[Paper Review] BUSSTEPP Lectures on Supersymmetry

José Figueroa-O’Farrill|ArXiv.org|Sep 21, 2001

Particle physics theoretical and experimental studies26 citations

TL;DR

This paper provides a comprehensive, pedagogical introduction to four-dimensional rigid N=1 supersymmetry, focusing on the Wess–Zumino model and supersymmetric Yang–Mills theory. It systematically develops the formalism using the Poincaré superalgebra, superspace, and superfields, demonstrating on-shell closure of the supersymmetry algebra and constructing gauge-invariant actions via chiral and vector superfields, with key results including the derivation of the superpotential action and analysis of spontaneous supersymmetry breaking via the Witten index and O’Raifeartaigh model.

ABSTRACT

This is the written version of the supersymmetry lectures delivered at the 30th and 31st British Universities Summer Schools in Theoretical Elementary Particle Physics (BUSSTEPP) held in Oxford in September 2000 and in Manchester in August-September 2001.

Motivation & Objective

• To provide a self-contained, graduate-level introduction to N=1 supersymmetry in four dimensions for students with foundational knowledge in relativistic field theory.
• To establish the invariance of the Wess–Zumino and supersymmetric Yang–Mills theories under the N=1 Poincaré superalgebra, proving on-shell closure of the algebra.
• To develop the superspace and superfield formalism as a systematic framework for constructing supersymmetric actions and understanding multiplet structures.
• To analyze spontaneous supersymmetry breaking, including the role of vacuum energy, auxiliary field VEVs, and the Witten index, using models like O’Raifeartaigh and Fayet–Iliopoulos terms.

Proposed method

• Constructs the Wess–Zumino model as the simplest supersymmetric theory, proving its invariance under supersymmetry transformations and closure of the superalgebra on-shell.
• Applies the two-component spinor formalism to derive the supersymmetry algebra, including the anticommutation relations between supercharges and the momentum generators.
• Introduces superspace with Grassmann coordinates (θ, θ̄) and defines chiral and vector superfields, using differential operators D̄α, Dα, and Qα to project physical components.
• Derives the gauge-invariant action for supersymmetric Yang–Mills theory using the vector superfield V and the field strength superfield Wα, with the action written as a superspace integral.
• Utilizes the superpotential W(Φ) as a holomorphic function of chiral superfields to construct renormalizable interactions, with the full Lagrangian given by integrals over d²θ d²θ̄ and d²θ.
• Analyzes spontaneous supersymmetry breaking via the vacuum expectation value of auxiliary fields, the Witten index, and the O’Raifeartaigh model, showing that non-degenerate ground states signal breaking.

Experimental results

Research questions

• RQ1 How does the Wess–Zumino model realize linearly realized N=1 supersymmetry, and what is the structure of its on-shell supersymmetry algebra closure?
• RQ2 How can the supersymmetric Yang–Mills Lagrangian be constructed using vector superfields and gauge-invariant actions in superspace?
• RQ3 What is the role of the superpotential in determining the renormalizable interactions of chiral superfields in four-dimensional supersymmetric theories?
• RQ4 How does spontaneous supersymmetry breaking manifest through vacuum energy, auxiliary field VEVs, and the Witten index?
• RQ5 What are the implications of central charges and the BPS bound in unitary representations of the Poincaré superalgebra?

Key findings

• The Wess–Zumino model is invariant under the N=1 Poincaré superalgebra, with the algebra closing on-shell up to equations of motion and gauge transformations.
• The supersymmetric Yang–Mills theory action is constructed via superspace integrals over d²θ d²θ̄ and d²θ, with the field strength superfield Wα ensuring gauge invariance and supersymmetry.
• The most general renormalizable supersymmetric Lagrangian in four dimensions consists of a Kähler sigma model coupled to a holomorphic superpotential W(Φ), with terms up to cubic order in chiral superfields.
• Spontaneous supersymmetry breaking is signaled by a non-zero vacuum energy and non-vanishing VEVs of auxiliary fields, as seen in the O’Raifeartaigh model and Fayet–Iliopoulos terms.
• The Witten index is non-zero in unbroken supersymmetry and vanishes when supersymmetry is spontaneously broken, providing a topological criterion for the phase of the theory.
• In the WZ gauge, the vector superfield V takes the form V = θ̄σμθ vμ + θ²θ̄λ + θθ̄²λ̄ + θ²θ̄²D, with D being the auxiliary field responsible for breaking supersymmetry when ⟨D⟩ ≠ 0.