The number of master integrals as Euler characteristic

Abstract

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial $\mathcal{G}$. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial $\mathcal{G}$.

arXiv commentContribution to the proceedings of Loops and Legs in Quantum Field Theory (LL2018), 29 April - 04 May 2018, St. Goar (Germany)

References in corpus (7)

• Explicit solutions of the 3--loop vacuum integral recurrence relations
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Algebraic algorithms for multiloop calculations. The first 15 years. What's next?
• Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation
• Modern techniques of multiloop calculations
• The number of master integrals is finite
• Constructive $D$-module Theory with \textsc{Singular}