Chiral condensate in nf=2 QCD from the Banks-Casher relation
aa r X i v : . [ h e p - l a t ] D ec Chiral condensate in n f = QCD from theBanks–Casher relation
Georg P. Engel ∗ Dipartimento di Fisica, Università Milano-Bicocca,and INFN, Sezione di Milano-Bicocca,Piazza della Scienza 3, 20126 Milano, ItalyE-mail: [email protected]
Exploiting the Banks-Casher relation, we present a direct determination of the chiral condensatein two-flavor QCD, computing the mode number of the O ( a ) -improved Wilson-Dirac operatorbelow various cutoffs. We make use of CLS-configurations with three different lattice spacings inthe range of 0.05-0.08 fm and pion masses down to 190 MeV. Our data indicate a non-zero densityof eigenmodes near the origin and hence points to spontaneous chiral symmetry breaking. Weextrapolate our results to the continuum and chiral limit to give a result for the chiral condensate. The XXXII International Symposium on Lattice Field Theory, Lattice 2014June 23 - 28, 2014Columbia University, New York, USA ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engel
1. Introduction
The chiral condensate, defined as expectation value of a quark-antiquark pair, S ≡ − h ¯ yy i , (1.1)plays a central rôle in QCD. It provides an order parameter for chiral symmetry breaking, a leading-order low-energy constant of Chiral Perturbation Theory (ChPT), and it naturally appears in theOperator Product Expansion. Recent related lattice QCD results are collected in the FLAG review[1]. The present work discusses a determination of S , exploiting the Banks-Casher relation [2], S = p lim l → lim m → lim V → ¥ r ( l , m ) , (1.2)with r ( l , m ) = V ¥ (cid:229) k = h d ( l − l k ) i , (1.3)where m is the current quark mass, i l k are the eigenvalues of the massless Dirac operator and V is the four-volume. The spectral density r is renormalizable and can be computed on the latticenumerically [3]. In lattice QCD with Wilson-type quarks, it turns out to be convenient to considerthe mode number n ( L , m ) of the massive hermitian operator D † m D m with eigenvalues a ≤ M = √ L + m , which is renormalization-group invariant, n ( L , m ) = V Z L − L d lr ( l , m ) (1.4) n R ( L R , m R ) = n ( L , m ) . (1.5)The method was shown to work in Ref. [3] and applied to twisted–mass fermions in Ref. [4]. Wedefine the effective spectral density, ˜ r R = p V n , R − n , R L , R − L , R , (1.6)which agrees with S after taking the appropriate order of limits as in the Banks-Casher relation.Note that any threshold effects are removed from ˜ r R as long as all L i , R are chosen large enough.Preliminary results have been presented in Ref. [5], the main physics results are published inRef. [6], while for a detailed discussion we refer to Ref. [7].
2. Chiral Perturbation Theory
At next-to-leading-order (NLO), continuum ChPT predicts for n f = r NLOR ( L , R , L , R , m R ) = S n + m R S ( p ) F h l + − ln ( ) − (cid:16) S m R F m (cid:17) + ˜ g n (cid:18) L , R m R , L , R m R (cid:19) io , (2.1)where ˜ g n ( x , x ) , explicitly given in Ref. [5], appears to be a mild function in the considered rangeof parameters [7]. F is the pseudo-scalar decay constant in the chiral limit, ¯ l an NLO low-energyconstant (LEC) and m is a fixed scale. It is noteworthy that there are no chiral logarithms at fixed2 hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engelid L / a m p [MeV] m p L a [fm] R t exp R t int ( m p ) R t int ( n ) Rn it ( n ) N cnfg A3 32 496 ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . Table 1: Parameters of the simulation. L is the linear spatial extent of the lattice, a the latticespacing [12], m p the pion mass, R the ratio of active links in DD-HMC [13] ( R = t exp and t int denote the exponential and integrated autocorrelation time, resp., given in unitsof molecular dynamics, n it the separation of configurations between subsequent measurements and N cnfg the number of configurations on which n is measured. L R , that ˜ r NLOR is a decreasing function of L R = ( L , R + L , R ) / O ( a ) improvement and the generically-small-quark-massregime (GSM) [9], gives an additional term of the form m R / ( L , R L , R ) added to Eq. (2.1). The signof this term was argued to be positive [10, 11], which implies that ˜ r NLO , latR is a decreasing functionof L R also at finite lattice spacing (in the GSM-regime). We remark that those NLO discretizationeffects, and any L R -dependence, are still absent in the chiral limit. The formalism of ChPT can beused also to address finite-volume effects, which increase towards light L R .
3. Details of the simulation
We measure the mode number on configurations with two flavors of O ( a ) -improved Wilsonquarks, provided by the CLS-collaboration. The most relevant details for the present study aredepicted in Tab. 1, further information is detailed in Refs. [12, 15]. Finite-size effects and autocor-relations are under control for all measurements. The mode number is computed for nine values of L R in the range 20-120 MeV with a statistical precision of a few percent on all ensembles. Rationalpolynomials are used to approximate the the spectral projector P M to the low modes of the Diracoperator. Its expectation value is then evaluated stochastically with pseudo-fermion fields h k , n = N N (cid:229) k = h ( h k , P M h k ) i , M = p L + m . (3.1)
4. Results
Fig. 1, left, shows the results for the mode number for all ensembles. It exhibits a roughly lineardependence on aM in all cases up to approximately 100-150 MeV. A phenomenological low-order3 hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engel polynomial fit indicates that in the considered range roughly 90% of n is given by the linear term.The effective spectral density ˜ r R shows a non-zero and flat behavior in L R at fine lattice spacingsand light quark masses. As an example, the results of ensemble O7 are shown in Fig. 1, right.To extrapolate to the continuum and chiral limit, some analytic guidance is needed. In thisrespect, first studies indicated that higher-order effects of ChPT are apparent in the data and corre-spondingly the functional form at finite lattice spacing is not entirely clear in the considered rangeof parameters [5]. For this reason, we attempt to build a clean fitting strategy where different effectscan be distinguished clearly. Such a strategy is based on performing first the continuum limit, andonly then removing the small corrections stemming from finite m R and L R . To do so, we interpo-late ˜ r R to three values of the quark mass ( m R = . , . , . ( L R , m R ) , examples of whichare shown in Fig. 2. The linear a -dependence, expected from Symanzik effective theory for the O ( a ) -improved theory, is respected well by the data. It is noteworthy that the discretization effectsexhibit a non-trivial dependence on ( L R , m R ) , but appear fairly mild at the lightest points.As a result of the extrapolation, we obtain ˜ r R in the continuum, where its non-zero values atlight ( L R , m R ) point to dynamical chiral symmetry breaking. This motivates to use ChPT to removethe remaining small corrections. We consider a fit function based on generalized NLO ChPT,˜ r R = c ( L R ) + c m R + c m R (cid:20) ˜ g n (cid:18) L , R m R , L , R m R (cid:19) − (cid:18) m R m (cid:19)(cid:21) , (4.1)where c ( L R ) = S = const . at NLO. The continuum data is described well by this ansatz (thecorrelated fit gives c / dof=16.4/14), the extrapolation is of the order of the statistical error, and weobtain the results for c ( L R ) shown in Fig. 3. The plateau-like behavior for L R ≤
80 MeV indicatesthe NLO range, and a corresponding fit gives S / = ( ) MeV in the MS scheme at 2 GeV.To substantiate the result, we consider a second strategy to extract the chiral condensate fromthe data on the effective spectral density. After the separate treatment of different effects in thefirst strategy, we now attempt to perform a combined fit in ( L R , m R , a ) at once. The advantages arethat this approach does not require an interpolation in m R but includes all data, and furthermoreneeds fewer fit parameters compared to the first strategy. However, ChPT is used from the start andthe discretization effects have to be modelled. We assume a linear dependence in a and m R , butstill allow for an arbitrary L R -dependence, inspired by Symanzik effective theory and the chiralpower expansion. It is worth noting that the model complies with the results of the first strategyand includes NLO Wilson-ChPT [9] as a special case. We find that the fit describes the data welland that the results agree very well with the ones of the first strategy. Having abundant degreesof freedom in the fit, we use the second strategy to estimate the systematic uncertainty of the finalresult by performing various different fits. An upward shift is found when neglecting data at coarselattices, while a downward shift is found when including higher-order terms O ( L , m ) in the fit.
5. Conclusions
We presented a determination of the chiral condensate based on an extensive discussion of thespectral density of the hermitean Wilson Dirac operator. Our final result is [ S MS ( )] / = ( )( ) MeV , (5.1)4 hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engel n L R [GeV] O7˜ r R [GeV ] m R = 12.9 MeV a = 0.048 fm Figure 1: First look at the numerical data. Left: Mode number n for all ensembles vs. the baredimensionless cutoff aM . Note the approximate linearity and the high number of modes achievedfor small quark masses. Right: Effective spectral density ˜ r R vs. the cutoff L R for the ensemble withthe lightest quark mass at the finest lattice spacing (O7). Note the non-zero flat behavior, whichcan be interpreted as a first hint for dynamical chiral symmetry breaking. a [fm ] L R = 108 MeV L R = 48 MeV L R = 23 MeV˜ r R [GeV ] m R = 12.9 MeV 00.010.020.030.040.050.06 0 0.002 0.004 0.006 a [fm ] L R = 108 MeV L R = 48 MeV L R = 23 MeV˜ r R [GeV ] m R = 32.0 MeV Figure 2: Continuum extrapolation of ˜ r R , performed individually for each pair ( L R , m R ) . Shownfor three values of L R covering the entire range, and for the lightest (left) and the heaviest referencequark mass (right). Note that the data agrees well with the linear a -dependence expected in the O ( a ) -improved theory. The discretization effects exhibit a non-trivial dependence on ( L R , m R ) , butappear mild at the lightest point.where the first error is statistical and the second one systematic. As a consistency test of dynamicalsymmetry breaking and ChPT, we consider its NLO prediction for the quark mass dependence ofthe pion mass. The latter is known as GMOR-relation and we show its prediction based on ourmeasurement of the chiral condensate together with the direct measurements of the quark and pionmasses in Fig. 4. The relation appears to be fulfilled to very good precision in the range considered. Acknowledgments
Simulations have been performed on BlueGene/Q at CINECA (CINECA-INFN agreement, ISCRA5 hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engel L R [GeV] m R = 0 MeV a = 0 fm Figure 3: The effective spectral density ˜ r R in the continuum and chiral limit. The flat and non-zerobehavior observed for L ≤
80 MeV is consistent with NLO ChPT, a plateau fit in this range yieldsa prediction for the chiral condensate given in the text. M p / ( p F ) m RGI /(4 p F)Banks-Casher + GMORContinuum data p /(2m RGI F) Figure 4: Consistency of the determined chiral condensate with the quark mass dependence ofthe pion mass as expected from the GMOR-relation. The pion mass squared M p is shown vs. therenormalizion-group-independent (RGI) quark mass, normalized to 4 p F ( ≈ F isthe pseudo-scalar decay constant in the chiral limit (taken from [6]). The direct measurements(red symbols) are extrapolated to the continuum as described in Ref. [6], while the (central) solidline represents the GMOR contribution to the pion mass squared, computed by taking the directdetermination of the chiral condensate through the spectral density. The thinner solid lines denotethe statistical error, the dotted-dashed ones the sum of statistical and systematic one.project IsB08_Condnf2), on HLRN, on JUROPA/JUQUEEN at Jülich JSC, on PAX at DESY,Zeuthen, and on Wilson at Milano–Bicocca. We thank these institutions for the computer resources6 hiral condensate in n f = QCD from the Banks–Casher relation
Georg P. Engel and the technical support. We are grateful to our colleagues within the CLS initiative for sharing theensembles of gauge configurations. We acknowledge partial support by the MIUR-PRIN contract20093BMNNPR.
References [1] S. Aoki, Y. Aoki, C. Bernard, T. Blum, G. Colangelo, et. al. , Review of lattice results concerning lowenergy particle physics , arXiv:1310.8555 .[2] T. Banks and A. Casher, Chiral Symmetry Breaking in Confining Theories , Nucl.Phys.
B169 (1980)103.[3] L. Giusti and M. Lüscher,
Chiral symmetry breaking and the Banks-Casher relation in lattice QCDwith Wilson quarks , JHEP (2009) 013, [ arXiv:0812.3638 ].[4] K. Cichy, E. Garcia-Ramos, and K. Jansen,
Chiral condensate from the twisted mass Dirac operatorspectrum , PoS
LATTICE2013 (2013) 128, [ arXiv:1303.1954 ].[5] G. P. Engel, L. Giusti, S. Lottini, and R. Sommer,
Chiral condensate from the Banks-Casher relation , PoS
LATTICE2013 (2013) 119, [ arXiv:1309.4537 ].[6] G. P. Engel, L. Giusti, S. Lottini, and R. Sommer,
Chiral symmetry breaking in QCD Lite , arXiv:1406.4987 .[7] G. P. Engel, L. Giusti, S. Lottini, and R. Sommer, Spectral density of the Dirac operator intwo-flavour QCD , arXiv:1411.6386 .[8] A. V. Smilga and J. Stern, On the spectral density of Euclidean Dirac operator in QCD , Phys.Lett.
B318 (1993) 531–536.[9] S. Necco and A. Shindler,
Spectral density of the Hermitean Wilson Dirac operator: a NLOcomputation in chiral perturbation theory , JHEP (2011) 031, [ arXiv:1101.1778 ].[10] M. T. Hansen and S. R. Sharpe,
Constraint on the Low Energy Constants of Wilson ChiralPerturbation Theory , Phys.Rev.
D85 (2012) 014503, [ arXiv:1111.2404 ].[11] K. Splittorff and J. Verbaarschot,
The Microscopic Twisted Mass Dirac Spectrum , Phys.Rev.
D85 (2012) 105008, [ arXiv:1201.1361 ].[12] P. Fritzsch, F. Knechtli, B. Leder, M. Marinkovic, S. Schaefer, et. al. , The strange quark mass andLambda parameter of two flavor QCD , Nucl.Phys.
B865 (2012) 397–429, [ arXiv:1205.5380 ].[13] M. Lüscher,
Solution of the Dirac equation in lattice QCD using a domain decomposition method , Comput.Phys.Commun. (2004) 209–220, [ hep-lat/0310048 ].[14] M. Marinkovic and S. Schaefer,
Comparison of the mass preconditioned HMC and the DD-HMCalgorithm for two-flavour QCD , PoS
LATTICE2010 (2010) 031, [ arXiv:1011.0911 ].[15] M. Marinkovic, S. Schaefer, R. Sommer, and F. Virotta,
Strange quark mass and Lambda parameterby the ALPHA collaboration , PoS
LATTICE2011 (2011) 232, [ arXiv:1112.4163 ].].