Classification of NLO operators for composite Higgs models

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 SU(N-F)/Sp(N-F) and SU(N-F)/SO(N-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 SU(4)/Sp(4) congruent to SO(6)/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., showing that it cannot develop a vacuum expectation value without explicit CP violation in the underlying theory.