Analytical continuation of 2,3,4-point integrals

Question

I was reading a paper that gives a nice collection of all scalar integrals that crop up in QCD loop calculations. Such integrals are computed in some kinematic region and then the authors say the results may be analytically continued if so desired. I just wonder how is this analytic continuation done in practice? It's a relatively short paper and the url is https://arxiv.org/abs/0712.1851

The authors state that a particular kinematic region allows for the $i\epsilon$ to be dropped and then one can analytically continue results via the prescription $p_i^2 \rightarrow p_i^2 + i\varepsilon, s_{ij} \rightarrow s_{ij} + i\varepsilon, m_i \rightarrow m_i - i\varepsilon$. I just wonder why this is the case and if the sign choices here are significant?

As a simple example, the analytic continuation of the massive tadpole is given as $I_1^D(m^2) = -\mu^{2\epsilon} \Gamma(\epsilon-1) (m^2-i\varepsilon)^{1-\epsilon}$ but what should I do with this result as it contains an explicit $\varepsilon$ in $m^2-i\varepsilon$?

Answer 1

This is significant: they are originate from the $+i\varepsilon$ prescription in the Feynman propagators of the original integral. The sign choices are important to obtain the correct sign for the imaginary part of the loop functions.

Note that these prescriptions are relevant only if evaluations are being made on a branch cut. Therefore, for your example, if $m^2 > 0$ (which is usually the case), the $-i\varepsilon$ can be dropped.