FCFeynmanProjectiveQ[int, x]
checks if the given Feynman parameter integral (without prefactors) depending on x[1], x[2], … is a projective form.
It is similar to FCFeynmanProjectivize
but unlike the former it simply returns True
or False
depending on whether the integral is projective or not.
Overview, FCFeynmanParametrize, FCFeynmanPrepare, FCFeynmanProjectivize.
= SFAD[{p3, mg^2}] SFAD[{p3 - p1, mg^2}] SFAD[{{0, -2 p1 . q}}] int
= FCFeynmanParametrize[int, {p1, p3}, Names -> x, Indexed -> True, FCReplaceD -> {D -> 4 - 2 ep},
fp Simplify -> True, Assumptions -> {mg > 0, ep > 0}, FinalSubstitutions -> {SPD[q] -> qq, mg^2 -> mg2}]
[fp[[1]], x] FCFeynmanProjectiveQ
[(x[1] + x[2])^(-2 + 2*ep)/(mb2*(x[1]^2 + x[1]*x[2] + x[2]^2))^ep, x] FCFeynmanProjectiveQ
Feynman parametrization derived from propagator representation should be projective in most cases. However, arbitrary Feynman parameter integral do not necessarily have this property.
[x[1]^(x - 1) (x[2])^(y - 1), x] FCFeynmanProjectiveQ