Quantum Lifts of Noninteger Power Law Field Theories

Field theories whose potentials have noninteger power laws $\alpha$ have found many applications, but are often claimed to have no lift to quantum field theory except as effective models. We define quantum lifts by expanding the classical potential in Hermite polynomials and then normal ordering at a mass scale shifted by a parameter $\beta$. We find that when $\alpha>2$, for sufficiently large $\beta$, the vacuum state can be perturbatively expanded in usual Fock states. We apply this to the following problem. The $\sigma=4$ P\"oschl-Teller model has a $\phi^{5/2}$ potential. As the third derivative of the potential diverges in each vacuum, one expects the three point interactions to diverge in the vacuum. The model's kink has three shape modes and the least bound mode extends so far into the vacuum that its probability of being excited by radiation apparently diverges. We show that a deformation $\beta$ of order the meson mass or larger is sufficient to tame this divergence, although it nonetheless results in an excitation probability which is enhanced by a $\beta$-dependent fractional power of the inverse coupling.