Topological susceptibility from the twisted mass Dirac operator spectrum
aa r X i v : . [ h e p - l a t ] M a r Prepared for submission to JHEP
DESY 13-208HU-EP-13/64SFB-CPP-13-114
Topological susceptibility from the twisted massDirac operator spectrum
Krzysztof Cichy, a,b
Elena Garcia-Ramos, a,c
Karl Jansen a,d a NIC, DESY, Platanenallee 6, 15738 Zeuthen, Germany b Adam Mickiewicz University, Faculty of Physics, Umultowska 85, 61-614 Poznan, Poland c Humboldt Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany d Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We present results of our computation of the topological susceptibility with N f = 2 and N f = 2 + 1 + 1 flavours of maximally twisted mass fermions, using the methodof spectral projectors. We perform a detailed study of the quark mass dependence anddiscretization effects. We make an attempt to confront our data with chiral perturbationtheory and extract the chiral condensate from the quark mass dependence of the topologicalsusceptibility. We compare the value with the results of our direct computation from theslope of the mode number. We emphasize the role of autocorrelations and the necessity oflong Monte Carlo runs to obtain results with good precision. We also show our results forthe spectral projector computation of the ratio of renormalization constants Z P /Z S . ontents Z P /Z S N f = 2 N f = 2 + 1 + 1 N f = 2 results 115.4 N f = 2 + 1 + 1 results 13 The topological susceptibility in gauge theories, e.g. in QCD, expresses the fluctuationsof the topological charge. As such, it describes non-trivial topological properties of theunderlying gauge field configurations. Such properties have far-reaching phenomenologicalconsequences, in particular topological effects are to a large extent responsible for the massof the flavour-singlet pseudoscalar η ′ meson, making it distinct from the octet of pions, kaonsand η . The relation between the topological susceptibility and the η ′ mass is expressed inthe Witten-Veneziano formula [1, 2].There exist many definitions of the topological charge on the lattice and there hasbeen a debate in the literature about the validity of different approaches. One of the mainproblems is the appearance of non-integrable short distance singularities in some definitions,which require regularization.To avoid such theoretical problems, a possible solution is to use the definition of thetopological charge as the index of the overlap Dirac operator [4], which is by constructioninteger-valued. However, this is very demanding in terms of computing time and henceimpractical when large lattice sizes are used. Using Ginsparg-Wilson fermions, it is also For a short review and discussion of different definitions and for further references, we refer to Ref. [3]. – 1 –ossible to derive an expression for the topological susceptibility which does not have anypower divergences [5, 6]. This has been further generalized by Lüscher, leading to a def-inition employing the so-called density chain correlation functions [7]. The latter can beevaluated efficiently using the method of spectral projectors [8]. This definition of the topo-logical susceptibility was subject to numerical analysis in the quenched case [9] and it isthe aim of the present paper to analyze the results of its usage in the case with N f = 2 and N f = 2 + 1 + 1 active flavours of twisted mass fermions.The outline of the paper is as follows. In section 2, we describe the theoretical principlesof the adopted approach. Section 3 presents our lattice setup. In section 4, we show ourresults for the renormalization constants ratio Z P /Z S and in section 5 for the topologicalsusceptibility. We conclude in section 6. In an appendix, we show our tests concerning thenumber of stochastic sources. The method that we follow in this paper was introduced in Refs. [8, 9] and we refer tothese papers for a comprehensive description. Here, we summarize only the main points torender the paper self-contained.Let us define an orthogonal projector P M to the subspace of fermion fields spanned bythe lowest lying eigenmodes of the operator D † D with eigenvalues below some thresholdvalue M . In practice, if the projector P M is approximated by a rational function of D † D ,denoted by R M (see Refs. [8, 9] for the details of this approximation), the following equationfor the topological susceptibility χ holds: χ = h Tr { R M }ih Tr { γ R M γ R M }i h Tr { γ R M } Tr { γ R M }i V . (2.1)The calculation of the topological susceptibility from this expression requires an evaluationof three gauge field ensemble averages. However, if the value of the scheme- and scale-independent ratio Z P /Z S is available from another computation, the above expression canbe rewritten as: χ = Z S Z P h Tr { γ R M } Tr { γ R M }i V , (2.2)where the numerator can be expressed using two stochastic observables defined in Ref. [9]: χ = Z S Z P hC i − hBi N V , (2.3)where N is the number of randomly generated pseudofermion fields η i added to the theory and C = 1 N N X k =1 ( R M η k , γ R M η k ) , (2.4) We use the Z(2) random noise, i.e. ( η i ) r = ( ± ± i ) / √ , where r spans the set of source degrees offreedom (space-time, colour, spin) and all signs ± are chosen randomly. – 2 – = 1 N N X k =1 ( R M γ R M η k , R M γ R M η k ) . (2.5)The term hBi /N is a correction to the result given by hC i needed if the number of stochasticsources N is finite and if one computes: C ≡ C { η k } · C { η l } ≡ N N X k =1 ( R M η k , γ R M η k ) 1 N N X l =1 ( R M η l , γ R M η l ) (2.6)using the same stochastic sources for C { η k } and C { η l } . In chiral symmetry preserving for-mulations of Lattice QCD (e.g. using overlap fermions), the observable C is just the index Q of the Dirac operator, i.e. the difference in the number of zero modes with positive andnegative chirality, since ( η, γ η ) = ± if η is a zero mode and 0 otherwise. Moreover, insuch theories Z P = Z S and in the limit N → ∞ Eq. (2.3) becomes just the well-knownformula χ = h Q i /V . The distribution of Q is expected to be of the Gaussian type (with h Q i = 0 ) and the topological susceptibility is then alternatively given by the width of thisdistribution. In theories where chiral symmetry is explicitly broken at finite lattice spacing,e.g. for Wilson fermions, the observable C is in general non-integer and counts the numberof zero modes only approximately (up to cut-off effects). However, as we will show, C is stillcompatible with a Gaussian-shaped distribution and the renormalized C ren ≡ Z S Z P C can bethought of as a proxy for the topological charge. As it is well known, the topological chargeis an observable which is particularly susceptible to autocorrelations in Monte Carlo (MC)time [10]. Hence, to obtain reliable estimates of the topological susceptibility, it is essentialthat MC histories are long enough, such that all topological sectors are correctly probed.Since the observable C is strongly related to the topological charge, its autocorrelation timeand the quality of its distribution provides a criterion of MC history being “long enough”.In particular, we demand the distribution of C to be compatible with a Gaussian and hCi should be compatible with zero.We have mentioned above that the full renormalized topological susceptibility can beobtained from expression (2.1). This means that the ratio of renormalization constants Z P /Z S can be calculated with spectral projectors, as first noticed in Ref. [8]. The formulareads: Z P Z S = s hBihAi , (2.7)where B is given by Eq. (2.5) and A is: A = 1 N N X k =1 (cid:0) R M η k , R M η k (cid:1) , (2.8)i.e. it is the mode number ν ( M ) – the number of eigenmodes of the operator D † D witheigenvalues below the threshold value M . Our computations were performed using dynamical twisted mass configurations gener-ated by the European Twisted Mass Collaboration (ETMC), with N f = 2 [11–13] or– 3 – f = 2 + 1 + 1 [14–16] dynamical quark flavours. In the gauge sector, the action is: S G [ U ] = β X x (cid:16) b X µ,ν =11 ≤ µ<ν Re Tr (cid:0) − P × x ; µ,ν (cid:1) + b X µ,ν =1 µ = ν Re Tr (cid:0) − P × x ; µ,ν (cid:1)(cid:17) , (3.1)with β = 6 /g , g the bare coupling and P × , P × are the plaquette and rectangularWilson loops, respectively. For the N f = 2 case, the tree-level Symanzik improved action[17] was used, i.e. b = − (with the normalization condition b = 1 − b ), while in the N f = 2 + 1 + 1 case, the Iwasaki action [18, 19] was employed, i.e. b = − . .The Wilson twisted mass fermion action for the light, up and down quarks for boththe N f = 2 and N f = 2 + 1 + 1 cases, is given in the twisted basis by: [20–23] S l [ ψ, ¯ ψ, U ] = a X x ¯ χ l ( x ) (cid:0) D W + m + iµ l γ τ (cid:1) χ l ( x ) , (3.2)where τ acts in flavour space and χ l = ( χ u , χ d ) is a two-component vector in flavourspace, related to the one in the physical basis by a chiral rotation. m and µ l are the bareuntwisted and twisted light quark masses, respectively. The renormalized light quark massis µ R = Z − P µ l . The standard massless Wilson-Dirac operator D W reads: D W = 12 (cid:0) γ µ ( ∇ µ + ∇ ∗ µ ) − a ∇ ∗ µ ∇ µ (cid:1) , (3.3)where ∇ µ and ∇ ∗ µ are the forward and backward covariant derivatives.The twisted mass action for the heavy doublet is: [22, 24] S h [ ψ, ¯ ψ, U ] = a X x ¯ χ h ( x ) (cid:0) D W + m + iµ σ γ τ + µ δ τ (cid:1) χ h ( x ) , (3.4)where µ σ is the bare twisted mass with the twist along the τ direction and µ δ the mass split-ting along the τ direction that makes the strange and charm quark masses non-degenerate.The physical renormalized strange m sR and charm m cR quark masses are related to the bareparameters µ σ and µ δ via m sR = Z − P ( µ σ − ( Z P /Z S ) µ δ ) and m cR = Z − P ( µ σ + ( Z P /Z S ) µ δ ) .The heavy quark doublet in the twisted basis χ h = ( χ c , χ s ) is related to the one in thephysical basis by a chiral rotation.The twisted mass formulation yields an automatic O ( a ) improvement of R -parity-even quantities if the twist angle is set to π/ (maximal twist). This is achieved by non-perturbative tuning of the hopping parameter κ = (8 + 2 am ) − to its critical value, atwhich the PCAC quark mass vanishes [21, 25–29].The details of the gauge field ensembles considered for this work are presented in Tab. 1for N f = 2 and Tab. 2 for N f = 2 + 1 + 1 . They include lattice spacings from a ≈ . fm to a ≈ . fm and up to 5 quark masses at a given lattice spacing. The renormalizedlight quark masses µ R are in the range from around 15 to 50 MeV. The values of therenormalization constant Z P for different ensembles [33, 34, 36], used to convert bare light For N f = 2 + 1 + 1 , the mass-independent renormalization constant Z P is extracted as a chiral limit ofa dedicated computation with 4 mass-degenerate flavours – see Refs. [37, 38] for details. – 4 –nsemble β lattice aµ l µ R [MeV] κ c L [fm] m π L b . × . × . × . × . × . × . × . × . × Table 1 . Parameters of the N f = 2 gauge field ensembles [11–13]. We show the inverse barecoupling β , lattice size ( L/a ) × ( T /a ) , bare twisted light quark mass aµ l , renormalized quark mass µ R in MeV, critical value of the hopping parameter at which the PCAC mass vanishes and physicalextent of the lattice L in fm and the product m π L . Ensemble β lattice aµ l µ l,R [MeV] κ c L [fm] m π L A30.32 1.90 × × × × × × × × × × × × × × × Table 2 . Parameters of the N f = 2 + 1 + 1 gauge field ensembles [14–16]. We show the inverse barecoupling β , lattice size ( L/a ) × ( T /a ) , bare twisted light quark mass µ l , renormalized quark mass µ l,R in MeV, critical value of the hopping parameter at which the PCAC mass vanishes, physicalextent of the lattice L in fm and the product m π L . quark masses µ l and bare spectral threshold parameters M to their renormalized values inthe MS scheme (at the scale of 2 GeV), are given in Tab. 3. There we also show the valuesof r /a (in the chiral limit), used to express our results for the topological susceptibility asa dimensionless product r χ . Our physical lattice extents L for extracting physical resultsrange from 2 fm to 3 fm (in the temporal direction, we always have T = 2 L ). To check forthe size of finite volume effects, we included different lattice sizes for β = 3 . , aµ l = 0 . – 5 – f β a [fm] Z P ( MS , Z P /Z S r /a Table 3 . The approximate values of the lattice spacing a [16, 30, 31], r /a [14, 30–32], the scheme-and scale-independent renormalization constants ratio Z P /Z S and the renormalization constant Z P in the MS scheme at the scale of 2 GeV [33–37], for different values of β and N f = 2 and N f = 2 + 1 + 1 flavours. ( N f = 2 ) and β = 1 . , aµ l = 0 . ( N f = 2 + 1 + 1 ). Z P /Z S We first present our results for the renormalization constants ratio Z P /Z S , which is a scale-and scheme-independent quantity. Nevertheless, in order to avoid problems with e.g. cut-off effects or dependence on the threshold parameter M R , it is necessary to determine awindow Λ ≪ M R ≪ a − for the computation of Z P /Z S , with Λ of O (Λ QCD ) . N f = 2 We perform our N f = 2 analysis using small volume ensembles (b40.16, c30.20, d20.24 ande17.32) at a fixed pion mass of around 300 MeV (in infinite volume). In this way, we cankeep the computational cost rather low and at the same time investigate a wide range ofvalues of M R to control the systematic effects of varying M R . For all these ensembles, wecan compare the values of Z P /Z S with an alternative computation – in the framework of theRI-MOM renormalization scheme ( β = 3 . , . , . [34]) or the X-space renormalizationscheme ( β = 4 . [35]).The dependence of Z P /Z S on M R is shown in Fig. 1. For small values of M R , we observea significant dependence of Z P /Z S on the threshold parameter M R . For larger values of M R , the dependence of Z P /Z S on M R flattens and we observe a tendency to approach aplateau. This signals that we obtain the above discussed lattice window, where we canextract the scale- and scheme-independent value. However, the lattice data shows that thestrong dependence of Z P /Z S on M R below about 1 GeV is related to the value of the latticespacing a involved. In addition, also the size of variation of Z P /Z S for GeV < M R < GeVis getting smaller for decreasing values of the lattice spacing. For β = 3 . , the change in Z P /Z S when going from around 1 to 2 GeV is approx. 6%, while for finer lattice spacingsthis change decreases to approx. 3%, 2% and below 1% (for β = 4 . , β = 4 . and β = 4 . ,respectively). A reassuring observation is, however, that the plateau value is consistent withvalues from the RI-MOM or X-space renormalization schemes, shown in Fig. 1 too. We– 6 – = 4 . β =4.2 β =4.05 β = 3 . methodalternative M R [MeV] Z P / Z S Figure 1 . Dependence of the renormalization constants ratio Z P /Z S on the renormalized threshold M R . The data points correspond to the computation from spectral projectors. The horizontal bandsare our estimates of the scale-independent values of Z P /Z S that correspond to the value at M R = 1 . GeV (solid lines) and the spread of results between M R = 1 GeV and 2 GeV as our estimate ofthe systematic error (bands). The values on the right of the vertical line correspond to RI-MOMresults at β = 3 . , . , . [34] and the X-space result at β = 4 . [35]. remark also that at large values of M R , finite volume effects are small. We have explicitlychecked that the gauge ensemble averages of the observables A and B are always compatiblewith each other for ensembles b40.16 and b40.24, provided that M R & GeV. Moreover, theratio of these two quantities, which gives Z P /Z S , is compatible between the two ensemblesat all values of M R that we investigated – see Fig. 2. Therefore, the conclusions from oursmall volume results are valid in general.To summarize, the method of spectral projectors allows in principle a computationof the ratio of Z P /Z S . However, to obtain the universal scale- and scheme-independentvalue, the calculation of the observables A and B (with N = 1 stochastic source) has to beperformed at a rather large number of threshold parameters M R to be able to explore thesignificant M R -dependence we observe. To account for this M R -dependence, we followedthe strategy to take the central value of Z P /Z S at some fixed (in physical units) value of M R , e.g. 1.5 GeV (sufficiently far away from the low-energy scales and sufficiently belowthe inverse lattice spacing for typical parameters of contemporary simulations), and assigna systematic error related to the difference of Z P /Z S across a range of scales (e.g. M R between 1 and 2 GeV). If we follow this strategy, we obtain the results of Tab. 4 and the– 7 – /a=24L/a=16 M R [MeV] Z P / Z S Figure 2 . Dependence of the renormalization constants ratio Z P /Z S on the renormalized threshold M R for N f = 2 , β = 3 . , aµ l = 0 . and two linear lattice extents: L/a = 16 and
L/a = 24 .Within errors, all values of Z P /Z S are compatible between the two ensembles. horizontal bands in Fig. 1. The first given error of the spectral projectors result is statisticaland the second one comes from the residual M R -dependence of Z P /Z S . Note that only theRI-MOM results, given for comparison in Tab. 4, were chirally extrapolated. However,non-zero quark mass corrections to the chiral limit value were found to be small in oursetup, both in the RI-MOM scheme and in the X-space scheme [33–35]. In particular, thevalues of Z P /Z S in the chiral limit and at the pion mass of 300 MeV are always compatiblein both of these schemes. Given our experience that Z P /Z S has a very mild quark massdependence [33–35], we consider the values obtained here at a fixed but small pion mass tobe an appropriate estimate. The overall agreement between the spectral projector methodand other renormalization schemes is certainly reassuring. However, it would be very goodto understand the M R -dependence of Z P /Z S better and to disentangle effects that lead toit. For example, a lattice perturbative calculation within the framework used would be veryhelpful to learn about the role of cut-off effects. N f = 2 + 1 + 1 We repeated the computation of the M R -dependence of Z P /Z S also for one chosen ensemblewith N f = 2 + 1 + 1 flavours (B55.32). The chiral limit value from RI-MOM is 0.697(7)[36]. The residual M R -dependence originating from spectral projectors is rather large inthis case and the prescription from the previous subsection leads to the value 0.637(1)(21).The systematic error is comparable to the one for β = 3 . with N f = 2 , which corresponds– 8 – Z P /Z S (spec.proj.) Z P /Z S (RI-MOM) Z P /Z S (X-space)3.9 0.635(1)(23) 0.639(3) 0.609(6)4.05 0.679(2)(12) 0.682(2) 0.671(9)4.2 0.717(2)(5) 0.713(3) 0.707(14)4.35 0.749(2)(2) – 0.740(3) Table 4 . The values of the scale- and scheme-independent ratio Z P /Z S for N f = 2 ensembles,extracted from spectral projectors (the first error given is statistical and the second one systematicfrom varying the threshold value M R ), as compared to RI-MOM [34] and X-space results [35]. AllRI-MOM results and the X-space results at β = 3 . and β = 4 . were chirally extrapolated. -10-8-6-4-2 0 2 4 6 8 10 5000 10000 15000 20000 C HMC trajectory number -10-8-6-4-2 0 2 4 6 8 10 0 200 400 600 800 1000 1200 1400 1600 C HMC trajectory number
Figure 3 . Monte Carlo history of the observable C for ensemble B55.32 (left) and B75.32 (right). to a similar lattice spacing. Although there is some tension between the spectral projectorresult and RI-MOM, the observed difference between the two results is still plausible, givena finite and rather large value of the lattice spacing. Note, however, that in the following wedo not rely on the values of Z P /Z S from spectral projectors – we rather use the RI-MOMvalues to evaluate the topological susceptibility. In this section, we discuss our results for the topological susceptibility. We first show thedetails of our analysis for two of the 2+1+1-flavour ensembles – B55.32 and B75.32. Then,we investigate finite volume effects and finally we present results for the cases with N f = 2 and N f = 2+1+1 flavours of twisted mass fermions and perform chiral perturbation theoryfits to the quark mass dependence of the topological susceptibility. We start with the ensemble B55.32, see Tab. 6, for which we performed measurements on538 independent gauge field configurations separated by 40 MC trajectories, using N = 6 stochastic sources for each configuration. For a discussion about the optimal number ofstochastic sources per configuration, we refer to Appendix A.– 9 – nu m be r o f c on f s C 0 10 20 30 40 50 60-15 -10 -5 0 5 10 15 nu m be r o f c on f s C Figure 4 . Histogram of the observable C for ensemble B55.32 (left) and B75.32 (right). The errorfor each box comes from a bootstrap analysis with blocking. The solid line is a Gaussian fit to thehistogram. The MC history of the observable C (whose fluctuations determine the topologicalsusceptibility) is shown in the left panel of Fig. 3. We observe that different topologicalsectors are sampled and the magnitude of fluctuations seems to be rather uniform fordifferent regions of MC time. As we have stated above, the sampling is correct if thehistogram of C is close to Gaussian and if the ensemble average hCi = 0 . The histogramof the observable C for the ensemble B55.32 is shown in Fig. 4 (left). It is almost ideallysymmetric and it is almost perfectly Gaussian. We have therefore fitted the followingGaussian ansatz: f ( C ) = N exp( − ( C − hCi ) / σ ) , (5.1)where N is a normalization constant and σ is related to the topological susceptibility: χ = ( Z S /Z P ) ( σ − hBi /N ) , i.e. σ = hC i . The 3 fitting parameters are then: N , hCi and σ . There is very good agreement between hCi extracted from the histogram and computeddirectly by averaging – the former yields 0.02(20) and the latter -0.06(16), which impliesthat both the negative and positive topological charge sectors are sampled equally often.The bare topological susceptibility extracted from the direct computation and using σ from the fit of the histogram is . · − (histogram) and . · − (direct).This agreement implies that indeed the observable C is Gaussian distributed and can beinterpreted to play the role of the topological charge. We also note that the constructedhistograms and the extracted values of hCi and a χ depend very little on the chosen binsize. Using bin sizes of 0.5, 1, 2 and 3, our results for a χ change only by a few percent andare fully compatible within errors. Hence, we decided to use such bin size that the numberof bins with non-zero number of gauge field configurations is around 10. We emphasizethat the good properties of the histogram ( hCi ≈ and Gaussian shape) hold only if theMC history is long enough. We think that both properties can provide a good benchmarkwhether the topological charge sectors are sampled in a correct way.The statistics that we have for ensemble B55.32 is significantly higher than for otherensembles. Let us show the details for a more typical ensemble B75.32 with around 100– 10 –ndependent measurements. The Monte Carlo history (right panel of Fig. 3) indicatesa correct sampling of topological sectors, however it is not long enough to build a fullysymmetric histogram (Fig. 4 (right)). For example, the number of configurations for which − ≤ C < − and ≤ C < is, respectively, 47(7) and 31(5), where the error comes frombootstrap with blocking analysis and takes into account autocorrelations. Hence, in thegenerated ensemble, the samples with slightly negative topological charge are somewhatoverrepresented with respect to the ones with slightly positive topological charge, althoughstatistically they are still compatible. As a consequence, the peak of the Gaussian fit isfor C below zero. Nevertheless, within the computed errors we observe that the shape ofthe histogram is close to Gaussian and the topological susceptibility and hCi are withinlarge errors compatible between the fit and the direct computation and read for ensembleB75.32: hCi = 0 . (direct) and -0.20(37) (histogram), bare topological susceptibility: a χ = 4 . · − (direct), a χ = 4 . . · − (histogram). However, we would liketo give a warning that the rather low statistics we have for the ensemble B75.32 may leadto an underestimation of the error, i.e. the error of the error might be large. To reach fullconfidence for the obtained results, statistics of at least the size we have for the ensembleB55.32 would be necessary. Before we show results for all our ensembles, we shortly discuss finite volume effects (FVE)in our simulations. We show the bare topological susceptibility for three N f = 2 + 1 + 1 ensembles at β = 1 . , aµ l = 0 . and four N f = 2 ensembles at β = 3 . , aµ l = 0 . in Fig. 5. All ensembles give compatible results (with some tension between A40.20 andA40.32), but given the precision we have for these ensembles, i.e. statistical errors of theorder of 10-20%, we can not conclude about the size of FVE from numerical data. However,general arguments involving the size of FVE imply that they should be exponentially smallif m π L & (see e.g. Ref. [39]). Since this condition is satisfied for almost all of our N f = 2 + 1 + 1 ensembles (see the last column of Tab. 2), we are confident that FVEare much smaller than our statistical errors. For N f = 2 , we analyze the quark massdependence of the topological susceptibility only at one lattice spacing ( β = 3 . ) and theproduct m π L > for all of them. However, even in a small volume ( L ≈ . fm, with m π L ≈ . ), FVE are not larger than the statistical errors (cf. β = 3 . , L/a = 16 and
L/a = 32 in Fig. 5). N f = 2 results Tab. 5 provides our results for the observables hAi , hBi , hCi and the topological suscep-tibility in the case of N f = 2 flavours. In Fig. 6, we show our results at a single latticespacing corresponding to β = 3 . and a physical volume such that the condition m π L > is satisfied. In order to test whether the obtained values of the topological susceptibilitycould, in principle, be used to obtain a value for the chiral condensate, we apply the leadingorder (LO) Chiral Perturbation Theory ( χ PT) expression for N f flavours of light quarks: χ = Σ µ l N f , (5.2)– 11 – a χ m π LL/a=16 L/a=20 L/a=24 L/a=32N f =2+1+1, β =1.90, a µ l =0.004N f =2, β =3.90, a µ l =0.004 Figure 5 . Bare topological susceptibility for ensembles A40.20, A40.24 and A40.32 ( N f = 2+1+1 )and b40.16, b40.20, b40.24 and b40.32 ( N f = 2 ). where Σ is the chiral condensate. In particular, we impose that the topological susceptibilityvanishes at zero quark mass. Working with the assumption that LO χ PT can be applied,the slope of this fit gives the following result for the renormalized condensate (MS schemeat 2 GeV): r Σ / = 0 . , where the error is mostly statistical, but takes into account also the uncertainties of Z P /Z S , Z P and r /a . The error decomposition is as follows: r Σ / = 0 . , wherethe first error is statistical, the second comes from the uncertainty of Z P /Z S (enteringvia Eq. (2.3)), the third one from the uncertainty of Z P (entering the renormalized quarkmass µ l ) and the fourth one from the final conversion of a Σ / to r Σ / . The used valuesof Z P /Z S , Z P and r /a , together with their uncertainties, are shown in Tab. 3. Thefinal error quoted is the sum of the individual errors, combined in quadrature. We recallhere respective values from our direct determination from the mode number of the Diracoperator: 0.696(20) (at β = 3 . in the chiral limit) or 0.689(33) (in the continuum limitand in the chiral limit) [40]. The fact that we observe an agreement indicates a posteriorithe validity of our assumption about the applicability of LO χ PT, at least within the largeerrors of our present results . We mention that we attempted NLO χ PT fits, but the resulting errors on the fit parameters were toolarge to say whether higher order corrections are statistically significant. – 12 – igure 6 . Renormalized quark mass dependence of the renormalized topological susceptibility(normalized with r ) for N f = 2 ensembles at β = 3 . . The fit is to a LO χ PT expression, χ / d . o . f . ≈ . .Ens. N cnfs step h A i τ int h B i τ int h C i τ int r χ b40.16 12 272 20 5.29(14) 1.9(6) 0.92(4) 2.0(6) -0.19(9) 1.6(5) 0.0097(16)(1)(3)b40.20 6 264 20 14.61(38) 3.5(1.3) 2.59(8) 3.1(1.1) -0.10(12) 0.9(2) 0.0092(11)(1)(3)b40.24 6 454 20 32.07(19) 1.0(2) 5.72(5) 1.1(2) -0.13(17) 1.5(4) 0.0096(11)(1)(3)b40.32 12 217 16 100.5(5) 1.7(6) 17.76(11) 1.4(4) -0.38(37) 1.5(5) 0.0082(13)(1)(3)b64.24 6 219 20 30.94(28) 1.0(3) 5.39(7) 1.2(4) -0.02(27) 1.8(6) 0.0106(17)(1)(3)b85.24 6 160 20 29.30(24) 0.6(1) 5.03(6) 0.8(2) 0.47(29) 1.4(5) 0.0118(25)(1)(3) Table 5 . Our results for N f = 2 flavours. We give the ensemble label, the number of stochasticsources N , the number of configurations used (cnfs), the step between measurements (in units ofmolecular dynamics trajectories) and the values of hAi , hBi , hCi and the topological susceptibility,together with integrated autocorrelation times τ int (in units of measurements). The error for r χ is, respectively, statistical, resulting from the uncertainty of Z P /Z S (from the RI-MOM method)and resulting from the uncertainty of r /a . In all other cases the error is statistical only. N f = 2 + 1 + 1 results In this subsection, we discuss our data for the case with 2+1+1 active flavours. Ourresults for the observables hAi , hBi , hCi and the topological susceptibility are collected inTab. 6 and Fig. 7 shows the results for the topological susceptibility. We required that theautocorrelations for the topological charge are kept under control, i.e. can be measured withreasonable accuracy using the method proposed in Ref. [41] (UW method). This methodallows for an estimate of the integrated autocorrelation time τ int and of its error. We alsomade an independent error analysis using the method of bootstrap with blocking. In allcases, we found results compatible with the UW method, given in Tab. 6. In particular, we– 13 – ns. N cnfs step h A i τ int h B i τ int h C i τ int r χ A30.32 6 223 20 167.9(2.3) 4.8(2.1) 30.28(40) 3.8(1.5) -0.18(26) 0.5(1) 0.0072(10)(3)(2)A40.20 6 200 16 29.91(56) 2.6(1.0) 5.40(10) 1.9(6) -0.10(20) 1.1(3) 0.0130(22)(5)(4)A40.24 6 198 20 53.77(1.35) 6.8(3.1) 9.78(21) 3.7(1.6) -0.03(21) 0.8(2) 0.0086(13)(3)(2)A40.32 6 190 16 170.5(1.2) 1.9(6) 30.74(20) 1.3(4) 0.25(34) 0.7(2) 0.0074(10)(3)(2)A50.32 6 201 20 175.6(1.1) 1.9(7) 31.73(27) 2.2(8) 0.36(31) 0.6(1) 0.0081(12)(3)(2)A60.24 6 163 8 54.46(59) 1.6(5) 9.87(13) 1.4(5) -0.26(25) 0.9(3) 0.0092(14)(3)(3)A80.24 6 201 8 53.70(44) 1.6(5) 9.73(10) 1.5(5) 0.76(24) 1.0(3) 0.0114(17)(4)(3)B25.32 8 199 20 91.88(1.27) 1.7(6) 18.95(23) 1.3(4) -0.57(32) 1.2(3) 0.0070(11)(1)(2)B35.32 8 198 20 95.58(87) 2.3(9) 19.52(15) 1.2(4) -0.55(23) 0.6(2) 0.0067(9)(1)(2)B55.32 6 538 40 95.59(34) 1.1(2) 19.47(7) 0.9(2) -0.06(16) 0.6(1) 0.0080(7)(2)(2)B75.32 8 201 8 92.57(51) 1.4(5) 18.75(13) 1.3(4) 0.04(35) 1.1(3) 0.0090(11)(2)(3)B85.24 12 236 20 31.48(20) 0.8(2) 6.52(5) 0.6(1) -0.09(15) 0.7(1) 0.0106(14)(2)(3)D20.48 6 97 20 157.1(1.3) 1.5(6) 49.73(38) 1.3(6) -0.42(48) 0.7(2) 0.0041(11)(1)(1)D30.48 6 101 20 158.2(9) 1.0(4) 50.18(24) 0.6(2) -0.44(64) 0.9(3) 0.0073(24)(1)(2)D45.32 6 96 40 29.44(36) 0.8(3) 9.34(9) 0.4(1) -0.11(36) 1.1(4) 0.0125(21)(2)(4)
Table 6 . Our results for N f = 2 + 1 + 1 flavours. We give the ensemble label, the number ofstochastic sources N , the number of configurations used (cnfs), the step between measurements(in units of molecular dynamics trajectories) and the values of hAi , hBi , hCi and the topologicalsusceptibility, together with integrated autocorrelation times τ int (in units of measurements). Theerror for r χ is, respectively, statistical, resulting from the uncertainty of Z P /Z S (from the RI-MOMmethod) and resulting from the uncertainty of r /a . In all other cases the error is statistical only. (cid:1)(cid:2)(cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:4)(cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:5)(cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:6)(cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:7)(cid:1)(cid:2)(cid:3)(cid:2)(cid:8)(cid:1)(cid:2)(cid:3)(cid:2)(cid:8)(cid:4)(cid:1)(cid:2)(cid:3)(cid:2)(cid:8)(cid:5)(cid:1)(cid:2)(cid:3)(cid:2)(cid:8)(cid:6) (cid:1)(cid:2) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4) (cid:1)(cid:2)(cid:3)(cid:2)(cid:5) (cid:1)(cid:2)(cid:3)(cid:2)(cid:6) (cid:1)(cid:2)(cid:3)(cid:2)(cid:7) (cid:1)(cid:2)(cid:3)(cid:8) (cid:9) (cid:2)(cid:5) (cid:1) (cid:9) (cid:2) (cid:2) (cid:10) (cid:3) (cid:11)(cid:8)(cid:3)(cid:12)(cid:2) (cid:3) (cid:11)(cid:8)(cid:3)(cid:12)(cid:13) (cid:3) (cid:11)(cid:4)(cid:3)(cid:8)(cid:2) Figure 7 . The dependence of the dimensionless quantity r χ on the renormalized quark mass r µ R . We show all ensembles used for the analysis of the N f = 2 + 1 + 1 flavour case. – 14 – a) β = 1 . (b) β = 1 . (c) β = 2 . (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:1)(cid:2)(cid:3)(cid:5)(cid:1)(cid:2)(cid:3)(cid:5)(cid:4)(cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:2)(cid:3)(cid:6)(cid:4)(cid:1)(cid:2)(cid:3)(cid:7) (cid:1)(cid:2) (cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:4) (cid:1)(cid:2)(cid:3)(cid:2)(cid:8) (cid:1)(cid:2)(cid:3)(cid:2)(cid:8)(cid:4) (cid:1)(cid:2)(cid:3)(cid:2)(cid:9) (cid:1)(cid:2)(cid:3)(cid:2)(cid:9)(cid:4) (cid:1)(cid:2)(cid:3)(cid:2)(cid:10) (cid:1)(cid:2)(cid:3)(cid:2)(cid:10)(cid:4) (cid:1)(cid:2)(cid:3)(cid:2)(cid:11) (cid:12) (cid:2) (cid:1) (cid:8) (cid:13) (cid:10) (cid:14)(cid:15)(cid:13)(cid:12) (cid:2) (cid:16) (cid:9) (d) continuum limit Figure 8 . (a,b,c) Renormalized quark mass dependence of the topological susceptibility for N f =2 + 1 + 1 . The straight line corresponds to a fit of LO SU(2) χ PT. Only ensembles with m π ≤ MeV are included. χ / d . o . f . values are: 1.45 (a), 5.02 (b), 0.49 (d). The continuum limit of r Σ / extracted from fits shown in (a,b,c). found that the autocorrelation time for the observable C is τ int . .Typically, we have O (200) configurations per ensemble, although for our ensembles atthe finest lattice spacing, we only have around 100 configurations. Thus, the histogramsthat we can build have large statistical errors and within these large errors the deviationfrom a zero-centered Gaussian is insignificant. Few exceptions to this rule occur – e.g. forensemble A80.24 hCi is more than 3 σ away from zero. The typical error of the computedtopological susceptibility is of the order of 15% and we manage to go below 10% only for en-semble B55.32. In this way, we conclude that the precision one can reach for the topologicalsusceptibility is only modest. However, we want to emphasize here that this is a conse-quence of too short lengths of typical Monte Carlo simulations in Lattice QCD and do notoriginate from the spectral projector method itself. Especially with finer lattice spacings,autocorrelations are such that to obtain truly independent gauge field configurations onehas to perform measurements skipping several trajectories. Our experience shows that toobtain a 10% precision in the computation of the topological susceptibility, we need around300-400 truly independent configurations, which implies Monte Carlo runs of 10000-20000trajectories, which is somewhat longer than is typically needed for most other applications.In order to describe the quark mass dependence of the topological susceptibility, we– 15 –ollow the same strategy as discussed above for N f = 2 flavours. Using only the LO χ PTformula, we decided to apply a mass cut on our data, excluding points for which the pionmass is larger than 400 MeV, i.e. keeping points for which r µ R < . . The fits of the LOformula to our data are shown in Fig. 8(a,b,c).As in the N f = 2 case, we can extract the chiral condensate from the dependence of χ onthe quark mass. We have performed an analysis separately for each lattice spacing, takingthe individual uncertainties of Z P /Z S , Z P and r /a into account (in the way described inthe previous section) to propagate them to the values of r Σ / at finite lattice spacings(shown in Fig. 8(d)). Such obtained values are then extrapolated to the continuum limit,yielding the value: r Σ / = 0 . . We mention here that it is possible to prove that the topological susceptibility computedusing the spectral projector method and twisted mass fermions at maximal twist is O ( a ) -improved. This is not guaranteed a priori by standard arguments for the automatic O ( a ) -improvement at maximal twist [21], since the topological susceptibility is defined via densitychains that include integrals (in the continuum) or sums (on the lattice) over all space timepoints, which leads to contact terms with short distance singularities. Such contact termscan, in principle, spoil automatic O ( a ) -improvement. The proof that this is not the caseis sketched in Refs. [42, 43], while for the details of this proof we refer to an upcomingpublication [44].In general, the quality of our LO χ PT fits is reasonable (see the values of χ / d . o . f . in the caption of Fig. 8), with the exception of β = 1 . , for which χ / d . o . f . ≈ . Thismay signal the presence of effects beyond the ones captured in our LO χ PT fitting ansatz.However, with our current precision we are not able to address this issue. The fact thatsome data points are off the fit line may well be a statistical fluctuation at this level ofprecision. As a check of the robustness of our result, we performed also another LO χ PTfit including all our data, i.e. also pion masses between 400 and 500 MeV. This leads toa value for the chiral condensate in the continuum limit: r Σ / = 0 . . The resultfrom this additional fit is slightly lower, although still compatible with the one from fitsapplying a mass cut. The values of χ / d . o . f . for the LO χ PT fits without pion mass cutsare: 1.70 ( β = 1 . ), 4.52 ( β = 1 . ), 0.49 ( β = 2 . ), i.e. they are comparable to the onesfor fits without pion mass cuts (see the caption of Fig. 8).The error that we give is dominated by statistical uncertainties, but the contributionfrom the systematic errors related to r /a and Z P /Z S is also included. However, it does notinclude the main source of systematic effects coming from χ PT: the use of only the leadingorder expression. As we mentioned above, our precision is not enough to use an NLOfitting ansatz. Still, our result is in agreement with the direct determination from the modenumber on the same set of gauge field ensembles – r Σ / = 0 . [40], indicatingthat LO χ PT describes the quark mass dependence of the topological susceptibility at leastwithin the rather large errors of our results.It is worth emphasizing that at β = 1 . and β = 1 . the data for r χ do not show aclear tendency to assume a zero value when the quark mass is decreased. Only at β = 2 . – 16 –nd hence closer to the continuum limit, the data seem to approach zero linearly in thequark mass. Thus, in order to cleanly identify this expected behaviour of the topologicalsusceptibility, smaller quark masses and a significantly increased precision are required. We have computed the topological susceptibility in dynamical Lattice QCD simulationsusing the method of spectral projectors. This method has two important advantages thatwe want to emphasize here: • it relies on a theoretically sound definition of the topological susceptibility from den-sity chain correlators that is free of short distance singularities, • it is significantly less computer time expensive than the topological susceptibilitycomputation from the index of the overlap Dirac operator.One main result of our work is that the topological susceptibility is affected by substantialstatistical fluctuations necessitating long Monte Carlo histories. With typical parameter val-ues of Lattice QCD simulations nowadays, i.e. lattice spacings of 0.05 fm . a . . fm andlengths of Monte Carlo runs of O (5000) trajectories with autocorrelation times τ int = O (10) trajectories, it is very difficult to obtain errors smaller than 10-15% for a given ensemble.We emphasize that this is not a property of the method used here, but of the gauge fieldconfigurations themselves and as such can not be easily overcome, i.e. without running verylong simulations. In addition, the topological properties of gauge fields – here characterizedby the quantity C of Eq. (2.4), which is closely related to the topological charge – tend tobe particularly susceptible to autocorrelation effects, which increase with decreasing lat-tice spacing. This is indeed observed with the present method and implies that very highstatistics is needed (in particular at small lattice spacings) to overcome this problem, unlessone works with open boundary conditions that naturally allow to move the problem to atpresent unachievably small lattice spacings [10].Despite these difficulties, we were able to demonstrate that by imposing LO chiralperturbation theory as a description of our data for the topological susceptibility, values ofthe chiral condensate could be determined, which read: r Σ / = 0 . ( N f = 2 , nocontinuum extrapolation) and r Σ / = 0 . ( N f = 2 + 1 + 1 ). These results, althoughhaving large errors for the reasons discussed above, are fully compatible with the ones of ourdirect calculation using spectral projectors [40]. We estimate that a meaningful test of theNLO chiral perturbation theory prediction for the quark mass dependence of the topologicalsusceptibility would require a factor 3-10 longer runs (than typical ones, as specified above),which would bring the errors down below 10%. Nevertheless, we have shown that suchcalculations are becoming feasible with present-day computing resources and the advantagesof computations with a theoretically sound definition of the topological susceptibility usingdensity chains, promote the here used method to one of the most promising ways to addresstopological properties of QCD in the future.– 17 – cknowledgments We thank the European Twisted Mass Collaboration for generatinggauge field ensembles used in this work. We are grateful to A. Shindler for collabora-tion and discussions concerning O ( a ) improvement of the topological susceptibility. Weacknowledge useful discussions with V. Drach, G. Herdoiza, M. Müller-Preussker, K. Ot-tnad, G.C. Rossi, C. Urbach, F. Zimmermann. K.C. was supported by Foundation forPolish Science fellowship “Kolumb”. This work was supported in part by the DFG Son-derforschungsbereich/Transregio SFB/TR9. K.J. was supported in part by the CyprusResearch Promotion Foundation under contract Π PO Σ E Λ KY Σ H/EM Π EIPO Σ /0311/16.The computer time for this project was made available to us by the Jülich SupercomputingCenter, LRZ in Munich, the PC cluster in Zeuthen, Poznan Supercomputing and Network-ing Center (PCSS). We thank these computer centers and their staff for all technical adviceand help. A Number of stochastic sources
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