Extended Soliton Solutions in an Effective Action for $\mathrm{SU}(2)$ Yang–Mills Theory

The Skyrme–Faddeev–Niemi (SFN) model which is an $O(3)$ $\sigma$ model in three dimensional space up to fourth-order in the first derivative is regarded as a low-energy effective theory of $SU(2)$ Yang–Mills theory. One can show from the Wilsonian renormalization group argument that the effective action of Yang–Mills theory recovers the SFN in the infrared region. However, the theory contains an additional fourth-order term which destabilizes the soliton solution. We apply the perturbative treatment to the second derivative term in order to exclude (or reduce) the ill behavior of the original action and show that the SFN model with the second derivative term possesses soliton solutions.