Reciprocal symmetry and Koba-Nielsen-Olesen scaling violation in proton-proton collisions

We analyze the charged particle multiplicity distributions in p − p collisions and discuss the violation of the Koba-Nielsen-Olesen (KNO) scaling. We extract the deviations from the leading exponential behavior of the KNO scaled probability and identify a reciprocal symmetry z ↔ 1 / z in the KNO violating corrections observed in the ATLAS and CMS data at s = 7 , 8 , 13 TeV . The symmetry imposes a local constraint on the multiplicity distribution at n = ⟨ n ⟩ , namely P ′ ( ⟨ n ⟩ ) = − P ( ⟨ n ⟩ ) / ⟨ n ⟩ , which we verify directly in the data. We use this constraint to extract the entanglement entropy from the well-measured region n ≃ ⟨ n ⟩ , avoiding the large uncertainties associated with the distribution tail.