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Production of the Higgs boson, $H$ in association with a massive vectorboson, $V$, i.e., the $VH$ process, plays an important role in the explorationsof Higgs physics at the Large Hadron Collider, both for a precise study ofHiggs' Standard Model couplings and for probing New Physics. In thispublication we present the two-loop corrections in massless quantumchromodynamics (QCD) to the amplitude of the Higgs production associated with a$Z$ boson via the bottom quark-antiquark annihilation channel with anon-vanishing bottom-quark Yukawa coupling, which is a necessary ingredient ofthe full next-to-next-to-leading-order QCD corrections to the $VH$ process inthe five-flavour scheme. The computation is performed by projecting theD-dimensional scattering amplitude directly onto an appropriate set of Lorentzstructures related to the linear polarisation states of the $Z$ boson. Weprovide analytic expressions of the complete set of renormalised polarisedamplitudes in terms of polylogarithms of maximum weight four. To give anestimation of the size of contributions from amplitudes considered in thiswork, we compute numerically the resulting cross sections under thesoft-virtual approximation. We also take the opportunity to make a dedicateddiscussion regarding an interesting subtlety appearing in the conventional formfactor decomposition of amplitudes involving axial currents regularised in Ddimensions.

Polarised Amplitudes and Soft-Virtual Cross Sections for $b\bar b \rightarrow ZH$ at NNLO in QCD