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 L=-12Tr(Gμν(x)Gμν(x))+ψ¯(x)γμ(∂μ+igAμa(x)TaA)ψ(x),(1)where the field strength tensor Gμν is ∂μAν-∂νAμ+ig[Aμ,Aν]. The generators TaA of the gauge group SU(2) act on the Dirac fermion field ψ in the adjoint representation.

As in ordinary QCD, AdjQCD has a conserved UV(1) vector symmetry and a UA(1) axial symmetry broken by anomaly. The first difference with respect to QCD is that axial anomaly leaves the partition function invariant under a discrete Z2Nc subgroup. A second difference is a peculiar flavor symmetry appearing already with a single adjoint Dirac fermion. In fact, as a gauge group element in the adjoint representation is real, the real and imaginary parts of the Dirac spinor do not mix, which means it decouples into two Majorana spinors in Minkowski space [11] ψ=12(λ++iλ-),(2)where λ+=ψ+Cψ¯T2,λ-=ψ-Cψ¯T2i.(3)The two components fulfill the Majorana condition by construction and they can be combined into a Dirac fermion field λ≡(λ+,λ-). After this decomposition, the Lagrangian of Eq. (1) can be rewritten as L=12∑kλk(x)(iD)λk(x)-12Tr(Gμν(x)Gμν(x)),(4)where k=+,-. A “two-flavor” SU(2) symmetry of the Lagrangian appears in terms of the two Majorana components. Therefore, chiral rotations belonging to the group U(1)A⊗SU(2) are a symmetry at the classical level of the Nf=1 AdjQCD action.

At the quantum level, the group of axial symmetry transformations leaving the partition function invariant is Z2Nc⊗SU(2). This remaining symmetry can be spontaneously broken by a nonvanishing expectation value of the chiral condensate. In this case, pions emerge as massless Goldstone bosons associated to the breaking of the continuous SU(2) symmetry, while the remaining discrete part implies Nc degenerate coexisting manifolds of vacua. For our specific choice of the gauge group SU(2) we have Nc=2, and the final unbroken symmetry group would be Z2⊗SO(2).

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 [16]. As demonstrated in Ref. [17], a modified chiral symmetry can be realized on the lattice if the continuum anticommutator of the massless Dirac operator D=γμ(∂μ+Aμa(x)TaA)) with γ5 Dγ5+γ5D=0,(5)is modified by the addition of an irrelevant term (a denotes the lattice spacing) Dγ5+γ5D=2aDγ5D.(6)The modified anticommutator, known as Ginsparg-Wilson relation, translates to a modified lattice chiral rotation of the Dirac field, ψ′=exp(iαγ5(1-aD))ψ,ψ′¯=ψ¯ exp(iα(1-aD)γ5).(7)Using the Ginsparg-Wilson relation Eq. (6) one can verify that the transformations Eq. (7) leave the Lagrangian Eq. (4) invariant. However, the chiral condensate for Ginsparg-Wilson fermions, Σ≡⟨ψ¯(1-D)ψ⟩,(8)transforms nontrivially under Eq. (7) and can be considered as an order parameter for chiral symmetry breaking.

A possible solution of the Ginsparg-Wilson relation is the massless overlap operator Dov=12+12γ5 sign(DH(κ)),(9)defined through the Hermitian Dirac-Wilson operator DH=γ5DW(κ), DW=1-κ[(1-γμ)(Vμ(x))δx+μ,y+(1+γμ)(V†μ(x-μ))δx-μ,y].Vμ(x) are the links in the adjoint representation and the parameter κ of the Dirac-Wilson operator used inside the sign function is an extra parameter taking values κ∈[0.125,0.25]. It appears as a freedom in choosing the overlap operator and can be tuned to improve locality. A more practical way to write the sign function is through the inverse square root sign(DH)=DH(κ)DH(κ)DH(κ).Unfortunately the evaluation of the square root of DH is computationally demanding. Furthermore, the force is ill-defined around the origin of the spectrum, which can lead to numerical problems when integrating the equation of motion for the hybrid Monte Carlo (HMC) algorithm [18,19].

In our simulations we implement overlap fermions using a polynomial approximation of order N of the sign function, following the algorithm described in Ref. [20]. At finite N the Ginsparg-Wilson equation Eq. (6) is only approximately fulfilled which introduces an explicit breaking of the chiral symmetry. The quality of the polynomial approximation can be visually seen when studying the eigenvalues of the overlap operator, see Fig. 1. An advantage of this approximation is that it introduces a gap on the spectrum, which acts as an IR regulator, preventing the forces on the HMC algorithm to diverge. As N is increased, the approximation converges to the exact one and in the limit N→∞, the spectral gap disappears, the chiral point is reached and the (modified) chiral symmetry of Eq. (6) is restored. There are several advantages in our approach compared to standard Wilson fermions: (i)

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

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 Nf is below the critical threshold where asymptotic freedom is lost. The fermion mass is a relevant direction even in the vicinity of the infrared fixed point inside the conformal window. In the Wilsonian low energy effective action, a mass term is generated near a fixed point from the violation of the Ginsparg-Wilson relation induced by our polynomial approximation, a mass that is going to vanish in the limit N→∞.

We have set the hopping parameter κ of the Dirac-Wilson operator inside the sign function to 0.2, and we apply one level of stout smearing to the corresponding link in the adjoint representation with a parameter ρ=0.15. We have verified that the overlap operator has zero eigenvalues, and our polynomial approximation converges toward the expected circle while keeping all eigenvalues inside it, see Fig. 1. The spectrum of the Dirac-Wilson operator is quite scattered and dense at small β, while converging toward the expected shape with four holes, one of them lying around the origin of the complex plane, see Fig. 2. The projection to the unit circle leads therefore to a single Dirac fermion interacting with the gauge fields [22].

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 64, see Table I, and free from possible bulk phase transitions.

The extrapolation of the order of the polynomial approximation to the limit N→∞ allows to recover the Ginsparg-Wilson relation exactly. We expect this limit to be equivalent to an extrapolation m→0 using an exact sign function and a positive nonvanishing explicit fermion mass parameter to regularize the overlap determinant. In other words, we expect to observe a relation between 1/N and fermion mass determined by the partially conserved axial current (PCAC) relation. It is important to estimate the dependence of the fermion mass on the order of the polynomial approximation, to ensure that our simulations are spanning a sufficiently wide range of pion masses so that the chiral limit can be reliably estimated and as a guideline for chiral extrapolations.

We have included simulation on lattices of size 83×72, which, together with the ensembles of size 184, have a sufficiently large time extent for a reliable measurement of the exponential decay of the axial and pseudoscalar current correlators. Our results are summarized in Table II. As shown in Fig. 3, we observe that the bare fermion mass amPCAC as a function of 1/N can be well described by a linear dependence close to the chiral limit, with an additional quadratic correction for our two smallest polynomial approximations (N=32 and N=40). These observations lead us to the hypothesis that the fermion mass is just proportional to 1/N close enough to chiral limit, although more simulations are required in order to verify whether this relation extends to different polynomial approximations defined for instance from a minimization of the sup-norm in a given interval. It would be interesting to test whether one particular choice of the polynomial approximation leads to a faster approach of the chiral limit, i.e., to a smaller pion mass when comparing polynomials of the same order. Further evidence for the linear dependence is provided in Sec. III B below.

From the results in Table II, we see that our simulations are spanning a reasonable range of pion and PCAC masses. Given the empirical evidence provided by the scaling of the fermion mass as a function of the inverse of the polynomial order, we will extrapolate in the following our measurements to the chiral point using fits as a function of 1/N including also ensembles for whose parameters we have not directly measured the PCAC mass.

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 0+ glueball mass toward the chiral limit.

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 N→∞ limit. We have measured the chiral condensate directly as defined from Eq. (8) using 20 noise stochastic estimators for each gauge-field configuration, skipping at least eight consecutive configurations between each measurement to improve the tradeoff between computational cost and autocorrelation.

First, we have explored the behavior of the chiral condensate Σ as a function of the volume. We have found that in the region of the bare couplings up to β=1.75 there is no clear evidence of finite volume effects up to a lattice of size 84, while at β=1.8 larger lattice sizes are required. On all runs used for the extrapolation of the chiral condensate to the massless limit, the Polyakov loop expectation value is zero and the theory is therefore in the confined phase.

We have extrapolated the bare chiral condensate as a function of 1/N to N=∞ limit at three different β. We have fitted a quadratic function with a χ2/d.o.f. smaller than two. The extrapolated condensate is nonzero indicating that at zero temperature chiral symmetry is spontaneously broken, see Fig. 4. As we have observed in the case of N=1 super-Yang-Mills, the main contribution for the nonvanishing expectation value of Σ comes from a smaller and smaller number of configurations as one approaches N→∞. Considering Fig. 5, we see that as N increases, the overall expectation value of the chiral condensate measured on each gauge-field configuration from stochastic estimators decreases, with the notably exception of some exceptional configuration providing a compensating nonvanishing contribution to the ensemble average. This pattern means that we have been able to effectively regularize the “zero over zero” problem of simulating massless fermions using a polynomial approximation.

We plan to complete our simulations at β=1.8 and to extend them at β=1.85, in order to extrapolate the renormalized value of the chiral condensate to the continuum limit in future studies. We also plan to add a full analysis of the chiral condensate as a function of the volume, which, together with the measurement of the chiral susceptibility, would enable us to test the order and the universality class of the chiral phase transition in the massless limit. In this study, we can just note that the bare value of Σ in dimensionless units is increasing as a function of the bare gauge coupling. Assuming that chiral symmetry breaking persists in the continuum limit, this changing of ⟨ψ¯ψ⟩ implies a large anomalous dimension, as it has been observed in previous investigations with Dirac-Wilson operator [11].

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 Z(2)⊗SO(2). The discrete part Z(2) corresponds to two manifolds of vacuum states distinguished by a positive and negative values of the chiral condensate. It is quite challenging to observe the coexistence of these phases directly in our simulations, as we are approaching the chiral limit from small but positive effective fermion masses. Nevertheless, if we perform a chiral rotation by an angle π, we expect to map in the limit N→∞ a configuration from positive to negative value of the chiral condensate, leaving its absolute value invariant. We can use the deviation at fixed N of the absolute value of the chiral rotated condensate as a measure of how much chiral symmetry is broken by the polynomial approximation of the sign function. Alternatively, we can view the rotation of the chiral condensate as a chiral Ward identity where the angle is chosen in order to avoid to take into account the effects of axial anomaly.

In order to perform a chiral rotation numerically, we approximate Eq. (7) for a small chiral rotation as ψ→ψ′=ψ+iωγ5(1-aD)ψ,ψ¯→ψ′¯=ψ¯+iωψ¯((1-aD)γ5),(10)such that we can decompose a chiral rotation of an angle α into n small steps α=∑nω by applying repeatedly Eq. (10) repeatedly as many times as needed. We have computed the chiral condensate Σ0 for a random source ψ and a rotated chiral condensate Σπ using the same sources rotated by an angle α=π. The difference in absolute value between the two condensates ΔΣ=|Σ0|-|Σπ| is shown in Fig. 6.

As we can see, the combination ΔΣ gives a nonzero value for a finite N. As we increase the order of the polynomial approximation, the distance to the chiral point decreases. In the limit N→∞ the chiral condensate changes by a sign flip and ΔΣ extrapolates to a value compatible with zero, pointing out that the Z2 symmetry is recovered and the chiral symmetry has been restored.

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 τ [23]. In particular, the measure of the flowed gauge energy density ⟨E(τ)⟩=14GμνaGμνa(τ),(11)allows to define a scale t0/a2 as the flow time τ where τ2⟨E(τ)⟩ reaches the reference value 0.3. The lattice spacing is then proportional to 1/t0, and its value in physical units, such as femtometers, could be defined in principle in terms of an experimentally measurable quantity.

We measured the scale t0 using the clover plaquette discretization of the energy density and the Wilson action for the definition of the flow equations. After ensuring that finite size effects are under control, see Fig. 7 the extrapolation of t0/a2 to the limit N→∞ is already an indication whether Nf=1 AdjQCD is infrared conformal or not. In the first case, the theory does not possess any scale other than the fermion mass, and the value of t0/a2 must be zero in the chiral limit.

The scale measured from our ensemble can be fitted by a quadratic function of 1/N. The scale data included in the fits are summarized in Table III and two examples of our fits are plotted in Fig. 8. The extrapolated value of t0/a is clearly different from zero, providing a first evidence that the theory is not infrared conformal. We see that t0/a grows as β is increasing, i.e. the lattice spacing is decreasing in the weak coupling limit as expected for a confining theory. Indeed, in the simplest way to compute the strong coupling constant scale dependence on the lattice, αs is defined in terms of the bare lattice gauge coupling as αs=2Nc4πβ, and the scale μ is equal to 1/a. As shown in Fig. 9, there are no evidence of an infrared fixed point in the region of bare coupling we have explored. However, a measure of the running of αs in the low-energy regime from the bare lattice coupling requires to perform simulations in the strong coupling region where possible lattice phases might prevent us from observing the true nature of the infrared fixed point. Further, this definition of the running coupling does depend on the lattice discretization of the continuum action. Fortunately, we can provide a further and cleaner evidence of the absence of an infrared fixed point by just exploiting the full dependence of the flowed gauge energy density on the flow time τ.

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 MS¯ scheme reads [23] ⟨E(τ)⟩=3(N2-1)16τ2π2gMS¯2(μ)(1+c1gMS¯2(μ)+O(gMS¯4)),(12)where we have set the scale μ to be equal to 1/8τ [23–25]. This relation can be truncated to the lowest order and inverted to define a renormalized gauge coupling gGF2(μ)=16π23(N2-1)τ2⟨E(τ)⟩|τ2=1/8μ.(13)The gradient flow scheme we employ requires first to extrapolate the coupling to the infinite volume limit, then to the chiral limit and finally to the continuum limit a→0 [26]. This nonperturbative determination of the running of strong coupling does not depend on the lattice discretization of the continuum action we have chosen, while only the first two coefficients of the β-function are universal and scheme independent.

The scale t0 is computed for each ensemble separately. Then, the running coupling gGF2(μ) is extrapolated to the chiral and continuum limit. We have considered a combined fit including all available ensembles in the scaling region using an ansatz of the form g(μ)2=g0+c0t0-1+c1t0-32+d1t0N+d21N+d3(t0N)2.(14)The fit is performed for each scale μ independently, interpolating the flow and its error in order to be able to include all ensembles at different lattice spacings in a single fit. The first term g0 represents the continuum limit value of the square of the running coupling extrapolated in the limit N→∞. The terms c0 and c1 are lattice artefact corrections proportional to a2 and a3, respectively. Finally, terms corresponding to the coefficient di are corrections proportional to a mass term, to its square and to a lattice discretization error equal to the bare fermion mass itself. At high energy, lattice artefacts are dominant, while at low energy the terms proportional to N becomes relevant, see Fig. 10. The inclusion of a finite-volume correction term, or of a logarithmic correction in the lattice spacing to the leading asymptotic scaling [27], does not improve the χ2 nor does change the final extrapolation significantly.

The final extrapolation to the chiral and continuum limit of the strong coupling determined from the Wilson flow is presented in Fig. 11 and compared to the perturbative running in Fig. 12. There are no evidence of an infrared fixed point in the running of the strong coupling, nor a signal of a slowing of the growth of the strong coupling constant in the infrared regime. Our present calculation is in agreement with our previous study of the β-function in the Mini-MOM scheme, where no evidence of an infrared fixed point have been found in the region of momenta we have been able to explore [28]. Further studies close to the continuum limit and deeper in the infrared region will be required to confirm this result.