Large-N analysis of three dimensional nonlinear sigma models
Large-N analysis of three dimensional nonlinear sigma models
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三维非线性 sigma 模型的大 N 分析
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
M. Tsuzuki
中科院分区：
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作者：
K. Higashijima;E. Itou;M. Tsuzuki
非微扰重整化群方法表明一大类非线性西格玛模型在三维时空中是可重整化的，而在微扰理论中它们是不可重整化的。 ${\cal N}=2$ 超对称非线性 sigma 模型的目标空间是具有正标量曲率的 Einstein-K\"{a}hler 流形就属于这一类。厄密对称空间是齐次的，是这些流形的特别简单的例子。为了找到这些模型的非微扰可重整性的独立证据，大 N 方法（另一种非微扰方法）被应用于 3 维 ${\cal N}=2$ 超对称非线性目标空间 $CP^{N-1}=SU(N)/[SU(N-1)\times U(1)]$ 和 $Q^{N-2}=SO(N)/[SO(N-2)\times SO(2)]$ 上的 sigma 模型是埃尔米特对称空间的两个典型示例。
我们发现这些模型中的 $\beta$ 函数与 1/N 展开的次前导阶中的非微扰重正化群方法的结果一致，并且具有非平凡的 UV 不动点。 $Q^{N-2}$ 模型的 $\beta$ 函数在 1/N 展开的次前序中接收非零修正。
我们还研究了模型的相结构。 $CP^{N-1}$ 模型有两个阶段； SU(N) 对称和不对称相。 $Q^{N-2}$ 模型分为三个阶段；陈-西蒙斯、希格斯和 SO(N) 破碎相。在陈-西蒙斯和希格斯相中，SO(N) 对称性保持不变，所有动力场都变得巨大。辅助规范场还通过陈-西蒙斯相中的诱导陈-西蒙斯项以及希格斯相中双夸克束缚态的真空期望值获得质量。
Non-perturbative renormalization group approach suggests that a large class of nonlinear sigma models are renormalizable in three dimensional space-time, while they are non-renormalizable in perturbation theory. ${\cal N}=2$ supersymmetric nonlinear sigma models whose target spaces are Einstein-K\"{a}hler manifolds with positive scalar curvature belongs to this class. hermitian symmetric spaces, being homogeneous, are specially simple examples of these manifolds. To find an independent evidence of the nonperturbative renormalizability of these models, the large N method, another nonperturbative method, is applied to 3-dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models on the target spaces $CP^{N-1}=SU(N)/[SU(N-1)\times U(1)]$ and $Q^{N-2}=SO(N)/[SO(N-2)\times SO(2)]$, two typical examples of hermitian symmetric spaces.
We find that $\beta$ functions in these models agree with the results of the nonperturbative renormalization group approach in the next-to-leading order of 1/N expansion, and have non-trivial UV fixed points. The $\beta$ function of the $Q^{N-2}$ model receives a nonzero correction in the next-to-leading order of the 1/N expansion.
We also investigate the phase structures of our models. The $CP^{N-1}$ model has two phases; SU(N) symmetric and asymmetric phase. The $Q^{N-2}$ model has three phases; Chern-Simons, Higgs and SO(N) broken phases. In the Chern-Simons and Higgs phase, SO(N) symmetry remains unbroken and all dynamical fields becomes massive. An auxiliary gauge field also acquires mass, through an induced Chern-Simons term in the Chern-Simons phase, and through the vacuum expectation value of a di-quark bound state in the Higgs phase.