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November 22, 2024

Fermion masses

- sage -

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.

二零零五年六月十五日