A real triplet-singlet extended Standard Model: dark matter and collider phenomenology

We examine the collider and dark matter phenomenology of the Standard Model extended by a hypercharge-zero SU(2) triplet scalar and gauge singlet scalar. In particular, we study the scenario where the singlet and triplet are both charged under a single ℤ2 symmetry. We find that such an extension is capable of generating the observed dark matter density, while also modifying the collider phenomenology such that the lower bound on the mass of the triplet is smaller than in minimal triplet scalar extensions to the Standard Model. A high triplet mass is in tension with the parameter space that leads to novel electroweak phase transitions in the early universe. Therefore, the lower triplet masses that are permitted in this extended model are of particular importance for the prospects of successful electroweak baryogenesis and the generation of gravitational waves from early universe phase transitions.

Scalar extensions for polar topologies in locally convex cones

We extend the scalar multiplications for dual pairs of cones and define the corresponding modular neighborhoods and linear polar topologies in locally convex cones. Endowed with the polar topology, every cone may be embedded in a larger cone carrying a linear polar topology over the extended scalars and the embedding is an isomorphism.

Disformal Transformations in Scalar–Torsion Gravity

We study disformal transformations in the context of scalar extensions to teleparallel gravity, in which the gravitational interaction is mediated by the torsion of a flat, metric compatible connection. We find a generic class of scalar–torsion actions which is invariant under disformal transformations, and which possesses different invariant subclasses. For the most simple of these subclasses we explicitly derive all terms that may appear in the action. We propose to study actions from this class as possible teleparallel analogues of healthy beyond Horndeski theories.

Quadratic Forms

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.