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Fermion masses
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Here is a somewhat more technical introduction In our space time, i.e., D=4, every fermion could be described by a 2 component spinor (Weyl spinor). Let me call it \chi. Now, fermion could either carry charge (not just electric charge, but other gauge charge as well) or not. If they do carry conserved charge, they are dirac fermions. for dirac fermions, because they carry charge, they are different from their antiparticle. Therefore, for each dirac fermion \chi, we should also include a NEW fermion \eta which is the anti-particle. If they do not carry conserved charge, they are called majorana fermions. There is no difference between them and their anti-particles. Therefore, we do not have to introduce new degrees of freedom to describe them. Let me call majorana fermions \xi Now, let's write mass terms for the fermions. We can see immediately that a term like m \chi \chi is not allowed. Such a term does not conserved the charge. The only term we could write down is m \chi \eta Therefore, mass term for the dirac fermions mix fermion and anti-fermion. Therefore, Dirac decided that it is more convenient to group the dirac fermions into 4-component spinors which include both the fermion and anti-fermion. Dirac equation is written in terms of these fermions. For majorana fermion, the issue is more subtle. Of course we could just write a mass term m \xi \xi this is usually called majorana mass. On the other hand, we could also introduce another majorana fermion \psi and write a mass term m \xi \psi. This is sometimes called (probably a misnomer) dirac mass for majorana fermions because it looks like as if \psi is playing the role of antifermion of \xi. Of course, fermions could also charge under some global symmetry (the charge does not have to be conserved). For example, I could assign \xi a charge 'fermion number' +1. m \xi \xi obviously violate the charge by 2 units. In the case of neutino, this charge will be the so called lepton number. On the other hand, a dirac mass term does not have to violate 'fermion number' if we choose \psi to be of charge -1. now we consider the neutrino in the standard model. So far, we have only see the left-handed neutrinos. After electroweak symmetry breaking, they are singlet under U(1)_EM. Therefore, they could behave like majorana fermions at low energy. As we discussed above, it could have both majorana or dirac mass terms (if we also introduce another majorana fermion, in this case the right-handed neutrino), or both. Before electroweak symmtry breaking, left-handed neutrino in the standard model is part of the weak doublet (e_L, nu_L)=L It is charged. In Standard Model (in its very stract definition), there is no right-handed neutrino, therefore, we could not write a mass term for the left-handed neutrino. However, remember that standard model is only an effective field theory. This means we should also include higher dimensional operators. For example, we could have the following dimension 5 operator (LH)^2/M where H is the doublet Higgs. After electroweak symmetry breaking, higgs will have a VEV (0,v). This will turn into a majorana mass term for the left-handed neutrinos. If mass M is large, this term could help explaining the smallness of neutrino mass. Such a term could arise from the so called see-saw mechanism. As we said above, such a majorana mass term will violate lepton number by 2 units. |