Characterizing the hadronization of parton showers using the HOMER method

TL;DR

This work extends the HOMER method to the hadronization of strings with arbitrary numbers of gluons, enabling extraction of the Lund string fragmentation function $f(z)$ from data using only observable information. It introduces a gluon-aware modification to the three-step HOMER workflow and a smearing procedure over fragmentation-history neighborhoods to cope with the non-bijective mapping between fragmentation chains and observable events. Across four increasingly complex string configurations, the study demonstrates that $f(z)$ can be recovered with percent-level fidelity in simpler cases and with about $5\%$ deviations in the most realistic multi-gluon scenario, provided that the smearing hyperparameter $\sigma_{\mathrm{s}}$ is optimally chosen. The approach offers a data-driven path to constrain hadronization directly from measurements, with potential extensions to unbinned observables and full parton-shower simulations in real data analyses.

Abstract

We update the HOMER method, a technique to solve a restricted version of the inverse problem of hadronization -- extracting the Lund string fragmentation function $f(z)$ from data using only observable information. Here, we demonstrate its utility by extracting $f(z)$ from synthetic Pythia simulations using high-level observables constructed on an event-by-event basis, such as multiplicities and shape variables. Four cases of increasing complexity are considered, corresponding to $e^+e^-$ collisions at a center-of-mass energy of $90$ GeV producing either a string stretched between a $q$ and $\bar{q}$ containing no gluons; the same string containing one gluon $g$ with fixed kinematics; the same but the gluon has varying kinematics; and the most realistic case, strings with an unrestricted number of gluons that is the end-result of a parton shower. We demonstrate the extraction of $f(z)$ in each case, with the result of only a relatively modest degradation in performance of the HOMER method with the increased complexity of the string system.

Figures (34)

• Figure 1: A string system at times $t = \{0, t_1, t_2, ...\}$, in three different configurations. The quark and anti-quark have momenta $\vec{p}_q$ and $\vec{p}_{\bar{q}}$, respectively. (a) A gluon with momentum $\vec{p}_g$ has enough energy that it will not be lost to the string before the system hadronizes. (b) The gluon loses all energy at a time between $t_2$ and $t_3$, resulting in a third string region in light green, parallel to the $q\bar{q}$ axis. (c) In the limit $E_g \rightarrow 0$ the gluon kink vanishes, and we are left with a normal $q\bar{q}$ string.
• Figure 2: Schematic from ref. Bierlich:2024xzg, detailing different components of a simulated run. String breaks are grouped into fragmentation chains, while collections of rejected and accepted fragmentation chains form fragmentation histories. Observable events are obtained from the last, accepted fragmentation chain. A collection of multiple events is a run.
• Figure 3: Flowchart of the modified Step 2 for the $\text{HOMER}$ method with gluons.
• Figure 4: Goodness-of-fit defined by \ref{['eq:chi2_gof']}, shown as a function of ${\sigma_{\text{s}}^{}}$ for the four different string scenarios with $N_\text{bins}\xspace = 50$.
• Figure 5: (left) Reweighted distributions for the fragmentation function averaged over all string break variables except $z$ and (right) fixing the transverse mass bin. All weights are from a model trained with the unbinned high-level observables for the fixed $qg\bar{q}$ scenario.