412
R. Stock
at the SPS. Characteristic observables are radial flow, hadro-chemical freeze-out,
and chiral symmetry restoration effects in dilepton vector meson spectra. Focusing
on this domain, we discuss fluctuations potentially associated with the existence of
a critical point [8–11, 146, 147].
At the end of Sect. 7.6.3 we mentioned the conclusion from chiral symmetry
restoration models [15, 259] that at high μ B the phase transformation occurring at
T c (μ B ) should be a chiral first order phase transition. On the other hand, lattice
QCD has characterized [16] the phase transformation at μ B → 0, to be merely a
rapid cross-over. Thus, the first order nature of the phase coexistence line in Fig. 7.1
has to end, with decreasing μ B , in a QCD critical point, tentatively located by recent
lattice QCD calculations [9–11] in the interval μ B = 300–500 MeV. The existence
of such a point in the [T , μ B ] plane would imply fluctuations analogous to critical
opalescence in QED [146, 147, 263]. Beyond this second order phase transition
point the coexistence line would be the site of a rapid cross-over [16]. This overall
theoretical proposal places potential observations related to the critical point itself,
and/or to the onset of first order phase transition conditions at higher μ B , within the
domain of the lower SPS energies,
√
s ≤ 10 GeV. Note that, at such low energies,
the initialization of thermal equilibrium conditions should occur in the vicinity of
T c , unlike at RHIC and LHC, and that the central fireball spends considerable time
near the coexistence line, at 300 ≤ μ B ≤ 500 MeV.
To analyze potential observable effects of a critical point, we recall briefly the
procedure in finite μ B lattice theory that led to its discovery. One method to compute
thermodynamic functions at μ B > 0 from the grand canonical partition function
Z(V , T , μ q ) at μ q = 0 is to employ a Taylor expansion with respect to the chemical
quark potential [10, 11, 270], defined by the derivatives of Z at μ = 0. Of particular
interest is the quark number density susceptibility,
χ u,d = T
2
δ 2
δ(μ/T ) 2
p
T 4
(7.74)
which can also be written as
χ q = T
2
δ
δ(μ u /T )
+
δ
δ(μ d /T )
n u + n d
T 3
(7.75)
with χ q = (χ u + χ d )/2 and quark number densities n u , n d . We see that the
susceptibility refers to the quark number density fluctuation. The lattice result
[10, 270] is shown in Fig. 7.62, a calculation with two dynamical flavors assuming
T c = 150 MeV and three choices of chemical quark potential, μ q = 0, 75 and
150 MeV, respectively, corresponding to μ B = 3 μ q = 0, 225 and 450 MeV. These
choices correspond to LHC/RHIC energy, top SPS energy and
√
s ≈ 6.5 GeV,
respectively. At μ B = 0 one sees a typical smooth cross-over transition at T = T c
whereas a steep maximum of susceptibility occurs with μ B = 450 MeV. This
suggests the presence of a critical point in the (T , μ B ) plane [270] in the vicinity of
(150 MeV, 450 MeV). For final confirmation one would like to see this maximum
R. Stock
at the SPS. Characteristic observables are radial flow, hadro-chemical freeze-out,
and chiral symmetry restoration effects in dilepton vector meson spectra. Focusing
on this domain, we discuss fluctuations potentially associated with the existence of
a critical point [8–11, 146, 147].
At the end of Sect. 7.6.3 we mentioned the conclusion from chiral symmetry
restoration models [15, 259] that at high μ B the phase transformation occurring at
T c (μ B ) should be a chiral first order phase transition. On the other hand, lattice
QCD has characterized [16] the phase transformation at μ B → 0, to be merely a
rapid cross-over. Thus, the first order nature of the phase coexistence line in Fig. 7.1
has to end, with decreasing μ B , in a QCD critical point, tentatively located by recent
lattice QCD calculations [9–11] in the interval μ B = 300–500 MeV. The existence
of such a point in the [T , μ B ] plane would imply fluctuations analogous to critical
opalescence in QED [146, 147, 263]. Beyond this second order phase transition
point the coexistence line would be the site of a rapid cross-over [16]. This overall
theoretical proposal places potential observations related to the critical point itself,
and/or to the onset of first order phase transition conditions at higher μ B , within the
domain of the lower SPS energies,
√
s ≤ 10 GeV. Note that, at such low energies,
the initialization of thermal equilibrium conditions should occur in the vicinity of
T c , unlike at RHIC and LHC, and that the central fireball spends considerable time
near the coexistence line, at 300 ≤ μ B ≤ 500 MeV.
To analyze potential observable effects of a critical point, we recall briefly the
procedure in finite μ B lattice theory that led to its discovery. One method to compute
thermodynamic functions at μ B > 0 from the grand canonical partition function
Z(V , T , μ q ) at μ q = 0 is to employ a Taylor expansion with respect to the chemical
quark potential [10, 11, 270], defined by the derivatives of Z at μ = 0. Of particular
interest is the quark number density susceptibility,
χ u,d = T
2
δ 2
δ(μ/T ) 2
p
T 4
(7.74)
which can also be written as
χ q = T
2
δ
δ(μ u /T )
+
δ
δ(μ d /T )
n u + n d
T 3
(7.75)
with χ q = (χ u + χ d )/2 and quark number densities n u , n d . We see that the
susceptibility refers to the quark number density fluctuation. The lattice result
[10, 270] is shown in Fig. 7.62, a calculation with two dynamical flavors assuming
T c = 150 MeV and three choices of chemical quark potential, μ q = 0, 75 and
150 MeV, respectively, corresponding to μ B = 3 μ q = 0, 225 and 450 MeV. These
choices correspond to LHC/RHIC energy, top SPS energy and
√
s ≈ 6.5 GeV,
respectively. At μ B = 0 one sees a typical smooth cross-over transition at T = T c
whereas a steep maximum of susceptibility occurs with μ B = 450 MeV. This
suggests the presence of a critical point in the (T , μ B ) plane [270] in the vicinity of
(150 MeV, 450 MeV). For final confirmation one would like to see this maximum