A Systematic Approach to the SILH Lagrangian

TL;DR

This paper systematically derives the SILH Lagrangian from the electroweak chiral Lagrangian, clarifying operator basis, power counting, and higher-order effects, with applications to composite Higgs models.

Contribution

It provides a complete and systematic derivation of the SILH Lagrangian from the electroweak chiral Lagrangian, including higher-order terms and conceptual clarifications.

Findings

The SILH Lagrangian is a limiting case of the electroweak chiral Lagrangian.

Terms of order ξ^2 can be larger than those of order ξ/16π^2 for ξ > 1/16π^2.

The minimal composite Higgs model is embedded within the general framework.

Abstract

We consider the electroweak chiral Lagrangian, including a light scalar boson, in the limit of small $\xi=v^2/f^2$. Here $v$ is the electroweak scale and $f$ is the corresponding scale of the new strong dynamics. We show how the conventional SILH Lagrangian, defined as the effective theory of a strongly-interacting light Higgs (SILH) to first order in $\xi$, can be obtained as a limiting case of the complete electroweak chiral Lagrangian. The approach presented here ensures the completeness of the operator basis at the considered order, it clarifies the systematics of the effective Lagrangian, guarantees a consistent and unambiguous power counting, and it shows how the generalization of the effective field theory to higher orders in $\xi$ has to be performed. We point out that terms of order $\xi^2$, which are usually not included in the SILH Lagrangian, are parametrically larger than terms of order $\xi/16\pi^2$ that are retained, as long as $\xi > 1/16\pi^2$. Conceptual issues such as custodial symmetry and its breaking are also discussed. For illustration, the minimal composite Higgs model based on the coset $SO(5)/SO(4)$ is considered at next-to-leading order in the chiral expansion. It is shown how the effective Lagrangian for this model is contained as a special case in the electroweak chiral Lagrangian based on $SU(2)_L\otimes SU(2)_R/SU(2)_V$.