An Optimal Observable Machine for reinterpretable measurements in high-energy physics

TL;DR

The paper addresses the challenge of obtaining high-precision, unfolding-friendly observables for parameter extraction in high-energy physics. It introduces the Optimal Observable Machine (OOM), which learns generator-level distributions $\frac{\mathrm{d}\sigma}{\mathrm{d}\mathcal{O}}$ and detector-level distributions $x_{\mathcal{O}}$ through differentiable mappings that are optimized via a likelihood-based loss incorporating detector response $R$ and nuisance parameters $\omega$, with precision quantified by $\Delta c=\sqrt{H_{cc}^{-1}}$. The approach is demonstrated in a top-quark toponium context, where a pseudoscalar excess near the $t\bar t$ threshold is used to constrain the signal strength $r$ by jointly training generator- and detector-level observables; a crucial enhancement is the introduction of a response-matrix constraint parameterized by $\lambda$ to mitigate $c$-dependent unfolding biases. The results show improved sensitivity and controlled bias, offering a path toward long-term reinterpretability of unfolded results and broad applicability to precision measurements and new-physics searches, with potential future work in deriving analytic forms via symbolic regression.

Abstract

A machine-learning-based framework for constructing generator-level observables optimized for parameter extraction in particle physics analyses is introduced, referred to as the Optimal Observable Machine (OOM). Unfoldable differential distributions are learned that maximize sensitivity to a parameter of interest while remaining robust against detector effects, systematic uncertainties, and biases introduced by the unfolding procedure. Detector response and systematic uncertainties are explicitly incorporated into the training through a likelihood-based loss function, enabling a direct optimization of the expected measurement precision while minimizing the bias from any assumption on the parameter of interest itself. The approach is demonstrated in an application to top quark physics, focusing on the measurement of a recently observed pseudoscalar excess at the top quark pair production threshold in dilepton final states. It is shown that a generator-level observable with enhanced sensitivity and long-term reinterpretability can be constructed using this method.

Figures (5)

• Figure 1: Results of different detector-level MLP training strategies. Left: Negative log-likelihood scans for the toponium signal strength $r$ using CE training with three input variables (blue), CE training with 29 input variables (green), and systematic-uncertainty-aware training with 29 input variables (yellow). Shown are the scans using only statistical uncertainties (triangles) and using both statistical and systematic uncertainties (lines). Right: Feature importance ranking, evaluated using Ref. Wunsch:2018oxb, for the ten most important input features, compared between the CE training (green) and the systematic-uncertainty-aware training (yellow), both using 29 input variables.
• Figure 2: Response matrix $R$ (main panel, with generator level on the $x$ axis and detector level on the $y$ axis), normalized distribution of $\mathrm{d}\sigma/\mathrm{d}\mathcal{O}\xspace$ at generator level (upper panel), and normalized distribution of $x_{\mathcal{O}\xspace}$ at detector level (right panel) for the OOM training without response-matrix constraint.
• Figure 3: Comparison of the ${ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace{ \mathup{{ \overline{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}}}{} _{ {}} ^{ {}}} }\xspace$ (left, blue) and $\mathup{{{\eta}}{} _{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}} ^{ {}}}$ (right, red) yields between $x_{\mathcal{O}\xspace}$ (dashed) and the folded distribution $R\,\mathrm{d}\sigma/\mathrm{d}\mathcal{O}\xspace\xspace$ (solid). The lower panel displays the ratio between the two distributions. Shown are the results for the training without response-matrix constraint, i.e., with $\lambda=0$. The captions list the Kolmogorov--Smirnov distance values.
• Figure 4: Comparison of the ${ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace{ \mathup{{ \overline{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}}}{} _{ {}} ^{ {}}} }\xspace$ (left, blue) and $\mathup{{{\eta}}{} _{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}} ^{ {}}}$ (right, red) yields between $x_{\mathcal{O}\xspace}$ (dashed) and the folded distribution $R\,\mathrm{d}\sigma/\mathrm{d}\mathcal{O}\xspace\xspace$ (solid). The lower panel displays the ratio between the two distributions. Shown are the results for the training using the response-matrix constraint with $\lambda=0.25$. The captions list the Kolmogorov--Smirnov distance values.
• Figure 5: Results for different values of the response-matrix constraint strength $\lambda$. The violin plots show the distribution of the 50 trainings per $\lambda$ value, the dots the median value, and the error bars the 25 and 75% quantiles. Left: Kolmogorov--Smirnov distance for the agreement between $x_{\mathcal{O}\xspace}$ and the folded distribution $R\,\mathrm{d}\sigma/\mathrm{d}\mathcal{O}\xspace\xspace$ shown for the $\mathup{{{\eta}}{} _{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}} ^{ {}}}$ (right) yields. Right: Uncertainty in the $\mathup{{{\eta}}{} _{ {{ \mathup{{{t}}{} _{ {}} ^{ {}}} }\xspace}} ^{ {}}}$ signal strength. The dotted red line indicates the uncertainty as obtained by the systematic-uncertainty-aware training shown in Fig. \ref{['fig:detectorlevelresults']}.