Question

Inserting a trace property into a divergent loop integral - what exactly is being done here?

Answer 1

the trace property referred to be used as integral and escort the theorem of grace property the loop integral is divergent.

Answer 2

I'm reading through "H. Kleinert and V. Schulte-Frohlinde" notes for "Critical Properties of ϕ4-Theories", and I've reached this point in the lecture notes:

enter image description here

The trace property being referred to is using ∑μδμμ=D to write integrals like:

∫dDp(2π)Dpμpνf(p2)=1D∫dDp(2π)Dδμνp2f(p2)

I am really confused how the authors use this "trace property" to go from (8.73) to (8.75). I get lost after I replace a 1 underneath the integral with a "12D(∂pμ1∂pμ1+∂pμ2∂pμ2)". How do we get a factor of 1D−3 in the end? Are we differentiating somehow?

Can someone break down for me what is happening there? I've been scratching my head at this for too long.