[Paper Review] Implications of sigma models in the standard model and beyond

C. S. Lim|ArXiv.org|Oct 11, 2000

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper demonstrates that sigma models provide a powerful effective field theory framework for analyzing low-energy weak processes and New Physics effects in the Standard Model, particularly through chiral symmetry and gauge-invariant operators. It shows how the sigma model description reveals non-decoupling New Physics contributions to gauge boson self-interactions beyond the standard oblique parameters, with key insights into the $\Delta I=1/2$ rule and $S$-, $T$-, $U$-parameters via higher-dimensional operators.

ABSTRACT

After a brief introduction to the sigma model in QCD, we discuss how the sigma model can be relevant in the Standard Model. It is shown to be useful in the analysis of weak processes, such as the study of $Δ{I} = 1/2$ rule, and in the analysis of the Higgs sector as well. The sigma model description is also shown to be quite useful in searching for the effects of New Physics via gauge boson self-interactions.

Motivation & Objective

• To understand the relevance of sigma models in describing low-energy hadronic weak decays within the Standard Model.
• To analyze the $\Delta I=1/2$ rule in kaon decays using sigma model techniques and effective field theory.
• To extend the sigma model framework to describe New Physics effects in gauge boson self-interactions, including oblique corrections and triple gauge vertices.
• To distinguish between decoupling and non-decoupling New Physics scenarios using effective operators in the non-linear sigma model formalism.

Proposed method

• Utilize the linear and non-linear sigma models to describe chiral symmetry breaking in QCD and its extension to the Standard Model.
• Apply the Wilsonian renormalization group approach to integrate out heavy particles (e.g., top quark, $W$ bosons) and derive effective operators at low energies.
• Construct a gauged sigma model with the Higgs doublet represented via a non-linear realization $\phi \to U \cdot v$, preserving gauge invariance and simplifying higher-dimensional operator analysis.
• Derive effective Lagrangians involving $d=6$ and higher-dimensional gauge-invariant operators, such as $\text{Tr}\{(U^\dagger iD^\mu U)\sigma_3 (U^\dagger iD_\mu U)\sigma_3\}$, to describe $T$-parameter and other oblique corrections.
• Analyze the $S$-parameter via the operator $c_s \phi^\dagger (W_{\mu\nu}^a \sigma^a)\phi \cdot B^{\mu\nu}$, showing its relation to Higgs vacuum expectation value $v$.
• Distinguish decoupling (suppressed by $v^2/M^2$) and non-decoupling (no suppression) New Physics by examining contributions from higher-dimensional operators.

Experimental results

Research questions

• RQ1 How can the sigma model framework be used to describe the $\Delta I=1/2$ rule in kaon decays within the Standard Model?
• RQ2 What is the role of the sigma model in encoding non-perturbative QCD effects in low-energy weak processes involving light quarks?
• RQ3 How do New Physics contributions to gauge boson self-interactions manifest in the sigma model formalism beyond the standard $S$, $T$, $U$ parameters?
• RQ4 Why do non-decoupling New Physics scenarios, such as fourth-generation quarks, lead to unsuppressed contributions in higher-dimensional operators?
• RQ5 Can the non-linear sigma model formalism simplify the analysis of higher-dimensional gauge-invariant operators in the context of New Physics?

Key findings

• The $\Delta I=1/2$ rule in kaon decays is effectively described by replacing quark currents with hadronic currents in the sigma model, preserving chiral symmetry.
• The $S$-parameter arises from a $d=6$ gauge-invariant operator $c_s \phi^\dagger (W_{\mu\nu}^a \sigma^a)\phi \cdot B^{\mu\nu}$, with $S \sim c_s v^2$, linking it directly to the Higgs vacuum expectation value.
• New Physics contributions to triple gauge boson vertices (TGV) are described by 4 independent parameters, distinct from $S$, $T$, $U$, and arise from the non-Abelian structure of $W_{\mu\nu}^a$.
• In the non-decoupling case (e.g., 4th generation quarks), higher-dimensional operators like $\phi^\dagger (W_{\mu\nu}^a \sigma^a)\phi \cdot B^{\mu\nu} \cdot (\phi^\dagger \phi)^n$ contribute significantly, without suppression by $v^2/M^2$.
• The non-linear sigma model representation $\phi \to U \cdot v$ simplifies the analysis of higher-dimensional operators and avoids complications from explicit Higgs field components.
• All New Physics effects on gauge boson self-interactions are fully captured by 2-, 3-, and 4-point functions, with no further contributions beyond these, as shown via Wilsonian effective action.