Sharp distinction between Higgs and confining phases with fundamental matter
In plain words
With quarks present, the usual test for confinement fails, and the confining phase and the Higgs phase (where a condensed field gives the force carriers mass) seem smoothly connected. Whether some generalized symmetry or order parameter still separates them is unknown.
Precise statement
With dynamical matter in the fundamental representation, the $Z_N$ one-form center symmetry is explicitly broken and the Wilson-loop area law is no longer an order parameter; Fradkin and Shenker (1979) showed the phases can be connected. Determine whether, in theories with additional global symmetries (for example a $U(1)$ broken in both regimes, as in dense QCD), a vortex-holonomy or other non-local order parameter sharply separates Higgs and confining phases, and whether this holds generally.
What would settle it
A proof that the proposed order parameter is quantized and differs between the phases, confirmed by lattice simulation of a model with a phase transition along every path.
Status in the literature
Unverified note
Cherman, Jacobson, Sen and Yaffe (2020-2024) proposed a vortex-holonomy order parameter; its validity is disputed, and lattice studies (2023-2025) report gauge-independent transitions in SU(2) gauge-scalar models.