Elliptische multiskalen Feynmanintegrale

• Antragsteller: Professor Dr. Stefan Weinzierl
• Fachliche Zuordnung: Kern- und Elementarteilchenphysik, Quantenmechanik, Relativitätstheorie, Felder ; Mathematik
• Förderung: Förderung von 2015 bis 2023
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 274006707

• Erstellungsjahr: 2023

Zusammenfassung der Projektergebnisse

The central theme of this project was to investigate elliptic Feynman integrals, which depend on more than one kinematic variable. The simplest Feynman integrals are the ones, which evaluate to multiple polylogarithms. Elliptic Feynman integrals are the next more complicated Feynman integrals. At the start of the project, elliptic Feynman integrals which only depend on one kinematic variable were understood. Typical examples are the equal-mass sunrise integral or the kite integral, depending on a single dimensionless kinematic variable x = p2 /m2. In phenomenological applications one encounters also elliptic Feynman integrals depending on more kinematic variables, and the goal of the project was to understand these better. We may divide elliptic multi-scale Feynman integrals into two categories: The ones, which only depend on a single elliptic curve and the ones, which depend on more than one elliptic curve. In both categories we obtained nice results. The case of a single elliptic curve is very clean: We identified the integration kernels in the differential equation as kernels related to modular forms and kernels related to the Kronecker function. We could show that the Feynman integrals evaluate to iterated integrals on the covering space of the moduli space of a genus one curve with a certain number of marked points, where the number of marked points depends on the concrete Feynman integral. We also showed that the unequal-mass sunrise integral admits an ε-factorised differential equation and we worked out the transformation behaviour under the full modular group SL2 (Z). This picture essentially carries over to Feynman integrals associated with more than one elliptic curve. Of course, there is an additional complication of cross-talk between the various curves. Also in this case we were able to show in a concrete example that one may achieve an ε-factorised differential equation and we made the modular transformation properties with respect to the two elliptic curves transparent. We also provided within GiNaC numerical routines for the evaluation of elliptic Feynman integrals. The results of this project allowed us to go well beyond the original goals of this project: Calabi-Yau manifolds are higher-dimensional generalisations of elliptic curves and with the methods developed within this project we could show that the l-loop equal mass banana integrals admit an ε-factorised differential equation. This implies that they can be calculated to any order in the dimensional regularisation parameter ε. The key result was the understanding of the four-loop equal mass banana integral, corresponding to a Calabi-Yau three-fold, as beyond four loop the further pattern is completely regular. This result was also highlighted as a press release on the main page of our university.

Projektbezogene Publikationen (Auswahl)

• The unequal mass sunrise integral expressed through iterated integrals on M‾1,3. Nuclear Physics B, 954, 114991.
  Bogner, Christian; Müller-Stach, Stefan & Weinzierl, Stefan
  
• Modular transformations of elliptic Feynman integrals. Nuclear Physics B, 964, 115309.
  Weinzierl, Stefan
  
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals. Computer Physics Communications, 265, 108020.
  Walden, Moritz & Weinzierl, Stefan
  
• A Feynman integral depending on two elliptic curves. Journal of High Energy Physics, 2022(7).
  Müller, Hildegard & Weinzierl, Stefan
  
• Feynman Integrals. UNITEXT for Physics. Springer International Publishing.
  Weinzierl, Stefan
  
• The three-loop equal-mass banana integral in ε-factorised form with meromorphic modular forms. Journal of High Energy Physics, 2022(9).
  Pögel, Sebastian; Wang, Xing & Weinzierl, Stefan
  
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter. Journal of High Energy Physics, 2023(4).
  Pögel, Sebastian; Wang, Xing & Weinzierl, Stefan
  
• Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral. Physical Review Letters, 130(10).
  Pögel, Sebastian; Wang, Xing & Weinzierl, Stefan