Finite Volume Effects on Transverse Momentum Spectra at LHC and RHIC Using a Blast-Wave Model with Planck Transformed Temperatures

This study demonstrates that employing a finite-volume blast-wave model with Planck-transformed thermodynamic variables is essential for extracting physically realistic freeze-out parameters from heavy-ion collision data, as the conventional infinite-volume approach yields unphysical results such as infinite volumes and light-speed flow velocities.

The Gist

Imagine you are at a massive, chaotic concert where a huge crowd of people (representing subatomic particles) is packed into a small room. Suddenly, the doors burst open, and the crowd rushes out in all directions. Physicists call this "heavy-ion collision," and they want to understand exactly how the crowd moves, how hot it was when the doors opened, and how big the room was.

For decades, scientists have used a tool called the Blast-Wave Model to describe this explosion. Think of this model as a way to predict the speed and direction of the crowd based on how hot the room was and how fast the walls were pushing them out.

However, this paper argues that the old way of using this tool has a few major flaws, like trying to measure the size of a puddle by assuming the ocean is infinite. Here is a simple breakdown of what the authors did and why it matters.

1. The Old Way: The "Infinite Room" Problem

In the traditional Blast-Wave model, scientists made a simplifying assumption: they treated the explosion as if it happened in an infinite room.

• The Analogy: Imagine trying to calculate how much water is in a cup, but your calculator assumes the cup is actually an infinite ocean.
• The Result: Because the model assumes the "room" (the fireball of particles) is infinitely long, it produces some weird, impossible answers. It suggests the explosion stretches forever and that the particles at the very edge are moving at the speed of light. It also treats the "temperature" of the room as if it's measured by someone standing still inside the moving crowd, rather than by an observer watching from the outside.

2. The New Way: The "Finite Cylinder" with a Twist

The authors of this paper introduced a smarter version of the model. They made two key changes:

A. The Room Has Walls (Finite Volume)
Instead of an infinite ocean, they realized the explosion happens in a finite cylinder (like a soda can).

• Why it matters: In real heavy-ion collisions, the "fireball" is only a few times larger than the original nuclei smashing together. By acknowledging the walls exist, the model stops predicting impossible infinite lengths and speeds. It gives a realistic size for the "room" where the explosion happened.

B. The "Planck Transformation" (The Moving Observer)
This is the trickiest part, but here is the analogy:

• Imagine you are on a fast-moving train (the fluid element) holding a thermometer. You read the temperature as 100°C.
• A person standing on the platform (the laboratory frame) looks at your thermometer. Because you are moving so fast, relativity says they will see a different temperature.
• The Old Model: It mixed up the two perspectives. It took the temperature from the moving train but tried to apply it to the stationary platform without adjusting for the speed.
• The New Model: It uses Planck Transformations. This is like a mathematical "translation tool" that correctly converts the temperature and pressure from the moving train's perspective to the stationary observer's perspective. This ensures the laws of physics (specifically relativity) are respected.

3. What They Found

The team tested both the "Old Infinite Room" model and their "New Finite Cylinder" model against real data from massive particle colliders (RHIC and LHC).

• The "Infinite" Model: It gave results that were physically impossible. It predicted the explosion had infinite volume and that the "temperature" was higher than it should be because it didn't account for the relativistic speed of the particles.
• The "Finite" Model: This model worked much better.
  • It calculated a realistic size for the explosion (several times larger than the original nuclei, but definitely not infinite).
  • It found the temperature was lower and more consistent with other theories (like the Tsallis statistics used by other physicists).
  • It showed that the "fire cylinder" stretches out as the collision energy increases, but it always stays within the speed of light.

4. The "Anomaly"

There was one strange hiccup. At two specific energy levels (193 and 200 GeV), the new model showed a weird flip where the "moving" temperature seemed higher than the "stationary" one. The authors admit this is an anomaly that needs more study, but for almost all other energies, the new model is a clear winner.

The Bottom Line

Think of this paper as upgrading a map. The old map assumed the world was flat and infinite, which worked okay for short trips but failed for long journeys. The authors drew a new map that accounts for the actual curvature and size of the world (the finite volume) and correctly adjusts for how fast you are traveling (the Planck transformation).

Why should you care?
By fixing these mathematical errors, scientists can now extract the true "freeze-out" parameters of the universe's earliest moments. This helps us understand the state of matter just microseconds after the Big Bang, ensuring that when we say "the temperature was X," we are actually talking about the real temperature, not a mathematical illusion caused by ignoring the size of the explosion.

Technical Summary

1. Problem Statement

The study addresses two fundamental theoretical inconsistencies in the conventional Blast-Wave (BW) model used to analyze heavy-ion collision data:

• Infinite Volume Approximation: Standard phenomenological fits assume an infinite emission volume (infinite longitudinal extent). This is physically unrealistic, as the fireball at kinetic freeze-out has a finite geometric size, typically only a few times larger than the initial nuclear overlap. Consequently, the system volume is treated as an external normalization parameter rather than an intrinsic physical quantity.
• Frame Inconsistency in Thermodynamics: In conventional BW models, thermodynamic parameters (temperature $T_0$ and chemical potential $\mu_0$) are defined in the local rest frame of fluid elements, while particle momenta are expressed in the laboratory (global) frame. This violates the standard principle of statistical mechanics where distribution functions should be defined within a single reference frame. This inconsistency complicates comparisons with other frameworks (e.g., Tsallis statistics, Lattice QCD, and statistical thermal models) which operate in the laboratory frame.

2. Methodology

The authors employ an extended Boltzmann-Gibbs Blast-Wave (BGBW) model that incorporates two critical improvements:

• Finite Volume Geometry: The model utilizes a cylindrically symmetric fire cylinder with a finite longitudinal extent, bounded by a maximum space-time rapidity ($\eta_{max}$). This contrasts with the infinite volume limit ($\eta_{max} \to \infty$) used in standard models.
• Planck Transformations: To ensure full Lorentz covariance, the model applies Planck transformations to convert the local rest frame thermodynamic variables ($T_0, \mu_0$) into laboratory frame variables ($T, \mu$) for each fluid element.
  • Transformation: $T = T_0 / \gamma$ and $\mu = \mu_0 / \gamma$, where $\gamma$ is the Lorentz factor of the fluid element.
• Comparative Analysis: The authors fit experimental transverse momentum ($p_T$) spectra of charged pions using two variants:
  1. Finite Volume Model: Cylindrical symmetry, finite $\eta_{max}$, Planck-transformed $T$ and $\mu$ (Laboratory frame).
  2. Conventional Infinite Volume Model: Infinite $\eta_{max}$, rest-frame $T_0$ and $\mu_0$.
• Data Scope: Fits were performed on data from the HADES, STAR, PHENIX, and ALICE collaborations, covering a center-of-mass energy range of $\sqrt{s_{NN}} = 2.4$ GeV to $5.44$ TeV.

3. Key Contributions

• Derivation of Lorentz-Invariant Spectra: The paper derives the transverse momentum distribution for a finite volume fire cylinder where the temperature and chemical potential are explicitly defined in the laboratory frame. This results in a spectrum that depends on particle rapidity ($y$), breaking the boost-invariance assumption of infinite models.
• Intrinsic Volume Determination: Unlike conventional models where volume is a free normalization parameter, this framework allows for the intrinsic calculation of the fire cylinder's volume ($V$), maximum half-length ($z_{max}$), and maximum longitudinal flow velocity ($v_{z,max}$) directly from fit parameters.
• Resolution of Frame Discrepancies: By using Planck transformations, the study bridges the gap between hydrodynamic blast-wave models and statistical models (like Tsallis distributions) that define temperature in the laboratory frame.

4. Key Results

• Thermodynamic Parameters:
  • The Finite Volume Model yields laboratory frame temperatures ($T$) and chemical potentials ($\mu$) that are systematically lower than the rest-frame parameters ($T_0, \mu_0$) extracted by the infinite volume model. This aligns with relativistic thermodynamics.
  • Anomaly: At $\sqrt{s_{NN}} = 193$ GeV (STAR U+U) and $200$ GeV (PHENIX Au+Au), the extracted rest-frame temperature in the finite volume model was found to be slightly lower than the laboratory frame temperature, an anomaly requiring further investigation.
• Physical Realism of Volume:
  • The finite volume model produces realistic fire cylinder volumes (e.g., $\sim 4000\text{--}8000$ fm$^3$ at LHC energies) that are several times larger than the initial nuclear overlap volume but remain finite.
  • The Infinite Volume Model yields unphysical results: infinite volume, infinite half-length, and a maximum longitudinal flow velocity equal to the speed of light ($c$) at all energies.
• Rapidity Dependence:
  • The finite volume model predicts $p_T$ spectra that vary with rapidity ($y$), reflecting the finite length of the fire cylinder.
  • The infinite volume model is boost-invariant, producing spectra independent of rapidity, which is unrealistic for finite systems.
• Model Limitations: The classical Boltzmann-Gibbs distribution fails to describe the "hard" part of the spectra (high $p_T$) at both very low (HADES) and very high (ALICE) energies, necessitating $p_T$ cut-offs (e.g., $p_T < 3.1$ GeV) for accurate fitting.

5. Significance

• Reliability of Freeze-Out Parameters: The study demonstrates that ignoring finite volume effects and using inconsistent reference frames leads to unphysical parameter extraction (infinite volumes, light-speed expansion). Proper treatment is essential for reliable extraction of freeze-out conditions.
• Unification of Frameworks: The results explain why temperatures extracted from conventional blast-wave models are typically higher than those from Tsallis-based statistical models. The difference arises because the former uses rest-frame temperatures while the latter uses laboratory-frame temperatures. The Planck-transformed finite volume model reconciles these values.
• New Observables: The methodology provides access to previously inaccessible kinematic quantities, such as the maximum longitudinal flow velocity and the specific space-time rapidity of the freeze-out surface, offering a more complete picture of the hydrodynamic evolution in heavy-ion collisions.

In conclusion, the paper establishes that a finite-volume, Lorentz-covariant blast-wave model is necessary for a physically consistent description of heavy-ion collisions, correcting the unphysical artifacts of the traditional infinite-volume approach.