The quantitative behavior of $α$-Yang-Mills-Higgs fields on surfaces

We investigate the blow-up behavior of $α$-Yang--Mills--Higgs ($α$-YMH) fields over closed Riemannian surfaces with the target fiber F = SK-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 L2,∞ to the optimal L2,1 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.