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

by Florian U. Bernlochner, Daniel C. Fry, Stephen B. Menary, Eric Persson

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Submission summary

Authors (as registered SciPost users): Florian Bernlochner
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-11-23
Date submitted: 2022-08-30 14:06
Submitted by: Bernlochner, Florian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Experiment
  • High-Energy Physics - Phenomenology
Approaches: Experimental, Phenomenological


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.

Published as SciPost Phys. Core 6, 013 (2023)

Reports on this Submission

Anonymous Report 1 on 2022-10-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2207.01350v2, delivered 2022-10-19, doi: 10.21468/SciPost.Report.5932


1- clarity


1- limited originality and applicability


The Asymptotic distribution of the Profile Likelihood Ratio (PLR) for quadratic dependence on the Wilson coefficient is obviously identical to the one derived (in half page) in Ref [12] of the manuscript, corresponding to the case in which the parameter of interest (\mu, in the notation of Ref [12]) is positive, by the identification \mu=c^2. The distribution in the case of linear dependence is the textbook Chi^2 result by Wilks. The two-parameters solutions described in Section 4 are original, as far as I can tell, but they constitute a rather trivial generalisation.

The most interesting part of the paper is Section 5, that identifies a strategy for the calculation of the Asymptotic distribution in the general case where both linear and quadratic terms contribute. However the study is limited to one single Wilson coefficient, making the resulting algorithm hardly useful in real EFT fits. Furthermore, computing the distribution in the idealised setup considered in the manuscript (Gaussian-distributed measurements of binned cross-sections) by pseudo-experiments is extremely fast as the Toy data consist in a bunch of Gaussians and the maximisation of the Likelihood a simple quadratic problem. Therefore it is unclear that an alternative strategy to compute the distribution along the line of the manuscript would be of practical relevance.

  • validity: high
  • significance: low
  • originality: low
  • clarity: top
  • formatting: excellent
  • grammar: -

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