  $S$-matrix elements are invariant under field redefinitions of the
Lagrangian. They are determined by geometric quantities such as the curvature
of the field-space manifold of scalar and gauge fields. We present a formalism
where scalar and gauge fields are treated together, with a metric on the
combined space of both types of fields. Scalar and gauge scattering amplitudes
are given by the Riemann curvature $R_{ijkl}$ of this combined space, with
indices $i,j,k,l$ chosen to be scalar or gauge indices depending on the type of
external particle. One-loop divergences can also be computed in terms of
geometric invariants of the combined space, which greatly simplifies the
computation of renormalization group equations. We apply our formalism to the
Standard Model Effective Field Theory (SMEFT), and compute the renormalization
group equations for even-parity bosonic operators to mass dimension eight.
