In QFT writtern by Peskin and Schroeder, in page 293, PS wick rotate both time axis and momentum axis of correlation function of Klein-Gordon field, ie $$D_F=<0|T\phi(x_1)\phi(x_2)|0>=\int\frac{d^4k}{(2\pi)^4}\frac{ie^{-ik(x_1-x_2)}}{k^2-m^2+i\epsilon}\tag{9.27}$$ Denote $\Delta x \equiv x_1-x_2$ and assume the time order is $x^o_1>x^0_2$ such that $\Delta x^o>0$ After time-axis $x^o$ is wick rotated clockwisely, $x^o=-ix^o_E$, $(9.27)$ becomes $$D_F=\int\frac{d^4k}{(2\pi)^4}\frac{ie^{-ik^o(-i\Delta x^o_E)+i\vec{k}\vec{\Delta x}}}{k^2-m^2+i\epsilon}=\int\frac{d^4k}{(2\pi)^4}\frac{ie^{-k^o(\Delta x^o_E)+i\vec{k}\vec{\Delta x}}}{k^2-m^2+i\epsilon}\tag{1}$$ For this expression, I have my first question, $e^{-k^o(\Delta x^o_E)}$ diverge on the negative region of $k^o$, is it valid?
If we neglect this and proceed, since the pole of $k^o$ are $\pm E_p \mp i\epsilon$, we try to wick rotate $k^o$ anticlockwisely,
$$D_F=\int_{-i\infty}^{+i\infty}\frac{dk^o}{2\pi}\int\frac{d^3\vec{k}}{(2\pi)^3}\frac{ie^{-k^o(\Delta x^o_E)+i\vec{k}\vec{\Delta x}}}{(k^o)^2-\vec{k}^2-m^2+i\epsilon}\tag{2}$$ For this wick rotation, I have my second question, it involves two contours, one in first Quad, one in third Quad, let $k^o=\rho(cos\theta+isin\theta)$, $e^{-\rho(cos\theta+isin\theta)\Delta x^o_E}\gt\gt \rho^2 $ for $\rho\rightarrow\infty$ which diverge in the third Quad, why the rotation is valid?
Finally, if we still neglect that, let $k^o=ik^o_E$ $$D_F=(i)\int_{-\infty}^{+\infty}\frac{dk^o_E}{2\pi}\int\frac{d^3\vec{k}}{(2\pi)^3}\frac{ie^{-ik^o_E(\Delta x^o_E)+i\vec{k}\vec{\Delta x}}}{-(k^o_E)^2-\vec{k}^2-m^2+i\epsilon}\tag{3}$$ $$D_F=\int_{-\infty}^{+\infty}\frac{dk^o_E}{2\pi}\int\frac{d^3\vec{k}}{(2\pi)^3}\frac{e^{-ik^o_E(\Delta x^o_E)+i\vec{k}\vec{\Delta x}}}{k_E^2+m^2}\tag{4}$$ $k_E\Delta x_E$ has (-+++) metric, which should be wrong, but in general, if we wick rotate $x^o$ clockwisely for $\theta$ and $k^o$ anticlockwisely for $\theta$, then $x^o\rightarrow e^{-i\theta}x^o$, while $k^o \rightarrow e^{i\theta}k^o$, $x^ok^o$ is somehow invariant under simultaneous rotation, thus it is somehow make sense, and the metric can change only if we rotate both $x^o, k^o$ clockwisely or anticlockwiaely... What's wrong with my augument?
