Sunday, 1 November 2020

quantum field theory - A certain regularization and renormalization scheme


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(lnΛμln|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.

The other integrals should follow from this one. Introduce the Pauli-Villars regulator, d2k(2π)21k2+α2d2k(2π)21k2+α2d2k(2π)21k2+Λ2=(Λ2α2)d2k(2π)21(k2+α2)(k2+Λ2)=(Λ2α2)10dxd2k(2π)21(k2+β2)2=(Λ2α2)10dx122π(2π)20dk21(k2+β2)2=(Λ2α2)14π10dx1β2=(Λ2α2)14π10dx1Λ2x(Λ2α2)=12πln|α|Λ
Where we have combined denominators with the Feynman parameter x, with the intermediate variable β2=Λ2x(Λ2α2). Of course, this could also be approached with dimensional regularization with the same result.


Addendum: After regularization we must renormalize. Using the minimal subtraction prescription we find d2k(2π)21k2+α212πln|α|μ,

as required.


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