GenEvA (II): A phase space generator from a reweighted parton shower

TL;DR

GenEvA introduces a novel phase-space generator that uses an analytic parton shower to sample complete final-state phase space across multiplicities and flavors, then reweights events to arbitrary target distributions σ(Φ). The framework assembles a master weight w(Σ) with components for the shower probability P(Σ), a Lorentz-invariant Jacobian J(Σ), an overcounting factor ˆα(Σ), and the desired matrix element σ[Φ(Σ)], with a concrete ALPHA-based method to manage multiple shower histories. It demonstrates that LO and LO/LL (leading-log improved) matrix elements can be reproduced and extended within GenEvA, achieving competitive or superior efficiency compared with MadEvent, while inherently incorporating Sudakov resummation through the shower. The paper also details truncation and matching strategies that preserve probability and enable phase-space projection, paving the way for hadronic applications, heavy resonances, and deeper LL/SCET integrations in a unified, reweightable framework.

Abstract

We introduce a new efficient algorithm for phase space generation. A parton shower is used to distribute events across all of multiplicity, flavor, and phase space, and these events can then be reweighted to any desired analytic distribution. To verify this method, we reproduce the e+e- -> n jets tree-level result of traditional matrix element tools. We also show how to improve tree-level matrix elements automatically with leading-logarithmic resummation. This algorithm is particularly useful in the context of a new framework for event generation called GenEvA. In a companion paper [arXiv:0801.4026], we show how the GenEvA framework can address contemporary issues in event generation.

Figures (20)

• Figure 1: The simplest version of reweighting. Events are distributed according to a normalized probability distribution $\mathcal{P}(\Phi)$ and given weights $w(\Phi) = \sigma(\Phi)/\mathcal{P}(\Phi)$ such that the resulting weighted events are effectively distributed according to $\sigma(\Phi)$. In GenEvA, $\Phi$ corresponds to Lorentz-invariant phase space variables and $\sigma(\Phi)$ is a differential cross section.
• Figure 2: Reweighting using a different distribution variable ${\Sigma}$, where there is a one-to-one and onto map ${\Sigma} \to \Phi({\Sigma})$. Events are distributed according to a normalized probability distribution $\mathcal{P}({\Sigma})$, which together with the Jacobian $J({\Sigma})$ from the variable transformation ${\Sigma}\to \Phi({\Sigma})$ defines an effective probability distribution $\mathcal{P}(\Phi) \equiv \mathcal{P}({\Sigma}) J({\Sigma})$. The weights $w(\Phi)$ can now be defined analogously to Fig. \ref{['fig:reweighta']}.
• Figure 3: Reweighting when there are multiple values ${\Sigma}_i$ that map to the same point $\Phi({\Sigma}_i)$. If all ${\Sigma}_i$ that map to a given $\Phi$ are known, then there is again an effective probability distribution $\mathcal{P}(\Phi) \equiv \sum_i \mathcal{P}({\Sigma}_i) J({\Sigma}_i)$ that can be used to define an appropriate weight $w(\Phi)$. In GenEvA, the parton shower provides a $\mathcal{P}({\Sigma})$, and it is well-known that multiple parton shower histories ${\Sigma}_i$ can correspond to the same phase space point $\Phi({\Sigma}_i)$. For this reweighting strategy to be computationally feasible, there must be an efficient way to calculate the sum over all parton shower histories $\sum_i \mathcal{P}({\Sigma}_i) J({\Sigma}_i)$.
• Figure 4: The reweighting approach used by GenEvA when the map $\Phi({\Sigma})$ is not one-to-one. Instead of taking the event weight to depend on $\Phi$ only, the event weight $w({\Sigma})$ is effectively a function of ${\Sigma}$. Using the overcounting factor $\hat{\alpha}({\Sigma}_i)$, the desired distribution $\sigma(\Phi)$ is split up among the various ${\Sigma}_i$ that map to the same $\Phi$, including the Jacobian factor $J({\Sigma}_i)$. As long as $\sum_i \hat{\alpha}({\Sigma}_i) = 1$, the resulting events will be distributed according to $\sigma(\Phi)$ without the need to calculate $\mathcal{P}({\Sigma}_i) J({\Sigma}_i)$ for every ${\Sigma}_i$. In practice, there is a small computational cost to make sure that $\hat{\alpha}({\Sigma}_i)$ is properly normalized.
• Figure 5: The structure of the parton shower. A solid line indicates an already processed particle, while a dashed line indicates a particle that has not yet been processed. A dot at the end of a solid line indicates a particle that did not branch above the shower cutoff, i.e. a final state particle. Starting from an existing branch $M \to LR$, where a mother particle $M$ has branched into daughter particles $L$ and $R$, GenEvA's parton shower generates the double branch $M \to LR \to (LL/LR)(RL/RR)$, where it is understood that there is also a finite probability for $L$ or $R$ to remain unbranched. Understanding how the $L$ and $R$ branches are coupled is essential for being able to exactly compute the double-branch probability and to truncate the parton shower.