How to do wick rotated surface int (srednicki ch75)

[LAHLH]

Hi,

I have the integral:

$$\int\,\frac{d^4 l}{(2\pi)^4)} \frac{\partial}{\partial l^{\beta}} f_{\alpha}(l)$$

Now apparently this can be written (using a Wick rotation and converting to a surface integral) as:

$$i \lim_{l\to\infty}\int\,\frac{\mathrm{d}S_{\beta}}{(2\pi)^4}\,f_{\alpha}(l)$$ where $$\mathrm{d}S_{\beta}=l^2 l_{\beta} d\Omega$$ is a surface-area element and $$d\Omega$$ is the differential solid angle in 4d.

Can anyone see how exactly? (if context is needed this in (75.41) of Srednicki's QFT available free online)

 

[LAHLH]

What I have already, well I understand the concept of a Wick rotation, but is this in the $l^{0}$ plane, such that we would write the Euclideanized variables as $l^{0}=i \bar{l^{0}}$, $l^{j}=\bar{l}^{j}$ and the Minkowski square $-(q^{0})+(q^{1})^2+...$ becomes the Euclidean +++... square or something else?

Then exactly how are we converting this integral to a surface integral? and how do we still have $f_{\alpha}(l)$ etc, not say $f_{\alpha}(i l)$ or something like that. Finally where does this limit outside come from?

 

[LAHLH]

anyone?