sage
发表文章数: 1125
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(第六重)
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neutrino mass in standard Model, etc.
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.
| 发表时间：2005-05-17, 22:33:43 |
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可见光
发表文章数: 421
武功等级: 拈花指
(第八重)
内力值: 405/405
Re: neutrino mass in standard Model, etc.
sage哥和昌海哥总是让我学到很多，还有Walk_f, No-Go....
如果中微子有质量，那宇称不守恒是如何实现的呢？目前我没有时间自学了，只有在这里顺便学到一些。
生活充满七彩阳光，是为可见光
宇宙无限，爱心永恒！
| 发表时间：2005-05-18, 01:03:16 |
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卢昌海
发表文章数: 1617
武功等级: 北冥神功
(第一重)
内力值: 602/602
Re: neutrino mass in standard Model, etc.
可可来得越来越少了。
中微子质量与宇称不守恒并不矛盾，但与宇称１００％破缺有一定的冲突。不过所谓宇称１００％破缺只不过表明V+A型的相互作用在实验精度之内未被检测到，因此这种冲突并不绝对。在中微子有质量的情况下，物理学家们提出了一些模型来说明为什么在实验中宇称几乎１００％破缺。比如有一种模型叫做：left-right symmetric model，在其中Lagrangian具有SU(2)_LxSU(2)_R对称性，该对称性在很高的能量下自发破缺，使得右旋中微子具有很大的质量，同时也抑制了V+A型的相互作用，因此在低能实验中我们只能观测到左旋中微子及V-A，宇称看起来是１００％破缺的。
宠辱不惊，看庭前花开花落
去留无意，望天空云卷云舒
| 发表时间：2005-05-18, 13:19:45 |
作者资料
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可见光
发表文章数: 421
武功等级: 拈花指
(第八重)
内力值: 405/405
Re: neutrino mass in standard Model, etc.
谢谢昌海哥！
现在杂事多，一时来得少了
生活充满七彩阳光，是为可见光
宇宙无限，爱心永恒！
| 发表时间：2005-05-20, 00:50:45 |
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