The Scale Dependence of the Hadron Multiplicity in Quark and Gluon Jets and a Precise Determination of $C_{A}/C_{F}$

TL;DR

DELPHI analyzed hadron multiplicities in quark and gluon jets at the $Z$ pole to test Quantum Chromodynamics via a scale-dependent approach. By defining a hardness scale $\kappa = E_{jet} \sin(\theta/2)$ and studying both single-jet and three-jet topologies, the authors extract the colour-factor ratio $C_A/C_F$ from the relative growth of multiplicities, incorporating non-perturbative offsets $N_0$ and coherence effects through the MLLA/DKT framework. The three-jet analysis yields $C_A/C_F = 2.246 \pm 0.062$ (stat.) $\pm 0.080$ (syst.) $\pm 0.095$ (theo.), the most precise hadronic determination to date and in agreement with the QCD value $9/4$. These results support Local Parton-Hadron Duality and provide direct evidence for the triple-gluon coupling, while emphasizing leading-particle fragmentation differences in jet formation.

Abstract

Data collected at the Z resonance using the DELPHI detector at LEP are used to determine the charged hadron multiplicity in gluon and quark jets as a function of a transverse momentum-like scale. The colour factor ratio, \cacf, is directly observed in the increase of multiplicities with that scale. The smaller than expected multiplicity ratio in gluon to quark jets is understood by differences in the hadronization of the leading quark or gluon. From the dependence of the charged hadron multiplicity on the opening angle in symmetric three-jet events the colour factor ratio is measured to be: C_A/C_F = 2.246 \pm 0.062 (stat.) \pm 0.080 (syst.) \pm 0.095 (theo.)

Figures (4)

• Figure 1: Definition of event topologies and angles used throughout this analysis. The length of the jet lines indicates the energies. In the symmetric (Y) events (see Fig. \ref{['eventtopol']}b)) $\theta_2\approx\theta_3$.
• Figure 2: a) Average charged particle multiplicity for light quark and gluon jets as function of $\kappa$ fitted with Eqn. \ref{['eqn:offset_ansatz']}; b) ratio of the gluon to quark jet multiplicity; the full line shows the ratio of the functions fitted to the data in a), the dashed curve is the ratio of the slopes of the fits in a). All curves are extrapolated to the edges of the plot by the dotted lines. Also included are measurements of the multiplicity ratio of some other experiments cleoopal. The grey band shown with the slope ratio indicates the error estimated by varying all fit parameters within their errors.
• Figure 3: a)Variation of the scales $2E^*$ and $p_1^{\perp}$ as function of the opening angle $\theta_1$ in symmetric three-jet events. The functions are the analytic expectation. The points include a correction (calculated with JETSET 7.3) for the cases where the gluon forms the most energetic jet. The lines matching the points are polynomials fitted to obtain continuous values. b) Charged hadron multiplicity as a function of the centre-of-mass energy of the $q \bar{q}$-pair fitted with the perturbative predictions Eqs. \ref{['eqn:mul_webber']} or \ref{['eqn:mul_khoze']}. c) Charged hadron multiplicity in symmetric three-jet events as a function of the opening angle. The dashed curve is the prediction using the ansatz Eqn. \ref{['eqn:3mul']} setting $C_A/C_F$ to its default value and omitting the constant offset, $N_0$. The full curve is a fit of the full ansatz Eqn. \ref{['eqn:3mul']} to the data treating $C_A/C_F$ and $N_0$ as free parameters. d) Stability of the result for $C_A/C_F$ against variation of the smallest opening angle used in the fit as well as $\chi^2/N_{df}$ and the $\chi^2$ probability of these fits. The dash-dotted horizontal line in the upper half shows the QCD expectation for $C_A/C_F$ with the dotted lines representing variations of $\pm 10\%$. The DELPHI data of Fig. \ref{['result_plot']} b) and c) will be made available in the Durham/RAL database durhamdb.
• Figure 4: Comparison of the charged hadron multiplicity for an initial $\rm q\bar{q}$ and a $gg$ pair as function of the scale. The dashed curve is a fit according to Eqns. \ref{['eqn:mul_webber']} or \ref{['eqn:mul_khoze']}, the full line is twice the second term of Eqn. \ref{['eqn:3mul']}. The grey band indicates the uncertainty due to the error of $N_0$. The DELPHI $gg$ data will be made available in the Durham/RAL database durhamdb.