On the stability of Born-Infeld-regularised electroweak monopoles

The Cho-Maison monopole provides a monopole solution of the electroweak field equations, but possesses an infinite classical energy due to the Maxwell form of the hypercharge sector. Motivated by string-inspired effective field theories, we study the perturbative stability of the Cho-Maison monopole when the hypercharge kinetic term is regularised by a Born-Infeld extension, which renders the monopole energy finite. Focusing on the bosonic electroweak theory with an unmodified $SU(2)_L$ sector and a Born-Infeld U(1)_Y sector, we analyze linear fluctuations about the regularised monopole background. Using a complex tetrad and a spin-weighted harmonic decomposition, we reduce the fluctuation equations to coupled radial Schroedinger-type eigenvalue problems and examine the spectrum of the resulting operators. We extend the separation-of-variables framework developed by Gervalle and Volkov to this non-linear gauge-field setting. We show that, after appropriate gauge fixing and constraint elimination, the Born-Infeld deformation preserves the angular channel structure of the Maxwell theory and leads to a self-adjoint Sturm-Liouville type problem for the stability of the radial modes, with modified radial coefficients determined by the background Born-Infeld profile. The resulting operator represents a smooth deformation of the Maxwell case and retains positive kinetic weight. Our results provide plausible evidence for the stability of the Born-Infeld deformed monopole and, most importantly, a systematic framework for future numerical or variational studies aimed at a definitive spectral analysis.

💡 Research Summary

The paper addresses a long‑standing issue in electroweak monopole physics: the Cho‑Maison (CM) monopole is a genuine solution of the Standard Model field equations but possesses an infinite classical energy because the hypercharge (U(1)Y) sector is governed by a Maxwell kinetic term, which yields a 1/r² magnetic field and a divergent r⁻⁴ energy density near the origin. The authors propose to regularise this divergence by replacing the Maxwell term with a Born‑Infeld (BI) non‑linear electrodynamics term, a modification that naturally arises in string‑inspired effective field theories. The BI Lagrangian introduces a new scale β; phenomenological constraints require β to be large enough that low‑energy electroweak processes remain essentially Maxwellian, yet small enough to soften the hypercharge field at short distances and render the monopole energy finite.

The central goal of the work is not to re‑derive the finite‑energy monopole solution (which has been established in earlier papers) but to investigate whether the BI regularisation preserves the linear stability properties of the original CM monopole. To this end the authors adopt the perturbative framework developed by Gervalle and Volkov (GV) for the Maxwell case. The GV method employs a complex tetrad decomposition of fluctuations, assigns spin‑weights, and expands all angular dependence in spin‑weighted spherical harmonics Y_{jm}^{s}(θ,φ). This separates the problem into independent angular‑momentum channels labelled by the total angular momentum j and parity.

In the Maxwell theory the hypercharge fluctuation equation is simply ∂μ δB^{μν}=J^ν_Y. In the BI‑deformed theory this is replaced by ∂μ δG^{μν}=J^ν_Y with G^{μν}=−2∂L{BI}/∂B{μν}. Linearising around the static, purely magnetic monopole background yields δG^{μν}=−\bar L_X(r) δB^{μν}−½ \bar L_{XX}(r) \bar B_{ρσ}δB^{ρσ} \bar B^{μν}, where the over‑bars denote background quantities and the functions \bar L_X(r)=∂L_{BI}/∂X|{background}, \bar L{XX}(r)=∂²L_{BI}/∂X²|{background} depend only on the radial coordinate through the background hypermagnetic field. In the Maxwell limit β→∞ one recovers \bar L_X→1 and \bar L{XX}→0, so the BI deformation is a smooth, positive‑definite modification of the kinetic term.

After fixing the background‑covariant gauge and eliminating non‑dynamical constraint variables (exactly as in GV), each j‑channel reduces to a set of coupled radial equations that can be written in Sturm‑Liouville form: − d/dr