How Invisible: Regressing The Key Model Parameter for Semi-visible Jet Searches

This paper introduces a robust regression model that utilizes high-level physics objects to precisely reconstruct the key parameter $r_{\mathrm{inv}}$ in semi-visible jet events, offering a superior alternative to analytical methods and a unified approach for enhancing sensitivity in both $s$-channel and $t$-channel dark sector searches.

The Gist

Imagine you are a detective trying to solve a crime at a massive, chaotic party (the Large Hadron Collider). The "criminals" are invisible particles called Dark Matter. Usually, when these particles are created, they vanish immediately, leaving behind nothing but a gap in the energy balance of the room.

But in this specific scenario, the criminals are wearing invisibility cloaks that are slightly see-through. They don't vanish completely; they leave behind a messy trail of "semi-visible" debris. Physicists call these Semi-Visible Jets (SVJs).

The problem is that the trail is messy. Sometimes the cloak is very thick (most of the energy is invisible), and sometimes it's thin (most of the energy is visible). The key to solving the case is figuring out exactly how thick the cloak is. In physics terms, this is a number called $r_{inv}$ (the invisible fraction).

The Old Way: Guessing with a Ruler

Previously, scientists tried to measure this "cloak thickness" using a simple formula, like trying to guess the weight of a suitcase by looking at how much it bends a scale.

• The Flaw: This method relied on a very strict assumption: that the invisible particles fly off in a perfectly straight line with the visible ones. In the chaotic environment of a particle collision, this is rarely true. It's like trying to measure the wind speed by watching a single leaf; if the leaf gets caught in a gust, your measurement is wrong.
• The Result: The old method was often inaccurate, giving a wide range of guesses rather than a precise answer.

The New Way: The "Super-Smart" AI Detective

This paper introduces a new tool: a Machine Learning (AI) model trained to be a much better detective. Instead of using a simple formula, the AI looks at the entire scene of the crime.

Here is how the new method works, using simple analogies:

1. The "Flashlight" (The ISR Photon)
To make the crime scene easier to see, the researchers focus on events where a bright "flashlight" (a high-energy photon) is shone into the room. In physics, this is called Initial State Radiation (ISR).

• Analogy: Imagine the criminals are running away in the dark. If someone shines a bright spotlight on them, you can see their path much more clearly. The spotlight pushes the criminals, making their movement more predictable.

2. The "Smart Eye" (The Neural Network)
The AI doesn't just look at one thing. It looks at the "flashlight," the two main trails of debris (jets), and the missing energy all at once.

• Analogy: A human detective might look at the footprints and guess the suspect's weight. The AI is like a detective who looks at the footprints, the wind direction, the time of day, the type of mud, and the suspect's gait all simultaneously to calculate the weight with incredible precision.

3. The Training (Practice Makes Perfect)
The researchers "trained" this AI using millions of simulated crime scenes. They showed it thousands of examples where they knew the exact "cloak thickness" ($r_{inv}$) and asked the AI to guess it.

• The Trick: They taught the AI to ignore the specific details of what the criminals were wearing (the specific mass of the particles) and focus only on the pattern of the chase. This makes the AI robust. Even if the criminals change their costumes (different particle masses), the AI still knows how to measure the cloak thickness.

Why This is a Big Deal

1. It's Much More Accurate
The paper shows that the AI can guess the "cloak thickness" with much higher precision than the old ruler method. It's the difference between guessing a person's height is "somewhere between 5 and 7 feet" versus "5 feet 10 inches."

2. It Unifies Two Different Crimes
Previously, scientists had to investigate two different types of "crimes" (called s-channel and t-channel) using completely different strategies.

• The Breakthrough: The AI found that the "pattern of the chase" looks almost the same for both types of crimes. This means scientists can now use one single tool to hunt for both types of Dark Matter production, making the search much more efficient.

3. It's Future-Proof
Because the AI only looks at the "big picture" (the high-level objects like jets and photons) and not the tiny, messy details inside them, it can be easily adapted to new theories. If new types of Dark Matter are discovered that have different internal structures, this AI can likely still solve the case without needing to be completely rebuilt.

The Bottom Line

This paper is about upgrading our detective toolkit. By using a smart AI trained on the "flashlight" events, physicists can now measure the invisible properties of Dark Matter with a precision that was previously impossible. It turns a blurry, confusing picture into a sharp, clear image, bringing us one step closer to understanding the invisible universe that makes up most of our reality.

Technical Summary

1. Problem Statement

Semi-visible jets (SVJs) are a distinctive collider signature arising from strongly interacting dark sectors (Dark QCD). In these scenarios, dark hadrons decay partially into stable dark matter (invisible) and partially into Standard Model (SM) particles (visible). The key phenomenological parameter governing this behavior is $r_{inv}$, which defines the fraction of dark hadrons decaying into invisible particles.

• The Challenge: Accurately reconstructing $r_{inv}$ from experimental data is difficult. Previous methods relied on an analytical reconstruction ($r_{inv}^{analytic}$) based on the assumption that invisible momentum aligns collinearly with the jet axis.
• Limitations of Current Methods: The analytical approach suffers from:
  • Poor resolution, especially at lower Initial State Radiation (ISR) transverse momentum ($p_T$).
  • Unphysical results (negative values or values $>1$).
  • Strong dependence on specific model assumptions (e.g., mediator mass, dark sector confinement scale).
  • Inability to easily unify searches across different production channels (s-channel resonant vs. t-channel non-resonant).

2. Methodology

The authors propose a Machine Learning (ML) regression model to reconstruct $r_{inv}$ directly from high-level physics objects, bypassing the need for low-level jet substructure information.

A. Data Generation & Simulation

• Framework: Monte Carlo simulations using MadGraph5_aMC@NLO, Pythia 8 (with the Hidden Valley module for dark sector dynamics), and Delphes 3 (CMS detector simulation).
• Signal Processes:
  • s-channel: $q\bar{q} \to Z' + \gamma_{ISR}$, followed by $Z' \to \chi\bar{\chi}$.
  • t-channel: $q\bar{q} \to \chi\bar{\chi} + \gamma_{ISR}$ mediated by a scalar $\Phi$.
• Background: $\gamma$+jets events, generated with a slicing strategy on the leading jet $p_T$ to ensure statistical precision in high-momentum regions.
• Training Target: The true event-level $r_{inv}$ is calculated using generator-level information:
  $$r_{inv} = \frac{N_{DM}}{2(N_{\pi_D} + N_{\rho_D})}$$
  where $N_{DM}$ is the number of dark matter particles and the denominator represents the total number of dark hadrons.

B. Input Features & Pre-processing

The model uses a Fully Connected Feed-Forward Neural Network.

• Inputs: High-level observables only (no substructure):
  • ISR Photon: $p_T, \eta, \phi$.
  • Leading Jet ($j_1$): $\eta, \phi, m$.
  • Sub-leading Jet ($j_2$): $p_T, \eta, \phi, m$.
  • Missing Transverse Momentum: $E_T^{miss}, \phi^{miss}$.
• Normalization: A physics-motivated normalization is applied where energy/momentum quantities are divided by the leading jet's $p_T$. This reduces dependence on the absolute mass scale of the mediator ($Z'$ or $\Phi$).
• Sampling: To handle the imbalance in $r_{inv}$ distribution, events are binned (0 to 1 in 20 bins), and oversampled/undersampled to ensure equal statistical weight across the $r_{inv}$ range.

C. Training Strategy

Four independent models were trained to cover different scenarios:

1. ISR Thresholds: $p_T^{ISR} > 150$ GeV (moderate boost) and $p_T^{ISR} > 500$ GeV (high boost).
2. Sample Composition:
   • Signal-only: To test ideal regression performance.
   • Signal + Background: To simulate realistic analysis conditions. For background events, $r_{inv}$ is assigned the value of the analytical reconstruction ($r_{inv}^{analytic}$) to maintain a continuous target variable.

3. Key Contributions

1. Superior Precision: Demonstrated that a neural network can reconstruct $r_{inv}$ with significantly higher precision than the analytical method, particularly in the low-ISR regime where analytical assumptions fail.
2. Robustness: The model is robust against variations in dark sector parameters (dark pion mass, confinement scale, vector meson probability) and mediator masses ($Z'$), provided the event topology (ISR + SVJ) remains consistent.
3. Channel Unification: Showed that a model trained exclusively on s-channel data generalizes effectively to t-channel production, despite the different underlying physics (scalar mediator vs. vector mediator). This suggests a unified search strategy is feasible.
4. Practical Applicability: The method relies solely on high-level objects, making it suitable for reinterpretation of existing data and applicable to various SVJ variations (e.g., those involving leptons or heavy-flavor quarks).

4. Results

• Regression Performance:
  • The ML model ($r_{inv}^{ML}$) produces a monotonic, ordered response to the true $r_{inv}$.
  • Resolution: The distribution of $r_{inv}^{ML}$ is much narrower than $r_{inv}^{analytic}$. For $p_T^{ISR} > 150$ GeV, the analytical method shows broad distributions and unphysical tails, whereas the ML method remains tightly clustered around the truth.
  • Correlation: Spearman correlation coefficients between true and predicted $r_{inv}$ reach 0.841 (Signal-only, 150 GeV) and 0.872 (Signal-only, 500 GeV).
• Robustness Tests:
  • Varying dark sector parameters (e.g., $m_{\pi_D}$ from 20 to 200 GeV, $\Lambda_D$ from 10 to 50 GeV) resulted in only mild changes in the regression output, confirming the model does not overfit specific dark shower details.
  • t-channel Transfer: When applied to t-channel samples (trained on s-channel), the model maintained high performance, correctly separating different $r_{inv}$ hypotheses.
• Search Sensitivity (Impact):
  • The authors evaluated the signal significance ($S/\sqrt{B}$) in a 2D plane of ($r_{inv}, m_{MAOS}$).
  • Table IV Results: The ML-based $r_{inv}$ provided massive improvements in sensitivity compared to the analytical method:
    • $r_{inv} = 0.1$: 20.7% improvement.
    • $r_{inv} = 0.5$: 81.8% improvement.
    • $r_{inv} = 0.9$: 146.8% improvement.

5. Significance

This work offers a paradigm shift in how SVJ searches are conducted at the LHC:

• Unified Search Strategy: By providing a robust estimator for $r_{inv}$ that works for both s-channel and t-channel, experimentalists can potentially combine these channels, significantly enhancing discovery potential.
• Model-Agnostic Reinterpretation: Since the model uses only high-level kinematic variables, it can be applied to reinterpret data for a wide range of dark sector models without retraining on specific low-level substructure features.
• Enhanced Sensitivity: The dramatic increase in $S/\sqrt{B}$, especially for high $r_{inv}$ scenarios (where invisible components dominate), opens up parameter space that was previously inaccessible or had low sensitivity using analytical methods.
• Future Outlook: The authors suggest this approach could be extended to other ISR objects (like jets) and combined with SVJ tagging algorithms to further refine dark sector searches.