量子色動力學 (quantum chromodynamics, QCD) 是一描述夸克和膠子交互作用的基本理論。在零溫度下，N_f 個無質量夸克的手則對稱因 QCD 真空而破缺，且軸 U(1) 對稱因軸畸異 (axial anomaly) 而破缺。手則對稱與軸 U(1) 對稱兩者在高溫時均期望會被還原。在此論文中，我們以 QCD 的等效場論，也就是 N_f = 2 之 SU(N_f) × SU(N_f) 線性 σ 模型，計算其包含所有耦合項的 β 函數至一階迴圈來研究手則對稱與軸 U(1) 對稱的還原。

Quantum chromodynamics (QCD) is the fundamental theory for the interaction between quarks and gluons. At zero temperature, the chiral symmetry of N_f massless quarks is broken by the vacuum of QCD, and the axial U(1) symmetry is broken by the axial anomaly. It is expected that the chiral symmetry and the axial U(1) symmetry both are restored at high temperature. In this thesis, we study the restorations of the chiral symmetry and the axial U(1) symmetry in the effective field theory of QCD, namely, the SU(N_f)_L x SU(N_f)_R linear σ model for N_f = 2, by computing the β functions of all couplings to the one-loop order.

口試委員審定書 i
誌謝 ii
摘要 iii
Abstract iv
Introduction 1
Chiral Symmetry in QCD 4
Spontaneous Breaking of Chiral Symmetry 7
The U(1)_A Problem 8
The σ Model 9
The U(N_f)_L x U(N_f)_R Linear σ Model 13
The SU(N_f)_L x SU(N_f)_R Linear σ Model 16
Order of The Chiral Phase Transition 19
The β Function of the SU(2)_L x SU(2)_R Linear σ Model 23
Effects of Approximate U(1)_A Restoration 36
Conclusions 40
Bibliography 41
Appendix 44
The Structure of the β Function 44
The One-Loop Structure of φ4 Theory 46
Dimensional Regularization: The Form of Renormalization Constants 51
Minimal Subtraction Scheme 51
The β Functions with Several Couplings 53
Renormalization Group Analysis 54
The Fixed Point 55
The Stability Matrix 57

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