The c and a-Theorems and the Local Renormalisation Group
The c and a-Theorems and the Local Renormalisation Group
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c 和 a 定理以及局部重正化群
DOI：
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发表时间：
2016
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通讯作者：
G. Shore
中科院分区：
文献类型：
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作者：
G. Shore
Zamolodchikov c-定理使我们对重整化群和准傅立叶变换空间的几何有了新的认识。在这里，我们回顾了寻找c-定理的高维推广和局部重正化群的平行发展。
在局部Weyl标度下运行的位置依赖耦合的重整化的想法可以追溯到它的早期实现，可以追溯到局部重整化小组优雅的现代形式主义。详细解释了关联的Weyl相容条件在建立弯曲时空中道异常系数的RG流动方程中的关键作用，以及它们与c定理和四维a定理的关系。
利用谱函数、能量动量张量的格林函数的RG分析和色散关系，给出了c-定理在二维的几种不同的推导方法，并推广到四维。讨论了建立与四维β_c和β_b道异常系数有关的单调C-函数的障碍。通过建立极大对称空间上的QFT，初步探讨了在道异常中导出涉及Euler-Gauss-Bonnet密度系数β_a的a-定理的可能性。在此基础上，利用四点函数的色散关系，给出了弱a-定理的表述。
最后，我们描述了局部重正化群在具有整体对称性的理论中的极限环问题中的应用，并展示了这是如何揭示QFT中耦合空间的几何问题的。
The Zamolodchikov c-theorem has led to important new insights in our understanding of the renormalisation group and the geometry of the space of QFTs. Here, we review the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local Renormalisation Group.
The idea of renormalisation with position-dependent couplings, running under local Weyl scaling, is traced from its early realisations to the elegant modern formalism of the local renormalisation group. The key role of the associated Weyl consistency conditions in establishing RG flow equations for the coefficients of the trace anomaly in curved spacetime, and their relation to the c-theorem and four-dimensional a-theorem, is explained in detail.
A number of different derivations of the c-theorem in two dimensions are presented -- using spectral functions, RG analysis of Green functions of the energy-momentum tensor T_{mu nu}, and dispersion relations -- and are generalised to four dimensions. The obstruction to establishing monotonic C-functions related to the beta_c and beta_b trace anomaly coefficients in four dimensions is discussed. The possibility of deriving an a-theorem, involving the coefficient beta_a of the Euler-Gauss-Bonnet density in the trace anomaly, is explored initially by formulating the QFT on maximally symmetric spaces. Then the formulation of the weak a-theorem using a dispersion relation for four-point functions of T^mu_mu is presented.
Finally, we describe the application of the local renormalisation group to the issue of limit cycles in theories with a global symmetry and it is shown how this sheds new light on the geometry of the space of couplings in QFT.