A double integral of dlog forms which is not polylogarithmic

TL;DR

The paper constructs explicit double iterated $d\log$-form integrals in perturbative quantum field theory that defy expression in terms of polylogarithms at algebraic arguments. By embedding the problem in a modular elliptic-Dirichlet motive and computing its relative cohomology, the authors isolate a nontrivial mixed elliptic extension whose period matches a nonzero regulator related to the weight-two cusp form $f$ via $\Lambda(f,2)$. They show the motivic periods $I_{\mathcal{E}}^{\mathfrak{m}}$ and $I^{\mathfrak{m}}$ are algebraically independent from polylogarithmic periods, with a precise decomposition $I^{\mathfrak{m}} = (1/6)I_{\mathcal{E}}^{\mathfrak{m}}+I^{\mathfrak{m}}_{\mathrm{Pol}}$, where the remaining part reduces to polylog constants such as $\mathrm{Cl}_2(\pi/3)$. The results underscore that integration cycles and non-rational arguments can produce genuinely elliptic or modular periods in Feynman integrals, challenging the universality of polylogarithmic descriptions and prompting refined perspectives on differential equations and motivic structures in perturbative QFT.

Abstract

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.

Figures (4)

• Figure 1: A local picture of the divisors $B$ (red) and the four components of $A_k$ (black) in affine coordinates $(x,y)= (x:y:1)$.
• Figure 2: A local picture of the blow-up $P$ at the point at infinity. Not all intersections are shown. The divisor $\widetilde{B}$ is pictured in red. The divisor $D_1$ (resp. $D_2$) is an open in the strict transform of $\mathcal{E}$, (resp. $x=2z$), and $D_3$ is an open in the exceptional locus.
• Figure 3: The path $\gamma_{\infty,-1}$ in $\mathbb C \backslash \{0,-1, \rho, \overline{\rho}\}$ relative to the two points $\{2, \infty\}$ (red). The punctured elliptic curve is a double cover, ramified at the additional point $-1$.
• Figure 4: A singular chain in $\mathbb{A}^1\backslash \{\pm 1\}$ given by the sum of the classes of the two paths shown. It is the image of a Frobenius-invariant path on $\mathcal{E}(\mathbb C)$ under the projection $\pi_y: \mathcal{E}\rightarrow \mathbb P^1$.

Theorems & Definitions (18)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4
• proof