Quadratic gravity is a theory that extends Einstein's gravity by including terms quadratic in curvature. The paper argues that the beta functions of quadratic gravity do not correctly describe the physical dependence of scattering amplitudes on external momenta. The authors derive the correct physical beta functions, showing that asymptotic freedom can be achieved without tachyons. Quadratic gravity has a massless graviton and a massive spin-2 particle that is a ghost and potentially a tachyon. It also has a massive spin-0 particle that is a tachyon for positive ξ. Despite these pathologies, the theory has attracted renewed interest. Previous calculations of beta functions missed contributions from Nakanishi-Lautrup ghosts, which were corrected in later studies. The beta functions derived in the paper are: βλ = -1/(4π)² * 133λ²/10 βξ = -1/(4π)² * 5(72λ² - 36λξ + ξ²)/36 These beta functions have been confirmed in several calculations. However, the authors argue that these functions do not correctly describe physical running, which is the dependence of couplings on external momenta, not the renormalization scale μ. Physical running is different from μ-running, which is the dependence of couplings on the renormalization scale. The authors compute the physical beta functions by considering the dependence of couplings on external momenta. They find that the physical beta functions are: βλ = -4bλλ² βξ = -2bξξ² where bλ and bξ are coefficients determined from the two-point function of the background fluctuation f. The physical beta functions are derived by considering the contributions from bubble diagrams and tadpole diagrams. The authors show that the physical beta functions are different from the standard beta functions and allow for asymptotic freedom without tachyons. The paper also discusses the implications of these results for the renormalization of quadratic gravity. The authors show that the physical beta functions allow for asymptotic freedom without tachyons, making quadratic gravity a more plausible candidate for a complete theory of quantum gravity. The results are supported by calculations of the beta functions using the background field method and the heat kernel technique. The authors conclude that the physical beta functions are necessary for a correct description of the theory.Quadratic gravity is a theory that extends Einstein's gravity by including terms quadratic in curvature. The paper argues that the beta functions of quadratic gravity do not correctly describe the physical dependence of scattering amplitudes on external momenta. The authors derive the correct physical beta functions, showing that asymptotic freedom can be achieved without tachyons. Quadratic gravity has a massless graviton and a massive spin-2 particle that is a ghost and potentially a tachyon. It also has a massive spin-0 particle that is a tachyon for positive ξ. Despite these pathologies, the theory has attracted renewed interest. Previous calculations of beta functions missed contributions from Nakanishi-Lautrup ghosts, which were corrected in later studies. The beta functions derived in the paper are: βλ = -1/(4π)² * 133λ²/10 βξ = -1/(4π)² * 5(72λ² - 36λξ + ξ²)/36 These beta functions have been confirmed in several calculations. However, the authors argue that these functions do not correctly describe physical running, which is the dependence of couplings on external momenta, not the renormalization scale μ. Physical running is different from μ-running, which is the dependence of couplings on the renormalization scale. The authors compute the physical beta functions by considering the dependence of couplings on external momenta. They find that the physical beta functions are: βλ = -4bλλ² βξ = -2bξξ² where bλ and bξ are coefficients determined from the two-point function of the background fluctuation f. The physical beta functions are derived by considering the contributions from bubble diagrams and tadpole diagrams. The authors show that the physical beta functions are different from the standard beta functions and allow for asymptotic freedom without tachyons. The paper also discusses the implications of these results for the renormalization of quadratic gravity. The authors show that the physical beta functions allow for asymptotic freedom without tachyons, making quadratic gravity a more plausible candidate for a complete theory of quantum gravity. The results are supported by calculations of the beta functions using the background field method and the heat kernel technique. The authors conclude that the physical beta functions are necessary for a correct description of the theory.