[Paper Review] Supersymmetry, Supercurrent and Scale Invariance

Olivier Piguet|ArXiv.org|Nov 3, 1996

Earth Systems and Cosmic Evolution6 references9 citations

TL;DR

This paper provides a comprehensive theoretical framework for understanding supersymmetry, supercurrents, and scale invariance in quantum field theories using superspace techniques. It derives the supercurrent current algebra, establishes its consistency with scale invariance, and demonstrates how anomalies in scale symmetry are linked to the supercurrent's trace, offering a systematic approach to finite and renormalizable supersymmetric theories in four dimensions.

ABSTRACT

Contents: Generalities, Chiral supermultiplets, Super Yang-Mills theory, Superspace Feynman graphs, Renormalization, Supercurrent, Finite theories.

Motivation & Objective

• To develop a consistent formulation of supercurrents in supersymmetric field theories using superspace techniques.
• To analyze the interplay between supersymmetry, scale invariance, and anomalies in chiral and super Yang-Mills theories.
• To clarify the role of the supercurrent in the trace anomaly and its implications for finite and renormalizable theories.
• To provide a pedagogical and rigorous treatment of supersymmetric quantum field theories with applications to renormalization and Feynman graph calculations.
• To establish the conditions under which scale invariance is preserved or anomalous in supersymmetric models, particularly in the context of supercurrent conservation.

Proposed method

• Utilizes the superspace formalism to construct chiral and vector supermultiplets and their dynamics.
• Applies the supercurrent formalism to derive the current algebra and its conservation laws in supersymmetric theories.
• Analyzes the trace of the energy-momentum tensor to identify anomalies in scale invariance, linking them to the supercurrent.
• Employs Feynman graph techniques in superspace to study renormalization and quantum corrections.
• Derives the consistency conditions for scale and supersymmetry anomalies using cohomological methods in the context of supercurrent anomalies.
• Examines finite theories by analyzing the absence of divergences and the structure of the supercurrent in such models.

Experimental results

Research questions

• RQ1 How is the supercurrent defined in supersymmetric field theories, and what are its transformation properties?
• RQ2 What is the role of the supercurrent in the anomaly structure of scale invariance?
• RQ3 How do supersymmetry and scale invariance constrain the renormalization of chiral and super Yang-Mills theories?
• RQ4 Under what conditions is a supersymmetric theory finite, and how is this reflected in the supercurrent and trace anomaly?
• RQ5 What is the relationship between the supercurrent and the trace anomaly in four-dimensional supersymmetric quantum field theories?

Key findings

• The supercurrent forms a conserved current in supersymmetric theories, and its divergence is determined by the supercurrent algebra and the superpotential.
• The trace of the energy-momentum tensor is proportional to the supercurrent's trace, which vanishes only if scale invariance is unbroken.
• Anomalies in scale invariance are directly related to the non-conservation of the supercurrent's trace, particularly in chiral theories.
• Finite supersymmetric theories are characterized by the absence of divergences in the supercurrent and trace anomaly, implying a specific structure in the superpotential and gauge kinetic functions.
• Superspace Feynman rules lead to a consistent renormalization procedure that preserves both supersymmetry and scale symmetry at the quantum level.
• The supercurrent formalism provides a unified framework to analyze both anomalies and renormalization in four-dimensional supersymmetric field theories.