^{1,2}

^{,*}

^{1}

^{3}

^{,†}

^{1}

^{3}.

In this work we investigate the infrared behavior of a Yang-Mills theory coupled to a massless fermion in the adjoint representation of the gauge group SU(2). This model has many interesting properties, corresponding to the

Non-Abelian Yang-Mills theories coupled to fermions in the adjoint representation of the gauge group (AdjQCD) have properties similar to ordinary QCD, while featuring many additional symmetries which are absent or broken in gauge theories interacting with fundamental fermions. The most notable examples are supersymmetry, center symmetry, and discrete-axial symmetry. For instance, the model of strong interactions between a gauge field and a massless adjoint Majorana fermion corresponds to

If the theory lies below the conformal window, it is also interesting to ask whether confinement and chiral symmetry breaking are both simultaneously present. The interplay between these two nonperturbative phenomena is already alone a good motivation to study

If the gauge group is

’t Hooft anomaly matching arguments suggest also a third alternative scenario for the low-energy effective theory of

Distinguishing a genuine conformal theory from a confining theory near the lower edge of the conformal window is a challenging task. Nonperturbative lattice simulations can explore a regime where supersymmetry is broken, and in general strong interactions outside the perturbative regime. They are, however, limited to a certain range of scales. In this contribution we will provide strong numerical evidence that the theory has a scale provided by the breaking of chiral symmetry in the range of considered parameters. In Sec.

In this section we begin by recalling the most important symmetries of the SU(2) gauge theory coupled to one massless fermion in the adjoint representation. The Lagrangian in the continuum reads

As in ordinary QCD, AdjQCD has a conserved

At the quantum level, the group of axial symmetry transformations leaving the partition function invariant is

Preserving chiral symmetry is crucial for our numerical study of the fermion condensate. However, a lattice discretization of fermion fields preserving chiral symmetry is challenging due to the limits imposed by the Nielsen-Ninomiya theorem

A possible solution of the Ginsparg-Wilson relation is the massless overlap operator

In our simulations we implement overlap fermions using a polynomial approximation of order

there is no need for fine tuning of the fermion mass, as the chiral limit is reached after a simple

as such, we can study chiral symmetry breaking in the massless limit directly using the chiral condensate as order parameter without having to worry about additive renormalization terms,

the lattice action is automatically

Spectrum of the exact and the approximated overlap operator for

For the discretization of the gauge part of continuum action we use a tree level Symanzik improved action. Even though the strongest source for lattice artifacts is the fermionic action, the Symanzik improvement helps to evade spurious phase transitions that could potentially appear when studying gauge observables like the Polyakov loop.

The critical behavior of a renormalization group transformation of AdjQCD near the Gaussian fixed point is dominated by two relevant parameters, namely the gauge coupling and the fermion mass, if the number of flavors

We have set the hopping parameter

Spectrum of the Dirac-Wilson operator (

We have also verified that all our simulations are in the confined phase in the region of bare couplings we have explored even at volumes as small as

Chiral condensate, plaquette and Polyakov loop of the small volumes runs used for tuning the bare lattice gauge coupling. Our lattice volume is

The extrapolation of the order of the polynomial approximation to the limit

We have included simulation on lattices of size

Pion and bare PCAC fermion masses measured on our lattices with the largest time extent.

(a) Bare fermion mass

From the results in Table

It is worth noting that the pion mass squared as a function of bare PCAC fermion mass can be described by a linear function with a small quadratic correction, as expected by the Gell-Mann-Oakes-Rennes (GMOR) relation, without any clear evidence of hyperscaling. We plan to investigate the full spectrum in a future forthcoming work, in particular it will be important to study the scaling of the

The chiral condensate is equal to the derivative of the partition function with respect to the quark mass. As we are interested in understanding whether the theory is chirally broken, an important indication is a nonzero value of the chiral condensate extrapolated to the

First, we have explored the behavior of the chiral condensate

We have extrapolated the bare chiral condensate as a function of

Extrapolation of the bare chiral condensate in dimensionless units to the chiral limit, including in the fit the volumes

Chiral condensate measured on individual gauge-field configurations at

We plan to complete our simulations at

As shown in the previous section, chiral symmetry is spontaneously broken by a nonvanishing vacuum expectation value of the chiral condensate in our simulations. The remaining symmetry corresponds to

In order to perform a chiral rotation numerically, we approximate Eq.

Sum of the bare fermion condensate

As we can see, the combination

We can show further evidence on the absence of a conformal behavior if we consider the running of the strong coupling constant in the infrared limit. The strong coupling constant

The gradient flow is a continuous smoothing applied to gauge fields, defined from a partial differential equation which is solved numerically to determine the evolution of certain gauge-invariant observables as a function of the flow-time

We measured the scale

Comparison of the flowed energy density for the volume

The scale measured from our ensemble can be fitted by a quadratic function of

Chiral condensate of all large volume simulations, with in addition the scale

Extrapolation to the chiral limit of the scale

Running of the bare lattice coupling squared as a function of the inverse lattice spacing measured from the scale

An efficient method to compute the running of the strong coupling constant from Monte Carlo simulations is to find an appropriate scheme which can be defined both on the lattice and in the continuum, in such a way that different computation methods and lattice discretizations can be easily compared among each other. The gradient flow can be consistently defined independently from the regularization used, and a perturbative expansion of the flowed gauge energy density in the

The scale

Extrapolation to the chiral and continuum limit of the strong coupling constant. The red lines are obtained from the global fit by fixing the order of the polynomial approximation

The final extrapolation to the chiral and continuum limit of the strong coupling determined from the Wilson flow is presented in Fig.

Extrapolation to the continuum limit of the strong coupling constant as a function of the scale (red band), extrapolated from the measured curves at the different

Comparison of the perturbative two-loop perturbative running of the

We have presented a numerical investigation of the

Our results also support the breaking of chiral symmetry induced by a nonvanishing expectation value of the chiral condensate. Consequently, we would predict two light pions emerging as Goldstone bosons in the massless limit. In the previous investigations of Ref.

Reference

We thank G. Münster for helpful comments. G. B. and I. S. C are funded by the Deutsche Forschungsgemeinschaft (DFG) under Grants No. 432299911 and No. 431842497. The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V. (