Twisted Riemann bilinear relations and Feynman integrals

TL;DR

We develop a twisted-cohomology framework to study quadratic relations among dimensionally regulated Feynman integrals via twisted Riemann bilinear relations ($TRBRs$). By formulating a period matrix built from maximal-cut integrals and analyzing its dual along with twisted-intersection data, we show that $TRBRs$ generate genuine quadratic relations for maximal cuts in a non-relative setting, while such relations do not generally extend to non-maximal cuts or uncut integrals due to the separate linear dependence on the period matrix and its dual. The maximal-cut relations connect to Calabi–Yau geometry at $\varepsilon=0$ (recovering classical Riemann bilinear relations, Legendre-type identities) and to one-variable $x$-dependent results in the literature, with explicit demonstrations on the unequal-mass sunrise and the non-planar crossed box. Relative twisted cohomology is essential for handling poles in non-maximal cases, clarifying why TRBRs fail to yield simple quadratic relations there. Overall, the work provides a geometry-inspired mechanism to express complex Feynman integrals in terms of simpler ones and highlights deep connections between Feynman integral relations, hyperelliptic/CY geometry, and the Gauss–Manin connection.

Abstract

Using the framework of twisted cohomology, we study twisted Riemann bilinear relations (TRBRs) satisfied by multi-loop Feynman integrals and their cuts in dimensional regularisation. After showing how to associate to a given family of Feynman integrals a period matrix whose entries are cuts, we investigate the TRBRs satisfied by this period matrix, its dual and the intersection matrices for twisted cycles and co-cycles. For maximal cuts, the non-relative framework is applicable, and the period matrix and its dual are related in a simple manner. We then find that the TRBRs give rise to quadratic relations that generalise quadratic relations that have previously appeared in the literature. However, we find that the TRBRs do not allow us to obtain quadratic relations for non-maximal cuts or completely uncut Feynman integrals. This can be traced back to the fact that the TRBRs are not quadratic in the period matrix, but separately linear in the period matrix and its dual, and the two are not simply related in the case of a relative cohomology theory, which is required for non-maximal cuts.

Figures (5)

• Figure 1: A six-loop integral obtained from a one-loop triangle integral by inserting banana integrals depending on at most one mass. Solid internal lines denote massless propagators, while the dashed line represents a massive propagator. The contraction of any internal line leads to a graph with a detachable massless tadpole integral, which vanishes in dimensional regularisation.
• Figure 2: Our choice of (dual) twisted cycles is inspired by the geometric picture for the elliptic curve defined by eq. \ref{['curvesunrise']}. The two canonical cycles on the elliptic curve, $a$ and $b$, inspire our choice of the dual twisted cycles $\check{\gamma}_1$ and $\check{\gamma}_2$. In the $\varepsilon \rightarrow 0$ limit, the cycle $\check{\gamma}_3$ corresponds to the cycle around infinity, so that all differentials without residue integrate to zero, simplifying the period matrix. For generic $\varepsilon$ the twist has an additional factor $z^\varepsilon$ which requires us to add the dual twisted cycle $\check{\gamma}_4$.
• Figure 3: Non-planar crossed box diagram. All internal propagators are massive.
• Figure 4: A canonical choice of cycles for a hyperellitpic curve of genus two with branch points $\lambda_1,\dots,\lambda_6$.
• Figure :

Theorems & Definitions (10)

• Example 1: Gauss' hypergeometric ${{}_2F_1}$ function
• Example 2: Gauss hypergeometric ${}_2F_1$ function in relative twisted cohomology
• Example 3: The period matrix for the one-loop bubble integral
• Example 4: The period matrix for the unequal-mass sunrise integral
• Example 5: The massive bubble in $D=2-2\varepsilon$ dimensions
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof