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Associated production of a Higgs boson decaying into bottom quarks and a weak vector boson decaying leptonically at NNLO in QCD

We present the calculation of next-to-next-to-leading order (NNLO) corrections in perturbative QCD for the production of a Higgs boson decaying into a pair of bottom quarks in association with a leptonically decaying weak vector boson: $\mathrm{pp} \to V \mathrm{H} + X \to \ell\bar{\ell}\;\mathrm{b\bar{b}} + X$. We consider the corrections to both the production and decay sub-processes, retaining a fully differential description of the final state including off-shell propagators of the Higgs and vector boson. The calculation is carried out using the antenna subtraction formalism and is implemented in the NNLOJET framework. Clustering and identification of $\mathrm{b}$-jets is performed with the flavour-$k_t$ algorithm and results for fiducial cross sections and distributions are presented for the LHC at $\sqrt{s}=13\;\text{TeV}$. We assess the residual theory uncertainty by varying the production and decay scales independently and provide scale uncertainty bands in our results, yielding percent-level accurate predictions for observables in this Higgs production mode computed at NNLO. Confronting a naïve perturbative expansion of the cross section against the customary re-scaling procedure to a fixed branching ratio reveals that starting from NNLO, the latter could be inadequate in estimating missing higher-order effects through scale variations.