FCFeynmanFindDivergences[exp, vars] identifies UV and IR divergences of the given Feynman parametric integral that arise when different parametric variables approach zero or infinity.
This function employs the analytic regularization algorithm introduced by Erik Panzer in 1403.3385, 1401.4361 and 1506.07243. Its current implementation is very much based on the code of the findDivergences routine from the Maple package HyperInt by Erik Panzer.
The function returns a list of lists of the form {{{x[i], x[j], ...}, {x[k], x[l], ...}, sdd}, ...}, where {x[i],x[j], ...} need to approach zero, while {x[k], x[l], ...} must tend towards infinity to generate the superficial degree of divergence sdd.
It is important to apply the function directly to the Feynman parametric integrand obtained e.g. from FCFeynmanParametrize. If the integrand has already been modified using variable transformations or the Cheng-Wu theorem, the algorithm may not work properly.
Furthermore, divergences that arise inside the integration domain cannot be identified using this method.
The identified divergences can be regularized using the function FCFeynmanRegularizeDivergence.
Overview, FCFeynmanParametrize, FCFeynmanProjectivize, FCFeynmanRegularizeDivergence.
int = SFAD[l, k + l, {{k, -2 k . q}}]
fpar = FCFeynmanParametrize[int, {k, l}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}]\frac{1}{(l^2+i \eta ).((k+l)^2+i \eta ).(k^2-2 (k\cdot q)+i \eta )}
\left\{(x(1) x(2)+x(3) x(2)+x(1) x(3))^{3 \varepsilon -3} \left(q^2 x(1)^2 (x(2)+x(3))\right)^{1-2 \varepsilon },-\Gamma (2 \varepsilon -1),\{x(1),x(2),x(3)\}\right\}
This Feynman parametric integral contains logarithmic divergences for x_1 \to \infty and x_{2,3} \to 0
FCFeynmanFindDivergences[fpar[[1]], x]\left( \begin{array}{cc} \{\{\},\{x(1)\}\} & \varepsilon \\ \{\{x(2),x(3)\},\{\}\} & \varepsilon \\ \end{array} \right)