I am working through the very good notes by Dr Schwartz in his book: QFT and the Standard Model and have hit a stumbling block.
I don't know how he has completed the integral in equation $(14)$ in the notes linked below. I have tried various ways of using the fact that $m_\gamma \ll q^2$, as well as Mathematica, but I can't seem to get the right answer. If anyone can help I would greatly appreciate it.
http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-6-InfraredDivergences.pdf
The second term in Eq. $(14)$ is IR divergent but UV finite. Moreover, for real $Q^2$ there is a pole in the integration region. Fortunately, there is a small imaginary part in the denominator (due to the $i\epsilon$ prescription) which makes the integral converge. Since $x$ and $y$ are positive we can perform the integral taking $Q^2\to Q^2+i\epsilon$, which gives$$\int_0^1{\rm d}x\int_0^{1-x}{\rm d}y \frac{Q^2(1-x)(1-y)}{-xyQ^2 + (1-x-y)m_\gamma^2} = -\frac{1}{2}\ln^2\left(\frac{m_\gamma^2}{-Q^2-i\epsilon}\right) - 2\ln\left(\frac{m_\gamma^2}{-Q^2-i\epsilon}\right) \\- \frac{\pi^2}{3} - \frac{5}{2} + \mathcal{O}(m_\gamma)$$