KNO scaling, memorylessness and maximal entanglement at universal fixed point

TL;DR

The paper investigates KNO scaling in $p$-$p$ collisions and identifies a universal fixed point at $z=2$ where KNO-violating corrections are suppressed. Applying the AGK cutting rules within a Markov-chain framework and the KL model yields expressions for $(\\langle n\\rangle P_n)$ that near large $\\langle n\\rangle$ behave like $\\Psi(z)\\approx e^{-z}$, with first-order corrections vanishing at $z=2$. ATLAS/Kulchitsky data show crossing of the KNO curves at $z=2$ with $\\Psi(z) \to e^{-2}$, supporting a memoryless high-energy limit. The authors relate this universal point to statistical saturation of the optical theorem, $\\sigma=2\\Im A$, and maximal entanglement of the final-state hadrons, implying a model-independent feature for unitarity-respecting dynamics.

Abstract

We analyze the experimental data of $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions by the ATLAS and confront it with the AGK model developed by two of the authors, the Kharzeev-Levin~(KL) model and the simple exponential behavior for the Koba, Nielsen and Olesen~(KNO) scaling function. We show that all three models virtually coincide with all available experimental curves crossed at value $z=n/\langle n \rangle$ of the KNO scaling parameter in the broad range of the center-of-mass energies. The theoretical results and the experimental data show that the KNO violating terms are highly suppressed in the vicinity of $z=2$. The special status of the universal scaling point $z=2$, where the KNO scaling is restored for $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions, suggests that it originates from statistical saturation of the optical theorem. We claim that any unitary model that approximates memoryless distribution in the high energy limit must have the same KNO scaling function $e^{-z}$ in the vicinity of $z=2$ with KNO violating corrections of the order of one over average multiplicity squared. This reflects the maximal entanglement of the final states.

Figures (3)

• Figure 1: The graphs Kulchitsky:2022gkm show the experimental data of $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions by the ATLAS ATLAS:2010zmcATLAS:2010jvhATLAS:2016quxATLAS:2016zkpATLAS:2016zba for the KNO scaled primary charged-particle multiplicity distributions as a function of $z$ for events with $n_{\mathrm{ch}} \ge 2$ and $\mid\eta\mid < 2.5$ measurement at the center-of-mass energies $0.9$, $2.36$, $7$, $8$ and $13$$\mathtt{TeV}$. The graphs show the KNO scaled probability $\Psi_{KNO}(z)=\langle n \rangle P_n=\langle n_{ch} \rangle (1/N_{ev}) (d N_{ev}/dn_{ch})$ as a function of the scaled variable $z=n/ \langle n \rangle$ Each graph shows the comparison of the asymptotic KNO behavior $\Psi_{KNO}(z) \simeq e^{-z}$ to the experimental data at different energies for .$p_{\mathrm{T}} >500$$\mathtt{MeV}$ (on the left) and $p_{\mathrm{T}} >100$$\mathtt{MeV}$ (on the right) .
• Figure 2: Experimental data of $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions by the ATLAS ATLAS:2010zmcATLAS:2010jvhATLAS:2016quxATLAS:2016zkpATLAS:2016zba for the KNO scaled primary charged-particle multiplicity distributions as a function of $z$ for events with $n_{\mathrm{ch}} \ge 2$, $p_{\mathrm{T}} >100$$\mathtt{MeV}$ and $\mid\eta\mid < 2.5$ measurement at the center-of-mass energies $0.9$, $7$, $8$ and $13$$\mathtt{TeV}$. Each graph shows the comparison of the Levin-Kharzeev model, the AGK model and the asymptotic KNO behavior $e^{-z}$ to the experimental data at different energies. All three models are very good in describing experimental data in the $1 \leq z \leq 3$ region, while the AGK model describes better the experimental data for large $z$.
• Figure 3: Experimental data of $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions by the ATLAS ATLAS:2010zmcATLAS:2010jvhATLAS:2016quxATLAS:2016zkpATLAS:2016zba for the KNO scaled primary charged-particle multiplicity distributions as a function of $z$ for events with $n_{\mathrm{ch}} \ge 2$, $p_{\mathrm{T}} >500$$\mathtt{MeV}$ and $\mid\eta\mid < 2.5$ measurement at the center-of-mass energies $0.9$, $7$, $8$ and $13$$\mathtt{TeV}$. Each graph shows the comparison of the Levin-Kharzeev model, the AGK model and the asymptotic KNO behavior $e^{-z}$ to the experimental data at different energies. All three models are very good in describing experimental data in the $1 \leq z \leq 3$ region, while the AGK model describes better the experimental data for large $z$.