$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

TL;DR

This work compares two paradigms for solving linear PDEs with polynomial coefficients that arise from Feynman integrals: $D$-module methods in the Weyl algebra and Wasow-style asymptotics for Fuchsian systems. Focusing on a conformal triangle integral, it shows that the associated $D$-ideal $I_3$ is regular holonomic of rank 4, and demonstrates how canonical Nilsson series can be computed via the SST algorithm from Gröbner data and independently via a Pfaffian-Wasow framework. The SST approach yields four independent canonical series $f_1,..,f_4$ that span the solution space, while Wasow’s method produces explicit asymptotic series matching these canonical solutions through a manifestly Fuchsian Pfaffian system and an auxiliary weight-tracking parameter. The results illustrate a productive dialogue between algebraic analysis and physics-oriented differential equations, with potential extensions to more complex (e.g., four-loop) Feynman integrals and broader applicability to Picard–Fuchs equations.

Abstract

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare $D$-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic $D$-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.

Figures (3)

• Figure 1: The Feynman graph representing the one-loop triangle Feynman integrals defined in \ref{['eq:triangle_mom']}. Due to momentum conservation, $p_1+p_2+p_3=0$. Next to the internal edges, we record the corresponding exponent $\nu_i$ as well as the loop momentum.
• Figure 2: The fan $\Sigma'$ in the basis $(v_1,v_2)$, where $v_1 = (1,0,-1)$ and $v_2 = (-1,2,-1)$. Its rays are $\rho_1=\mathbb{R}_{\geq 0}\cdot (0,1)$, $\rho_2=\mathbb{R}_{\geq 0}\cdot(-3, -1)$, and $\rho_3=\mathbb{R}_{\geq 0}\cdot(3,-1)$. In $\mathbb{R}^3$, the corresponding hyperplanes are $\rho_1 = [\{w_1 - w_3 = 0\}],$$\rho_2 =[ \{w_1 - w_2 = 0\}],$ and $\rho_3 = [\{w_2 - w_3 = 0\}].$ The cones $C_1$, $C_2$, and $C_3$ correspond to the following cones in $\mathbb{R}^3$: $C_1 =[\{w_1<w_2,w_3\}]$, $C_2=[\{ w_2<w_1,w_3\}]$, and $C_3 =[\{w_3<w_1,w_2\}]$.
• Figure 3: The Feynman graph representing the four-loop "ladder" Feynman integral defined in \ref{['eq:ladder4L']}. Due to momentum conservation, $p_1+p_2+p_3+p_4=0$. The momenta corresponding to thin (thick) external legs are on-shell (off-shell), i.e., we have that $|p_i|^2 = 0$ for $i=1,2,3$, and $|p_4|^2 \neq 0$.

Theorems & Definitions (17)

• Definition 3.1
• Theorem 3.2: Cauchy--Kovalevskaya--Kashiwara
• Definition 3.3
• Theorem 3.4
• Theorem 3.5: SST00
• Definition 3.6
• Proposition 3.7
• Proposition 3.8
• Definition 3.9
• Proposition 3.10: SST00