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 flavour patterns of spontaneous symmetry breaking SU($N_F$)/Sp($N_F$) and SU($N_F$)/SO($N_F$). All possible explicit-breaking sources parametrised by spurions transforming in the fundamental and in the two-index representations of the flavour symmetry are included. While our general framework can be applied to any model of strong dynamics, we specialise to composite-Higgs models, where the main explicit breaking sources are a current mass, the gauging of flavour 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)$\cong$ 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 $\eta$, showing that it cannot develop a vacuum expectation value without explicit CP violation in the underlying theory.