One-loop soliton mass correction ($\Delta M_{\mathrm{osc}}=\frac{1}{2}{\sum }_n({\Omega }_n-{\Omega }_n^{(0)})$)

In a common finite-volume regularization in quantum field theory, the oscillator contribution is $\Delta M_{\mathrm{osc}}=\frac{1}{2}{\sum }_n({\Omega }_n-{\Omega }_n^{(0)})$, including every discrete and continuum-derived mode and treating translation as a collective coordinate. Physical frequencies are ${\Omega }_n=m{\omega }_n$ when coordinates are measured in units of $m^{-1}$. A mass counterterm is added only afterwards. The vacuum subtraction removes the extensive vacuum energy but can leave an ultraviolet divergence.

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.