Fractional dynamics, Cantorian space–time and the gauge hierarchy problem

The gauge hierarchy problem in particle physics refers to the large numerical disparity between the value of the Planck mass (MPl 1.22×1019 GeV) and the mass scale of the electroweak interaction (MEW≈102 GeV). Explaining the hierarchy paradox has been attempted so far in quantum field models based on supersymmetry or higher dimensional space–time with a large number of extra dimensions (brane theories). Despite several years of experimental search, there is currently no validation for either one of these models. We approach the hierarchy paradox using the methodology of fractal operators in four-dimensional space–time. It is found that departure from the inverse-square gravity in the high-energy regime emerges naturally from the fractional Helmholtz equation and offers a simple resolution to the problem. Our work makes an explicit connection between the hierarchy problem and Cantorian geometry of space–time on energy scales comparable to the Planck mass.