# 臺灣博碩士論文加值系統

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 本篇論文討論弱動量定理的兩個延伸：將規範對稱性所蘊含的弱動 量定理推展到無限階的弱動量展開，以及將自發性對稱性破缺所隱含 的定理推廣到多動量的弱動量定理。其目的為探討近年新發現的兩組 弱動量定理，是否蘊含比原本的弱動量定理更多的資訊: 由殘餘規範對 稱性推導出的無限階弱動量定理，以及在以 CHY 代表式所推導出在具 有自發性對稱破缺之理論所發現的雙點弱動量定理。前者的部分，藉 由將發散與為發散的費曼圖關聯起來，我們可以證明一般的規範對稱 即可得到前述的無限階弱動量定理。因此殘餘規範對稱在散射幅度上 並未提供多餘的資訊，但有可能暗示散射幅度的自由度在質量為零時 可能需要進一部探討。後者的部分，我們發展出一套能夠系統性推導 雙點弱動量定理的方法，並據此得出前述的雙點弱動量定理，以及其 是否蘊含比單點弱動量定理更多的資訊。我們同時探討此兩者在環圖 位階以及高階運算符時受修正的形式。
 In this thesis, we discuss the derivation of extending existing soft theorems in two aspect: pushing single soft theorems from on-shell gauge invariance into infinite order, and deriving, for general theories, double soft theorems from single soft theorems. The motivation is to investigate whether new information can be extracted from two sets of new soft theorems: infinite order single soft theorems for gauge bosons derived from large gauge transformations, and double soft theorems for several theories of Goldstone bosons from CHY representations of tree level amplitudes. In the former case, we show that on-shell gauge invariance reproduce those infinite soft theorems. This indicates that the residual gauge symmetries produces no new constraint on amplitudes, although they might hint the usual choice of asymptotic states for S-matrices might need modification. For the latter case, we developed a general scheme to derive double soft theorems from single soft theorems, adding the information of four-point vertex, where some difficulties in existing methods has been solved. This allows us to reproduce the new double soft theorems, and what additional information is contained compared with single soft theorems. In both scenarios, we also discuss how loop correction or effective operators will modify the soft theorems.
 1 Introduction 12 Overview of Soft Theorems 73 Infinite Soft Theorems From Gauge Symmetry 224 Double Soft Theorems from Single Soft Theorems 355 Fixing amplitudes by soft theorems 626 Conclusion 72A Type A and Type B Expansions for Pole Diagrams 77Bibliography 86
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