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 $\mathrm{SU}(N_{\mathrm{F}})/\mathrm{Sp}(N_{\mathrm{F}})$ 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 à la partial compositeness. Our templates provide a basis for lattice calculations in specific models. As a special example, we consider the $\mathrm{SU}(4)/\mathrm{Sp}(4)\cong \mathrm{SO}(6)/\mathrm{SO}(5)$ 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 $\eta$, showing that it cannot develop a vacuum expectation value without explicit $CP$ violation in the underlying theory.

Figure 1

Right panel: trilinear Higgs coupling normalized to the SM value as a function of $m_{\eta }$ and $s_{\theta }$ (for $c_4=0$). Left panel: contours of the di-Higgs cross section via gluon fusion at the LHC@14 TeV, normalized to the SM one. The continuous lines correspond to bilinear top couplings and linear couplings in the fundamental representation, and the dotted lines correspond to linear couplings in two-index representations.

Figure 2

Right panel: Higgs coupling to $\eta$ normalized to the SM Higgs trilinear coupling as a function of $m_{\eta }$ and $s_{\theta }$ (for $c_4=0$). Left panel: contours of the branching ratio $\mathrm{BR}(h\to \eta \eta )$ for singlet mass below threshold. The continuous lines correspond to bilinear top couplings and linear couplings in the fundamental representation, and the dotted lines correspond to linear couplings in two-index representations.