Adventures of the Coupled Yang-Mills Oscillators: II. YM-Higgs Quantum Mechanics

We continue our study of the quantum mechanical motion in the $x^2y^2$ potentials for $n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM) equations. In the present paper, we develop a new approach to the calculation of the partition function $Z(t)$ beyond the Thomas-Fermi (TF) approximation by adding a harmonic (Higgs) potential and taking the limit $v\to 0$, where $v$ is the vacuum expectation value of the Higgs field. Using the Wigner-Kirkwood method to calculate higher-order corrections in $\hbar$, we show that the limit $v\to 0$ leads to power-like singularities of the type $v^{-n}$, which reflect the possibility of escape of the particle along the channels in the classical limit. We show how these singularities can be eliminated by taking into account the quantum fluctuations dictated by the form of the potential.