# Why is the mass of even the upper limit of the possible Higgs boson is less than the top quark?

On 7 March 2012, the and CDF collaborations announced that, after analyzing the full data set from the Tevatron accelerator, they found excesses in their data that might be interpreted as coming from a Higgs boson with a mass in the region of 115 to 135 GeV.

But the top quark has a mass of 172.9 ±1.5 GeV which is about the same mass as an atom of tungsten.

1. why is the mass of even the upper limit of possible Higgs boson less than the top quark!? $$135 \frac {\mathrm{GeV}}{c^2} < \left( 172.9 \pm 1.5 \right) \frac {\mathrm{GeV}}{c^2}$$

There is no contradiction in Higgs' being lighter than the top quark. The top quark mass arises from the Yukawa interaction between the Higgs and the top quark Dirac fields $${\mathcal L}_{\rm Yukawa} = y\cdot h\bar\Psi_t\Psi_t$$ We may substitute $h=v + \Delta h$ where $v=246\,{\rm GeV}$. The top quark mass is then $m_t = vy$ in my conventions where $y$ is the dimensionless Yukawa coupling. This shouldn't be much greater than one so the top quark mass shouldn't be much greater than $v=246\,{\rm GeV}$. However, the precise threshold that the quark mass can't exceed isn't necessarily $v$; there may be a purely numerical constant I don't want to discuss now.
On the other hand, the Higgs boson itself has a mass that may be much ligher than $v$ because it comes from the quartic potential $$V(h) = \frac{\lambda}{4} h^2 - \mu^2 h^2$$ that has some minima and the second derivative near these minima determines the Higgs mass. At any rate, if the quartic dimensionless coupling $\lambda$ were very small, the potential could be just rescaled to have the same minima at $h=\pm v$, however the Higgs mass could be made arbitrarily smaller. In reality, $\lambda$ is safely smaller than one (and even more safely lower than the upper allowed bound which is actually something like $\pi$) but it is not "insanely" smaller than one which is why the Higgs mass $(125\pm 1)\,{\rm GeV}$ measured by the LHC ends up being about one-half of the vev $v=246\,{\rm GeV}$ and it is allowed to be lower than a quark mass. However, there would be no contradiction if the Higgs mass were much lower than that, either.