散射振幅與其對稱性

Abstract

近年來，由於散射振幅上計算方法的發展，使得我們在不利用任何Lagrangian的資訊下可以得到散射振幅的結果。在這篇論文中，我們將討論關於散射振幅中不同面向的對稱性。在第一部分，我們討論三維的'陳-Simons-物質'理論中隱含的對偶保角超對稱性並利用其性質建構第一圈及第二圈四點散射振幅。在第二部分，我們討論了手徵規範理論中，一圈散射振幅中的奇宇稱有理項性質。 有理項的存在深切地關聯於局域性條件(locality)。並且我們可發現規範對稱的異常可解釋為局域性與歸一性(unitarity)條件間的相互矛盾。在最後一部分，我們證明了三維及四維超重力理論中雙純量場之弱動量定理。 然而更出乎意料地，散射振幅在雙費米子之弱動量極限之下給出了規律性的結果，此暗示著新對稱性存在的可能性。 除此之外，我們利用了弱動量定理分析可能的三維極超重力(maximal supergravity)重整抵消項(counter term)。在直接維度縮減(dimension reduction)的假設之下，我們排除了D^8R^4作為重整抵消項的可能性。

The development of modern on-shell methods for scattering amplitudes allows us to construct scattering amplitudes without the knowledge of a Lagrangian. In this thesis, various aspects of symmetry in scattering amplitudes are studied in this content. In Part 1, we review the hidden dual superconformal symmetry in three-dimensional Chern-Simons-matter theory and utilize it to construct the amplitudes up to two-loop. In Part 2, we discuss the rational terms of the parity-odd part of one-loop amplitudes in chiral gauge theories. The presence of rational terms is connected to the requirement of locality. Anomalies are then recast as the result of a inconsistent rational term with both locality and unitarity. Finally, we establish the soft theorems for double-soft-scalar-limit in three- and four-dimensional supergravity theories. Surprisingly, the amplitudes also have a universal pattern under double-soft-fermion-limit, there exists implicitly a new symmetry. As a preliminary application of the soft theorem to three-dimensional maximal supergravity, we rule out the direct dimensional reduction of D^8R^4 out as a valid counter term.