Riccion from higher-dimensional space-time with D-dimensional sphere as a compact manifold and one-loop renormalization

Lagrangian density of riccions is obtained with the quartic self-interacting potential using higher-derivative gravitational action in (4 +D)-dimensional space-time withS ^D as a compact manifold. It is found that the resulting four-dimensional theory for riccions is one-loop multiplicatively renormalizable. Renormalization group equations are solved and its solutions yield many interesting results such as (i) dependence of extra dimensions on the enegy mass scale showing that these dimensions increase with the increasing mass scale up toD = 6, (ii) phase transition at 3.05 × 10¹⁶ GeV and (iii) dependence of gravitational and other coupling constants on energy scale. Results also suggest that space-time above 3.05 × 10¹⁶ GeV should be fractal. Moreover, dimension of the compact manifold decreases with the decreasing energy mass scale such thatD = 1 at the scale of the phase transition. Results imply invisiblity of S¹ at this scale (which is 3.05 × 10¹⁶ GeV).