I want to prove that the the scalar field of the yang-mills lagrangian tends to some constant value which is a function of theta at infinity and that this value is a zero of the potential, when we consider a soliton ( finite energy, non dissipative) solution . Here is an attempt. From the energy expression obtained by splitting the yang-mills lagrangian into a kinetic energy and potential energy part, we can see that $$E=\int_1^\infty dr\, r^2\int_0^{2\pi} d\Omega \frac{1}{2}(D_i \phi)^\dagger(D_i \phi)+ U $$
Here $\Omega$ is the solide angle, D is the covariant derivative, this expression can be obtained by gauging away the time component of the gauge fields. We can apply a further gauge transformation so that the component of the gauge field $A_i$ in a radial direction say $\hat{\textbf{x}}$ is 0. Then if I take a dot product of the above expression I get that $\lim_{r\to \infty}r^2 U$ and $\lim_{r \to \infty} r^2\,\partial_i \phi$ is 0, as for finite energy the integral should be identically 0. From this how do I prove that the actual limits of U and $\partial_i \phi$ are 0?
