Exploring the QCD Phase diagram with imaginary chemical potential - - PowerPoint PPT Presentation

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 𝜈 =0
• Status on chiral phase transition in the RW plane.
• Results and discussions.

2

Introduction

3

2nd 3d Ising mu,d ms ∞ 1st tr. 1st tr. ∞ N

f

= 1 Nf = 2

−

✓π

3

◆2 ✓µ

T

◆2

tric tric 1st Z(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 𝜈=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(𝜈/T=(2k+1)𝜌/3). Thus critical mass in the RW plane puts a bound on the critical mass at 𝜈=0. Does a 1st order chiral symmetry restoring transition exist at 𝜈=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 > m𝜌 >80 MeV. Bound on critical pion mass is given as, m𝜌cr ≲50 MeV from the scaling analysis.

μ = 0

mud ms Pure Gauge

1st 1st

crossover

Nf = 3

phys. point

Z ( 2 ) Z(2) Nf = 2 mtric

s

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 m𝜌 >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)

m𝜌﹥1 GeV, for the ‘heavy quark mass RW transition’ Standard staggered action: m𝜌～400 MeV (N𝝊=4) Standard Wilson action: m𝜌～930 MeV(N𝝊=4) m𝜌～680 MeV (N𝝊=6) 1st order end point (of the line of 1st order RW transitions) exist already for 𝜈/T=𝜌/3 and mcri >mphy. The results are strongly fermion discretization scheme and cut-off(N𝝊) 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(m𝜌～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 m𝜌 >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

Nτ

ml ms

mπ

We vary 𝛾 in the range [5.850-6.038],

Nf = 2 + 1, μ T = π 3

We work on the 2nd RW plane

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

Generally we generated 20k trajectory per 𝛾 value away from 𝛾c and 80k trajectory near 𝛾c

∼ Tc ± 0.1Tc

corresponds to,

Ising endpoint of a first order line

8

Heff(t, ξ) = tℰ + hℳ

Effective Ising Hamiltonian which defines the universal critical behaviour of the system temperature like field energy like

• perator

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 → ℰ

ℳ → − ℳ Im L → − Im L

Re L → Re L

• rder parameter

energy like i.e. at

μ = μRW

lim

h→0 lim V →∞hIm Li ⌘ lim V →∞h|Im L|i =

( 0, if β < βc non-zero, if β > βc

9

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

Im L → − Im L

Re L → Re L

• rder parameter

energy like i.e. at

μ = μRW

lim

h→0 lim V →∞hIm Li ⌘ lim V →∞h|Im L|i =

( 0, if β < βc non-zero, if β > βc

9

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

Im L → − Im L

Re L → Re L

• rder parameter

energy like i.e. at

μ = μRW

lim

h→0 lim V →∞hIm Li ⌘ lim V →∞h|Im L|i =

( 0, if β < βc non-zero, if β > βc

−0.02 −0.01 0.00 0.01 0.02 δ(ReL) −0.04 −0.02 0.00 0.02 0.04 δ(ImL) β = 5.925

T<Tc

9

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

Im L → − Im L

Re L → Re L

• rder parameter

energy like i.e. at

μ = μRW

lim

h→0 lim V →∞hIm Li ⌘ lim V →∞h|Im L|i =

( 0, if β < βc non-zero, if β > βc

−0.02 −0.01 0.00 0.01 0.02 δ(ReL) −0.04 −0.02 0.00 0.02 0.04 δ(ImL) β = 5.925

T<Tc

−0.02 −0.01 0.00 0.01 0.02 δ(ReL) −0.08 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 δ(ImL) β = 5.990

T>Tc

Finite size scaling and Z(2) universality class

10

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

χh = z1Nγ/ν

σ fh(z0tN1/ν σ )

χt = z2Nα/ν

σ

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 ∂t2 fs( . . . )|h→o χh = ( . . . . ) ∂2 ∂h2 fs( . . . )|h→o susceptibility of op

• rder parameter

specific heat universal functions

Finite size scaling and Z(2) universality class

10

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

χh = z1Nγ/ν

σ fh(z0tN1/ν σ )

χt = z2Nα/ν

σ

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 ∂t2 fs( . . . )|h→o χh = ( . . . . ) ∂2 ∂h2 fs( . . . )|h→o susceptibility of op

• rder parameter

specific heat universal functions

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 0.2 0.4 0.6 0.8 1 1.2 1.4 175 180 185 190 195 200 205 210

mπ ∼ 135 MeV

χh

Finite size scaling and Z(2) universality class

10

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

χh = z1Nγ/ν

σ fh(z0tN1/ν σ )

χt = z2Nα/ν

σ

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 ∂t2 fs( . . . )|h→o χh = ( . . . . ) ∂2 ∂h2 fs( . . . )|h→o susceptibility of op

• rder parameter

specific heat universal functions

175 190 200 215 T[MeV] 0.075 0.080 0.085 0.090 0.095 0.100

Ns = 12 Ns = 16 Ns = 24

mπ ∼ 135 MeV

χt

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 0.2 0.4 0.6 0.8 1 1.2 1.4 175 180 185 190 195 200 205 210

mπ ∼ 135 MeV

χh

Finite size scaling and Z(2) universality class

11

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

f = fns + b−dfs(bytut, byhuh, b−1Nσ)

Free energy, Responsible for the universal critical behaviour are 3d Ising Exponents

α, γ, ν

“Susceptibility” and “specific heat” scale with corresponding Z(2) finite size universal scaling functions

Finite size scaling and Z(2) universality class

11

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

f = fns + b−dfs(bytut, byhuh, b−1Nσ)

Free energy, Responsible for the universal critical behaviour are 3d Ising Exponents

α, γ, ν

χh = z1Nγ/ν

σ fh(z0tN1/ν σ )

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

• 6
• 4
• 2

2 4

line: Z(2) scaling curve Tc = 202.6(4) MeV

χh Nσ

−γ/ν

z=z0 Nσ

1/ν (T-Tc)/Tc

Nσ=24 16 12 8 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

• 6
• 4
• 2

2 4

mπ ∼ 135 MeV

“Susceptibility” and “specific heat” scale with corresponding Z(2) finite size universal scaling functions

Finite size scaling and Z(2) universality class

11

ut ∼ ctt, ut ∼ chh

near, T → Tc

t = T − Tc Tc ∼ β − βc

f = fns + b−dfs(bytut, byhuh, b−1Nσ)

Free energy, Responsible for the universal critical behaviour are 3d Ising Exponents

α, γ, ν

χh = z1Nγ/ν

σ fh(z0tN1/ν σ )

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

• 6
• 4
• 2

2 4

line: Z(2) scaling curve Tc = 202.6(4) MeV

χh Nσ

−γ/ν

z=z0 Nσ

1/ν (T-Tc)/Tc

Nσ=24 16 12 8 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

• 6
• 4
• 2

2 4

mπ ∼ 135 MeV

χt = z2Nα/ν

σ

ft(z0tN1/ν

σ )

−10 −5 5

(T−Tc) Tc

N1/ν

σ

0.0450 0.0475 0.0500 0.0525 0.0550 0.0575 0.0600 0.0625

χtN−α/ν

σ

Nσ=12 Nσ=16 Nσ=24

mπ ∼ 135 MeV

“Susceptibility” and “specific heat” scale with corresponding Z(2) finite size universal scaling functions

12

Binder cumulants of order parameter

|Im L|

175 190 200 215 2 3 4 BImL

Ns = 8 Ns = 12

190 200 215 T(MeV) binder cumulant of the imaginary part of L

Ns = 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

0.01 0.02 0.03 0.04 0.05 175 180 185 190 195 200 205 210 215

lines: Z(2) scaling fit Tc = 200.8(3) MeV

<|ImL|> T [MeV] Nσ=24 16 12 0.01 0.02 0.03 0.04 0.05 175 180 185 190 195 200 205 210 215

mπ ∼ 135 MeV

0.05 0.1 0.15 0.2 0.25

• 6
• 4
• 2

2 4 6

line: Z(2) scaling curve Tc = 200.8(3) MeV

<|ImL|> Nσ

β/ν

z=z0 Nσ

1/ν (T-Tc)/Tc

Nσ=24 16 12 8 0.05 0.1 0.15 0.2 0.25

• 6
• 4
• 2

2 4 6

Universal Z(2) scaling of the

• rder 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

• rder 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|

175 190 200 215 T(MeV) 0.1 0.2 0.3 0.4 0.5 0.6 χ|ImL|

ml/ms = 1/27 ml/ms = 1/40 ml/ms = 1/60

185 190 195 200 205 210 T(MeV) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 χ|ImL|

ml/ms = 1/27 ml/ms = 1/40

Nσ = 24 Nσ = 16

Chiral limit and RW transition

14

⟨ ¯ ψψ⟩ ∼ const + m1/2

χdisc

¯ ψψ ∼ m−1/2

For, T < Tc Susceptibility of ⟨¯ ll⟩

• Sub. Chiral condensate

Δls = (ms/f 4

k)(⟨¯

ll⟩ − (ml/ms)⟨¯ ss⟩)

175 190 200 220 T[MeV] 10 20 30 40 50 χdisc

¯ ll

/T 2

ml/ms = 1/27 ml/ms = 1/40 ml/ms = 1/60

Goldstone effect (square root singularity) in below TRW is evident.

chiral symmetry restoration at TRW ?

χdisc

¯ ll

175 180 185 190 195 200 205 210 215 T[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, m𝜌 ～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 δ(h ¯ ψψi) 0.06 0.04 0.02 0.00 0.02 0.04 0.06 δ(ImL) β = 5.975 175 180 185 190 195 200 205 210 215 T(MeV) 5 10 15 20 25 30 ∆ls

Nσ=12 Nσ=16 Nσ=24

175 180 185 190 195 200 205 210 215 T[MeV] 0.0 0.5 1.0 1.5 2.0 −d∆ls

dT

Nσ=12 Nσ=16 Nσ=24

mixed chiral susceptibility sensitive to transition at the RW endpoint

T=201.85 MeV

ml/ms = 1/27

mπ ∼ 135 MeV

mπ ∼ 135 MeV

Nσ = 16

mπ ∼ 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 m𝜌～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

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 m𝜌～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 π 3 µ T

T

Z(2) 2nd order (3d Ising) Tpc 1st order

Intermediate quark mass

RW trans. crossover π 3 µ T

T

deconfinement 1st

• rder

Tc 1st order Triple point 1st order RW trans.

Heavy quark mass

π 3 µ T

T

1st order Triple point Tpc

small quark mass

RW trans. crossover 2nd order end 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 𝜈=0 can be 1st
• r 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

Nf = 2

ms

mud

1st

• rder

1st

• rder

physical point Z(2) Z(2) Crossover !/T=0 1st order tr. 1st order tr. 2nd order 3d Ising tri critical line

ms

tri critical line

mud

Nf = 2

Nf = 3

!/T=i"/3

mud ms Pure Gauge

1st 1st

crossover

N

f

= 3

phys. point

Z(2) Z(2) Nf = 2 mtric

s

Nf = 1 O(4)? U(2)L ⊗ U(2)R/U(2)V ?

chiral limit Nf = 2

Columbia plot in 𝜈=0 and RW plane

20

𝜈/T=𝜌/3 𝜈/T=0