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We study the asymptotic safety conjecture for quantum gravity in the presenceof matter fields. A general line of reasoning is put forward explaining whygravitons dominate the high-energy behaviour, largely independently of thematter fields as long as these remain sufficiently weakly coupled. Ourconsiderations are put to work for gravity coupled to Yang-Mills theories withthe help of the functional renormalisation group. In an expansion about flatbackgrounds, explicit results for beta functions, fixed points, universalexponents, and scaling solutions are given in systematic approximationsexploiting running propagators, vertices, and background couplings. Invariably,we find that the gauge coupling becomes asymptotically free while thegravitational sector becomes asymptotically safe. The dependence on matterfield multiplicities is weak. We also explain how the scheme dependence, whichis more pronounced, can be handled without changing the physics. Our findingsoffer a new interpretation of many earlier results, which is explained indetail. The results generalise to theories with minimally coupled scalar andfermionic matter. Some implications for the ultraviolet closure of the StandardModel or its extensions are given.

One force to rule them all: asymptotic safety of gravity with matter