It's an old idea that the Higgs boson might be a top-antitop bound state. The standard form of the idea is that this is due to some new force, "topcolor".
It's also well-known that a QCD quark condensate can in certain respects resemble a Higgs field. In a higgsless standard model, the W and Z bosons would still acquire (MeV) masses, by absorbing the pions. But there would be nothing like yukawa couplings and so the fermions would remain massless.
There have nonetheless been attempts to obtain the Higgs specifically as a QCD bound state (John Moffat, Bruno Machet), but they don't seem to work.
Meanwhile, back in the standard model with a Higgs, there is actually a little-noticed "Higgs force" between fermions, due to Higgs boson exchange.
In most situations, it's too weak to matter. But for top quarks up close, the Higgs force can actually rival the strong force, because the top yukawa is so large (of order 1).
When I saw this just now, in a plot by Matt Strassler, I thought, Wow! This Higgs force could be the "something extra" that binds top and antitop into a Higgs... oh, wait...
So the idea sounds circular. But the real problem, even if we still suppose that the Higgs is somehow just QCD toponium, is the lack of yukawa couplings. We need yukawa couplings for the fermions to get their masses, and we specifically need a top yukawa coupling for a toponium-Higgs to emerge.
Here I wonder if an answer could come from quantum gravity. Could integrating out gravitational interactions create effective yukawa couplings? And there's also the fact that Shaposhnikov and Wetterich were able to predict the Higgs mass, by supposing that the quartic coupling and its beta function both go to zero at the Planck scale.
The big picture implied is that electroweak symmetry breaking, and all the fermion masses and mixings, derive from a quantum-gravitational perturbation of the SM QCD vacuum.
A nice concept, but how to test its viability? I can think of two ways.
First, study something simpler, like top + top/bottom doublet + QCD + gravity, and see if a large effective top yukawa coupling can be obtained.
Second, study the reasoning which has led most particle physicists to conclude from the precision measurements of the Higgs, that it is elementary, because a composite Higgs would need to be highly fine-tuned. How does the "toponium bootstrap Higgs" look from that perspective?
Obviously, I welcome comments.