[Paper Review] Forests, Groves and Higgs Bosons: Gauge Invariance Classes in Spontaneously Broken Gauge Theories

Thorsten Ohl, Christian Schwinn|ArXiv.org|May 30, 2003

Particle physics theoretical and experimental studies12 references3 citations

TL;DR

This paper establishes a diagrammatic proof of gauge invariance classes (GICs) in spontaneously broken gauge theories, demonstrating that Higgs bosons lead to new, nontrivial GICs in nonlinearly realized scalar sectors. It shows that in unitarity gauge, these classes satisfy Ward identities individually, and unlike linear realizations, nonlinear models generate additional GICs—such as separating Higgsstrahlung and vector boson fusion diagrams—providing a gauge-invariant decomposition for multi-particle amplitudes in Higgs and top quark physics at colliders.

ABSTRACT

We determine the gauge invariance classes of tree level Feynman diagrams in spontaneously broken gauge theories, providing a proof for the formalism of gauge and flavor flips. We find new gauge invariance classes in theories with a nonlinearly realized scalar sector. In unitarity gauge, the same gauge invariance classes correspond to a decomposition of the scattering amplitude into pieces that satisfy the relevant Ward Identities individually. In theories with a linear realized scalar sector in R_ξgauge, no additional nontrivial gauge invariance classes exist compared to the unbroken case.

Motivation & Objective

• To systematically classify gauge invariance classes (GICs) in tree-level Feynman diagrams of spontaneously broken gauge theories (SBGTs), especially involving Higgs bosons.
• To resolve the long-standing challenge of identifying gauge-invariant subsets of diagrams for high-multiplicity final states in electroweak processes.
• To prove the formalism of 'gauge and flavor flips' from Slavnov-Taylor identities, extending it to SBGTs with both linear and nonlinear scalar sector realizations.
• To clarify why Higgs bosons lead to new, nontrivial GICs in nonlinearly realized models, while only extending existing classes in linearly realized ones.

Proposed method

• A diagrammatic analysis of Slavnov-Taylor identities (STIs) is used to derive the structure of GICs in SBGTs, ensuring gauge invariance at the level of individual diagrams.
• The formalism of 'gauge and flavor flips' is extended to SBGTs, with a rigorous derivation of correct flip rules for both linear and nonlinear scalar sector realizations.
• The analysis distinguishes between unitarity gauge and $R_\xi$ gauge, showing that GICs in unitarity gauge satisfy Ward identities individually.
• The method classifies GICs via topological decomposition of diagrams, identifying minimal subsets that are gauge-invariant and independent of gauge-fixing parameters.
• Explicit construction of GICs is performed for processes like $e^+e^- \to t\bar{t}H$ and eight-fermion final states, using graphical notation for STIs.
• The paper proves that in nonlinearly realized models, Higgsstrahlung and vector boson fusion diagrams belong to separate GICs, while in linear models, they are grouped into larger GICs with gauge boson exchange diagrams.

Experimental results

Research questions

• RQ1 Do Higgs bosons introduce new, nontrivial gauge invariance classes (GICs) in spontaneously broken gauge theories with nonlinearly realized scalar sectors?
• RQ2 How do the GICs in unitarity gauge relate to the satisfaction of Ward identities at the individual diagram level?
• RQ3 Why do nonlinearly realized scalar sectors generate additional GICs not present in linearly realized models?
• RQ4 Can the formalism of gauge and flavor flips be rigorously extended to SBGTs using Slavnov-Taylor identities?
• RQ5 Are Higgsstrahlung and vector boson fusion diagrams part of the same GIC in both linear and nonlinear realizations of the Higgs sector?

Key findings

• In nonlinearly realized scalar sectors, Higgsstrahlung and vector boson fusion diagrams belong to distinct, individually gauge-invariant GICs, even when the Higgs is massless.
• In unitarity gauge, each GIC satisfies the relevant Ward identities independently, ensuring gauge invariance at the diagram level.
• For linearly realized scalar sectors in $R_\xi$ gauge, no new nontrivial GICs emerge beyond those in the unbroken theory.
• The formalism of gauge and flavor flips is rigorously justified via diagrammatic analysis of Slavnov-Taylor identities, proving its validity in SBGTs.
• The Higgsstrahlung and vector boson fusion diagrams are not gauge-invariant by themselves in the linear realization, requiring inclusion of additional diagrams (e.g., with gauge boson exchanges) to form a GIC.
• The study provides a complete classification of GICs for four-fermion and eight-fermion processes, enabling gauge-invariant decomposition of amplitudes for signal-background separation in high-multiplicity final states.