= Cutoff surface term for a Sine-Gordon kink
{title2=$-\omega(\Lambda)\delta(\Lambda)/(2\pi)\longrightarrow-m/\pi$}
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>.
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