Sigma Decomposition: The CP-Odd Lagrangian

TL;DR

This paper extends the effective chiral Lagrangian framework to include CP-odd couplings in composite Higgs models, identifying independent operators and analyzing their implications across various models and the low-energy Higgs sector.

Contribution

It introduces a systematic construction of CP-odd operators in the chiral Lagrangian for composite Higgs models, including their relations and differences with elementary Higgs scenarios.

Findings

Identifies up to six independent CP-odd operators in the effective Lagrangian.

Analyzes three models: $SU(5)/SO(5)$, $SO(5)/SO(4)$, and $SU(3)/(SU(2)\times U(1))$.

Establishes relations between high-energy operator coefficients and low-energy phenomenology.

Abstract

In Alonso et al., JHEP 12 (2014) 034, the CP-even sector of the effective chiral Lagrangian for a generic composite Higgs model with a symmetric coset has been constructed, up to four momenta. In this paper, the CP-odd couplings are studied within the same context. If only the Standard Model bosonic sources of custodial symmetry breaking are considered, then at most six independent operators form a basis. One of them is the weak-$\theta$ term linked to non-perturbative sources of CP viola- tion, while the others describe CP-odd perturbative couplings between the Standard Model gauge bosons and an Higgs-like scalar belonging to the Goldstone boson sector. The procedure is then applied to three distinct exemplifying 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 cus- todial symmetry. Moreover, the projection of the high-energy electroweak effective theory to the low-energy chiral effective Lagrangian for a dynamical Higgs is per- formed, uncovering strong relations between the operator coefficients and pinpointing the differences with the elementary Higgs scenario.