格點量子色動力學: Domain-Wall 夸克之探討

Abstract

量子色動力學(QCD)是研究夸克和膠子間交互作用的最根本理論。它不但可描述核子間的強交互作用力，更在研究早期宇宙的演進中(由夸克-膠子相位到強子相位)扮演重要的角色。要在四維時空的離散晶格點上完全地計算解答量子色動力學是相當困難的挑戰，因為它需要非常大尺度的數值模擬計算。此外，為了使無質量極限下的夸克在有限晶格區間上能保持確切的手則對稱，額外的第五維空間將被引入並定義夸克在第五維空間的邊界上，這就是所謂的domain-wall費米子。在這篇論文中，我們討論其擁有確切手則對稱的格點量子色動力學，並研究如何用蒙地卡羅演算法來做包含動態的u, d, s 和c 夸克的量子色動力學的數值模擬計算。我們推導了domain-wall 費米子在格點量子色動力學的軸向Ward 恆等式，並從中獲得“殘留質量”的數學式，其可以用來測量由於有限晶格點數的第五維空間所帶來的手則對稱破壞。更進一步，我們獲得格點量子色動力學上最佳手則對稱domain-wall費米子的 ”殘留質量”的數值上限。

Quantum Chromodynamics (QCD) is the fundamental theory for the interaction between quarks and gluons. It manifests as the short-range strong interaction inside the nucleus, and plays an important role in the evolution of the early universe, from the quark-gluon phase to the hadron phase. To solve QCD is a grand challenge, since it requires very large-scale numerical simulations of the discretized action of QCD on the 4-dimensional space-time lattice. Moreover, since quarks are relativistic fermions, the 5-th dimension is introduced such that massless quarks with exact chiral symmetry can be realized at finite lattice spacing, on the boundaries of the 5-th dimension, the so-called domain-wall fermion (DWF). In this thesis, we discuss the formulation of lattice QCD with exact chiral symmetry, and the algorithms to perform Monte Carlo simulation of QCD with dynamical u, d, s, and c quarks. We also derive the axial Ward identity for lattice QCD with domain-wall fermion, and from which we obtain a formula for the residual mass, that can be used to measure the chiral symmetry breaking due to the finite extension $ N_s $ in the fifth dimension. Furthermore, we obtain an upper bound for the residual mass in lattice QCD with the optimal domain-wall fermion.