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ファインマン積分は、場の量子論で粒子の挙動を記述するために使用される数学的ツールです。 これらの積分は、1940 年代に初めてそれらを導入した物理学者のリチャード ファインマンにちなんで名付けられました。 それらは、粒子が特定の経路をたどる確率を計算する方法を提供し、亜原子の世界を理解する上で不可欠な要素となっています。


世界は最小のスケールでどのように見えますか? これは、研究者が大ハドロン衝突型加速器などの粒子加速器実験を通じて解決しようとしている問題です。[{” attribute=””>CERN in Switzerland. In order to assess the outcomes of these experiments, theoretical physicists require increasingly accurate predictions based on the standard model, our current understanding of the interactions between fundamental particles.

A key ingredient in these predictions are so called Feynman integrals. Recently, a team of the PRISMA+ Cluster of Excellence at Mainz University, consisting of Dr. Sebastian Pögel, Dr. Xing Wang and Prof. Dr. Stefan Weinzierl developed a method to efficiently compute a new class of these Feynman integrals, associated to Calabi–Yau geometries.

This research is now published in the renowned journal Physical Review Letters and opens the path to high-precision theoretical predictions of particle interactions, and to better understand the elegant mathematical structure underpinning the world of particle physics.

“During the interaction of subatomic particles something special happens: Any number of additional particles can temporarily pop in and out of existence”, Prof. Dr. Stefan Weinzierl explains. “When making theoretical predictions of such interactions, the more of the these additional particles are taken into account, the more precise the computation will be to the real result.”

Feynman integrals are mathematical objects which describe this effect, summing in effect all possible ways particles can appear and immediately disappear again.

Calabi–Yau Geometries: An Interplay of Mathematics and Physics

An important property determining the complexity of a Feynman integral is its geometry. Many of the simplest Feynman integrals have the geometry of a sphere or a torus – the mathematical term for a donut shape. Such integrals are nowadays well understood.

However, there exist entire families of geometries, so-called Calabi–Yau geometries, which are generalizations of the donut case to higher dimensions. These have proven to be a rich field of research in pure Mathematics, and have found extensive application in string theory in the last decades.

In recent years it was discovered that also many Feynman integrals are associated with Calabi–Yau geometries. However, due to the complexity of the geometry, the efficient evaluation of such integrals has remained a challenge.

In their recent publication, Dr. Sebastian Pögel, Dr. Xing Wang, and Prof. Dr. Stefan Weinzierl present a method that allows them to tackle integrals of Calabi–Yau geometries. In their research, they studied a simple family of Calabi–Yau Feynman integrals, called banana integrals.

The name is derived from the Feynman graph (see picture). Thereby they could find for the first time a so-called “epsilon-factorized form” for these integrals. This form allows us to quickly evaluate the integral to nearly arbitrary precision, making them accessible for future experimental predictions. “It opens the door to a wide variety of hitherto unreachable Feynman integrals,” says Dr. Xing Wang.

According to Dr. Sebastian Pögel, it is a nice example of how pure mathematics feeds into phenomenological predictions for high-energy experiments. “We are grateful to our colleagues in mathematics, and in particular to the group of Prof. Dr. Duco van Straten, as we build on their work and now were able to achieve this exciting result”, summarizes Prof. Dr. Stefan Weinzierl.

Reference: “Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral” by Sebastian Pögel, Xing Wang and Stefan Weinzierl, 8 March 2023, Physical Review Letters.
DOI: 10.1103/PhysRevLett.130.101601


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