I've read in a paperin a paper
- E. Guadagnini, M. Martellini, M. Mintchev, Wilson Lines in Chern-Simons Theory and Link Invariants, Nucl. Phys. B 330 (1990) pp 575–607 https://doi.org/10.1016/0550-3213(90)90124-V, Inspirehep
that the following integral:
$I^i(\textbf{a},\textbf{b})=\int_{\textbf{w}\in\mathbb{R}^3}\frac{w^i}{\|\textbf{w}\|^3\|\textbf{w}-\textbf{a}\|\|\textbf{w}-\textbf{b}\|}$
$$ I^i(\textbf{a},\textbf{b})=\int_{\textbf{w}\in\mathbb{R}^3}\frac{w^i}{\|\textbf{w}\|^3\|\textbf{w}-\textbf{a}\|\|\textbf{w}-\textbf{b}\|} $$ ...where $\textbf{w}, \textbf{a}, \textbf{b}\in\mathbb{R}^3$ are vectors, $w^i$, $a^i$, $b^i$ denote the $i^{th}$ component of $\textbf{w}$, $\textbf{a}$ and $\textbf{b}$ respectively, and $\|\|$ is the Euclidean norm, can be simplified by using Feynman Parameters $s$ and $t$ and integrating over $\textbf{w}$ to get the result:
$I^i(\textbf{a},\textbf{b})=2\int_0^1ds\int_0^1dt\sqrt{\frac{t}{s(1-s)}}\frac{sa^i+(1-s)b^i}{\{ s(1-s)\|\textbf{a}-\textbf{b}\|^2 + t\| s\textbf{a}+(1-s)\textbf{b} \|^2 \}}$ $$I^i(\textbf{a},\textbf{b})=2\int_0^1ds\int_0^1dt\sqrt{\frac{t}{s(1-s)}}\frac{sa^i+(1-s)b^i}{\{ s(1-s)\|\textbf{a}-\textbf{b}\|^2 + t\| s\textbf{a}+(1-s)\textbf{b} \|^2 \}}$$
I have two questions on this result:
- Because i'mI'm not 100% versed in these techniques, can someone show me the exact steps to get between the first equation above to the second equation?
- With these techniques, is there ever a concern about convergence of the integral? I ask this because the above integral comes up in Bott-Taubes integration, and there is a lot of work done with respect to compactifying integration domains to make sure integrals are finite. Does the Feynman parametrization technique implicitly address these issues?
(Because question 2. is a bit open-ended, I will happily accept an answer that addresses 1.)
Thanks in advance!
