geoSCET: Soft Theorems from Power Counting

We apply the framework of Soft Collinear Effective Theory (SCET) to the theory of the geometric scalar field. The resulting ``geoSCET'' manifests an emergent geometry in the soft sector, mirroring the emergent soft gauge invariance of QCD SCET. We use geoSCET to derive the geometric soft theorems as straightforward consequences of effective-field-theory power counting, including extensions to multiple soft emissions and loop corrections. We demonstrate that theories without a potential satisfy universal geometric soft theorems for any number of soft emissions to all orders in perturbation theory. We also show that we can turn on a potential at the soft scale without spoiling the factorization of the soft physics. This work demonstrates the underlying field-space diffeomorphism origin of the geometric soft theorem.