^{*}E-mail:

^{3}

We investigate the validity of the limiting-fragmentation hypothesis in relativistic heavy-ion collisions at energies reached at the Large Hadron Collider (LHC). A phenomenological analysis of central AuAu and PbPb collisions based on a three-source relativistic diffusion model is used to extrapolate the pseudorapidity distributions of produced charged hadrons from RHIC to LHC energies into the fragmentation region. Data in this region are not yet available at LHC energies, but our results are compatible with the limiting-fragmentation conjecture in the full energy range

The significance of the fragmentation region in relativistic heavy-ion collisions was realized when data on AuAu collisions in the energy range

The existence of the phenomenon had been predicted for hadron–hadron and electron–proton collisions by Benecke et al. [

It is not currently clear, however, whether limiting fragmentation will persist at the much higher incident energies that are available at the CERN Large Hadron Collider (LHC), namely

Given the lack of LHC data in the fragmentation regions, one has to rely on either microscopic approaches such as the multiphase transport model AMPT by Ko et al. [

Regarding microscopic approaches, the AMPT code [

The ALICE collaboration has argued in Ref. [

This procedure leads ALICE to conclude that the 2.76 TeV PbPb data are consistent with the LF hypothesis. Related, but different, extrapolation schemes give similar conclusions. As an example, we have fitted the ALICE midrapidity data with a sum of two Gaussians that are peaked at the experimental maxima, used the proper Jacobian transformation from

In this work, we investigate whether limiting fragmentation in heavy-ion collisions at LHC energies can be expected to be fulfilled in yet another phenomenological model, the three-source relativistic diffusion model (RDM) [

In relativistic heavy-ion collisions, the relevant observable in stopping and particle production is the Lorentz-invariant cross section

We shall first investigate rapidity distributions of protons minus produced antiprotons, which are indicative of the stopping process as described phenomenologically in a relativistic two-source diffusion model (RDM) [

Schematic representation of the three-source model for particle production in relativistic heavy-ion collisions at RHIC and LHC energies in the center-of-mass system. Following the collision of the two Lorentz-contracted slabs (blue), the fireball region (center, yellow) expands anisotropically in longitudinal and transverse direction. At midrapidity, it represents the main source of particle production. The two fragmentation sources (red) contribute to particle production, albeit mostly in the forward and backward rapidity regions.

The rapidity distributions for all three sources

The incoherent addition of the three sources also applies to the model with sinh drift that we consider in this work, because the FPE is a linear partial differential equation, allowing for linear superposition of independent solutions. For a symmetric system, one can further simplify the problem by only considering the solution for the positive rapidity region and mirroring the result at

The parameters of the three-source model, which will be detailed in the following, are then determined via

We rely on Boltzmann–Gibbs statistics and hence adopt the Maxwell–Jüttner distribution as the thermodynamic equilibrium distribution for

The nonequilibrium evolution of all three partial distribution functions

It should be noted that interpenetration and stopping (or more precisely, slowing down) of the Lorentz-contracted, highly transparent nuclei occurs before the quark–gluon plasma medium with quarks and gluons in the fireball is fully formed. Hence, there exists no medium or heat bath that could act as a solvent providing friction and noise due to thermal fluctuations, as is the case in the diffusion model for Brownian motion, or for heavy quarks in a quark–gluon plasma. Instead, the incident baryons lose their momentum (rapidity) without any globally static medium, but through random partonic two-body collisions between valence quarks and low-

The FPE in Eq. (

The equilibrium limit of the FPE solution for constant diffusion and linear drift is, however, found to deviate slightly from the Maxwell–Jüttner distribution. Although the discrepancies are small and become visible only for sufficiently large times, we use the RDM with the sinh drift,

The strength of the drift force in the fragmentation sources

Whereas the RDM with linear drift has analytical solutions that can be used directly in

Since

To recover the drift and diffusion coefficients, one has to specify a time scale (or the other way round). Considering that it is only the drift term that is responsible for determining the peak position, we choose the time-like variable

We calculate the numerical solution using

To compare the simulation to experimental data, we have to insert relevant values for

For the temperature, we take the critical value

The results are then transformed to rapidity distributions [

The constant

To emphasize the relevance of the fragmentation sources, we first investigate stopping and calculate net-proton rapidity distributions in central AuAu collisions at RHIC energies of 200 GeV, where data are available from Ref. [

For AuAu at an RHIC energy of 200 GeV it was not possible to measure the fragmentation peaks because the forward spectrometer hit the beam pipe at large rapidities. At LHC energies, a forward spectrometer with particle identification in the region of the expected fragmentation peaks [

The fragmentation sources are visible in net-proton (proton minus antiproton, or stopping) rapidity density functions for central AuAu collisions at

We now compare these AuAu RHIC data with the fragmentation distributions that arise from the three-source model with both linear and sinh drift. In the case of a linear drift, the average positions of the fragmentation peaks agree with the maximum-value positions,

The results of the RDM calculation with the sinh drift are shown as dashed curves in

In

Equilibrium distributions with physical values for the diffusion strengths that include collective expansion would be much broader, but they do not exhibit fragmentation peaks with a midrapidity valley, and hence equilibrium models are not suited to describing stopping distributions.

In charged-hadron production, we consider the sum of produced charged particles and antiparticles. Hence, the fireball source has to be added, and yields the essential contribution to charged-hadron production in heavy-ion collisions at LHC energies. Particles that are produced from the fragmentation sources are not directly distinguishable from those originating from the fireball, but still the fragmentation sources are relevant and must be included in a phenomenological model. In particular, when regarding the limiting-fragmentation conjecture, the role of the fragmentation distributions will turn out to be decisive since they determine the behavior of the rapidity distribution functions at large values of rapidity.

For unidentified charged particles, we first have to transform from rapidity to pseudorapidity space in order to directly compare to data. The pseudorapidity variable

Pseudorapidity density distribution in the three-source RDM with linear drift (solid curve, top) resulting from a

Parameters in the RDM with linear drift for central (0%–5%) PbPb collisions at 2.76 TeV and 5.02 TeV: particle content

2.76 | 3505 | 10681 | 4.98 | 6.38 | 2.44 | 0.07 | ||

5.02 | 4113 | 14326 | 4.99 | 6.38 | 1.17 | 0.04 |

The Jacobian has almost no effect in the fragmentation region, which we are emphasizing in this work. In

When plotted as function of

We now proceed to investigate the consequences of the model with sinh drift, with emphasis on the fragmentation region. We solve Eq. (

Parameters in the RDM with sinh drift for central (0%–5%) PbPb collisions at 2.76 TeV and 5.02 TeV: particle content

2.76 | 2700 | 12000 | 1000 | 115 | 5.89 | 0.16 | |||

5.02 | 2800 | 15800 | 2000 | 205 | 7.50 | 0.26 |

Pseudorapidity density distribution of produced charged hadrons for central (0%–5%) 2.76 TeV (top) and 5.02 TeV PbPb (bottom) in the three-source RDM with sinh drift (solid curves) from

To confirm that the RDM with sinh drift is consistent with the observed LF at the available RHIC energies, we compare the PHOBOS AuAu data [

Three-source RDM distributions with sinh drift compared to central (0%–3%) PHOBOS AuAu data [

Parameters in the RDM with sinh drift for central (0%–3%) AuAu collisions at 19.6 GeV to 200 GeV: particle content

19.6 | 870 | 60 | 6 | 1 | 157.07 | 3.02 | |||

62.4 | 1280 | 540 | 18 | 4 | 18.11 | 0.37 | |||

130 | 1350 | 1800 | 42 | 13 | 4.07 | 0.08 | |||

200 | 1400 | 2650 | 52 | 24 | 3.39 | 0.07 |

Interestingly, the value for the diffusion strength

Our overall results from the relativistic diffusion model with linear and sinh drift are summarized in

Comparison of the three-source RDM distributions with linear and sinh drift, ALICE data [

We have investigated charged-hadron pseudorapidity distributions in central PbPb collisions within a three-source relativistic diffusion model with nonlinear drift, which ensures the correct Maxwell–Jüttner equilibrium distribution. Our analysis indicates that the phenomenon of limiting-fragmentation scaling can be expected to hold at RHIC and LHC energies, spanning a factor of almost 260 in collision energy. This conclusion is in line with results from microscopic numerical models such as AMPT, but it disagrees with expectations from simple parametrizations of the rapidity distributions such as the difference of two Gaussians, and also with predictions from the thermal model. The latter does not explicitly treat the fragmentation sources, it refers only to particles produced from the hot fireball. In contrast, the fragmentation sources play an essential role in our approach. It remains to be seen whether future upgrades of the detectors will make it possible to actually test the limiting-fragmentation conjecture experimentally at LHC energies.

Discussions with multiparticle dynamics group members Johannes Hölck, Philipp Schulz (both ITP, Heidelberg), and Alessandro Simon (now Sophia University, Tokyo) are gratefully acknowledged.

Open Access funding: SCOAP