Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in $pp$ collisions

TL;DR

This paper investigates the reciprocal symmetry in proton-proton multiplicity distributions, deriving local constraints, testing them with ATLAS data, and exploring implications for entanglement entropy, revealing symmetry breaking at high energies.

Contribution

It derives new algebraic constraints from reciprocal symmetry, tests them with experimental data, and formulates a model-independent expression for entanglement entropy in high-energy collisions.

Findings

Symmetry holds at leading orders near z=1 but breaks down at 13 TeV.

The derived constraints are consistent with data at 7 and 8 TeV but not at 13 TeV.

A new expression for entanglement entropy is evaluated using experimental data.

Abstract

The reciprocal symmetry $f_s(z)=f_s(1/z)$ of the KNO-violating term in proton--proton charged-multiplicity distributions, observed at $\sqrt{s}=7$, $8$ and $13$ TeV, implies that the function $h(u)\equiv f_s(e^u)$ is even in $u=\ln z$. Each odd derivative of $h$ at $u=0$ then provides a local algebraic constraint on the multiplicity distribution at $n=\langle n\rangle$. The $k=0$ constraint $P'(\langle n\rangle)=-P(\langle n\rangle)/\langle n\rangle$ has been verified previously. We derive the $k=1$ constraint, $\langle n\rangle^3 P'''(\langle n\rangle)+6\langle n\rangle^2 P''(\langle n\rangle)=5\,P(\langle n\rangle)$, equivalent to the unconditional residual $\delta_3\equiv 1-2\rho_0+\rho_1=0$, and test it in the ATLAS data: at $13$~TeV with the largest fit window we find $\delta_3=-0.02\pm 0.11$, consistent with the leading-order symmetric value, while the $7$ and $8$~TeV results are inconclusive at the available precision. A $\chi^2$ test of the global symmetry across the window $1/3<z<3$ is consistent with the symmetry at $7$ and $8$~TeV but rejects it decisively at $13$~TeV, where the smaller experimental uncertainties expose a residual departure from $z\to 1/z$ invariance away from $z=1$. The overall picture is therefore that the symmetry holds at the leading two non-trivial orders near $z=1$ but breaks down globally at the precision afforded by the $13$~TeV data, suggesting that $f_s(z)=f_s(1/z)$ is at best an approximate symmetry. We derive a model-independent expression for the entanglement entropy $S=\ln\langle n\rangle+1-\tfrac{1}{2}\int_0^\infty e^{-z}f_s^2(z)\,dz+\mathcal{O}(f_s^3)$, in which the linear term in $f_s$ cancels by virtue of normalisation and the constraint $\langle z\rangle=1$, and evaluate it numerically on the ATLAS data.