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First of all, Quantum Field Theory is not my field of research. However, I have to investigate on some problems in QFT and I'm trying to get familiar with it again.

I'm basically working with scalar fields and I encounter some problems in dealing with renormalization, counterterms and so on.

To make my point clear, I will cite the two loop correction to the propagator in phi 4 theory, the so called "sunset diagram" (see for example, Peskin and Schroeder problem 10.3). To solve it, we proceed as usual, we make use of the Feynman trick to join the denominators, we integrate over momenta and we are left with an integral in the parameter $x$ which runs from 0 to 1 and have a momentum dependence of (p^2)^(n-3) (where $n$ denotes the number of dimensions including time).

So, in $n=4$, we get a correction proportional to $p^2$. The divergent part of it can be removed by means of the field strength renormalization counterterm. What happens in $n=3$ and $n=2$ ?

Explicitely, in $n=2$, we get a momentum dependence of $(p^2)^{-1}$. What does this correction represent? If we do not use counterterm renormalization, what is the meaning of such a contribution?

I'm aware that the divergence of a diagram depends strongly on the number of dimensions we are working in, but I simply don't understand what the resulting corrections to the propagator mean - if they renormalize the field strength, mass, etc.

I would really like to hear your opinions and suggestions.

Kindest regards !