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I know the electromagnetic force is mediated by a photon and the weak nuclear force is mediated by two massive bosons. Are there any other insights into why the masses are so different?

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I think that the Weak bosons are three: W+, W- and Z, am I right? –  neutrino Mar 22 '13 at 13:33
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@neutrino is right, there are three. The short answer to this question is the Higgs mechanism. Maybe I'll post about it later, but you can start by looking at this. –  David Z Mar 22 '13 at 14:53
    
If your question is just why the W and Z have different masses, then it may be helpful to know that at tree-level the ratio of the masses is determined by the Weinberg angle. In the Standard Model, the Weinberg angle is an unknown/measured parameter, so the SM does not explain why exactly the difference is specifically ~10 GeV. –  sujeet Mar 24 '13 at 20:41
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The details of this is a somewhat convoluted set of ideas that would be a whole chapter of a quantum field theory textbook when done correctly. What I offer below is a brief, popular-level overview. I gloss over several details, and yadda yadda through many things. I tried to be as accurate as possible, but given the brevity of the discussion, I'm sure that some of the things I say below are wrong under certain circumstances.

That said, let's go.

The core idea here is that the fundamental equation governing the electroweak force is a theory coupling three SU(2) bosons with matter. This equation is invariant under an arbitrary rotation of the SU(2) bosons into each other-- you could replace $A_{1}$ and $A_{2}$ with $A_{a} = n A_{1} + m A_{2}$ and $A_{b} = kA_{1} + \ell A_{2}$, and the fundamental equation would be the same, so long as $n^{2} + m^{2} = k^{2} + \ell^{2} =1$ and $nk + m\ell = 0$ Similarly, thanks to the SU(2) symmetry, the interacting matter particles can be arbitrarily replaced with each other, and the theory doesn't know the difference--the "left-handed" component of the neutrino can be replaced with the "left handed" component of the electron, and the SU(2) interaction would be none the wiser.

The electroweak interaction also includes a U(1) interaction that behaves like ordinary electrodynamics. If there is no Higgs particle, the whole thing behaves like this, and none of the bosons has mass.

Then, the higgs comes into the picture. It turns out that the Higgs boson has a potential energy function such that its potential energy function is higher when it is absent than when it is present. Since the Higgs field interacts with the SU(2) field and the U(1) field, it takes values in these spaces, and therefore, the nonzero way that its potential energy is minimized means that if a Higgs particle is present and near this minimum value, real-world physics does NOT respect the rotations talked about above. This lack of respect for rotations picks out very special physical values of the rotation coeficcients that respect the value taken by the Higgs. When all of this is said and done, the four vector bosons (three SU(2) ones and the one U(1) one) have combined in a very particular way to form three three weakly interacting bosons, which acquire a mass as a result of the symmetry breaking, and the beginning symmetry group of rotations gets to maintain one single U(1) symmetry, which produces a photon, which, since it is still related to a symmetry, gets to retain its masslessness.

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you must mean "gets to retain its zero mass". Also maybe one should be talking of the Higgs field here, even if it is simplified, since so much publicity exists for the 126GeV Higgs boson. –  anna v Mar 23 '13 at 12:36
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I will +1 this simplified tale, but stress that the Higgs field permeates everything and the recently observed 126GeV candidate for the Higgs boson is the validation of the existence of the permeating field. –  anna v Mar 25 '13 at 11:44
    
@ Jerry Schirmer: How can you answer this question without a discussion of Spontaneous Symmetry Breaking. I believe it is spontaneous or void of a mechanism because the logic which produced those equations does not support a mechanism. Example: Newtonian mechanics has no arrow of time so the fact that one exist in nature means time symmetry is spontaneously broken. Thermodynamics supports an arrow of time. How do you just insert a mechanism into a theories logic to get a desired result? Something this fundamental should be a natural consequence of the theory not an insert. –  user4884 Mar 27 '13 at 18:17
    
@user4884: The whole last paragraph is a discussion of spontaneous symmetry breaking. I just don't use the term. And historically, this is close to how the theory was developed--something with the correct mass spectrum needed to show up, but theories that had massive vector bosons we nonrenormalizable. The Higgs and spontaneous symmetry breaking created a mechanism for giving masses to the weak vector bosons while having renormalizability still guaranteed. –  Jerry Schirmer Mar 27 '13 at 18:53
    
You can have spontaneous symmetry breaking without the Higgs Mechanism. (i.e. The definition of spontaneous symmetry breaking does not include the Higgs Mechanism). Why does everything move at the speed of light without the Higgs Mechanism says something is fundamentally wrong with the theory? I thought the Higgs Mechanism was just one way to explain what was wrong and try to correct it. I am assuming some other mechanism could have done a better job. –  user4884 Mar 27 '13 at 20:18
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