Higher-order effects in amplitude-assisted polarisation extraction with machine-learning techniques

Technical Summary: Higher-order effects in amplitude-assisted polarisation extraction with machine-learning techniques

Problem Statement
The extraction of electroweak (EW) gauge boson polarisation fractions from collider data is a critical probe of the Electroweak Symmetry Breaking (EWSB) mechanism. While recent LHC analyses (Run-2 and Run-3) have successfully measured these fractions using template fits based on Leading Order (LO) simulations, the increasing experimental precision demands theoretical predictions at higher orders. Previous machine-learning (ML) approaches for polarisation tagging, such as the amplitude-assisted method proposed in Ref. [36], were restricted to LO accuracy. A significant gap exists in understanding how Next-to-Leading Order (NLO) QCD corrections and parton-shower (PS) effects impact the training and performance of ML models designed to extract polarisation fractions. Specifically, it was unclear whether ML models trained on LO data could accurately regress polarisation ratios when applied to NLO+PS data, or if the training itself required higher-order accuracy to account for kinematic shifts induced by QCD radiation.

Methodology
The authors address this gap by extending the amplitude-assisted polarisation extraction framework to NLO QCD accuracy matched with parton showers (NLO+PS). The methodology consists of three main components:

1. Theoretical Framework and Reweighting:

   • The study focuses on the di-boson process $pp \to ZZ \to e^+e^-\mu^+\mu^-$ at $\sqrt{s} = 13$ TeV.
   • To generate polarised samples from unpolarised Monte Carlo (MC) events, the authors define a reweighting ratio $r_{LL} = |\mathcal{M}_{LL}|^2 / |\mathcal{M}|^2$, representing the squared amplitude for doubly longitudinal bosons over the unpolarised amplitude.
   • At NLO, this ratio is constructed using the $\bar{B}$ function from the Powheg method, which incorporates Born, virtual, and integrated real-radiation contributions. The reweighting is applied event-by-event to unpolarised NLO+PS samples generated with the Powheg-Box-Res framework and the VV_pol package.
   • The validity of this reweighting approach is verified through "closure tests," comparing reweighted distributions against direct simulations of polarised signals.

2. Machine Learning Architectures:
   The paper benchmarks four distinct regression approaches to predict the ratio $r_{LL}$ (or its proxy $\tilde{r}_{LL}$) using only experimentally accessible observables (final-state lepton momenta):

   • Feed-Forward Neural Network (FFNN): A deep network with 10 hidden layers and a reduced width-to-depth ratio to handle the increased kinematic complexity of the ZZ process compared to previous Z+j studies.
   • Autoencoder-like NN (AE): A network with a bottleneck structure designed to compress the 16 input features (lepton four-momenta) into a latent representation before reconstructing the target.
   • Physics-Informed Network (PN): A model based on ParticleNet, utilizing EdgeConv layers and a Lorentz-inspired preprocessing layer to ensure Lorentz invariance and reduce the number of trainable parameters.
   • Random Forest Regressor (RFR): A tree-based ensemble method using a specific subset of kinematic features (boson transverse momenta, rapidities, and decay angles in the rest frame).

3. Training Strategy:

   • Models are trained on datasets generated at LO, LO+Sudakov, and NLO+PS accuracies.
   • A log-transformation of the target variable is applied to stabilize gradients and prevent the network from collapsing to a constant mean prediction.
   • Hyperparameters (learning rate, batch size) are optimized using the Optuna framework.

Key Contributions

• Extension to NLO+PS: This work presents the first implementation of amplitude-assisted polarisation extraction at NLO QCD accuracy matched to parton showers. It demonstrates that the reweighting technique, while approximate in the context of fixed-order matching, reproduces direct polarised simulations with high fidelity (within $\sim 1\%$ for integrated cross-sections).
• Impact of Higher-Order Corrections on ML: The study systematically evaluates the necessity of training data accuracy. It finds that training ML models on LO data and testing on NLO+PS data leads to significant degradations (deviations of 2–6% in cross-sections), whereas training and testing at matching NLO+PS accuracy yields sub-per-mille precision.
• Architecture Benchmarking: The authors compare four distinct ML architectures. They find that while all models perform well in the bulk of the phase space, the Physics-Informed Network (PN) and the Random Forest Regressor (RFR) exhibit superior robustness across the full kinematic range, particularly in the tails of distributions. The PN model is highlighted for achieving high accuracy with a significantly smaller number of trainable parameters ($\sim 60$k) compared to the FFNN ($\sim 4$M) and AE ($\sim 1.4$M).

Results

• Closure Tests: The reweighting procedure reproduces direct NLO+PS simulations for longitudinal-longitudinal ($LL$) and transverse-transverse ($TT$) states with deviations of $-0.9\%$ and $+0.3\%$, respectively. Mixed states show a $\sim 7\%$ underestimation, attributed to mismatches in the Sudakov exponential for hard radiation.
• Differential Distributions:
  • For the positron decay angle ($\cos \theta^*_{e^+}$), all models agree with the truth within a few percent in the central region. In the tails, the RFR and PN models perform best.
  • For the positron transverse momentum ($p_{T, e^+}$), models trained on LO data fail to describe the NLO+PS tails accurately. Models trained on NLO+PS data show good agreement up to $\sim 200$ GeV.
  • The RFR model, when using a specific set of input features including decay angles in the rest frame, provides the most consistent description across the full $r_{LL}$ distribution range.
• Cross-Section Extraction: When trained and tested at NLO+PS accuracy, the PN and RFR models predict the fiducial cross-section for $Z_L Z_L$ production with deviations of less than $0.2\%$ from the truth.

Significance and Claims
The paper claims that the proposed framework successfully bridges the gap between high-precision theoretical predictions and experimental polarisation extraction. By generalizing the amplitude-assisted method to NLO+PS, the authors demonstrate that:

1. Higher-order corrections are non-negligible: Training ML models on LO data is insufficient for precision physics at the LHC; NLO+PS training data is required to capture the kinematic shifts induced by QCD radiation.
2. Efficiency and Robustness: The Physics-Informed Network (PN) offers a compelling balance between performance and computational cost, making it a preferred choice for phenomenological studies.
3. General Applicability: While the study focuses on leptonic $ZZ$ decays, the methodology is applicable to other di-boson processes and hadronic decays, provided a suitable event generator is available.

The authors conclude that their results capture the bulk of relevant effects for polarisation tagging at NLO, enabling future extensions to higher perturbative orders without conceptual bottlenecks. The implementation of the reweighting procedure is made public via the VV_pol package, and all ML predictions are available in a public repository to ensure reproducibility.