Deformations and spectral properties of merons

Abstract

We consider a meron–antimeron pair located at a, b, ∈ R⁴, and show that the spectrum of its stability operator is not bounded below [in precise mathematical terms: The stability operator defined on C^∞₀(R⁴−{a,b}) has a self‐adjoint extension, possibly many, all of which are unbounded below]. We regularize a single meron located at the origin by replacing it inside a sphere of radius R₀ and outside a sphere of radius R by ’’half instantons,’’ and show that for R≫R₀ the regularized configuration continues to be unstable. For R₀ finite and R=∞, we show that the spectrum of the stability operator continues to extend to −∞. We employ a singular transformation to embed R⁴ into S³×R where the meron pair takes a simple form and its stability operator L becomes L=−d²/dτ²+V, where τ∈R, and the potential V can be diagonalized in terms of the angular momenta, spin, and isospin of the vector field. The spectrum of L is continuous and extends from −2 to +∞. We determine the number of (generalized) zero eigenmodes of L, and calculate its spectrum explicitly.