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

Production of the Higgs boson, $H$ in association with a massive vector boson, $V$, i.e., the $VH$ process, plays an important role in the explorations of Higgs physics at the Large Hadron Collider, both for a precise study of Higgs' Standard Model couplings and for probing New Physics. In this publication we present the two-loop corrections in massless quantum chromodynamics (QCD) to the amplitude of the Higgs production associated with a $Z$ boson via the bottom quark-antiquark annihilation channel with a non-vanishing bottom-quark Yukawa coupling, which is a necessary ingredient of the full next-to-next-to-leading-order QCD corrections to the $VH$ process in the five-flavour scheme. The computation is performed by projecting the D-dimensional scattering amplitude directly onto an appropriate set of Lorentz structures related to the linear polarisation states of the $Z$ boson. We provide analytic expressions of the complete set of renormalised polarised amplitudes in terms of polylogarithms of maximum weight four. To give an estimation of the size of contributions from amplitudes considered in this work, we compute numerically the resulting cross sections under the soft-virtual approximation. We also take the opportunity to make a dedicated discussion regarding an interesting subtlety appearing in the conventional form factor decomposition of amplitudes involving axial currents regularised in D dimensions.