Is there a systematic procedure to approximating Feynman parameter integrals that Peskin uses?

Question

On page 199 in 'An Introduction to Quantum Field Theory' by Peskin and Schroeder they use an approximation for an integral that is relevant for the electron vertex function. In the \(\mu \rightarrow 0\)limit they have:

\(F_1(q^2) \approx \frac{\alpha}{2 \pi} \int^1_0 dx \ dy \ dz \ \delta (x+y+z-1) \left[ \frac{m^2(1-4 z + z^2) + q^2 (1-x)(1-y)}{m^2 (1-z)^2 - q^2 x y + \mu^2 z} - \frac{m^2 (1-4 z +z^2)}{m^2(1-z)^2 + \mu^2 z} \right] \)

He then states that 'First note that the divergence occurs in the corner of the Feynman-parameter space where \(z \approx 1\)(and therefore \(x \approx y \approx 0\)). In this region we can set \(z = 1\)and \(x = y = 0 \)in the numerators of [the above equation]. We can also set \(z \approx 1\)in the \(\mu^2\)terms in the denominators. Using the delta functions to evaluate the x - integral we then have': 

\(F_1(q^2) \approx \frac{\alpha}{2 \pi} \int^1_0 dz \int^{1-z}_0 dy \left[ \frac{-2 m^2 + q^2 }{m^2 (1-z)^2 - q^2 y(1-z-y) + \mu^2 } - \frac{ - 2m^2}{m^2(1-z)^2 + \mu^2} \right] \)

How does Peskin justify these approximations? i.e. - how do you make this process systematic so that you can expand in something, or calculate higher order corrections to Peskin's 'zero-th order' calculation?

Answer 1

To get higher order corrections, eliminate $z=1-x-y$ and expand the terms deleted in the above derivation into a power series in $x$ and $y$.