FCFeynmanProjectivize[int, x] checks if the given
Feynman parameter integral (without prefactors) depending on x[1], x[2],
… is a projective form. If this is not the case, the integral will be
projectivized.
Projectivity is a necessary condition for computing the integral with the aid of the Cheng-Wu theorem
Overview, FCFeynmanParametrize, FCFeynmanPrepare, FCFeynmanProjectiveQ.
int = SFAD[{p3, mg^2}] SFAD[{p3 - p1, mg^2}] SFAD[{{0, -2 p1 . q}}]\frac{1}{(\text{p3}^2-\text{mg}^2+i \eta ) ((\text{p3}-\text{p1})^2-\text{mg}^2+i \eta ) (-2 (\text{p1}\cdot q)+i \eta )}
fp = FCFeynmanParametrize[int, {p1, p3}, Names -> x, Indexed -> True, FCReplaceD -> {D -> 4 - 2 ep},
Simplify -> True, Assumptions -> {mg > 0, ep > 0}, FinalSubstitutions -> {SPD[q] -> qq, mg^2 -> mg2}]\left\{(x(2) x(3))^{3 \;\text{ep}-3} \left((x(2)+x(3)) \left(\text{mg2} x(2) x(3)+\text{qq} x(1)^2\right)\right)^{1-2 \;\text{ep}},-\Gamma (2 \;\text{ep}-1),\{x(1),x(2),x(3)\}\right\}
FCFeynmanProjectivize[fp[[1]], x]\text{FCFeynmanProjectivize: The integral is already projective, no further transformations are required.}
(x(2) x(3))^{3 \;\text{ep}-3} \left((x(2)+x(3)) \left(\text{mg2} x(2) x(3)+\text{qq} x(1)^2\right)\right)^{1-2 \;\text{ep}}
FCFeynmanProjectivize[(x[1] + x[2])^(-2 + 2*ep)/(mb2*(x[1]^2 + x[1]*x[2] +
x[2]^2))^ep, x]\text{FCFeynmanProjectivize: The integral is already projective, no further transformations are required.}
(x(1)+x(2))^{2 \;\text{ep}-2} \left(\text{mb2} \left(x(1)^2+x(2) x(1)+x(2)^2\right)\right)^{-\text{ep}}
Feynman parametrizations derived from propagator representations should be projective in most cases. However, arbitrary Feynman parameter integrals do not necessarily have this property.
FCFeynmanProjectivize[x[1]^(x - 1) (x[2])^(y - 1), x]\text{FCFeynmanProjectivize: The integral is not projective, trying to projectivize.}
\text{FCFeynmanProjectivize: Projective transformation successful, the integral is now projective.}
\frac{\left(\frac{x(1)}{x(1)+x(2)}\right)^{x-1} \left(\frac{x(2)}{x(1)+x(2)}\right)^{y-1}}{(x(1)+x(2))^2}