Existence of instanton-like solutions mediating tunnelling between degenerate flat Yang–Mills vacua

Determine whether there exist instanton-like finite-action solutions of the SU(2) Yang–Mills equations that induce tunnelling transitions between the degenerate zero-energy flat connections with vanishing field strength G_{ij} = 0 that arise when the Abelian background field H and the constant vectors a and b satisfy g H = a × b, where (H, a, b) form an orthogonal right-handed frame.

Background

The paper constructs exact non-perturbative solutions of the sourceless SU(2) Yang–Mills equations using the Cho decomposition, yielding configurations characterized by two constant spatial vectors a and b and an Abelian background magnetic field H. The magnetic energy density takes the form ε = ((g H − a × b)^2)/(2 g^2). When g H = a × b, the field strength vanishes and the gauge potentials reduce to flat (pure-gauge) connections, producing a highly degenerate set of zero-energy vacua separated by finite potential barriers.

The authors argue that tunnelling between these degenerate flat connections could restore Lorentz invariance at the quantum level by superposing vacua with different orientations of g H = a × b. However, they explicitly note that it is currently unknown whether suitable instanton-like solutions exist to mediate such tunnelling transitions, leaving this as an open problem.