Internal multiplicity distributions of jets from nonlinear evolution within the jet function framework

Abstract

Jets selected with high internal charged-particle multiplicity exhibit markedly different substructure patterns compared to inclusive jet samples. Such correlations motivate a systematic study of jet observables as a function of the normalized multiplicity, $ν= N_{\rm ch}/\langle N_{\rm ch}\rangle$. In this work, we develop a theoretical framework for the full charged-particle multiplicity distribution of exclusive and inclusive jets, formulated within the jet-function approach. The hard production and jet function are evaluated at NLO+LL$_R$ accuracy. The internal parton dynamics governing the multiplicity distribution are described by coupled nonlinear branching equations with angular ordering, supplemented by a nonperturbative modeling term that accounts for hadron-level effects. The resulting predictions are validated against \textsc{Pythia8} simulations and compared with CMS data. We examine the effects of both nonperturbative and perturbative components in shaping the multiplicity distribution, and show that Koba--Nielsen--Olesen (KNO) scaling is notably violated in the region $ν> 2$ in the full solution, with a trend consistent with Monte Carlo results. This framework that numerically solves the nonlinear multiplicity evolution goes beyond DLA-like approximations and reproduces key features seen in event generators, providing a solid foundation for future investigations of multiplicity -- conditioned jet substructure within the jet function formalism.

Figures (13)

• Figure 1: Left: the LO diagram for $Z_g(s,\omega, \mu)$. Middle: the real emission amplitude. Its amplitude square contribute to the NLO correction $Z_g(s,\omega, \mu)$. Right: the loop diagram that interfere with the LO diagram, which is scaleless under dimensional regularization. Diagrams with collinear emission from the Wilson line is not present for light-cone gauge.
• Figure 2: On the log-log plot of energy and angle, evolution equations are initialized at an angle that determined by the largest jet energy under consideration. Kinematics within the gray shaded region is removed from the evolution such that the initial condition is preserved at the boundary $\theta\omega = Q_0$.
• Figure 3: Charged-particle multiplicity distributions for exclusive quark and gluon jets with $p_T=500$ GeV and $R=0.4$. Left: absolute distributions $P(N_{\rm ch})$; right: rescaled distributions $\langle N_{\rm ch}\rangle P(N_{\rm ch})$ as a function of $\nu = N_{\rm ch}/\langle N_{\rm ch}\rangle$. Blue open circles indicate gluon jets in the pure-gluon system. In the quark–gluon system, quark jets are shown by orange lines (solid: LO+LL, dashed: NLO+LL) and gluon jets are shown by blue symbols/lines (solid triangles: LO+LL, solid line: NLO+LL).
• Figure 4: Ratio of the mean charged particle multiplicity between exclusive gluon and quark jets, $\langle N_{ch} \rangle_g / \langle N_{ch} \rangle_q$. Left: dependence on the jet radius $R$ for fixed $p_T = 500$ GeV. Right: dependence on the jet transverse momentum $p_T$ for fixed $R = 0.6$. Results are shown for LO+LL (blue dots) and NLO+LL (orange dots).
• Figure 5: Rescaled charged-particle multiplicity distributions for exclusive gluon jets in the pure-gluon system, plotted as $\langle N_{\rm ch}\rangle P(\nu)$ versus $\nu = N_{\rm ch}/\langle N_{\rm ch}\rangle$ for $p_T=500$ GeV and $R=0.4$. Distributions are shown for different initial binomial parameters $n_{\max} = 1, 2, 3$, with $n_{\max}\to\infty$ corresponding to the Poisson limit.