Gauge-covariant Hadamard descent in Yang-Mills theory

Abstract

We study the dimensional reduction of non-Abelian Yang-Mills theory from five to four dimensions. In this framework, Hadamard descent is formulated as a restriction to gauge-equivalence classes that are invariant, up to gauge transformations, under translations along the reduced direction. The central result is that, prior to gauge fixing, the adjoint scalar of the reduced theory is not the connection component $A_u$ itself, but the gauge-covariant combination $\widetildeΦ=ε-ι_YA$, which follows naturally from the covariant Cartan identity. The usual identification with $A_u$ is recovered, up to the sign convention adopted here, in the $u$-independent gauge, while global holonomy data remain additional information to be specified separately. We show that the reduced theory takes the form of a Yang--Mills--Higgs theory with the Higgs field in the adjoint representation. As an example, we focus on the $U(2)$ gauge theory and analyze the Higgs mechanism, together with the associated symmetry breaking pattern, when the reduced theory is expanded around the ground state of the parent theory. We then include fermions and show that, in the $u$-independent gauge, the extra-dimensional component of the gauge connection generates a pseudoscalar Yukawa coupling which, on the vacuum configuration, can be converted into a fermion mass term by a chiral field redefinition, while the gauge interaction remains vectorial.