一圈之幾何結構

Abstract

散射振幅的幾何表達式相較於傳統公式更為簡潔且計算效率更高，學界對於這類結構也已經有了不少研究。我們的目標是透過具體計算，探討這類的幾何表示是否也適用於一圈的結構。我們計算了純量有效場論的所有一圈振幅，在前人已建立樹圖階級振幅幾何結構的基礎上，進一步推導了一圈階級的結果。我們採用了費曼圖方法和廣義么正性方法，成功得到一個自洽的幾何表達式。兩種不同方法的結果完全吻合，證實了我們研究結果的可靠性。

This work is motivated by the observation that geometric representations of amplitudes are often more compact and computationally efficient than traditional formulations. Given the growing interest in exploring such structures, we investigate whether a similar representation can be achieved at the one-loop level through explicit calculation. In this thesis, we compute the one-loop amplitude of a scalar effective field theory (EFT) Lagrangian. Previous studies have established the geometric structure of tree-level amplitudes. By employing both Feynman diagram techniques and the generalized unitaritymethod, we derive a compact geometric expression for the one-loop result. We explicitly verify the consistency between the two approaches, confirming the correctness of our result.