Multiplicity distributions in QCD jets and jet topics

Abstract

We evaluate the Koba-Nielsen-Olesen (KNO) scaling functions for quark- and gluon-initiated jets by incorporating energy conservation into the Double Logarithmic Approximation (DLA). The resulting modified DLA (MDLA) expressions differ substantially from the DLA predictions and qualitatively align with the recently proposed QCD-inspired expressions, albeit with some quantitative differences. By fixing the two parameters in the MDLA expressions, we show that the inclusive charged-particle multiplicity distributions of the two leading jets in $pp$ collisions at $\sqrt{s} = 13$ TeV, measured by ATLAS over a wide jet $p_T$ range of $0.1$-$2.5$ TeV, are well described within experimental uncertainties and consistent with PYTHIA simulations. This conclusion is further supported by direct comparisons with quark- and gluon-initiated jet distributions extracted via jet topics, though the propagated uncertainties from experimental data remain sizable.

Figures (9)

• Figure 1: KNO scaling functions for quark and gluon jets. Left: Verification of KNO scaling within DLA by directly evaluating $P_a(n)$ according to eq. \ref{['eq:Pqg']} with $Q_0 = 0.5$ GeV, using a fixed coupling $\alpha_s = 0.2$, over a $p_T$ range from 0.1 to 2.5 TeV (solid), and comparing them to the DLA KNO scaling functions derived in the asymptotic limit $Q\to\infty$ (dashed). Within this $p_T$ range, we observe that the curves of $\bar{n}_a P_a(n)$ for different $p_T$ still show some dependence on the value of $\alpha_s$. With the chosen value, they all converge to the DLA KNO scaling functions. Right: Comparison of the MDLA KNO scaling functions evaluated from eq. (\ref{['eq:nmaMDLA']}) (solid) with the QCD-inspired forms in ref. Dokshitzer:2025fky (dashed) for $\gamma_0 = 0.4=\gamma$ and $c_q=2/3$.
• Figure 2: KNO scaling functions for quark and gluon jets in PYTHIA and MDLA. Left: Charged-particle multiplicity distributions for quark and gluon jets in PYTHIADuan:2025ngi (solid) versus MDLA results (dashed) with $\gamma_0 = 0.43$ and $c_q = 0.8$. Right: Negative binomial distributions (NBD, solid), with $k = 5.4$ for quark jets and $k = 8.5$ for gluon jets (see eq. \ref{['eq:NBD']}) compared to the same MDLA results (dashed).
• Figure 3: Mean charged-particle multiplicity within QCD jets at the LHC. Left: Inclusive mean charged-particle multiplicity as a function of jet transverse momentum $p_T$, comparing ATLAS data ATLAS:2019rqw, PYTHIA simulations, DLA, and N$^3$LO results. Right: Mean charged-particle multiplicity in quark and gluon jets from PYTHIA, DLA, and N$^3$LO predictions. For the DLA results, $K_{\text{LPHD}}=0.80$ is obtained by fitting the inclusive mean multiplicity data shown in the left panel with $Q_0=0.5$ GeV held fixed, while for the N$^3$LO results $K=0.0353$ and $Q_0=0.053$ GeV are both determined from fits to the same data.
• Figure 4: Ratio of mean multiplicities between gluon and quark jets. In the case of DLA, this ratio is not constant at the expected value of $C_A/C_F = 9/4$, indicated by the horizontal dashed line, because the leading partons are included in the mean multiplicity given in eq. (\ref{['eq:nbarQG']}).
• Figure 5: Investigation of KNO scaling via inclusive multiplicity distributions in $pp$ collisions. Shown here are our theoretical results using the MDLA KNO scaling functions ($\Psi_{\text{MDLA}}$), the LO cross section ($\sigma_{\text{LO}}$), and the ratios of mean multiplicities between gluon and quark jets in DLA ($\bar{n}_{\text{DLA}}$) and N$^3$LO ($\bar{n}_{\text{N$^3$LO}}$), compared with the ATLAS measurements of $\bar{n}_{ch} P(n_{ch})$ reported in ref. ATLAS:2019rqw across eight $p_T$ bins.