Vector Space of Feynman Integrals and Multivariate Intersection Numbers

TL;DR

The paper develops a recursive algorithm to compute multivariate intersection numbers in twisted cohomology, enabling projection of any Feynman integral onto a master-integral basis without heavy linear-elimination. It shows how ν, the number of master integrals, is determined topologically via χ(P_ω) and critical points of log u, and demonstrates the approach with a hypergeometric contiguity relation and explicit one- and two-loop Feynman integral decompositions, including the massless box. The results align with traditional IBP reductions while offering a conceptually transparent, geometry-driven framework. By linking intersection theory to Morse theory and Lefschetz thimbles, the work suggests broad applicability to complex diagrams and representations beyond standard perturbative techniques.

Abstract

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the derivation of contiguity relations for special functions admitting multi-fold integral representations, and to the decomposition of a few Feynman integrals at one- and two-loops, as first steps towards potential applications to generic multi-loop integrals.

Figures (4)

• Figure 1: Massless box with massless external legs ($p_{i}^{2}=0$, for $i=1,2,3,4$). The invariants are $s=(p_{1}+p_{2})^2$ and $t=(p_{2}+p_{3})^2$.
• Figure 2: Other examples of one- and two-loop integrals reduced to MIs with the technique proposed in this work.
• Figure 3: Morse--Smale complex associated to the Morse function $h(z)=\Re (\log u(z))$ with eq. \ref{['appendix-u-example']} and $\rho{>}s{>}0$, $\gamma{>}0$. The set of filled dots corresponds to $\mathcal{P}_\omega = \{\pm \rho, \pm s,\infty \}$ removed from $X$. Empty dots at $z_{(\alpha)}^\ast$ represent critical points of the Morse function, with paths of steepest descent $\mathcal{J}_\alpha$ (solid lines) and ascent $\mathcal{K}_\alpha$ (dashed lines) extending from them. They give a triangulation of $X = \mathbb{CP}^1 {-} \mathcal{P}_\omega$. The arrows indicate the direction of the flow towards lower values of $h(z)$.
• Figure 4: Massless box with a self-energy insertion diagram ($p_{i}$ with $p_{i}^{2}=0$, for $i=1,2,3,4$). The invariants are $s=(p_{1}+p_{2})^2$ and $t=(p_{2}+p_{3})^2$.