[Paper Review] Violation of the Feynman scaling law as a manifestation of nonextensivity

F. S. Navarra, O. V. Utyuzh+2 more|arXiv (Cornell University)|Sep 14, 2000

Statistical Mechanics and Entropy1 references3 citations

TL;DR

The paper proposes that violations of Feynman scaling in particle production spectra arise from nonextensive statistical mechanics, using Tsallis entropy to derive a one-parameter fit (q) that reproduces empirical spectra from cosmic ray and high-energy collider data. The key result is that the widely used ad hoc parametrization (dN/dx ∝ (1 − a′x)^4/x) emerges naturally from information-theoretic principles when nonextensivity is introduced via fluctuating Lagrange multipliers.

ABSTRACT

We demonstrate that the apparently ad hoc parametrization of the particle production spectra discussed in the literature and used in description of cosmic ray data can be derived from the information theory approach to multiparticle production processes. In particular, the violation of the Feynman scaling law can be interpreted as a manifestation of nonextensivity of the production processes.

Motivation & Objective

• To explain the empirical parametrization of particle production spectra in cosmic ray and high-energy physics as a consequence of nonextensive statistical mechanics.
• To demonstrate that the violation of Feynman scaling is not ad hoc but a manifestation of long-range correlations or memory effects in multiparticle production.
• To replace complex convolution models of fluctuating inelasticity and multiplicity with a single-parameter nonextensive distribution.
• To show that the q-parameter in Tsallis statistics captures the essential physics of spectrum broadening and central enhancement.

Proposed method

• Derive the extensive particle distribution using Shannon entropy maximization under energy and normalization constraints, yielding an exponential form in rapidity.
• Introduce fluctuations in the Lagrange multiplier β via gamma-distributed fluctuations, transforming the exponential into a power-law distribution.
• Apply Tsallis q-entropy formalism to replace Shannon entropy, leading to a nonextensive distribution with a q-dependent power-law form.
• Construct a simplified one-parameter fit (eq. 5) using the mean multiplicity ⟨N(s)⟩ and q-parameter to describe spectra across different energies.
• Account for transverse mass evolution with energy using an empirical interpolation formula: μ_T = 0.3 + 0.044 ln(√s/20) GeV.
• Fit the model to UA5, UA7, and P238 data, adjusting q and multiplicity normalization to match charged vs. neutral particle samples.

Experimental results

Research questions

• RQ1 Can the empirically observed violation of Feynman scaling in particle spectra be derived from a fundamental statistical mechanics principle?
• RQ2 Does the observed power-law form (1 − a′x)^4/x in particle spectra emerge naturally from information-theoretic principles?
• RQ3 Is the nonextensivity parameter q related to physical fluctuations in inelasticity and particle multiplicity in high-energy hadronization?
• RQ4 Can a single-parameter q-distribution replace complex convolution models of fluctuating initial conditions?

Key findings

• The empirical formula dN/dx ∝ (1 − a′x)^4/x, widely used in cosmic ray physics, is reproduced as a natural consequence of nonextensive Tsallis statistics with q ≈ 0.72.
• The best-fit nonextensivity parameter q = 0.72 for UA5 data yields a power-law exponent of 1/(1−q) = 4, matching the exponent in the empirical formula.
• For P238 and UA7 data (neutral particles), a higher q = 0.85 is required, reflecting differences in particle composition and multiplicity normalization.
• The model successfully describes spectra at 53 and 200 GeV, though it becomes too broad at higher energies, indicating limitations in the current q-constant assumption.
• The q-parameter effectively summarizes the impact of fluctuations in inelasticity and multiplicity, suggesting it may be x-dependent across rapidity regions.