Cutoff surface term for a Sine-Gordon kink ($-\omega (\Lambda )\delta (\Lambda )/(2\pi )⟶-m/\pi$)

The scattering phase shift $\delta (k)=2\arctan (m/k)$ of the Sine-Gordon kink fluctuation operator gives a continuum frequency sum. In mode-number regularization of soliton masses, the zero-frequency translation mode replaces the vacuum oscillator at $k=0$. The bare difference is $-m/2-(2\pi {)}^{-1}{\int }_0^{\Lambda }\delta (k)k/\sqrt{k^2+m^2}dk$. Integration by parts cancels the lower endpoint against $-m/2$ and produces the displayed upper surface term. Although $\delta (\Lambda )$ vanishes, its product with $\omega (\Lambda )$ tends to $2m$. Vacuum normal ordering cancels the logarithmic bulk ultraviolet divergence, leaving this finite one-loop soliton mass correction. Parameters and finite counterterms must be fixed by a stated renormalization condition.