Strangeness and QGP Freeze-Out Dynamics
In the nucleus - nucleus collision, the emerging ﬁnal state particles remember relatively little of their primordial source, since they had been subject to many rescattering processes during the hadronic gas stage. Moreover for what concerns nucleus-nucleus collisions at relativistic energies, the major fraction of ﬁnally observed particles comes from decays of resonances (mesonic or baryonic) which have undergone many collisions from their point of production to the detection. The ﬁnal hadron yields seem to be compatible with a hadronic gas described by the baryo-chemical potential µB and a temperature parameter T in a statistical model. Models take into account two sequential diﬀerent kinds of freeze-out:
- A chemical freeze-out, were the inelastic flavor changing collisions processes cease, roughly at an energy of 1 GeV per particle.
- A later kinetic/thermal freeze-out, were also elastic processes have come to an end and the system decouples.
At the SPS energies chemical and thermal freeze-out happen sequentially at different temperatures (Tch ≈ 160 − 170 MeV, Tth ≈ 120 MeV) and the system becomes colder with time. To investigate the sequential freeze-out in heavy-ion reactions at SPS the Ultra-relativistic Quantum Molecular Dynamics model (UrQMD) is applied. This microscopic transport approach is based on the covariant propagation of constituent quarks and di-quarks accompanied by mesonic and baryonic degrees of freedom. The leading hadrons of the fragmenting strings contain the valence-quarks of the original excited hadron and represent a simpliﬁed picture of the leading (di)quarks of the fragmenting string.
In Figure 1 the time evolution of the elastic and inelastic collision rates in Pb+Pb at 160 AGeV of the SPS beams are depicted. The inelastic collision rate (full line) is defined as the number of collisions with ﬂavor changing processes. The elastic collision rate consists of two components, true elastic processes and pseudo-elastic processes. While elastic collision do not change ﬂavor, in the pseudo-elastic collisions the ingoing hadrons are destroyed and a resonance is formed. If this resonance decays later into the same ﬂavors as its parent hadrons, this scattering is called pseudo-elastic.
Figure 1: Inelastic and (pseudo-)elastic collision rates in Pb + Pb at 160 AGeV. τch and τth denote the chemical and thermal/kinetic freeze-out as given by the microscopic reaction dynamics of UrQMD.
Even if the main features revealed by this microscopic study do not contradict the idea of a chemical and thermal break-up of the source as shown in Figure 1, however the detailed freeze-out dynamics is much richer and by far more complicated as expected in simplified models. Below a simplified schematic sequence is sketched:
1. t < 2 fm/c: in the initial stage of the nucleus-nucleus reaction, non-equilibrium dynamics leads to strong baryon stopping in multiple inelastic interactions, shown by huge and strongly time dependent collision rates. This stage deposits a large amount of (non-thermalized) energy and creates the ﬁrst generation particles.
2. 2 fm/c < t < 6 fm/c: due to the high particle densities and energies, inelastic scattering processes dominate this stage of the reaction. Chemical equilibrium might be achieved due to a large number of ﬂavour and hadro-chemistry changing processes until chemical freeze-out.
3. 6 fm/c < t < 11 fm/c: after the system has expanded and cooled down, elastic and pseudo-elastic collisions take over. Here, only the momenta of the hadrons change, but the chemistry of the system is mainly unaltered, leading to the kinetic freeze-out of the system.
4. t > 11 fm/c: ﬁnally, the reactions cease and the scattering rates drop drastically. The system breaks up.
The spectra and abundances of the resonances (such as Λ(1520), K0 (892), Σ(1385)...) can be used to study the break-up dynamics of the source between chemical and thermal freeze-out. If chemical and thermal freeze-out are not separated - e.g. due to an explosive break-up of the source - all initially produced resonances are reconstructable by invariant mass analysis of the ﬁnal state hadrons. However, if there is a separation between the diﬀerent freeze-outs, a part of the resonance daughters rescatter, making this resonance unobservable in the ﬁnal state. Thus, the relative suppression of resonances in the final state compared to the behaviour expected from thermal estimates provides a chronometer for the time interval between the different reaction stages.
Dynamical Freeze-Out Constraint with Resonances
In recent years some studies were performed in order to explore if it is possible to experimentally determine the period of time between the ﬁreball chemical and thermal freeze-out using the strange hadron resonances behaviour. The short-lived resonances, detectable through invariant mass reconstruction are natural candidates for freeze-out diagnostic since their lifetime is comparable to the hadronization timescale and to the lifetime of the interacting HG. Resonances usually have the same quark numbers as light particles, making their yield compared to the light particle independent of chemical potential. The rich variety of detected resonances includes particles with very diﬀerent masses and widths, allowing us to probe both production temperatures and interaction lifetimes in detail. Figure 2 shows what percentage of observed light particles comes from the decays of heavier resonances (quite a few of them experimentally observable). As can be seen this resonance contribution is signiﬁcant, and varies appreciably with both particle type and temperature of the production reaction.
Figure 2: Relative resonances contribution to individual stable hadrons for three particle freeze-out temperatures.
The resonance Λ(1520) (ΓΛ(1520) = 15.6 MeV) has been observed in heavy-ion reactions at SPS energies. Both SPS and RHIC experiments report measurement of the K*0(892) signal, which has a much greater width, ΓK∗ = 50 MeV. The Λ(1520) abundance yield is found about 2 times smaller than expectations based on the yield extrapolated from nucleon-nucleon reactions, scaled with hadron yield. This has to be compared with an increased Λ production by factor 2.5. A possible explanation for this relative suppression by a factor 5 is that the decay products (π,Λ) have re-scattered and thus their momenta did not allow to reconstruct this state in an invariant mass analysis. However, the observation of a strong K∗0 yield signal contradicts this point of view. Another explanation is that in the nuclear matter Λ(1520) decays faster and there is much more opportunity for the rescattering of decay products, and fewer observable resonances. The observable yield of the resonances is thus controlled by several physical properties, such as the freeze-out temperature T, the decay width in nuclear matter Γ, and the time spent in the hadron phase after freeze-out τ. The suppressed yield can mean either a low temperature chemical freeze-out, or a long interacting phase with substantial re-scattering.
A model based on the width of the resonances K0(892), Λ(1520), and also the (more difficult) Σ(1385) and the decay products reaction cross-sections within an expanding fireball of nuclear matter, in order to explore their production and suppression observability, could be a further way of distinguishing between different reaction scenarios. The Σ(1385) is expected to be produced more abundantly than Λ(1520) in a hadronic fireball due to its high degeneracy factor and smaller mass. Because of its shorter life-time (ΓΣ∗ = 36-39 MeV > ΓΛ∗ = 15.6), the Σ∗ signal is more strongly inﬂuenced by final state interactions than that of Λ(1520). Like for K0(892), one would naively expect that the observable yield of Σ∗ should be suppressed by a factor 10.
In this model hadrons produced directly from a medium at temperature T ﬁll the available statistical phase space which has the relativistic Boltzmann distribution shape:
d2N/dmt2∝ g Πni=1 λi γi mt cosh(y) e−E/T
where g is the statistical degeneracy, λi and γi are the fugacity and equilibrium parameters of each valence quarks, and E is the energy. When the ﬁreball is expanding at a relativistic speed, equation 3.1 describes the energy distribution of an element of the fireball in a reference frame at rest with respect to the expansion (ﬂow). However in this model were evaluated ratios of particles with similar masses, and interaction modes, often considered in full phase space. For this reasons to a good approximation, the ﬂow effects largely cancel out. Similarly, for ratios of particles with the same valence quark composition, such as Σ∗/Λ, Λ(1520)/Λ and, in the limit of λu = λd, K∗0(892)/K−(=K∗0/K+) the chemical factors (λ’s and γ’s) cancel out between the two states compared. In Figure 3 we show the relative thermal production ratios at chemical freeze-out over the entire spectrum of rapidity and mt (solid lines) as well a central rapidity range deﬁned by the y − mt region covered by the WA97 experiment (|y| < 0.5 in the center of mass frame)(dashed lines). Looking at the graph it is possible to observe the possibility to measure the chemical freeze-out temperature by a measurement of the relative resonance yields.
Figure 3: Temperature dependence of ratios of Σ∗, K∗0 and Λ(1520) to the total number of observed K+’s, Λ’s, Ξ’s and Ω’s. Branching ratios are included. Dashed lines show the result for a measurement at central rapidity Δy = ± 0.5.
A simple test of hadronization model consists in measurement of the ratio Σ∗/Ξ. If it is significantly smaller than unity, we should expect a re-equilibration mechanism to be present. Otherwise sudden hadronization applies. The ratios of observed particles, however, can be considerably diﬀerent from production ratios, since if the decay products rescatter before detection their identiﬁcation by reconstructing their invariant mass will generally not be possible. While the lifetime of the Ξ and Ω are large enough to ensure that only a negligible portion of particles decay near enough to the ﬁreball for rescattering to be a possibility, the lifetime of K∗0,Σ∗ and even Λ(1520) is instead within the same order of magnitude of the ﬁreballs dimensions i.e. 2R/c ≈ 1/Γ. For this reason, a considerable number of decay products will undergo rescattering, and the estimation of this percentage is required before any meaningful parameters are extracted from the data. Figures 4 and 5 show the dependence of the Λ(1520) Λ, Σ∗/Λ and K∗0(892)/K+ on the temperature and lifetime of the interacting phase.
Figure 4: Relative Λ(1520)/(all Λ) yield as function of freeze-out temperature T. Dashed - thermal yield, solid lines: observable yield for evolution lasting the time shown (1...20 fm/c) in an opaque medium. Horizontal lines: experimental limits of NA49. Up: natural resonance width Γ Λ(1520) = 15.6 MeV. Down: quenched Γ Λ(1520) = 150 MeV.
Figure 5: Produced (dashed line) and observable (solid lines) ratios Σ∗/(total Λ) (up) and K∗0/K (down). The solid lines correspond to evolution after chemical freeze-out of 1, 2, 3, 4, 5, 7, 10, 15, 20 fm/c, respectively.
It is clear that, given a determination of the respective signals to a reasonable precision, a qualitative distinction between the high temperature chemical freeze-out scenario followed by a rescattering phase and the low temperature sudden hadronization scenario can be performed. Figures 4 and 5 demonstrate the sensitivity of strange hadron resonance production to the interaction period in the hadron phase, i.e. the phase between the thermal and chemical freeze-out. What we learn from them is that while the suppression of one of these ratios considered has generally two interpretations, as it can mean either a low temperature chemical freeze-out or a long interacting phase with substantial rescattering, the comparison of two resonances with considerably different lifetimes can be used to constrain both the temperature of chemical freeze-out and the lifetime of the interacting phase.
We can reexpress the results presented in Figures 4 and 5 representing one ratio against the other as is seen in Figures 6, 7 and 8.
Figure 6: Dependence of the combined Σ∗/(all Λ) with K 0 (892)/(all K) signals on the chemical freeze-out temperature and interacting phase lifetime. Up: quenched ΓΣ∗ = 150 MeV. Down: natural widths. Vertical lines: experimental limits of NA49 and STAR.
Figure 7: Dependence of the combined Σ∗/(all Λ) with K*0(892)/(all K) signals on the chemical freeze-out temperature and interacting phase lifetime. Up: quenched ΓΣ∗ = 150 MeV. Down: natural widths. Vertical lines: experimental limits of NA49 and STAR.
Figure 8: Dependence of the combined Σ∗/(all Λ) and Σ∗/(all Ξ) signals on the chemical freeze-out temperature and interacting phase lifetime.
In all these figures we can see that from top to bottom in the grid, the lifespan in fireball increases, while from left to right the temperature of chemical particle freeze-out increases. The medium is eﬀectively opaque, all resonances that decay in medium become unobservable. A remarkable prediction is found for the resonances Σ∗: when both the Σ∗/Λ and Σ∗/Ξ ratios become available both the temperature and the lifetime can be inferred from the Σ∗ alone.
Moreover, further studies were performed in order to analyze the mt and pt dependence of these ratios for diﬀerent freeze out temperatures. Figures 9 shows the prediction obtained for the ratio Σ(1385)/Λ, at two freeze-out temperatures and ﬂows: T = 140 MeV, vmax/c = 0.55 on left and T = 170 MeV, vmax/c = 0.3 on the right. Signiﬁcant deviations from simple constant values are observed, showing the sensitivity of the ratio to the diﬀerent model applied. The graphs takes into account the feed down from resonances (i.e. the decay products from reconstructed Σ(1385)).
Figure 9: Dependence of the Σ(1385)/Λ ratio on the Freeze-out model including feed down from resonances.
To further study the sensitivity of resonance-particle ratio to freeze-out dynamics, the (feed down corrected) ratios as a function of pt were also considered. Comparing them with those in the Figure 3.14 it is possible to see grossly diﬀerent behaviors, with many of the results coalescing. This is an expression of the fact that Σ(1385) and Λ have dramatically different pt at the same mt and vice versa.
Figure 10: pt dependence of Σ(1385)/(all Λ) ratios, including feed down from resonances.
From the Figures 9 and 10 is evident that the mt and pt dependence of the ratios depends on the freeze-out model and, in particular, changes in temperature and flow velocity alter the ratios shape. Moreover, as previously mentioned, the presence of a long living hadronic gas rescattering phase can distort this freeze-out probe. Infact the Σ(1385)/Λ ratio will be altered due to the depletion of the detectable resonances through the rescattering of their decay products. Its dependence on mt will be affected in a non-trivial way, since the faster resonances will have a greater chance to escape the fireball without decaying, thus avoiding the rescattering phase. Regeneration of the resonances in hadron scattering may add another mt dependence. A long rescattering phase would affect the Σ∗ distribution and the effect would beeasy to detect experimentally: 95% of Σ∗ decay through the p-wave Σ∗ → Λπ channel. However, regenerating Σ∗ in a gas of Λ and π is considerably more diﬃcult, since Λ π scattering will be dominated by the s-wave Λ π → Σ±. This situation will not occur for K∗ ↔ Kπ, since both decay and regeneration happen through the same process, leading to a very fast reequilibration time. Since both Σ∗/Λ and K∗/K have been calculated within the thermal model (neglecting rescattering), a strongly depleted Σ∗/Λ (compared with K∗/K ) suggests that a statistical freeze-out description is incomplete and an interacting hadron gas phase is also necessary.