What is the interaction with Higgs field(s) that give the quarks so much different masses?

The masses of quarks are:

 mu 2∼3 MeV    md 4∼6 MeV
mc 1.3 GeV    ms 80∼130 MeV
mt 173 GeV    mb 4∼5 GeV

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Could you pelase elaborate? –  Manishearth Mar 29 '12 at 12:18
(this comment has no relation to the question)--You seem to be a zoologist/biologist judging by your username--you may want to check out biology.stackexchange.com –  Manishearth Mar 29 '12 at 12:19

The relevant interactions are the cubic "Yukawa" interactions of the form $${\mathcal L}_{\rm yukawa} = y \cdot h \bar\Psi \Psi$$ where $y$ is dimensionless (no units) Yukawa coupling constant, $h$ is the Higgs scalar field, and $\Psi$ or its complex conjugate is the field that describes the fermions (quarks in your case).

Because it has three fields in the product, the Higgs field and two copies of the fermionic field, it's cubic. This interaction is able to decay the Higgs boson into a quark-antiquark pair. If the energy conservation were not imposed, it would also allow a quark to emit or absorb a Higgs boson.

At any rate, the masses arise because in the vacuum, the average "expectation" value of the Higgs field $h$ isn't zero. Instead, it is $$\langle h\rangle = v$$ which is approximately 246 GeV. Different quark flavors have very different magnitude of the Yukawa coupling $y$ and the products $yv$ manifest themselves as different masses $m$.

In the Standard Model, $y$ are parameters that can be measured and pretty much all small enough values are allowed. To explain why some $y$ are much smaller than others, one needs string theory or at least some models beyond the Standard Model. Several proposed mechanisms for the "hierarchy of Yukawa couplings" are known; which of them is right, if any, isn't known.

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