= Kink–antikink attraction from the stress tensor
{title2=$F\sim32m^2\kappa^2e^{-2\kappa d}$}
For $V(\phi)=\lambda(m^2-\phi^2)^2$ with $\lambda,m>0$, set $\kappa=\sqrt{2\lambda}m$. A well-separated <kink>–<antikink> pair at $-d/2,d/2$ is approximated by $\phi=m\tanh\kappa(x+d/2)-m\tanh\kappa(x-d/2)-m$. At its midpoint, $\phi_x=0$ and $\phi=m-4m e^{-\kappa d}+O(e^{-2\kappa d})$. The static <stress-energy tensor> component $T^{11}=\phi_x^2/2-V$ is therefore negative there. <Momentum> conservation gives the force on the left half-line as $T^{11}(-\infty)-T^{11}(0)$, which is positive. Its leading magnitude is $F\sim32m^2\kappa^2e^{-2\kappa d}$. Thus the pair attracts. This is a controlled leading-tail estimate at large separation, not an exact superposed solution or a theorem about all subsequent collision outcomes.
Back to article page
