The effective chiral Lagrangian for low-energy pion-Skyrmion physics is considered as an expansion in powers of (∂Φ), where Φ is the chiral field. The O((∂Φ${)}^2$) term is the nonlinear σ-model Lagrangian $L_2$. Two theoretical approaches to the higher-order terms are examined. The first involves integration over fermion degrees of freedom. The O((∂Φ${)}^4$) terms obtained this way, together with $L_2$, are found to be consistent with low-energy ππ phase shifts—in particular, in the dominant ρ and σ channels. In the second approach, an effective pion Lagrangian is generated by integrating some simple chiral meson-field Lagrangians over the heavy mesons. At O((∂Φ${)}^4$) two terms are found: a term $L_4$,ρ of Skyrme form, with a model-independent coefficient determined by the ρ mass, and a non-Skyrme term $L_4$,σ, the coefficient of which is more model dependent and determined by the σ mass; both terms are compatible with the fermion integration results. It is concluded that the O((∂π${)}^4$) terms are well established theoretically, and in good qualitative agreement with pion data. These terms do not, however, stabilize the soliton. Some discussion of the possible stabilizing effects of O((∂π${)}^6$) terms is given.

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