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We present the two-loop QCD corrections to the amplitude of the Higgsproduction associated with a $Z$ boson via the bottom quark-antiquarkannihilation channel with a non-vanishing bottom-quark Yukawa coupling. Thecomputation is performed by projecting the D-dimensional scattering amplitudedirectly onto a set of Lorentz structures related to the linear polarisationstates of the $Z$ boson. We cross-check the finite remainders through acomputation based on conventional form factor decomposition. We show that forphysical observables, an ultimate D-dimensional form factor decomposition ofamplitudes is not necessary which has a huge potential to simplify a multiloopcomputation. We compute numerically the resulting cross sections under thesoft-virtual approximation to NNLO and find it three orders of magnitudesmaller than that of the s-channel.

Two-Loop QCD Helicity Amplitudes for Higgs Production Associated with a Vector Boson through Bottom Quark Annihilation