SciPost Submission Page
Cover Your Bases: Asymptotic Distributions of the Profile Likelihood Ratio When Constraining Effective Field Theories in HighEnergy Physics
by Florian U. Bernlochner, Daniel C. Fry, Stephen B. Menary, Eric Persson
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Florian Bernlochner 
Submission information  

Preprint Link:  https://arxiv.org/abs/2207.01350v2 (pdf) 
Date accepted:  20221123 
Date submitted:  20220830 14:06 
Submitted by:  Bernlochner, Florian 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Experimental, Phenomenological 
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 partlynumerical solution for the oneparameter 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 realworld EFT constraints using the PLR as a teststatistic.
Published as SciPost Phys. Core 6, 013 (2023)
Reports on this Submission
Anonymous Report 1 on 20221019 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2207.01350v2, delivered 20221019, doi: 10.21468/SciPost.Report.5932
Strengths
1 clarity
Weaknesses
1 limited originality and applicability
Report
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 twoparameters 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 (Gaussiandistributed measurements of binned crosssections) by pseudoexperiments 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.