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Corresponding author: Raghunath.Sahoo@cern.ch

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The hypothesis of limiting fragmentation (LF), or, as it has been otherwise recently called, longitudinal scaling, is an interesting phenomena in the high-energy multiparticle production process. This paper discusses different regions of phase space and their importance in hadron production, giving special emphasis on the fragmentation region. Although it was conjectured as a universal phenomenon in high-energy physics, with the advent of higher center-of-mass energies, it has become prudent to analyze and understand the validity of such a hypothesis in view of the increasing inelastic nucleon-nucleon cross section (

Understanding the particle production in high-energy nuclear collisions is always fascinating. Particle production in high-energy collisions happens from three different regions: the projectile, the target, and the central region. Particles emitted from the outer region are called projectile/target fragments. There are various nuclear fragmentation mechanisms discussed in the literature

It is expected that a central plateau develops at higher energies, which clearly separates the central rapidity from the fragmentation region. However, as such, there is no separating boundary between the central rapidity and the fragmentation region. The width of the fragmentation region is around two units in rapidity

A schematic of (pseudo)rapidity distribution showing the pionization and fragmentation regions.

There have been several experimental efforts to understand particle production in both mid- and forward rapidities

Various theoretical works

Our main aim in this work is to study the phenomena of LF for

The angular distribution of the particles produced in high-energy collisions is described by the famous Landau model with relativistic hydrodynamics given by the conservation of energy momentum tensor,

The conclusion from Ref.

In this section, we study the limiting fragmentation phenomenon in the pseudorapidity distributions of differential cross sections of charged particles (

The inelastic cross section as a function of

Considering the crude approximation to the physical situation in the framework of the Landau hydrodynamical model of particle production, the relationship between the differential cross sections per unit pseudorapidity (

Here

Using Eq.

Now we proceed toward deriving the relationship between the differential cross section per unit pseudorapidity in

Using Eqs.

A large number of experimental data on the charged-particle pseudorapidity distribution are available at various center-of-mass energies ranging from RHIC energies like

In Fig.

The number of participant pair normalized pseudorapidity distribution of charged particles (

The values of parameters obtained from the fitting of experimental data of

Now we evaluate

The differential cross section per unit pseudorapidity (

The experimental data for pseudorapidity distributions of charged particles in the full phase space are not available at the LHC energies. In addition, a double Gaussian extrapolation of

The comparison of AMPT model predictions with experimental data on

We convert the AMPT results of

In this work, we have revisited the phenomenon of limiting fragmentation in the pseudorapidity distributions of differential cross sections of charged particles using the energy-dependent inelastic cross section. The findings of this analysis are as follows:

We have observed the limiting fragmentation phenomenon in the experimental data of

We have transformed experimental data of

We have also studied the phenomenon of longitudinal scaling using the AMPT model and employing the same procedure as used for the experimental data. Our studies suggest that AMPT seems to show a possible violation of the limiting fragmentation phenomenon for

The hypothesis of LF comes as a natural outcome when particle production follows the Landau hydrodynamics, with a Gaussian pseudorapidity profile.

LF works fine when the hadronic cross section is assumed to be almost independent of energy, which is not the case and hence it is expected to be violated at higher energies. We find that the limiting fragmentation appears to be violated at LHC energies while using the energy-dependent cross section.

The thermal model with Landau extrapolation to LHC energies for charged particles, predicts a violation of LF

It is expected that at higher energies, Landau hydrodynamics should fail and we should expect Bjorken boost-invariant hydrodynamics to work out, with the observation of a midrapidity plateau. If LF is a natural outcome of the Landau model, then LF should be violated at LHC for two reasons: (i) failure to see a Gaussian pseudorapidity distribution and (ii) cross sections vary substantially toward higher collision energies.

At lower collision energies, baryon stopping at the midrapidity is expected and the

Going from the top RHIC energy to the LHC energies, there is an order-of-magnitude increase in the collision energy. Considering at least two units of (pseudo)rapidity overlap for the LF to be valid, the observed

Theoretical models are mostly assumption dependent. In order to validate a model, one needs to confront a model to experimental data. We need forward charged particle and photon detectors at the LHC energies to validate the LF hypothesis. In the absence of this, extrapolation of any theoretical findings from midrapidity to extreme forward rapidity would be a speculation sometimes or a mere coincidence, as the physics of particle production is highly rapidity dependent. In view of this, in the present work we have taken the inelastic cross section with the collision energy to study the LF hypothesis. This is the novelty of the present work.

R.S. acknowledges stimulating discussions with Edward Sarkisyan-Grinbaum. Useful help from Aditya Nath Mishra while preparing the manuscript is highly appreciated. The authors acknowledge the financial support from ALICE Project No. SR/MF/PS-01/2014-IITI(G) of the Department of Science & Technology, Government of India. This research work used resources of the LHC grid computing center at the Variable Energy Cyclotron Center, Kolkata.