δls = (ms/f 4k )(⟨l̄l⟩ − (ml /ms)⟨s̄s⟩)

175 190 200 220T [MeV]

10

20

30

40

50

�di

scl̄l

/T2

ml/ms = 1/27

ml/ms = 1/40

ml/ms = 1/60Goldstone effect (square root singularity) in below TRW is evident. chiral symmetry restoration at TRW ?

χdiscl̄l

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

5

10

15

20

25

30

�ls

ml/ms = 1/27

ml/ms = 1/40

ml/ms = 1/60

Nσ = 16

Nσ = 16

!15

Nf =2+1, m𝜋 ～135 MeV

Strong volume dependence of “subtracted chiral condensate” at fixed ml/ms below TRW

�0.06 �0.02 0.02 0.06 0.10�(h ̄ i)

�0.06�0.04�0.02

0.000.020.040.06

�(Im

L)

� = 5.975

175 180 185 190 195 200 205 210 215T(MeV)

5

10

15

20

25

30

�ls

N�=12

N�=16

N�=24

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

0.0

0.5

1.0

1.5

2.0

�d�

lsdT

N�=12

N�=16

N�=24

mixed chiral susceptibility sensitive to transition at the RW endpoint

T=201.85 MeVml /ms = 1/27mπ ∼ 135 MeV

mπ ∼ 135 MeV

Nσ = 16mπ ∼ 135 MeV

Conclusions• Our preliminary findings from calculations with the HISQ

action with physical pion mass suggest that the RW-end point is 2nd order and belongs to the Z(2) universality class. This is consistent with the earlier result found with the stout-improved staggered action.

• preliminary trends of m𝜋～110, 90 MeV results are also consistent with a 2nd order phase transition.

• RW transition and chiral phase transition may coincide in the chiral limit.

Calculations on larger lattices and smaller quark masses are ongoing.

!16

Back up slides

!17

!18

TRW

crossover

0⇡3µT

T Z(2) 2nd order(3d Ising)

Tpc

1st order

Intermediate quark mass

RW trans.

crossover

0⇡3µT

T

deconfinement 1st

orderTc

1st orderTriple point

1st orderRW trans.

Heavy quark mass

0⇡3µT

T 1st orderTriple point

Tpc

small quark mass

RW trans.

crossover

2nd orderend point

Chiral 1st order??

Different scenarios for different quark masses

TRW

TRW

TRW

Conjectured phase diagrams in the imaginary

chemical potential plane

1st order

Phases in the RW plane

• RW transition happens between two Z(3) sectors of the Polyakov loop. Hence, the order parameter can be the phase or the imaginary part of the Polyakov loop.

• In the RW plane, the 1st order region (for small mass) consists of three 1st order transitions, where high temperature RW transition meets two chiral phase transitions.

• The physical point which is crossover for 𝜇=0 can be 1st or 2nd order in the RW plane. So, our first goal is to confirm this issue and then going to the chiral limit to “search for a 1st order” transition.

!19

Nf = 3

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ms<latexit sha1_base64="fimh0WjPAqaAw4/gbE6Qd7027/k=">AAAB7XicdVA9SwNBEJ2LXzFGjVpqsRgEq+PuDEnsAhZaRjAfkBxxb7OXrNn7YHdPCEdKexsLRWz9P3b+G/cSBRV9MPB4b4aZeV7MmVSW9W7klpZXVtfy64WN4ubWdmlnty2jRBDaIhGPRNfDknIW0pZiitNuLCgOPE473uQs8zu3VEgWhVdqGlM3wKOQ+YxgpaV2MEjlrDAolS3ztF51KlVkmZZVsx07I06tclJBtlYylBsHd8VrAGgOSm/9YUSSgIaKcCxlz7Zi5aZYKEY4nRX6iaQxJhM8oj1NQxxQ6abza2foSCtD5EdCV6jQXP0+keJAymng6c4Aq7H87WXiX14vUX7dTVkYJ4qGZLHITzhSEcpeR0MmKFF8qgkmgulbERljgYnSAWUhfH2K/idtx7Qt077UaZzDAnnYh0M4Bhtq0IALaEILCNzAPTzCkxEZD8az8bJozRmfM3vwA8brB3xikIc=</latexit><latexit sha1_base64="QQJwveTIyWi+w+V9E8bmr+9O2gc=">AAAB7XicdVA9SwNBEN3zM8ao0ZRaLAbB6tiLIYldwELLCOYDkjPsbfaSNXt7x+6eEI6U9jYWitj6A+z9EXb+G/cSBRV9MPB4b4aZeV7EmdIIvVsLi0vLK6uZtex6bmNzK7+901JhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2Nz5J/fY1lYqF4kJPIuoGeCiYzwjWRmoF/URNs/18EdnHtUqpXIHIRqjqlJyUlKrlozJ0jJKiWN+7yV2+FF4b/fxbbxCSOKBCE46V6joo0m6CpWaE02m2FysaYTLGQ9o1VOCAKjeZXTuFB0YZQD+UpoSGM/X7RIIDpSaBZzoDrEfqt5eKf3ndWPs1N2EiijUVZL7IjznUIUxfhwMmKdF8YggmkplbIRlhiYk2AaUhfH0K/yetku0g2zk3aZyCOTJgF+yDQ+CAKqiDM9AATUDAFbgF9+DBCq0769F6mrcuWJ8zBfAD1vMHgB2SCQ==</latexit><latexit sha1_base64="QQJwveTIyWi+w+V9E8bmr+9O2gc=">AAAB7XicdVA9SwNBEN3zM8ao0ZRaLAbB6tiLIYldwELLCOYDkjPsbfaSNXt7x+6eEI6U9jYWitj6A+z9EXb+G/cSBRV9MPB4b4aZeV7EmdIIvVsLi0vLK6uZtex6bmNzK7+901JhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2Nz5J/fY1lYqF4kJPIuoGeCiYzwjWRmoF/URNs/18EdnHtUqpXIHIRqjqlJyUlKrlozJ0jJKiWN+7yV2+FF4b/fxbbxCSOKBCE46V6joo0m6CpWaE02m2FysaYTLGQ9o1VOCAKjeZXTuFB0YZQD+UpoSGM/X7RIIDpSaBZzoDrEfqt5eKf3ndWPs1N2EiijUVZL7IjznUIUxfhwMmKdF8YggmkplbIRlhiYk2AaUhfH0K/yetku0g2zk3aZyCOTJgF+yDQ+CAKqiDM9AATUDAFbgF9+DBCq0769F6mrcuWJ8zBfAD1vMHgB2SCQ==</latexit><latexit sha1_base64="cNBVvQQBwsEOvudlTZEsw54D8kM=">AAAB7XicdVDLSgMxFM34rPVVdekmWARXQ2Ysbd0VXOiygn1AO5RMmmljk8yQZIQy9B/cuFDErf/jzr8x01ZQ0QMXDufcy733hAln2iD04aysrq1vbBa2its7u3v7pYPDto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5DL3O/dUaRbLWzNNaCDwSLKIEWys1BaDTM+Kg1IZuRf1ql+pQuQiVPN8Lyd+rXJegZ5VcpTBEs1B6b0/jEkqqDSEY617HkpMkGFlGOF0VuynmiaYTPCI9iyVWFAdZPNrZ/DUKkMYxcqWNHCufp/IsNB6KkLbKbAZ699eLv7l9VIT1YOMySQ1VJLFoijl0MQwfx0OmaLE8KklmChmb4VkjBUmxgaUh/D1KfyftH3XQ653g8qNq2UcBXAMTsAZ8EANNMA1aIIWIOAOPIAn8OzEzqPz4rwuWlec5cwR+AHn7RO6Yo86</latexit>

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1st order

1st order

physical point

Z(2)

Z(2)

Crossover

!/T=0

1st order tr.

1st order tr.

2nd order 3d Ising

tri critical line

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tri cr

itica

l line

mud<latexit sha1_base64="0x4aBwaMF94gOwOxwplmtAl1wQ0=">AAAB7nicdVBNS8NAEJ3Ur1qrVj3qYbEInkISS1tvBQ96rGBtoQ11s9m0Szcf7G6EEnr0B3jxoIhXf483/42bVkFFHww83pthZp6XcCaVZb0bhaXlldW14nppo7y5tV3Z2b2WcSoI7ZCYx6LnYUk5i2hHMcVpLxEUhx6nXW9ylvvdWyoki6MrNU2oG+JRxAJGsNJSNxxmqT8rDStVyzxt1p1aHVmmZTVsx86J06id1JCtlRzV1sFd+QYA2sPK28CPSRrSSBGOpezbVqLcDAvFCKez0iCVNMFkgke0r2mEQyrdbH7uDB1pxUdBLHRFCs3V7xMZDqWchp7uDLEay99eLv7l9VMVNN2MRUmqaEQWi4KUIxWj/HfkM0GJ4lNNMBFM34rIGAtMlE4oD+HrU/Q/uXZM2zLtS53GOSxQhH04hGOwoQEtuIA2dIDABO7hEZ6MxHgwno2XRWvB+JzZgx8wXj8APn+Q9w==</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="x7NVE2FJJIQx4Ac2iCaHilpJfv8=">AAAB7nicdVDLSgMxFM3UV62vqks3wSK4GjK1tHVXcKHLCvYB7VAymUwbmmSGJCOUoR/hxoUibv0ed/6NmbaCih64cDjnXu69J0g40wahD6ewtr6xuVXcLu3s7u0flA+PujpOFaEdEvNY9QOsKWeSdgwznPYTRbEIOO0F06vc791TpVks78wsob7AY8kiRrCxUk+MsjScl0blCnIvm/VqrQ6Ri1DDq3o5qTZqFzXoWSVHBazQHpXfh2FMUkGlIRxrPfBQYvwMK8MIp/PSMNU0wWSKx3RgqcSCaj9bnDuHZ1YJYRQrW9LAhfp9IsNC65kIbKfAZqJ/e7n4lzdITdT0MyaT1FBJlouilEMTw/x3GDJFieEzSzBRzN4KyQQrTIxNKA/h61P4P+lWXQ+53i2qtK5XcRTBCTgF58ADDdACN6ANOoCAKXgAT+DZSZxH58V5XbYWnNXMMfgB5+0TfH+Pqg==</latexit>

Nf = 2<latexit sha1_base64="lDZ6GTcUZ43Dbrzla29lS8np/dM=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoMgHpbdNSTxIAQ86EkimAckS5idzCZjZh/MzAphyT948aCIV3/Fszf/xtlEQUULGoqqbrq7vJgzqSzr3cgtLC4tr+RXC2vrG5tbxe2dlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG1747PMb99SIVkUXqtJTN0AD0PmM4KVllqXff/UKfSLJcs8qVWccgVZpmVVbcfOiFMtH5eRrZUMpfrRq40AoNEvvvUGEUkCGirCsZRd24qVm2KhGOF0WuglksaYjPGQdjUNcUClm86unaIDrQyQHwldoUIz9ftEigMpJ4GnOwOsRvK3l4l/ed1E+TU3ZWGcKBqS+SI/4UhFKHsdDZigRPGJJpgIpm9FZIQFJkoHlIXw9Sn6n7Qc07ZM+0qncQ5z5GEP9uEQbKhCHS6gAU0gcAN38ACPRmTcG0/G87w1Z3zO7MIPGC8fsdOQCA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="/1K3hh3FBI5sBQ080byuw0vfeW8=">AAAB7XicdVBNSwMxEM3Wr1q/qh69BIvgacmupa0HoeBBT1LB1kK7lGyabWOzyZJkhVL6H7x4UMSr/8eb/8ZsW0FFHww83pthZl6YcKYNQh9Obml5ZXUtv17Y2Nza3inu7rW0TBWhTSK5VO0Qa8qZoE3DDKftRFEch5zehqPzzL+9p0ozKW7MOKFBjAeCRYxgY6XWVS868wu9Ygm5p7WKX65A5CJU9XwvI361fFKGnlUylMACjV7xvduXJI2pMIRjrTseSkwwwcowwum00E01TTAZ4QHtWCpwTHUwmV07hUdW6cNIKlvCwJn6fWKCY63HcWg7Y2yG+reXiX95ndREtWDCRJIaKsh8UZRyaCTMXod9pigxfGwJJorZWyEZYoWJsQFlIXx9Cv8nLd/1kOtdo1L9YhFHHhyAQ3AMPFAFdXAJGqAJCLgDD+AJPDvSeXRenNd5a85ZzOyDH3DePgGmbo6F</latexit>

Nf = 3

!/T=i"/3

mud

ms

PureGauge

1st

1st

crossover

Nf=

3phys.point

Z(2)

Z(2)

Nf = 2

mtrics

Nf

=1

O(4)?U(2)L ¢ U(2)R/U(2)V ?

chiral limit Nf = 2

Columbia plot in 𝜇=0 and RW plane

!20

𝜇/T=𝜋/3

𝜇/T=0

exploring the qcd phase diagram with imaginary chemical ......anirban lahiri 1 jishnu goswami...

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