Department of Physics,
Faculty of Science and Technology,
Tokyo University of Science,
2641, Noda,
Chiba 278-8510,
JAPAN
A modified O(3) $\sigma$ model (by adding special fourth-order term with first derivative) in three space dimensions, called Skyrme-Faddeev-Niemi (SFN) model, has a class of topological soliton solutions emerging remarkable ring-, or knot-like structure [1-3]. The model is regarded as the low-energy effective model of SU(2) Yang-Mills (YM) theory. Wilsonian renormalization group arguments suggest that the effective action of YM achieves that of SFN in the infrared region [4]. Unfortunately, in this recipe another fourth-order term appears at the same time and it destabilizes the soliton. In order to recover the stability, we take into account effects of the higher (second)-derivative term and finally obtain the stable soliton solutions.