Tuesday, 28 November 2017

quantum field theory - Two math methods apply the same loop integral lead different results! Why?


I tried to adopt the cut-off regulator to calculate a simple one-loop Feynman diagram in ϕ4-theory with two different math tricks. But in the end, I got two different results and was wondering if there is a reasonable explanation.



The integral I'm considering is the following I=Λd4k(2π)4ik2m2+iϵwhereημν=diag(1,1,1,1)

Λ is the cu-off energy scale and ϵ>0. Then I do the calculations.


Method #1 - Residue Theorem:


Since I=id3k(2π)3+dk02π[(2k0)1k0+z0+(2k0)1k0z0]wherez0=|k|2+m2iϵ

choosing the upper contour in k0-complex plane which encloses the pole, z0, we have I=d3k(2π)312πidk0(2k0)1k0+z0=12d3k(2π)31k2+m2=14π2Λ0k2dkk2+m2=18π2[Λ21+m2Λ2m2ln(Λm)m2ln(1+1+m2Λ2)]18π2[Λ2m2ln(Λm)m2ln2]


Method #2 - Wick Rotation:


Drawing the poles, z0,z0, one finds the integration contour can be rotated anticlockwise so that, I=id3k(2π)3+iidk02π1k2m2+iϵ=id3k(2π)3+idk42π1k2E+m2

where k4=ik0 and k2E=k2, which are 4d Euclidean variables. So we have I=d4kE(2π)41k2E+m2=116π2Λ20k2Ed(k2E)k2E+m2=18π2[Λ22m2ln(Λm)m22ln(1+m2Λ2)]
Comparing the results obtained from the above two methods, we will find only the lnΛ dependent parts are the same; Other two parts (Λ2-dependence and finite piece) are different. Since I use the same regulator, it's a bit wired to me how could the math tricks affect the results.



Answer



I don't think it is exactly the same regulator: In the first method, you integrate dk0Λd3k, but in the second calculation you integrate Λd4kE.


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