# 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!? $$135 \frac {\mathrm{GeV}}{c^2} < \left( 172.9 \pm 1.5 \right) \frac {\mathrm{GeV}}{c^2}$$

## 1 Answer

Some particles are heavier than other particles but it is fair because other particles are lighter than some particles.

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.

• "Some particles are heavier than other particles but it is fair because other particles are lighter than some particles." but there is just one god particle and that is lighter than top quark!. god particle is origin of mass but top quark is heavier!. – user8784 Jun 8 '12 at 15:29
• More importantly, your comment suggests that you think that god particle gives mass to others "from a fixed reservoir of bread". But this doesn't work like this, there's no limitation on how much mass the particles may get. You're apparently taking the religious and kindergarten metaphors too literally, interpreting them in ways that have absolutely nothing to do with the physics and you had no chance to understand my answer. If I am right, you should give up studying particle physics because you're light years away from understanding even the high-school level of how particle physics works. – Luboš Motl Jun 9 '12 at 6:56
• your offensive comment and harsh angry votedown shows everyone you even doesn't know what is the first things about physics. – user8784 Jun 9 '12 at 8:34
• @David Zaslavsky The loss of users Due to Downvotes "That doesn't make sense. Could you explain more clearly?" you wants more explain i have proof Luboš Motl harsh speech: "You're apparently taking the religious and kindergarten metaphors too literally," "yo light years away from understanding even the high school level of how particle physics works." – user8784 Jun 9 '12 at 9:59
• @MrAres Lubos's "religious and kindergarten metaphors" line is not more that a tiny bit harsh as it is a plain spoken bit of truth. The language of physics is mathematics, and trying to understand these concepts in terms of word pictures constructed for the laymen is fraught with opportunities for misunderstanding. You appear to have latched onto a big misunderstanding. Nor do you have any idea which of 95+ unique visitors to this question cast the downvote, and it does not do to presume. – dmckee Jun 9 '12 at 15:54