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

The reciprocal symmetry $f_s(z)=f_s(1\mathrm{/}z)$ 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 $n=⟨n⟩$. The $k=0$ constraint $P^{\mathrm{\prime }}(⟨n⟩)=-P(⟨n⟩)\mathrm{/}⟨n⟩$ has been verified previously. We derive the $k=1$ constraint, $⟨n{⟩}^3{P}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(⟨n⟩)+6⟨n{⟩}^2{P}^{\mathrm{\prime }\mathrm{\prime }}(⟨n⟩)=5P(⟨n⟩)$, 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 $1\mathrm{/}3<z<3$ 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 $z\to 1\mathrm{/}z$ 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 $f_s(z)=f_s(1\mathrm{/}z)$ is at best an approximate symmetry. We derive a model-independent expression for the entanglement entropy $S=\ln ⟨n⟩+1-\frac{1}{2}{\int }_0^{\mathrm{\infty }}{e}^{-z}{f}_s^2(z)dz+\mathcal{O}(f_s^3)$, in which the linear term in $f_s$ cancels by virtue of normalisation and the constraint $⟨z⟩=1$, and evaluate it numerically on the ATLAS data.