What gives the higgs boson mass?

Question

In light of the discovery of the Higgs boson. The Higgs Boson is a force particle which interacts with matter particles. My question is what does the Higgs Boson interact with to give itself mass.

Answer 1

Yes, it is correct to say that the Higgs boson, just like other elementary particles, get its mass from the interaction with the Higgs boson – which means "with itself" in the case of this particle.

More concretely, the mass may be derived from the Higgs potential (energy density) $$ V(h) = \frac a4 h^4 - \frac b2 h^2 + c$$ where the additive shift $c$ isn't important (in the absence of curved spacetime). The constants $a,b$ determine the shape of the potential or, if you wish, the coefficients of the "quadratic" and "quartic" self-interactions of the Higgs field.

The function $V(h)$ has minima at $$ h_\pm = \pm \sqrt{\frac{b}{a}} $$ and near these minima, the potential may be approximated as $$ V(h_\pm + \Delta h ) = c' + \frac{m^2}{2} (h-h_\pm)^2 $$ and because of this reinterpretation, the parameter $m$ in the coefficient $m^2/2$ above (which is a simple function of $a,b$ again) may be interpreted as the Higgs mass.

This quadratic+quartic self-interaction plays the role of the gauge interaction $h-h-Z$ or $h-h-W$ which gives masses to the Z-bosons or W-bosons; or the $h\bar{\psi}\psi$ Yukawa interactions that give masses to the fermions.

There is one sense in which the "God particle giving mass to itself" is somewhat more vacuous: we had to use the quadratic term, $-bh^2/2$, itself, and quadratic terms are of course nothing else than mass terms in disguise (although the sign of $b$ was the opposite one than for a "direct" Higgs mass). So we could only derive the mass term around the physical vacuum $h_\pm$ because we inserted some (tachyonic) mass term to the initial formula. But that doesn't change the fact that even this mass term of the Higgs boson may be interpreted as a self-interaction of the Higgs with itself.