  We investigate the non-perturbative renormalization group flow of the scalar
Galileon model in flat space. We discuss different expansion schemes of the
Galileon truncation, including a heat-kernel based derivative expansion, a
vertex expansion in momentum space and a curvature expansion in terms of a
covariant geometric formulation. We find that the Galileon symmetry prevents a
quantum induced renormalization group running of the Galileon couplings.
Consequently, the Galileon truncation only features a trivial Gaussian fixed
point.
