$CPT$ symmetry is at the heart of the standard model of particle physics and experimentally very well tested, but expected to be broken in some approaches to quantum gravity. It thus becomes pertinent to explore which of the two alternatives is realized: (i) $CPT$ symmetry is emergent, so that it is restored in the low-energy theory, even if it is broken beyond the Planck scale, (ii) $CPT$ symmetry cannot be emergent and must be fundamental, so that any approach to quantum gravity, in which $CPT$ is broken, is ruled out. We explore this by calculating the renormalization group flow of $CPT$-violating interactions under the impact of quantum fluctuations of the metric. We find that $CPT$ symmetry cannot be emergent and conclude that quantum-gravity approaches must avoid the breaking of $CPT$ symmetry. As a specific example, we discover that in asymptotically safe quantum gravity $CPT$ symmetry remains intact, if it is imposed as a fundamental symmetry, but it is badly broken at low energies if a tiny amount of $CPT$ violation is present in the transplanckian regime.

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• Quantum gravity
• Discrete symmetries