CERN-PH-TH-2012-119

DESY 12-078

HU-EP-12/15

SFB/CPP-12-24

WUB/12-11

The strange quark mass and Lambda parameter of two flavor QCD

Patrick Fritzsch, Francesco Knechtli, Björn Leder, Marina Marinkovic,

Stefan Schaefer,
Rainer Sommer and
Francesco Virotta

Humboldt Universität zu Berlin, Institut für Physik,

Newtonstr. 15, 12489 Berlin, Germany Bergische Universität Wuppertal,

Fachbereich C – Mathematik und Naturwissenschaften,

Gaussstr. 20, 42119 Wuppertal, Germany
CERN, Physics Department, 1211 Geneva 23, Switzerland
NIC, DESY, Platanenallee 6, 15738 Zeuthen, Germany

Abstract

We complete the non-perturbative calculations of the strange quark mass and the Lambda parameter in two flavor QCD by the ALPHA collaboration. The missing lattice scale is determined via the kaon decay constant, for whose chiral extrapolation complementary strategies are compared. We also give a value for the scale in physical units as well as an improved determination of the renormalization constant .

Key words: Lattice QCD; Lambda parameter; strange quark mass

PACS: 12.38.Gc; 12.38.Aw; 14.65.Bt

## 1 Introduction

The parameters of the standard model of particle physics have to be determined by matching theory to experimental data. For two flavor QCD we present results for two of these parameters, the scale parameter and the mass of the strange quark.

These results are the outcome of a long project by the ALPHA collaboration. The general strategy using the Schrödinger functional to define the coupling constant has been laid out in Refs. [1, 2], and results for the parameter in pure gauge theory have been published in [3]. In the two flavor theory for [4, 5] and the strange quark mass[6], however, the determination of the lattice scale, which allows for their conversion to physical units, has been lacking. This determination is the subject of the present paper.

In lattice computations, the physical mass scale is set by picking one
dimensionful observable and identifying its value at the point where the
quark mass ratios correspond to the physical situation with the
experimental input. If we have done the calculation with all physical
effects taken into account (and if the theory is correct), it does not
matter, which observable we take. Here, however, we restrict ourselves
to QCD with two dynamical flavors of light quarks leading to a
systematic uncertainty which is hard to determine. In our computation,
we use the kaon decay constant to set the scale. Over the
pion decay constant it has the advantage of a chiral
extrapolation which is milder and therefore better under
control.^{1}^{1}1This property depends on how one actually
approaches the physical point. It is in particular true
for the strategy 1 which we introduce below.
However, we need a quenched strange quark. Also the
mass of the Omega baryon is popular to set the scale
and first results indicate that similar numbers are
obtained from this observable[7].

In previous publications the results have been converted to physical units using the scale parameter [8], defined via the force between static quarks. The conversion relied on measurements of by QCDSF[9] and the assumption that fm. Below, we present our own results for , which differ substantially from the previous values and lead to an update in .

The paper is organized as follows. The lattice action, an overview of the ensembles and details of the error analysis are given in Sect. 2, followed by the definition of the hadronic observables in Sect. 3 and results for the scale parameter in Sect. 4. The strategies for the chiral extrapolation are discussed in Sect. 5 leading to the scale determination from the kaon decay constant. The results for the Lambda parameter and the strange quark mass are contained in Sects. 6 and 7, respectively.

The appendices contain updates of many quantities, whose analysis has been subject of previous publications. The renormalization constants and are discussed in App. B and App. C, the hadronic scale of the Schrödinger functional calculations is subject of App. D, followed by a determination of the critical mass of the improved Wilson fermions and the singlet renormalization factor in App. E.

## 2 Lattice parameters and simulation algorithms

In this computation we use the Wilson plaquette gauge action for the gluon fields together with two degenerate flavors of improved Wilson fermions[10]. The action

(1) |

has three parameters: , and . The coupling constant is given by . The fermions with bare mass , usually substituted by the hopping parameter , are implemented by the lattice Dirac operator

(2) |

It includes the covariant forward and backward derivatives, and , and the Sheikholeslami–Wohlert[11] improvement term involving the standard discretization of the field strength tensor[12]. Its coefficient has been determined non-perturbatively[13].

We have generated ensembles at three values of , 5.3 and 5.5 which correspond roughly to lattice spacings of fm, fm and fm, respectively, with details given in Section 5.6. The ensembles are listed in Table 1. All lattices have size and the pion mass is always large enough such that . We therefore expect finite size effects to be small.

id | [MeV] | ||||||
---|---|---|---|---|---|---|---|

A2 | |||||||

A3 | |||||||

A4 | |||||||

A5 | |||||||

E4 | — | ||||||

E5 | |||||||

F6 | |||||||

F7 | |||||||

N4 | |||||||

N5 | |||||||

N6 | |||||||

O7 |

### 2.1 Simulation algorithms

For most of the ensembles, generated within the
CLS effort,^{2}^{2}2https://twiki.cern.ch/twiki/bin/view/CLS/
the DD-HMC algorithm[14, 15] has been used as implemented
in the software package by M. Lüscher[16]. It is based on
a domain decomposition to separate the infrared from the ultraviolet
modes of the fermion determinant. A main feature is the locally
deflated, Schwarz preconditioned GCR
solver[17, 18] which significantly reduces
the increase in computational cost as the quark mass is lowered.

The drawback of this algorithm is that due to the block decomposition only a fraction of gauge links is updated during a trajectory. In pure gauge theory the autocorrelation times are inversely proportional to this fraction of active links[19]; we expect this behavior also in the theory with fermions. Typical domain decompositions lead to active link ratios between and and therefore a factor between 2 and 3 increased autocorrelation times.

For some lattices, we therefore used a Hybrid Monte Carlo algorithm[20] with a mass preconditioned fermion determinant[21, 22]. Our implementation[23], MP-HMC from here on, is based on the DD-HMC package and in particular takes over the deflated solver because of its efficient light quark inversions. This algorithm was employed for ensembles A5, N6 and O7 given in Table 1; all other ensembles were generated with the DD-HMC. Appendix A gives details about these algorithms and the values of the parameters used for the gauge field generation.

### 2.2 Autocorrelations

In Monte Carlo data the effect of the autocorrelations has to be accounted for in the error analysis. For all observables , functions of expectation values of primary observables , we therefore compute an estimator of the autocorrelation function

(3) |

where following the procedures detailed in Ref. [24]. The argument indicates the Monte Carlo time. The integrated autocorrelation time is then

(4) |

which then enters the statistical error of the observable from measurements

(5) |

The sum in eq. (4) is normally truncated at a “window” [25] which balances the statistical uncertainty due to the limited sample size and the systematic error coming from neglecting the tail for . The value of is determined from the measurement of alone and for each separately. Neglecting the tail above leads — at least on average — to an underestimation of and the statistical error of the observable. It is particularly problematic in the presence of slow modes of the Monte Carlo transition matrix which only couple weakly to the observable in question. To account for them we use the method outlined in LABEL:Schaefer:2010hu, estimating their time constants from observables to which the slow modes couple strongly. Using them, we can then estimate the tails of the autocorrelation functions of the observables we are interested in and arrive at a more conservative error estimate.

Experience tells us that for small lattice spacing the topological charge is particularly sensitive to slow modes[26, 19], for which we use the field theoretical definition after smoothing the field by the Wilson flow integrated up to as defined in LABEL:Luscher:2010iy. Actually, only the square of the charge needs to be considered, because we are only interested in parity even observables. Unfortunately we are not in the position to accurately determine its autocorrelation time for most of our ensembles. We therefore combine the scaling laws found in pure gauge theory[19] with the measurement for our high statistics ensembles E5 and arrive at the estimate

(6) |

in units of molecular dynamics time with for trajectories of length and for and 4. The values of can be found in Table 9 for the DD-HMC algorithm and is equal to one for the MP-HMC.

An example of the procedure is given in Fig. 1, showing the autocorrelation function of the kaon decay constant on the O7 ensemble. Using the standard procedure[24, 25], the sum in eq. (4) is truncated at the window from which we would get . When the contribution of the tail is included, the improved estimate gives , which translates to a more than doubled error estimate.

## 3 Observables

The kaon decay constant necessarily requires the addition of a quenched strange quark to the theory. We denote the hopping parameter of this third flavor as and have for the two sea quarks. For the purpose of a definition of the strange quark mass from the PCAC relation, we do in fact add a fourth quenched flavor with .

The computation of our pseudoscalar observables is based on two-point functions of the pseudoscalar density and the time component of the axialvector current. At a fixed they are constructed from two valence quarks and

(7) | ||||

(8) |

with and . This notation and the analysis that follows is
similar to the one presented in Ref. [28].
Using the PCAC relation, average quark masses
of flavors and can then be defined as^{3}^{3}3This definition
differs by a factor of two from Ref. [28].

(9) |

In this formula, and denote the forward and backward difference operators in time direction. The improvement coefficient has been determined non-perturbatively[29].

For sufficiently large the mass will have a plateau over which we can average. From its value the renormalized quark mass is obtained[30]

(10) |

with . We will use one-loop perturbation theory for the improvement coefficients , , and , noting that they multiply very small terms. At this order in perturbation theory , while and computed from the perturbative coefficients of [31]. An update of the non-perturbative determination of [32] and [6] is given in App. B and App. C, respectively.

The renormalization and improvement of the PCAC quark masses is much simpler than the corresponding expression in terms of the bare subtracted quark masses , where terms proportional to are present already at the leading order in [30]. In our analysis we therefore only use the renormalized PCAC relation eq. (10). The alternative definition of renormalized quark masses as well as the determination of additive renormalization and the multiplicative renormalization factor is discussed in App. E.

### 3.1 Computation of the two-point functions

We compute the two-point functions eq. (8) using noise sources located on randomly chosen time slices [33, 34]. Solving the Dirac equation once for each noise vector is sufficient to get an estimator for the two-point functions projected to zero momentum

(11) | |||||

(12) |

where the average is over noise sources and gauge configurations. For our lattices, we use 10 noise sources per configuration, balancing the numerical cost and the accuracy which we wanted to reach on the given ensembles.

### 3.2 Analysis of the data

The following presentation applies to any flavor combination “” and we drop this sub/superscript for the sake of brevity. The mass of the pseudoscalar meson and its decay constant can be extracted from and the PCAC mass. For infinite time extent , the spectral decomposition gives an expansion in terms of functions which decay exponentially for large time separations

(13) |

with the energy of the ground state and the excited state contribution. For large time separations, we can then extract the decay constant from the leading coefficient

(14) | |||||

(15) |

In the analysis with finite time extent and time separation we have to deal with particles running backwards in time and excited states. Since our lattices are large and the statistical precision of pseudoscalar correlators does not deteriorate at large , we use the following procedure to fix the region , in which we can neglect the excited state contribution: we first perform a fit to the data using the first two terms in the expansion eq. (13), now including the finite effects

(16) |

to a range where this function describes the data well, given the accuracy of the data. We then determine : it is the smallest value of where the statistical uncertainty on the effective mass is four times larger than the contribution of the excited state as given by the result of the fit using eq. (16). In a second step, only the first term of eq. (16) is fitted to the data restricted to this region. Figure 2 illustrates the procedure on our largest lattice. Formally at large and small sea quark mass the leading correction to eq. (13) comes from states which additionally to the ground state have two pions, . However, for small quark masses and large , the coefficient can be computed in chiral perturbation theory. It turns out to be very small [35] (at least for our large volumes) and such a contribution is invisible within our precision. In this sense the value of determined by the fit may actually be a higher state, which is one reason for us to use this excited state fit only in order to determine a safe .

## 4 Scale parameter

The analysis strategy for the scale [8] is based on [36] and we refer to this work for more detailed explanations and notation. The procedure consists of measuring on-axis Wilson loops on smeared gauge configurations, extracting the static potential and finally solving the equation

(17) |

where is the static force. For the latter we use an improved definition which eliminates cut-off effects at tree level. Wilson loops are measured on gauge configurations after all links are replaced by HYP smeared [37] links. We take the HYP2 parameter choice [38] , and and do one HYP-smearing level. The Wilson loops can be exactly represented as an observable in a theory including static quarks and this first smearing step corresponds to the choice of the static quark action (as far as the time-like links are concerned). On the HYP2-smeared gauge link configurations we measure a correlation matrix of Wilson loops for fixed by smearing the space-like links using numbers and of spatial HYP smearing iterations. Spatial HYP smearing means that only staples restricted to spatial directions are used and we therefore need only two parameters, which we set to and . This second smearing step corresponds in the Hamiltonian formalism to the construction of a variational basis of operators that create a state consisting of a static quark and anti-quark pair. We use a basis of the operators, labelled by . The numbers of smearing iterations at each level are listed in Table 2. They are chosen such that the physical extensions of the operators are approximately constant as the lattice spacing changes. Finally we use the generalized eigenvalue method to extract the static potential from the correlation matrix , the details of this can be found in [36].

5.2 | 6 | 9 | 15 |
---|---|---|---|

5.3 | 8 | 12 | 20 |

5.5 | 16 | 24 | 40 |

, | , | |||
---|---|---|---|---|

The solution of eq. (17) is found by interpolation of the force , using a 2-point interpolation . In order to control the systematic error we compare the result with a 3-point interpolation adding a term. We find the systematic error to be negligible. The error analysis takes into account the coupling to the slow modes as explained in Sect. 2.2 and we neglect the systematic error due to excited state contributions to the potential, because we ensure that it is much smaller than the statistical one. The values of for each ensemble are listed in Table 1.

In order to define the lattice spacing in a mass independent way we need to perform a chiral extrapolation of at fixed . The chiral fits are done in the variable

(18) |

Before discussing the chiral fits we study the cut-off effects in the mass dependence of . For this purpose we define a reference value , which corresponds to the value of at the pseudoscalar mass

(19) |

The reference point eq. (19) corresponds to a pseudoscalar mass of MeV. In the left plot of Fig. 3, we plot versus at our three different values. The value of is obtained by linear interpolation and the error analysis of takes into account the correlations between the data. We only consider data with . Using these data we determine the first coefficient in the Taylor expansion of around independently for each value of

(20) |

We do not find significant cut-off effects in the slope [39] and in the continuum limit we obtain the value by fitting to a constant.

Motivated by the results in the left plot of Fig. 3, we perform a global fit for all values simultaneously of the form

(21) |

and with cuts

(22) |

on the pseudoscalar mass. The cuts correspond to , . The fit takes into account both errors on and on . The chirally extrapolated values are listed in Table 3 for various possibilities of the global fit. Columns two and three are linear fits while columns four and five are quadratic fits. We quote as our final numbers for the results in the second column of Table 3 from the linear fit with (which has and ):

(23) |

The data for and the linear fit with (solid lines) are shown in the right plot of Fig. 3. The numbers in eq. (23) cover all the fit results of Table 3 within errors. In particular they are perfectly consistent with the results in the fifth column of Table 3 from a quadratic fit applying the cut (which has , and ). This quadratic fit is represented by dashed lines in the right plot of Fig. 3. The quadratic term is not significant even with .

## 5 Chiral extrapolation of and strange quark mass

In this section we describe our central determination of the scale. For a number of reasons, it is based on the kaon decay constant . First, is experimentally accessible once the CKM-matrix element is considered known, which is a good assumption within the envisaged precision. Second, chiral perturbation theory (ChPT) provides a theory for the quark mass dependence of at small masses of the light quarks, i.e., our sea quarks. Third, as already mentioned, we remain within the pseudoscalar sector of the theory, where ground state properties can be determined without doubt.

### 5.1 Our strategies

The main difficulty and source of a systematic error is the extrapolation to the proper quark masses, the “physical point”. Once we decide to set the scale through , this point is naturally defined by

(24) |

where

(25) |

and are the values of these ratios in Nature. In an attempt to minimize uncertainties, we take the physical masses and decay constants to be the ones in the isospin symmetric limit with QED effects removed as discussed in [42]. We use

(26) |

The two conditions eq. (24) define a point in the plane spanned by , , or equivalently , . We are presently not able to simulate at or very close to this physical point, especially not for the smallest lattice spacing, where huge lattices would be needed to keep finite size effects under control. The physical point has to be approached from unphysically large values of . Along which trajectory in the plane of bare parameters one approaches the physical point is in principle arbitrary. However, one would like the quantity that is to be computed — here — to depend very little on the distance to the physical point, allowing for an easy extrapolation. Secondly, one would like a theoretically motivated extrapolation formula. In ChPT both and depend on the sum of the quark masses at leading order in the systematic expansion in small quark masses. Keeping

(27) |

thus defines a trajectory where varies little in ChPT. We will discuss this quantitatively below. An additional advantage is that all along this trajectory we have , while for a more conventional trajectory, where is kept constant, the mass is significantly heavier than . Since the ChPT expansion is written in terms of and , having no larger then increases our chance of being inside the expansion’s domain of applicability. To our knowledge this strategy has not been used so far, somewhat surprisingly.

As an alternative we use a second strategy, where we keep constant, however instead of using the expansion in and , we use SU(2) ChPT for kaon observables [43, 44], where only is considered a small parameter (in units of the chiral scale ). This strategy provides a suitable definition for future investigations of mesons and baryons with strange quark content.

The trajectories belonging to the two strategies, whose details are the subject of the following sections, are schematically shown in Fig. 4 on the left.

Since the following section deals with the scale setting, we want to distinguish between the kaon decay constant in physical units and in lattice units

(28) |

### 5.2 Strategy 1 and partially quenched SU(3) ChPT

As discussed above, the trajectory is defined by

(29) |

where is considered a function . In practice we take a fixed sea quark hopping parameter and a few values of for which is close to and then interpolate in to find . Also is interpolated to . We give more details on the various interpolations in in Sect. 5.4.

It remains to extrapolate in to the physical point eq. (24). We use the partially quenched chiral perturbation theory results by Sharpe [45], implement our condition eq. (27), which expresses in terms of , and find

(30) | ||||

(31) | ||||

(32) |

The variables are proportional to (averages of) quark masses up to quadratic terms

Because of eq. (27), we have and does not appear in eq. (30).

At this order in the chiral expansion we can also replace

(34) |

with the corresponding replacement , which we will use as a check of the typical size of effects.

In the right panel of Fig. 4 we compare the chiral log function to the one describing the chiral behavior of ,

(35) | ||||

(36) |

Our condition eq. (29) leads to the specific combination of chiral logarithms eq. (32), which has very little curvature and is overall much smaller than ; the suppression of the light quark mass dependence thus extends also to the NLO chiral logarithms. This suggests that the chiral extrapolation is much easier than for and was one of our reasons to select to set the scale. Of course, the counter terms do not contribute in Fig. 4 (right), but as they are linear in introduce no curvature.

### 5.3 Strategy 2 and SU(2) ChPT

Here we extrapolate in the light quark mass at fixed mass of the strange quark, namely
we tune for each sea quark mass the strange quark’s hopping parameter such that the
PCAC mass has a prescribed value , which is independent of
.^{5}^{5}5In principle we should keep fixed,
but the ratio is independent of
since we set , see eq. (10).
This defines the function
. In practice, we again
interpolate the data for in and then solve

(37) |

for , with the left hand side represented by the interpolation formula.

To find the value of corresponding to the physical point, we employ SU(2) ChPT [43, 44] to first extrapolate and (both interpolated in to the point ) in to at fixed value of ,

These expressions represent the asymptotic expansions for small at fixed
correct up to error terms of order .^{6}^{6}6We note that
SU ChPT for kaons does not take into account
kaon loops as opposed to SU(3) ChPT. This corresponds to the production
of two kaons, i.e., states with an energy of around one
GeV. From this point of view it is not really worse than ChPT
in the pion sector where intermediate states are dropped.
We thank Gilberto Colangelo for emphasizing this point.

From eq. (LABEL:e:hm), and are computable for arbitrary values of . The requirement that attains its physical value at the physical light quark mass then defines ,

(39) |

This equation is solved numerically for and the lattice spacing is then given by

(40) |

As before, the constants have common values for all three in the fits to eq. (LABEL:e:hm).

### 5.4 Interpolations

In various places we need observables, such as as a continuous function of , not just for a few numerical values. In all cases we chose four different numerical values for , close to the required one, namely those which have the smallest distance defined by . We then determine an interpolation polynomial through a fit to the data with weights . The regulating term is chosen as We always checked that uncertainties due to the specific interpolation are negligible by considering natural variants.

### 5.5 Cutoff effects

So far the discussion neglects lattice artifacts completely. We will see that these are very small in our formulation. Since both and are small, it is natural to keep terms of order (as done above) and those of order which are independent of but drop terms of order etc. In this approximation we can use eq. (30) for the decay constants in lattice units , and with a global, -independent low energy constant . The straight term does not contribute to eq. (30), since is used to set the scale. Such a term is, however, present in general. An example is the combination . It is shown in Fig. 5 with open symbols, where both and are evaluated at the finite quark mass. Since a ChPT expansion of does not exist, we use

(41) |

to describe the data and to extrapolate to the physical point and continuum limit . Taking into account that we have , the lattice artifact parametrized by is rather small, see Table 4. In these fits we take the chirally extrapolated value in the last term of eq. (41). Replacing it with at the given value of , we get an insignificant shift of in the continuum value, which tests the smallness of terms.

Alternatively, we use the already chirally extrapolated values from Sect. 4 and

(42) |

The dashed dotted lines in Fig. 5 show the corresponding fit with the parameters listed in Table 4.

The just discussed fits mainly illustrate that cut-off effects are small and provide a motivation that indeed terms such as can be dropped. We then determine the values of the kaon decay constant in lattice units at the physical point by a direct application of

(43) |

Here the data at the three different are combined in a global fit with one common value of , but, of course, with different for the three different . The lattice spacings are then obtained from

(44) |

Results are discussed in Sect. 5.6 together with our alternative strategy.

We finish the discussion of cut-off effects by mentioning
an investigation of the PCAC relation. In a properly
improved theory we have
.
We have computed for all our combinations of ,
, and found
and in physical units.^{7}^{7}7The improvement term
proportional to was
inserted non-perturbatively using and from [46].
It is very small in practice.
This confirms again the
smallness of cut-off effects.

### 5.6 Numerical results

We now apply the above formulae to a determination of the kaon decay constant and the strange quark mass. The renormalization of the decay constant, eq. (14), starts from computed in [32] and improved in statistical precision in App. B. For the mass-dependent improvement terms, which yield very small corrections, we use and from one-loop perturbation theory [31]. The extrapolations of through eq. (43), strategy 1, and eq. (LABEL:e:hm), strategy 2, are shown in Fig. 6. In the case of strategy 2, the parameter is fixed to the strange quark mass through eq. (39), with the values of given in Table 1. The fits include data for with . We observe that the two rather different chiral effective theory extrapolations yield results in close agreement at the physical point. Table 5 lists the fit parameters and also includes fits with , simple linear fits in and as well as results including data out to .

fit | fits with | fits with | ||||||
---|---|---|---|---|---|---|---|---|

Strategy 1 | [fm] | [fm] | ||||||

(43) | 0.15 | 5.2 | 5.92(7) | 0.67(6) | 7.53(9) | 5.81(7) | 0.98(7) | 7.40(9) |

5.3 | 5.15(5) | 6.55(7) | 5.05(5) | 6.43(7) | ||||

5.5 | 3.80(3) | 4.84(4) | 3.73(3) | 4.75(4) | ||||

0.1 | 5.2 | 5.93(7) | 0.57(12) | 7.55(9) | 5.87(7) | 0.71(12) | 7.47(10) | |

5.3 | 5.17(6) | 6.58(7) | 5.12(6) | 6.52(7) | ||||

5.5 | 3.82(4) | 4.86(4) | 3.78(4) | 4.82(5) | ||||

linear | 0.1 | 5.2 | 5.99(7) | 1.24(12) | 7.62(9) | 5.93(7) | 1.26(12) | 7.55(9) |

5.3 | 5.21(6) | 6.64(7) | 5.17(6) | 6.58(7) | ||||

5.5 | 3.85(3) | 4.91(4) | 3.82(4) | 4.86(5) | ||||

Strategy 2 | [fm] | [fm] | ||||||

(LABEL:e:hm) | 0.15 | 5.2 | 5.86(7) | 1.30(6) | 7.45(9) | 5.77(7) | 1.42(6) | 7.35(9) |

5.3 | 5.09(5) | 6.48(7) | 5.01(5) | 6.38(7) | ||||

5.5 | 3.76(3) | 4.79(4) | 3.71(3) | 4.72(4) | ||||

0.1 | 5.2 | 5.88(7) | 1.13(8) | 7.49(9) | 5.83(7) | 1.15(8) | 7.42(9) | |

5.3 | 5.11(5) | 6.51(7) | 5.06(6) | 6.44(7) | ||||

5.5 | 3.79(3) | 4.82(4) | 3.75(3) | 4.77(4) | ||||

linear | 0.1 | 5.2 | 5.97(7) | 1.78(8) | 7.61(9) | 5.92(7) | 1.72(8) | 7.54(9) |

5.3 | 5.19(5) | 6.61(7) | 5.15(5) | 6.55(7) | ||||

5.5 | 3.84(3) | 4.89(4) | 3.81(3) | 4.85(4) |

A few observations are worth pointing out. Using as the chiral variable leads to a stronger dependence on the cut . Still, the extrapolation in terms of shows a tendency to converge to the one in terms of when the cut is lowered, as it should be. Results of linear extrapolations are also very close.

We therefore take our central values from the results of strategy 1 with cut . As a systematic error we take into account the deviations to the fit following strategy 2 and to a simple linear extrapolation; for our smallest lattice spacing these different extrapolations are compared more closely in Fig. 7. The final numbers for at the physical point and the associated lattice spacings are shown in Table 6.

[fm] | ||
---|---|---|