Digit anomalies in the hadronic mass spectrum, Shannon information entropy, and the dynamical QCD scale

TL;DR

This work applies Shannon information entropy to the leading-digit distribution of the hadronic mass spectrum to probe scale-invariance breaking in QCD. It demonstrates that, while a scale-invariant ensemble would maximize entropy via Newcomb–Benford statistics (with $P(d)=\log_{10}(1+1/d)$), the emergent QCD scale $Λ_{\text{QCD}}$ induces mass clustering and yields a measurable entropy deficit relative to Benford. Quantitative analysis of PDG data for mesons, baryons, and their combination yields consistent deficits (e.g., $ΔS\approx0.427$ nat for mesons, $ΔS\approx0.524$ nat for baryons, $ΔS\approx0.430$ nat for the combined set) and highly significant $\chi^2$ deviations from Benford’s law, signaling scale-invariance breaking. The findings connect confinement physics, the mass gap, and the Hagedorn density of states to an information-theoretic signature, providing a model-independent diagnostic of dynamical scale generation in hadronic QCD. This approach offers a novel, data-driven lens on how $Λ_{\text{QCD}}$ shapes the hadronic spectrum.

Abstract

Quantum Chromodynamics (QCD) has an emergent dynamical energy scale $Λ_{\rm QCD}$ which sets the threshold between perturbative and nonperturbative regimes. This characteristic scale causes hadronic masses to cluster within certain mass ranges, instead of following a uniform distribution. Analyzing the Shannon information entropy underlying the hadronic mass spectrum provides novel insight into this phenomenon, revealing a pronounced deviation from the law of anomalous numbers. This deviation quantifies the emergence of the dynamical scale in strongly interacting systems, also encoding the information-entropy cost associated with the breaking of scale invariance in QCD.

Figures (5)

• Figure 1: Distribution of hadronic mass spectrum in PDG pdg.
• Figure 2: Distribution of meson (with further mesonic states) [left panel] and baryon [right panel] mass spectrum in PDG pdg.
• Figure 3: Distribution of the meson mass spectrum with further mesonic states in PDG pdg, according to the leading digit.
• Figure 4: Distribution of the baryon mass spectrum in PDG pdg, according to the leading digit.
• Figure 5: Distribution of the meson mass spectrum with further mesonic states and baryons in PDG pdg, according to the leading digit.