Classifying BPS states in supersymmetric gauge theories coupled to higher derivative chiral models

We study N=1 supersymmetric gauge theories coupled with higher derivative chiral models in four dimensions in the off-shell superfield formalism. We solve the equation of motion for the auxiliary fields and find two distinct on-shell structures of the Lagrangian that we call the canonical and noncanonical branches characterized by zero and nonzero auxiliary fields, respectively. We classify Bogomol'nyi-Prasado-Sommerfield (BPS) states of the models in Minkowski and Euclidean spaces. In Minkowski space, we find Abelian and non-Abelian vortices, vortex lumps (or gauged lumps with fractional lump charges) as 1/2 BPS states in the canonical branch, and higher derivative generalization of vortices and vortex-(BPS)baby Skyrmions (or gauged BPS baby Skyrmions with fractional baby Skyrme charges) as 1/4 BPS states in the noncanonical branch. In four-dimensional Euclidean space, we find Yang-Mills instantons trapped inside a non-Abelian vortex, intersecting vortices, and intersecting vortex-(BPS)baby Skyrmions as 1/4 BPS states in the canonical branch but no BPS states in the noncanonical branch other than those in the Minkowski space.