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November 22, 2024

Standard Model as Effective Field Theory

- sage -

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

二零零五年六月十五日