Gaining analytic control of parton showers

TL;DR

<3-5 sentence high-level summary> The paper addresses the loss of analytic control in standard parton showers caused by enforcing four-momentum conservation at each vertex. It introduces an analytic parton shower that conserves momentum at every vertex while preserving a closed-form probability distribution, enabling exact event weights and post-generation reweighting. This framework facilitates direct matching to matrix elements, systematic inclusion of power corrections, and efficient uncertainty estimation, with practical implementation details demonstrated in Sherpa. The approach yields improved control over the IR structure and provides flexible tools for tuning and updating simulations without re-simulation of detector effects.

Abstract

Parton showers are widely used to generate fully exclusive final states needed to compare theoretical models to experimental observations. While, in general, parton showers give a good description of the experimental data, the precise functional form of the probability distribution underlying the event generation is generally not known. The reason is that realistic parton showers are required to conserve four-momentum at each vertex. In this paper we investigate in detail how four-momentum conservation is enforced in a standard parton shower and why this destroys the analytic control of the probability distribution. We show how to modify a parton shower algorithm such that it conserves four-momentum at each vertex, but for which the full analytic form of the probability distribution is known. We then comment how this analytic control can be used to match matrix element calculations with parton showers, and to estimate effects of power corrections and other uncertainties in parton showers.

Figures (6)

• Figure 1: Diagrammatic representation of a tree of branches.
• Figure 2: Diagrammatic representation of a standard parton shower algorithm. Solid lines represent off-shell partons with nonzero invariant mass, dashed lines unbranched, on-shell partons.
• Figure 3: Diagrammatic representation of the analytic algorithm. Solid lines represent off-shell partons with nonzero invariant mass, dashed lines unbranched, on-shell partons. The evolution of $t_L$ in step 1 starts at $t_\mathrm{ini} = t_0$, and that of $t_R$ in step 2 at $t_\mathrm{ini} = (\sqrt{t_0} - \sqrt{t_L})^2$.
• Figure 4: Comparison of the analytic parton shower [medium (orange) dots] and Sherpa's parton shower [dark (blue) triangles]. Left: The function $\bar{P}(t)$ as defined in Eq. \ref{['Pbar_def']}. The solid (orange) line shows the result obtained from Eq. \ref{['Pdouble']}, and the gray shading gives an estimate of the expected size of power corrections. Right: The integrated thrust distribution, where the gray shading shows the expected size of finite perturbative corrections. The error bars show the statistical uncertainties.
• Figure 5: Pull distribution for the double branch probability $P(t_L,t_R)$ defined in Eq. \ref{['Pdouble_t']}. See text for further explanation.