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Internal energy of the medium formed in pp collisions

This paper employs a thermostatistical analysis of ALICE's charged particle data from LHC pp collisions to demonstrate that the internal energy of the produced medium exhibits nonlinear behavior, indicating a complex degree-of-freedom structure that departs from the ideal gas model and explains the persistent production of high- $\pT$ hadrons.

Original authors: J. Alonso Tlali, D. Rosales Herrera, J. R. Alvarado García, A. Fernández Téllez, C. Pajares, J. E. Ramírez

Published 2026-10-16

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CC BY 4.0

Original authors: J. Alonso Tlali, D. Rosales Herrera, J. R. Alvarado García, A. Fernández Téllez, C. Pajares, J. E. Ramírez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

High-energy physics is the study of what happens when matter is smashed together at speeds approaching the speed of light. When two protons collide, they do not simply bounce off one another; they create a fleeting, chaotic burst of new particles. Scientists have long tried to understand the rules governing this burst, treating the collection of particles like a hot gas. In a standard gas, the energy and temperature are linked in a simple, predictable way: if you heat it up, the particles move faster in a straightforward manner, and the total energy rises in direct proportion to the heat. This "ideal gas" picture has been the default way to describe these collisions for decades. However, the collisions happening at the Large Hadron Collider are far more complex than a simple gas, and the question remains whether the old rules still apply when the energy is this high.

A team of researchers from Mexico and Spain has taken a fresh look at this problem by analyzing data from the ALICE experiment at CERN. Instead of trying to model every single particle interaction from the bottom up, they treated the entire spray of particles produced in the collision as a single statistical system. They focused on the transverse momentum of the particles, which is a measure of how fast they are moving sideways relative to the direction of the collision. By looking at the distribution of these speeds across many different collision energies, the team applied a method of statistical analysis usually reserved for complex systems that do not behave like simple gases. They used two specific mathematical tools, known as the Hagedorn and Tricomi functions, to describe the data. These tools are designed to handle situations where extreme events—particles moving much faster than average—are more common than a simple gas would predict.

The researchers found that the medium created in these collisions does not behave like an ideal gas. When they calculated the internal energy of the system—the total energy stored within the collection of particles—they discovered that it does not rise in a straight line as the temperature increases. Instead, the relationship is curved and nonlinear. This means that as the collision energy gets higher, the system gains energy in a more complex way than simply adding up the number of particle types present. The study suggests that the number of ways the system can store energy, or its "degrees of freedom," becomes more intricate at high energies. It is not just a matter of having more flavors of particles; the system itself becomes more complicated, with new ways of organizing energy that only appear when the collisions are violent enough.

This finding helps explain a long-standing mystery in particle physics: why scientists have always observed the production of high-speed particles, even in collisions with relatively low energy. If the system were a simple gas, such high-speed particles would be rare and difficult to produce. The new analysis shows that the system is inherently "nonthermal," meaning it is not in a state of perfect equilibrium where everything is uniform. Instead, the temperature fluctuates in small regions within the collision zone. These fluctuations allow for the creation of high-speed particles naturally, without needing special conditions. The researchers determined that this behavior is consistent across all the energies tested at the Large Hadron Collider, from the lowest to the highest.

The study also calculated the entropy of the system, which is a measure of its disorder or the amount of information needed to describe it. They found that the entropy also follows a nonlinear path, diverging from the predictions of simple thermal models. This confirms that the system is not just a hot gas but a complex medium where the rules of standard thermodynamics need to be adjusted. The researchers noted that this behavior is likely not unique to proton-proton collisions but may also apply to other types of particle collisions, such as those involving electrons and protons. By using the experimental data directly to define the temperature and energy, the team avoided making many assumptions about the microscopic details of the strong nuclear force. Their work provides a clearer, data-driven picture of how matter behaves under extreme conditions, showing that the universe's most energetic collisions create a medium that is far richer and more complex than previously thought.

Technical Summary: Internal Energy of the Medium Formed in pp Collisions

Problem Statement
The study of quark matter properties and the thermodynamics of strongly interacting systems remains a central challenge in high-energy physics. While lattice QCD and various theoretical frameworks (e.g., Landau thermodynamics, holographic approaches) successfully describe the Equation of State (EOS) for large systems like heavy-ion collisions, extending these descriptions to small collision systems, such as minimum bias proton-proton (pp) collisions, presents significant difficulties. In small systems, the lack of a well-defined nuclear geometry and the presence of finite-size effects invalidate standard volume-scaling assumptions used in the thermodynamic limit. Furthermore, constructing an EOS for these systems is complicated by the need to account for non-extensive behaviors and the production of high transverse momentum ( $p_T$ ) hadrons, which suggest deviations from an ideal gas picture. The paper addresses the need for a consistent thermostatistical framework to describe the medium formed in pp collisions using only experimental $p_T$ spectrum data, without relying on detailed microscopic QCD assumptions or uncertain transverse area estimations.

Methodology
The authors perform a thermostatistical analysis of charged particle production in minimum bias pp collisions across various center-of-mass energies ( $\sqrt{s}$ ) reported by the ALICE Collaboration at the LHC. The methodology proceeds as follows:

Data and Models: The analysis utilizes the normalized $p_T$ spectrum as a probability density function (PDF). The spectrum is modeled using two functions derived from the string fragmentation framework: the Hagedorn function ( $W_{Hag}$ ) and the Tricomi function ( $W_U$ ). These functions arise from convolving the Schwinger mechanism (describing pair production via quantum tunneling) with heavy-tailed distributions of color string tension fluctuations (specifically, a Pareto-type distribution and a $q$ -Gaussian distribution, respectively).
Parameter Extraction: The models are fitted to ALICE data for $\sqrt{s} = 0.9, 2.76, 5.02, 7, 8,$ and $13$ TeV. The fitting procedure excludes the intermediate $p_T$ region (1–6 GeV) where collective effects may dominate, focusing on low and high $p_T$ regions. The quality of fits is validated using Kolmogorov-Smirnov (KS) tests.
Thermodynamic Definitions:
- Temperature ( $T$ ): Defined via the slope of the $p_T$ spectrum at low $p_T$ values, derived from the asymptotic exponential decay of the Hagedorn and Tricomi functions.
- Entropy ( $H$ ): Calculated using the Shannon entropy definition applied to the normalized $p_T$ spectrum, treating $p_T$ states as microstates.
- Internal Energy ( $E$ ): Determined by solving the fundamental thermodynamic relation $dE/dT = C$ , where $C = T(dH/dT)$ is the heat capacity.
Alternative Approach: To contrast with the Shannon entropy approach, the authors also estimate a thermodynamically consistent entropy ( $S$ ) by assuming the average transverse momentum $\langle p_T \rangle$ acts as the internal energy per particle (a Bjorken-type estimator), retaining the temperature definition from the spectrum slope.

Key Contributions and Results

Non-Extensive Framework: The study confirms that the Hagedorn and Tricomi functions, derived from non-extensive string tension fluctuations, provide a consistent description of the $p_T$ spectrum across all analyzed energies. The non-extensivity parameters ( $m$ and $q$ ) follow a power-law dependence on the center-of-mass energy.
Thermal Limit and Critical Temperature: The analysis identifies a critical temperature ( $T^*$ ) below which the system would behave as an ideal gas ( $q \to 1$ ). Extrapolation yields $T^*_U \approx 0.103$ GeV and $T^*_{Hag} \approx 0.065$ GeV, corresponding to a center-of-mass energy $\sqrt{s}^* \approx 1.876$ GeV. Since all LHC energies exceed this threshold, the systems studied are firmly in a non-thermal regime.
Nonlinear Internal Energy: The primary result is that the internal energy $E$ behaves nonlinearly with temperature at LHC energies. For the Hagedorn and Tricomi models, $E$ follows a power law $E \propto T^\alpha$ with exponents $\alpha_{Hag} \approx 1.27$ and $\alpha_U \approx 1.39$ . This indicates that the count of degrees of freedom is not constant but increases with temperature, more complex than simply summing the number of quark flavors.
Heat Capacity and Stability: The heat capacity is found to be a positive, increasing function of temperature, indicating thermodynamic stability and a rising number of effective degrees of freedom.
Bjorken-Type Analysis: When using $\langle p_T \rangle$ as the energy estimator, the resulting entropy ( $S$ ) also deviates from the ideal gas logarithmic behavior, reinforcing the conclusion that the system possesses a complex, non-additive structure of degrees of freedom.

Significance
The paper claims that its results demonstrate the medium formed in pp collisions departs significantly from the ideal gas picture. The nonlinear behavior of internal energy and the power-law tail of the $p_T$ spectrum are direct consequences of the non-extensive nature of the system, driven by intense color interactions and hard partonic scattering.

The authors argue that this thermostatistical approach explains why high- $p_T$ hadron production is consistently observed in pp collisions, even at lower energies (such as ISR energies), without requiring the system to be in thermal equilibrium. The framework suggests that the "hard" part of the spectrum is an intrinsic feature of the non-extensive initial state, arising from large fluctuations in color string tension. Furthermore, the authors posit that these findings may extend to other collision systems, such as $e^-p$ or $e^+e^-$ , where high- $p_T$ hadron production is also observed. Crucially, the internal energy is determined directly from experimental data via thermodynamic relations, bypassing the need for detailed microscopic assumptions about QCD interactions or the problematic estimation of transverse interaction areas in small systems.

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