Abstract
We propose a candidate $c$-function in arbitrary dimensional quantum field theories with broken Lorentz and rotational symmetry. For holographic theories we derive the necessary and sufficient conditions on the geometric background for these $c$-functions to satisfy the $c$-theorem. We obtain the null energy conditions for anisotropic background to show that do not themselves assure the $c$-theorem. By employing them, we find that is possible to impose conditions on the UV data that are enough to guarantee at least one monotonic $c$-function along the RG flow. These UV conditions can be used as building blocks for the construction of anisotropic monotonic RG flows. Finally, we apply our results to several known anisotropic theories and identify the region in the parameters space of the metric where the $c$-theorem holds for our proposed $c$-function.
Highlights
The renormalization group (RG) is a powerful method for constructing relations between theories at different length scales
The function is stationary at the fixed point of the RG flow, with value given by the central charge of the conformal field theory (CFT)
We show that for anisotropic theories, the monotonicity of the c-function depends on the satisfaction of certain geometric background conditions which are complementary to the null energy conditions
Summary
The renormalization group (RG) is a powerful method for constructing relations between theories at different length scales. The c-theorem of Zamolodchikov [1] is a remarkable result of this kind It states, for two-dimensional quantum field theories (QFTs), the existence of a positive real function c that decreases monotonically along the RG flow from the ultraviolet (UV) to the infrared (IR). We remark that in relativistic holographic CFTs, the monotonicity of the c-function is assured as long as the null energy conditions are satisfied for the bulk gravity theory [13]. This is not necessarily the case for anisotropic or Lorentz violating holographic QFT.
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