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
Standard Model and Renormalizability
Let me open a new thread to discuss the renormalizability of the Standard Model as raised by sage. Since sage has clarified his opinion in the other post, and his description is self-consistent (according to HIS definition of standard model which we will discuss below), this post is only about a few confusing issues I think sage's statement "standard model is non-renormalizable" may cause.
It is certainly true that a theory beyond standard model will be needed for many reasons, and a theory that looks like "321 + non-renormalizable terms" will necessarily emerge if one looks from quantum gravity point of view, or other point of view that is based on physical knowledge beyond standard model. But that's not what "renormalizibility" (therefore "non-renormalizibility") means. The term "renormalizability" is used to refer to theories that by THEMSELVES don't contain and will NOT induce non-renormalizable terms.
For instance, Fermi theory DOES contain non-renormalizable term, and DOES induce more such terms as energy goes up. That's why we call Fermi theory non-renormalizable. Standard Model (as defined in the traditional sense, to be discussed below), on the other hand, does NOT contain such term and will NOT induce such terms BY ITSELF. That's what we mean when we say Standard Model is renormalizable, and that is independent of whether standard model should be a final theory or part of an effective theory.
Of course, as sage correctly pointed out, there is the Landau pole that predicts the break down of perturbation method at a high energy scale even WITHOUT physical knowledge about anything beyond standard model (such as gravity). But Landau pole is NOT an unambiguious indication of non-renormalizability. In fact, it's mere existence is not without debating, many physicists believe the existence of such a pole in perturbation theory at a ridiculously high energy (10^280 GeV, Planck scale is nothing compare with it) only means that perturbation method itself has limitation. Renormalizability, on the other hand, is refering to the fact that cut-off dependent features that arise in perturbation theory can be obsorbed into a finite number of parameters WITHIN the regime of perturbation theory. So the two are not really talking about the things in the same area.
:: I call standard model: 321+non-renormalizable terms (with 321
:: symmetry). In this sense, it is the same as QED+Fermi theory
:: (or more general non-renormalizable terms).
:: You probably want to call the 321 piece Standard Model. historically
:: this was the case. I just don't think it is a very useful statement since
:: as a full theory, there is no such a separation.
That (the 123 piece) is indeed what I mean by standard model, and I think it is still what most people, most articles and textbooks mean when they use the term standard model, therefore it is still a meaningful and non-obsolete concept. It's just like although we may have QED+Fermi to describe low energy electroweak phonomena, it doesn't mean when we talk about QED we should include Fermi part and call QED a non-renormalizable theory because of this. Pure QED remains to be a meaningful and widely used concept, and is a renormalizable theory. So does standard model (in the sense of 123).
Sage's statements are all correct if we call "321+non-renormalizable terms" standard model. But I don't think it is appropriate to define the term standard model in this way. We do say that standard model is only part of an effective theory that contains non-renormalizable terms, but those terms are usually treated as corrections to the standard model (which, by definition, means they are not part of the standard model), we usually don't extend the meaning of standard model to INCLUDE those terms. Therefore I feel statement such as "standard model is non-renormalizable" may cause confusions.
:: It is true that we can consider standard model as a low energy effective theory of a more
:: general theory, in which we will probably see terms - IN ADDITION TO standard model terms
:: - like 1/M_P^2, but the standard model piece is still renormalizable.
:: ==============================================================
:: I don't know in what sense you could talk about those pieces separately. They mix under the
:: renormalization.
If you start with Standard model Lagrangian, renormalization will never induce terms like 1/M_P^2. There is no mix with 1/M_P^2 terms or other non-renormalizable terms under renormalization.
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：2005-04-07, 07:08:40 |
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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: Standard Model and Renormalizability
A few more words about treating SM as an effective field theory.
Another reason that I don't think we should define standard model as "123+non-renormalizable terms" even if we treat standard model as an effective field theory (which IS indeed the modern view), is because non-renormalizable terms are NOT the only terms that will emerge into the traditional standard model Lagrangian (the "123" piece). The whole MSSM, if exists, will appear as renormalizable corrections to standard model far before any non-renormalizable terms representing quantum gravity effects might become visible. Since we certainly don't call MSSM as standard model, therefore it doesn't make much sense to extend the meaning of standard model merely to include non-renormalizable terms, while omitting renormalizable MSSM corrections that are far more important in the regime standard model applies.
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：2005-04-07, 10:24:05 |
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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: Standard Model and Renormalizability
我这里谈一点跟上面论题关系不大的，但希望能够看到各位看法：
我今天早上在家里，临时把S.Weinberg的quantum field theory第一卷找出来翻了一下，那里有一节专门讲重整化不是必要的，没有来得及细看，但感觉这或许是一家之言——他的逻辑好像是：由于自然界存在引力这样的不可重整化的场，所以说重整化不是必要的（照这种逻辑，那干脆就不要什么Higgs机制了，直接说规范不变不是必要的，因为矢量场质量项破坏规范不变性，呵呵，多半是我这个地方没来得及看清楚）。另外他认为，如果拉氏量里面包含的作用项有无穷多项（这样就会包含很多不可重整项，如高阶导数项），那么即使有无穷多个基本发散图，也可能找到无穷多个抵消项（甚至不同作用项之间的发散有时候可能相互抵消？）。我觉得这个想法倒是有趣——
函数有一般函数和特殊函数，后者没有解析表达式，而是用一般函数做无穷多项的展开（例如Bessel函数）；拉氏量相当于场量及其导数的泛函，一般的都是“普通泛函”，一个简单的解析表达式就可以搞定。如果同样有“特殊泛函”的拉氏量，需要用无穷多项的场量及其导数的级数展开，那么或许可以推广传统的重整化方法，使得引力问题也得到解决（当然这样需要改写引力场方程，至少需要改变表达形式）。我先前说引力量子化的困难与其说是不可重整，不如说我们对重整化的认识不够，就包含这种意思：也许重整化还有待进一步发展，从而使得引力在新的重整化意义上成为可重整的；而不是削足适履，改变引力理论来迁就现有的重整化理论。
还有Weinberg好象在说，传统可重整化的理论之所以正确，因为我们的实验能量标度没有到达那么高的水平，否则仍然是不可重整的。这些跟前面sage兄讲的好像类似。
我猜想Weinberg上面会不会是这种意思：例如，完全的Feynman自能图，利用链近似来求的时候，采用了公式1/(1-x)=1+x+x^2+x^3＋x^4+...，其中x在高能极限或尺度趋于零时是趋于无穷大的量。但是在远远超出我们目前实验所能达到的能量标度上，x仍然还是可以作为小量来处理，从而公式1/(1-x)=1+x+x^2+x^3＋x^4+...的采用是合理的，结果必然与当下的实验结果相符合；但如果趋于Planck尺度，那公式不适用，理论结果必然与实验不符。这种公式的采用，是当年我的最大迷惑。当然直接用维数重整化和抵消项方法，重整化给人的感觉很简单，数学上找不出漏洞，物理意义上，无非就是引进裸量的假设，裸量中的无穷大与辐射修正中的无穷大相抵消，给出我们观察到的有限结果。这样重整化不仅仅是数学处理，也是物理的东西。而且裸量假设意味着一个物理量的大小依赖于我们观测的方式，例如对电荷我们观测的距离越近，结果越大，因为越接近裸量。
对重整化的认识，我经历了三个过程：初期－中期－返回初期，初期的看法就是我上个帖子里表达的；中期是我认为我都理解了，重整化其实不难，无论物理还是数学上没有任何问题。量子场论的书，应该我学的主要还是“现代”的，基于路径积分和维数重整化的。
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2005-04-08, 01:09:02 |
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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: Standard Model and Renormalizability
S. Weinberg 及 sage 兄介绍的对有效场论及重整化的看法是现代的观点。我上面试图说明的只有一点，那就是由传统的拉氏量所定义的标准模型符合可重整理论的定义，因此直接说“标准模型不可重整”容易引起误会。除此之外，我对他们的说法并无异议。我想，Weinberg 指的是由于引力的存在，一个完整（或比较完整）的有效场论必然包含被标准模型所忽视的不可重整的附加项。这样的有效理论在那些附加项被 suppress 的能区象是一个可重整理论，其实却不是。但我不认为这等价于说“标准模型不可重整”，因为“标准模型”并不等价于该有效场论，而且也不宜把标准模型重新定义为该有效场论，理由如我上面的帖子（尤其是第二个）中所叙述的。
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：2005-04-08, 07:01:09 |
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kanex
发表文章数: 860
武功等级: 弹指神通
(第六重)
内力值: 343/343
Re: Standard Model and Renormalizability
其实我们也可以换种角度思考重整化，那就是重整化不外乎：要求Lagrangian中不出现过高次导数，且耦合参数的单位也不能超过一定限度，总之就是一过mass^4就爆炸了。
如果这样来看，“好的理论都是可重整化的”也许就并不太神秘。
换个话题，我觉得2这一块并没有完全建立起来。没有人能解释weak eigenstate和mass eigenstate之间的神秘混和，就如同没有人能解释弱力的三个vector boson的质量起源－－我对higgs机制抱怀疑态度，总觉得它更像是数学游戏，因为照它这样方法做的话什么对称都可以硬来自发破缺，而我们却从未观测到。
江畔何人初见月`江月何年初照人`
| 发表时间：2005-04-10, 16:03:51 |
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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: Standard Model and Renormalizability
First, to avoid repetition, let me say what I do agree with you.
1) I completely agree with your technical definition of renormalizability,
i.e., what a renormalizable theory is.
2) I agree that the Standard Model as you defined is renormalizable.
Next, I do want to clarify a couple further points (I am not implying that you
will not agree with these).
1) let's forget about what should be called the Standard Model (It is just
a name anyway). Let's instead ask what is the Lagrangian we will write down
which best summarizes our current state of knowledge. It will of course
include the renormalizable standard Model terms. On the other hand, we know
we cannot stop here, for the following reasons
First, we know there are higher scales, at least the
Planck scale. Second, just our ignorance of physics at higher energy scales
will force us to include higher dimension (non-renormalizable) operators with
undetermined scales. Third, all the theories we have developed so far are
indeed just effective field theories applicable to a finite range of scales.
Fourth, renormalization itself implies the existence of
higher scales (more on this later in the post).
Therefore, a Lagrangian which represents our current understanding of particle
physics will be Standard Model+non-renormalizable terms. Knowing the gauge
symmetries of the Standard Model means such symmetries are embedded in
the theory
of higher scales, which in turn means that the non-renormalizable terms in the
effective Lagrangian sould respect the gauge symmetries of the Standard Model.
2) Historically, renormalizable Standard Model has been a very useful
concept. Just renormalizable part of the Lagrangian seems to describe the
experimental results quite well.
A purely renormalizable theory is only useful (and only in an
approximate sense) when there is such separation of mass scales.
This is an important point. A truncation of
the Lagrangian down to renormalizable terms is in some sense the leading order
in the effective field theory calculation. The level of correctness of such
a calculation will depends on the value of the higher mass scale. It
is correct in the limit that the next scale in physics is much much higher.
From the point of view of a effective field theory, this shows the
existence of a separation of mass scales between the scale of Standard
Model (~100 GeV) and the scale of next layer of physics (at least
higher by the order of an one-loop ~16 pi^2 factor).
We have been lucky with the Standard Model in the sense that there is
such a separation of scale in nature around electroweak scale. On the
other hand, there is absolutely no reason to think this is indeed
going to be the case. The next layer of physics around the next scale
does not have to have such a big separation.
In this case, the next effective Lagrangian we write to summarize our
knowledge of high energy physics does not have to be, and possibly can
not be, just renormalizable at all. This is similar to the situation
historically after we saw beta-decay but before we saw the W,Z
bosons. Then, all we can write down is QED+Fermi theory.
Of course, because we are imaginative, we can always imagine there might be some
separation of scales further up so that we could try to write a
approximate renormalizable Lagrangian. This is what
happened for the Standard Model. However, this does not have to happen again.
3) Is there anything wrong with a effective field theory with some
non-renormalizable terms?
The answer is absolutely not if we want a predictive theory which can
produce accurately predictions within experimental limits. This is
the best one can hope for anyway for any theory before of final theory of everything.
Effective field theory is predictive not because it is renormalizable,
it is NOT. However, at the applicable scale of a effective field
theory, it gives a well defined power expansion of Lagrangian in terms
of a higher mass scale. Knowing the expansion of the Lagrangian to a
certain order, we could compute observables to the accuracy of this
order. One of the prime example of effective field theory is
actually the Fermi theory (I mean in a general sense all the terms at
the order of G_F is included). It can give accurate result for all
observables measured at scales below 100 GeV.
4) In a more general sense, renormalization only makes sense when
there is a high scale.
A common claim is that an effective field theory is inferior
to a purely renormalizable one since it breaks down at
certain scale. However, nothing can be further away from the
truth.
Renormalizable theory by themselves is not consistent without the
existence of high scales. As very well illustrated in Wilson's
approach to renormalization, renormalization procedure makes
perfect sense only when there is a high scale at which the bare
couplings are defined. The subtraction procedure corresponds to
integrating out high momentum scales down to renormalization
scale. (for those of you who have not seen this, the field theory
book by M. Peskin has a nice introduction to it).
if we are really looking for a final theory from which everything
can be derived and nothing left to be explained, we should look for
a finite theory (such as something like string theory), not a
renormalizable one.
Again, due to the existence of higher scale, we have to include
higher dimensional operators in the Lagrangian if we want to make
more precise predictions. Therefore, with some twist of the
argument, renormalization is only useful if the theory is
fundamentally non-renormalizable.
Now, a couple slight more technical points.
1) MSSM is not fully renormalizable since we do not include in it the
degrees of freedoms which breaks supersymmetry (while in
renormalizable Standard Model we include the Higgs). They are assumed to
be integrated out at some higher scale. And, we do not know where
the supersymmetry breaking scale is. In principle, we should include
all the operators suppressed by the SUSY breaking scale. One of the
common practice is assuming that SUSY breaking scale is high and we
can safely through away all the other operators. However, this is
again just a hope for the existence of scale separation.
There is nothing wrong to hope and we do it all the time.
However, strictly speaking, MSSM is not analogous to the
renormalizable Standard Model.
2) By renormalizable Standard Model, I assume you mean 321 piece with
a Higgs. This is indeed the common assumption. However, we have not
discovered Higgs yet. All we can confidently write down is indeed an
non-renormalizable Lagrangian with Higgs (or whatever responsible
for electroweak symmetry breaking) integrated out. This is an
non-renormalizable theory. Again, nothing wrong with it. The only
thing we know is that something has to come in below 1 TeV to
unitarize the WW->WW scattering amplitude.
3) Landau pole for the top Yukawa coupling is much closer.
4) I mention an example where renormalizable theory is
useless. Consider the alternative theory of electroweak symmetry
breaking called technicolor. They postulates the existence of other
gauge interaction at high scales which become strongly coupled at
around electroweak scale (at least in one version of technicolor),
very much like QCD confinement. However, because of the strong
interaction, its low energy physics cannot be computed directly
from the renormalizable theory. the only way to do it is to build a
effective field theory based on symmetries and try to make
predictions. (This is completely analogous to hadron spectrum
cannot be computed from QCD directly, although lattice calculation
is getting close. the only analytical handle we have is chiral
perturbation theory)
For those of you who are interested in effective field theory, I
include here a couple of references
1) a review article by Howard Georgi which could be found on
http://schwinger.harvard.edu/%7Egeorgi/index.htm
2) Aneesh V. Manohar, e-Print Archive: hep-ph/9508245
| 发表时间：2005-04-11, 10:27:14 |
作者资料
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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: Standard Model and Renormalizability
This is certainly the most quality and valuable post on the forum. Thanks a lot for putting together, sage! :)
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：2005-04-11, 12:05:25 |
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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: Standard Model and Renormalizability
呵呵，好的论坛如同第二课堂，而且有些东西是课堂上学不到的。每个人都有自己的心得、体会，有自己的诀窍（trick），两人交流后一份变两份，N个人交流之后每人获得N份。
有时候，甚至一个90％在胡说的人，他也可能给出一些能启发别人的东西，或者让原来自己没有仔细思考过的东西变得更加深思熟虑，更加严密。有些东西原以为自己很了解了，结果通过讨论，让别人步步追问，才知道自己原来有些遗漏。总之学术交流和讨论，是及其有价值的。
我把这个页面给下载了。这个是繁星论坛两大最顶尖高手的到目前为止的最顶尖对话:-)（可惜的是，没法把中间我那个帖子删掉，不过作为扶鲜花之绿叶、捧明月之群星，也未尝不可，呵呵）。
唯有与时间赛跑，方可维持一息尚存
| 发表时间：2005-04-11, 21:36:53 |
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