I'm trying to understand the following example of analytic continuation given in a physics paper. In the last paragraph of page 16 the authors define the function $$f(z,\bar z)=(z\bar z)^{-\Delta}$$
where $x+i\tau$ and $\bar z = x-i\tau$. Initially both $x,\tau\in \mathbb{R}$ but they want to analytically continue this function to $\tau=it$ with $t\in \mathbb{R}$ with $x$ held fixed as a real number. To that end they define a path $\tau(\phi)=t e^{i\phi}$. Then they say that if you follow this path for $\phi\in [0,\frac{\pi}{2}]$ you get the analytic continuation $$(t^2-x^2)^{-\Delta}e^{-i\pi\Delta}$$
while if you follow the corresponding clockwise path in which $\phi$ decreases on the interval $[-\frac{3\pi}{2},0]$ you get $$(t^2-x^2)^{-\Delta}e^{i\pi\Delta}.$$
I don't understand very well. I feel that it has to do with the fact that if we place the branch cut of the log on $(-\infty,0)$ then on the first path we don't cross it while in the second we do, but I don't know how to make it precise. After searching a bit I found the concept of analytic continuation along a path, which should be the rigorous version of this, but it involves prescribing several open discs along the path with analytic continuations in them and I feel a bit confused on how that works in practice.
My question here is: how do we use analytic continuation along a path rigorously in this simple example to derive the result the authors mentioned?