Critical end points in (2+1)-flavor QCD with imaginary

chemical potentialIn collaboration with: Frithjof Karsch,

Christian Schmidt, and Anirban Lahiri

Jishnu Goswami

Universität Bielefeld

"Hirschegg 2019”

From QCD matter to hadrons

QCD phase diagram

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

crossover μB /T < 2

1st order transition

Quark gluon plasma

Hadron gas

CEP??

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

2nd order O(4)??

1st order transitionPossibility of a

1st order transition??

TCP??

Pisarski, Wilczek, PRD 29 (1984) !2

Nature of dense QCD phase diagram is not yet conclusive from Lattice QCD calculations.

However the crossover nature is verified and the search for CEP with physical pion mass is ongoing.

QCD phase diagram

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

crossover μB /T < 2

1st order transition

Quark gluon plasma

Hadron gas

CEP??

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

2nd order O(4)??

1st order transitionPossibility of a

1st order transition??

TCP??

Pisarski, Wilczek, PRD 29 (1984) !2

Nature of dense QCD phase diagram is not yet conclusive from Lattice QCD calculations.

However the crossover nature is verified and the search for CEP with physical pion mass is ongoing.

Crossover is related to the chiral symmetry restoration, can be realized from the fact that in the chiral limit chiral symmetry is restored via a phase transition.

QCD phase diagram

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

crossover μB /T < 2

1st order transition

Quark gluon plasma

Hadron gas

CEP??

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

2nd order O(4)??

1st order transitionPossibility of a

1st order transition??

TCP??

Pisarski, Wilczek, PRD 29 (1984) !2

Nature of dense QCD phase diagram is not yet conclusive from Lattice QCD calculations.

However the crossover nature is verified and the search for CEP with physical pion mass is ongoing.

Crossover is related to the chiral symmetry restoration, can be realized from the fact that in the chiral limit chiral symmetry is restored via a phase transition.

Work from Pisarski and Wilczek suggests that the chiral transition (Nf =2+1, 𝜇=0) belongs to the O(4) universality class if U(1)A is still broken at Tc(~130-135 MeV) . [See talk by Frithjof]

QCD phase diagram

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

crossover μB /T < 2

1st order transition

Quark gluon plasma

Hadron gas

CEP??

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

2nd order O(4)??

1st order transitionPossibility of a

1st order transition??

TCP??

Pisarski, Wilczek, PRD 29 (1984) !2

Nature of dense QCD phase diagram is not yet conclusive from Lattice QCD calculations.

However the crossover nature is verified and the search for CEP with physical pion mass is ongoing.

Crossover is related to the chiral symmetry restoration, can be realized from the fact that in the chiral limit chiral symmetry is restored via a phase transition.

Work from Pisarski and Wilczek suggests that the chiral transition (Nf =2+1, 𝜇=0) belongs to the O(4) universality class if U(1)A is still broken at Tc(~130-135 MeV) . [See talk by Frithjof]Although, if U(1)A restoration happens at Tc then there is a possibility of a fluctuation induced 1st order transition in the chiral limit.[See talk by Lukas]

QCD phase diagram

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

crossover μB /T < 2

1st order transition

Quark gluon plasma

Hadron gas

CEP??

0 500 1000 1500µB(MeV)

0

50

100

150

200

250

T(M

eV)

2nd order O(4)??

1st order transitionPossibility of a

1st order transition??

TCP??

Pisarski, Wilczek, PRD 29 (1984) !2

Nature of dense QCD phase diagram is not yet conclusive from Lattice QCD calculations.

However the crossover nature is verified and the search for CEP with physical pion mass is ongoing.

Crossover is related to the chiral symmetry restoration, can be realized from the fact that in the chiral limit chiral symmetry is restored via a phase transition.

Work from Pisarski and Wilczek suggests that the chiral transition (Nf =2+1, 𝜇=0) belongs to the O(4) universality class if U(1)A is still broken at Tc(~130-135 MeV) . [See talk by Frithjof]Although, if U(1)A restoration happens at Tc then there is a possibility of a fluctuation induced 1st order transition in the chiral limit.[See talk by Lukas]Order of the phase transition in the chiral limit is also of relevance for studies of fluctuations at the LHC.

Chiral transition for zero chemical potential with HISQ

!3

Chiral transition for zero chemical potential with HISQ

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

!3

Chiral transition for zero chemical potential with HISQ

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.

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

!3

Chiral transition for zero chemical potential with HISQ

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.

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

!3

Chiral transition for zero chemical potential with HISQ

Nf =2+1 : No hint of 1st order phase transition for m𝜋 >55 MeV. Chiral transition is most likely 2nd order O(4) rather than 1st order.

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

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.

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

See talk by Frithjof Karsch

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

!3

Chiral transition for zero chemical potential with HISQ

!4

μ = 0• HotQCD results on chiral phase transition, [ ]

Chiral transition for zero chemical potential with HISQ

!4

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

HotQCD: Based on studies with HISQ action

μ = 0• HotQCD results on chiral phase transition, [ ]

Chiral transition for zero chemical potential with HISQ

!4

However, still one may argue that 1st order transition can be possible for,mπ ≪ 55 MeV

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

HotQCD: Based on studies with HISQ action

μ = 0

mud

ms

PureGauge

1st

1st

crossover

N f=

3phys.point

Z(2)Z(2)

Nf = 2

N f=

1

chiral limit Nf = 2

mπ ≪ 55MeV

• HotQCD results on chiral phase transition, [ ]

Chiral transition for zero chemical potential

!5

Chiral transition for zero chemical potential

!5

mud

ms

PureGauge

1st

1st

crossover

N f=

3phys.point

Z(2)

Z(2)

Nf = 2

N f=

1

chiral limit Nf = 2

Philipsen, Pinke, PRD 93 (2016)

A few Lattice actions show a first order chiral transition but…..

1.These results are done on fixed N𝝉. Not continuum extrapolated.

2. They use unimproved actions which have large cut-off effects, means the calculations are done away from the continuum.

3. The order of the transition can change if calculations are done close to continuum.

Chiral transition for zero chemical potential

!5

mud

ms

PureGauge

1st

1st

crossover

N f=

3phys.point

Z(2)

Z(2)

Nf = 2

N f=

1

chiral limit Nf = 2

Philipsen, Pinke, PRD 93 (2016)

A few Lattice actions show a first order chiral transition but…..

1.These results are done on fixed N𝝉. Not continuum extrapolated.

2. They use unimproved actions which have large cut-off effects, means the calculations are done away from the continuum.

3. The order of the transition can change if calculations are done close to continuum.

Possible Solutions:1. Continuum extrapolate the results (requires

decent effort).2. Verify with actions which allows us to do the

calculations close to the continuum i.e. with improved actions

!6

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

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

!6

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

Possible scenario of extended 3d Columbia plot

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

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

Another possible way to examine the nature of chiral transition(1st order??) at 𝜇=0 is to study the phase diagram with imaginary chemical potential. It is expected that the first order region, if exists, increases in the imaginary chemical potential direction.

!6

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

Possible scenario of extended 3d Columbia plot

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

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

Another possible way to examine the nature of chiral transition(1st order??) at 𝜇=0 is to study the phase diagram with imaginary chemical potential. It is expected that the first order region, if exists, increases in the imaginary chemical potential direction.

Largest 1st order region:Roberge-Weiss (RW) plane, i𝜇/T=i𝜋/3

!6

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

Possible scenario of extended 3d Columbia plot

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

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

Another possible way to examine the nature of chiral transition(1st order??) at 𝜇=0 is to study the phase diagram with imaginary chemical potential. It is expected that the first order region, if exists, increases in the imaginary chemical potential direction.

Largest 1st order region:Roberge-Weiss (RW) plane, i𝜇/T=i𝜋/3

!6

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

Possible scenario of extended 3d Columbia plot

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

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

Another possible way to examine the nature of chiral transition(1st order??) at 𝜇=0 is to study the phase diagram with imaginary chemical potential. It is expected that the first order region, if exists, increases in the imaginary chemical potential direction.

mπ ∼ 135 MeV

General strategy:Locate the physical point for Nf =2+1, approach the chiral limit while keeping the strange quark mass fix to its physical value. Extrapolate the result to 𝜇=0.

Largest 1st order region:Roberge-Weiss (RW) plane, i𝜇/T=i𝜋/3

Details about the RW plane

!7

• If (i𝜇) is purely imaginary then we can compare the results with the real chemical potential as,

• The partition function is periodic in 𝜇/T as, Z(𝜇/T)=Z(𝜇/T+2k𝜋/3), known as , Roberge-Weiss(RW) periodicity. And 𝜇/T=(2k+1)𝜋/3, is known as RW plane.

hOi =X

n

cn

✓iµ

T

◆n

, n = 2, 4, 6.....<latexit sha1_base64="0uHAIq3K2enkP8F97X4F/aP+OEw=">AAACL3icdZBdS8MwFIZTv51fUy+9CQ5BYZS2lk0vBFEQ71RwKqyzpFnaBZO0JKkwyv6RN/4Vb0QU8dZ/YTonqOgLgZfnnMPJeaOMUaUd58kaG5+YnJqema3MzS8sLlWXVy5UmktMWjhlqbyKkCKMCtLSVDNylUmCeMTIZXRzWNYvb4lUNBXnup+RDkeJoDHFSBsUVo8ChkTCCDwJ5NDsBSrnoYA4FEFEkwRuBt1YIlzQgOeD4nwwpFvXoi72vLpfb9ilwmrNsXd3Gp7fgI7tOE3Xc0vjNf1tH7qGlKqBkU7D6kPQTXHOidCYIaXarpPpToGkppiRQSXIFckQvkEJaRsrECeqUwzvHcANQ7owTqV5QsMh/T5RIK5Un0emkyPdU79rJfyr1s51vNMpqMhyTQT+XBTnDOoUluHBLpUEa9Y3BmFJzV8h7iGTjjYRV0wIX5fC/82FZ7uO7Z75tf2DURwzYA2sg03ggibYB8fgFLQABnfgATyDF+veerRerbfP1jFrNLMKfsh6/wBRz6fm</latexit><latexit sha1_base64="0uHAIq3K2enkP8F97X4F/aP+OEw=">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</latexit><latexit sha1_base64="0uHAIq3K2enkP8F97X4F/aP+OEw=">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</latexit><latexit sha1_base64="0uHAIq3K2enkP8F97X4F/aP+OEw=">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</latexit>

0 ⇡/3 2⇡/3 ⇡ 4⇡/3µ/T

T RW endpoint

2⇡

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Possible fate of the RW end point

!8

1storderTc

1st orderTriple point

1st orderRW trans.

TRW

Approaching Chiral limit

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

Tpc

1st order

Intermediate quark mass

RW trans.

crossover

TRW

For, smaller pion mass RW end point may become a 1st order triple point.

For, intermediate pion mass RW end point is of Z(2) 2nd order.

This could indicate that the chiral transition

at 𝜇=0 is 1st order.

0 ⇡/3 2⇡/3µ/T

T

0 ⇡/3 2⇡/3µ/T

T

Studies in the RW plane

!9

Nf =2: 1st order triple point (at the end of the line of 1st order RW transitions) exist for 𝜇/T=𝜋/3 and mcri

>mphy.

Standard staggered action: m𝜋～400 MeV(N𝝉=4) Standard Wilson action: m𝜋～930 MeV(N𝝉=4) m𝜋～680 MeV (N𝝉=6)The results are strongly fermion discretization scheme and cut-off(N𝝉) dependent.

Mostly with unimproved actions

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)

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)

Studies in the RW plane

!9

Nf =2: 1st order triple point (at the end of the line of 1st order RW transitions) exist for 𝜇/T=𝜋/3 and mcri

>mphy.

Standard staggered action: m𝜋～400 MeV(N𝝉=4) Standard Wilson action: m𝜋～930 MeV(N𝝉=4) m𝜋～680 MeV (N𝝉=6)The results are strongly fermion discretization scheme and cut-off(N𝝉) dependent.

Stout improved staggered fermions(Nf =2+1): For physical quark mass the RW endpoint is belongs to the Z(2) universality class.

No, 1st order triple point exist at least for m𝜋 >50 MeV.

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

Mostly with unimproved actionsVery recent studies with improved

actions,

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)

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)

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

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

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

Studies with HISQ in the RW plane

!10

Mq = DHISQ(μf ) + mq ,Action,

Simulation details, Nf = 2 + 1,μf

T=

π3

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

Generally we generated 20k trajectory per T value away

from Tc and 80k trajectory near Tc

T is varied in the range, 176-215 MeV, corresponds to,T ∼ Tc ± 0.1Tc

(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, 1/160,1/320

135, 110,55,40

Nσ Nτml

msmπ

μf is purely imaginary

Ising endpoint of a first order line

!11

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|ih=0 =

(0, if � < �c

non-zero, if � > �c<latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit>

!12

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

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|ih=0 =

(0, if � < �c

non-zero, if � > �c<latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit><latexit sha1_base64="4ShRgRbrXpPeJh5BsYTfLpvM2Yc=">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</latexit>

0.00 0.05”(ImL)

≠0.01

0.00

0.01

”(ReL

)

≠0.02 0.00 0.02 0.04”(ImL)

≠0.01

0.00

0.01

”(ReL

)

T<Tc T>Tc

Finite size scaling and Z(2) universality class

!13

Free energy and the universal functions for second order transition near the critical point can be written as,[ t=(T-Tc)/Tc]

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

⟨ | Im L |⟩ =∂f∂h

h→0

∼ N−β/νσ fh(z0tN1/ν

σ )

χh =∂2f∂h2

h→0

∼ Nγ/νσ fχ(z0tN1/ν

σ )

B4 − (1/Ndσ )Bns ∼ fB(z0tN1/ν

σ )

ut ∼ ctt, ut ∼ chh

χt =∂2f∂t2

h→0

∼ Nα/νσ fc(z0tN1/ν

σ )

universal scaling function of order parameter

universal scaling function of order parameter susceptibility

universal scaling function of Binder cumulant

universal scaling function of specific heat

In our case 𝛽,𝜈,𝛼 and 𝛾 are Z(2) critical exponents

Finite size scaling of order parameter and its susceptibility

!14

β = 0.32 γ = 1.24 ν = 0.62

180 190 200 210T [MeV]

0.0

0.5

1.0

1.5

‰h

N‡=12N‡=16N‡=24

180 190 200 210T [MeV]

0.01

0.02

0.03

0.04

È|ImL

|Í

N‡=12N‡=16N‡=24

≠6 ≠4 ≠2 0 2 4z = z0N1/‹

‡ (T ≠ Tc)/Tc

1

2

3

4

N≠“/‹

‡‰h

◊10≠3

N‡=8N‡=12N‡=16N‡=24

≠6 ≠4 ≠2 0 2 4z = z0N1/‹

‡ (T ≠ Tc)/Tc

0.05

0.10

0.15

0.20

0.25

N—/‹

‡È|ImL

|Í

N‡=8N‡=12N‡=16N‡=24

mπ ∼ 135 MeV

Finite size scaling of specific heat and Binder cumulantmπ ∼ 135 MeV

!15

α = 0.11

175 190 200 215T [MeV]

0.075

0.080

0.085

0.090

0.095

0.100

�t

Ns = 12

Ns = 16

Ns = 24

�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

180 190 200 210T [MeV]

1

2

3

B4

Tc=202.4(3)MeV

lines:Z(2) scaling curve

N‡=8N‡=12N‡=16N‡=24

≠5 0 5z = z0N1/‹

‡ (T ≠ Tc)/Tc

1

2

3B

4≠

(1/N

3 ‡)B

ns

line:Z(2) scaling curve

Tc=202.4(3)MeV

N‡=8N‡=12N‡=16N‡=24

Quark mass dependence of RW transition

!16

No significant rise in the peak of the susceptibility of the order parameter with respect to pion mass down to, MeV mπ ∼ 40

Order of the RW transition seems to be unchanged ??

Susceptibility of |ImL|

ml m⇡(MeV)ms/27 135ms/40 110ms/160 55ms/320 40

<latexit sha1_base64="Fdc1elXpOzgZMpC70bHkQBDporw=">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</latexit><latexit sha1_base64="Fdc1elXpOzgZMpC70bHkQBDporw=">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</latexit><latexit sha1_base64="Fdc1elXpOzgZMpC70bHkQBDporw=">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</latexit><latexit sha1_base64="Fdc1elXpOzgZMpC70bHkQBDporw=">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</latexit>

Nσ = 24

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

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

�h

ml = ms/27

ml = ms/40

ml = ms/160

ml = ms/320

Preliminary finite size scaling study with is consistent with Z(2) second order transition.

mπ ∼ 55 MeV

Finite size effects on chiral observables

!17

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

Δls = (ms/f 4k )(⟨ll⟩ − (ml /ms)⟨ss⟩)

≠0.05 0.00 0.05”(ImL)

≠0.05

0.00

0.05

0.10

”(ÂÂ

)

T > TRW

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

5

10

15

20

25

30

�ls

N�=12

N�=16

N�=24

mπ ∼ 135 MeV

180 190 200 210T [MeV]

50

100

150

m2 s‰disc

ÂlÂ

l/f

4 K

Ns = 12Ns = 16Ns = 24

mπ ∼ 135 MeV

Finite size effects on chiral observables

!18

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

mixed chiral susceptibility sensitive to transition at the RW endpoint

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

100

200

300

400

�T

d�ls

dT

N�=12

N�=16

N�=24

180 190 200 210T [MeV]

50

100

150

m2 s‰disc

ÂlÂ

l/f

4 K

Ns = 12Ns = 16Ns = 24

mπ ∼ 135 MeV

mπ ∼ 135 MeV

mπ ∼ 135 MeV

Chiral limit and RW transition

!19

⟨ψ ψ⟩ ∼ c0(T ) + cg(T ) m1/2

χdiscψψ ∼ m−1/2 For, T < Tc

Sub. Chiral condensate

Goldstone effect (square root singularity) in below TRW is evident. chiral symmetry restoration at TRW ?

χdiscll

Nσ = 24

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

0

100

200

300

400

�T

d�ls/d

T

ml = ms/27

ml = ms/40

ml = ms/160

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]

050

100150200250300350400

m2 s�

disc

/f4 K

ml = ms/27

ml = ms/40

ml = ms/160

ml = ms/320

Outlook on the fate of the RW end point

!20

1storderTc

1st orderTriple point

1st orderRW trans.

TRW

Approaching Chiral limit

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

Tpc

1st order

Intermediate quark mass

RW trans.

crossover

TRW

For, intermediate pion mass RW end point stays as Z(2) 2nd order.

Tc

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

1st orderRW trans.

TRW

Approaching Chiral limit

O(4)chiral

?? ??

0 ⇡/3 2⇡/3µ/T

T

0 ⇡/3 2⇡/3µ/T

T

0 ⇡/3 2⇡/3µ/T

T

55 MeV ≤ mπ ≤ 135 MeV

Conclusions

!21

Preliminary results down to m𝜋～

40 MeV suggests that the RW end point remains as Z(2) second order.

Nature of the chiral transition in the RW plane needs to be examined further. Favours 2nd order(O(4)) transition at 𝜇=0.

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

Calculations on larger lattices are ongoing.

no 1st order ≥ 40 MeV

phys. point

Conclusions

!21

Preliminary results down to m𝜋～

40 MeV suggests that the RW end point remains as Z(2) second order.

Nature of the chiral transition in the RW plane needs to be examined further. Favours 2nd order(O(4)) transition at 𝜇=0.

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

Calculations on larger lattices are ongoing.

no 1st order ≥ 40 MeV

phys. point

Conclusions

!21

Preliminary results down to m𝜋～

40 MeV suggests that the RW end point remains as Z(2) second order.

Nature of the chiral transition in the RW plane needs to be examined further. Favours 2nd order(O(4)) transition at 𝜇=0.

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

Calculations on larger lattices are ongoing.

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

no 1st order ≥ 40 MeV

phys. point

Conclusions

!21

Preliminary results down to m𝜋～

40 MeV suggests that the RW end point remains as Z(2) second order.

Nature of the chiral transition in the RW plane needs to be examined further. Favours 2nd order(O(4)) transition at 𝜇=0.

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

Calculations on larger lattices are ongoing.

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

no 1st order ≥ 40 MeV

phys. point

Conclusions

!21

Preliminary results down to m𝜋～

40 MeV suggests that the RW end point remains as Z(2) second order.

Nature of the chiral transition in the RW plane needs to be examined further. Favours 2nd order(O(4)) transition at 𝜇=0.

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

Calculations on larger lattices are ongoing.

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

no 1st order ≥ 40 MeV

phys. point

mπ ∼ 135 MeV

Phases in the RW plane

!22

• 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.

J. Goswami, Hirschegg 2018 !22