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)