Sigma Decomposition

TL;DR

This paper constructs a comprehensive effective chiral Lagrangian for composite Higgs models, analyzing different symmetry-breaking frameworks and connecting high-energy theories to low-energy Standard Model descriptions.

Contribution

It develops a general framework for the effective chiral Lagrangian in composite Higgs models, including custodial symmetry considerations and specific model cases.

Findings

Derived the operator basis for symmetric cosets up to four momenta.

Connected high-energy effective theories to the low-energy Standard Model Higgs sector.

Uncovered relations between operator coefficients across different models.

Abstract

In composite Higgs models the Higgs is a pseudo-Goldstone boson of a high-energy strong dynamics. We have constructed the effective chiral Lagrangian for a generic symmetric coset, restricting to CP-even bosonic operators up to four momenta which turn out to depend on seven parameters, aside from kinetic terms. Once the same sources of custodial symmetry breaking as in the Standard Model are considered, the total number of operators in the basis increases up to ten, again aside from kinetic terms. Under these assumptions, we have then particularised the discussion to three distinct frameworks: the original $SU(5)/SO(5)$ Georgi-Kaplan model, the minimal custodial-preserving $SO(5)/SO(4)$ model and the minimal $SU(3)/(SU(2)\times U(1))$ model, which intrinsically breaks custodial symmetry. The projection of the high-energy electroweak effective theory into the bosonic sector of the Standard Model is shown to match the low-energy chiral effective Lagrangian for a dynamical Higgs, and it uncovers strong relations between the operator coefficients. Finally, the relation with the bosonic basis of operators describing linear realisations of electroweak symmetry breaking is clarified.