= Sine-Gordon vacuum tadpole counterterm
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{title2=$\Delta M_{\rm ct}=4\delta m^2/(m\beta^2)$}
With canonical field $\varphi=\phi/\beta$, 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 $\Delta M_{\rm ct}=4\delta m^2/(m\beta^2)$ because $\int(1-\cos\phi_K)dx=4$. This cancels the logarithmic <ultraviolet divergence> in the <one-loop soliton mass correction>. Finite parts depend on the <renormalization condition>.
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