Observable Optimization for Precision Theory: Machine Learning Energy Correlators

Abstract

The practice of collider physics typically involves the marginalization of multi-dimensional collider data to uni-dimensional observables relevant for some physics task. In any cases, such as classification or anomaly detection, the observable can be arbitrarily complicated, such as the output of a neural network. However, for precision measurements, the observable must correspond to something computable systematically beyond the level of current simulation tools. In this work, we demonstrate that precision-theory-compatible observable space exploration can be systematized by using neural simulation-based inference techniques from machine learning. We illustrate this approach by exploring the space of marginalizations of the energy 3-point correlator to optimize sensitivity to the the top quark mass. We first learn the energy-weighted probability density from simulation, then search in the space of marginalizations for an optimal triangle shape. Although simulations and machine learning are used in the process of observable optimization, the output is an observable definition which can be then computed to high precision and compared directly to data without any memory of the computations which produced it. We find that the optimal marginalization is isosceles triangles on the sphere with a side ratio approximately $1:1:\sqrt{2}$ (i.e. right triangles) within the set of marginalizations we consider.

Figures (11)

• Figure 1: Schematic of the ML workflow which can explore the space of precision-compatible observables. One first learns an analytic surrogate of the multidimensional observable as a function of physical parameters such as couplings and masses, followed by solving the inverse problem of inferring those parameters given marginals of or uni-dimensional observables from those distributions. The latter step is where observable quality can be quantified by the fidelity of the inferred physical parameters.
• Figure 2: Comparison of the EEEC$_{\bm \phi}$ distribution learned by the DNN to that computed directly from Pythia data as a function of the perimeter of the triangle $\zeta_p = \zeta_1+\zeta_2+\zeta_3$ for 'equilateral' triangles with the asymmetry parameter $\delta_{a} \leq 0.02$. The EEEC is normalized to integrate to 1 in the range shown.
• Figure 3: Comparison of a representative two dimensional differential EEEC distribution as a function of $\zeta_p$ (triangle perimeter) and $\delta_{a}$ (asymmetry) between the learned EEEC$_{\bm \phi}$ and Pythia histograms. Histograms are normalized to integrate to 1 over the plotted region. We find good agreement.
• Figure 4: Comparing observables computed directly from Pythia data to marginals from the learnt EEEC. On the left, we plot the projected EEEC as a function of $\zeta_\text{max} = \zeta_3$, and on the right we plot the distribution of right-angled isosceles triangles. The learnt EEEC density model indeed produces an analytic surrogate that captures features of the underlying simulated data.
• Figure 5: Shape distributions learned by the normalizing flow. Left: Two dimensional marginal to asymmetry and perimeter $\zeta_p = \zeta_1 + \zeta_2 + \zeta_3$. Right: Shape from Eq. \ref{['eqn:shape']} with $(n_1, n_2, \delta_a) = (1, \sqrt{2}, 0.01)$. Both offer reasonable agreement with data, but the agreement with data is not as good as for the DNN (see Figs. \ref{['fig:perim_asym_2d']} and \ref{['fig:subshape2']}).