[Paper Review] Supersymmetry breaking on the lattice: the N=1 Wess-Zumino model

David Baumgartner, Kyle Steinhauer+1 more|arXiv (Cornell University)|Nov 25, 2011

Quantum Chromodynamics and Particle Interactions8 references10 citations

TL;DR

This paper presents a lattice simulation of the N=1 Wess-Zumino model in two dimensions using Wilson fermions and the fermion loop formulation to study spontaneous supersymmetry breaking. By circumventing the fermion sign problem via the open fermion string algorithm, the authors identify a phase transition between unbroken and spontaneously broken supersymmetry, and estimate a renormalized critical coupling in the continuum limit, providing a non-perturbative framework for studying supersymmetry breaking in lattice field theory.

ABSTRACT

We discuss spontaneous supersymmetry breaking in the N=1 Wess-Zumino model in two dimensions on the lattice using Wilson fermions and the fermion loop formulation. In that formulation the fermion sign problem related to the vanishing of the Witten index can be circumvented and the model can be simulated very efficiently using the recently introduced open fermion string algorithm. We present first results for the supersymmetry breaking phase transition and sketch the preliminary determination of a renormalised critical coupling in the continuum limit.

Motivation & Objective

• To investigate spontaneous supersymmetry breaking in the N=1 Wess-Zumino model using non-perturbative lattice methods.
• To overcome the fermion sign problem associated with a vanishing Witten index in supersymmetric theories.
• To determine the critical coupling for the supersymmetry breaking phase transition in the continuum limit.
• To explore the interplay between supersymmetry breaking and Z(2) chiral symmetry breaking in the model.
• To establish a non-perturbative framework for studying supersymmetry breaking using fermion loop formulations on the lattice.

Proposed method

• The model is discretized using Wilson fermions and formulated in terms of fermion loops to avoid the sign problem.
• The open fermion string algorithm is employed to efficiently simulate the fermion loop partition function.
• The Witten index is computed via the ratio of Pfaffians for periodic and antiperiodic fermion boundary conditions.
• The Z(2) chiral symmetry is used as a probe: its spontaneous breaking correlates with the choice of ground state (bosonic or fermionic).
• The critical coupling is extracted by scanning the bare mass parameter at fixed lattice spacing and varying system size.
• Renormalization is performed via the relation between bare mass m and renormalized mass m_R, with the continuum limit taken by extrapolating to vanishing lattice spacing.

Experimental results

Research questions

• RQ1 What is the nature of the phase transition between unbroken and spontaneously broken supersymmetry in the N=1 Wess-Zumino model on the lattice?
• RQ2 How does the Z(2) chiral symmetry breaking correlate with supersymmetry breaking in this model?
• RQ3 Can the fermion loop formulation with the open fermion string algorithm successfully simulate the model despite the vanishing Witten index?
• RQ4 What is the value of the renormalized critical coupling for supersymmetry breaking in the continuum limit?
• RQ5 How do the bosonic and fermionic ground states emerge in the different phases, and what is their relative weight?

Key findings

• For large m/g (e.g., m/g = 4), the Z(2) symmetry is spontaneously broken, and the system selects a unique ground state: either bosonic (⟨ϕ̄⟩ ≈ +2) or fermionic (⟨ϕ̄⟩ ≈ -2), with Z_pp ≈ ±Z_pa, indicating unbroken supersymmetry.
• For small m/g (e.g., m/g = 0.16), the Z(2) symmetry is unbroken (⟨ϕ̄⟩ ≈ 0), and the contributions from bosonic and fermionic ground states cancel in the Witten index, yielding Z_pp ≈ 0, indicating spontaneous supersymmetry breaking.
• The ratio Z_pp / Z and the sign of the average bosonic field ⟨s_ϕ⟩ serve as (pseudo-)order parameters for the supersymmetry and Z(2) symmetry breaking transitions, respectively.
• The critical mass m_c/g is determined at fixed lattice spacing ag = 0.03125, with evidence suggesting a second-order phase transition in the Z(2) order parameter.
• The continuum limit of the renormalized critical coupling f_crit = (g/m_R)_crit is extracted by extrapolating to zero lattice spacing, with the result expected to be compared to previous non-perturbative determinations.
• The fermion loop formulation allows for efficient simulation and direct access to the boson and fermion mass spectra, enabling further non-perturbative analysis.