Trouble Understanding Computation In Nucleon Scattering Example in David Tong Lecture Notes

Question

I am struggling to understand the following computation from page 59 of Tong's QFT notes

http://www.damtp.cam.ac.uk/user/tong/qft.html

The expression

$$ (-ig)^{2} \int \frac{d^{4}k}{(2\pi)^{4}} \frac{i(2\pi)^{8}}{k^{2} - m^{2} + i\epsilon} [\delta^{4}(p'_{1}-p_{1} + k)\delta^{4}(p'_{2}-p_{2} - k) \\ \hspace{50mm} + \delta^{4}(p'_{2}-p_{1} + k)\delta^{4}(p'_{1}-p_{2} - k)] \tag{3.51} $$

after integrating is said to yield

$$ i(-ig)^{2} \left[\frac{1}{(p_{1}-p'_{1})^{2} - m^{2} + i\epsilon} + \frac{1}{(p_{1}-p'_{2})^{2} - m^{2} + i\epsilon} \right] (2\pi)^{4} \delta^{4}(p_{1} + p_{2} - p'_{1} - p'_{2}) .\tag{p.59}$$

I am fairly certain the "sifting property" of the delta function is being used but I can't seem to convince myself. By using the sifting property I mean applying the following property to the above scenario

$$ f(x) = \int dy \hspace{1mm} f(y) \hspace{1mm} \delta(x-y). $$

Answer 1

It should be sufficient to show just how the first term is obtained, namely $$\int\frac{\mathrm d^4k}{(2\pi)^4}\frac{(2\pi)^8}{k^2-m^2+i\varepsilon}\delta^{(4)}(k-(p_1-p_1^\prime))\delta^{(4)}(k+p_2-p_2^\prime)=\frac{(2\pi)^4\delta^{(4)}(p_1-p_1^\prime+p_2-p_2^\prime)}{(p_1-p_1^\prime)^2-m^2+i\varepsilon}$$ where you missed a factor of $(2\pi)^8$ in your rendition of Equation (3.51). You can check Tong's original for that.

But with my way of writing this integral, the usage of that property is maximally obvious: We are getting rid of the leftmost Dirac delta distribution, thereby swapping $k$ for $p_1-p_1^\prime$ everywhere else.

You are free to do the same with the other term, and then assemble for yourself the final result.