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)4ik2−m2+iϵwhereημν=diag(−1,1,1,1)
Method #1 - Residue Theorem:
Since I=i∫d3→k(2π)3∫+∞−∞dk02π[(2k0)−1k0+z0+(2k0)−1k0−z0]wherez0=√|→k|2+m2−iϵ
Method #2 - Wick Rotation:
Drawing the poles, −z0,z0, one finds the integration contour can be rotated anticlockwise so that, I=i∫d3→k(2π)3∫+i∞−i∞dk02π1k2−m2+iϵ=−i∫d3→k(2π)3∫+∞−∞idk42π1k2E+m2
Answer
I don't think it is exactly the same regulator: In the first method, you integrate ∫∞−∞dk0∫Λd3k, but in the second calculation you integrate ∫Λd4kE.
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