Sampling NNLO QCD phase space with normalizing flows

TL;DR

The paper addresses the computational challenge of NNLO QCD phase-space integration by applying neural importance sampling with Normalizing Flows to the sector-improved residue subtraction framework. It develops Coupling-Layer and Continuous Normalizing Flow architectures, with iterative, label-conditioned training, to map phase-space variables to target sampling densities and reduce cross-section variance. By stratifying integrands into positive and negative contributions and training discrete-flow-conditioned samplers, the study achieves substantial variance reduction and unweighting efficiency gains, translating to roughly a factor of 2–8 in computational cost savings for gluonic $t\bar{t}$ production at NNLO QCD. The results demonstrate accurate differential distributions and improved efficiency, indicating strong potential for applying flow-based sampling to broader NNLO QCD calculations and other complex phase-space integrals.

Abstract

We showcase the application of neural importance sampling for the evaluation of NNLO QCD scattering cross sections. We consider Normalizing Flows in the form of discrete Coupling Layers and time continuous flows for the integration of the various cross-section contributions when using the sector-improved residue subtraction scheme. We thereby consider the stratification of the integrands into their positive and negative contributions, and separately optimize the phase-space sampler. We exemplify the novel methods for the case of gluonic top-quark pair production at the LHC at NNLO QCD accuracy. We find significant gains with respect to the current default methods used in STRIPPER in terms of reduced cross-section variances and increased unweighting efficiencies. In turn, the computational costs for evaluations of the integrand needed to achieve a certain statistical uncertainty for the cross section can be reduced by a factor 8.

Figures (6)

• Figure 1: Representative Feynman diagrams contributing to gluonic top-quark pair production at NNLO QCD.
• Figure 2: The distribution of weights for the $\sigma^{\rm RF}$ (top row) and $\sigma^{\rm RRF}$ contributions (bottom row). The first column shows the absolute weight, and the second and third columns show the positive and negative weight distributions. The distributions are shown for the Vegas, Cl and ODE integrators, all trained on the absolute integrand.
• Figure 3: Example of non-factorizable correlations between two phase-space parameters for the $\sigma^{\rm RRF}$ contribution. The left panel shows the learned sampling distribution of Vegas, and the right panel that of the Cl integrator.
• Figure 4: The sign (positive - yellow, negative - blue) of the binned cross section of the $\sigma^{\rm RVF}$ contribution as a function of two input parameters is shown. All other dimensions are integrated out.
• Figure 5: Differential cross section with respect to the top-pair invariant mass $m_{t\bar{t}}$. The upper panel shows the absolute LO QCD result obtained with the Vegas, ODE and Cl integrator. The middle and lower panels show the NLO and NNLO QCD results as a ratio to the LO result, respectively. The vertical bars indicate the statistical uncertainty.