We investigate the blow-up behavior of α-Yang--Mills--Higgs (α-YMH) fields over closed Riemannian surfaces with the target fiber F = S^K-1 RK being the round sphere, focusing on the establishment of the α-energy identity and the no-neck property during the bubbling process. A central innovation is the identification of a hidden Jacobian structure through Hodge decomposition and a new conservation law. Furthermore, we derive a Pohozaev-type identity for α-YMH fields, which enables refined control of the energy density. Together, these advances ensure the validity of the α-energy identity as α 1. Our analysis further sharpens the Lorentz space estimates from the L^2, to the optimal L^2, 1 scale, ultimately yielding the no-neck property in the blow-up regime. These results provide a unified and quantitative framework for understanding singularity formation in variational gauge theories.
Ai et al. (2025) studied this question.