Question about this interaction in QFT

[giova7_89]

Question about this "interaction" in QFT

Hi, I started two months ago my course in QFT, and since I heard about the fact that the bare mass appearing in the Lagrangian of a theory isn't the physical mass of a particle (due to self interaction, I guess), I tried to find an example explicitly solvable where one could see this effect directly. That is find the one particle states of the theory and their energy, E^2= p^2+m^2_physical.

To do so, the simplest model i could think of was the "generalization" of the external current problem presented at the end of the second chapter of Peskin & Schroeder. That is I added to that Lagrangian the free term for the scalar field j(x).

I'm attaching a short .pdf (5 pages) with my calculations and conclusions. The "conclusions section" of my .pdf was a bit rushed, since I hoped that we could discuss it directly on the forum (the very last line, however, seemed very important to me, since it could help to answer some of my other questions i wrote in the .pdf)...

 

[fzero]

Your intuition in the last sentence is correct. This is not an interacting theory at all. Once you rewrite the Lagrangian in terms of the eigenvectors of the mass matrix, you find two free bosons.

If you really wanted to see mass renormalization, you could read on later in the book where a boson with the self-interaction [itex]V = \lambda \phi^4[/itex] is considered. After reading that, you could work out the case of two bosons coupled by [itex]V = \lambda \phi_1^2\phi_2^2[/itex] as an example.

 

[giova7_89]

Ok, but I tried that example because I was able to treat it nonperturbatively (after all we agreed that it is a free theory!). We just started feynman diagrams in our course and didn't get to renormalization theory yet (I don't even knopw if we will do it in this course) so my knowledge of these things is limited.

Thanks for the reply, however.

May I ask if someone could tell me if there are some articles or references where there is an exact solution for a theory, presented in a "simple way" (that is, a way that I can follow with my current knowledge)? I'd like to have that because I always understand things better if I have a "exact model" to refer to...

 

[fzero]

Ok, but I tried that example because I was able to treat it nonperturbatively (after all we agreed that it is a free theory!). We just started feynman diagrams in our course and didn't get to renormalization theory yet (I don't even knopw if we will do it in this course) so my knowledge of these things is limited.

Thanks for the reply, however.

May I ask if someone could tell me if there are some articles or references where there is an exact solution for a theory, presented in a "simple way" (that is, a way that I can follow with my current knowledge)? I'd like to have that because I always understand things better if I have a "exact model" to refer to...

As far as I know, there aren't any exactly solvable interacting QFTs in four dimensions.

In two dimensions (one space + time), there are exactly solvable models, at least in the sense that exact correlation functions can be written down. The best studied examples are the Thirring and sine-Gordon models, which can actually be shown to be different, "dual," formulations of the same physics. There is a very brief review in ch 2 of this dissertation, but I'm not sure how easy it will be to follow along. Other references are the chapter on "lumps" in Coleman's "Aspects of Symmetry," as well as Rajaraman, "Solitons and Instantons."