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

Abstract

Abstract The KNO-violating term $$f_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> of the charged-particle multiplicity distribution in pp collisions measures the relative deviation of $$\langle n\rangle P_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> from $$e^{-z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:msup> </mml:math> , with $$z=n/\langle n\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> , and is reported to obey the reciprocal symmetry $$f_s(z)=f_s(1/z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , taken here as input. Being an evenness condition in $$\ln z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>ln</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> , it generates a tower of local constraints on the derivatives of $$P_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> at the mean. The lowest member holds for the ATLAS data at 7, 8 and 13 TeV at the few-per-cent level, while the third-derivative residual testing the next member is not determined with a controlled uncertainty by the present binning and stays inconclusive. The global test is consistent at 7 and 8 TeV, while at 13 TeV a residual deviation remains, which a closure test shows is not a binning artefact. Multiplicative noise in the Mueller colour-dipole cascade, the negative binomial and the dipole cascades with recombination each give an $$f_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> that is not invariant under $$z\rightarrow 1/z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> , so none of them carries the symmetry. We further obtain a dynamics-independent entropy in the KNO continuum, $$S=\ln \langle n\rangle +I_0-\tfrac{1}{2}\int e^{-z}f_s^2\,dz$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo>ln</mml:mo> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>-</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:math> , with <jats:tex-mat