Beta Function and Asymptotic Safety in Three-dimensional Higher Derivative Gravity

We study the quantum properties of the three-dimensional higher derivative gravity. In particular we calculate the running of the gravitational and cosmological constants. The flow of these couplings shows that there exist both Gaussian and nontrivial fixed points in the theory, thus confirming that the theory is asymptotically safe. It is shown that the new massive gravity or $f(R)$ gravity in three dimensions do not correspond to the fixed point within the approximation that the coefficients of the higher curvature terms are not subject to the flow. The fixed point value of the cosmological constant is found to be gauge-independent, positive and small. We also find that if we start with Einstein term with negative sign, the fixed point only exists when the coefficient of the Einstein term has positive sign.