  Asymptotic Safety provides an elegant mechanism for obtaining a consistent
high-energy completion of gravity and gravity-matter systems. Following the
initial idea by Steven Weinberg, the construction builds on an interacting
fixed point of the theories renormalization group (RG) flow. In this work we
use the Wetterich equation for the effective average action to investigate the
RG flow of gravity supplemented by a real scalar field. We give a
non-perturbative proof that the subspace of interactions respecting the global
shift-symmetry of the scalar kinetic term is closed under RG transformations.
Subsequently, we compute the beta functions in an approximation comprising the
EinsteinHilbert action supplemented by the shift-symmetric quartic scalar
self-interaction and the two lowest order shift-symmetric interactions coupling
scalar-bilinears to the spacetime curvature. The computation utilizes the
background field method with an arbitrary background, demonstrating that the
results are manifestly background independent. Our beta functions exhibit an
interacting fixed point suitable for Asymptotic Safety, where all matter
interactions are non-vanishing. The presence of this fixed point is rooted in
the interplay of the matter couplings which our work tracks for the first time.
The relation of our findings with previous results in the literature is
discussed in detail and we conclude with a brief outlook on potential
phenomenological applications.
