N=1 Supersymmetric Gauge Theories on the Lattice

**Principal Investigator:**

André Sternbeck

**Affiliation:**

Institute for Theoretical Physics, Friedrich-Schiller-University Jena

**Local Project ID:**

pr48ji

**HPC Platform used:**

SuperMUC and SuperMUC-NG of LRZ

**Date published:**

**Introduction**

The standard model (SM) of particle physics is experimentally well established. It very successfully describes all processes mediated by the electromagnetic, weak and strong forces. Nonetheless, several questions remain unanswered and the SM is seen as an effective theory of a more fundamental theory.

Supersymmetric extensions are an interesting step beyond the standard model. A straightforward extension of the SM is the minimal supersymmetric standard model. This project addresses the strongly coupled subsector of this model, also known as N=1 Super-Yang-Mills (SYM) theory. It describes interactions of gluons and their fermionic superpartners, the massless gluinos, and allows for a numerical treatment using lattice Monte Carlo simulations, similar as for lattice regulations of the strong interaction. This non-perturbative treatment, however, is non-trivial. Lattice formulations explicitly break supersymmetry and consequently the mass degeneration within a super-multiplet at any finite lattice spacing.

**Results and Methods**

For our project we use the lattice formulation introduced by Curci and Veneziano [1] which is based on Wilson-type Majorana fermions. The Wilson term explicitly breaks supersymmetry and chiral symmetry at finite lattice spacing, but this breaking leads to a counter-term which is proportional to the gluino mass term. By adding an explicit gluino mass term this can be compensated, such that the renormalized gluino becomes massless in the continuum limit, and both supersymmetry and chiral symmetry will be restored.

This approach is straightforward, but it requires a careful treatment of the continuum limit and implies studying a wide range of lattice sizes and spacings. Furthermore, due to the adjoint SU(3) representation of the gluino, most bound-state correlators have connected and disconnected contributions. In particular the latter require large gauge ensembles to reduce the statistical noise. But also connected contributions of correlator functions with gauge fields, like for the gluino-glue, are afflicted by large statistical uncertainties. The calculations are therefore numerically demanding and any improvement of the lattice regularization, as well as of the numerical methods, are highly welcome.

We introduce and test two novel concepts [2]:

- We modify the original lattice formulation by adding a mass-like term to the fermionic part of the action, similar to an one-flavor formulation of twisted-mass QCD. By tuning the two mass parameters of the lattice action, we achieve a considerably improved mass degeneracy of the chiral partners in the Veneziano-Yankielowicz supermultiplet already at finite lattice spacing (see Figure 1). We can reproduce this for different gauge couplings and gluino masses. Furthermore, the superpartner, the so-called gluino-glue, has a similar mass at finite lattice spacing. Our findings promise that both chiral and supersymmetry can be improved when using a twisted Wilson-Dirac operator at a special twist and an improved continuum extrapolation may be possible.
- We apply an adaptive aggregation-based domain decomposition multigrid (DDαAMG) algorithm [3] for the calculation of two-point correlator functions. This algorithm has been successfully employed in lattice QCD studies and similar performance gains may be possible for a N=1 SYM theory. We study the performance gain for some benchmark scenarios and achieve a speed-up factor of 9 to 20, which is an enormous speed up (see Figure 2). Consequently, we use this algorithm for the calculation of two-point functions of the adjoint-
*η*’ and adjoint-*f*0, where many stochastic estimators are required. It has much reduced the amount of required CPU time.

**On-going Research / Outlook**

The results of our project have been published recently [4]. The character of our study remains exploratory, however. With the granted CPU time we were able to analyze the benefits of using a twisted-mass Dirac operator for lattice studies of N=1 SYM theory and to demonstrate the performance gains when using the DDαAMG algorithm. For a reasonable estimate of the associated supermultiplets, however, a follow-up study is required. In particular larger volumes and ensembles sizes are needed to reduce systematical and statistical errors.

**References and Links**

[1] Curci and Veneziano, Nucl. Phys. B292 (1987) 555

[2] Steinhauser et al., PoS LATTICE2019 (2019) 201 [eprint: 1912.09979]; PoS LATTICE2018 (2018) 211 [eprint: 1811.01785]; EPJ Web Conf. 175 (2018) 08022 [eprint: 1711.05086].

[3] Alexandrou et al, Phy.Rev. D94 (2016) 114509; Bacchio et al., DDalphaAMG library including twisted mass fermions, github.com/sbacchio/DDalphaAMG

[4] Steinhauser, Sternbeck, Wellegehausen, Wipf, JHEP01 (2021) 154 [eprint: 2010.00946].

**Research Team**

Marc Steinhauser, André Sternbeck, Björn Wellegehausen and Andreas Wipf. All: Institute for Theoretical Physics, Friedrich-Schiller-University Jena.

**Scientific Contact**

Dr. André Sternbeck

Institute for Theoretical Physics, Friedrich-Schiller-University Jena

D-07743 Jena (Germany)

e-mail: andre.sternbeck [@] uni-jena.de

*Local project ID: pr48ji*

*February 2021*