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

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 probing the 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 from very large irreducible backgrounds from other Higgs production channels, notably gluon-fusion ($gg$F). In this paper we calculate for the first time the so-called QCD and electroweak complete-NLO predictions for $H b \bar b$ production, using the four-flavour scheme. Our calculation shows that not only the $gg$F but also the $ZH$ and even the vector-boson fusion channels are sizeable 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 suppression occurs at the differential level and therefore backgrounds survive typical analysis cuts. This fact further jeopardises the chances of measuring at the LHC 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 for beyond-the SM scenarios the direct determination of $y_b$ via this process seems to be hopeless at the LHC.