Emerging Translational Variance: Vacuum Polarization Energy of the $\mathbf{\phi^6}$ kink

We propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow a decomposition into symmetric and anti-symmetric channels in one space dimension. In particular we compute the vacuum polarization energy of the kink soliton in the $\phi^6$ model. We link the dependence of this energy on the position of the center of the soliton to the different masses of the quantum fluctuations at negative and positive spatial infinity.