For the generation of the gauge field configurations we use as a basis the Hybrid Monte Carlo (HMC) algorithm [22,23] as described in Refs. [24,25]. For the light quark sector Hasenbusch mass preconditioning [26,27] is applied. In particular, we employ four determinant ratios with mass shifts ρ={0.0;0.0003;0.0012;0.01;0.1}. The heavy quark determinant is treated by a rational approximation [28,29] with ten terms tuned such that the (eigenvalue) interval [0.000065, 4.7] is covered. For the molecular dynamics integration we use a nested second order minimal norm integrator. This results in 12 integration steps for the smallest mass term in the light and heavy quark sector and 192 steps for the gluonic sector [20]. We use the software package tmLQCD [25] which incorporates the multigrid algorithm DDalphaAMG for the inversion of the Dirac matrix [30]. The force calculation in the light quark sector is accelerated by a 3-level multigrid approach optimized for twisted mass fermions [31]. Moreover, we extended the DDalphaAMG method for the mass nondegenerate twisted mass operator. The multigrid solver used in the rational approximation [32,33] is particularly helpful for the lowest terms of the rational approximation, as well as for the rational approximation corrections in the acceptance steps, where it yields a speed up of 2 over the standard multimass shifted conjugate gradient(MMS-CG) solver. We checked the size of reversibility violation of this setup yielding a standard deviation <0.01 for δΔH and |1-⟨ΔH⟩|<0.02 fulfilling the criteria discussed in [34]. Here, δΔH is the difference of the Hamiltonian at integration time t=0, and the Hamiltonian of the reversed integrated field variables after one trajectory is performed.

As shown in Refs. [35,36] a most suitable and theoretically sound condition for the desired automatic O(a) improvement for twisted mass fermions is achieved by demanding a vanishing of the partially conserved axial current (PCAC) quark mass mPCAC=∑x⟨∂0A0a(x,t)Pa(0)⟩2∑x⟨Pa(x,t)Pa(0)⟩,a=1,2,(6)with Aμa the axial vector current and Pa the pseudoscalar current. In the twisted basis and for light, mass degenerate quarks, the axial and pseudoscalar currents can be calculated via Aμ+(x)=χ¯ℓ(x)γμγ5τ+2χℓ(x),P+(x)=χ¯ℓ(x)γ5τ+2χℓ(x),using τ+=(τ1+iτ2)/2 where τi are the Pauli matrices. The tuning procedure to maximal twist requires a value of the hopping parameter κ=κcrit where mPCAC(κcrit)=0. Note that the corresponding definition of the critical mass amcrit=1/(2κcrit)-4 is a function of aμℓ, aμσ, aμδ. Thus, even if the 1/a divergence in mcrit is independent from μℓ, μσ and μδ, determining amcrit at the μσ and μδ values of interest is important in order to keep lattice artifacts small which are introduced by the heavy quark doublet [37,38]. Instead the dependence of amcrit on μℓ reflects much milder discretization errors. In practice, we allow for some tolerance to this strict condition and following Ref. [39] we impose that ZAmPCACμℓ<0.1(7)within errors. In Eq. (7) ZA is the renormalization constant of the axial current. Fulfilling the condition Eq. (7) is numerically consistent with O(a)-improvement of physical observables, where it entails only an error of order O((ZA·mPCAC/μℓ)2). Hence for <0.1 follows for the targeted lattice spacing the error is comparable to other O([aΛQCD]2) discretization errors. This allows an O(a2) scaling of physical observables towards the continuum limit.

In order to tune to κcrit, we have generated several ensembles with fixed volumes of size 243·48 and 323·64, as listed in Table II. For a fixed twisted mass parameter of the up and down doublet, we scan over several values of the hopping parameter κ, see Table II. After fixing κcrit in this manner we proceed by tuning the light and heavy twisted mass parameters to realize physical pion, kaon and D-meson masses and decay constants. This procedure, which is described in more detail below will provide the input parameters for the target large volume simulations, denoted as the ensembles cB211.072.64.r1 and cB211.072.64.r2 in Table II.

Initially, we had attempted to start our Nf=2+1+1 simulations at a smaller value of β=1.726 that would correspond to the lattice spacing of our Nf=2 ensemble with a∼0.095  fm [5]. However, it turned out that tuning to maximal twist for a physical value of the pion mass for this β-value was not feasible. Nevertheless, our simulations at β=1.726 for pion masses in the range between 170 MeV and 350 MeV allowed us to develop a tuning strategy to realize the situation of maximal twist and also to reach the physical kaon and D-meson masses. This tuning strategy was then used at the finer lattice spacing as discussed in the present paper. The occurrence of instabilities of the simulations at β=1.726 when approaching the physical pion mass is, in fact, not unexpected. With twisted mass fermions, going to sufficiently small values of the light twisted mass parameter at a fixed lattice spacing one either enters the Aoki [40] or the Sharpe-Singleton [41] regime, see for an recent overview [42]. For the Sharpe-Singleton case, which is realized in our unquenched simulations, a sizable O(a2) negative shift of the neutral pion mass occurs.

Let us consider the region close to maximal twist, where |ω-π/2|≪1 or, equivalently, mℓ=m0-mcrit≪μℓ. Here the pion mass splitting can be related the PCAC quark mass by [40,43] amPCAC∼Zamℓmπ2mπ(0)2+⋯,(8)where Z=ZmZP/ZA is a combination of the untwisted quark mass (Zm), the pseudoscalar (ZP) and the axial (ZA) renormalization factors and am0 denotes the bare quark mass. The charged pion mass is denoted throughout this paper by mπ, while the neutral pion is given by mπ(0). The twisted mass angle ω can be defined via the gap equation, see [40,43]. From Eq. (8) it is clear that the tuning necessary to satisfy Eq. (7) becomes very hard for a large pion mass difference mπ2-mπ(0)2≫0.

In the Sharpe-Singleton scenario a first order phase transition is predicted from chiral perturbation theory. In simulations on finite lattices this leads to large fluctuations and jumps of physical observables [44–49], driving the simulations to become unstable. This makes it very hard to tune successfully to maximal twist. In our simulations at β=1.726 we observed a strong dependence of the PCAC quark mass on the bare mass parameter m0, which made it difficult to tune to the critical hopping parameter for a pion mass below 170 MeV. Although at β=1.726 we did not investigate in detail which of the lattice χPT scenario is realized, the fact that at β=1.778 we find (see Sec. III C) a neutral pion mass ∼20% smaller than the charged one suggests that a Singleton-Sharpe lattice scenario occurs in the scaling region with our chosen action (see Sec. II).

In order to avoid the aforementioned difficulties, we therefore decided to choose a finer value of the lattice spacing that would facilitate tuning to critical mass at the physical point. We found that a value of β=1.778, corresponding to a≈0.08  fm, allows us to tune to maximal twist successfully. In the following, we consider therefore a lattice volume of size 643·128, which is sufficiently large to suppress finite size effects but at the same time can be simulated with reasonable computational resources, given the algorithmic improvements that were discussed in Sec. II A.

For the tuning process of κ, which is a function of the light, strange, and charm quark mass parameters, we use the 243·48 and 323·64 lattices, see Sec. III. b. for more details. The dependence of the PCAC quark mass on κ at fixed light twisted mass parameter is shown in Fig. 1. Note that it can be assumed that Eq. (8) is valid here for the range -0.4141≲amℓ≲-0.4135 i.e., |a(mℓ-mcrit)|<0.0003. Using simple linear fits for the L=32 ensembles, we determine a critical value of κ, κcrit=0.1394265. We then employ this κ-value for our large volume ensembles.

For the simulations on the 643·128 lattices we first thermalize one configuration using 500 trajectories. We then use this configuration as a starting point for two replicas, each having a final statistics of about 1500 MDUs. In Fig. 1 we depict the Monte Carlo history of the PCAC quark mass for these two replicas, where we show, for better visibility, one history plotted by reversed history. The PCAC quark mass fluctuates around zero and does not show particularly large autocorrelation times nor any indication of a first order Sharpe-Singleton transition. Performing the average over the two replica runs, we find mPCAC/μ=0.03(2). Thus, the condition of Eq. (7) is nicely fulfilled. Note that here we do not include the renormalization factor ZA. However, our first estimate is that ZA≈0.8 and anyhow smaller than one, making the condition even better fulfilled. We therefore conclude that the tuning to maximal twist is achieved for the Nf=2+1+1 setup. And, as we will demonstrate below, the parameters of the cB211.072.64 runs are chosen such that we indeed simulate at, or very close to the physical values of the pion, the kaon and the D-meson masses.

In tuning the mass parameters of the heavy quark sector we exploit the fact that the value of the critical hopping parameter, as determined in the light quark sector, can be employed also for the heavy quark action while preserving automatic O(a)-improvement of all physical observables [3,50]. Nevertheless, tuning the heavy twisted mass parameters to reproduce the physical values of the strange and charm quark masses is a nontrivial task, owing to the O(a2) flavor violation [15] inherent to the heavy sector fermion action in Eq. (4). In order to tackle the problem, it is convenient to employ in an intermediate step the so-called Osterwalder Seiler (OS) fermions [51] in the valence which avoids these mixing effects. The OS-fermions can be used in a well-defined mixed action setup as valence fermions at maximal twist with the same critical mass, mcrit, as determined in the unitary setup [3]. The flavor diagonal action, denoted as Osterwalder Seiler fermion action, is given by SOSf=∑f=s,c{∑xχ¯f(x)[DW[U]+i4cSWσμνFμν(U)+mcrit+iμfOSγ5]χf(x)},(9)with χf a single-flavor fermion field. The renormalized valence masses μc,sOS,ren=μc,sOS/ZP can be matched to the corresponding renormalized quark masses via μc,sOS,ren=1ZP(μσ±ZPZSμδ),(10)with ZP and ZS denoting the nonsinglet pseudoscalar and scalar Wilson fermion quark bilinear renormalization constants. Then correlation functions using OS or unitary valence quarks are equivalent in the continuum. Moreover they still yield O(a) improved physical observables.

The general idea to tune the heavy quark twisted mass parameters is to start with an educated guess in the unitary setup and to tune the OS charm and strange valence masses by imposing two suitably chosen physical renormalization conditions. The so determined parameters of the OS action, i.e., aμsOS for the strange quark and aμcOS for the charm quark, can then be translated to new heavy quark twisted mass parameters [aμσ and aμδ of Eq. (4)] via Eq. (10), in the unitary setup and, together with a slight retuning of κcrit, a new unitary simulation can be performed. With a convenient choice of the physical renormalization conditions, here C1 and C2 (see below), this parameter tuning procedure can be carried out on a nonlarge lattice (in the present case, 323·64) and at a larger than a physical up/down quark mass.

In this work, we follow the above described strategy. As physical conditions we choose C1≡μcOSμsOS=11.8andC2≡mDsfDs=7.9,(11)where mDs is the Ds-meson mass and fDs the Ds-meson decay constant. The condition C2 has a strong sensitivity to the charm quark mass while C1 fixes the strange-to-charm mass ratio. They show only small residual light quark mass dependence arising from sea quark effects. We expect these conditions to be essentially free from finite-size effects due to the heavy Ds-meson mass. This setup leads indeed to an only small error for the final parameter choices. Details on our measurements of meson masses and decay constants for twisted mass fermions are given in the Appendix A.

As a first step, we work on gauge ensembles produced with μℓ around 3 times larger than the physical up-down average quark mass and with educated guess values of μσ, μδ and m0. We choose the OS quark masses μcOS and μsOS such that condition C1 is fulfilled. We then vary the OS quark masses, while maintaining condition C1, over a broad enough range such that also condition C2 is satisfied within errors.

In a second step, we match the heavy charm and strange twisted mass of the unitary action (4) to the OS fermion quark mass parameters via Eq. (10). The value of aμσ is directly determined from aμsOS and aμcOS, while aμδ is fixed by the ratio ZP/ZS. The latter can be estimated by adjusting aμδ such that the kaon mass evaluated in the unitary formulation (mKtm) and its counterpart computed with valence OS fermions (mKOS) are equal. Although the kaon mass value can be unphysical due to having a too large value of μℓ and possible finite size effects, the matching condition actually relates only heavy quark action parameters. It fixes the relation of aμδ to aμsOS and aμcOS, or equivalently the ratio ZP/ZS. In that way it is insensitive to both the finite lattice size and the actual value of μℓ up to O(a2) artifacts. Since the matching steps described so far were implemented only on the valence quark mass parameters of the unitary and OS actions using gauge ensembles with so far different values of the sea quark mass parameters, one still needs to generate new gauge configurations at the so-determined values of aμσ and aμδ. Now on these new ensembles it can be rechecked whether the condition C2 and the matching condition mKtm=mKOS, as well as the maximal twist condition Eq. (7) in the light quark sectors, are fulfilled with sufficient accuracy. If this happens not to be the case, the procedure has to be iterated.

More concretely, we start with an initial guess for the heavy quark mass parameters given by aμδ=0.1162 and aμσ=0.1223, which we deduce from a number of tuning runs on a lattice of size 243×48 and 323×64 along the lines of Ref. [52]. These parameters are realized for the ensemble Th1.200.32.k2, which is moreover very close to maximal twist. We then employ OS fermions in the valence sector and vary the values of μsOS and μcOS—while maintaining condition C1—such that condition C2 is fulfilled. This is illustrated in Fig. 2 for the Th1.200.32.k2 ensemble. By requiring that condition C2 is exactly fulfilled, we then fix the values of aμsOS and aμcOS, finding aμsOS=0.01948andaμcOS=0.2299.(12)

As explained above, the values in Eq. (12) already determine aμσ. To determine aμδ, we first compute the kaon mass in the OS setup at aμℓ used in the unitary setup and aμsOS from Eq. (12). Having found the OS kaon mass, we go back to the ensemble Th1.200.32.k2 and tune in the unitary heavy quark valence sector μδ such that we match the OS kaon mass. We then take the so found value of μσ and μδ for our simulations on the target large volume lattice. In this process a useful guidance is provided by assuming ZP/ZS=0.8 known to be a typical value from our previous simulations. As we will discuss later, this assumption for ZP/ZS turns out to be rather close to the values we determine on the cB211.072.64 ensembles. Our final result for the action parameters in the heavy sector of maximally twisted mass fermions then read aμσ=0.12469andaμδ=0.13151.(13)Due to the retuning of the heavy quark masses κcrit has to be retuned as well. To this end, several ensembles with volumes of 323×64 at light twisted mass values of aμℓ=0.002 and aμℓ=0.00125 were generated to determine the critical hopping parameter for the simulation at aμℓ=0.00072 resulting in κcrit=0.1394265.

In this work, it turned out that we only needed one iteration of the above procedure using the Th1.200.32.k2 ensemble. After this first step, the tuning conditions for the heavy quark masses were checked again on the Th2.200.32.k2 ensemble (see Table II) and found to hold to a good accuracy within statistical errors. A similar finding holds also on our target ensemble cB211.072.64 ensembles. If we impose again an exact matching between mKOS and the unitary mKtm on the two cB211.072.64 ensembles we find the ratio of the pseudoscalar to the scalar renormalization constants to be ZPZS=0.813(1).(14)Using this value of ZP/ZS the values of μσ,δ of Eq. (13) are close to the corresponding parameters at the physical point (the cB211.072.64 ensembles) that match our tuning conditions. Indeed the actually employed sea quark mass parameters correspond to a sea strange (charm) quark mass 6% lighter (4% heavier) than those derived a posteriori from imposing the same tuning and matching conditions on the physical point ensembles. It is also very nice to observe that by enforcing these conditions with very high precision one would obtain at the physical point with a∼0.08  fm a kaon mass in isosymmetric QCD less than 1% smaller than its experimental value.

An important aspect when working with twisted mass fermions at maximal twist is to keep the size of isospin violations small. This isospin breaking manifests itself by the fact that the neutral pion mass becomes lighter than the one of the charged pion. In leading order (LO) of chiral perturbation theory this effect is described by a2(mπ2-mπ02)=-4c2a2sin2(ω),(15)with the twisted mass angle given by ω=atan(μℓ/ZAmPCAC) and c2 a low energy constant characterizing the strength of O(a2)-effects of twisted mass fermions. As shown in Refs. [5,18], using a clover term the value of the low energy constant c2 decreases. Indeed, employing a clover term, simulations at physical quark masses become possible as demonstrated in Ref. [5]. It turns out that c2<0 for twisted mass fermions [53] leading to the Sharpe-Singleton scenario [41].

In order to calculate the neutral pion mass one needs to compute disconnected two-point functions that are notoriously noisy. To suppress the noise in the computation of the two-point functions we use a combination of exact deflation, projecting out the 200 lowest lying eigenvalues, and 6144 stochastic volume sources corresponding to an eight-distance hierarchical probing [54,55]. The disconnected correlator needed is given by Cdisc(t0)=⟨O^(0)O^(t0)⟩withO^(t0)=D-1(t0,t0)-⟨D-1(t,t)⟩,(16)where the ensemble and time average of the vacuum contribution is subtracted from the disconnected operator. Note that we used global volume noise sources to extract the disconnected contribution; however, methods which do not subtract the vacuum expectation value explicitly could be more effective as pointed out in [5,6,56]. We have found that the disconnected contribution dominates the correlator for time distances t/a>10, as can be seen in Fig. 3. However we include the connected contribution in the plateau average, leading to a neutral pion mass given by amπ(0)=0.044(9).(17)Note, that for the connected contribution small statistics of around 250 measurements are used, which results in a relatively large statistical error. The charged pion mass is straight forward to compute, and we find for the charged pion mass amπ=0.05658(6). This gives an isospin splitting in the pion mass of 22(16)% and the low energy constant c2 of Eq. (15) reads 4c2a2=-0.0013(8),(18)assuming ω=π/2. Thus, introducing a clover term for Nf=2+1+1 twisted mass fermions suppresses isospin breaking effects effectively, i.e., by a factor of 6 compared to an Nf=2+1+1 ensembles with twisted mass fermions without a clover term and a pion mass of 260 MeV at a similar lattice spacing of a=0.078(1)  fm [53,57], where it was found that the mass splitting is given by (amπ(0))2-(amπ)2=-0.0077(4). The suppression of the pion isospin breaking effects, thanks to the use of the clover term, is the underlying reason why we can perform our simulations at the physical point with Nf=2+1+1 flavors of quarks.

The first goal of this section is to determine the value of the lattice spacing within the pion sector. The extracted value will then be compared to the one from a similar investigation in the nucleon sector in Sec. V. In principle, the lattice spacing could be determined already from our cB211.072.64 target ensembles given in Table II, having a twisted mass parameter of μℓ=0.00072 and yielding a pion mass to decay constant ratio of mπ/fπ=1.073(3), which is rather close to the physical one. However, it is helpful to also use other ensembles, listed in Table II, which are all tuned to maximal twist, namely Th1.350.24.k2, Th2.200.32.k2, Th2.150.32.k2 in addition to the cB211.072.64 ensembles. By employing chiral perturbation theory (χPT) to describe the quark mass dependence of the pion decay constant and pion mass, we obtain a robust result for the value of the lattice spacing. Since the ensembles that are not at the physical point have partly only a small volume, we include finite volume corrections from chiral perturbation theory to the χPT formulas used [58]. We depict in Fig. 4 the ratio mπ2/fπ2 and the pion decay constant itself as function of the light bare twisted quark mass.

In Fig. 4 we also show the fits to NLO χPT [60–62], which for the ratio mπ2/fπ2 read mπ2fπ2=16π2ξℓ(1+Pξℓ+5ξℓlog(ξℓ))FfπFVE2FmπFVE2(19)and for the pion decay constant afπ=af0(1+Rξℓ-2ξℓlog(ξℓ))1/FfπFVE,(20)with the finite volume correction terms FfπFVE,FmπFVE [58]. Here ξℓ=2B0μℓ/ZP[(4πf0)2] where B0 and f0 are low energy constants. From the fits, we determine the values of 2B0/ZP=4.52(6) and af0=0.0502(3). The fitting constants P, R are related to the NLO low energy constants by P=-l¯3-4l¯4-5log(mπphys4πf0)2andR=2l¯4+2log(mπphys4πf0)2.(21)We determine the finite volume correction terms by fixing the low energy constants using the results of Ref. [38]. For our target ensemble cB211.072.64 with mπL=3.62 we find that the finite volume effects yield corrections of less than 0.5% for the pion mass and less than 0.5% for the pion decay constant. By using the fit functions from χPT and fixing the ratio mπ,phys2/fπ,phys2≡1.034 we find for the light twisted mass parameter aμℓ,phys=0.00067(1). We then use this value in Eq. (20) to determine the lattice spacing by using the phenomenological value of the pion decay constant, fπ,phen=130.41(20)  MeV [63]. We get afπ=0.07986(15)(35)  fm,(22)where the first error is the statistical and the second the systematic. We follow the procedure adopted in Ref. [39] for determining a systematic error by performing several different fits, adding or neglecting finite volume terms. Such fits employ e.g., the finite volume corrections of Ref. [59] using the calculated low energy constant c2 of Eq. (18) different orders in chiral perturbation theory and including or excluding the ensemble Th2.150.32.k2 due to larger finite size effects. The systematic error is then given by the deviations of these different fits from the central value given in Eq. (22). Although we include ensembles like Th2.150.32.k2 or Th1.350.24.k2 which have large finite size effects of up to 8% in the pion decay constant, the systematic uncertainties are suppressed due to the fact that we are using ensembles close to physical quark masses which stabilize the fits. Thus this demonstrates the importance of working at physical quark masses. Moreover this is confirmed by an estimation of the lattice spacing which takes only the pion mass and decay constant from cB211.072.64 into account. Requiring a vanishing pion mass in the chiral limit, the lattice spacing and the physical twisted mass value can be fixed by assuming a linear dependence of μ on a2mπ2 and mπ2/fπ2. The so determined lattice spacing agrees with Eq. (22) and reads a=0.0801(2)  fm.

As discussed in Sec. III B, the heavy sea quark parameters used in the simulation are tuned by employing the ensemble Th1.200.32.k2. With these parameters the kaon mass on the cB211.072.64 ensembles is smaller as compared to the OS kaon mass using the parameters of Eq. (12). By employing the tuning condition of Eq. (11) we therefore readjust the OS-parameters aμsOS and aμcOS following the tuning procedure of Sec. III B, to take the values aμsOS=0.01892(13)andaμcOS=0.2233(16)(23)for the cB211.072.64 lattices. The OS valence quark parameters are lower by around 2.4% compared to the values determined using the Th1.200.32.k2 ensemble [see Eq. (12)]. By using aμℓ,phys=0.000674 the strange to light quark mass ratio reads μsOSμl=0.01892(13)0.00067(1)=28.1(5).(24)

The kaon and D-meson masses and the respective decay constants as well as the corresponding quantities for the Ds-meson are all computed at three different values of μsOS and μcOS. We use a linear interpolation of mK2, mD and mDs with respect to the heavy OS quark masses. Using the values for μsOS and μcOS of Eq. (24) this allows us to determine the masses and decay constants for these mesons. In Fig. 5 we show the decay constants of the kaon and the D-meson and compare them with the results extracted from the Nf=2 clover ensembles [5]. We employ 244 measurements for the cB211.072.64 and 100 for the Th2.200.32.k1 ensemble. The ratios of the kaon and D-meson masses to decay constants for the cB211.072.64 ensembles are found to be mKfK=3.188(7)andmDfD=8.88(11),(25)where the former ratio has a central value slightly larger than the physical ratio mKphys/fKphys=3.162(18) [64], while the latter agrees well within errors with the value mDphys/fDphys=9.11(22) [63]. These results indicate that discretization effects for our setup are small in the heavy quark sector. For a more rigorous check, a direct calculation at different values of the lattice spacing will be carried out.

As an alternative way to determine the lattice spacing, one can use the nucleon mass. A direct way would be to use the physical ratio from which, by using the (lattice) pion mass determined above, the lattice spacing can be estimated directly by the value of the lattice nucleon mass. Indeed, with the pion mass amπ=0.05658(6) the nucleon to pion mass ratio 0.3864(9)/0.05658(6)=6.83(2) is close to its physical value of mNphys/mπphys=0.9389/0.1348=6.965 where we take the average of neutron and proton mass [63] and the pion mass in the isospin symmetric limit [64]. However, as in the case of the meson sector, using more data points at heavier pion masses and χPT to describe their quark mass dependence, a more robust result can be obtained. More concretely, we have employed chiral perturbation theory at O(p3) [68,69] for the nucleon mass dependence on the pion mass, i.e., mN=mNphys-4b1mπ2-3gA216πfπ2mπ3.(28)Similar to [70], where the authors observed that using previous Nf=2+1+1 ensembles showed no detectable lattice cutoff effects in the nucleon and pion mass, we use the nucleon masses of the Nf=2+1+1 ETMC ensembles without a clover term, determined in [71] to perform the chiral fit of Eq. (28). In our analysis we neglect cutoff effects, which appear to be small and not visible within our statistical errors. The same holds true for finite volume effects, see [70]. We fixed fπ=0.1304(2)  GeV and gA=1.2723(23) [63] in Eq. (28). The resulting fit to Eq. (28) is shown in Fig. 6 (right) and allows to determine the lattice spacing as amN(β=1.778)=0.08087(20)(37)  fm.(29)The first error is statistical while the second error is the deviation between the estimate obtained from Eq. (28) and taken the mass from the two-state fit. Note that the statistics of the nucleon correlator, thanks to the use of multiple inversion sources per gauge configuration, is 2 orders of magnitude larger than the one for the pion correlator. In this way the lattice spacing determination from the nucleon mass in Eq. (29) turns out to have a statistical error comparable to the one from the pion sector, see Eq. (22).