Cumulants ratios of conserved charge fluctuations: A comparison of - - PowerPoint PPT Presentation

SLIDE 1

Christian Schmidt Sign2015, Debrecen, Hungary

Cumulants ratios of conserved charge fluctuations: A comparison of lattice QCD and experimental results

Christian Schmidt

1

BNL-Bi-CCNU Collaboration:

• A. Bazavov, H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, E. Laermann, S. Mukherjee, P. Petreczky,
• C. Schmidt, W. Soeldner, M. Wagner

SLIDE 2

Christian Schmidt Sign2015, Debrecen, Hungary 2

Motivation: The QCD phase diagram QCD

hadron gas

nuclear matter neutron stars vacuum

quark-gluon-plasma

154(9)

chemical potential µB

Expected phase diagram of QCD:

critical end-point Critical end-point? T [MeV]

SLIDE 3

Christian Schmidt Sign2015, Debrecen, Hungary 2

Motivation: The QCD phase diagram ⇒ Diverging correlation length and fluctuations.

Universal behavior within a scaling region.

QCD

hadron gas

nuclear matter neutron stars vacuum

quark-gluon-plasma

154(9)

chemical potential µB

Expected phase diagram of QCD:

critical end-point Critical end-point? T [MeV]

SLIDE 4

Christian Schmidt Sign2015, Debrecen, Hungary 3

T [MeV]

QCD

nuclear matter neutron stars vacuum

154(9)

chemical potential µB

Expected phase diagram of QCD:

what we really know...

Critical end-point?

Motivation: The QCD phase diagram

SLIDE 5

Christian Schmidt Sign2015, Debrecen, Hungary 3

T [MeV]

QCD

nuclear matter neutron stars vacuum

154(9)

chemical potential µB

Expected phase diagram of QCD:

what we really know...

⇒ Diverging correlation length and fluctuations.

Universal behavior within a scaling region. Critical end-point?

Motivation: The QCD phase diagram

SLIDE 6

Christian Schmidt Sign2015, Debrecen, Hungary 4

2nd order, O(4) 2nd order, Z(2) 1st order crossover

T

Lattice Experiment (freeze-out)

mu,d mphys

u,d

µB µCEP

B

µtri

B

Motivation: The QCD phase diagram

Quark mass dependance of the phase diagram:

Is physics on the freeze-out line sensitive to QCD critical behavior? More critical points!

SLIDE 7

Christian Schmidt Sign2015, Debrecen, Hungary 5

Experimental efforts: Beam Energy Scan

Chemical freeze-out: defines the moment from where particle abundance are fixed (up to particle decays), parametrized by

Tf(√s), µf(√s), Vf(√s)

dN/dy

1

10 1 10

2

10 Data STAR PHENIX BRAHMS

=29.7/11

df

/N

2

• Model,

3

= 30 MeV, V=1950 fm

b

µ T=164 MeV,

=200 GeV

NN

s

+

• +

K

• K p p
• +

d d K* *

• *
• Andronic, Braun-Munzinger,

Stachel, PLB 673 (2009) 142.

√sNN

Initial conditions: depend

• n collision energy ,

the system size (type of ion), the impact parameter, ... hydrodynamic evolution

SLIDE 8

• κσ2 = χ4/χ2
• X. Luo, CPOD’14

Christian Schmidt Sign2015, Debrecen, Hungary 6

Experimental efforts: Beam Energy Scan

intriguing non-monotonic behavior in the cumulant ratio of net-proton number fluctuations

⇒

√s

SLIDE 9

Christian Schmidt Sign2015, Debrecen, Hungary 7

Experimental efforts: Beam Energy Scan

• κσ2 = χ4/χ2
• X. Luo, CPOD’14

Can this data be understood in terms of equilibrium thermodynamics? √s

SLIDE 10

Christian Schmidt Sign2015, Debrecen, Hungary 8

• κσ2 = χ4/χ2
• X. Luo, CPOD’14

Can this data be understood in terms of equilibrium thermodynamics? How far do we get with a low order Taylor expansion?

Experimental efforts: Beam Energy Scan

SLIDE 11

Christian Schmidt Sign2015, Debrecen, Hungary 9

Motivation: The QCD phase diagram

T [MeV]

nuclear matter neutron stars vacuum

154(9)

chemical potential µB Are the curvature of the crossover temperature and the freeze-out curve considerably different ?

0.005 0.01 0.015 0.02 0.025 0.03

Lattice 2015 -- Curvature of the phase diagram

Bielefeld-CCNU HISQ Nt=6 Taylor Cea et al HISQ Nt=6,8,10,12 Analytical [1508.07599] Pisa 2stout Nt=6,8,10,12 Analytical [1507.03571] Wuppertal 4stout Nt=10,12,16 Analytical [1507.07510]

• J. Cleymans et al., PRC 73, 034905 (2006).

Bielefeld-BNL P4, Nt=8, PRD 83 (2011) 014504 Figure taken from S. Borsanyi, QM2015 (modified)

SLIDE 12

Christian Schmidt Sign2015, Debrecen, Hungary 10

Content

1) Introduction and Motivation 2) Taylor expansion of pressure

• definitions, state-of-the-art, convergence estimate

2) Cumulant ratios at nonzero baryon number density

• determination of freeze-out parameter, expressing by
• RHIC data vs. QCD equilibrium thermodynamics
• constraints: strangeness neutrality, constant baryon number to

electric charge ratio

3) Conclusions and Summary

MB/σ2

B

µf

B

SLIDE 13

Christian Schmidt 2nd Heavy Ion Collisions in the LHC era and Beyond 11

Taylor expansion of the pressure

SLIDE 14

Christian Schmidt Sign2015, Debrecen, Hungary 12

Conserved charge fluctuations

Expansion of the pressure: X = B, Q, S: conserved charges Lattice Experiment generalized susceptibilities cumulants of net-charge fluctuations

δNX ⌘ NX hNXi

⇒

• nly at !

µX = 0

⇒ only at freeze-out ( )!

µf(√s), Tf(√s)

V T 3 χX

2

= ⌦ (δNX)2↵ V T 3 χX

4

= ⌦ (δNX)4↵ − 3 ⌦ (δNX)2↵2 V T 3 χX

6

= ⌦ (δNX)4↵ −15 ⌦ (δNX)4↵ ⌦ (δNX)2↵ +30 ⌦ (δNX)2↵3

p T 4 =

∞

X

i,j,k=0

1 i!j!k!χBQS

ijk,0

✓µB T ◆i ✓µQ T ◆j ✓µS T ◆k

χX

n =

∂n[p/T 4] ∂(µX/T )n

• µX=0

SLIDE 15

χX

1 (µB, T )

χX

2 (µB, T )

= MX σ2

X

χX

3 (µB, T )

χX

2 (µB, T )

χX

4 (µB, T )

χX

2 (µB, T )

SXσX κXσ2

X

Christian Schmidt Sign2015, Debrecen, Hungary 13

Conserved charge fluctuations

Expansion of the pressure: X = B, Q, S: conserved charges Lattice Experiment

p T 4 =

∞

X

i,j,k=0

1 i!j!k!χBQS

ijk,0

✓µB T ◆i ✓µQ T ◆j ✓µS T ◆k

= = κ := S := σ2 := M := mean variance skewness kurtosis consider cumulant ratios to eliminate the freeze-out volume

SLIDE 16

Christian Schmidt Sign2015, Debrecen, Hungary 14

pressure , energy density and entropy density , at : Bazavov et al. [HotQCD], Phys. Rev. D90 (2014) 094503.

• improves over earlier HotQCD

calculation Bazavov et al. [HotQCD],

• Phys. Rev. D80 (2009) 014504.
• consistent with results from

Budapest-Wuppertal (stout)

• S. Borsanyi et al. [WB] Phys. Lett.

B730 (2014) 99

State-of-the-art equation of state for (2+1)-flavor

p ε s µB = µQ = µS = 0

• up to the crossover region the QCD

EoS agrees well with the HRG EoS, however, QCD results are systematically above HRG evidence for additional hadronic states?

⇒

SLIDE 17

Christian Schmidt Sign2015, Debrecen, Hungary 15

The equation of state at

chemical potential dependent part:

T [MeV] 2

B

free continuum extrap. N=12 8 6 PDG-HRG 0.05 0.1 0.15 0.2 0.25 0.3 0.35 120 140 160 180 200 220 240 260 280

T [MeV] χ4

B/χ2 B

hadron resonance gas free quark gas

Nτ=6 8 0.2 0.4 0.6 0.8 1 1.2 120 140 160 180 200 220 240 260 280 BNL-Bielefeld preliminary

∆

• P/T 4

= P(T, µB) P(T, 0) T 4 =

∞

X

n=1

χB

2n(T)

(2n)! ⇣µB T ⌘2n = 1 2χB

2 (T)ˆ

µ2

B

✓ 1 + 1 12 χB

4 (T)

χB

2 (T) ˆ

µ2

B +

1 360 χB

6 (T)

χB

2 (T) ˆ

µ4

B + ...

◆

LO NLO NNLO LO NLO

ratios are unity in the HRG

µB > 0

ˆ µB = µB/T with

SLIDE 18

Christian Schmidt Sign2015, Debrecen, Hungary 16

The equation of state at µB > 0

chemical potential dependent part: ∆

• P/T 4

= P(T, µB) P(T, 0) T 4 =

∞

X

n=1

χB

2n(T)

(2n)! ⇣µB T ⌘2n = 1 2χB

2 (T)ˆ

µ2

B

✓ 1 + 1 12 χB

4 (T)

χB

2 (T) ˆ

µ2

B +

1 360 χB

6 (T)

χB

2 (T) ˆ

µ4

B + ...

◆

LO NLO NNLO

ratios are unity in the HRG ˆ µB = µB/T with

SLIDE 19

. 30% . 20% √s & 20 GeV ˆ µB < 2

Christian Schmidt Sign2015, Debrecen, Hungary 17

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6 8

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6 8

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6 8

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6 8

<5%

• 4
• 2

2 4 6 160 180 200 220 240 260 280 T [MeV]

χ6

B/χ2 B

Nτ=6 8

<5%

• 0.4
• 0.2

0.2 0.4 0.6 180 220 260 300 340

<1%

• 0.4
• 0.2

0.2 0.4 0.6 180 220 260 300 340

<1%

• 0.4
• 0.2

0.2 0.4 0.6 180 220 260 300 340

<1%

NLO NNLO

relative contributions to LO for T<150 MeV at : ˆ µB = 2

(relative to LO) (relative to NLO)

. 5% translates into a contribution to the total pressure: NNLO

⇒ the 4th order EoS is well

controlled for , corresponding to BNL-Bi-CCNU preliminary

• ongoing calculation, need to control the

statistical error further estimating the convergence:

The equation of state at µB > 0

SLIDE 20

Christian Schmidt 2nd Heavy Ion Collisions in the LHC era and Beyond 18

Cumulant ratios at µB > 0

SLIDE 21

µQ = µS = 0

MQ σ2

Q

= µB T χBQ

11

χQ

2

1 + 1

6 χBQ

31

χBQ

11

µB

T

2 1 + 1

2 χBQ

22

χQ

2

µB

T

2

Christian Schmidt Sign2015, Debrecen, Hungary 19

Conserved charge fluctutations and freeze-out

expanding ratios of baryon number fluctuations χB

n+2/χB n

• in QCD all the ratios are

a function of temperature, in HRG they are unity electric charge fluctuations:

• for simplicity at

SLIDE 22

√sNN µB → MB/σ2

B = χB 1 /χB 2 = RB 12

RP

12

RB

12

µB/T RB

12

µB/T = mB

1 RB 12

+ mB

3

• RB

12

3 + · · · RB/P

12

∼ µB/T

Christian Schmidt Sign2015, Debrecen, Hungary 20

Conserved charge fluctutations and freeze-out

how to translate into without making further approximations? trick: express all ratios as function of

• use as proxy for

1/√s µB

• is found to be monotonic

in and thus also in

RP

12

→ can be inverted, for

we use the expansion HRG: 1 2 3

• in HRG we find

for small (as long as )

µB/T

µQ = µS = 0

SLIDE 23

χB

1,µ = χB 2 ˆ

µB + 1 6χ4ˆ µ3

B + · · ·

χB

2,µ = χB 2 + 1

2χ4ˆ µ2

B + · · ·

¯ χB

1,µ =

⇣ χB

2 + χBS 11 s1 + χBQ 11 q1

⌘ ˆ µB + · · · ¯ χB

2,µ = χB 2 +

✓1 2χB

4 + χBS 31 s1 + χBQ 31 q1

◆ ˆ µ2

B + · ·

Christian Schmidt Sign2015, Debrecen, Hungary 21

Introducing and µS > 0 µQ > 0

ˆ µB = µB/T

• strangeness neutrality: hNSi = 0
• isospin assymetry: hNQi = r hNBi

Apply: initial conditions in HIC r ≈ 0.4 for Au-Au and Pb-Pb expand in powers of solve for µB, µQ, µS µQ, µS LO NLO µQ(T, µB) = q1(T )ˆ µB + q3(T )ˆ µ3

B

µS(T, µB) = s1(T )ˆ µB + s3(T )ˆ µ3

B

define strangeness neutral coefficients

→ →

et cetera

SLIDE 24

RQB

12

≡ RQ

12/RB 12 =

MQ/σ2

Q

MB/σ2

B

RQB

12

= a12 ⇣ 1 + c12

• RB

12

2⌘ RQB,0

12

(T ) = r χQ

2 (T )

χB

2 (T )

Tf(µB) = Tf,0 ⇣ 1 − κf

2 (µB/Tf,0)2⌘

c12(T, κf

2) = c0 12(T ) − κf 2D12(T )

Christian Schmidt Sign2015, Debrecen, Hungary 22

The curvature of the freeze-out line

Ansatz: Expand: along the freeze-out line

a12 Tf,0

the constant fixes the freeze-out temperature

c12

the coefficient fixes the curvature of the freeze-out line

RQB,0

12

(T ), c0

12(T ), D12(T )

can be calculated in QCD

a12, c12

can be extracted from RHIC data

SLIDE 25

Christian Schmidt Sign2015, Debrecen, Hungary 23

The curvature of the freeze-out line

eV V V

lattice data on RQB

12

RQP

12

RHIC data on BNL-Bi-CCNU, arXiv: 1509:05786

Tf,0

155 MeV 147 MeV 145 MeV

κf

2

• nce is fixed,

the slope yields

Tf,0

SLIDE 26

κf

2 < 0.011

pmax

t

= 0.8 pmax

t

= 2.0 κf

2 = 0.023(3)

Christian Schmidt Sign2015, Debrecen, Hungary 24

The curvature of the freeze-out line

s1/2

NN [GeV]

(MQ/σQ

2 )/(MP/σP 2 )

(MP/σP

2)2

QCD: (Tf,0, κf

2=0)

(Tf,0,κf

2=0.02)

STAR: pt

max=2.0 GeV

pt

max=0.8 GeV

PHENIX/STAR2.0

0.05 0.1 0.15 0.2 0.25 0.2 0.4 0.6 0.8 1 200 62.4 39 27 19.6 11.5 7.7

STAR0.8 STAR2.0 PHENIX/STAR2.0 a12 0.079(3) 0.087(2) 0.110(9) c12 0.858(101) 0.329(74) 0.559(352) Tf,0 [MeV] 145(2) 147(2) 155(4) c0

12(Tf,0)

0.343(31) 0.326(32) 0.265(52) D12(Tf,0) 7.04(44) 6.62(36) 5.27(78) κf

2

• 0.073(16) -0.001(12)
• 0.056(67)

BNL-Bi-CCNU, arXiv: 1509:05786 we obtain an upper bound of

• the published STAR data

( GeV) are too steep, yield a negative curvature

• the preliminary STAR data

( GeV) favor a small curvature in contrast: a parametrization of HRG model fit results gave

• J. Cleymans et al.,

PRC 73, 034905 (2006).

SLIDE 27

RB

31 ≡ SBσ3 B

MB = RB,0

31

+ RB,2

31 RB 12 + O

⇣ RB

12

4⌘ SBσB = χB

4

χB

2

MB σ2

B

+ 1 6 @χB

6

χB

2

− χB

4

χB

2

!21 A ✓MB σ2

B

◆3 + · · · ⇐ ⇒

Christian Schmidt Sign2015, Debrecen, Hungary 25

Skewness at µB > 0

use from fit

Tf,0 RQP

12

fit to STAR data LO consistent with QCD result

SLIDE 28

Christian Schmidt Sign2015, Debrecen, Hungary 26

Skewness at µB > 0

RB

31 ≡ SBσ3 B

MB = RB,0

31

+ RB,2

31 RB 12 + O

⇣ RB

12

4⌘ SBσB = χB

4

χB

2

MB σ2

B

+ 1 6 @χB

6

χB

2

− χB

4

χB

2

!21 A ✓MB σ2

B

◆3 + · · · ⇐ ⇒

• 0.2

use from fit

Tf,0 RQP

12

fit to STAR data NLO consistent with QCD result

SLIDE 29

RB

31 ≡ SBσ3 B/MB

RB

42 ≡ κBσ2 B

RB

42 = RB,0 42

+ RB,2

42

• RB

12

2 RB

31 = RB,0 31

+ RB,2

31

• RB

12

2

Christian Schmidt Sign2015, Debrecen, Hungary 27

Kurtosis at µB > 0

in NLO Taylor expansion, and are closely related the NLO expansion:

RB,0

42

= RB,0

31

= χB

4

χB

2

RB,2

42

= 3RB,2

31

= 1 2 @χB

6

χ2 − χB

4

χB

2

!21 A

at we find µQ = µS = 0

µQ > 0, µS > 0

at

SLIDE 30

Christian Schmidt Sign2015, Debrecen, Hungary 28

Kurtosis at µB > 0

RB

31 ≡ SBσ3 B/MB

RB

42 ≡ κBσ2 B

RB

42 = RB,0 42

+ RB,2

42

• RB

12

2 RB

31 = RB,0 31

+ RB,2

31

• RB

12

2

in NLO Taylor expansion, and are closely related the NLO expansion:

RB,0

42

= RB,0

31

= χB

4

χB

2

RB,2

42

= 3RB,2

31

= 1 2 @χB

6

χ2 − χB

4

χB

2

!21 A

at we find µQ = µS = 0 this is can be seen also in the STAR data

SLIDE 31

Christian Schmidt 2nd Heavy Ion Collisions in the LHC era and Beyond 29

Conclusions and Summary

SLIDE 32

Christian Schmidt Sign2015, Debrecen, Hungary 30

• Cumulants of conserved charge fluctuations are interesting quantities to compute in

(lattice) QCD, they can also be measured in heavy ion collision.

Conclusions and Summary

• The freeze-out temperature is smaller than the crossover temperature.
• The fourth-order expansion of the EOS is suitable for the modeling of dense matter

created in heavy ion collisions with center-of-mass energies down to . √s & 20 GeV

• The curvature of the freeze-out line is compatible with zero

(upper bound is )

κf

2 < 0.011

• In the range the pattern seen in the beam energy

dependance of up to 4th order cumulants of net-proton (baryon) number and electric charge fluctuations can be understood in terms of QCD equilibrium thermodynamics.

27 GeV ≤ √sNN ≤ 200 GeV

SLIDE 33

Christian Schmidt Sign2015, Debrecen, Hungary 31

B: net baryon number

⇒ statistical errors are under control for all 4th order cum ⇒ In general: find good agreement with HRG model for T<155 MeV The lattice setup

Nτ = 6, 8, 12 mπ = 140, 160 MeV Lattice parameters:

• (2+1)-flavor of highly improved

staggered fermions (HISQ-action)

• a set of different lattice spacings

( )

• two different pion masses:
• high statistics:

configurations

0.1 0.2 0.3 2

B

Tc=154(9)MeV

N=6 8 phys,8 0.05 0.1 140 160 180 200 220 240 4

B

T [MeV] BNL-Bielefeld preliminary

(10 − 30) × 103

SLIDE 34

Christian Schmidt Sign2015, Debrecen, Hungary 32

The lattice setup

N = 1500

∂ ln Z ∂µ = 1 Z Z DU Tr ⇥ M 1M 0⇤ eTr lnMeβSG = ⌦ Tr ⇥ M 1M 0⇤↵

Observables: traces of combinations of and Method: stochastic estimators with random vectors

Tr [Q] ≈ 1 N

N

X

i=1

η†

i Qηi

lim

N→∞

1 N

N

X

i=1

η†

i,xηi,y = δx,y

with

M (n) = ∂nM/∂µn

M −1

∂2 ln Z ∂µ2 = ⌦ Tr ⇥ M 1M 00⇤↵ − ⌦ Tr ⇥ M 1M 0M 1M 0⇤↵ + D Tr ⇥ M 1M 0⇤2E

SLIDE 35

Christian Schmidt Sign2015, Debrecen, Hungary 33

The lattice setup

0.001 0.01 0.1 1 10 100 1 10 100 1000 10000 σr( ) Nsrc χ2

gauss Z2 h128 Z2 boundary µ t-slice dil. color dil.

N = 1500 Method: stochastic estimators with random vectors

Tr [Q] ≈ 1 N

N

X

i=1

η†

i Qηi

lim

N→∞

1 N

N

X

i=1

η†

i,xηi,y = δx,y

with

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 1 10 100 1000 10000 χ2 Nsrc

boundary µ gauss

unbiased estimator on a single configuration:

courtesy P. Steinbrecher

SLIDE 36

T f ' 150(5)

Christian Schmidt Sign2015, Debrecen, Hungary 34

Freeze-out parameter from QCD

from fluctuations of electric charge constrains freeze-out temperature determines freeze-out chemical potential BI-BNL, PRL 109, 192302 (2012);

• S. Mukerhjee CPOD 2014