Tsallis fits trace multiplicity trends in 13 TeV proton-collision spectra

Proton collisions at 13 TeV produced hadron spectra that shifted systematically with event multiplicity, according to a model-based analysis using a thermodynamically consistent Tsallis distribution. Independent fits to pion, kaon and proton transverse-momentum spectra indicated a denser, more collision-dominated emitting system in high-multiplicity events and a progressively more dilute system in low-multiplicity events. These results describe spectral shape within the model; they are not direct measurements of temperature, physical volume or the time at which individual particles stopped interacting.

The study examined ALICE measurements from the 2016 Large Hadron Collider run, comprising about 143 million proton-proton collision events at a center-of-mass energy of 13 TeV. The selected spectra covered pions from approximately 0.1 to 20 GeV per particle in transverse momentum, kaons from 0.2 to 20 GeV per particle and protons from 0.3 to 20 GeV per particle, with the exact limits varying slightly among multiplicity classes. Events were divided into 10 classes using percentiles of visible inelastic cross section measured by the V0 detector.

Mean charged-particle multiplicity density ranged from about 2.55 to 26.02, with MC-1 representing the highest multiplicity and MC-10 the lowest. Each particle species and multiplicity class was fitted independently with the Tsallis distribution, which includes an effective temperature, a non-extensivity parameter and a normalization volume. The chemical potential was fixed at zero because the authors considered it negligible at these high energies.

The effective temperature controls the broad slope of a spectrum, while the non-extensivity parameter controls its curvature and transition toward a harder, power-law-like tail. A value of one for the latter recovers the Boltzmann-Gibbs form. Because no common temperature or non-extensivity parameter was imposed across the three species, their fitted values characterize different spectral shapes and should not be interpreted as properties of one thermal source with a shared temperature.

The effective temperature displayed a clear mass ordering: proton values were highest, kaon values intermediate and pion values lowest. That ordering follows from the use of transverse mass in the distribution, because the larger rest mass of a heavier particle shifts its spectrum toward higher transverse momentum and can produce a steeper fitted slope when species are fitted separately. From MC-1 through MC-9, all three temperatures decreased as multiplicity declined, consistent with less energy being deposited in smaller collision systems.

In MC-10, however, pion and proton temperatures rose again while the kaon temperature continued downward. The authors describe this reversal as tentative evidence that a growing contribution from hard scattering was flattening spectral tails and forcing the single-component fit to compensate. The non-extensivity parameter showed a different species pattern.

Pions carried the largest values, kaons occupied an intermediate range and protons had the smallest values in every multiplicity class. For pions, the parameter declined from 1.1675 in MC-1 to 1.1571 in MC-9 and then dropped sharply to 1.1333 in MC-10. Kaons remained narrowly grouped around 1.154 to 1.157, while protons stayed near 1.10 to 1.12 through MC-9 before falling to 1.09 in MC-10.

The ordering is consistent with lighter particles being more sensitive to hard processes and jet fragmentation, but the parameter remains phenomenological: it cannot by itself distinguish temperature fluctuations, correlations or other causes of spectral curvature. The different trend relative to an earlier Tsallis-blast-wave analysis also demonstrates that its extracted value depends on the parametrization and on whether flow is represented explicitly. Normalization volumes fell monotonically as the events became less populous, with the pion effective volume declining from 2122 in MC-1 to 1167 in MC-10.

Kaon volumes decreased from 1965 to 1045, while proton volumes fell from 1700 to 711. At every multiplicity, the extracted hierarchy was pion volume above kaon volume above proton volume. This pattern resembles a picture in which lighter particles remain coupled to the expanding medium longer, whereas heavier particles decouple earlier.

Nevertheless, the volume is an effective normalization quantity rather than a directly measured geometric volume, and the three species were fitted independently. The ordering is therefore suggestive of species-dependent decoupling, not direct evidence for a chronological freeze-out sequence. Resonance feed-down, particularly relevant to pions, may also affect the apparent volume.

Derived thermodynamic quantities followed comparable mass and multiplicity patterns. The volumetric thermal expansion coefficient, defined as the fractional volume change per unit temperature increase at constant pressure, was most negative for pions, intermediate for kaons and closest to zero for protons. Mean free path was longest for pions, intermediate for kaons and shortest for protons when a common hard-sphere cross section was used.

As multiplicity fell, the mean free path increased for all three species, from 48.9 to 143 femtometers for pions, from 24.8 to 162.6 femtometers for kaons and from 3.95 to 48.7 femtometers for protons. The increase reflects lower modeled densities: particles traveled farther between encounters as the emitting system became more dilute. Because the cross section was identical for all species, the ordering arose from their separately fitted densities and masses rather than from different interaction strengths.

The transport analysis placed the same ordering on the Knudsen number, the ratio of mean free path to effective system size. Pions had the largest values, kaons were intermediate and protons the smallest, while all three increased toward lower multiplicity. The values remained well below one, meaning the system did not approach free-streaming.

Even in MC-10, however, the Knudsen number rose to about 0.6, and the proton value increased sharply from 0.40 in MC-8 to 0.64 in MC-10. This suggests that collisions became less frequent relative to system size and that a hydrodynamic description became less reliable, even though the modeled hadrons continued to interact. The study did not extract a transverse-flow parameter, so its results do not establish whether the low-multiplicity events were weakly collective.

Isothermal compressibility also increased as multiplicity declined, consistent with the modeled emitting medium becoming softer and more responsive to changes in chemical potential. Proton compressibility displayed the clearest late-class departure, rising from 68.2 in MC-7 to 126.7 in MC-8 and 239.3 in MC-9 before easing to 233 in MC-10. Because compressibility is calculated from densities derived from the fitted parameters, distortions in those parameters propagate into the response function.

MC-10 also produced departures in effective temperature, mean free path and expansion coefficient. The authors considered a larger hard-scattering contribution plausible, but statistical limitations and fit instability remain possible, and the analysis did not use a two-component soft-plus-hard model to separate the mechanisms. Overall, the fits depict a progression from an interacting, relatively dense proton-collision system at high multiplicity to a more dilute one at low multiplicity.

Species-dependent effective volumes, mean free paths, Knudsen numbers and compressibilities were consistent with lighter hadrons remaining coupled to the medium longer than heavier ones, although the independent fits prevent that hierarchy from proving a time-ordered freeze-out sequence. The evidence is not uniform: proton spectra were described less accurately than pion and kaon spectra in several classes, some fit-quality measures were unusually small, and the published bin uncertainties did not allow a full covariance-matrix treatment. The central conclusion lies in the coherent multiplicity trend across several derived quantities, while the physical cause of the MC-10 deviations remains an open question requiring better statistics and a dedicated two-component analysis.

Subject of Research: Multiplicity-dependent thermodynamic and transport properties of pion, kaon and proton spectra in 13 TeV proton-proton collisions

Article Title: Exploring the thermodynamic and transport properties of hadron spectra in proton-proton collisions at s=13 TeV

Article References: Haj Ismail, A., & Ben Slimane, J. (2026). Exploring the thermodynamic and transport properties of hadron spectra in proton-proton collisions at s=13 TeV. Frontiers in Physics, 14, Article 1990813. https://doi.org/10.3389/fphy.2026.1990813

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1990813

Keywords: proton-proton collisions, 13 TeV, Tsallis distribution, hadron spectra, event multiplicity, freeze-out parameters, Knudsen number, mean free path, isothermal compressibility, ALICE, Exploring, thermodynamic