In a certain lecture of Witten's about some QFT in 1+1 dimensions, I came across these two statements of regularization and renormalization, which I could not prove,
(1) ∫Λd2k(2π)21k2+q2i|σ|2=−12πln|qi|−12πln|σ|μ
(..there was an overall ∑iqi in the above but I don't think that is germane to the point..)
(2) ∫Λd2k(2π)21k2+|σ|2=12π(lnΛμ−ln|σ|μ)
I tried doing dimensional regularization and Pauli-Villar's (motivated by seeing that μ which looks like an IR cut-off) but nothing helped me reproduce the above equations.
I would glad if someone can help prove these above two equations.
Answer
Let's just look at the integral ∫d2k(2π)21k2+α2.
Addendum: After regularization we must renormalize. Using the minimal subtraction prescription we find ∫d2k(2π)21k2+α2→−12πln|α|μ,
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