Mode-number regularization of soliton masses

Compare the same total number of finite-box modes in the soliton and vacuum sectors before removing the ultraviolet cutoff. A localized bound state replaces a continuum mode, and its contribution must be retained. For the Sine-Gordon kink fluctuation operator and the odd phase branch $\delta (k)=2\arctan (1/k)$, the bound translation mode replaces the free $k=0$ oscillator. Thus the complete oscillator correction is $-m/2-(m/2)\int dk\delta (k)k/[2\pi \sqrt{k^2+1}]$ in this convention. The continuum integral alone misses the finite $-m/2$ contribution.

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.