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. Focussing on the bosonic electroweak theory with an unmodified $SU{\left(2\right)}_L$ sector and a Born-Infeld $U{\left(1\right)}_Y$ sector, we analyse 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 Schrödinger-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 nonlinear 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.