Cover Your Bases: Asymptotic Distributions of the Profile Likelihood Ratio When Constraining Effective Field Theories in High-Energy Physics

TL;DR

This paper studies the asymptotic behavior of the profile likelihood ratio in constraining effective field theories in high-energy physics, revealing deviations from Wilks' theorem and providing new methods for accurate statistical inference.

Contribution

It derives the correct asymptotic distributions of the PLR for EFTs with linear and quadratic couplings, and introduces a numerical approach for complex cases.

Findings

Wilks' theorem often violated in EFT constraints

Derived asymptotic distributions for linear/quadratic EFT couplings

Provided a numerical method for complex PLR distributions

Abstract

We investigate the asymptotic distribution of the profile likelihood ratio (PLR) when constraining effective field theories (EFTs) and show that Wilks' theorem is often violated, meaning that we should not assume the PLR to follow a $\chi^2$-distribution. We derive the correct asymptotic distributions when either one or two real EFT couplings modulate observable cross sections with a purely linear or quadratic dependence. We then discover that when both the linear and quadratic terms contribute, the PLR distribution does not have a simple form. In this case we provide a partly-numerical solution for the one-parameter case. Using a novel approach, we find that the constants which define our asymptotic distributions may be obtained experimentally using a profile of the Asimov likelihood contour. Our results may be immediately used to obtain the correct coverage when deriving real-world EFT constraints using the PLR as a test-statistic.