What kinds of divergences for a given interaction

[copernicus1]

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.)?

 

[Einj]

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.

 

[copernicus1]

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?

 

[Einj]

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.