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

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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?

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 $y$.
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