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The problem of the gauge hierarchy is brought up in a hypercomplex scheme fora U(1) field theory; in such a scheme a compact gauge group is deformed througha \gamma-parameter that varies along a non-compact internal direction,transverse to the U(1) compact one, and thus an additional SO(1,1) gaugesymmetry is incorporated. This transverse direction can be understood as anextra internal dimension, which will control the spontaneous symmetrybreakdown, and will allow us to establish a mass hierarchy. In this mechanismthere is no brane separation to be stabilized as in the braneworld paradigm,however, a different kind of fine-tuning is needed in order to generate thewished electroweak/Planck hierarchy. By analyzing the effectiveself-interactions and mass terms of the theory, an interesting duality isrevealed between the real and hybrid parts of the effective potential. Thisduality relates the weak and strong self-interaction regimes of the theory, dueto the fact that both mass terms and self-coupling constants appear asone-parameter flows in \gamma. Additionally the \gamma-deformation willestablish a flow for the electromagnetic coupling that mimics therenormalization group flow for the charge in QED.

Hyperbolic deformation of a gauge field theory and the hierarchy problem