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

Quantum Chromodynamics (QCD) has an emergent dynamical energy scale ${\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}$ 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.