A comparison of particle-collision data finds that the “horn” in the K+/π+ ratio differs across collision-system sizes, but no single tested model explains the pattern from p+p to Pb+Pb. The result points to different descriptions working in different size ranges, rather than one settled account of strangeness production.
The study examines how often positively charged kaons appear relative to pions, using measurements from p+p, Be+Be, Ar+Sc, Xe+La, Pb+Pb and Au+Au collisions. The authors compare those data with hadronic and partonic transport calculations, a core-corona model, canonical suppression models and the Statistical Model of the Early Stage, or SMES, within a common goodness-of-fit analysis.
Different systems favor different pictures
The transport calculations provide the clearest warning against a one-model explanation. SMASH, a hadronic transport model, produces a steady rise but falls short of the data by factors of 1.6 to 2.3. PHSD raises the predicted Pb+Pb ratio to about 0.19 ± 0.01, roughly 60% to 80% above SMASH, yet its prediction remains flat and still does not reproduce the heavy-system horn.
The core-corona approach, a model that combines core and corona contributions, gives intermediate systems a mixed description. For Pb+Pb, adding that mixture reduces the fit statistic per degree of freedom from 42.0 with SMASH to 1.69, with a reported p value of 0.15. The improvement is smaller for Ar+Sc, where the value falls from 14.9 to 11.2. The fitted core fraction is about 0.22 in Be+Be and 0.58 in Ar+Sc.
A separate quantity, the strangeness saturation factor γs, shows the same broad pattern in two stages. The extracted value rises from about 0.22 to 0.65 between p+p and Ar+Sc, then from 0.65 to 0.95 between Ar+Sc and Pb+Pb. At the horn energy, the mean is 0.34 ± 0.04 for light systems and 0.66 ± 0.04 for heavy systems, a difference of 0.32 ± 0.05, reported as 6.3 standard deviations. Because γs is a model-dependent thermal representation, the pattern is evidence within that framework, not a direct measurement of a new state of matter.
The canonical-suppression model, which treats strangeness conservation within a finite volume, does not provide a single bridge across the full size range. Its best-fit volume is 0.20 cubic femtometres, while the global fit has a chi-squared per degree of freedom of 48.7 across 25 points. Using a different collision-scaling prescription raises that value to 59.6, and leave-one-out refits keep the volume at 0.20 cubic femtometres. The analysis therefore finds that one global canonical volume cannot fit both small and large systems.
SMES with canonical strangeness conservation performs best for the heaviest systems, giving chi-squared per degree of freedom of 0.67 for Au+Au and 2.46 for Pb+Pb. But the same family of models overpredicts Ar+Sc, with values ranging from 86.8 to 132. Taken together, the comparisons favor core-corona for intermediate systems and SMES for heavy systems, while leaving the small-system data without an adequate tested description.
A step, with important qualifications
Fitting the core fraction with a sigmoid, or S-shaped curve, places the midpoint of the system-size change near Ac = 18 and gives a chi-squared per degree of freedom of 0.16. That number describes the midpoint of the fitted curve, not a precise universal boundary. With published data alone, the uncertainty on the midpoint diverges, so the estimate depends on including preliminary Xe+La input and on the chosen corona baseline.
The analysis also tests whether the threshold changes with collision energy. A logarithmic energy-dependent threshold lowers the Pb+Pb fit statistic from 1.69 to 0.25, an improvement in chi-squared of 6.0. Wilks testing gives p = 0.014, while 1,000 bootstrap pseudo-experiments give p = 0.010 ± 0.003. The authors caution that the test uses only five points and three degrees of freedom, which weakens the usual large-sample interpretation.
A related double ratio compares the K+/π+ horn with the corresponding K−/π− behavior. Near a collision energy of about 7.6 GeV, it rises from roughly 1.8 in p+p and Be+Be to about 2.8 in Ar+Sc and Pb+Pb. The combined light-heavy test gives a global significance of 3.7 standard deviations. That headline significance is reduced when correlations between measurements at different energies are included, and the interpretation depends on treating the baryon chemical potential as approximately system-independent at fixed energy.
The next check is still preliminary
The analysis holds out preliminary, digitized Xe+La data from the core-corona threshold calibration and from the canonical-model ranking, using them only for comparison. On that check, the fit statistic is 5.8 for core-corona, 6.4 for fitted SMES and 23.3 for SMASH. Core-corona and fitted SMES predictions differ by 27% at the horn energy, so finalized Xe+La measurements could help distinguish the two descriptions.
The model comparison therefore does not by itself select a unique microscopic mechanism. It does not establish a universal critical system size, show that a threshold calibrated with Pb+Pb transfers to every collision geometry or provide final Xe+La evidence.
The work is an arXiv version 1 preprint dated 28 August 2026. The authors report computational support from the Indian Institute of Technology Mandi, acknowledge public data from NA49, NA61/SHINE and STAR, and say the analysis code is available from the corresponding author on reasonable request.
Paper data and sources
Original title: System-size dependence of strangeness production from p+p to Pb+Pb: quantitative tests of the $K^+/π^+$ horn
Authors: Neeraj, Amal Sarkar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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