Further reduction of the complexity of the 2HDM stability condition

Determine whether the quantifier complexity of the stability condition for the two Higgs doublet model (2HDM) potential can be reduced below the authors’ best-known formulation: there exists a real number c ≥ 0 such that for all vectors k ∈ R^3 with ||k||^2 ≤ 1, 0 ≤ J4(k) and (J2(k) < 0 implies J2(k)^2 ≤ 4 c J4(k)), where J2(k) = ξ0 + Σ_{a=1}^3 ξ_a k_a and J4(k) = η00 + 2 Σ_{a=1}^3 η0a k_a + Σ_{a,b=1}^3 ηab k_a k_b, with ξ and η determined by the standard reparameterization of the 2HDM potential parameters.

Background

The paper studies the logical complexity (in terms of quantifier structure) of the stability condition for the two Higgs doublet model (2HDM) potential. The authors translate the potential into a form involving a Gram vector K and then into functions J2 and J4 of a normalized 3-vector k.

They disprove a widely cited claim (Condition C from Maniatis et al. 2006) that would have yielded a purely universal stability criterion, and instead provide a formally verified best-known reduction: an existential quantifier over a constant c followed by a universal quantifier over k with ||k|| ≤ 1 and constraints involving J2 and J4.

After establishing this reduction, the authors explicitly state that whether the complexity can be reduced further remains unknown, framing it as an open question about achieving an even simpler (e.g., fewer quantifiers) equivalent formulation.