^{1,2}

^{3,1}

^{1}

^{4}

^{1}

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^{3}.

The physical particles in supersymmetric Yang-Mills (SYM) theory are bound states of gluons and gluinos. We have determined the masses of the lightest bound states in SU(3)

Supersymmetry (SUSY) plays a fundamental role in the physics of elementary particles beyond the standard model. The understanding of the nonperturbative phenomena of SUSY theories is important since they might explain the supersymmetry breaking at low energies. Besides the relevance for extensions of the standard model, supersymmetric gauge theories also provide insights into nonperturbative phenomena that also occur in QCD, such as confinement of color charges, at least in certain regimes since supersymmetry constrains the nonperturbative contributions. Nonperturbative numerical methods such as lattice simulations are essential to complement and extend the obtained analytical understanding from SUSY models to theories with less or no supersymmetry.

Supersymmetric extensions of the standard model must include the superpartners of the gluons, the so-called gluinos, which are Majorana fermions transforming under the adjoint (octet) representation of SU(3). The gluino would interact strongly, and the minimal theory describing the interactions between gluons and gluinos is

SU(3) SYM theory is of a complexity comparable to QCD, and Monte Carlo lattice simulations are an ideal

In this Letter, we focus on the spectrum of bound states of the

We have presented our first data at a single lattice spacing in Ref.

In the continuum the (on shell) Lagrangian of SU(3) supersymmetric Yang-Mills theory, containing the gluon fields

The technical details of our approach for the numerical simulations of SU(3) SYM theory have been described in our previous publication

Alternative approaches have been investigated for the simulation of SYM theory

The complexity and the cost of the numerical lattice simulations for this theory is at least as challenging as in corresponding investigations of QCD. Additionally, there are more specific challenges for the technical realization of numerical simulations of SYM theory, such as the unavoidable explicit breaking of supersymmetry on the lattice. Therefore, the most important task of our project is to demonstrate that the infrared physics emerging from the numerical simulations is consistent with restoration of supersymmetry in the continuum limit.

A further specific challenge is related to the integration of Majorana fermions, which leads to an additional sign factor in the simulation

The scale, i. e., the determination of the lattice spacings in physical units in terms of a common observable, is measured from gluonic observables. We are using chirally extrapolated values of the scale

An improvement with respect to our work on SU(2) SYM theory, where we extrapolated the observables first to the chiral limit and in a second step to the continuum limit, is that we now use a combined fit towards the chiral and continuum limit. The chiral continuum values

The main indication for restoration of supersymmetry in lattice simulations presented in this Letter is the formation of mass degenerate supermultiplets. An alternative indication is given by the supersymmetric Ward identities. The violation of the supersymmetric Ward identities in the chiral limit is an indication of lattice artifacts, since chiral symmetry and supersymmetry should be restored at the same point in the continuum theory, if there is no unexpected supersymmetry breaking. The Ward identities also provide a cross check for the tuning of the bare gluino mass parameter. We have found that the Ward identities are consistent with a restoration of supersymmetry, and the leading lattice artifacts are

We have performed simulations at a large range of values of the inverse gauge coupling

In the current Letter, we present the final results for the lightest particles of SU(3) SYM theory. We are now able to combine several different lattice spacings in an extrapolation to the continuum limit. In comparison to Ref.

The considered states and corresponding interpolating operators are the scalar meson

The chiral extrapolations to the point of vanishing adjoint pion mass

The chiral extrapolations of the particle masses at the different lattice spacings using the fit function

At our two finest lattice spacings (

A particular problem with our first data at the finest lattice spacing (

The three different lattice spacings allow for the first time a complete extrapolation of the lightest states of SU(3) SYM theory to the continuum. Compared to our previous work with an unimproved Wilson fermion action for the investigations of SU(2) SYM theory, the differences of the masses in units of

The extrapolation of the bound states masses to the continuum using the fit function [Eq.

We have finalized our first continuum extrapolation of the lightest bound states in supersymmetric SU(3) Yang-Mills theory. We have found a formation of a chiral supermultiplet in the continuum limit. In combination with the results from an analysis of the supersymmetric Ward identities, this is a good indication for the absence of supersymmetry breaking by the nonperturbative dynamics of the theory. It also shows that the unavoidable breaking of supersymmetry by the lattice discretization is under control in this nontrivial theory.

This important observation opens the way towards several further investigations of SU(3) SYM theory, in particular concerning the phase transitions and chiral dynamics of the theory. In addition, it is the first step towards investigations of supersymmetric QCD and other supersymmetric gauge theories that can not be accomplished without control of the supersymmetry breaking in the pure gauge sector.

Our investigation is based on the approach proposed in Ref.

Our results can be compared to the our previous analysis of SU(2) SYM theory, presented in Refs.

One interesting additional aspect for further investigations is the continuum limit of the splitting of the multiplet as a function of the soft supersymmetry breaking. Our current data in Fig.

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time on the GCS Supercomputers JUQUEEN, JURECA, and JUWELS at Jülich Supercomputing Centre (JSC) and SuperMUC at Leibniz Supercomputing Centre (LRZ). Further computing time has been provided the compute cluster PALMA of the University of Münster. This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group “GRK 2149: Strong and Weak Interactions—from Hadrons to Dark Matter.” G. Bergner acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) Grant No. BE 5942/2-1. S. A. acknowledges financial support from the Deutsche Akademische Austauschdienst (DAAD).

We have chosen the common reference value of

We have neglected the ensemble with the largest adjoint pion mass at