Classification of NLO operators for composite Higgs models

Abstract

We provide a general classification of template operators, up to next-to-leading order, that appear in chiral perturbation theories based on the two flavor patterns of spontaneous symmetry breaking $\mathrm{SU}({N}_{\mathrm{F}})/\mathrm{Sp}({N}_{\mathrm{F}})$ and $\mathrm{SU}({N}_{\mathrm{F}})/\mathrm{SO}({N}_{\mathrm{F}})$. All possible explicit-breaking sources parametrized by spurions transforming in the fundamental and in the two-index representations of the flavor symmetry are included. While our general framework can be applied to any model of strong dynamics, we specialize to composite-Higgs models, where the main explicit breaking sources are a current mass, the gauging of flavor symmetries, and the Yukawa couplings (for the top). For the top, we consider both bilinear couplings and linear ones \`a la partial compositeness. Our templates provide a basis for lattice calculations in specific models. As a special example, we consider the $\mathrm{SU}(4)/\mathrm{Sp}(4)\ensuremath{\cong}\mathrm{SO}(6)/\mathrm{SO}(5)$ pattern which corresponds to the minimal fundamental composite-Higgs model. We further revisit issues related to the misalignment of the vacuum. In particular, we shed light on the physical properties of the singlet $\ensuremath{\eta}$, showing that it cannot develop a vacuum expectation value without explicit $CP$ violation in the underlying theory.