What kinds of divergences for a given interaction

  • Thread starter copernicus1
  • Start date
  • Tags
    Interaction
In summary, the level of divergence in diagrams depends on the particular interaction term in the Lagrangian, with higher numbers of loops resulting in higher degrees of divergence. The dimensions of the coupling in the interaction term can determine if the theory is renormalizable or not. There is a relationship between the divergence in a single-vertex interaction and interactions with higher numbers of vertices, which can be determined through a power counting procedure that considers the number of derivatives, internal lines, vertices, loops, and external lines. This method is further explained in T. Muta's "Foundation of Quantum Chromodynamics" in chapter 2.5.
  • #1
copernicus1
99
0
Can you look at an interaction term in your lagrangian or hamiltonian, like L_{\rm int} or H_{\rm int}, and say immediately how its diagrams will diverge (as in quartic, quadratic, linear, log, etc.)?
 
Physics news on Phys.org
  • #2
The level of divergence depends on the particular diagrams. An interaction term in the Lagrangian is just one vertex that may compose your diagram. For example, in general, diagrams with higher number of loops have higher degree of divergence.

From the interaction term you can usually determine if you theory is renormalizable or not by looking at the dimensions of the coupling. For example, theories with dimensionless couplings are renormalizable.
 
  • #3
Great thanks. Is there a relationship though between the divergence in the single-vertex interaction and the interactions with higher numbers of vertices? Like, if a single-vertex diagram has a quadratic divergence, would a two-vertex diagram have a quartic divergence?
 
  • #4
If you assume that your theory only has one kind of interaction vertex then you can always perform a power counting procedure in a general fashion. This procedure clearly depends on you vertex but in order to do that you need to consider:

1) The number of derivatives contained in your vertex
2) The number of internal lines
3) The number of vertices in a given diagram
4) The number of loops
5) The number of external lines

Note that some of these quantities can be related with each other. If you want to see a very neat application of these kind of methods you can look into T. Muta - "Foundation of Quantum Chromodynamics", in particular Ch. 2.5.
 

1. What are the different types of divergences for a given interaction?

There are three main types of divergences for a given interaction: UV (ultraviolet), IR (infrared), and Logarithmic. UV divergences arise when the energy of a particle becomes very high, causing the interaction to become infinite. IR divergences occur when the energy of a particle becomes very low, leading to a similar infinite behavior. Logarithmic divergences arise when the energy is neither too high nor too low, but rather at an intermediate scale.

2. How do these divergences affect our understanding of the interaction?

The presence of divergences in a given interaction can make it difficult to accurately describe and predict the behavior of particles involved. They can also cause theoretical predictions to become infinite, making it necessary to use mathematical techniques such as renormalization to remove these divergences and obtain meaningful results.

3. Can divergences be avoided in interactions?

While some interactions may naturally produce divergences, it is possible to avoid them by choosing specific values for the parameters involved in the interaction. This is known as fine-tuning and can be used to eliminate or reduce the effects of divergences.

4. How do we deal with divergences in our calculations?

To deal with divergences in calculations, scientists use techniques such as dimensional regularization and renormalization. These methods involve redefining the parameters involved in the interaction to eliminate or reduce the effects of divergences. This allows for more accurate predictions and descriptions of the interaction.

5. How do divergences impact the development of new theories?

Divergences play a crucial role in the development of new theories in physics. By studying the divergences that arise in different interactions, scientists can gain insights into the underlying principles and mechanisms at work. This can lead to the development of new theories and models that better describe and predict the behavior of particles and interactions.

Similar threads

Replies
4
Views
785
  • Quantum Physics
Replies
10
Views
1K
  • Quantum Physics
Replies
6
Views
734
Replies
4
Views
1K
Replies
1
Views
633
Replies
2
Views
893
Replies
33
Views
3K
  • Quantum Physics
Replies
2
Views
702
Replies
16
Views
1K
Replies
1
Views
1K
Back
Top