Reciprocal symmetry and KNO 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 $z\leftrightarrow 1\mathrm{/}z$ in the KNO violating corrections observed in the ATLAS and CMS data at $\sqrt{s}=7,8,13$~TeV. The symmetry imposes a local constraint on the multiplicity distribution at $n=⟨n⟩$, namely $P^{\mathrm{\prime }}(⟨n⟩)=-P(⟨n⟩)\mathrm{/}⟨n⟩$, which we verify directly in the data. We use this constraint to extract the entanglement entropy from the well-measured region $n\simeq ⟨n⟩$, avoiding the large uncertainties associated with the distribution tail.