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A thrust to trust minimum thrust

Matteo Cacciari

TL;DR

The paper investigates the minimum thrust $T_N$ for $N$ momentum configurations under momentum conservation, equivalently maximizing $\tau_N^{\max} = 1-T_N^{\min}$. It combines an exact thrust computation based on sign-pattern maximization with stochastic global optimisers (Differential Evolution and CMA-ES) for the external momentum search, and applies Extreme Value Theory with MLE and Bayesian analysis to handle cases where the true maximum is not found. The authors report exact results for $N=3,4$ in 3D and for $N=5$, and provide numerical and EVT-based bounds for larger $N$ up to $N=20$ across 3D, 2D, and higher dimensions, noting that some maximal configurations are less isotropic than naively expected. This work delivers the best current estimates for $\tau_N^{\max}$ and demonstrates a robust statistical framework to validate maxima, informing event-shape analyses in collider physics.

Abstract

We determine the minimum value of thrust for a number of N-particle configurations. For N=5 in three dimensions an exact result is found for the first time. For larger N we obtain numerical results through optimisation. When a definite value cannot be reliably identified, the results are analysed in the context of Extreme Value Theory, using a Maximum Likelihood Estimate and a Bayesian analysis. Results are given for three spatial dimensions, two dimensions, and selected cases in d dimensions.

A thrust to trust minimum thrust

TL;DR

The paper investigates the minimum thrust for momentum configurations under momentum conservation, equivalently maximizing . It combines an exact thrust computation based on sign-pattern maximization with stochastic global optimisers (Differential Evolution and CMA-ES) for the external momentum search, and applies Extreme Value Theory with MLE and Bayesian analysis to handle cases where the true maximum is not found. The authors report exact results for in 3D and for , and provide numerical and EVT-based bounds for larger up to across 3D, 2D, and higher dimensions, noting that some maximal configurations are less isotropic than naively expected. This work delivers the best current estimates for and demonstrates a robust statistical framework to validate maxima, informing event-shape analyses in collider physics.

Abstract

We determine the minimum value of thrust for a number of N-particle configurations. For N=5 in three dimensions an exact result is found for the first time. For larger N we obtain numerical results through optimisation. When a definite value cannot be reliably identified, the results are analysed in the context of Extreme Value Theory, using a Maximum Likelihood Estimate and a Bayesian analysis. Results are given for three spatial dimensions, two dimensions, and selected cases in d dimensions.

Paper Structure

This paper contains 11 sections, 17 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The numerical set-up
  3. The calculation of $T_N$
  4. The calculation of $\tau^{max}_N$
  5. Statistical analysis and results
  6. Conclusions
  7. The exact thrust algorithm
  8. Minimum thrust for $N=5$ particles in three dimensions
  9. Tests of EVT
  10. Thrust in two dimensions
  11. Thrust in $d$ dimensions

Figures (2)

  • Figure 1: Observed and posterior predictive distribution for the maxima (top row), and prior and posterior distribution for the endpoint of the inferred Weibull distribution (bottom row). We have used 1000 runs of the CMA-ES minimiser for $N=11$, 100 runs of DE for $N=12$, and 500 runs of CMA-ES for $N=15$.
  • Figure 2: Configuration of minimum thrust for five particles. The second and third plot show a front and a top view respectively. The same configuration gives the same valua for the thrust with many other thrust axis directions.