Critical endpoints of the finite temperature QCD Yoshifumi Nakamura - - PowerPoint PPT Presentation

Critical endpoints of the finite temperature QCD

Yoshifumi Nakamura

RIKEN Center for Computational Science

in collaboration with

• Y. Kuramashi, H. Ohno & S. Takeda

International Molecule-type Workshop Frontiers in Lattice QCD and related topics April 15 - April 26 2019 Yukawa Institute for Theoretical Physics, Kyoto University

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Contents

• introduction
• Columbia plot
• previous studies for critical endpoint
• previous studies for critical endline
• preliminary results
• New Nf = 3 simulations at Nt = 12
• New Nf = 2 + 1 simulations away from symmetric point
• summary

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Columbia plot

nature of finite temperature phase transition of 2+1 flavor QCD at µ = 0 in the plane of mu,d and ms

• 1st order : small mq region

[Pisarski, Wilczek, ’84]

• 1st order : heavy mq region
• crossover : medium mq
• 2nd order (Z2) : boundary

between 1st and crossover At small mq region

• crossover at the physical point
• critical end point at SU(3) flavor symmetric point, msym,

has not been determined yet

• critical end line has not been well determined yet

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Previous studies for critical endpoint

found Nf SG SF Nt endpoint ref y 3 S W 4 mPS > 1GeV Iwasaki et al, ’96 y 3 S KS 4 mPS ∼ 290 MeV Karsch et al, ’01 y 3 I p4 4 mPS ∼ 190 MeV Karsch et al, ’01 y 3 S KS 4 mPS = 290(20) MeV Karsch et al, ’04 y 3 I p4 4 mPS = 67(17) MeV Karsch et al, ’04 y 3 S KS 4 amq ≈ 0.033 Karsch et al, ’04 y 3 S KS 4 amq = 0.0260(5) de Forcrand et al, ’07 y 3 S KS 4 mPS/T = 1.680(4) de Forcrand et al, ’07 y 3 S KS 6 mPS/T = 0954(12) de Forcrand et al, ’07 n 2+1 I stout KS 4 mq/mphy ≤ 0.07 Endrodi et al, ’07 n 2+1 I stout KS 6 mq/mphy ≤ 0.12 Endrodi et al, ’07 n 3 I HISQ 6 mPS ≲ 50 MeV Bazavov et al, ’17 y 3 I

• imp. W

4,6,8 mPS ∼ 300 MeV

• ur, ’14

y 3 I

• imp. W

8,10 mPS ≲ 170 MeV

• ur, ’17

n 2+1 I HISQ 6,8(,12) mπ ≤ 80 MeV Ding et al, ’18, ’19 Endrodi et al, ’07 : ml/ms = mphy

l

/mphy

s

Ding et al, ’18, ’19 : ms = mphy

s 4 / 34

Our study for critical endpoints

• Iwasaki gauge + NP O(a) improved Wilson fermions
• chiral condensate (10 - 20 noises for TrD−1,−2.−3,−4)
• kurtosis intersection method to determine the critical

endpoint

• reweighting method to obtain more critical endpoints

E 1st order crossover Kt

x x K K

• 2

light heavy

V>V>V

• O(100) zero temperature runs for physical scale setting

are covering almost critical endpoints and also transition points of finite temperature simulations

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Nf = 3 simulation parameters

• Nt = 4 (a ≈ 0.23fm) , Nl = 6, 8, 10, 12, 16
• β = 1.60, κ = 0.1431 − 0.1439
• β = 1.65, κ = 0.1410 − 0.1415
• β = 1.70, κ = 0.1371 − 0.1390
• Nt = 6 (a ≈ 0.19fm) , Nl = 10, 12, 16, 24
• β = 1.715, κ = 0.140900 − 0.141100
• β = 1.73, κ = 0.140420 − 0.140450
• β = 1.75, κ = 0.139620 − 0.139700
• Nt = 8 (a ≈ 0.16fm) , Nl = 16, 20, 24, 28
• β = 1.745, κ = 0.140371 − 0.140393
• β = 1.74995, κ = 0.140240
• β = 1.76, κ = 0.139950
• Nt = 10 (a ≈ 0.13fm), Nl = 16, 20, 24, 28
• β = 1.77, κ = 0.139800 − 0.139900
• β = 1.78, κ = 0.139550 − 0.139650
• β = 1.79, κ = 0.139300 − 0.139400

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Nf = 3 susceptibility and kurtosis (example 1)

50 100 150 200 250 300 350 400 450 susceptibility 0.14037 0.14038 0.14038 0.14039 0.14039 0.14040 0.14040

• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 kurtosis κ Nt=8, β=1.745 Ns=16 Ns=20 Ns=24 Ns=28 7 / 34

Nf = 3 susceptibility and kurtosis (example 2)

10 20 30 40 50 60 70 80 90 susceptibility 0.13956 0.13958 0.13960 0.13962 0.13964 0.13966

• 1.0
• 0.5

0.0 0.5 1.0 kurtosis κ Nt=10, β=1.78 Ns=16 Ns=20 Ns=24 Ns=28 8 / 34

Nf = 3 kurtosis intersection (Nt = 4)

• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 kurtosis β Nt=4 Ns=10 Ns=12 Ns=16 CEP 3-dim Z2

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Nf = 3 kurtosis intersection (Nt = 6)

• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.71 1.72 1.73 1.74 1.75 1.76 kurtosis β Nt=6 Ns=10 Ns=12 Ns=16 Ns=24 CEP 3-dim Z2

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Nf = 3 kurtosis intersection (Nt = 8)

• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.740 1.745 1.750 1.755 1.760 1.765 1.770 kurtosis β Nt=8 Ns=16 Ns=20 Ns=24 Ns=28 CEP 3-dim Z2

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Nf = 3 kurtosis intersection (Nt = 10)

• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.75 1.76 1.77 1.78 1.79 1.80 1.81 kurtosis β Nt=10 Ns=16 Ns=20 Ns=24 Ns=28 CEP 3-dim Z2

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Nf = 3 kurtosis intersection fitting

K =

[

KE + AN1/ν

l

(β − βE) ] (1 + BNyt−yh

l

)

Nt Fit βE KE ν A B yt − yh χ2/dof 4 1 1.6115(26) −1.383(48) 0.84(13) 0.88(42) × × 1.75 2 1.61065(61) −1.396 0.63 0.313(12) × × 3.05 3 1.6099(17) −1.396 0.63 0.311(14) 0.10(21) −0.894 3.77 6 1 1.72518(71) −1.373(17) 0.683(54) 0.58(17) × × 0.68 2 1.72431(24) −1.396 0.63 0.418(11) × × 0.70 3 1.72462(40) −1.396 0.63 0.422(12) −0.052(52) −0.894 0.70 8 1 1.75049(57) −1.219(25) 0.527(55) 0.146(88) × × 0.73 2 1.74721(42) −1.396 0.63 0.404(36) × × 5.99 3 1.74953(33) −1.396 0.63 0.414(13) −1.33(15) −0.894 0.73 10 1 1.77796(48) −0.974(25) 0.466(45) 0.084(52) × × 0.22 2 1.7694(16) −1.396 0.63 0.421(95) × × 10.03 3 1.77545(53) −1.396 0.63 0.559(29) −2.97(25) −0.894 0.43 Fit-1: no correction term (B = yt − yh = 0) and all other parameters are used as fit parameter. Fit-2: no correction term and assuming the 3D Z2 universality class for KE and ν. Fit-3: including correction term and assuming the 3D Z2 universality class for KE, ν and yt − yh.

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Nf = 3, b = γ/ν of χmax ∝ Nb

l

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 b β Nt=4 Nt=6 Nt=8 Nt=10 3-dim Z2

critical endpoints determined by kurtosis intersection and critical exponent are consistent

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Critical endpoint (mPS)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑛PS, E 1⁄𝑂 2

𝑢

crossover 1st order SU(3) symmetric point fit: 𝑏0 + 𝑏1⁄𝑂 2

𝑢 + 𝑏2⁄𝑂 3 𝑢

solve: 𝑏0 + 𝑏1⁄𝑂 2

𝑢

solve: 𝑏0 + 𝑏1⁄𝑂 2

𝑢 + 𝑏2⁄𝑂 3 𝑢

Continuum extrapolation for √ t0mPS,E is still uncertain

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Critical endpoint (T)

0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑈E 1⁄𝑂 2

𝑢

fit: 𝑏0 + 𝑏1⁄𝑂 2

𝑢 16 / 34

Critical endpoint

In the continuum limit mPS,E ≲ 170MeV TE = 134(3)MeV mPS,E/TE ≲ 1.3 We shall extend our study at Nt = 12

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Previous study for critical end line

[de Forcrand, Philipsen, ’07]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.01 0.02 0.03 0.04 ams amu,d Nf=2+1 physical point ms

tric - C mud 2/5

• staggered fermions
• Nt = 4, a ≈ 0.3 fm
• data exhibits that slope at msym is not - 2
• amcrit

s

≈ 0.7 (roughly 5 times larger than mphy

s

)

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Our study for critical end line(1/2)

around msym

0.20 0.25 0.30 0.35 0.40 0.45 (󰠌𝑢0𝑛𝜃𝑡)2 endpoints 2nd degree poly. fit 3rd degree poly. fit 𝑂𝑔 = 3 0.20 0.25 0.30 0.35 0.40 0.45

• 2.4
• 2.2
• 2.0
• 1.8
• 1.6

slope (󰠌𝑢0𝑛𝜌)2

• Wilson-clover fermion
• Nt = 6
• a ≈ 0.19 fm
• we confirmed that

slope at msym is - 2

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Our study for critical end line(2/2)

We shall extend our study for critical end line away form msym

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Preliminary results

• New Nf = 3 simulation at Nt = 12
• New Nf = 2 + 1 simulation away from symmetric point

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New Nf = 3 simulation at Nt = 12

• Nt = 12 (a ≈ 0.1fm ?), Nl = 16, 20, 24, 28, 32
• β = 1.78, κ = 0.1396 − 0.1397
• β = 1.79, κ = 0.1393 − 0.1395
• β = 1.80, κ = 0.1390 − 0.1394
• β = 1.81, κ = 0.13895 − 0.13910
• β = 1.82, κ = 0.13875 − 0.13895
• β = 1.83, κ = 0.13855 − 0.13880
• β = 1.84, κ = 0.13835 − 0.13870
• β = 1.85, κ = 0.13815 − 0.13860

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Nf = 3 susceptibility (Nt = 12)

20 40 60 80 100 120 0.1391 0.13915 0.1392 0.13925 0.1393 0.13935 pbpz sus κ β=1.80 Ns=16 β=1.80 Ns=20 β=1.80 Ns=24 23 / 34

Nf = 3 kurtosis (Nt = 12)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 0.1391 0.13915 0.1392 0.13925 0.1393 0.13935 pbpz krt κ Z2 β=1.80 Ns=16 β=1.80 Ns=20 β=1.80 Ns=24 24 / 34

Nf = 3 kurtosis intersection (Nt = 12)

• 2
• 1.5
• 1
• 0.5

1.79 1.795 1.8 1.805 1.81 1.815 1.82 1.825 1.83 kurtosis β Nt=12 (Fit 3) Ns=24 Ns=28 Ns=32 CEP 3-dim Z2

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Nf = 3, b = γ/ν of χmax ∝ Nb

l

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 b β Nt=4 Nt=6 Nt=8 Nt=10 Nt=12 3-dim Z2

critical endpoints determined by kurtosis intersection and critical exponent are consistent including Nt = 12

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New Nf = 2 + 1 simulation away from symmetric point

• Nt = 6 (a ≈ 0.19fm), Nl = 10, 12, 16, 20, 24
• symmetric runs
• β = 1.715, κ = 0.140900 − 0.141100
• β = 1.725, κ = 0.140600 − 0.140618
• β = 1.73, κ = 0.140420 − 0.140450
• very heavy ms runs (κs = 0.128)
• β = 1.73, κl = 0.143365 − 0.143390
• β = 1.74, κl = 0.142970 − 0.143042
• β = 1.745, κl = 0.142733 − 0.142790
• heavy ms runs (κs = 0.1328)
• β = 1.72, κl = 0.143160
• β = 1.73, κl = 0.142702 − 0.142750
• β = 1.735, κl = 0.142508

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Nf = 2 + 1 expectation, susceptibility, skewness and kurtosis (example)

6.7 6.75 6.8 6.85 6.9 6.95 7 7.05 0.1425 0.143 0.1435 0.144 pbpz exp κl κs=0.128000 β=1.72 Ns=10 β=1.73 Ns=10 β=1.73 Ns=12 β=1.73 Ns=16 β=1.74 Ns=10 β=1.74 Ns=12 β=1.74 Ns=16 β=1.745 Ns=10 β=1.745 Ns=12 β=1.745 Ns=16 20 40 60 80 100 120 140 160 180 0.1425 0.143 0.1435 0.144 pbpz sus κl κs=0.128000 β=1.72 Ns=10 β=1.73 Ns=10 β=1.73 Ns=12 β=1.73 Ns=16 β=1.74 Ns=10 β=1.74 Ns=12 β=1.74 Ns=16 β=1.745 Ns=10 β=1.745 Ns=12 β=1.745 Ns=16

• 4
• 3
• 2
• 1

1 2 3 4 0.1425 0.143 0.1435 0.144 pbpz skw κl κs=0.128000 β=1.72 Ns=10 β=1.73 Ns=10 β=1.73 Ns=12 β=1.73 Ns=16 β=1.74 Ns=10 β=1.74 Ns=12 β=1.74 Ns=16 β=1.745 Ns=10 β=1.745 Ns=12 β=1.745 Ns=16

• 2
• 1.5
• 1
• 0.5

0.1425 0.143 0.1435 0.144 pbpz krt κl κs=0.128000 Z2 β=1.72 Ns=10 β=1.73 Ns=10 β=1.73 Ns=12 β=1.73 Ns=16 β=1.74 Ns=10 β=1.74 Ns=12 β=1.74 Ns=16 β=1.745 Ns=10 β=1.745 Ns=12 β=1.745 Ns=16

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Nf = 2 + 1 kurtosis intersection (example)

• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1

1.715 1.72 1.725 1.73 1.735 1.74 1.745 1.75 1.755 kurtosis β NT=6 κs=0.128000 Ns=10 Ns=12 Ns=16 CEP 3-dim Z2

• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1

1.715 1.72 1.725 1.73 1.735 1.74 1.745 1.75 1.755 kurtosis β NT=6 κs=0.128800 Ns=10 Ns=12 Ns=16 CEP 3-dim Z2

• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1

1.715 1.72 1.725 1.73 1.735 1.74 1.745 1.75 1.755 kurtosis β NT=6 κs=0.131300 Ns=10 Ns=12 Ns=16 CEP 3-dim Z2

• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1

1.715 1.72 1.725 1.73 1.735 1.74 1.745 1.75 1.755 kurtosis β NT=6 κs=0.132800 Ns=10 Ns=12 Ns=16 CEP 3-dim Z2

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Critical endpoints in bare parameter plane

7.00 7.20 7.40 7.60 7.80 8.00 6.90 7.00 7.10 7.20 7.30 7.40 1⁄𝜆s 1⁄𝜆l endpoints (very large 𝑛s runs) endpoints (large 𝑛s runs) endpoints (symmetric runs) 7.00 7.20 7.40 7.60 7.80 8.00 1.720 1.725 1.730 1.735 1.740 1.745 1.750 1⁄𝜆s 𝛾 endpoints (very large 𝑛s runs) endpoints (large 𝑛s runs) endpoints (symmetric runs)

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Critical endline at Nt = 6

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 (󰠌𝑢0𝑛𝜃𝑡)2 (󰠌𝑢0𝑛𝜌)2 CEP at 𝑂T = 6 CEP at 𝑂T = 8 CEP at 𝑂T = 10 CEP cont. upper lim. const 𝑛𝑡⁄𝑛𝑚 physical point 𝑂𝑔 = 3 fit range 1 fit range 2 fit range 3 fit range 4

ms − mtric

s

∼ m2/5

l

[Rajagopal ’95] Fitting endpoints for tri-critical point x = ( √ t0mπ)2 ∝ ml y = ( √ t0mηs)2 ∝ ms y = b0 + b1x2/5 b0 b1 χ2/dof range (x <) 6.71(8)

• 13.3(2)

0.54

• f. r. 1

6.44(2)

• 12.62(4)

0.97

• f. r. 2

6.34(3)

• 12.39(6)

3.26

• f. r. 3

3.28(9)

• 5.1(2)

1655

• f. r. 4

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Critical endline at Nt = 6

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 (󰠌𝑢0𝑛𝜃𝑡)2 (󰠌𝑢0𝑛𝜌)2 CEP at 𝑂T = 6 CEP at 𝑂T = 8 CEP at 𝑂T = 10 CEP cont. upper lim. const 𝑛𝑡⁄𝑛𝑚 physical point 𝑂𝑔 = 3

Fiting endpoints for all range with b0 = 6.71 Fit 1 y = b0 + a0x2/5 + a1x + a2x2 + a3x3 χ2/dof = 158 Fit 2 y = b0 + a0x2/5 + a1x + a2x2 + a3x3 + a4x4 χ2/dof = 6.6 Fit 3 y = b0 + a0x2/5 + a1x + a2x2 + a3x3 + a4x4 + a5x5 χ2/dof = 1.3

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Critical endline at Nt = 8, 10, ∞ (estimated upper bound)

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 (󰠌𝑢0𝑛𝜃𝑡)2 (󰠌𝑢0𝑛𝜌)2 CEP at 𝑂T = 6 CEP at 𝑂T = 8 CEP at 𝑂T = 10 CEP cont. upper lim. const 𝑛𝑡⁄𝑛𝑚 physical point 𝑂𝑔 = 3

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 (󰠌𝑢0𝑛𝜃𝑡)2 (󰠌𝑢0𝑛𝜌)2 upper bound of CEL upper bound of tricritical point physical point const 𝑛𝑡⁄𝑛𝑚 𝑂𝑔 = 3

assuming same shape

33 / 34

Summary

We are updating the critical endpint in the continuum limit and the critical endline away from the SU(3)-flavor symmetric point at Nt = 6 and presented preliminary results for the critical end lines at Nt = 8, 10 and in the continuum limit with NP O(a) improved Wilson fermions. We find

• 3 series of multi-ensemble, multi-parameter re-weighting

determines well the critical end line

• critical end line at Nt = 6 is nice agreement with ms − mtri

s ∼ m2/5 l

in small ml region

• mtri

s ≲ 1.5 mphy s

(very preliminary!!)

• tentative tri-critical scaling region : m2

ηs ≳ 5m2 π

34 / 34