5 QCD on the Lattice
213
behaviour sets in. Similar findings have been reported for QCD with N f = 2 flavours
of dynamical overlap quarks [113].
The confirmation of the RMT prediction for the distribution of the low-lying
eigenvalues supports the Banks–Casher scenario of spontaneous chiral symmetry
breaking. In a subsequent step one may therefore extract the LEC Σ via the relation
k || ν ΣV = =μ k N ν ≡
∞
0
dz z p
(ν)
k (z).
(5.205)
If Σ is identified with the expectation value of the scalar density, as suggested
by the effective low-energy description of QCD, it must be related to a particular
continuum scheme, like the MS-scheme of dimensional regularization. If the regularization prescription obeys chiral symmetry, the corresponding renormalization
factor, Z S , satisfies
Z S = Z P = 1/Z m .
(5.206)
where Z m relates the bare quark mass to the chosen continuum scheme (for instance,
MS). Provided that Z S , or equivalently, Z m has been computed for a range of bare
couplings, the lattice estimates for Σ can be used to determine the renormalized
condensate in units of some scale, e.g.
r
3
0 Σ MS (μ) = Z S (g 0 , aμ)r
3
0 Σ + O(a
2 ).
(5.207)
For the Neuberger-Dirac operator, Z S has been computed non-perturbatively in the
quenched approximation [114], employing the technique outlined in Ref. [115]. The
resulting values for Z S could then be combined with the results for Σ extracted from
the matching to RMT from [111]. A subsequent extrapolation to vanishing lattice
spacing yields the results for the renormalized condensate in the continuum limit:
Σ MS (2 GeV) = (285 ± 9 MeV)
3 ,
(scale set by f K ).
(5.208)
The quoted error represents the total uncertainty arising from statistics, the uncertainty in the renormalization factor, and the continuum extrapolation. If the nucleon
mass is used to set the scale the central value drops to 261 MeV, as a consequence
of the scale ambiguity encountered in the quenched approximation. We stress once
more that the chiral condensate is ill-defined in the quenched theory, and thus great
care must be taken when the results are interpreted in the context of the full theory.
Nevertheless, it is encouraging that for N f = 2 flavours of dynamical quarks, a
similar calculation [113] finds Σ MS (2 GeV) = (251±7±11 MeV) 3 at a 0.11 fm,
in good agreement with the quenched result, given the inherent ambiguities and
inconsistencies of the latter.
Lattice results for the condensate have been reported by many other authors (e.g.
[116–125]), employing a variety of approaches. Although the various calculations
213
behaviour sets in. Similar findings have been reported for QCD with N f = 2 flavours
of dynamical overlap quarks [113].
The confirmation of the RMT prediction for the distribution of the low-lying
eigenvalues supports the Banks–Casher scenario of spontaneous chiral symmetry
breaking. In a subsequent step one may therefore extract the LEC Σ via the relation
k || ν ΣV = =μ k N ν ≡
∞
0
dz z p
(ν)
k (z).
(5.205)
If Σ is identified with the expectation value of the scalar density, as suggested
by the effective low-energy description of QCD, it must be related to a particular
continuum scheme, like the MS-scheme of dimensional regularization. If the regularization prescription obeys chiral symmetry, the corresponding renormalization
factor, Z S , satisfies
Z S = Z P = 1/Z m .
(5.206)
where Z m relates the bare quark mass to the chosen continuum scheme (for instance,
MS). Provided that Z S , or equivalently, Z m has been computed for a range of bare
couplings, the lattice estimates for Σ can be used to determine the renormalized
condensate in units of some scale, e.g.
r
3
0 Σ MS (μ) = Z S (g 0 , aμ)r
3
0 Σ + O(a
2 ).
(5.207)
For the Neuberger-Dirac operator, Z S has been computed non-perturbatively in the
quenched approximation [114], employing the technique outlined in Ref. [115]. The
resulting values for Z S could then be combined with the results for Σ extracted from
the matching to RMT from [111]. A subsequent extrapolation to vanishing lattice
spacing yields the results for the renormalized condensate in the continuum limit:
Σ MS (2 GeV) = (285 ± 9 MeV)
3 ,
(scale set by f K ).
(5.208)
The quoted error represents the total uncertainty arising from statistics, the uncertainty in the renormalization factor, and the continuum extrapolation. If the nucleon
mass is used to set the scale the central value drops to 261 MeV, as a consequence
of the scale ambiguity encountered in the quenched approximation. We stress once
more that the chiral condensate is ill-defined in the quenched theory, and thus great
care must be taken when the results are interpreted in the context of the full theory.
Nevertheless, it is encouraging that for N f = 2 flavours of dynamical quarks, a
similar calculation [113] finds Σ MS (2 GeV) = (251±7±11 MeV) 3 at a 0.11 fm,
in good agreement with the quenched result, given the inherent ambiguities and
inconsistencies of the latter.
Lattice results for the condensate have been reported by many other authors (e.g.
[116–125]), employing a variety of approaches. Although the various calculations