Digit anomalies in the hadronic mass spectrum, classical and quantum information entropies, and the dynamical QCD scale

Abstract Quantum Chromodynamics (QCD) has an emergent dynamical energy scale $$\Lambda _{\textrm{QCD}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mtext>QCD</mml:mtext> </mml:msub> </mml:math> 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, and also other classical information entropies, 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. Quantum entanglement entropy also reveals correlations across energy scales through the Shannon entropy of the bipartitioned probability distribution, reflecting how the emergence of $$\Lambda _{\textsc {QCD}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>QCD</mml:mi> </mml:msub> </mml:math> shapes the hierarchical structure of the hadronic mass spectrum.