7 Relativistic Nucleus-Nucleus Collisions and the QCD Matter Phase Diagram
423
A first conclusion from this model is that it should not be essential to fine-tune
√
s to make the system pass near the CEP because the attractor character would
dominate over a substantial domain, e.g. at hadro-chemical freeze-out conditions
250 ≤ μ B ≤ 500 MeV in the example of Fig. 7.69. Please note that hadro-chemical
freeze-out is not treated correctly in this model, to occur at T H which is 160 ≤ T H ≤
130 MeV from Figs. 7.1 and 7.33. The trajectories shown here below T H are thus not
very realistic. The expected pattern could cause the “plateau” behavior of several
observables at hadronic freeze-out over the range of SPS energies that corresponds
to the above interval of μ B and T H , e.g. m T and T in Figs. 7.21 and 7.22, elliptic
flow v 2 at mid-rapidity in Fig. 7.36, and coherent hadronic freeze-out volume V f
from HBT in Fig. 7.67.
A consistent search for critical point effects implies, at first, a correct treatment
of the hadronic expansion phase in hydro-models as above [101, 296], properly
distinguishing chemical composition freeze-out, and eventual “thermal” freezeout at hadronic decoupling. From such a model predictions for the
√
s or S/n B
systematics could be provided for the HBT source parametrization implied by
Eqs. (7.84)–(7.86). I.e. the hydrodynamic model provides the correlator S(x, K) in
cases with, and without a critical point which, as Fig. 7.69 shows, leads to considerable changes in the system trajectory in the domain near hadronization and onward
to hadronic thermal freeze-out, at each given S/n or
√
s. On the experimental
side, this trajectory is documented by [T , μ B ] at hadronic freeze-out from grand
canonical analysis [140] (Sect. 7.3) which also yields S/n B . Furthermore, as we
shall show below, HBT in combination with the analysis of p T or m T spectra will
describe the situation at thermal hadron freeze-out yielding T f , β T at the surface
and the “true” transverse radius R geom of the source, in addition to the coherence
volume V f illustrated in Fig. 7.67 (which also documents the present, insufficient
data situation). Of course, we have to note that the lattice results concerning the
critical point are not yet final [270], and neither is the hydrodynamic treatment [296]
concerning the EOS in the vicinity of the CEP.
We turn to combined analysis of the two final processes of expansive evolution,
formation of p T spectra and decoupling to Bose-Einstein pair correlation, in order
to show how the experimental information mentioned above can be gathered. At
the level of “hydrodynamically inspired” analytic models for the emission function
S(x, K) several approaches [106, 297] have established the connection, arising from
radial flow, between hadronic m T spectra and the K T dependence of the HBT radii
R side , R out andR long , Eqs. (7.87), (7.92), which fall down steeply with increasing K T
[291–294]. A combined analysis lifts continuous ambiguities between the thermal
freeze-out temperature T f and the radial expansion velocity β T (for which a radial
profile β T = (r/R side ) β 0 is assumed), that exist in the blast wave model derivation
of both the p T spectra, Eq. (7.25), and the K T dependence of R s .
This was first demonstrated in a NA49 study of pion correlation in central Pb+Pb
collisions at
√
s = 17.3 GeV [90] that is shown in Fig. 7.70. The ambiguous anticorrelation of fit parameters T f and β 2
T can be somewhat constrained if several
hadronic species are fit concurrently. This is obvious in Fig. 7.70 from the overlap
of parametrization regions for negative hadron, and for deuterium m T spectra. An
423
A first conclusion from this model is that it should not be essential to fine-tune
√
s to make the system pass near the CEP because the attractor character would
dominate over a substantial domain, e.g. at hadro-chemical freeze-out conditions
250 ≤ μ B ≤ 500 MeV in the example of Fig. 7.69. Please note that hadro-chemical
freeze-out is not treated correctly in this model, to occur at T H which is 160 ≤ T H ≤
130 MeV from Figs. 7.1 and 7.33. The trajectories shown here below T H are thus not
very realistic. The expected pattern could cause the “plateau” behavior of several
observables at hadronic freeze-out over the range of SPS energies that corresponds
to the above interval of μ B and T H , e.g. m T and T in Figs. 7.21 and 7.22, elliptic
flow v 2 at mid-rapidity in Fig. 7.36, and coherent hadronic freeze-out volume V f
from HBT in Fig. 7.67.
A consistent search for critical point effects implies, at first, a correct treatment
of the hadronic expansion phase in hydro-models as above [101, 296], properly
distinguishing chemical composition freeze-out, and eventual “thermal” freezeout at hadronic decoupling. From such a model predictions for the
√
s or S/n B
systematics could be provided for the HBT source parametrization implied by
Eqs. (7.84)–(7.86). I.e. the hydrodynamic model provides the correlator S(x, K) in
cases with, and without a critical point which, as Fig. 7.69 shows, leads to considerable changes in the system trajectory in the domain near hadronization and onward
to hadronic thermal freeze-out, at each given S/n or
√
s. On the experimental
side, this trajectory is documented by [T , μ B ] at hadronic freeze-out from grand
canonical analysis [140] (Sect. 7.3) which also yields S/n B . Furthermore, as we
shall show below, HBT in combination with the analysis of p T or m T spectra will
describe the situation at thermal hadron freeze-out yielding T f , β T at the surface
and the “true” transverse radius R geom of the source, in addition to the coherence
volume V f illustrated in Fig. 7.67 (which also documents the present, insufficient
data situation). Of course, we have to note that the lattice results concerning the
critical point are not yet final [270], and neither is the hydrodynamic treatment [296]
concerning the EOS in the vicinity of the CEP.
We turn to combined analysis of the two final processes of expansive evolution,
formation of p T spectra and decoupling to Bose-Einstein pair correlation, in order
to show how the experimental information mentioned above can be gathered. At
the level of “hydrodynamically inspired” analytic models for the emission function
S(x, K) several approaches [106, 297] have established the connection, arising from
radial flow, between hadronic m T spectra and the K T dependence of the HBT radii
R side , R out andR long , Eqs. (7.87), (7.92), which fall down steeply with increasing K T
[291–294]. A combined analysis lifts continuous ambiguities between the thermal
freeze-out temperature T f and the radial expansion velocity β T (for which a radial
profile β T = (r/R side ) β 0 is assumed), that exist in the blast wave model derivation
of both the p T spectra, Eq. (7.25), and the K T dependence of R s .
This was first demonstrated in a NA49 study of pion correlation in central Pb+Pb
collisions at
√
s = 17.3 GeV [90] that is shown in Fig. 7.70. The ambiguous anticorrelation of fit parameters T f and β 2
T can be somewhat constrained if several
hadronic species are fit concurrently. This is obvious in Fig. 7.70 from the overlap
of parametrization regions for negative hadron, and for deuterium m T spectra. An