Question

does this expression appear in renormalization?

Answer 1

It's a good question. I don't know of any emergence, mainly because it's unusual for contributions from momenta nn to decrease like 1/n1/n etc. (it is easier to get positive powers of nn from derivatives etc.). The log-based generalization is interesting but slightly less well-defined. For a<0a<0, there has to be a discontinuity and branching due to logarithms of negative numbers etc. – Luboš Motl Apr