Chiral symmetry breaking in QCD Lite
A distinctive feature of the presence of spontaneous chiral symmetry breaking in QCD is the condensation of low modes of the Dirac operator near the origin. The rate of condensation must be equal to the slope of with respect to the quark mass in the chiral limit, where and are the mass and the decay constant of the Nambu-Goldstone bosons. We compute the spectral density of the (Hermitian) Dirac operator, the quark mass, the pseudoscalar meson mass and decay constant by numerical simulations of lattice QCD with two light degenerate Wilson quarks. We use CLS lattices at three values of the lattice spacing in the range – fm, and for several quark masses corresponding to pseudoscalar mesons masses down to MeV. Thanks to this coverage of parameters space, we can extrapolate all quantities to the chiral and continuum limits with confidence. The results show that the low quark modes do condense in the continuum as expected by the Banks–Casher mechanism, and the rate of condensation agrees with the Gell–Mann-âOakes-âRenner (GMOR) relation. For the renormalisation-group-invariant ratios we obtain and , which correspond to MeV and MeV if is used to set the scale by supplementing the theory with a quenched strange quark.
Introduction.— There is overwhelming evidence that the chiral symmetry group of Quantum Chromodynamics (QCD) with a small number of light flavours breaks spontaneously to . This progress became possible over the last decade thanks to the impressive speed-up of the numerical simulations of lattice QCD with light dynamical fermions, see Ref. Schaefer (2012) for a recent review and a comprehensive list of references. The impact on phenomenological analyses of chiral dynamics is already striking Leutwyler (2007).
The formation of a non-zero chiral condensate in the theory, , was conjectured to be the effect of the condensation of the low modes of the Dirac operator near the origin Banks and Casher (1980). The rate of condensation is indeed a renormalisable universal quantity in QCD, and is unambiguously defined once the bare parameters in the action of the theory have been renormalised Giusti and Luscher (2009). The Banks–Casher mechanism links the spectral density of the Dirac operator to the condensate as Banks and Casher (1980)
an identity which can be read in both directions: a non-zero spectral density implies that the symmetry is broken by a non-vanishing and vice versa.
The above conceptual and technical advances in lattice gauge theory paved the way for a quantitative study of the Banks–Casher mechanism from first principles. It is the aim of this letter to achieve a precise and reliable determination of the density of eigenvalues of the Euclidean Dirac operator near the origin at small quark masses in the continuum. As in any numerical computation, the limits in Eq. (1) inevitably require an extrapolation of the results with a pre-defined functional form. The distinctive signature for spontaneous symmetry breaking is the agreement between the chiral-limit value of the spectral density at the origin, reached by extrapolating the data with the functional form dictated by chiral perturbation theory (ChPT), and the slope of with respect to the quark mass Weinberg (1979); Gasser and Leutwyler (1985). We thus complement our study with the computations of , and .
To reach these goals, we use -improved Wilson fermions at several lattice spacings, and we extrapolate the numerical results to the universal continuum limit following the Symanzik effective theory analysis. For technical reasons we focus on the mode number of the Dirac operator Giusti and Luscher (2009)
which at the same time is the average number of eigenmodes of the massive Hermitian operator with eigenvalues .
It is a renormalisation-group-invariant quantity as it stands. Its (normalised) discrete derivative
carries the same information as , but the effective
density is a more convenient quantity to consider
in numerical computations. For practical
purposes we also extend the theory by introducing a quenched “strange”
quark so to have a graded chiral symmetry group . This is instrumental
to derive the ChPT formula for Osborn et al. (1999), and
allows us to fix the lattice spacing from the kaon decay constant
. The latter is a well-defined intermediate reference
scale which can be computed precisely on the lattice Fritzsch et al. (2012) and is
directly accessible to experiments once the CKM matrix element is known.
This scale is used here to convert all quantities in physical units, with
the scheme-dependent ones renormalised in the scheme at
GeV. The final results, however,
are independent of this intermediate step: they are expressed as ratios of quantities
of the two-flavour theory only.
It is worth noting that there were several exploratory studies of the spectral density of the Dirac operator in QCD, see for instance Giusti and Luscher (2009); Fukaya et al. (2011); Cichy et al. (2013). The approach pursued here is rather general, and it may be useful in order to study theories at non-zero temperature or strongly interacting models of electroweak symmetry breaking Patella (2012).
We have profited from CLS simulations of
two-flavour QCD with the -improved Wilson
action. On all the lattices listed in Table 1
we have computed the mode number and the two-point functions of
The ensembles have lattice spacings of fm
as measured from Fritzsch et al. (2012). The quark masses range
from to MeV and are small compared to the typical scale of the
theory from the condensate or the string tension of about - MeV.
All lattices are of size , and the pion mass is always
large enough so that . Finite-size effects are within the
statistical errors for all measured quantities, see Ref. Engel et al. ()
for more details. The error analysis takes care of autocorrelations Schaefer et al. (2011),
all Markov chains except for one (N5) being of length between and 74 .
The mode number has been computed for nine values of in the range – MeV with a statistical accuracy of a few percent on all lattices. Four larger values of in the range – MeV have also been analyzed for the ensemble E5, see Ref. Engel et al. () for tables with all results. In Fig. 1 (top-left) we show as a function of for the lattice O7, corresponding to the smallest quark mass at the smallest lattice spacing. On all other lattices an analogous qualitative behaviour is observed. The mode number is a nearly linear function in up to approximately – MeV. A clear departure from linearity is observed for MeV on the lattice E5. At the percent precision, however, the data show statistically significant deviations from the linear behavior already below MeV. To guide the eye, a quadratic fit in is shown in Fig. 1, and the values of the coefficients are given in the caption. The bulk of is given by the linear term, while the constant and the quadratic term represent corrections in the fitted range. The nearly linear behaviour of the mode number, expected if the Banks–Casher mechanism is at work, is manifest on the top-right plot of Fig. 1, where the cubic root of the discrete derivative defined in Eq. (3) is shown as a function of for each couple of consecutive values of . When the regularisation breaks chiral symmetry, discretization effects heavily distort the spectral density near Del Debbio et al. (2006); Damgaard et al. (2010): we thus focus on the effective spectral density rather than the mode number.
In general, shows quite a flat behaviour in
at fine lattice spacings and light quark masses, similar to the
one shown in Fig. 1 (top-right). Because the action and
the mode number are -improved, the Symanzik effective-theory analysis
predicts that discretization errors start at Giusti and Luscher (2009).
In order to remove them, we interpolate the effective spectral density to
three quark mass values (, , MeV) at each lattice spacing.
The values of show very mild discretization effects at light
and , while they differ by up to among the three
lattice spacings toward heavier . Within the statistical
errors all data sets are compatible with a linear dependence in , and
we thus independently extrapolate each triplet of points to the continuum
limit accordingly. The difference between the values of at the
finest lattice spacing and the continuum-extrapolated ones is within
for the lightest and , and it remains within few
standard deviations at
heavier values of and . This makes us confident that the
extrapolation removes the cutoff effects within the errors quoted.
The results for at MeV in the continuum limit are shown as a function of in the bottom-left plot of Fig. 1. A similar -dependence is observed at the two other reference masses. It is noteworthy that no assumption on the presence of spontaneous symmetry breaking was needed so far. These results point to the fact that the spectral density of the Dirac operator in two-flavour QCD is non-zero and (almost) constant in near the origin at small quark masses. This is consistent with the expectations from the Banks–Casher mechanism. In presence of spontaneous symmetry breaking, next-to-leading (NLO) ChPT indeed predicts Leutwyler and Smilga (1992); Smilga and Stern (1993); Osborn et al. (1999); Giusti and Luscher (2009)
i.e. an almost flat function in (small) at (small) finite quark
masses.111The parameter is a low-energy constant of
the chiral effective theory renormalised at the scale ,
while is a parameter-free function, see Ref. Engel et al. ().
Once the pion mass and decay constant
are measured, the (mild) parameter-free -dependence of
in Eq. (4) is compatible
with our data.
The extrapolation to the chiral limit requires an assumption on how the effective spectral density behaves when . In this respect it is interesting to notice that toward the chiral limit the function in Eq. (4) is the simplest possible one, i.e. it goes linearly in since there are no chiral logarithms at fixed Giusti and Luscher (2009). A fit of the data to Eq. (4) shows that they are compatible with that NLO formula. Eq. (4) predicts that in the chiral limit also at non-zero , since all NLO corrections vanish Smilga and Stern (1993). By extrapolating the effective spectral density with Eq. (4) but allowing for the constant term to depend on , we obtain the results shown in the bottom-right plot of Fig. 1 with a . Within errors the -dependence is clearly compatible with a constant up to MeV. Moreover the differences between the values of in the chiral limit and those at MeV are of the order of the statistical error, i.e. the extrapolation is very mild. A fit to a constant of the data gives MeV, where the first error is statistical and the second one is systematic. The latter is a conservative estimate obtained by performing various combined fits of all data suggested by NLO ChPT and the Symanzik effective theory analysis Engel et al. ().
To compare the value of the spectral density at the origin with the slope of with respect to the quark mass , we complement the computation of the mode number with those for the pion masses and the decay constants, and , as well as the quark mass . They are extracted from the two-point functions of the non-singlet pseudoscalar density and axial current as in Refs. Del Debbio et al. (2007); Fritzsch et al. (2012), see Ref. Engel et al. () for more details. The results are reported in Table 1, and those for the pseudoscalar decay constant in lattice units are shown in Fig. 2 versus . We fit to the function
where is common to all lattice spacings, restricted to the points with MeV (see Fig. 2). Apart for the NLO ChPT just given, we also perform a number of alternative extrapolations in . As a final result we quote , and at , and fm respectively, where the second (systematic) error takes into account the spread of the results from the various fits. By performing a continuum-limit extrapolation we obtain our final result MeV.
Once the value of is determined, we compute the ratio for all data points. We fit the data with MeV to
where again is common to all lattice spacings. Also in this case we checked several variants although the data look very flat up to the heaviest mass. The result for the condensate is MeV, where the errors are determined as for .
Discussion and conclusions.— From the previous analysis, our best results for the leading-order low-energy constants of QCD with two flavours are
where the renormalisation-group-invariant (RGI) condensate
is defined with the convention
of Refs. Capitani et al. (1999); Della Morte et al. (2005b).
Our results show that the spectral density of the Dirac operator in the continuum is non-zero at the origin and that its value agrees with the slope of with respect to the quark mass when both are extrapolated to the chiral limit. If expanded in , is dominated by the leading (GMOR) term proportional to the chiral condensate, see Fig. 3. The ratio is nearly constant within errors up to quark masses that are about one order of magnitude larger than in Nature.
Measurements have been performed on BlueGene/Q at CINECA (CINECA-INFN agreement, ISCRA 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 resources and the technical support. We are grateful to our colleagues within the CLS initiative for sharing the ensembles of gauge configurations. G.P.E. and L.G. acknowledge partial support by the MIUR-PRIN contract 20093BMNNPR. S.L. and R.S. acknowledge support by the DFG Sonderforschungsbereich/Transregio SFB/TR9.
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