If the Higgs field gives mass to particles, and the Higgs boson itself has mass, does this mean there is some kind of self-interaction?
Also, does the Higgs Boson have zero rest mass and so move at light-speed?
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Most of the popular science TV programmes and magazine articles give entirely the wrong idea about how the Higgs mechanism works. They tend to give the impression that there is a single Higgs boson that (a) causes particles masses and (b) will be found around 125GeV by the LHC.
The mass is generated by the Higgs field. See the Wikipedia article on the Higgs mechanism for details. To (over)simplify, the Higgs field has four degrees of freedom, three of which interact with the W and Z bosons and generate masses. The remaining degree of freedom is what we see as the 125Gev Higgs boson.
In a sense, the Higgs boson that the LHC is about to discover is just what's left over after the Higgs field has done it's work. The Higgs boson gets its mass from the Higgs mechanism just like the W and Z bosons: it's not the origin of the particle masses.
The Higgs boson doesn't have zero rest mass.
A quick footnote:
Matt Strassler's blog has an excellent article about this. The Higgs mass can be written as an interaction with the Higgs field just like e.g. the W boson. However Matt Strassler makes the point that this is a coincidence rather than anything fundamental and unlike the W and Z the Higgs boson could have a non-zero mass even if the Higgs field was zero everywhere.
does this mean there is some kind of self-interaction
Yes, the Higgs field is self-interacting and, to the extent I understand it, it is this self-interaction and particularly, its form, that allows the Higgs field to "condense" by giving the lowest energy states of the field a non-zero expectation value. But the Higgs fields have electroweak charge.
So, essentially, space, when the Higgs field is in a ground state, is an electroweak superconductor that "breaks" ("hides", "screens") the electroweak symmetry down to the observed electromagnetic symmetry.
In addition to the electroweak gauge field interactions with the Higgs field, the fermion (matter) fields also couple to the Higgs fields via a so-called Yukawa interaction which is, in The Standard Model, the origin of matter's mass.
The Higgs field can be thought of as a giant field of energy, with the Higgs Bosons being excitations in that field.
This field interacts with particles to give them mass, and this is similar to how the Higgs Boson is 'created'. When a particle causes lots of fluctuations in the Higgs field, it can be thought to be 'interacting' with it. This is not unlike the fluctuations in the field responsible for the existence of the Higgs Boson.
So yes, the Higgs Field can be considered 'self interacting'