[Paper Review] Propagation of Uncertainty in a Parton Shower

Philip Stephens, A. van Hameren|ArXiv.org|Mar 22, 2007

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper introduces a method to propagate uncertainties through parton shower Monte Carlo simulations by computing alternate event weights without modifying the underlying code. The technique enables systematic study of how variations in key components—such as the running coupling or splitting kernel—affect predictions, with numerical validation showing measurable differences in observables like thrust.

ABSTRACT

Presented here is a technique of propagating uncertainties through the parton shower by means of an alternate event weight. This technique provides a mechanism to systematically quantify the effect of variations of certain components of the parton shower leading to a novel approach to probing the physics implemented in a parton shower code and understanding its limitations. Further, this approach can be applied to a large class of parton shower algorithms and requires no changes to the underlying implementation.

Motivation & Objective

• To develop a systematic method for quantifying uncertainties in parton shower Monte Carlo predictions due to variations in physics inputs.
• To enable analysis of how changes in the running coupling, splitting functions, or kinematic ordering affect final-state observables without rewriting the shower code.
• To provide a framework for comparing different parton shower implementations by mapping their kinematic definitions via reweighting.
• To offer a tool for probing the limitations of existing parton shower algorithms in a model-independent way.

Proposed method

• The method uses variational calculus to compute small changes in the parton shower probability density function (PDF) due to perturbations in functional components like the running coupling or splitting kernel.
• It derives an analytical expression for the event weight as a ratio of modified to original shower probabilities, given by $ w = ig(1 + rac{ar{ ho}_R}{ ho_R}ig) imes ext{Sudakov factor} $, where $ ar{ ho}_R $ is the variation in the real emission density.
• The weight is decomposed into real emission and Sudakov components, allowing separate treatment of emission and virtuality evolution.
• The approach is applied to a toy model of $ e^+e^- \to q\bar{q} $ with final-state radiation, using Pythia-like and Herwig-like kinematic definitions.
• Reweighting is performed by comparing the Sudakov factor and real emission weights between two shower implementations with different kinematic ordering conventions.
• The method is validated numerically using the thrust observable, comparing the original shower with the reweighted version to quantify deviations.

Experimental results

Research questions

• RQ1 How can uncertainties in the running coupling or splitting function be systematically propagated through a parton shower without altering the event generation code?
• RQ2 To what extent do different kinematic ordering schemes (e.g., virtuality vs. angle) in parton showers affect final-state observables like thrust?
• RQ3 Can reweighting between two distinct parton shower implementations reveal systematic differences due to their underlying kinematic definitions?
• RQ4 How well does the variational method capture the effects of next-to-leading-logarithmic corrections in the shower evolution?

Key findings

• The reweighting technique successfully reproduces differences in the thrust distribution between Pythia-like and Herwig-like shower implementations, with deviations visible but moderate at $ \sqrt{s} = 1~\text{TeV} $.
• The method allows for the calculation of alternate event weights that reflect changes in the running coupling or splitting kernel, enabling sensitivity studies without re-running the full shower.
• The Sudakov factor is computed over the full evolution scale, not just between emissions, improving accuracy in the weight calculation.
• The approach is fully compatible with existing Monte Carlo codes and requires no changes to the underlying physics or event generation logic.
• The technique reveals that kinematic ordering conventions—such as virtuality or angle—can lead to observable differences in event shapes, even when the underlying physics is the same.
• The method provides a practical tool to assess the robustness of Monte Carlo predictions and to understand the limitations of current shower implementations.