# Classification of NLO operators for composite Higgs models

Alanne, Tommi (Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany) ; Bizot, Nicolas (Université de Lyon, Université Lyon 1, CNRS/IN2P3, UMR5822 IPNL, F-69622 Villeurbanne, France) ; Cacciapaglia, Giacomo (Université de Lyon, Université Lyon 1, CNRS/IN2P3, UMR5822 IPNL, F-69622 Villeurbanne, France) ; Sannino, Francesco (CP3-Origins and Danish IAS, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark)

24 April 2018

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 $\mathrm{SU}\left({N}_{F}\right)/\mathrm{Sp}\left({N}_{F}\right)$ and $\mathrm{SU}\left({N}_{F}\right)/\mathrm{SO}\left({N}_{F}\right)$. 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 $\mathrm{SU}\left(4\right)/\mathrm{Sp}\left(4\right)\cong \mathrm{SO}\left(6\right)/\mathrm{SO}\left(5\right)$ 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.

Published in: Physical Review D 97 (2018)