Confining kinks. ζ -regularized one-loop kink mass shifts in exotic field theories

Abstract By combining stability and the analysis of topological solitons in ( 1 + 1 ) -dimensional scalar field theories with the Darboux transformation technique, we create models featuring kink-like solutions whose perturbations are all bounded. The stability analysis of extended classical solutions in scalar field theories is determined from the spectrum of quantum mechanical Schrödinger operators whose eigenfunctions serve to expand quantum perturbations to the classical soliton. Our strategy is to invert the procedure: we start from the Schrödinger operator for a single harmonic oscillator and ask whether there is a field theory admitting kink solutions. Having identified the parent exotic field theory to the harmonic oscillator, the Darboux transformation allows for constructing new exotic but solvable Schrödinger equations. This framework relates the quantum harmonic oscillator and its rational deformations to exotic scalar theories featuring non-trivial potentials. Depending on the structure of the spectrum of perturbation frequencies, these potentials may have various local maxima, minima, and inflexion points. The stationary solutions take the form of a definite integral over a finite interval of a function times the Gaussian bell distribution, including the error and the Owen T functions. In such models, the spectrum of the Schrödinger operator governing kink fluctuations is purely discrete. Therefore, Riemann − ζ function regularization allows us to achieve finite one-loop quantum corrections to the classical mass.