Why are dimension > 4 operators non-renormalizable?

In summary, the Lagrangian terms with mass dimension greater than four are considered "nonrenormalizable" because the action must be dimensionless and the coupling must have negative dimension in effective field theories. This leads to a divergence in the renormalization group flow for such coupling constants, as explained in chapters 4 and 10 of Peskin and Schroeder's book.
  • #1
fliptomato
78
0
Hi everyone--I'm curious why terms in the Lagrangian with mass dimension greater than four are "nonrenormalizable."

I understand that the action must be dimensionless, hence the Lagrangian [density] has mass dimension 4. However, in effective field theories, we can end up with terms of dimension > 4, hence the coupling must have negative dimension. What's so bad about this?

(I guess somehow the renormalization group flow for such coupling constants diverges a mass scale given by the coupling?)

Thanks,
Flip
 
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  • #2
fliptomato said:
(I guess somehow the renormalization group flow for such coupling constants diverges a mass scale given by the coupling?)

It is exactly that ! It is explained for example in Ch 12 of Peskin and Schroeder, although quite sketchy.

cheers,
Patrick.
 
  • #3
vanesch said:
It is exactly that ! It is explained for example in Ch 12 of Peskin and Schroeder, although quite sketchy.

cheers,
Patrick.

Peskin and Schroeder first talks about this in chapter 4 after introducing
the φ4, QED and Yukawa interaction terms, See bottom of page 79.

Then there's more in chapter 10.

PS: Thanks to Google-Print we may hope to link directly to the appropriate
book pages like in this example here:

http://print.google.com/print?id=ZbTXdWsrsAEC&lpg=237&dq=renormalizable+higher+order+interaction+term&prev=http://print.google.com/print%3Fie%3DUTF-8%26q%3Drenormalizable%2Bhigher%2Border%2Binteraction%2Bterm%26btnG%3DSearch&pg=237&sig=5Vrhek6UvWRHsdqimJlTP0ACCrY [Broken]

Regards, Hans.
 
Last edited by a moderator:

1. What is the meaning of "dimension > 4 operators" in the context of renormalizability?

Dimension > 4 operators refer to terms in a quantum field theory Lagrangian that have a mass dimension greater than 4. These operators are typically associated with high energy or short distance physics and can lead to non-renormalizable theories.

2. Why are dimension > 4 operators non-renormalizable?

Non-renormalizable theories are those in which the interactions between particles become infinitely strong at high energies. This means that the theory breaks down and cannot make consistent predictions. Dimension > 4 operators contribute to this problem because they introduce terms that grow faster than any power of energy, making it impossible to properly renormalize the theory.

3. How do dimension > 4 operators affect the renormalization of a theory?

When dimension > 4 operators are present in a theory, the renormalization process becomes more complicated. These operators introduce new divergences that cannot be absorbed by the standard renormalization procedures, leading to non-renormalizable theories. This makes it difficult to make meaningful predictions about the behavior of the theory at high energies.

4. Can dimension > 4 operators be removed from a theory to make it renormalizable?

In some cases, it is possible to remove dimension > 4 operators from a theory by using the Higgs mechanism or other techniques. However, this is not always possible and can lead to significant changes in the behavior of the theory. In general, it is better to start with a renormalizable theory and then add higher dimensional operators if needed, rather than trying to remove them later.

5. How do dimension > 4 operators affect our understanding of the universe?

Dimension > 4 operators are important for describing high energy physics, such as the behavior of particles at the energies present in the early universe. However, the presence of these operators makes it difficult to make precise predictions about these high energy regimes. This means that our understanding of the universe is limited and could potentially change as we gain more knowledge about these operators and their effects on physical theories.

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