Hyperbolic deformation of a gauge field theory and the hierarchy problem

Abstract

The problem of the gauge hierarchy is brought up in a hypercomplex scheme for a U (1) field theory; in such a scheme a compact gauge group is deformed through a γ-parameter that varies along a non-compact internal direction, transverse to the U (1) compact one, and thus an additional SO(1, 1) gauge symmetry is incorporated. This transverse direction can be understood as an extra internal dimension, which will control the spontaneous symmetry breakdown, and will allow us to establish a mass hierarchy. In this mechanism there is no brane separation to be estabilized as in the braneworld paradigm, however, a different kind of fine-tuning is needed in order to generate the wished electroweak/Planck hierarchy. By analyzing the effective self-interactions and mass terms of the theory, an interesting duality is revealed between the real and hybrid parts of the effective potential. This duality relates the weak and strong self-interaction regimes of the theory, due to the fact that both mass terms and self-coupling constants appear as one-parameter flows in γ. Additionally the γ-deformation will establish a flow for the electromagnetic coupling that mimics the renormalization group flow for the charge in QED.

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

“…The physics of topological defects associated with continuous symmetries turns out to be radically different from that established for conventional gauge field theories. Furthermore, the effective value for the coupling constants in hypercomplex electrodynamics are comparable with those generated by quantum fluctuations [24].…”

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

“…The physics of topological defects associated with continuous symmetries turns out to be radically different from that established for conventional gauge field theories. Furthermore, the effective value for the coupling constants in hypercomplex electrodynamics are comparable with those generated by quantum fluctuations [22].…”

“…Considering that the real exponentials in Eq. ( 22) are solutions to the equations of motion, the analytically continued solutions can be obtained through the transformations ω k → iω k , and k → ik, within the standard scheme, and through ω k → jω k , and k → jk within the case at hand, which lead to the purely hyperbolic expression (22). In both cases the complexification does not change the relative sign between ω 2 k , and k 2 in the dispersion relations, but the relative sign of the dissipative term γ 2 is different in each case, since in the former i 2 = −1, and in the later j 2 = 1.…”

“…If the spectral decomposition of the field operator is constructed from the solution (22), considering the full range k ∈ (−∞, +∞), taking into account the usual Dirac delta functions resulting from the commutators between the annihilaton and creation operators,, then the field operator commutators will undergo UV divergences; therefore we have to re-consider the decomposition range and the distribution representing the Dirac delta on the hyperbolic ring.…”

“…In order to define a spectral decomposition of the fields, we consider first that the full field defined in Eq. ( 16), Ω √ 2 = J + φ + J − ψ, represents the complete system in the basis (J + , J − ), including the subsystem A (φ) , and the environment (ψ); thus, the full range of the spectral parameters k for the complete system, must be splitted into sub-ranges; such a splitting depends on the physical properties of the subsystems, and we propose here an equipartition of the momenta according to (−∞, 0) ←→ J − , (0, +∞) ←→ J + ; (28) hence the spectral decomposition of the field operator constructed from the solution (22) reads…”