What the paper is about: The authors present the first next-to-leading-order (NLO) calculation of electroweak corrections for vector‑boson scattering (VBS) that produces two Z bosons with well‑defined polarisation states, in the fully leptonic decay channel at the Large Hadron Collider (LHC). They study the process pp → e+ e− μ+ μ− jj at a Run‑3 energy of √s = 13.6 TeV, and compare results for doubly polarised Z pairs (longitudinal or transverse) and for the unpolarised signal. The Z‑pair channel with four charged leptons is experimentally very clean but rare, so precise theory predictions are needed to make the most of future data from Run‑3 and the High‑Luminosity LHC (HL‑LHC).

What the researchers did: The team computed the O(α6) leading‑order and the O(α7) electroweak corrections that affect both the production and decay of the intermediate Z bosons. They separate polarisation states at the amplitude level and use the double‑pole approximation (DPA) to treat the Z bosons as on‑shell resonances while keeping factorisable real and virtual electroweak corrections. The full off‑shell calculation is handled in the complex‑mass scheme, and single‑photon‑induced channels (γq) at NLO are included; photon–photon channels are negligible and were not considered. Two independent Monte Carlo codes (MoCaNLO and BBMC), together with Recola and Collier for amplitudes, were used and shown to agree to a few per mille.

How it works at a high level: The double‑pole approximation isolates the contributions where both intermediate Z bosons are near their mass shell and lets the authors define longitudinal and transverse polarisations in a consistent, frame‑dependent way (they use the diboson centre‑of‑mass frame). Factorisable corrections mean the calculable electroweak effects can be attached separately to the production and to the decay of each Z. Non‑factorisable electroweak corrections, which mix production and decay in a way that is hard to factorise, are neglected because they are expected to be small in the region dominated by Z resonances. The calculation applies realistic selection rules adapted from a CMS study, including lepton pT and rapidity cuts, invariant‑mass windows around the Z mass, and jet requirements.