Post-hoc reweighting of hadron production in the Lund string model

TL;DR

This work introduces a post-hoc reweighting framework for flavor selection in the Lund string model, enabling exact parameter variation on pre-generated events via event weights $w$. Two prescriptions, analytic and stochastic, are developed to handle reweighting through hadronization’s hierarchical Markov process, including rejection steps and diquark/baryon production. The approach is validated in Pythia8 across multiple observables, showing agreement with direct simulations while offering substantial timing gains for uncertainty estimation and tuning. By treating hadronization as a modular, transferable process, the method promises broader applicability and potential gradient-based optimization within high-energy physics and beyond.

Abstract

We present a method for reweighting flavor selection in the Lund string fragmentation model. This is the process of calculating and applying event weights enabling fast and exact variation of hadronization parameters on pre-generated event samples. The procedure is post hoc, requiring only a small amount of additional information stored per event, and allowing for efficient estimation of hadronization uncertainties without repeated simulation. Weight expressions are derived from the hadronization algorithm itself, and validated against direct simulation for a wide range of observables and parameter shifts. The hadronization algorithm can be viewed as a hierarchical Markov process with stochastic rejections, a structure common to many complex simulations outside of high-energy physics. This perspective makes the method modular, extensible, and potentially transferable to other domains. We demonstrate the approach in Pythia, including both numerical stability and timing benefits.

Figures (13)

• Figure 1: An illustration of the multi-step procedure required to go from strings to hadrons. Reading from bottom to top, a parton-level interaction (blue circle) produces a color singlet quark--antiquark system (lines with arrows). Additional (di)quark-anti(di)quark pairs (colored circles) are produced in the color field (brown area) of the original pair through a vacuum tunneling mechanism. The string is broken into smaller pieces by intersecting combinations of flavored partons. The string pieces will in turn be mapped to hadrons, with species selection steered by Clebsch--Gordan weights and model parameters. Note that the $u\xspace\bar{u\xspace}\xspace$ piece in the upper left part of the figure is allowed to form a $\phi$-meson. With default Pythia8 parameters (almost ideal mixing in the vector meson nonet) this happens for less than 1% of the $u\xspace\bar{u\xspace}\xspace$ string pieces.
• Figure 2: Illustration of the decision flow in the simple case of string breaks without diquarks. There are three possibilities, $u\xspace\bar{u\xspace}\xspace$, $d\xspace\bar{d\xspace}\xspace$, and $s\xspace\bar{s\xspace}\xspace$, the latter with a probability $\rho/(2+\rho)$, and the two former with a probability of $1/2$, given that no $s\xspace\bar{s\xspace}\xspace$ pair was produced. All probabilities can be read off directly from the decision flow chart.
• Figure 3: Flowchart depicting the introduction of rejection/filtering in a simplified example beginning with an $s$-quark end-point. In this toy example, vector mesons and the $\eta'$ meson are not included, so a string break following an $s$-quark can only result in an $\eta$ (for $s\xspace\bar{s\xspace}\xspace$ breaks) or kaons (for $u\xspace\bar{u\xspace}\xspace$ or $d\xspace\bar{d\xspace}\xspace$ breaks). Because of this simplification, this example does not conserve isospin. However, the full reweighting implementation of this paper does.
• Figure 4: Sketch of the full flavor selection part of the string breaking algorithm, for the simple model for diquark production introduced in \ref{['sec:lund-model']}. Note that the actual algorithm implemented in Pythia8 differs from the one sketched here (see main text for details).
• Figure 5: Comparison of $\ln(x_p(K^\pm))$ distributions, where $x_p$ is the beam momentum fraction (see eq. (4) of ref. Skands:2014pea), shown as fractions of the total number of kaons in an event, when the parameter $\rho$ is (top) explicitly set to different values or (bottom) varied using different methods. In the top panel, the lower row shows the ratios of the distributions generated with various values of $\rho$ to that generated with $\rho=0.190$. In the bottom panel, the distributions labeled $e$ were generated with the value of the parameter $\rho$ explicitly set to (left) $0.108$ and (right) $0.217$. The distributions labeled $w'$ are all taken from the same sample generated with $\rho=\rho^\text{base}\xspace=0.190$ but with different sets of alternative event weights $w'$ corresponding to the alternative values of $\rho$. The bottom row shows the ratios of the latter distributions to the former. Statistical error bars shown.