[Paper Review] Dynamical simulation of lattice 4d N=1 SYM

Kamel Demmouche, F. Farchioni+5 more|ArXiv.org|Nov 12, 2008

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This study presents large-scale lattice simulations of 4d N=1 supersymmetric Yang-Mills theory with SU(2) gauge group, using the Two-Step Multi-Boson and Two-Step Polynomial Hybrid Monte Carlo algorithms to simulate dynamical gluinos at small masses down to $am_{\tilde{g}} \approx 0.068$. The results indicate that finite-size effects strongly influence the low-energy spectrum, with mass degeneracy between $a-\eta'$ and $\chi$ states in small volumes, while larger volumes reveal a mass splitting inconsistent with soft SUSY breaking, suggesting the need for finer lattices to resolve the continuum SUSY limit.

ABSTRACT

The lattice provides a powerful tool to non-perturbatively investigate strongly coupled supersymmetric Yang-Mills (SYM) theories. The pure SU(2) SYM theory with one supercharge is simulated on large lattices with small Majorana gluino masses down to about $am_{ ilde g}=0.068$ with lattice spacing $a\simeq 0.125$ fm. The gluino dynamics is simulated by the Two-Step Multi-Boson (TSMB) and the Two-Step Polynomial Hybrid Monte Carlo (TS-PHMC) algorithms. Supersymmetry (SUSY) is broken explicitly by the lattice and the Wilson term and softly by the presence of a non-vanishing gluino mass. However, the recovery of SUSY is expected in the infinite volume continuum limit by tuning the bare parameters to the SUSY point in the parameter space. This scenario is studied by the determination of the low-energy mass spectrum and by means of lattice SUSY Ward-Identities (WIs).

Motivation & Objective

• To investigate the non-perturbative dynamics of 4d $\mathcal{N}=1$ SYM with SU(2) gauge group using lattice field theory.
• To study the recovery of supersymmetry (SUSY) and chiral symmetry in the infinite volume, continuum limit by tuning bare parameters to critical values.
• To determine the low-energy mass spectrum of bound states, including glueballs and gluino-bound states, and assess their supermultiplet structure.
• To test lattice SUSY Ward-Identities and OZI-based estimates for the critical hopping parameter $\kappa_{\rm cr}$ to locate the SUSY point.
• To assess the impact of finite volume and lattice spacing on the observed mass degeneracy and splitting in the spectrum.

Proposed method

• Simulations performed using the Wilson formulation for gluino and gauge fields, with improved tree-level Symanzik (tlSym) gauge action and stout-smeared gauge links to reduce eigenvalue fluctuations.
• Dynamical gluino updates implemented via Two-Step Multi-Boson (TSMB) and Two-Step Polynomial Hybrid Monte Carlo (TS-PHMC) algorithms, using polynomial approximations and noisy corrections for the fermion determinant.
• The static quark potential $V(r)$ was computed from APE-smeared Wilson loops to extract the string tension and set the physical scale via $r_0$.
• Lattice SUSY Ward-Identities and OZI-based estimates were used to determine the critical hopping parameter $\kappa_{\rm cr}$ for the chiral limit.
• Correlation functions of interpolating operators with definite quantum numbers were computed to extract masses of low-lying bound states, including $0^{+}$, $0^{-}$, and $1/2$ states.
• Finite-size effects were analyzed by comparing results on $16^3\cdot32$ and $24^3\cdot48$ lattices with different physical volumes.

Experimental results

Research questions

• RQ1 Does the low-energy spectrum of 4d $\mathcal{N}=1$ SYM on the lattice exhibit approximate degeneracy consistent with supermultiplet structure in the chiral limit?
• RQ2 To what extent do finite-volume effects distort the observed mass spectrum, particularly the degeneracy between $a-\eta'$ and $\chi$ states?
• RQ3 Can the critical hopping parameter $\kappa_{\rm cr}$ be consistently determined using both lattice SUSY Ward-Identities and OZI-based estimates?
• RQ4 Is the observed mass splitting between $a-\eta'$ and $\chi$ states in larger volumes consistent with the soft SUSY breaking induced by the gluino mass?
• RQ5 What is the physical scale of the simulations, and how does it affect the interpretation of the spectrum in the context of continuum SUSY?

Key findings

• The critical hopping parameter was estimated as $\kappa_{\rm cr} \approx 0.1969$ for TSMB runs and $\kappa_{\rm cr} \approx 0.2033$ for TS-PHMC runs, with good agreement between SUSY Ward-Identity and OZI-based methods.
• For TSMB runs on a $(1\,\text{fm})^3$ volume, the $a-\eta'$ and $\chi$ masses show near-degeneracy, suggesting possible supermultiplet structure in small volumes.
• In larger TS-PHMC volumes of $(2-3\,\text{fm})^3$, a significant mass splitting between $a-\eta'$ and $\chi$ states was observed, exceeding the expected splitting from the gluino mass.
• The masses of the $\chi$, $a-f_0$, and $0^+$ glueball states converge to a common point near $\kappa_{\rm cr}$, indicating a heavier supermultiplet.
• The $a-\eta'$ state remains lighter than the $\chi$ state in larger volumes, suggesting it may belong to the lightest supermultiplet.
• Finite-size effects are found to strongly influence the spectrum, particularly in small-volume TSMB simulations, indicating that larger volumes and finer lattices are essential for resolving the continuum SUSY limit.