Sine-Gordon vacuum tadpole counterterm ($\Delta M_{\mathrm{ct}}=4\delta m^2/(m{\beta }^2)$)

With canonical field $\phi =\varphi /\beta$, the quartic interaction has coupling $-m^2{\beta }^2$. Cancelling the vacuum tadpole diagram requires the divergent mass counterterm $\delta m^2=(m^2{\beta }^2/4){\int }_{-\Lambda }^{\Lambda }dk/[2\pi \sqrt{k^2+1}]$. Its kink energy is $\Delta M_{\mathrm{ct}}=4\delta m^2/(m{\beta }^2)$ because $\int (1-\cos {\varphi }_K)dx=4$. This cancels the logarithmic ultraviolet divergence in the one-loop soliton mass correction. Finite parts depend on the renormalization condition.