[Paper Review] Infrared Behaviour of Landau Gauge Yang-Mills Theory with a Fundamentally Charged Scalar Field

Leonard Fister|arXiv (Cornell University)|Feb 8, 2010

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This thesis investigates the infrared behavior of Landau gauge Yang-Mills theory coupled to a fundamentally charged scalar field using Dyson-Schwinger equations (DSEs). By assuming power-law scaling solutions in the skeleton expansion, it demonstrates that scalar Yang-Mills theory exhibits the same infrared scaling exponents as quenched QCD, with kinematic singularities playing a critical role in massive scalar cases, confirming the model's viability for studying confinement mechanisms without quark dynamics.

ABSTRACT

The infrared behaviour of the n-point functions of a Yang-Mills theory with a charged scalar field in the fundamental representation of SU(N) is studied in the formalism of Dyson-Schwinger equations. Assuming a stable skeleton expansion solutions in form of power laws for the Green functions are obtained. For a massless scalar field the uniform limit is sufficient to describe the infrared scaling behaviour of vertices. Not taking into account a possible Higgs-phase it turns out that kinematic singularities play an important role for the scaling solutions of massive scalars. On a qualitative level scalar Yang-Mills theory yields similar scaling solutions as recently obtained for QCD.

Motivation & Objective

• To understand the non-perturbative infrared behavior of Yang-Mills theory coupled to a fundamental scalar field using non-perturbative field theory methods.
• To determine whether scalar-Yang-Mills theory exhibits the same infrared scaling behavior as quenched QCD, particularly in the context of confinement and dynamical symmetry breaking.
• To assess the role of kinematic singularities in the scaling solutions of massive scalar fields within the DSE framework.
• To validate the use of scalar-Yang-Mills theory as a simplified model for studying confinement mechanisms without the complexity of fermionic degrees of freedom.

Proposed method

• Employing the formalism of Dyson-Schwinger equations (DSEs) in Landau gauge to study the n-point functions of the scalar-Yang-Mills system.
• Assuming a stable skeleton expansion with power-law ansätze for Green's functions to analyze infrared scaling behavior.
• Analyzing the scalar-gluon vertex DSE in different kinematic limits (soft-gluon, soft-scalar, uniform limit) to identify dominant terms.
• Using constraints from linear terms in DSEs to prove that non-QCD-like diagrams (e.g., 'sheep' diagrams) are subleading in the infrared.
• Comparing the resulting infrared exponents for the scalar propagator and vertex functions with those obtained in quenched QCD.
• Applying the uniform limit to simplify the analysis and show equivalence to QCD scaling solutions in the massless scalar case.

Experimental results

Research questions

• RQ1 Does scalar-Yang-Mills theory in Landau gauge exhibit the same infrared scaling behavior as quenched QCD?
• RQ2 What is the role of kinematic singularities in the scaling solutions for massive scalar fields?
• RQ3 Are the additional diagrams present in the scalar-gluon DSE (absent in QCD) subleading in the infrared?
• RQ4 Can the scalar-Yang-Mills system serve as a viable simplified model for studying confinement and dynamical symmetry breaking?
• RQ5 How do the infrared exponents of the scalar propagator and vertex functions compare to those in QCD under the same assumptions?

Key findings

• The scalar-Yang-Mills system yields the same infrared scaling exponents for the scalar propagator and scalar-gluon vertex as quenched QCD, confirming qualitative similarity in non-perturbative behavior.
• For a massless scalar field, the uniform limit is sufficient to describe the full infrared scaling behavior, simplifying the analysis.
• In the massive scalar case, kinematic singularities play a crucial role in determining the scaling solution, particularly in the partial and full scaling regimes.
• Non-QCD-like diagrams (e.g., 'sheep' diagrams) in the scalar-gluon DSE are proven to be subleading in the infrared due to positivity constraints and structural analogies with QCD terms.
• The system of DSEs for scalar-Yang-Mills theory leads to the same solution structure as in quenched QCD, with identical infrared exponents for δ′_s, α^u_sg, δ^g_sg, and δ^s_sg.
• The analysis confirms that the scalar-gluon vertex scaling behavior is robust under the inclusion of bosonic self-interactions and kinematic effects, supporting the model's utility for confinement studies.