AI-Driven Discovery of Information-Efficient Collider Observables for Interference Measurements

This paper demonstrates that AI-driven symbolic evolution can discover compact, interpretable event-level observables for collider interference measurements that retain substantially more statistical information than standard angular baselines by leveraging matrix-element reweighting scores.

The Gist

Imagine you are a detective trying to solve a very subtle mystery at a massive particle collider. The "mystery" is a tiny, hidden flaw in the Standard Model of physics—a slight twist in how the Higgs boson interacts with other particles. This twist is so small that it's like trying to hear a whisper in a hurricane.

In the past, physicists tried to catch this whisper by building specific, hand-crafted "microphones" (mathematical formulas) based on their intuition. Sometimes these worked, but often they missed the most important clues because the clues were hidden in complex patterns that human brains couldn't easily map out.

This paper introduces a new method: AI-driven symbolic discovery. Think of this as hiring a super-smart, creative AI detective that doesn't just listen to the noise, but actually writes its own microphones from scratch.

Here is how the paper breaks down, using simple analogies:

1. The Problem: The "Perfect" Microphone is Too Complicated

In physics, there is a theoretical "perfect" way to measure a tiny change. It's called the score function. Imagine this as the ultimate, crystal-clear recording of the whisper. However, this perfect recording is usually a messy, impossible-to-read mathematical monster. It's too complex to use in a real experiment.

Physicists usually have to settle for "good enough" microphones (simple angles and shapes) that are easy to understand but miss a lot of the whisper's details. They lose information in the process.

2. The Solution: AI as a Creative Architect

The authors used an AI system (specifically a type of evolutionary search) to act as an architect.

• The Goal: The AI was told, "Build a simple, readable mathematical formula that captures as much of the 'whisper' (the score) as possible."
• The Process: The AI didn't just guess. It started with basic building blocks (like sine waves, angles, and energy levels) and evolved them over thousands of generations, like natural selection. It kept the formulas that were better at catching the whisper and discarded the ones that weren't.
• The Result: The AI didn't just give a "black box" answer (like a neural network that says "I know the answer" but can't explain why). Instead, it produced compact, readable formulas that humans can actually read and understand.

3. The Two Test Cases: Two Different Rooms

The team tested this AI architect in two different "rooms" (collider scenarios) to see if it could find the same hidden pattern in different settings:

• Room A (The Electron Collider): They looked at particles smashing together to create a Higgs and a Z boson.

  • The Old Way: Physicists used a simple angle measurement (like looking at the angle of a spinning top). It caught about 6% of the available information.
  • The AI Way: The AI discovered a new formula that combined angles with the energy differences of the particles. It caught about 10% of the information.
  • The Analogy: The old microphone was like listening to a song with one ear. The AI built a new microphone that used both ears and adjusted for the room's echo, making the whisper much clearer.

• Room B (The Proton Collider): They looked at the Higgs boson decaying into four particles (electrons and muons).

  • The Old Way: The standard method was so weak it only caught a tiny fraction of the signal (0.02%). It was like trying to find a needle in a haystack with a blindfold.
  • The AI Way: The AI found a formula that organized the chaos, separating the "whisper" from the noise much more effectively, boosting the information catch to 1.9%.
  • The Analogy: The AI didn't just find a better needle; it figured out a way to sort the haystack so the needle stood out on its own.

4. What Did the AI Actually Find?

The most exciting part is what the AI wrote down. It didn't invent random math; it rediscovered the physics in a new way.

• The Core Pattern: In both rooms, the AI found a specific "rhythm" or "harmonic" in the data. This rhythm corresponds to the interference of particle spins (helicity). It's like finding the specific beat in a song that proves the singer is off-key.
• The "Wrapper": The AI added extra "wrappers" to this rhythm. In the first room, it wrapped the rhythm in a "laboratory map" (using energy differences) to make it easier to read. In the second room, it wrapped it in a "mass ratio" (a smooth factor based on particle weights) to stabilize the measurement.

5. The Big Takeaway

The paper claims that we can stop trying to guess the perfect formula by hand. Instead, we can treat the search for the best measurement tool as a symbolic discovery problem.

• Before: "I think the answer is this complicated equation."
• Now: "Let the AI explore the space of all possible simple equations, find the one that listens best to the data, and write it down in plain English (math)."

The result is a set of transparent, human-readable formulas that are much better at spotting the tiny, subtle deformations in physics than the old methods. It bridges the gap between the raw power of AI and the need for human understanding in science.

In short: The AI acted as a translator, turning the messy, high-dimensional "noise" of particle collisions into a clean, simple, and powerful mathematical sentence that tells us exactly where to look for new physics.

Technical Summary

Technical Summary: AI-Driven Discovery of Information-Efficient Collider Observables for Interference Measurements

Problem Statement
Future collider programs are entering a precision regime where the central challenge is discriminating subtle deformations of the Standard Model (SM) using high-dimensional event information. In this setting, sensitivity is often governed by quantum interference. While the optimal event-level probe is theoretically the score function (the derivative of the log-likelihood with respect to the parameter of interest), which saturates the Fisher-information bound, this score is rarely available in a compact, analytic form that is transparent, portable, and experimentally robust.

Current approaches face a trade-off:

1. Analytic observables: Physically interpretable and experimentally stable but often constructed by hand, potentially missing important correlations across phase space.
2. Machine Learning (ML) approaches: Can approach optimal statistical performance by exploiting high-dimensional features but typically rely on "black-box" architectures whose learned structures are difficult to interpret physically.

The goal is to construct compact, interpretable observables that retain parameter-relevant score information while remaining analytic and experimentally portable.

Methodology
The authors propose a framework that reformulates optimal-observable design as a symbolic discovery problem. The methodology employs AI-driven symbolic evolution to discover analytic surrogates for performant observables.

• Statistical Target: The optimization target is the local score function, estimated numerically via matrix-element reweighting. For a deformation parameter $\kappa$, the event weight is expanded as $w(x|\kappa) = c_0(x) + \kappa c_1(x) + \kappa^2 c_2(x)$. The locally optimal statistic is the ratio $t(x) = c_1(x)/c_0(x)$, which coincides with the score up to an $x$-independent shift.
• Search Strategy: An evolutionary search (guided by a Large Language Model for proposal generation, mutation, and crossover) explores a physics-motivated function space. This space is built from measured kinematic variables and simple analytic operations (sums, products, trigonometric functions, etc.).
• Fitness Function: The search maximizes the Fisher-information efficiency, defined as $\epsilon[f] = I[f] / I_{\text{full}}$, where $I[f]$ is the retained Fisher information after compressing the event kinematics into a one-dimensional observable $y=f(x)$.
• Constraints: The search does not assume analytic amplitude input; it uses the score only as a target direction. The resulting expressions are constrained to be compact analytic functions.

Key Contributions and Results
The study focuses on the CP-sensitive interaction $HZ_{\mu\nu}\tilde{Z}^{\mu\nu}$ and evaluates two distinct collider realizations:

1. Associated Production: $e^+e^- \to Z(\to \mu^-\mu^+)H$ at a lepton collider.
2. Decay Channel: $pp \to H \to ZZ^* \to e^-e^+\mu^-\mu^+$ at a hadron collider.

Findings for Associated Production ($e^+e^- \to ZH$):

• Baseline: The best simple angular baseline, $\sin(2\phi^*)$, achieves a Fisher efficiency of $\epsilon \simeq 0.059$.
• Discovered Observable: The symbolic search identified a compact analytic function $f_{\text{ext}}$ that combines:
  • A CP-sensitive interference harmonic (transverse-transverse and longitudinal-transverse).
  • Laboratory-frame asymmetry mappings (energy asymmetry $A_E$ and transverse-momentum asymmetry $A_{pT}$) which act as proxies for decay polar factors.
  • A smooth kinematic prefactor involving the $Z$ boson transverse momentum, $(p_T^Z)^2 / ((p_T^Z)^2 + p_0^2)$.
• Performance: The learned observable achieves $\epsilon \simeq 0.102$, representing a 74% improvement in information efficiency over the best angular baseline. The expression recovers characteristic helicity-interference harmonics and follows the score more monotonically.

Findings for Four-Lepton Decay ($H \to 4\ell$):

• Baseline: The standard angular baseline $\sin(\phi_1^* + \phi_2^*)$ has a very low efficiency of $\epsilon \simeq 2.3 \times 10^{-4}$.
• Discovered Observable: The search identified $O_{4\ell} = r_Z(1-r_Z) \sin(2\theta_1^*) \sin(2\theta_2^*) \sin(\phi_1^* + \phi_2^*)$, where $r_Z$ is a mass-ratio factor.
• Performance: The efficiency improves to $\epsilon \simeq 1.9 \times 10^{-2}$. The angular kernel dominates the information retention, while the mass-ratio factor serves as a bounded representative prefactor.
• Asymmetry: Applying a cut around $|O_{4\ell}| \simeq 0.05$ improves the asymmetry-based significance measure ($S_{\text{asym}}/\sqrt{D_{\text{SM}}}$) by approximately 30%.

General Structure of Discovered Observables
In both channels, the discovered observables follow a schematic structure:
$$f(x) \sim (\text{Interference Kernel}) \times (\text{Process-Dependent Mapping or Prefactor})$$
The "Interference Kernel" captures the CP-sensitive helicity interference geometry, while the "Mapping/Prefactor" utilizes process-specific kinematic variables (like lab-frame asymmetries or mass ratios) to stabilize the projection or enhance separation of interference regions.

Significance and Claims
The paper claims to demonstrate that:

1. Symbolic Discovery: Optimal-observable design can be successfully reformulated as a symbolic discovery problem, where interpretable analytic expressions emerge directly from the geometry of interference in phase space.
2. Information Efficiency: AI-driven symbolic evolution can discover observables that retain substantially more local Fisher information than standard angular baselines while remaining compact and analytic.
3. Transparency: The resulting expressions provide a transparent route to information-efficient probes, revealing the underlying interference geometry in analytic form. This bridges the gap between traditional analytic design and modern data-driven optimization.
4. Scope: The results presented are truth-level local Fisher benchmarks. The authors explicitly state that incorporating parton showers, detector response, and continuum backgrounds, as well as extending the search to likelihood-oriented objectives, are natural next steps not covered in this work.

The work suggests that the structure of statistically efficient measurements can itself be discovered and interpreted as a physical object, offering a systematic route toward information-efficient observables in precision collider studies.