350
R. Stock
parameter, T ≈ 230 MeV over the SPS energy regime, cannot be identified with the
fireball temperature at hadron formation which is T h ≈ 165 MeV from Fig. 7.1. The
latter is seen, however, to be well represented by the p+p spectral data exhibited in
the left panel of Fig. 7.22. There is, thus, not only thermal energy present in A+A
transverse expansion, but also hydrodynamical radial flow.
We note that the indications in Figs. 7.21 and 7.22, of a plateau in both m T
and T , extending over the domain of SPS energies, 6 ≤
√
s ≤ 17 GeV, have
not yet been explained by any fundamental expansive evolution model, including
hydrodynamics. Within the framework of the latter model, this is a consequence
of the initialization problem [96] which requires a detailed modeling, both of
primordial energy density vs. equilibration time scale, and of the appropriate
partonic matter equation of state (EOS) which relates expansion pressure to energy
density. At top RHIC energy, this initialization of hydro-flow occurs, both, at a
time scale t 0 ≈ 0.6 fm/c which is far smaller than the time scale of eventual bulk
hadronization (t ≈ 3 fm/c), and at a primordial energy density far in excess of the
critical QCD confinement density. After initialization, the partonic plasma phase
thus dominates the overall expansive evolution, over a time interval far exceeding
the formation and relaxation time scale.
Thus, at RHIC energy, parton transport [92] and relativistic hydrodynamic
[95, 96] models establish a well developed expansion mode that survives the
subsequent stages of hadronization and hadronic expansion. This is reflected in their
success in describing elliptic flow. On the other hand, the hydrodynamical model
far overestimates elliptic flow at SPS energy [96] at which, as we have shown in
Sect. 7.2.5, the initialization period may be not well separated from the confinement
(hadronization) stage. Thus, whereas the expansion evolution at
√
s = 200 GeV
(occurring at near-zero baryo-chemical potential in Fig. 7.1) “races” across the
parton-hadron phase boundary with fully established flow patterns, near μ B = 0
where lattice QCD predicts the phase transformation to be merely a soft cross-over
[16], the dynamics at
√
s = 10–20 GeV may originate from only slightly above,
or even at the phase boundary, thus sampling the domain 200 ≤ μ B ≤ 500 MeV
where the equation of state might exhibit a “softest point” [96]. The hydrodynamic
model thus faces formidable uncertainties regarding initialization at SPS energy.
The plateau in Figs. 7.21 and 7.22 may be the consequence of the fact that not
much flow is generated in, or transmitted from the partonic phase, at SPS energies,
because it is initialized close to the phase boundary [100] where the expected critical
point [9, 10] (Fig. 7.1), and the corresponding adjacent first order phase transition
might focus [101] or stall [96] the expansion trajectory, such that the observed radial
flow stems almost exclusively from the hadronic expansion phase. The SPS plateau,
which we shall subsequently encounter in other bulk hadron variables (elliptic flow,
HBT radii) might thus emerge as a consequence of the critical point or, in general,
of the flatness of the parton-hadron coexistence line. RHIC dynamics, on the other
hand, originates from far above this line.
Hadronic expansion is known to proceed isentropically [102]: commensurate
to expansive volume increase the momentum space volume must decrease, from
a random isotropic thermal distribution to a restricted momentum orientation
R. Stock
parameter, T ≈ 230 MeV over the SPS energy regime, cannot be identified with the
fireball temperature at hadron formation which is T h ≈ 165 MeV from Fig. 7.1. The
latter is seen, however, to be well represented by the p+p spectral data exhibited in
the left panel of Fig. 7.22. There is, thus, not only thermal energy present in A+A
transverse expansion, but also hydrodynamical radial flow.
We note that the indications in Figs. 7.21 and 7.22, of a plateau in both m T
and T , extending over the domain of SPS energies, 6 ≤
√
s ≤ 17 GeV, have
not yet been explained by any fundamental expansive evolution model, including
hydrodynamics. Within the framework of the latter model, this is a consequence
of the initialization problem [96] which requires a detailed modeling, both of
primordial energy density vs. equilibration time scale, and of the appropriate
partonic matter equation of state (EOS) which relates expansion pressure to energy
density. At top RHIC energy, this initialization of hydro-flow occurs, both, at a
time scale t 0 ≈ 0.6 fm/c which is far smaller than the time scale of eventual bulk
hadronization (t ≈ 3 fm/c), and at a primordial energy density far in excess of the
critical QCD confinement density. After initialization, the partonic plasma phase
thus dominates the overall expansive evolution, over a time interval far exceeding
the formation and relaxation time scale.
Thus, at RHIC energy, parton transport [92] and relativistic hydrodynamic
[95, 96] models establish a well developed expansion mode that survives the
subsequent stages of hadronization and hadronic expansion. This is reflected in their
success in describing elliptic flow. On the other hand, the hydrodynamical model
far overestimates elliptic flow at SPS energy [96] at which, as we have shown in
Sect. 7.2.5, the initialization period may be not well separated from the confinement
(hadronization) stage. Thus, whereas the expansion evolution at
√
s = 200 GeV
(occurring at near-zero baryo-chemical potential in Fig. 7.1) “races” across the
parton-hadron phase boundary with fully established flow patterns, near μ B = 0
where lattice QCD predicts the phase transformation to be merely a soft cross-over
[16], the dynamics at
√
s = 10–20 GeV may originate from only slightly above,
or even at the phase boundary, thus sampling the domain 200 ≤ μ B ≤ 500 MeV
where the equation of state might exhibit a “softest point” [96]. The hydrodynamic
model thus faces formidable uncertainties regarding initialization at SPS energy.
The plateau in Figs. 7.21 and 7.22 may be the consequence of the fact that not
much flow is generated in, or transmitted from the partonic phase, at SPS energies,
because it is initialized close to the phase boundary [100] where the expected critical
point [9, 10] (Fig. 7.1), and the corresponding adjacent first order phase transition
might focus [101] or stall [96] the expansion trajectory, such that the observed radial
flow stems almost exclusively from the hadronic expansion phase. The SPS plateau,
which we shall subsequently encounter in other bulk hadron variables (elliptic flow,
HBT radii) might thus emerge as a consequence of the critical point or, in general,
of the flatness of the parton-hadron coexistence line. RHIC dynamics, on the other
hand, originates from far above this line.
Hadronic expansion is known to proceed isentropically [102]: commensurate
to expansive volume increase the momentum space volume must decrease, from
a random isotropic thermal distribution to a restricted momentum orientation