FCFeynmanRegularizeDivergence[exp, div] regularizes the
divergence div in the Feynman parametric integral
exp. Provided that all divergences have been regularized in
this fashion, upon expanding the integrand around \varepsilon = 0 one can safely integrate in
the Feynman parameters.
Notice that div can be also a list made of divergences
found by FCFeynmanFindDivergences.
This function uses the method of analytic regularization introduced by Erik Panzer in 1403.3385, 1401.4361 and 1506.07243.
Its current implementation is very much based on the code of the
dimregPartial routine from the Maple package HyperInt by Erik
Panzer.
Here div must be 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.
Overview, FCFeynmanParametrize, FCFeynmanProjectivize, FCFeynmanFindDivergences.
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 integrand contains logarithmic divergences for x_1 \to \infty and x_{2,3} \to 0
divs = FCFeynmanFindDivergences[fpar[[1]], x]\left( \begin{array}{cc} \{\{\},\{x(1)\}\} & \varepsilon \\ \{\{x(2),x(3)\},\{\}\} & \varepsilon \\ \end{array} \right)
Regularizing the first divergence we obtain
intReg = FCFeynmanRegularizeDivergence[fpar[[1]], divs[[1]]]-\frac{3 (\varepsilon -1) q^2 x(1)^2 x(2) x(3) (x(2)+x(3)) (x(1) x(2)+x(3) x(2)+x(1) x(3))^{3 \varepsilon -4} \left(q^2 x(1)^2 (x(2)+x(3))\right)^{-2 \varepsilon }}{\varepsilon }
It turns out that there are no further divergences left
FCFeynmanFindDivergences[intReg, x]\{\}
Now one can expand the integrand in Epsilon and perform
the integration in Feynman parameters order by order in
Epsilon
Series[intReg, {Epsilon, 0, 0}] // Normal\frac{3 q^2 x(1)^2 x(2) x(3) (x(2)+x(3))}{\varepsilon (x(1) x(2)+x(3) x(2)+x(1) x(3))^4}-\frac{3 q^2 x(1)^2 x(2) x(3) (x(2)+x(3)) \left(2 \log \left(q^2 x(1)^2 (x(2)+x(3))\right)-3 \log (x(1) x(2)+x(3) x(2)+x(1) x(3))+1\right)}{(x(1) x(2)+x(3) x(2)+x(1) x(3))^4}
Here is an example of regularizing two divergences at a time
FCFeynmanRegularizeDivergence[(y[1]*(y[1] + y[2] + y[3])^(2*ep)*(y[1]^2 - 4*y[2]*y[3])^(-2 -
ep))/(x[1] + x[2])^2, {{{{y[2]}, {y[3]}}, -2*ep}, {{{y[3]}, {y[2]}}, -2*ep}}]\frac{1}{2 \;\text{ep} (x(1)+x(2))^2}y(1) (y(1)+y(2)+y(3))^{2 (\text{ep}-1)} \left(y(1)^2-4 y(2) y(3)\right)^{-\text{ep}-2} \left(2 \;\text{ep} y(1)^2+4 \;\text{ep} y(2) y(1)+4 \;\text{ep} y(3) y(1)+8 \;\text{ep} y(2) y(3)-y(2) y(1)-y(3) y(1)-4 y(2) y(3)\right)