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This question already has an answer here:

If the Higgs Boson is supposed to be the particle responsible for other particles having mass. How can it itself have a mass?

Is it not then a 'who came first, the chicken or the egg' situation?

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marked as duplicate by Danu, John Rennie, Ross Millikan, Alfred Centauri, BMS Jul 6 '14 at 16:51

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No.

The mass of most particles is not a problem. But for the force carriers, i.e. gauge bosons like the gluon or the W and Z bosons, it is a theorem that they must be (naively) massless. But we find that the W and Z bosons act as if they have a mass! The mechanism by which this mass arises is the Higgs mechanism, but we can have masses without it - just not on gauge bosons. Since the Higgs is no gauge boson, it can have mass without its own mechanism (fortunately).

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    $\begingroup$ Also the fermions in the SM need a Higgs field with a non-vanishing vev to get their masses since the SM is a chiral theory. No mass term is allowed by the gauge invariance which thus need to be broken spontaneously to generate the fermion masses. $\endgroup$ – TwoBs Jul 6 '14 at 16:12
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It is not the Higgs boson, but rather the Higgs field, that gives mass to the elementary particles, Higgs boson included. In fact, even in Higgsless theories, e.g. such as a technicolor, the W and Z get mass but there is no Higgs boson (although there is a composite Higgs field made of techniquarks).

Said this, the Higgs boson has a finite mass because its quartic selfcoupling $\lambda$, in the SM, is not vanishing, $V=\lambda(|H|^2-\frac{v^2}{2})^2$, which implies $m_h^2=2\lambda v^2$.

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