General proof of Symanzik-positivity for non-planar integrals (complete monotonicity)

Establish that for all non‑planar scalar Feynman integrals defined by the Feynman parameter representation I(x_i) in Eq. (FeynmanInt), there exists a choice of independent kinematic variables x_i in the Euclidean region such that the second Symanzik polynomial satisfies F = ∑_i A_i x_i with A_i ≥ 0; this would imply that the integrals are completely monotone functions of the variables x_i in the Euclidean region.

Background

Within the Euclidean region, scalar Feynman integrals depend on external kinematics only through the second Symanzik polynomial F(α;x_i), which is linear in the x_i. If F can be written as a non‑negative linear combination of the independent kinematic variables, then the integral inherits complete monotonicity from the integrand’s structure.

For planar graphs this structure is generally satisfied, but for non‑planar graphs it is only supported by examples. A general proof would extend the complete monotonicity result beyond the planar case.

References

For non-planar integrals, there is strong evidence from explicit examples that the above condition on the Symanzik polynomial continues to hold, although a general proof remains an open problem.

— Lecture Notes on Positivity Properties of Scattering Amplitudes  (2603.28454 - Raman, 30 Mar 2026) in Section 3, Subsection 3.2 (Scalar Feynman Integrals); tcolorbox: "Sufficient condition for scalar Feynman integrals to be completely monotone functions"