描述强相互作用的基本理论是量子色动力学（Quantum Chromodynamics，简称QCD）。在极高温度和/或密度条件下可以形成以强相互作用为主的大块物质形态，这类物质被称为QCD物质。理论预言高温、高重子数密度、强磁场等极端条件会改变QCD物质的对称性，即QCD相图中存在丰富的相结构。探究这些极端条件下的相结构，以及不同相之间对称性和自由度的转变是目前高能核物理实验的目标之一。因此，利用理论准确刻画QCD物质的性质是至关重要的。然而，渐近自由所带来的非微扰效应加大了研究低能标区域QCD物质性质的难度，低能有效模型提供了一种研究非微扰区域QCD问题的有效方法。本文将利用低能有效模型研究极端条件下QCD物质的对称性相关问题。手征对称性是QCD物质的重要对称性之一。手征对称性的转变往往伴随QCD物质自由度的转变。著名的低能有效模型---Nambu--Jona-Lasinio（简称NJL）模型可以唯象地描述手征相变，并利用夸克构造介子。NJL模型也常被用于研究磁场对QCD相图的影响。但在强磁场中，NJL模型中带电介子质量定义比较模糊。以pi介子为例，我们利用不依赖于空间平移不变性的虚时方法计算了磁场中pi介子的质量，并证明了以往工作的正确性及其质量结果的规范不依赖性。QCD物质的另一个重要对称性是U_A(1)对称性。我们利用夸克--介子模型在介子平均场层次研究了高重子数密度区域的U_A(1)对称性。我们发现U_A(1)对称性的破缺由夸克凝聚与介子热涨落共同决定。重子数密度效应会单调地恢复破缺的U_A(1)对称性，而温度效应所导致的恢复并不单调。U_A(1)对称性的恢复在低温区域更加明显，并近乎与手征相变同时发生。中子星因其自身性质一直被视为研究低温高重子数密度区域QCD物质的天然实验室，我们希望可以在中子星中找到上述工作所预言的对称性改变现象。是否可以利用平直时空状态方程描述具有如此大引力效应的星体？最近一些工作对此问题持否定态度，并认为要对平直时空状态方程作引力效应修正。我们利用局域能动量守恒和平衡态热力学证明中子星物质的状态方程应与平直时空一致，引力效应只体现在对中子星内部热力学变量的红移中。一般，相对论性致密物质状态方程是由场论方法所确定的，我们提出了一种更便于利用这类状态方程作为输入的Tolman--Oppenheimer--Volkoff方程求解方法。
The fundamental theory describing strong interaction is Quantum Chromodynamics (QCD). Under extremely high temperature and/or density, large blocks of matter dominated by strong interaction can be formed, which is called QCD matter. The theory predicts that extreme conditions such as high temperature, high baryon number density, and strong magnetic field can alter the symmetry of QCD matter, leading to a rich phase structure in the QCD phase diagram. Exploring these novel phases under extreme conditions, as well as the transition of symmetries and degrees of freedom among them are one of the goals of high-energy nuclear experiments. Therefore, characterizing the properties of QCD matter accurately with theory is of great essence. However, non-perturbative effects resulting from asymptotic freedom pose challenges to studying the properties of QCD matter in low-energy scale. The low-energy effective model provides an effective mean to study non-perturbative QCD issues. This thesis focuses on symmetry and related questions of QCD matter under extreme conditions within low-energy effective models. Chiral symmetry is one of the most important symmetries of QCD matter. The change of chiral symmetry commonly accompanies the transition of degrees of freedom in QCD matter. The well-known low-energy effective model---Nambu--Jona-Lasinio (NJL) model can phenomenologically describe the chiral phase transition, and construct mesons based on quarks. The NJL model has been usually used to study the influence of magnetic fields on the QCD phase diagram. But, in strong magnetic fields, the definition of the charged meson mass in the NJL model is rather ambiguous. Taking pion as an instance, we calculated the pion mass in a magnetic field using the imaginary time method, which does not depend on spatial transformation invariance, and we proved the validity and gauge independence of mass results given in previous work. Another important symmetry of QCD matter is U_A(1) symmetry. We applied the quark--meson model to study the U_A(1) symmetry at high baryon number density in the mesonic mean-field level. We found that the broken U_A(1) symmetry is determined by both quark condensates and meson thermal fluctuations. The baryon number density effect can restore the broken U_A(1) symmetry monotonically, while the restoration induced by temperature is not monotonic. In the low temperature region, there is an apparent restoration of U_A(1) symmetry, which may occur simultaneously with the chiral phase transition. Neutron stars are often considered as the natural laboratory for studying QCD matter at low temperature and high baryon number density due to their unique properties. We hope to find some phenomenons of symmetry changing predicted by the above work in neutron stars. If one can apply the equation of state in flat spacetime to depict such kind of stars with large gravity effects? Some recent work held negative opinions on this question and suggested that equation of state in flat spacetime needs to be modified to account for gravity. We used the local energy-momentum conservation law and the equilibrium thermodynamics to prove that the equation of state of the matter in neutron stars should be consistent with the one in flat spacetime. The gravity effect only leads to the redshifting of the thermal variables in neutron stars. Typically, the equation of state of relativistic dense matter is derived by quantum field methods. We proposed a solving method for the Tolman--Oppenheimer--Volkoff equation that is appropriate for such kind of equation of state as input.
