Demo · Tropical subtraction
Feynman integrals are how we turn quantum field theory into numbers — but they are riddled with infinities. Here is how tropical geometry sorts those infinities into a finite, computable list, and how the SubTropica package does it for you.
To turn quantum field theory into a number — a cross-section, a decay rate — we evaluate Feynman integrals. Take a one-loop box — four propagators closing into a square. Its value is not finite: somewhere inside, it diverges.
Rather than integrate over the loop momentum, rewrite the integral over Feynman parameters — one per propagator — living on a simplex. The whole integrand is built from two Symanzik polynomials, U and F.
The divergences are not scattered everywhere. They sit on the boundary of the simplex, where Feynman parameters vanish. Work in d = 4 − 2ε dimensions and each one surfaces as a pole in ε.
Which boundaries diverge, and how badly? The answer is combinatorial. Collect the exponents of the Symanzik polynomials as lattice points, take their convex hull, and you have a Newton polytope.
Each facet of the Newton polytope is one way the integral diverges. The scheme subtracts a counterterm for it — geometrically, peel that facet off. What remains is the polytope minus that piece.
A neighbouring facet diverges too, so subtract its counterterm as well. But along the edge where the two facets meet, that overlap has now been removed twice.
Inclusion–exclusion fixes it: add back their common edge — the piece counted twice. Carry on down the face lattice and every divergence is removed exactly once. The remainder, Iren, is finite.
SubTropica is open and documented. Point it at your own integral and watch the ε-expansion come back — no tropical bookkeeping by hand.
Tropical means keeping only the leading exponential behaviour — replacing sums by maxima and products by sums. For a Feynman integral that coarse shadow is exactly the data that controls its divergences. The two papers here turn the observation into a method: The Tropical Geometry of Subtraction Schemes sets up the framework, and SubTropica (with M. Giroux & S. Mizera) makes it run inside Mathematica.
IntersectionObserver
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