Emergent symmetry in a two-Higgs-doublet model from quantum information and nonstabilizerness

Studies of scattering processes in scalar models with two Higgs doublets have recently hinted at a connection between the absence of flavor-space entanglement in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi mathvariant="normal">Φ</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi mathvariant="normal">Φ</a:mi><a:mn>0</a:mn></a:msup></a:math> scattering and an emergent SO(8) symmetry in the scalar potential. We extend the analysis to all scattering channels with two particles in the external states by treating the process as a four-qubit system in the weak isospin and flavor subspaces of the two-particle state. We work with a generic quantum-information-theoretic principle encoded by the commutativity of the initial-state density matrix with the transition matrix (at leading order in perturbation theory). This yields a special case of the entanglement minimization conditions previously derived in the literature, and we interpret the principle in terms of the conservation of nonstabilizerness (or “magic”). Working at leading order in the quartic couplings, we find a consistent set of conditions that implies an SO(8) symmetry on the quartic part of the potential for scattering an arbitrary initial state but a smaller <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>SU</e:mi><e:mo stretchy="false">(</e:mo><e:mn>2</e:mn><e:msub><e:mo stretchy="false">)</e:mo><e:mi>R</e:mi></e:msub></e:math> symmetry when the initial state is chosen to have definite isospin. This follows by accounting for Bose symmetry in the initial state, which introduces entanglement between the isospin and flavor subspaces.