Spontaneous symmetry breaking, and strings defects in hypercomplex gauge field theories

Abstract

Inspired by the appearance of split-complex structures in the dimensional reduction of string theory, and in the theories emerging as byproducts, we study the hypercomplex formulation of Abelian gauge field theories by incorporating a new complex unit to the usual complex one. The hypercomplex version of the traditional Mexican hat potential associated with the U (1) gauge field theory, corresponds to a hybrid potential with two real components, and with U (1)× SO(1, 1) as symmetry group. Each component corresponds to a deformation of the hat potential, with the appearance of a new degenerate vacuum. Hypercomplex electrodynamics will show novel properties, such as spontaneous symmetry breaking scenarios with running masses for the vectorial and scalar Higgs fields, and such as AharonovBohm type strings defects as exact solutions; these topological defects may be detected only by quantum interference of charged particles through gauge invariant loop integrals. In a particular limit, the hyperbolic electrodynamics does not admit topological defects associated with continuous symmetries.

Hyperbolic deformation of a gauge field theory and the hierarchy problem

“…This extended complexification allows a new description of fermions and bosons, with the possibility of new interactions arising from hypercomplex gauge transformations [7]. From the mathematical point of view there exist formally three different complex units, namely, the (conventional) elliptic i 2 = −1, the hyperbolic j 2 = 1, and the parabolic, for which the square of the complex unit vanishes (see for example, [8]).With these antecedents, we have recently developed a hypercomplex formulation of Abelian gauge field theories, by incorporating the new complex unit to the usual complex one [9]. Physically the hypercomplex formulation allows us to accommodate hyperbolic complex counterparts for the usual U (1) interactions, and hence to explore their possible realizations beyond presently known energies.In the particular case of the hypercomplex electrodynamics, the results show exotic scenarios for spontaneous symmetry breaking, such as running masses for vectorial and scalar fields, that mimic flows of the renormalization group.…”

“…By using the commutative ring of hypercomplex numbers, the usual real objects such as Lagrangians, vector fields, the norm of a complex field, masses, coupling parameters, etc, are generalized to Hermitian objects, encoding two real quantities. Furthermore, in this scheme, a hypercomplex field will have four real components, instead of the two real components of an usual complex field; however, in the hypercomplex formulation developed in [9], those four components are identified to each other using a real dimensionless γ-parameter, leading to two real effective variables. Therefore, the new formulation is constructed as a γ-deformation of the U (1)-formulation of an Abelian gauge theory; the deformation implies the incorporation of a new symmetry, namely, the hyperbolic rotations as a complement of the circular U (1)-rotations; the full symmetry group will correspond at the end to SO(1, 1) × U (1), the product of a noncompact group and a compact one.…”

“…With these antecedents, we have recently developed a hypercomplex formulation of Abelian gauge field theories, by incorporating the new complex unit to the usual complex one [9]. Physically the hypercomplex formulation allows us to accommodate hyperbolic complex counterparts for the usual U (1) interactions, and hence to explore their possible realizations beyond presently known energies.…”

“…In this picture, the Higgs boson is light, while its SU(5) partners automatically get large masses through the vev of a complex field. of the theory, it played a passive role in the pattern of the spontaneous symmetry breakdown obtained in [9], since the diagonalization of the mass matrix was realized by choosing a gauge that involves only the circular and hyperbolic parameters of the symmetry group U (1) × SO(1, 1), according to the conventional procedure for such a purpose. However, as we shall see in the present work, it is possible to generate a qualitatively different pattern of SSB controlled by the same γparameter, introducing naturally a hierarchy of orders of magnitude in the mass spectrum.…”

“…However, as we shall see in the present work, it is possible to generate a qualitatively different pattern of SSB controlled by the same γparameter, introducing naturally a hierarchy of orders of magnitude in the mass spectrum. The γ-parameter varies smoothly within a range determined in [9] in order to induce a stable vacuum; the determination of the hierarchy is then transformed into the choice of the adjustable γ-parameter.…”