A recent study reports the first next‑to‑next‑to‑leading‑order (NNLO) quark‑pair correction to the decay of a neutral axial‑vector current (the Z boson) into massless quarks using Implicit Regularization (IReg). The calculation avoids dimensional continuation, keeping the Dirac algebra in physical four dimensions and introducing an auxiliary mass μ to separate ultraviolet (UV) divergences from infrared (IR) singularities. The UV poles are expressed as basic divergent integrals (BDIs) that are absorbed into local counterterms in a BPHZ‑like scheme; surface terms are set to zero to preserve momentum‑routing invariance and Ward identities.

After performing UV renormalization and phase‑space integration, the μ‑dependent logarithms cancel. The remaining finite terms match the known large‑N_f contribution to the inclusive Z→qar{q} rate, confirming the consistency of IReg for chiral processes at two loops. The result also tests the cancellation of IR divergences between virtual quark‑bubble diagrams and real four‑quark emission, as required by the Kinoshita–Lee–Nauenberg theorem.

The study highlights that, while IReg offers practical advantages for loop calculations involving γ^5, the method requires careful handling of chiral identities and may need additional prescriptions in cases where γ^5 appears inside propagators or at higher loops.

The calculation is a non‑trivial NNLO check of gauge invariance, renormalization consistency, and perturbative unitarity in a four‑dimensional framework.