FCFeynmanProjectivize
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
See also
Overview, FCFeynmanParametrize, FCFeynmanPrepare, FCFeynmanProjectiveQ.
Examples
int = SFAD[{p3, mg^2}] SFAD[{p3 - p1, mg^2}] SFAD[{{0, -2 p1 . q}}]
(p32−mg2+iη)((p3−p1)2−mg2+iη)(−2(p1⋅q)+iη)1
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}]
{(x(2)x(3))3ep−3((x(2)+x(3))(mg2x(2)x(3)+qqx(1)2))1−2ep,−Γ(2ep−1),{x(1),x(2),x(3)}}
FCFeynmanProjectivize[fp[[1]], x]
FCFeynmanProjectivize: The integral is already projective, no further transformations are required.
(x(2)x(3))3ep−3((x(2)+x(3))(mg2x(2)x(3)+qqx(1)2))1−2ep
FCFeynmanProjectivize[(x[1] + x[2])^(-2 + 2*ep)/(mb2*(x[1]^2 + x[1]*x[2] +
x[2]^2))^ep, x]
FCFeynmanProjectivize: The integral is already projective, no further transformations are required.
(x(1)+x(2))2ep−2(mb2(x(1)2+x(2)x(1)+x(2)2))−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]
FCFeynmanProjectivize: The integral is not projective, trying to projectivize.
FCFeynmanProjectivize: Projective transformation successful, the integral is now projective.
(x(1)+x(2))2(x(1)+x(2)x(1))x−1(x(1)+x(2)x(2))y−1