This paper shows how to bring the matrix element method up to next-to-leading order accuracy in quantum chromodynamics (QCD) using the POWHEG approach. The matrix element method (MEM) compares individual collision events directly to theoretical predictions and is powerful because it keeps all correlations encoded in the fundamental scattering probability. The authors present a practical path to do this at higher precision and test it on a clean example: fully leptonic W+W− production in the Standard Model effective field theory (SMEFT), focusing on a CP-even, dimension-six operator that changes how the W bosons are polarized.

At leading order (LO) the MEM is widely used because per-event weights are straightforward to compute. Moving to next-to-leading order (NLO) is important for higher precision but is technically hard. Problems include infrared divergences (mathematical singularities that cancel only between different pieces of the calculation), the appearance of extra final-state partons (additional particles from radiation), burdensome multi-dimensional integrations, and the emergence of negative event weights that complicate a probabilistic interpretation.

The authors exploit the POWHEG method and its central building block, the "tilde-B" or ˜B(Φ) function, to overcome these issues in practice. POWHEG generates events according to ˜B(Φ), which encodes the NLO-corrected weight at fixed underlying Born kinematics (the simpler configuration without the extra radiation). Real-emission events are projected back onto these Born kinematics using the sector maps that POWHEG already uses. This keeps the effect of the hardest QCD radiation while preserving the NLO-accurate total rate. The implementation uses the Frixione-Kunszt-Signer (FKS) subtraction scheme to handle infrared pieces and the POWHEG Sudakov procedure to generate radiation, which typically yields mostly positive-weight events. The authors note that negative weights can still appear if large negative virtual corrections make ˜B(Φ) negative, and that cancellations of such negatives occur only after integrating over the radiation variables.