RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-5. Hadron Physics (Theory)

Volume dependence of baryon number cumulants and their ratios†∗1Skokov,∗1V. Skokov†∗1

Condensed from the article, 1612.04416
RIKEN Nishina Center- 128 -60
50T , MeV40

1.6
2.0 2.4
1.8
2.2

3.0

20

3.2

3.4

100
100

2.8

2.6

30

3.6
5.0
4.6
4.2
4.8
3.8
4.4
4.0

3.8
150

200

250μ, MeV

300

350400Fig. 1. The location of the apparent critical points (ACP)
as a function of the system size, L. The squares (circles)
denote ACP I (II), see text.

1.5
1.4
1.3
1.2
χ4/χ2

Experiments with ultrarelativistic heavy-ion collisions at RHIC and LHC explore the phase structureof Quantum ChromoDynamics (QCD) at nonzero temperature and density, and so probe the phase transitions associated with deconﬁnement and the restoration of chiral symmetry. Two of the most promising observables are the ﬂuctuations of the net baryon numberand electric charge. The cumulants and related quantities of these ﬂuctuations may provide experimental
evidence for a chiral critical endpoint or chirally inhomogenous phases.
However, there are many other eﬀects besides thecritical dynamics which might be important in the interpretation of the data. Those include the conservation of baryon number, corrections for eﬃciency inthe detectors, hadronic rescattering, non-equilibriumeﬀects, and ﬁnally volume ﬂuctuations. The latterare important due to a ﬁnite size of a domain passing through the critical region during the evolution of
the ﬁreball. Usually one tries to minimize the eﬀects
of ﬂuctuations in the volume by considering the ratiosof cumulants. As we describe in the main text, in suchratios the explicit dependence on the volume cancels
out, making the analysis of volume ﬂuctuations trivial. However, we show that the implicit dependence on
the volume might be very strong if the characteristic
system size is below 5 fm.
In this study we use the quark-meson model as arealization of the chiral symmetry in QCD at low energies. The quark-meson (QM) model consists of aO(4) multiplet of mesons, φ = (σ, π ), coupled to quarkﬁelds through a Yukawa-type coupling.
In order to formulate a non-perturbative thermodynamics in the QM model we adopt a method basedon the functional renormalization group (FRG). The
FRG is based on an infrared regularization with themomentum scale parameter, where the full propagatoris derived from a corresponding eﬀective action.
We consider systems in which there is a true criticalpoint in inﬁnite volume. In ﬁnite volume, instead there
is an apparent critical point (ACP). There is some degree of arbitrariness in how one deﬁnes an apparent
critical point. We deﬁne the position of the apparent
critical point from the maximum in the correspondingchiral susceptibility, which is equivalent to the minimum in the sigma mass. We stress, however, that
unlike the case of inﬁnite volume, that in ﬁnite volume other deﬁnitions will give diﬀerent positions forthe apparent critical point.
With our deﬁnition, we show that at some interme-

1.11.0μ/T = 0.0; A = 1
L =1.5L =2.25
L =2.5
L =3
L =50.9
0.8
0.7
0.6

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T /TpcFig. 2. The ratio of the fourth to the second order susceptibilities as a function of temperature for diﬀerent
systems sizes; the results are computed at zero chemical potential.

diate system size, the system has two apparent criticalpoints, located at diﬀerent values of T and μ. One of
the apparent critical points, which we call ACP I, approaches the true critical point in the limit of inﬁnite
volume; we show that for the ACP I, it approaches thezero temperature axis as the volume decreases. The
second apparent critical point, which we call ACP II,
appears near the zero temperature axis, and evolves
to higher temperature as the volume decreases. The
location of the two apparent critical points is depicted
in Fig. 1. The emergence of a second apparent criticalpoint inﬂuences the cumulants of baryon number.
In Fig. 2, we show the dependence of the kurtosis
χ4 /χ2 on the temperature for diﬀerent system sizes
and diﬀerent anisotropy parameters. The calculations
are done for zero μ.