Hadron masses in QCD with one quark flavour
Abstract
Oneflavour QCD – a gauge theory with colour gauge group and a fermion in the fundamental representation – is studied by Monte Carlo simulations. The mass spectrum of hadronic bound states is investigated in a volume with extensions of at two different lattice spacings: and . The lattice action is Symanzik treelevelimproved Wilson action for the gauge field and (unimproved) Wilson action for the fermion.
1 Introduction
QCD with one flavour of quarks is an interesting theoretical laboratory to understand some aspects of the strong interaction dynamics, namely those not connected to spontaneous chiral symmetry breaking and to the existence of light pseudoGoldstone bosons. As a consequence of a quantum anomaly, the U(1) axial symmetry of the classical Lagrangian is broken and in the limit of vanishing quark mass no massless Goldstone boson exists.
An intriguing possibility at negative quark masses is the spontaneous breakdown of parity and charge conjugation symmetry – a phenomenon first conjectured by Dashen [1] in the threeflavour theory. This has to do with the possible negative sign of the fermion determinant at negative quark masses because under the assumption of the positiveness of the fermion determinant Vafa and Witten [2] proved the impossibility of this kind of spontaneous symmetry breaking.
A dramatic consequence of the absence of (broken) chiral symmetry is the difficulty to find a unique definition of the point with zero quark mass in parameter space [3]. (For an excellent summary and discussion of this problem see [4].)
Another line of recent theoretical developments is the relation between oneflavour () QCD and supersymmetric YangMills (SYM) theory with one supersymmetry charge () [5]. This connection is the consequence of orientifold planar equivalence in the limit of large number of colours (). This might imply approximate relations among hadron masses even at , for instance, the approximate degeneracy of scalar and pseudoscalar bound states of quarks [6] reflecting the properties of the VenezianoYankielovicz low energy effective action of SYM [7] in the mass spectrum of QCD. For instance, the mass ratio of the lowest pseudoscalar meson to scalar meson is predicted, including corrections, to be [8]. Another prediction of orientifold equivalence is the size of the quark condensate in oneflavour QCD which has recently been compared with numerical simulation results in Ref. [9].
In the present paper we start to explore the mass spectrum of hadronic states in oneflavour QCD by numerical Monte Carlo simulations. This requires reasonably large physical volumes at small quark masses and high statistics – especially for determining glueball masses and contributions of disconnected quark diagrams. We apply the Wilson lattice fermion action which has recently been shown by several collaborations [10, 11, 12, 13] to be well suited for such an investigation. We start our exploratory studies here on and lattices with lattice spacing and , respectively. This means that our present setup roughly corresponds to the earlier simulations of the qq+q Collaboration [10], but we hope to continue these investigations in the near future closer to the continuum limit as in Refs. [11, 12, 13].
For setting the scale we use the Sommer parameter [14] which we set by definition to be . In other words, whenever we speak about “” we always mean “” – having in mind that oneflavour QCD is a theory different from QCD realised in nature.
Since the sign of the quark determinant is a sensitive issue, we carefully determine it and take it into account in determining the expectation values. In the present paper we choose the quark mass to be sufficiently far away from zero on the positive side, where the effect of the determinant sign is not very strong. In spite of this, as we shall see, we can investigate quite small quark masses down to (that is ), corresponding to a pion mass .
Let us mention that keeping the quarks sufficiently heavy (choosing the hopping parameter in the Wilson fermion action (2) below ) the problem of negative quark determinants can be avoided. (The thermodynamics of QCD for heavy quarks have been investigated under this assumption in Ref. [15].) Our aim is, however, to reach small quark masses and therefore we have to deal with the possibly negative sign of the quark determinant.
For interpreting our results on the mass spectrum we find it useful to embed the QCD theory in a partially quenched theory with more quark flavours. This embedding is particularly useful if the additional quenched valence quark flavours have the same mass as the dynamical sea quark because of the exact flavour symmetry in the combined sea and valencesectors ( denotes here the total number of quenched and unquenched flavours). In most cases we consider the natural choice which is closest to the situation realised in nature. We also work out some of the predictions of partially quenched chiral perturbation theory (PQChPT) and compare them to the numerical data.
The plan of this paper is as follows: in the next section we define the lattice action and briefly discuss the updating algorithm. In Section 3 the partially quenched viewpoint is introduced and PQChPT is considered for it. Section 4 is devoted to the presentation of our numerical simulation data. The last section contains a discussion and summary.
2 Lattice action and simulation algorithm
2.1 Lattice action
For the SU(3) YangMills gauge field we apply, following Ref. [13], the treelevel improved Symanzik (tlSym) action which is a generalisation of the Wilson plaquette gauge action. It belongs to a oneparameter family of actions obtained by renormalisation group considerations and in the Symanzik improvement scheme [16]. Those actions also include, besides the usual Wilson loop plaquette term, planar rectangular Wilson loops:
(1) 
with the normalisation condition . For the tlSym action we have [17].
2.2 Simulation algorithm
For preparing the sequences of gauge configurations a Polynomial Hybrid Monte Carlo (PHMC) updating algorithm was used, which is wellsuited for theories with an odd number of fermion species. This algorithm is based on multistep (actually twostep) polynomial approximations of the inverse fermion matrix with stochastic correction in the update chain as described in Ref. [18]. The starting point is the PHMC algorithm as introduced in Ref. [19, 20]. The polynomial approximation scheme and the stochastic correction in the update chain are taken over from the twostep multiboson algorithm of Ref. [21]. For details of the updating algorithm and for notations related to it see Ref. [18].
In order to speed up the updating evenodd preconditioning was used which pushes the small eigenvalues of the (squared Hermitean) fermion matrix to larger values. The eigenvalues of are assumed to be covered on typical gauge configurations by the approximation interval . In exceptional cases some of the eigenvalues (typically just the smallest one) are outside this interval. In order to correct for this a correction factor is associated with such configurations. The exact value of this correction factor can be written as
(4) 
Here the product runs over the eigenvalues of , the polynomial is an approximation for , for . The positive integer defines the determinant breakup which means that in the path integral the fermions are represented by
(5) 
The part of the product in (4) where is inside the interval can be effectively replaced by a stochastic estimator and then
(6) 
Here the runs over the eigenvalues outside the interval , is a sufficiently good approximation of , is the arbitrary number of stochastic estimators and the ’s are Gaussian vectors in the subspace orthogonal to the eigenvectors corresponding to the eigenvalues . In practice, one can choose the polynomial to be such a good approximation that the stochastic part in (6) has no noticeable effect on the expectation values and therefore can completely be neglected. In this case the correction factor is simply given by
(7) 
Besides the correction factor , the sign of the fermion determinant has also to be included in the reweighting of the configurations and then the expectation value of a quantity is given by
(8) 
This formula shows the dangerous sign problem which can arise due to the fluctuation of the determinant sign because in case of strong fluctuations of both nominator and denominator on the right hand side may become small, spoiling the statistical accuracy. (Similarly, one can also loose statistics if the correction factors are much smaller than 1 on many configurations.)
Typical values of the approximation interval and of the polynomial orders at the lightest quark mass simulated on and lattices, respectively, are collected in Table 1. As in Ref. [18], the orders of the polynomials are denoted by and those of by , respectively. The simulations have been done with determinant breakup . (The polynomials are approximating . For more details see [18] and references therein.)
The last four columns of Table 1 show the values of the deviation norm which is minimised for a given polynomial order in the leastsquare approximation scheme we are using. Generically is defined as
(9) 
Here is the function to be approximated and is a positive weight function actually chosen in our case to be and , respectively. The values of in Table 1 are such that the average acceptance rate of the stochastic correction at the end of trajectory sequences is between . The other values are small enough to ensure practically infinite precision of the expectation values. For more details on the algorithmic setup in our runs see also Section 4.
2.6  350  550  1400  1600  

2.4  250  370  1000  1150 
3 Partially quenched viewpoint
Because the classical U(1) axial symmetry is anomalous, the singleflavour QCD theory does not have a continuous chiral symmetry apart from the U(1) quark number symmetry. Consequently it does not have spontaneous chiral symmetry breaking and hence no (pseudo) Goldstone bosons and no easy definition of the quark mass [3]. In the lattice regularisation it is, however, possible to enhance the symmetry artificially by adding extra valence quarks which are quenched, that is, are not taken into account in the Boltzmannweight of the gauge configurations by their fermion determinants. In principle, one might consider any number of quenched valence quarks with any mass values but, to remain close to QCD realised in nature, the most natural choice is to take two equalmass valence quarks and to call them and quarks. The original dynamical quark can then be called quark where “” may stand for sea or strange. The theory with dynamical quark and quenched and quarks is partially quenched. (Observe that this partially quenching is somewhat unconventional, since some of the valence quarks are quenched but taken degenerate with the sea quark.)
Using this terminology, for instance, the pseudoscalar bound state of and can be called . The corresponding scalar state is then . The lowest baryon state consisting of quarks, which has to have spin because of the Pauli principle, can be named or e.g. etc.
A theoretical description of partially quenched QCD can be obtained through the introduction of ghost quarks [24]. For each (quenched) valence quark a corresponding bosonic ghost quark is added to the model. The functional integral over the ghost quark fields then cancels the fermion determinant of the valence quarks and only the sea quark determinant remains in the measure. In our case there are 2 flavours of valence quarks and ghost quarks, each, with equal masses , and a single flavour of sea quarks with mass .
A particularly interesting point of the partially quenched theory is the one where all the three quark masses are equal. In this point there is an exact SU(3) vectorlike flavour symmetry in the valence + sea quark sector, and the hadronic bound states appear in exactly degenerate SU(3)symmetric multiplets. For instance, there is a degenerate octet of pseudoscalar mesons – the “pions” () satisfying an SU(3)symmetric PCAC relation. With the help of the divergence of the axialvector current and pseudoscalar density one can define, as usual, the bare PCAC quark mass in lattice units:
(10) 
Here the indices and refer to the “charged” components corresponding to (with some offdiagonal GellMann matrices) and denotes the backward lattice derivative. Due to the exact SU(3)symmetry, the renormalised quark mass corresponding to can be defined by an SU(3)symmetric multiplicative renormalisation:
(11) 
By tuning the bare quark mass on the lattice suitably, the masses of the “pions” can be made to vanish, as the numerical results indicate, and the renormalised quark mass vanishes, too. At this point the partially quenched theory has a graded symmetry, which is broken spontaneously to a “flavour” . (Here is the number of additional valence quark flavours and .) In our case, with flavours of valence quarks, the symmetry is thus SU(32). The “pions” are the Goldstone bosons of the broken SU(3) subgroup.
Adding generic quark masses and , the symmetry group is explicitly broken down to SU(22). In the special case , considered here, the symmetry is still SU(32), and its subgroup SU(3) is the flavour symmetry mentioned above.
The “pions” are, of course, not physical particles in the spectrum of QCD. Nevertheless, their properties such as masses and decay constants are well defined quantities which can be computed on the lattice. The same is true of the PCAC quark mass , which is therefore a potential candidate for a definition of a quark mass of this theory.
The relation between the pion masses and the quark masses can be considered in partially quenched chiral perturbation theory [25, 26], including effects of the lattice spacing [27, 28, 29, 30, 31]. The pseudoGoldstone fields are parameterized by a graded matrix
(12) 
in the supergroup SU(32). (Here the normalization of is such that its phenomenological value is .) The commuting elements of the graded matrix represent the pseudoGoldstone bosons made from a quark and an antiquark with equal statistics, and the anticommuting elements of represent pseudoGoldstone fermions which are built from one fermionic quark and one bosonic quark. The supertrace of has to vanish, which can be implemented by a suitable choice of generators [32].
We have calculated the masses of pseudoGoldstone bosons in nexttoleading order of partially quenched chiral perturbation theory along the lines of Ref. [32], including O() lattice effects [29]. The quark masses enter the expressions in the combinations
(13) 
with the usual lowenergy constant , and the lattice spacing occurs as
(14) 
where is another, latticespecific, lowenergy constant. For the pion masses we obtain
(15)  
where the usual lowenergy parameters appear, together with addtional ones () describing lattice artifacts.
The mixed mesons, whose masses we have also calculated, become degenerate with the pions in the special case . In this case the expression reduces to
(16)  
To leading order the PCAC quark mass obeys , and we recognize the GellMannOakesRenner relation
(17) 
Including terms in nexttoleading (NLO) order, we can express in terms of as
(18)  
where we define
(19) 
As a remark, in the case the masses can alternatively be obtained from the partially quenched theory with symmetry SU(21) by considering mixed pions made from a valence quark and a degenerate sea quark. Indeed, calculating the masses in this model reproduces (16).
The can be included in the analysis by relaxing the constraint of a vanishing supertrace [25, 32], and associating it with the field
(20) 
The effective Lagrangian then contains additional terms depending on :
(21) 
where and are free parameters in this context. We content ourselves with displaying only the leading order expression for the mass of the , which reads
(22) 
Our numerical results for allow to determine and .
4 Numerical simulations
After some preparatory search in the parameter space we concentrated our runs on the lattice to and those on to . The parameter values, the number of analysed configurations, the average plaquette, its integrated autocorrelation and the value of the Sommer scale parameter in lattice units are summarised in Table 2. As one can see, taking the values of at highest ’s (smallest quark masses), the extensions of the and lattices are and , respectively. Since we fix by definition, these correspond to lattice spacings and , respectively.
In the updatechain by the PHMC algorithm with stochastic correction [18] a sequence of PHMC trajectories is followed by a Metropolis acceptreject step with a higher precision polynomial. The total length of the trajectory sequence in the runs in Table 2 was between 1.5 and 1.8. The sequences consisted out of 36 individual trajectories. The precision of the first step of polynomial approximations was tuned such that the acceptance of the PHMC trajectories was about 0.800.85. The total length of the trajectory sequence was chosen such that the acceptance of the Metropolis test was again 0.800.85. This ensured a relatively high total acceptance of 0.640.72. During the runs we tried to optimise the parameters of PHMC. The different values of the integrated autocorrelation times for the average plaquette in Table 2 are, in fact, mainly due to increasingly better optimisations and not so much to the dependence on run parameters.
The second step approximations were more than good enough to ensure that the expectation values were completely unaffected by the remaining small imprecision. (See, for instance, the small relative deviations in Table 1.) This has also been explicitly checked by performing a final stochastic correction on a large sample of configurations with polynomials of order 2500 in the stochastic part of the right hand side of (6).
For the calculation of the expectation values the reweighting procedure according to (8) has to be carried out. For this, besides the correction factor from (7), also the sign of the fermion determinant is needed. This we calculated by the spectral flow method [22]. For the dependent computation of the lowlying eigenvalues of the hermitean fermion matrix we followed Ref. [23].
It turned out that the effect of the correction factors is in most cases negligible. For instance, in run of Table 2 the average value of in the denominator is 0.9982. In run it is 0.9842. In run there are 34 configurations out of 3403 where some eigenvalue is outside the approximation interval and out of them there is a single one with negative fermion determinant. In run there are 167 from 2884 outside and out of them there are 26 with negative correction factor due to .
Since the sign of the fermion determinant was not determined on every configuration, the question arises whether perhaps some negative signs were missed. This is very improbable because we determined the sign also on the neighbouring configurations in addition to those with small eigenvalues and out of the remaining configurations we have chosen 100 randomly for sign determination. None of these additional configurations turned out to have a negative determinant.
In the average plaquette and the effect of the correction factors is completely negligible. For instance, in runs and the correction has an effect in the average value of only in the fifth digit – whereas the statistical error is in the third digit. In all other runs besides and every eigenvalue is inside the approximation interval and therefore, according to (7), the correction factor is equal to 1 on every configuration.
label  plaquette  

3.80  0.1700  5424  0.546041(66)  12.5  2.66(4)  
3.80  0.1705  3403  0.546881(46)  4.6  2.67(5)  
3.80  0.1710  2884  0.547840(67)  7.6  2.69(5)  
4.00  0.1600  1201  0.581427(36)  4.3  3.56(5)  
4.00  0.1610  1035  0.582273(36)  4.1  3.61(5)  
4.00  0.1615  1005  0.582781(32)  3.3  3.73(5) 
4.1 Results for hadron masses
run  

0.462(13)  0.660(39)  0.777(11)  1.215(20)  
0.403(11)  0.629(29)  0.685(10)  1.116(38)  
0.398(28)  0.584(55)  0.842(16)  1.204(57)  
0.455(17)  0.607(57)  1.083(79)  1.006(15)  
0.380(18)  0.554(52)  1.032(66)  0.960(15)  
0.316(22)  0.613(67)  0.980(97)  0.876(26) 
Starting with the mesonic states, we consider the simplest interpolating operators in the pseudoscalar and scalar sectors:
(23)  
(24) 
We denote with and the corresponding hadron states at the lowest end of the energy spectrum (the usual notation is used for the respective quantum numbers). Corresponding states in the QCD spectrum with the same quantum numbers are the and (or ). (Note, however, that the states in QCD are linear combinations of , and components – in contrast to the states in QCD which are built out of a single quark flavour.)
In the case of the pseudoscalar mesons, invariance under the flavour group plays a special role when comparing with QCD states because of the U(1) axial anomaly. (This is not the case for baryons, see the following.)
Analogously to flavour singlet mesons in QCD, the correlators of the above interpolating operators contain disconnected diagrams. These were computed by applying stochastic estimator techniques (SET), and in particular the variant of [33] with noise and spin dilution. The method was already applied to the case of SYM [34] (as mentioned in the introduction, SYM shares many similarities with QCD). In order to optimize the computational load, taking also autocorrelations into account, every fifth configuration was typically analysed, with 20 stochastic estimates each.
Spin 0 states can be also build by purely gluonic operators. These are a well known object of investigation in lattice QCD were they should describe the glueballs. Due to the expected signalnoise ratio of their purely gluonic correlation they belong to the most notorious particles to measure. In particular the glueball has the same quantum numbers as the meson. As a consequence, these two states can also mix with each other but in this first investigation we neglect the mixing and consider only diagonal correlators for both states.
We used the single spatial plaquette to obtain the mass of the ground state. To increase the overlap of the operator with this state we used APE smearing and also performed variational methods to obtain optimal glueball operators from linear combinations of the basic operators.
We now come to the baryon sector. The simplest baryonic interpolating field which can be built out of one quark flavour is
(25) 
The above operator also contains a spin 1/2 component implying that the spin 3/2 component, on which we focus, must be projected out from the spinorial correlator
(26) 
We follow [35] and consider the spinprojected correlator
(27) 
The low lying hadron state contributing to the above correlator is expected to have positive parity . This corresponds to the of QCD if our dynamical fermion is interpreted as an quark. If the dynamical fermion is taken to be the quark then this would be the baryon. (However, spin and parity of the corresponding particle have not been yet measured, so the identification of this state with the baryon is still uncertain [36]). In correspondence to and , in what follows we call this state . (Here one can interpret the index as referring to the “sea” quark.)
It should be noted at this point that the above QCD states are not flavour singlets in QCD (and in the one flavor partially quenched theory). We recall here that interpolating fields corresponding to flavour singlet baryon states cannot be build in QCD if only quark fields are considered as ingredients.
The results of the hadron masses are reported in Table 3 and, as a function of the bare PCAC quark mass , in Fig. 1. In the figure the masses are multiplied by the Sommer scale parameter , therefore one can put the results for both lattice spacings in a single plot and check their scaling. (The expected small change of the multiplicative renormalisation factor of between and is neglected here.)
Only in the case of run the measurement correction has a sizeable effect on the mass estimates. In this case configurations with negative determinant where singled out: the sign of the determinant has the effect of pushing the masses up by %.
The errors on the glueball mass are rather large – especially on the lattice at – therefore they are not shown in the figure. Obviously, our statistics is not sufficient for this purpose. In general a larger number of configurations would improve the determinations in the glueball sector. Since the computational load is in this case negligible, for future runs we plan a more frequent storage of the gauge configuration.
Valence analysis
The connected contribution to the meson correlators can be interpreted as a non singlet meson made up of valence quarks in the partially quenched picture, see Sec. 3. The pseudoscalar channel corresponds in particular to the “valence” pion. Since the computation of the connected diagrams is less demanding, we could afford the analysis of the complete set of configurations.
In the baryon sector, one can define a “valence” nucleon, with the usual projector operator
(28) 
where can be interpreted as the field of the valence quark.
The results concerning valence hadron masses are reported in Table 4 and Fig. 2. In addition, the bare PCAC quark mass according to the definition in (10) and the bare pion decay constant in lattice units are also included. and its renormalised counterpart are defined as
(29) 
where is the axialvector current as in (10) and is a pion state with zero momentum. (The normalisation of is such that in nature we have .) The value of on the lattice is obtained by the method described in [37]. In Fig. 2 the masses are multiplied by the scale parameter in order to obtain dimensionless variables.
run  

0.02771(45)  0.3908(24)  0.1838(11)  1.0439(54)  
0.01951(39)  0.3292(25)  0.1730(15)  0.956(27)  
0.0108(12)  0.253(10)  0.156(10)  1.011(51)  
0.04290(36)  0.4132(21)  0.1449(9)  0.9018(44)  
0.02561(31)  0.3199(22)  0.1289(10)  0.7978(53)  
0.01700(30)  0.2635(24)  0.1188(12)  0.734(10) 
Chiral Perturbation Theory fits
The properties of the valence pion (pion mass and decay constant ) can be analysed in partially quenched ChPT. We fit and simultaneously as a function of including the data at both values of . There are not enough data in order to account for the lattice artifacts. Therefore the fit is done with the continuum formulae
(30) 
with the lowenergy constants
(31) 
The changes of the renormalisation constants , between the two values are neglected. The results are displayed in Figs. 3 and 4.
Owing to the fact that the number of degrees of freedom in the fit is small, the uncertainty of the fit parameters is relatively large. The determination of the universal lowenergy scales and can be improved by considering the ratios [40, 10]
(32) 
in which some of the coefficients cancel. We consider the data on the larger lattice at and take the quantities at as reference. The fit yields
(33)  
(34) 
which is compatible with the phenomenological values from ordinary QCD [38].
In order to estimate the parameters and , related to the mass of the (see Sec. 3), we made a fit of and at in leadingorder ChPT. The result is
(35) 
indicating the vanishing of . Fixing in the fit yields
(36) 
where the value of extrapolated to vanishing PCAC quark mass is used.
5 Discussion
This first Monte Carlo investigation of the hadron masses in QCD
with dynamical quark flavour reveals the qualitative features
of the low lying particle spectrum in this theory.
The spatial extensions of our and
lattices are about (see
Table 2).
The lightest hadron is the pseudoscalar meson bound state of a quark and an antiquark – the meson (see Table 3 and Figure 1). The corresponding scalar bound state – the meson – is in our points by about a factor 1.5 heavier. Compared to the estimate in [8] this result is too low but the situation could be better in the zero quark mass limit where the prediction of [8] applies to. The lightest baryon – the baryon – is by a factor of about 3 heavier than the meson. The lightest glueball lies between the meson and the baryon, but its mass could not be properly measured on the lattice with our statistics. In general, the mass measurements have relatively large errors – between 310% – and no infinite volume and continuum limit extrapolations could be performed with our present data. We hope to return to these questions and to give more precise results in future publications.
An interesting aspect of QCD is the possibility of a partially quenched extension with valence quarks. In particular, adding two valence quarks, the model has similarities to QCD in nature with its three light (, and ) quark flavours. A theoretically interesting special case is if all three quarks, the dynamical one and the two valence ones, have exactly equal masses. In this case there is an exact flavour symmetry. This can be exploited for the introduction of a quark mass by defining it as the PCAC quark mass in the partially quenched theory. In this extended model there exist the usual light hadron states well known from real QCD: the pseudoscalar pseudoGoldstone bosons (pions etc.), the nucleon etc. The results for the masses of the lightest states and the decay constant of the pseudoscalar bosons are collected in Table 4 and also shown in Figure 2.
Since the physical volumes of the and lattices are to a good approximation equal, the comparison of the results at the two different lattice spacings gives a hint for the magnitude of the deviations from the continuum limit. As one can see in Figs. 1 and 2, the scaling between and is reasonably good – especially for the lightest states and . However, for reliable continuum limit estimates more data at several lattice spacings are required.
In the pseudoscalar sector of the partially quenched model one can apply partially quenched Chiral Perturbation Theory for fitting the mass and the decay constant. As Figs. 3 and 4 show, the NLO formulae give good fits but the number of degrees of freedom in the fits is small and therefore the uncertainty of the fit parameters is relatively large.
Acknowledgments
We are grateful to Luigi Scorzato for valuable discussions and for helping us in the setup of the programs for investigating the eigenvalue spectrum of the fermion matrix. We thank the computer centers at DESY Hamburg and NIC at Forschungszentrum Jülich for providing us the necessary technical help and computer resources. This work is supported in part by the Deutsche Forschungsgemeinschaft under grant Mu757/131. E.S. is supported by the U.S. Dept. of Energy under contract DEAC0298CH10886.
Footnotes
 In order to have some relation to the scales in real QCD, we set the Sommer scale parameter by definition to be
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