Moments of charge fluctuations, pseudo-critical temperatures and freeze-out in heavy ion collisions

Abstract

We discuss universal properties of higher order cumulants of net baryon
number fluctuations and point out their relevance for the analysis of
freeze-out and critical conditions in heavy ion collisions at LHC and RHIC.

Full text

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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
3
-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.
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