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

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.