FCLoopScalelessQ[int, {p1, p2, ...}] checks whether the loop integral int depending on the loop momenta p1, p2, ... is scaleless. Only integrals that admit a Feynman parametrization with proper U and F polynomials are supported.
The function uses the algorithm of Alexey Pak arXiv:1111.0868. Cf. also the PhD thesis of Jens Hoff 10.5445/IR/1000047447 for the detailed description of a possible implementation.
Overview, FCTopology, GLI, FCLoopToPakForm, FCLoopPakOrder, FCLoopScalelessQ.
A scaleless 2-loop tadpole
FCLoopScalelessQ[FAD[p1, p2, p1 - p2], {p1, p2}]\text{True}
A 1-loop massless eikonal integral.
FCLoopScalelessQ[SFAD[{{0, 2 p . q}, 0}, p], {p}]\text{True}
A 1-loop massive eikonal integral.
FCLoopScalelessQ[SFAD[{{0, 2 p . q}, 0}, p], {p}]\text{True}
A scaleless topology
FCTopology[topo, {SFAD[{{I p3, 0}, {0, 1}, 1}], SFAD[{{I p1, 0}, {0, 1}, 1}],
SFAD[{{0, -2 p1 . q}, {0, 1}, 1}], SFAD[{{I p3 + I q, 0}, {-mb^2, 1}, 1}],
SFAD[{{0, p1 . p3}, {0, 1}, 1}]}, {p1, p3}, {q}, {}, {}]
FCLoopScalelessQ[%]\text{FCTopology}\left(\text{topo},\left\{\frac{1}{(-\text{p3}^2+i \eta )},\frac{1}{(-\text{p1}^2+i \eta )},\frac{1}{(-2 (\text{p1}\cdot q)+i \eta )},\frac{1}{((i \;\text{p3}+i q)^2+\text{mb}^2+i \eta )},\frac{1}{(\text{p1}\cdot \;\text{p3}+i \eta )}\right\},\{\text{p1},\text{p3}\},\{q\},\{\},\{\}\right)
\text{True}