Wick rotation and Minkowski/Euclidean space

Pablo88

Hello, I was wondering why in integrals such as
[tex]
\int d^4k F(k^2)
[/tex]
where [tex] k^2 = (k^0)^2 - |\vec{k}|^2 [/tex] ranges from -∞ to ∞, once the Wick-rotation is performed, we have [tex] -k^2_E = -(k^0_E)^2 - |\vec{k}_E|^2 [/tex] which lies in the (-∞,0) interval ... So, the contribution which lies in the part where [tex] k^2 >0[/tex] it seems that fades away. This function would have even poles. It is still a bit counter-intuitive for me that this part just fades away.

I guess we could shift the space-variables as [tex] \vec{k} \rightarrow \vec{k+k_0} [/tex] so this always lays in the euclidean space so to say. Anyway I am not sure, if this shift is always allowed.

Asides, things get more annoying for me if we have other four-vectors in the function, appart that the loop variable, which are Minkowskian, i.e. [tex] F((k+q)^2,(k-p)^2) [/tex] where [tex] q^2 = m^2, \ p^2 = M^2 [/tex] (both are positive quantities) ...

I find this kind of examples in triangle loops diagrams where the function F(k,q,..) is the form factor, and I this form factor is supposed to be the in the Spacelike or Euclidean region ...

Thanks in advance!