The QCD crossover from Lattice QCD July 25, 2018 Patrick - - PowerPoint PPT Presentation

The QCD crossover

from Lattice QCD

July 25, 2018 Patrick Steinbrecher HotQCD collaboration

The QCD phase diagram

July 25, 2018 Patrick Steinbrecher Slide 1

Quantum Chromodynamics

from first principles

Lattice QCD

HISQ action Nσ = 4Nτ

• sim. at µ = 0

physical quarks

2 light quarks 1 strange quark ms/ml = 27

mπ ≃ 138 MeV

10k 100k 1M 135 145 155 165 175 #configurations T [MeV] ms/ml=27, Nτ=16 12 8 6

July 25, 2018 Patrick Steinbrecher Slide 2

everything continuum extrapolated

Chiral observables in two-flavor formulation

subtracted condensate Σsub ≡ ms(Σu + Σd) − (mu + md)Σs with Σf = T V ∂ ∂mf ln Z subtracted susceptibility χsub ≡ T V ms ∂ ∂mu + ∂ ∂md

• Σsub

χdisc is defined as χsub without connected part

July 25, 2018 Patrick Steinbrecher Slide 3

Start of the QCD crossover line: T0

d2 dT 2 Σsub f 4

K

≡ 0 and d dT χsub f 4

K

≡ 0

July 25, 2018 Patrick Steinbrecher Slide 4

two crossover temperatures: T0 (Σsub) and T0 (χsub)

Pseudo-critical temperatures

for ml → 0: pseudo-critical temperatures converge to the chiral transition temperature T 0

c

at finite quark mass it is given by maximum of O(4) universal scaling functions (Thursday talk, Sheng-Tai Li, Chiral phase transition) χm = m1/δ−1

l

fχ(z) + reg. χt = m(β−1)/βδ

l

f ′

G(z) + reg.

for ml → 0

χt ∼ ∂TΣsub and χt ∼ ∂2

µBΣsub

χm ∼ χsub and χm ∼ χdisc

July 25, 2018 Patrick Steinbrecher Slide 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3 z ∼ (T-Tc

0)/ml 1/(β δ)

O(4): fχ(z) f’G(z)

The subtracted chiral susceptibility

50 100 150 200 250 135 145 155 165 175 χsub/fk

4

T [MeV] ms/ml=27, Nτ=16 12 8 6

July 25, 2018 Patrick Steinbrecher Slide 6

The subtracted chiral susceptibility

50 100 150 200 250 135 145 155 165 175 χsub/fk

4

T [MeV] ms/ml=27, Nτ=16 12 8 6

July 25, 2018 Patrick Steinbrecher Slide 6

The 2nd µB derivative of chiral condensate Σsub

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 135 145 155 165 175 µQ=µS=0

• c2

Σ/2

T [MeV] ms/ml=27, Nτ=12 8 6

July 25, 2018 Patrick Steinbrecher Slide 7

HotQCD preliminary

The 1st T derivative of chiral condensate Σsub

20 40 60 80 100 120 135 145 155 165 175

• T dc0

Σ/dT

T [MeV] ms/ml=27, Nτ=12 6 8

July 25, 2018 Patrick Steinbrecher Slide 8

HotQCD preliminary

The T0 continuum extrapolation

150 152 154 156 158 160 162 164 166 c

• n

t i n u u m N

τ

= 1 6 N

τ

= 1 2 N

τ

= 8 N

τ

= 6

HotQCD preliminary

Tc(µB=0) [MeV] 1/Nτ

2

χdisc χsub Σsub ∂µB

2 Σsub

∂µB

2 χdisc

156.5 ± 1.5 MeV

July 25, 2018 Patrick Steinbrecher Slide 9

Crossover temperature T0

140 145 150 155 160 165 170 Σsub χdisc χsub ∂µB

2 Σ s u b

∂µB

2 χ d i s c

Σsub, Bonati2015 χtot, Bazavov2012 Σsub, Borsanyi2010 T0 [MeV]

July 25, 2018 Patrick Steinbrecher Slide 10 HotQCD preliminary

The QCD crossover at µ = 0

d2 dT 2 Σsub(T, µB) f 4

K

≡ 0 and d dT χdisc(T, µB) f 4

K

≡ 0

July 25, 2018 Patrick Steinbrecher Slide 11

need Taylor expansion in T and µB around (T0, 0)

Taylor expansion in chemical potentials

(just notation)

simplest case µQ = µS = 0 subtracted condensate Σsub f 4

K

=

∞

• n=0

cΣ

n

n! ˆ µn

B

with cΣ

n = ∂Σsub/f 4 K

∂ˆ µn

B

• µ=0

disconnected susceptibility χdisc f 4

K

=

∞

• n=0

cχ

n

n! ˆ µn

B

with cχ

n = ∂χdisc/f 4 K

∂ˆ µn

B

• µ=0

July 25, 2018 Patrick Steinbrecher Slide 12

Coefficients for a strangeness neutral system

July 25, 2018 Patrick Steinbrecher Slide 13

• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

135 145 155 165 175 nS=0, nQ/nB=0.4 c2

Σ/2

T [MeV] ms/ml=27, Nτ=12 8 6

• 30
• 20
• 10

10 20 30 135 145 155 165 175 nS=0, nQ/nB=0.4 T dc2

Σ/dT

T [MeV] ms/ml=27, Nτ=12 6 8

• 15
• 10
• 5

5 10 15 135 145 155 165 175 nS=0, nQ/nB=0.4 c2

χ/2

T [MeV] ms/ml=27, Nτ=12 8 6

• 1000
• 500

500 1000 135 145 155 165 175 nS=0, nQ/nB=0.4 T dc2

χ/dT

T [MeV] ms/ml=27, Nτ=12 6 8 HotQCD preliminary HotQCD preliminary HotQCD preliminary HotQCD preliminary

The curvature of the crossover line

Tc(µB) T0 = 1 − κ2 µB T0 2 − κ4 µB T0 4 + O(µ6

B)

Taylor expansion in µ and T of: d dT χdisc(T, µB) f 4

K

= (...)µ2

B + (...)µ4 B + ... = 0

has to be zero order by order κ2 = 1 2T 2 T0

∂cχ

2

∂T

• (T0,0) − 2 cχ

2

• (T0,0)

∂2cχ ∂T 2

• (T0,0)

July 25, 2018 Patrick Steinbrecher Slide 14

• 0.01
• 0.005

0.005 0.01 0.015 0.02 0.025 0.005 0.01 0.015 0.02 0.025 0.03 0.035 nS=0, nQ/nB=0.4, ms/ml=27 1/Nτ

2

χdisc: κ2 κ4

HotQCD preliminary

The QCD crossover line

July 25, 2018 Patrick Steinbrecher Slide 15

130 135 140 145 150 155 160 165 170 50 100 150 200 250 300 350 400 Tc [MeV] µB [MeV] nS=0, nQ/nB=0.4

HotQCD preliminary

crossover line: O(µB

4)

constant: ε s freeze-out: STAR ALICE

STAR: arxiv:1701.07065 ALICE: arxiv:1408.6403

The curvature κn for strangeness neutral system

• 0.01
• 0.005

0.005 0.01 0.015 0.02 Σsub χdisc Σsub, Bellwied2015 κ2 κ4 nS=0, nQ/nB=0.4

July 25, 2018 Patrick Steinbrecher Slide 16 HotQCD preliminary

The crossover line

Tc(µX) T0 = 1 − κX

2

µX T0 2 − κX

4

µX T0 4 + O(µ6

X)

• 0.010

0.000 0.010 0.020 0.030 0.040 X = B B nS=0 nQ/nB=0.4 S I Q

HotQCD preliminary

κ2

X

κ4

X July 25, 2018 Patrick Steinbrecher Slide 17 Bonati 2018: X = B, µS = 0 κ2 = 0.0145(25)

Fluctuations along the QCD crossover Tc(µB)

Baryon-number fluctuations σ2

B

Vf 3

K

= 1 Vf 3

K

∂ ln Z ∂ˆ µ2

B

=

∞

• n=0

cB

n

n! ˆ µn

B

with cB

n =

1 Vf 3

K

∂ ln Z ∂ˆ µn+2

B

• µ=0

σ2

B couples to condensate −

→ diverges at a critical point study increase along the crossover line σ2

B(Tc(µB), µB) − σ2 B(T0, 0)

σ2

B(T0, 0)

= λ2 µB T0 2 + λ4 µB T0 4 + · · ·

July 25, 2018 Patrick Steinbrecher Slide 18

Baryon-number fluctuations along Tc(µB)

• 0.2

0.2 0.4 0.6 0.8 1 1.2 50 100 150 200 250 300 nS=0, nQ/nB=0.4 σB

2(Tc(µB),µB)/σB 2(T0,0) - 1

µB [MeV]

HotQCD preliminary

O(µB

4)

O(µB

2)

HRG

July 25, 2018 Patrick Steinbrecher Slide 19

Susceptibility fluctuations along Tc(µB)

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 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(µB

4)

O(µB

2)

20 40 60 80 100 135 145 155 165 175 185 195 nS=0, nQ/nB=0.4 χdisc/fk

4

T [MeV] Nτ=8, O(µB

6)

HotQCD preliminary

µB = 0.0 MeV 125.0 MeV 200.0 MeV

July 25, 2018 Patrick Steinbrecher Slide 20

σ2

B and χdisc show no indication for a narrowing crossover

Critical point from Taylor expansions

e.g. expansion of the pressure around µB =0

(for µQ ≡ µS ≡ 0)

P T 4 =

• n

1 n!χB

n ˆ

µn

B ,

χB

n =

1 VT 3 ∂n ln Z ∂ˆ µn

B

• µB=0

analysis of convergence radius can determine bound on the location of a critical point: r P

2n =

• (2n + 2)(2n + 1)χB

2n

χB

2n+2

• 1/2

, r χ

2n =

• 2n(2n − 1)χB

2n

χB

2n+2

• 1/2
• nly if coefficients are positive for all n ≥ n0

if not → no critical point on real axis

July 25, 2018 Patrick Steinbrecher Slide 21

Critical point from Taylor expansions

1 2 3 4 5 6 7 8 9 135 140 145 150 155 rn

χ -- estimator for µB crit/T

T [MeV]

2017: lower bound for r4

χ

estimator r2

χ

D’Elia et al., 2016, r4

χ

Datta et al., 2016 Fodor, Katz, 2004

1 2 3 4 5 6 7 8 9 135 140 145 150 155

r2

χ,HRG

disfavored region for the location of a critical point r4

χ,HRG

r6

χ,HRG

July 25, 2018 Patrick Steinbrecher Slide 22

Summary

crossover starts at T0 = 156.5 ± 1.5 MeV crossover curvature for strangeness neutral system

Tc(µB) T0

= 1 − κ2

• µB

T0

2 − κ4

• µB

T0

4 + O(µ6

B)

κ2 = 0.0123 ± 0.003 κ4 = 0.000131 ± 0.0041

for µB < 250 MeV and ns = 0, nQ/nB = 0.4

crossover along const. entropy density and energy density chemical freeze-out might be close to crossover no indication for critical point

July 25, 2018 Patrick Steinbrecher Slide 23

Thank you for your attention!

July 25, 2018 Patrick Steinbrecher Slide 24