# Feynman integral relations from parametric annihilators

@article{Bitoun2017FeynmanIR, title={Feynman integral relations from parametric annihilators}, author={Thomas Bitoun and Christian Bogner and Ren{\'e} Pascal Klausen and Erik Panzer}, journal={Letters in Mathematical Physics}, year={2017}, volume={109}, pages={497-564} }

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee–Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with…

## 63 Citations

### Reduction of Feynman integrals in the parametric representation III: integrals with cuts

- MathematicsThe European Physical Journal C
- 2020

Phase space cuts are implemented by inserting Heaviside theta functions in the integrands of momentum-space Feynman integrals. By directly parametrizing theta functions and constructing…

### Reduction of Feynman integrals in the parametric representation

- PhysicsJournal of High Energy Physics
- 2020

In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum…

### Reduction of Feynman integrals in the parametric representation

- PhysicsJournal of High Energy Physics
- 2020

In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum…

### Feynman integrals as A-hypergeometric functions

- MathematicsJournal of High Energy Physics
- 2019

Abstract
We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel’fand-Kapranov-Zelevinsky (GKZ) system. In order to define…

### Dodgson polynomial identities

- MathematicsCommunications in Number Theory and Physics
- 2019

Dodgson polynomials appear in Schwinger parametric Feynman integrals and are closely related to the well known Kirchhoff (or first Symanzik) polynomial. In this article a new combinatorial…

### Feynman Integrals and Mirror Symmetry

- Mathematics, Physics
- 2020

In this text we describe various approaches to the computation of Feynman integrals. One approach uses toric geometry to derive differential equations satisfied by the imaginary part of the Feynman…

### On epsilon factorized differential equations for elliptic Feynman integrals

- Mathematics, Computer ScienceJournal of High Energy Physics
- 2022

A method is developed and demonstrated to obtain epsilon factorized differential equations for elliptic Feynman integrals by choosing an integral basis with the property that the period matrix obtained by integrating the basis over a complete set of integration cycles is diagonal.

### Feynman integrals and intersection theory

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we…

### Toric geometry and regularization of Feynman integrals.

- Mathematics
- 2018

We study multivariate Mellin transforms of Laurent polynomials by considering special toric compactifications which make their singular structure apparent. This gives a precise description of their…

### Feynman integrals and intersection theory

- MathematicsJournal of High Energy Physics
- 2019

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider…

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