Beyond the thermal model in relativistic heavy-ion collisions

Deviations from thermal distribution functions of produced particles in relativistic heavy-ion collisions are discussed as indicators for nonequilibrium processes. The focus is on rapidity distributions of produced charged hadrons as functions of collision energy and centrality, which are used to infer the fraction of particles produced from a central fireball as compared with that from the fragmentation sources that are out of equilibrium with the rest of the system. Overall thermal equilibrium would only be reached for large times $t\to \infty$.

Figure 1

Jüttner distribution functions of a relativistic thermal Maxwell gas at temperature $T$ without collective expansion compared to measured transverse momentum distributions of produced charged hadrons in 2.76 TeV PbPb collisions from ALICE [17] for 0%–5%, 30%–40%, and 70%–80% centralities (top to bottom; $\left|\eta \right|<0.8$); see text [16].

Figure 2

Transverse momentum distributions of produced charged hadrons in 2.76 TeV PbPb collisions calculated from $f\left(p_{\mathrm{T}}\right)\propto {[1+(q-1)m_{\mathrm{T}}/T]}^{1/(1-q)}$ compared with ALICE data [17] for 0%–5%, 30%–40%, and 70%–80% centralities (top to bottom). Solid curves are for $q=1.10$, the dashed curve is for $q=1.12$. Peripheral spectra are scaled for better visibility; see Fig. 1 for absolute values.

Figure 3

Evidence for fragmentation sources: Rapidity distributions of net protons in central PbPb collisions at SPS energies of $\sqrt{s_{NN}}=17.3$ GeV (top frame) compared with NA49 data [31]. Solid curves correspond to a gluon saturation momentum $Q_{\mathrm{s}}=0.9$ GeV/$c$ at $x=0.01$, dashed curves to $Q_{\mathrm{s}}=1.2$ GeV/$c$. At RHIC energies of 62.4 GeV (middle frame) and 200 GeV (bottom frame) for central AuAu, theoretical results are compared with BRAHMS net proton data [11]. The fragmentation peaks move apart in rapidity space with increasing energy. Arrows indicate the beam rapidities. From Mehtar–Tani and Wolschin [12, 32].

Figure 4

Fragmentation sources for charged-hadron production in $\sqrt{s_{NN}}=2.76$ TeV PbPb collisions $(y_{\text{beam}}=\mp 7.987)$. Solid curves are from a ${\chi }^2$ minimization of analytical solutions in the relativistic diffusion model (RDM) [35] with respect to the ALICE data [8, 37] (upper curve includes the gluonic source, lower curve is for fragmentation sources only). Dashed curves indicate the time evolution of the fragmentation sources in the RDM. The fragmentation sources remain far from equilibrium at LHC energies.

Figure 5

The RDM pseudorapidity distribution functions for charged hadrons in central AuAu (RHIC) and PbPb (LHC) collisions at c.m. energies of 19.6, 130, 200 GeV, and 2.76 TeV shown here are optimized in ${\chi }^2$ fits with respect to the PHOBOS [44, 45] (bottom) and ALICE [8, 37] (top) data, with parameters from Ref. [35]. The upper distribution function at 5.02 TeV is an extrapolation within the relativistic diffusion model. The 5.02 TeV midrapidity data point is from ALICE [46].

Figure 6

The total charged-hadron production in central AuAu and PbPb collision in the energy region 19.6 GeV to 5.02 TeV is following a power law (solid upper line), whereas the particle content in the fragmentation sources is $N_{\mathrm{qg}}\propto ln(s_{NN}/s_0)$; dash-dotted curve. The particle content in the midrapidity source obeys $N_{gg}\propto {ln}^3(s_{NN}/s_0)$; dashed curve. The energy dependence of the measured midrapidity yields is shown as a dotted line, with PHOBOS data [45] at RHIC energies, and ALICE data [18, 46] at 2.76 and 5.02 TeV. The vertical line indicates 5.02 TeV.

Figure 7

Pseudorapidity distributions for produced charged hadrons in 5.02 TeV PbPb collisions $(y_{\text{beam}}=\mp 8.586)$ as functions of centrality, from bottom to top: 20%–30%, 10%–20%, 5%–10%, 0%–5%. Calculated RDM distributions (solid curves) are compared to midrapidity ALICE data from [46]. For 20%–30% centrality, fragmentation and gluonic distribution functions are shown separately.

Figure 8

The centrality-dependent RDM pseudorapidity distribution functions for charged hadrons in $p\mathrm{Pb}$ collisions at LHC c.m. energy of 5.02 TeV [47] are adjusted in the midrapidity region to the ALICE data [9] through ${\chi }^2$ minimizations. The underlying distributions in the three–source model are also shown, with the dashed curves arising from gluon-gluon collisions, the dash-dotted curves from valence quark-gluon events in the Pb-going region ($y>0$ in this plot), and the dotted curves in the $p$-going direction (fragmentation sources) [60].