=== SUMMARY ===

Technical Summary: Unbinning Global LHC Analyses

Problem Statement

The extraction of information from high-dimensional event data at the Large Hadron Collider (LHC) has become a bottleneck for precision physics. Traditional inference methods rely on reducing data to few-dimensional histograms, a process that discards significant kinematic information through dimensionality reduction and binning. While Neural Simulation-Based Inference (SBI) has demonstrated superior performance over histogram-based methods for individual processes, it remains unclear whether these advantages persist in global analyses where multiple processes are combined to constrain Standard Model Effective Field Theory (SMEFT) parameters. This paper addresses the gap in understanding how SBI performs when applied to a global combination of di-boson processes, specifically investigating whether the ability to disentangle degenerate theory parameters extends beyond single-process studies.

Methodology

The authors employ a neural simulation-based inference framework to estimate likelihood ratios without binning, comparing these results against traditional histogram-based limits and total rate constraints.

Likelihood Learning via Derivative Learning:
Instead of learning the likelihood ratio directly at specific benchmark points, the authors utilize a derivative-learning approach. Given the SMEFT Lagrangian structure $L_{SMEFT} = L_{SM} + \sum_i \theta_i O_i$, where $\theta_i$ are Wilson coefficients, the differential cross-section ratio $R(x|\theta, \theta_0)$ is expanded to second order in $\theta$. The neural network is trained to learn the coefficients of this expansion ($R_i(x)$ and $R_{ij}(x)$) using parton-level ratios as targets. This allows the reconstruction-level likelihood ratio to be reconstructed for any $\theta$ via:
$$r(x|\theta, \theta_0) \approx 1 + (\theta - \theta_0)_i R_{\phi,i}(x) + (\theta - \theta_0)_i (\theta - \theta_0)_j R_{\phi,ij}(x)$$
where $R_{\phi}$ represents the neural network estimators.

Background Handling:
To account for backgrounds, the authors split the squared matrix element into signal, background, and interference components. Since interference is negligible for the considered processes, the likelihood is modeled as a sum of signal and background probabilities. A signal-background classifier is trained to estimate the ratio $\omega(x) = p_{sig}(x)/p_{bkg}(x)$, which is then used to construct the full likelihood ratio. To ensure the likelihood minimum remains at the reference hypothesis ($\theta_0$) despite numerical imperfections, the authors analytically enforce the vanishing of the first derivative of the log-likelihood at $\theta_0$ by adjusting the signal cross-section derivative terms.

Fractional Smearing:
To mitigate numerical instability caused by outlier events with large derivatives, a "fractional smearing" technique is applied. Events with large derivatives are passed through the parton shower and detector simulation multiple times ($n$), with each resulting event assigned a weight of $1/n$. These weights are incorporated into the Mean Squared Error (MSE) loss function during training.

Analysis Setup:
The study focuses on four di-boson processes: $WW$, $WZ$, $WH$, and $ZH$ production.

• Operators: Six dimension-6 SMEFT operators are considered: $O_{\Phi D}$, $O_{\Phi W}$, $O_{\Phi B}$, $O_{\Phi W B}$, $O^{(3)}_{\Phi q}$, and $O_{WWW}$.
• Data: Events are generated at $\sqrt{s} = 13.6$ TeV using MADGRAPH5_AMC@NLO, PYTHIA8, and DELPHES. An integrated luminosity of $300 \text{ fb}^{-1}$ is assumed.
• Comparison: Results are compared against:
  1. Parton-level likelihoods (theoretical upper bound).
  2. Total rate limits (lower sensitivity bound).
  3. Reconstruction-level histograms using standard observables and binning (e.g., STXS stage 1.2).
  4. Reconstruction-level SBI limits.

Systematic uncertainties are neglected to isolate the kinematic information gain, though the authors note that SBI is expected to handle systematics better by decorrelating kinematic effects from theory parameters.

Key Contributions and Results

1. Single-Process Performance:
For each individual process ($WW$, $WZ$, $WH$, $ZH$), the SBI approach consistently yields tighter constraints on Wilson coefficients compared to histogram-based methods.

• Degeneracy Breaking: The primary advantage of SBI is its ability to break degeneracies between operators that affect total rates similarly but alter kinematic distributions differently. For example, in $WZ$ production, the histogram approach struggles to disentangle $c_{\Phi W B}$ and $c_{\Phi D}$, whereas SBI significantly tightens the limits in this plane.
• Polarization Sensitivity: In $ZH$ production, the histogram observable ($p_T^Z$) is insensitive to $Z$-boson polarization, leading to a complete degeneracy among $c_{\Phi W}$, $c_{\Phi B}$, $c_{\Phi W B}$, and $c_{\Phi D}$ in one-dimensional profiled limits. SBI, by utilizing the full phase space, breaks this degeneracy and constrains all four coefficients individually.
• Background Robustness: Even in channels with significant background contributions (e.g., $ZH$ with $Zb\bar{b}$ background), SBI outperforms histograms, providing constraints where histograms fail to extract information beyond the total rate.

2. Global Combined Analysis:
The paper demonstrates that the advantages of SBI persist and are amplified when combining all four processes.

• Complementarity: Different processes constrain different directions in the parameter space (e.g., $WZ$ constrains $c_{\Phi W B}$, $WH$ constrains $c_{\Phi W}$).
• Superiority in Combination: The combined SBI limits are significantly stronger than combined histogram limits. For coefficients like $c_{\Phi D}$, $c_{\Phi W}$, $c_{\Phi W B}$, and $c_{\Phi B}$, the SBI limits are approximately a factor of 2 stronger than histogram limits.
• Luminosity Equivalent: The authors estimate that to achieve the same sensitivity as the SBI approach using the histogram method, a factor of $\sim 4$ more data would be required for the most degenerate coefficients (assuming standard scaling with $\mathcal{L}^{-1/2}$). Even for coefficients where histograms are more sensitive (e.g., $c_{WWW}$), SBI provides a $\sim 30\%$ sensitivity gain.

3. Validation:
The authors verified the empirical coverage of the learned likelihoods. For most parameter planes, the empirical coverage matches the nominal confidence level. In a few cases (e.g., $c_{WWW}$ in $WW$ production), the likelihood was slightly underconfident (conservative), suggesting that with more training data, the constraints could be further tightened.

Significance and Claims

The paper claims to demonstrate that neural simulation-based inference is not merely an improvement for single-process analyses but is essential for unlocking the full potential of global LHC analyses. The authors argue that:

• Global Disentanglement: SBI is uniquely effective at disentangling multiple theory parameters in a global context where traditional binned histograms fail due to information loss and degeneracies.
• Scalability: The derivative-learning approach allows for the straightforward inclusion of multiple operators and processes without a combinatorial explosion in complexity.
• Future Readiness: As LHC datasets grow, the information loss inherent in binning will become increasingly problematic. SBI offers a pathway to utilize the full high-dimensional event information, potentially equivalent to doubling or quadrupling the effective luminosity for specific SMEFT constraints.

The authors conclude that while their analysis simplifies certain aspects (neglecting systematics and subleading backgrounds), the persistent advantage of SBI over histogram-based methods in a global SMEFT context suggests that SBI should be integrated into future global fits to maximize the physics reach of the LHC.