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In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated:

$$ F_{2}(0) = \frac{\alpha}{\pi} \int_{0}^{1} dz \int_{0}^{1} dy \int_{0}^{1} dx \frac{\delta(x+y+z-1) z}{1-z}$$

which gives the result

$$ F_{2}(0) = \frac{\alpha}{\pi} \int_{0}^{1} dz \int_{0}^{1-z} dy \frac{z}{1-z}$$

My question is, how do you fill in the gap of this calculation? I am not sure how to use the delta function on a definite integral, especially when there is more than 1 variable.

My attempt has been to integrate over $x$ using the delta function to set $x = 1-y-z$, and then evaluate this at the limits $0 \: \& \: 1$, which gives $y = 1-z$ for the upper limit and $y=z=0$ for the lower one. But I'm not sure why this factor appears in the limit of the integral over $y$ and not elsewhere. Any tips on how to approach these kinds of integrals would be appreciated, as they seem to be important when using Feynman parameter.

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Hint: Well, the condition on $x$ is $x= 1\!-\!y\!-\!z \in[0,1]$, so the condition on $y$ is $y\!+\!z \in[0,1]\Leftrightarrow y\in [-z,1\!-\!z]$. Together with the condition $y\in[0,1]$, we get $y\in [-z,1\!-\!z]\cap [0,1]=[0,1\!-\!z]$, which is the displayed integration limits.

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