  We investigate the effect of one-loop corrections of ${\cal
O}(\alpha_S^2\alpha_W)$ on the $t\bar t$ invariant mass spectrum at the Large
Hadron Collider (LHC) in presence of both resonant and non-resonant Higgs boson
effects. We show that corrections of ${\cal O}(\alpha_S^2\alpha_W)$ involving a
non-resonant Higgs boson are comparable to or even larger than those involving
interference with the s-channel resonant Higgs boson amplitude and that both of
these are subleading with respect to all other (non-Higgs) diagrams through
that order. We also compute the contribution through ${\cal
O}(\alpha_S^2\alpha_W^2)$ of resonant Higgs boson production (i.e. Higgs
production via $gg$ fusion) as well as the pure QCD ones of ${\cal
O}(\alpha_S^3)$. Altogether, we show that the well known peak-dip structure of
the $M_{t\bar t}$ spectrum emerging from interference effects between the
$t,u$-channel $gg$-induced Leading Order (LO) QCD diagrams and the one due to a
Higgs boson in s-channel via $gg$-fusion is drastically swamped by the
remainder of the terms of ${\cal O}(\alpha_S^2\alpha_W)$ discussed above.
