Geometry in Scattering Amplitudes

The geometry of field space governs on-shell scattering amplitudes. We formulate a geometric description of effective field theories with scalars, fermions, and gauge bosons. The field-space geometry simplifies the computation of quantum loop corrections. Using this geometric framework, we calculate the one-loop contributions to the renormalization group equations for the Standard Model Effective Field Theory up to mass dimension eight. We also derive novel relations between scattering amplitudes. The soft limit of amplitudes with a massless scalar is captured by a geometric soft theorem. Also, scattering amplitudes for Goldstone bosons with nonsymmetric cosets exhibit a duality between curvature and kinematics, analogous to the color-kinematics duality for Goldstone bosons with symmetric cosets.