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The hadroproduction of a Higgs boson in association with a bottom-quark pair($H b \bar b$) is commonly considered as the key process for directly probingthe Yukawa interaction between the Higgs boson and the bottom quark ($y_b$).However, in the Standard-Model (SM) this process is also known to suffer fromvery large irreducible backgrounds from other Higgs production channels,notably gluon-fusion ($gg$F). In this paper we calculate for the first time theso-called QCD and electroweak complete-NLO predictions for $H b \bar b$production, using the four-flavour scheme. Our calculation shows that not onlythe $gg$F but also the $ZH$ and even the vector-boson fusion channels aresizeable irreducible backgrounds. Moreover, we demonstrate that, at the LHC,the rates of these backgrounds are very large with respect to the "genuine" and$y_b$-dependent $H b \bar b$ production mode. In particular, no suppressionoccurs at the differential level and therefore backgrounds survive typicalanalysis cuts. This fact further jeopardises the chances of measuring at theLHC the $y_b$-dependent component of $H b \bar b$ production in the SM.Especially, unless $y_b$ is significantly enlarged by new physics, even forbeyond-the SM scenarios the direct determination of $y_b$ via this processseems to be hopeless at the LHC.

RIP $H b \bar b$: How other Higgs production modes conspire to kill a rare signal at the LHC