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arXiv:1108.0834v1 [hep-lat] 3 Aug 2011

Moments of charge fluctuations, pseudo-critical

temperatures and freeze-out in heavy ion collisions

Frithjof Karsch

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA;

Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, D-33615 Bielefeld, Germany

E-mail: karsch@bnl.gov

Abstract.

number fluctuations and point out their relevance for the analysis of freeze-out and

critical conditions in heavy ion collisions at LHC and RHIC.

We discuss universal properties of higher order cumulants of net baryon

1. Chemical freeze-out, the hadron resonance gas and lattice QCD

Higher order moments, or more accurately higher order cumulants, of conserved charges

are central observables analyzed in the ongoing low energy runs at RHIC. It also is

quite straightforward to analyze their thermal properties in equilibrium thermodynamics

of QCD, e.g. by performing lattice QCD calculations.

different order are particularly well suited for a comparison with experiment as they

are independent of the interaction volume. Lattice QCD calculations have shown that

such ratios, e.g. ratios of baryon number, electric charge or strangeness fluctuations,

are quite sensitive probes for detecting critical behavior in QCD. They are sensitive

to universal scaling properties at vanishing as well as non-vanishing baryon chemical

potential (µB) and directly reflect the internal degrees of freedom that are carriers of

the corresponding conserved charge [1, 2] in a thermal medium. These ratios change

rapidly in the crossover region corresponding to the chiral transition in QCD and reflect

the change from hadronic to partonic degrees of freedom [3].

In the low temperature phase of QCD ratios of cumulants seem to be well described

by a hadron resonance gas model (HRG) [1, 3, 4]. The HRG is also very successful in

describing the thermal conditions which characterize the chemical freeze-out of hadron

species. The freeze-out temperature and its dependence on µBis found to be close to

the pseudo-critical temperature Tpcfor the QCD transition. We recently pointed out

that also ratios of cumulants of net baryon number fluctuations, as measured by STAR

at different beam energies [5], agree well with HRG model calculations on the chemical

freeze-out curve [6]. This is shown in Fig. 1. At the same time these results are also

consistent with lattice QCD calculations when temperature and chemical potential in

these calculations is chosen to agree with the freeze-out parameters [7, 8]. This also is

shown in Fig. 1.

Ratios of cumulants of

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Charge fluctuations and freeze-out in heavy ion collisions

2

0.1

1.0

10.0

5 10 20 50 100200

sNN

1/2 [GeV]

σB

2/MB

κBσB

2

SBσB

STAR

HRG

LGT, NLO in µB

Figure 1.

STAR [5] and calculated in a HRG model [6]. The lattice QCD results [8] have been

determined from a next-to-leading order Taylor expansion of cumulants calculated at

the values of the freeze-out chemical potential determined from the HRG model. The

freeze-out temperature at µB = 0 has been assumed to coincide with the crossover

temperature determined in lattice calculations.

Ratios of cumulants of net baryon number fluctuations measured by

The HRG model, of course, is not sensitive to critical behavior. The agreement of

current lattice calculations with HRG model calculations of ratios of cumulants including

up to fourth order fluctuations thus also indicates that they present so far no compelling

evidence for critical behavior. We recently argued that a determination of sixth order

cumulants may be particularly helpful in this respect as they clearly deviate from HRG

model calculations in the crossover region of QCD and start to become sensitive to

critical behavior earlier in the hadronic phase [9]. In the following we will point out some

generic features of higher order cumulants in QCD that arise from universal properties

of QCD close to the chiral limit of vanishing light quark masses, mq≡ mu= md= 0.

2. O(4) universality

At vanishing baryon chemical potential (µB), as well as in a certain range 0 ≤ µB≤ µc

QCD is expected to undergo a second order phase transition for mq ≡ 0. In this

entire range of µB values the chiral phase transition belongs to the universality class

of 3-dimensional, O(4) symmetric spin models.

transition temperature (Tc(¯ µB)) thermodynamic quantities show universal properties

that are controlled by the singular part, ff, of the free energy. For a fixed value of the

chemical potential ¯ µB< µc

B,

In the vicinity of the chiral phase

B, close to Tc(¯ µB) the free energy may be parametrized as,

f(T,µB,mq) = h2−αff(z) + fr(T,µB,mq) ,(1)

with z ≡ t/h1/βδdenoting the particular scaling combination of the reduced temperature

t ∼ (T − Tc(¯ µB))/Tc(¯ µB) + κB((µB/T)2− (¯ µB/T)2) and the symmetry breaking

parameter h, which for QCD is taken to be the ratio of light and strange quark

masses, h ∼ mq/ms; α, β, δ denote critical exponents of the 3-d, O(4) universality

class. Cumulants of net baryon number fluctuations are then obtained as derivatives of

f(T,µB,mq) with respect to ˆ µB≡ µB/T, i.e., χB

n∼ ∂nf/∂ˆ µn

B. Close to the chiral limit

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Charge fluctuations and freeze-out in heavy ion collisions

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-0.1

0.0

0.1

0.2

0.3

0.4

-6-4-2 0 2 4 6

z=t/h1/βδ

ff

(3)(z)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

-4 -3-2 -1 0 1 2 3 4

-(h0h)−(1+α)/βδ ff

(3)(z)

z0(T-Tc)/Tc

h0h=1

0.5

0.2

0.1

0.0

Figure 2. The third derivative of the singular part of the free energy in theories

belonging to the 3-d, O(4) universality class (left) and its contribution to third or higher

order cumulants (see Eq. 2) (right). h0and z0are non-universal scale parameters.

and for n ≥ 3 these derivatives are dominated by the singular part,

χB

n∼

−m(2−α−n/2)/βδ

q

?¯ µB

T

f(n/2)

f

(z) , for ¯ µB/T = 0, and n even

−

?n

m(2−α−n)/βδ

q

f(n)

f (z), for ¯ µB/T > 0

(2)

The first derivative of the scaling function that leads in the chiral limit to a divergent

cumulant at Tc(¯ µB) and, in fact, leads to a change of sign of χB

from the third derivative of the scaling function f(3)

function for the 3-d, O(4) universality class which recently has been extracted from high

statistics spin model calculations [10]. The right hand part of this figure shows how the

singular contribution strengthens as the symmetry breaking parameter h is reduced.

It is evident from Fig. 2(right) that the universal scaling properties of cumulants

will, for sufficiently small values of the quark mass, induce negative values for cumulants

already for n = 3. However, as shown in Eq. 2, the contribution from the singular part is

weighted by a factor proportional to (µB/T)3. Its contribution to the total value of the

cumulant may thus be small for small values of µB/T and non-zero values of the quark

mass. Nonetheless, negative values for 3rdand 4thorder moments have been found in

model calculations [9, 11]‡

The first cumulant, which is not parametrically suppressed by powers of µB/T and

thus stays finite even at µB = 0 is the sixth order cumulant of net baryon number

fluctuations. This cumulant changes sign in the crossover region of the QCD transition.

Although the temperature at which the sixth order cumulant changes sign is not a

universal quantity there are strong indications from lattice [4] as well as model [9]

calculations that the change of sign occurs close to the chiral transition temperature.

This is shown in Fig. 3. In fact, for sufficiently small values of the quark mass, i.e.,

in the O(4) scaling regime, the location of the minimum of the sixth order cumulant

nat Tc(¯ µB) is obtained

f. In Fig. 2 (left) we show this scaling

‡ Close to the chiral critical point the negative values of fourth order cumulants that arise from O(4)

criticality compete with similar effects that arise from Z(2) critical behavior and also lead to negative

fourth order cumulants [12].

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Charge fluctuations and freeze-out in heavy ion collisions

4

-6

-4

-2

0

2

4

6

8

180 200 220 240 260 280 300 320

T [MeV]

SB

6χ6

2χ6

χ6

B

Q

S

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

T/Tpc

µB/T=0

χ6

B

Figure 3. Sixth order cumulants of net baryon number, electric charge and strangeness

fluctuations calculated in (2+1)-flavor QCD at µB = 0 [4, 8] (left) and in the PQM

model (right) [9].

and its position relative to the pseudo-critical temperature for the chiral transition is

controlled by the location of the maximum of f(3)

in the chiral susceptibility. The latter appears at a somewhat higher temperature [10].

For µB/T > 0 the onset of negative values for χB

the QCD transition [9, 13].

f

and its location relative to the peak

6indeed follows the crossover line for

3. Conclusions

Sixth order cumulants are thus expected to change sign at a temperature below the

(pseudo-critical) chiral transition temperature. This effect should become visible even

at the LHC and the highest RHIC energy if chemical freeze out indeed occurs close to the

QCD transition temperature and if higher moments probe these freeze out conditions.

Acknowledgments

This manuscript has been authored under contract number DE-AC02-98CH10886 with

the U.S. Department of Energy.

References

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[12] M. A. Stephanov, [arXiv:1104.1627 [hep-ph]].

[13] V. Skokov, these proceedings.