sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Mass Problem II: Fermion mass
II. Fermion mass
In principle, writing down a fermion mass is easy. We just write, for
dirac fermion,
m \bar{\psi} \psi. (1)
One such a example is just QED with massive electron. (1) will just be
the electron mass. If Dirac were right that QED is the complete
theory of our world (at least at low energy), the only task we will
have is to explain electron mass in terms of some fundamental
scale. This by itself is not s easy task. On the other hand, as we
will see later, the mass problem is much more complicated than
this. For now, before we leave the QED-only-toy-world, let's make one
more comment about electron mass.
Notice that without a electron mass, the QED Lagrangian has a symmetry
\psi --> exp(i \gamma_5 \alpha) \psi (2)
where \gamma_5 is the gamma matrix in the chiral projection
operator. This is the so called chiral symmetry. Mass term breaks this
symmetry. What this means is that:
starting from a Lagrangian without a mass term, calculate self
interaction as much as one wants, renormalize as much as one wants, a
mass term can never be generated, because nothing in the Lagrangian
withoutt mass term can break the chiral symmetry! Therefore, mass
purely from self-interaction is not possible. (an important exception
to this is when the interaction becomes very strong and fermions
starts to form condensates. the famous example is QCD. On the other
hand, this is not the case for QED).
Notice that such a claim is not true for bosons. As we will discuss in
detail later, a bosonic mass term breaks no such symmetry. Therefore,
if our electrons are bosons, it is possible to generate mass from pure
quantum corrections.
Now let's move on. It turns out that there are more interactions in
nature than just QED. There is of course the strong interaction,
QCD. As mentioned above, it forms bound states and condensates, which
breaks chiral symmetry and so on. Mass spectrum after QCD phase
transition, such as the masses of mesons, proton and neutron, etc. is
another big suject. Right now, let's focus on mass parameter in the
fundamental Lagrangian, in particular, the mass parameters of quarks
and leptons. For those parameters, it is believed that QCD condensate
plays a very minor role.
There is also, of course, the weak interactions. The non-trivial
feature about the weak interaction is that it has parity violation
built in. Left-handed and right-handed fermions transform differently
under SU(2). Left-handed fermions form doublets while the
right-handed ones are singlets. On the other hand, if we write the
mass term (1) in left-handed and right-handed components, it is
m(\psi_L \psi_R + \psi_R \psi_L).
It is not invariant under SU(2) since it is a product of a doublet
and a singlet! Therefore, weak-interaction would not allow us to write
down a mass term for fermions. Therefore, it seems that weak
interaction predicts that all the fermions are massless. Fortunately,
SU(2) weak symmetry is broken. Therefore, we could have a mass term
after the symmetry breaking. As a result, the fermion masses are
proportional to the size of such symmetry breaking. What we could do,
instead, is to write something like
y H f_L f_R + (conjugate)
where H is another doublet, a Higgs boson. Electroweak symmetry is
broken when Higgs acquire a vacuum expectation value (VEV)
<H>=v. Therefore, after symmetry breaking, we will have a mass term
y v f_L f_R + conjugate
the mass will be y \times v . As promised, it is proportional to the
size of symmetry breaking, v.
This is all fine. This is precisely the fermion sector of the Standard
Model. On the other hand, there are at least two unresolved problems.
1) the fermion mass is proportional to v, which is about 175 GeV. On
the other hand, this seems to be very far away from a fundamental
scale, such as Planck scale 10^19 GeV. The question is obviously
why v is where it is, how can we explain it in terms of the
fundamental scale. This is sometimes called the hierarchy problem,
which I will talk about in more detail when we discuss boson
masses.
2) Although all the fermion masses are of the form y v, in reality,
their sizes are very different. For exmple, electron is about 0.5
MeV, muon about 100 MeV, where top quark is 175 GeV. Therefore,
there must be different ys for each different fermions. Moreover,
the size of those ys are very different. How do we understand this?
This is the so called flavor problem. There are many attempts and
models to address the flavor problem. I will comment on them
briefly in the next section.
| 发表时间：2006-05-05, 23:35:41 |
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星空浩淼
发表文章数: 1743
武功等级: 九阳神功
(第五重)
内力值: 617/617
Re: Mass Problem II: Fermion mass
谢谢sage兄为我们献上精美大餐！等sage兄整个写完了，我再好好地请教几个问题。
在量子力学的角度上看，质量不为零的粒子才能定义位置算符（粒子可以局域到大于Compton波长的空间范围内）。
有人探讨过可变质量的理论，在那里质量不是常数而是一个变量，因此存在与之对应的质量算符。
在我个人看来，当前的物理，只有质量问题（质量的起源与本质）和时间问题，是最有趣、最有挑战性、最值得搞的两个超难问题。有个量子引力大牛认为：或许只有当量子引力问题得到解决之后，时间问题才可能得到解决。
我在故我寻，我寻故我痴；我痴故我傻，我傻故我贫；我贫故我苦，我苦故我悲；我悲故我思，我思故我在
| 发表时间：2006-05-06, 23:58:55 |
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季候风
发表文章数: 291
武功等级: 太极剑法
(第四重)
内力值: 370/370
Re: Mass Problem II: Fermion mass
在实验室怎样测量质量?
书山有路勤为径
学海无涯苦作舟
| 发表时间：2006-05-10, 16:04:23 |
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: Mass Problem II: Fermion mass
在实验室怎样测量质量?
what is your first guess? :-)
| 发表时间：2006-05-11, 01:27:16 |
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萍踪浪迹
发表文章数: 1983
武功等级: 深不可测
内力值: 645/645
Re: Mass Problem II: Fermion mass
：：what is your first guess? :-)
=======
弹簧测力计？哈哈，开个玩笑
漫漫长夜不知晓 日落云寒苦终宵
痴心未悟拈花笑 梦魂飞度同心桥
| 发表时间：2006-05-11, 13:36:23 |
作者资料
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季候风
发表文章数: 291
武功等级: 太极剑法
(第四重)
内力值: 370/370
Re: Mass Problem II: Fermion mass
我只能想到在经典力学范围内做这个测量---用另一个已知质量粒子来散射未知粒子. 可是从量子场论的观点来看, 这相当于引入新的相互作用项, 而散射概率给出这个作用项的耦合常数......迷惑中.
书山有路勤为径
学海无涯苦作舟
| 发表时间：2006-05-12, 01:59:04 |
作者资料
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: Mass Problem II: Fermion mass
我只能想到在经典力学范围内做这个测量---用另一个已知质量粒子来散射未知粒子. 可是从量子场论的观点来看, 这相当于引入新的相互作用项, 而散射概率给出这个作用项的耦合常数......迷惑中.
in principle, this is possible.
On other hand, in principle, mass is an asymptotic concept which is there for free particles as well. Therefore, we could measure it classically. In practice, we measure mass of a stable particle by classical means, such as bending of the trajectory in magnetic field and/or ionization rate in a some medium.
| 发表时间：2006-05-22, 14:47:53 |
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