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I am embarrassed to ask this but I can't figure it out. Say we are in scalar QFT and we have a Feynman diagram like the following (straight from my course notes) enter image description here

Apparently this integral blows up for $large$ $q$ and that's why we need to do something like introduce a momentum cutoff. My misunderstanding here is that I see this integral blowing up for $small$ q and so clearly I don't understand this integral at all. Can somebody please be kind and help me out? I know this is probably a dumb question but I really need to understand this and I'm not getting very far on my own.

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It blows up for large $q$ because of the integral measure, $\operatorname{d}^4q$. One way of writing that after Wick rotation is $$\Omega_4 q^3 \operatorname{d}q,$$ which grows as $q$ for $q\gg m$. Note that $\Omega_4$ is the solid angle in $4$-dimensional space.

That comes from the fact that in hyperspherical coordinates the volume element is given by $$\operatorname{d}^4q = q^3 \sin^2(\phi_1)\,\sin(\phi_2) \operatorname{d}q \operatorname{d}\phi_1 \operatorname{d}\phi_2 \operatorname{d}\phi_3.$$ Doing the angular integrals yields $\Omega_4$ because the integrand is invariant under $4$-d rotations.

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  • $\begingroup$ Thanks. I think I'm struggling with generic misunderstanding of how to do relativistic integrals. Is there a resource you can point me to? I understand multidimensional calc but there's some kind of missing piece here for me. I don't understand why your answer is correct, even though I know it is (it's basically the next line in my notes). $\endgroup$ – SabrinaChoice Dec 11 '17 at 3:26
  • $\begingroup$ I don't have a handy link to a reference. Do you understand Wick rotations? I ask because Wick rotations turn relativistic integrals into Euclidean ones, so once you've Wick rotated, it becomes a straightforward application of Jacobians, and the definition of solid angle. $\endgroup$ – Sean E. Lake Dec 11 '17 at 3:29
  • $\begingroup$ Yes, not perfectly, but I get the general idea and they make sense. I suppose if I just think about it after a Wick rotation it all makes sense, but I struggle with a physical intuition. Thanks so much - I think your answer is what I needed since I hadn't refreshed my memory on Wick rotations until now. $\endgroup$ – SabrinaChoice Dec 11 '17 at 3:34
  • $\begingroup$ A small followup: are Wick rotations the main way to deal with relativistic integrals? $\endgroup$ – SabrinaChoice Dec 11 '17 at 3:40
  • $\begingroup$ @SChoice In QFT, I believe that's the case. I am not actually an expert on Wick rotations, just how to handle things once the rotation is done. $\endgroup$ – Sean E. Lake Dec 11 '17 at 3:42

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