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Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are

Strong interactions are responsible for the confinement of quarks and for the generation of the hadron masses in QCD. The coupling (

The running of the coupling with the scale

Supersymmetric Yang-Mills theories (SYM) are a remarkable exception, since many nonperturbative features of these theories are known analytically. Instanton calculations lead to the conclusion that there exists a scheme where the

The number of fermions

In this contribution, we explore the nonperturbative dynamics of gluons and ghosts in the Landau gauge by means of numerical lattice simulations for an SU(2) gauge theory coupled to one, two, three and four Majorana fermions, corresponding to

There are several long-term goals of our investigations. The first one is to measure the running coupling of

The second important goal is a general comparison of the running coupling in the MOM scheme for a QCD-like theory and a theory in the conformal window in order to see how the signs for an infrared fixed point appear in this setup. MWT is a prime example for such a comparison since the indications for a conformal behavior of this theory have been found in a large number of numerical investigations. So far this comparison of the running coupling in the MOM scheme has only been done using Dyson-Schwinger equations Ref.

The final purpose is to determine a scale

The third goal is an identification of the onset of the conformal window for theories with

This work is organized as follows. In the first part, we start in the continuum and present the considered scheme for the running coupling in Sec.

In the continuum and in the Landau gauge, the form of gluon and ghost propagators, denoted in the following respectively as

There are various possible alternative prescriptions for the renormalization of the coupling in the MOM schemes, which can be defined from the three or four gluon vertexes

The “Taylor coupling” defined by

The interactions of gauge fields with matter fields changes nontrivially the behavior of ghost and gluon propagators

A four-loop perturbative calculation of the running of

We study on the lattice gauge theories with fermions in the adjoint representation of the gauge group SU(2), which action reads in the continuum

The general properties of adjoint QCD depend strongly on the number of fermions. There are three relevant possibilities:

Very little is known about the properties of

To the best of our knowledge, there has been no previous lattice investigations of AdjQCD with three Majorana fermions. The theory is predicted to be just close to the conformal window. The main interest for

Our numerical simulations are done on a four-dimensional lattice of

On the lattice, the Landau gauge is fixed by maximizing the functional

Once the gauge-fixed links are known, the gluon propagator in the momentum space can be easily computed from the Fourier transform of the gluon field in the real space. The computation of the ghost propagator requires an inversion of the Faddeev-Popov operator for each momentum of the ghost field. We solve the corresponding linear system using the BiCGStab biconjugate gradient algorithm. The definition of the Faddeev-Popov operator on the lattice is discussed in detail in Ref.

We compute the gluon and ghost propagators for each ensemble on

There are several possible equivalent definitions of the modulus squared of the lattice momentum

The running of the strong coupling is given by the

The fit to perturbation theory is performed including the functional form of the perturbative lattice correction

We employ in all our simulations for all theories the tree-level Symanzik improved gauge action and stout-smeared Wilson fermions, except for the

Summary table of the analyzed ensembles. In the case of

The study of

The bare gluon and the ghost dressing functions are shown at fixed

(a, b) Bare gluon and ghost dressing functions for

The running coupling defined in

(a, b) The running of the coupling

According to the methods presented in Sec.

The knowledge of the

Our main ensembles for

The gluon and the ghost dressing functions have clearly a flatter behavior than in

(a) The running of the coupling

Bare gluon and ghost dressing functions for

The running coupling is presented for our three different bare lattice couplings in Fig.

Strong coupling constant for

Besides this plateau, there is a considerable running of coupling the far infrared region at

We don’t see a significant running in the ultraviolet part. Due to the flat behavior of the strong coupling, the fit to perturbation theory is very unstable and it is impossible to estimate the ultraviolet scale

The general behavior with a nearly zero running of the coupling over a large energy scale is consistent with the expectations for an infrared fixed point. The fermion mass is a relevant direction at the fixed point and is responsible for the running in a region that moves further to the infrared the smaller the fermion mass is. Only the gluon dressing function at the smallest fermion masses of

Unfortunately we are not able to connect the running to the perturbative one since up to the highest energies we are able to explore the running seems to be dominated by the influence of the fixed point. There are two explanations, either our energy scales are still too low, or we are at couplings above the fixed point value and hence in a region not connected to the ultraviolet fixed point.

It is worth to investigate in more detail the dependence of the running coupling on the fermion mass and the deviations form

Mass dependence of the running coupling for

The fact that the fermion mass affects rather the far infrared part is quite expected and can be understood by comparison with perturbative QCD. The calculation of the first two coefficients of the

There are two possible interpretations of the form of the fermion mass dependence, which are both consistent with our data. The first one is to assume that there is a real and exact zero of the beta function already at a finite fermion mass. Our simulations at

The alternative interpretation is that the running goes to zero only in the massless limit as dictated by hyperscaling. An infrared conformal theory does not possess any dimensionful scale related to long-range physics. In the chiral limit, all masses of bound states, including decay constants and condensates, extrapolate to zero according to a universal scaling

“Hyperscaling” of the running coupling for

We have generated ensembles at three different

For the two ensembles simulated with the largest

Bare gluon and ghost dressing functions for

The running coupling has a downward tendency in the deep infrared region for our smallest

The running of the running coupling

In contrast to the

Fit of

Since moving from four to three Majorana fermions coupled to SU(2) gluodynamics has been sufficient for a significant change in the running of

Bare gluon and ghost dressing functions for

The running coupling is shown in Fig.

(a) The running of the running coupling

It is therefore interesting to compare the fits of

Comparison of the fit of

We have presented a lattice study of the gluodynamics in the Landau gauge for AdjQCD and observed that the properties of gluon propagators depend crucially on the number of adjoint fermions. We have determined the running coupling in the MiniMOM scheme and investigated how it changes when going from the QCD-like case to a theory in the conformal window.

For

The running coupling computed from the gluon and ghost dressing functions provides a clear evidence for the asymptotic freedom of adjoint QCD with

We have estimated the

The results for

At our smallest PCAC mass, the gluon dressing functions of

The study of the running of

We thank Gunnar S. Bali, Jacques C. R. Bloch, Biagio Lucini, Holger Gies, and Meinulf Göckeler for interesting discussions. We thank Gernot Münster and Pietro Giudice for reading and commenting on the first version of the manuscript. We thank Istvan Montvay for his support in performing the numerical simulations. The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V. (GCS) for providing computing time for a GCS Large-Scale Project on the GCS share of the supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC) and on the supercomputer SuperMUC at Leibniz Computing Centre (LRZ). GCS is the alliance of the three national supercomputing centres HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayerische Akademie der Wissenschaften), funded by the German Federal Ministry of Education and Research (BMBF) and the German State Ministries for Research of Baden-Württemberg (MWK), Bayern (StMWFK) and Nordrhein-Westfalen (MIWF). Further computing time has been provided by the iDataCool cluster of the Institute for Theoretical Physics at the University of Regensburg. S. P. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) Grant No. SFB/TRR 55, and G. B. received support from Grant No. BE 5942/2-1.