What is an effective field theory

In summary, step 2 of the effective field theory procedure is to redefine the coupling constants in the Lagrangian so that they include corrections from the cutoff to infinity. This allows the effective field theory to be derived without the need to integrate to infinity. By finding the cutoff and evaluating the parameters at that point, it is possible to determine if the theory is a complete or effective field theory.
  • #1
RedX
970
3
Can anyone tell me if this is the basic concept behind effective field theories:

1) You begin with your original Lagrangian, and you want to construct an effective Lagrangian out of it.

2) You redefine your coupling constants in your original Lagrangian so that they include the integral from a cutoff to infinity (i.e., you combine the higher order corrections into the coupling constants, but only from the cutoff to infinity). This way, when you make an effective Lagrangian, you no longer need to integrate to infinity, since this was already done in the coupling constants.

3) Step 2 isn't perfect for some reason (why?), so your effective Lagrangian needs to include not just a redefinition of the coupling constants in your original Lagrangian, but new terms (an infinite amount of them) whose coupling can have negative mass dimensions.

4) As of yet all your parameters are undefined - you didn't specify a mass scale or anything. You decide that the mass scale is the cutoff. At this point you could determine the value of all your parameters evaluated at the cutoff by comparing to experiment?

5) Now you integrate below your cutoff, and in doing so you derive a renormalization group equation for all your coupling constants.

6) Now you're done and can see how parameters in your effective Lagrangian changes as you vary the cutoff.

What's the point of all this? What's this used for that regular renormalization can't do?
 
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  • #2
I don't know this stuff very well, but I'll give it a try. As I understand it (and maybe I don't), step 2 should be more like this: Choose a cutoff and split the integrals into two parts, integration up to the cutoff, and integration above the cutoff. Then do the integrals above the cutoff so that only the integrals up to the cutoff remain. The result of the integration that was performed shows up as an adjustment to all of the coupling constants in the Lagrangian that appears in the integrals up to the cutoff. Note that it's not just the coupling constants in the renormalizable terms that get adjusted. The new Lagrangian contains all the terms that are consistent with the symmetries of the original Lagrangian, including an infinite number of non-renormalizable terms.

The fact that any adjustment of the cutoff changes all the coupling constants suggests that it doesn't really make sense to start with a renormalizable Lagrangian, but it can also be shown that non-renormalizable terms have a negligible effect in experiments performed at low energies. I don't remember the details of that argument, but it was based on order-of-magnitude estimates that you can make once you have figured out the dimensions (units) of the coupling constants.
 
  • #3
The infinite number of additional terms that you have to add in an effective Lagrangian do seem to be negligible at low energies.

Does this mean that the Standard Model Lagrangian might really be an effective field theory, and we just don't know it because all the experiments are at low energy? If this is true, how do we know if our theory is a complete theory, or an effective field theory? Take phi^4 theory. This theory is perfectly renormalizeable. But if you construct an effective field theory from it, you can find through a renormalization group equation that the cutoff can't be taken to infinity. Isn't that an inconsistency? Before performing an effective field theory analysis on it, you were integrating out to infinity perfectly fine, with no worries about a cutoff or anything. Anyways, does this finite cutoff that you find equal the symmetry breaking scale? Is this how unification is done?
 

Related to What is an effective field theory

1. What is an effective field theory?

An effective field theory (EFT) is a theoretical framework used in physics to describe interactions between particles. It is based on the idea that there are different energy scales in a system, and at each energy scale, a different set of particles and interactions dominate. EFTs are used to simplify complex systems and make predictions at a specific energy scale.

2. How does an effective field theory differ from other theories?

EFTs differ from other theories, such as quantum field theory, in that they are not concerned with describing the fundamental laws of nature, but rather the behavior of a system at a specific energy scale. EFTs are effective because they allow for simpler calculations and predictions without needing to know the underlying fundamental theory.

3. What are the advantages of using an effective field theory?

One advantage of using an effective field theory is that it allows for easier calculations and predictions at a specific energy scale. It also allows for the inclusion of new particles and interactions that may not be included in the fundamental theory. Additionally, EFTs are more flexible and can be used to describe a wide range of physical phenomena.

4. How are effective field theories tested and validated?

EFTs are tested and validated by comparing their predictions to experimental data. If the predictions match the experimental results, it provides evidence that the EFT is an accurate description of the system at that energy scale. EFTs can also be validated by comparing them to other theoretical frameworks and seeing if they produce similar results.

5. Can effective field theories be used in all areas of physics?

EFTs are primarily used in high energy physics, but they can also be applied to other areas of physics, such as nuclear and condensed matter physics. However, their effectiveness may vary depending on the specific system and energy scale being studied. EFTs are constantly evolving and improving, so their applications may expand in the future.

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