Stieltjes property for non‑planar integrals via linear‑in‑one‑variable Symanzik polynomial

Establish that for non‑planar scalar Feynman integrals defined by Eq. (FeynmanInt) in loop order L and spacetime dimension D, and satisfying 0 < ∑_i ν_i − (L D)/2 ≤ 1, there exists a choice of a single independent kinematic variable x (with all other variables fixed in the Euclidean region) such that the second Symanzik polynomial has the form F = A x + B with A ≥ 0 and B ≥ 0; this would imply that the integral is a Stieltjes function of x.

Background

Under the condition 0 < ∑_i ν_i − (L D)/2 ≤ 1 and with F linear in a single kinematic variable x with non‑negative coefficients, planar Feynman integrals are Stieltjes functions of x when other kinematics are held fixed in the Euclidean region.

For non‑planar integrals, this linear‑in‑one‑variable structure has been verified in examples but lacks a general proof; establishing it would extend Stieltjes properties 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 Stieltjes functions"

However, the verification of the Stieltjes representation required a case-by-case decomposition of the second Symanzik polynomial, and the question of a general proof---especially for non-planar graphs and arbitrary numbers of external legs---was left open.

— The Stieltjes Bootstrap for Feynman Integrals  (2609.37536 - Brochet et al., 29 Sep 2026) in Section 1, Introduction