Abstract

We study both the CP-even and CP-odd effective chiral Lagrangians of next minimal composite Higgs model {with symmetry breaking pattern depicted by the coset $SO(6)/SO(5)$} through the sigma/omega decomposition, in which the Goldstone matrix of the coset is decomposed in terms of the standard model Higgs doublet and an additional scalar singlet $s$ at the electroweak scale. The effective Lagrangian is described by the electroweak chiral Lagrangian up to $p^4$ order, with a function dependence on the Higgs boson and new scalar $s$, named Higgs function. This function in the effective Lagrangian incorporates the Higgs non-linearity or vacuum misalignment effects in the next minimal composite Higgs model, due to the Riemann curvature on the Nambu-Goldstone bosons scalar manifold, leads to various Higgs couplings deviated from the standard model ones, and also indicates the relations among different Higgs couplings in the low energy. Matching to the Higgs effective field theory below the electroweak scale, we obtain various low energy observables such as the electroweak oblique parameters, anomalous triple and quartic gauge couplings, anomalous couplings of Higgs to gauge bosons, in which the Higgs non-linearity effects are encoded in the ratio $\xi\equiv v^2/f^2$ of the electroweak scale and the new physics scale.

Highlights

  • The standard model (SM) of elementary particle physics provides a framework for all of the visible matters in nature among the three known fundamental interactions except gravity [1]

  • In a series expansion of h=v in the Higgs effective field theory (HEFT) with the parameter ξ fixed, we find the effective Wilson coefficients associated with the high-energy effective chiral Lagrangian in the next-to-minimal composite Higgs model (NMCHM)

  • VI, we study the connection of effective field theory to the corresponding physical observables

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Summary

INTRODUCTION

The standard model (SM) of elementary particle physics provides a framework for all of the visible matters in nature among the three known fundamental interactions except gravity [1]. The composite Higgs model (CHM) provides a scenario [5,6,7], that might be related to a strong dynamics in the high energy, to solve the electroweak hierarchy problem. The Higgs functions provide the connection between the low-energy EW EFT and the chiral Lagrangian for the CHM. In a series expansion of h=v in the Higgs effective field theory (HEFT) with the parameter ξ fixed, we find the effective Wilson coefficients associated with the high-energy effective chiral Lagrangian in the NMCHM. Observables such as the EW oblique parameters, anomalous triple and quartic gauge couplings, and anomalous couplings of Higgs to gauge bosons are obtained.

NEXT-TO-MINIMAL COMPOSITE
From Omega to Sigma parametrization
Rotational symmetry
Left-right parity symmetry
Vacuum misalignment
Embedding with rotation θ
CP-even sector
Nonlinear chiral Lagrangian in Ω parametrization
MATCHING TO EW CHIRAL
CP-even sector with the block components as
CP-odd sector
HIGGS FUNCTIONS
Higgs kinetic terms below EW scale hhi2
CP-odd case
Higgs functions in curved field space
B â bjφj2 f qffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi þ 2
CONNECTION TO PHYSICAL
Higgs kinetics and self-interactions
Low-energy EWPT oblique parameters
HVV anomalous couplings ð2Þ
HVVV anomalous couplings
CONCLUSIONS
Generators of the SOð5Þ group i B
Generators of the SOð4Þ group
Ω decomposition
Nonlinear chiral Lagrangian ðB98Þ
High-energy effective operators in literature
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