Renormalization of the Standard Model Effective Field Theory from Geometry

Understand

$S$-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.