The exact one-loop beta functions for the four-derivative terms (Weyl tensor squared, Ricci scalar squared, and the Gauss-Bonnet) are derived for the minimal six-derivative quantum gravity (QG) theory in four spacetime dimensions. The calculation is performed by means of the Barvinsky and Vilkovisky generalized Schwinger-DeWitt technique. With this result we gain, for the first time, the full set of the relevant beta functions in a super-renormalizable model of QG. The complete set of renormalization group (RG) equations, including also those for the Newton and the cosmological constant, is solved explicitly in the general case and for the six-derivative Lee-Wick (LW) quantum gravity proposed in a previous paper by two of the authors. In the ultraviolet regime, the minimal theory is shown to be asymptotically free and describes free gravitons in Minkowski or (anti–)de Sitter [(A)dS] backgrounds, depending on the initial conditions for the RG equations. The ghostlike states appear in complex conjugate pairs at any energy scale consistently with the LW prescription. However, owing to the running, these ghosts may become tachyons. We argue that an extension of the theory that involves operators cubic in Riemann tensor may change the beta functions and hence be capable of overcoming this problem.

中文翻译：

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六导数量子引力重整化群

四导数项（外尔张量平方、里奇标量平方和高斯-博内）的精确单环 beta 函数是为四个时空维度的最小六导数量子引力 (QG) 理论推导出来的。计算是通过 Barvinsky 和 ​​Vilkovisky 广义 Schwinger-DeWitt 技术进行的。有了这个结果，我们第一次在 QG 的超重整化模型中获得了完整的相关 beta 函数集。完整的重整化群 (RG) 方程组，包括牛顿和宇宙常数的方程组，在一般情况下和之前论文中提出的六阶导数 Lee-Wick (LW) 量子引力得到了明确的求解。的作者。在紫外线范围内，极小理论被证明是渐近自由的，并且根据 RG 方程的初始条件描述了 Minkowski 或 (anti-)de Sitter [(A)dS] 背景中的自由引力子。在与 LW 规定一致的任何能量尺度上，幽灵状状态以复共轭对的形式出现。但是，由于运行，这些鬼魂可能会变成速子。我们认为，涉及黎曼张量中三次算子的理论的扩展可能会改变 beta 函数，因此能够克服这个问题。