Matijašić, Antonela (2024): Singularity structure of Feynman integrals with applications to six-particle scattering processes. Dissertation, LMU München: Faculty of Physics |

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**DOI**: 10.5282/edoc.34154

### Abstract

Scattering amplitudes serve as a bridge between theoretical predictions and experimental data in Quantum Field Theories (QFTs). They are computed in perturbation theory as a series in the coupling constants. Beyond the leading order, computing the contributing Feynman integrals is one of the most critical and often complex steps required to draw predictions from the theory. A well--known method for addressing this involves deriving a system of ordinary differential equations in the canonical form that the Feynman integrals satisfy, where solutions in terms of known functions can be readily obtained. An important aspect of this method lies in understanding the singularity structure of Feynman integrals. We first introduce several representations of Feynman integrals used in different parts of this thesis. In particular, we introduce two parametric representations suitable for analyzing the singularities of Feynman integrals. Next, we address the question of determining the location of possible kinematic singularities of Feynman integrals. To this end, we first review the Landau equations and a modern approach for finding their solutions based on methods from nonlinear algebra. Next, we use the connection between the singularities of Feynman integrals and symbol alphabets to find the alphabets without ever solving the integrals. Furthermore, we review a method for the analytic computation of Feynman integrals based on differential equations. We argue that knowing the singularity structure and a good basis of Feynman integrals makes the computation more efficient and simplifies finding a solution. The main contribution of the thesis is an efficient algorithm for finding algebraic letters from the knowledge of the kinematic singularities of Feynman integrals. The algorithm is based on an observed factorization property of algebraic letters. In order to make the algorithm more efficient and ready to use in cutting--edge applications, we introduce a criterion that significantly reduces the size of the problem at hand, allowing us to handle large alphabets appearing in the high--multiplicity problems. Finally, we use the methods discussed in this thesis in a state--of--the--art computation of Feynman integrals that could not be computed without having insight into their singularity structure.

Item Type: | Theses (Dissertation, LMU Munich) |
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Subjects: | 500 Natural sciences and mathematics 500 Natural sciences and mathematics > 530 Physics |

Faculties: | Faculty of Physics |

Language: | English |

Date of oral examination: | 16. September 2024 |

1. Referee: | Henn, Johannes |

MD5 Checksum of the PDF-file: | 17335b1c5e2a1a17cecd4f0f8f8ea00f |

Signature of the printed copy: | 0001/UMC 30709 |

ID Code: | 34154 |

Deposited On: | 04. Oct 2024 12:34 |

Last Modified: | 04. Oct 2024 12:37 |