5 QCD on the Lattice
191
which is not afflicted with the bad convergence properties encountered in the direct
matching of hadronic and MS-schemes. Finally, for the whole method to work, one
must be able to fix the virtualities μ of the external fields such that
QCD μ 1/a.
(5.134)
In other words, the method relies on the existence of a “window” of scales in which
lattice artefacts in the numerical evaluation are controlled, μ 1/a, and where
μ is also large enough such that the perturbative matching to the MS scheme can
be performed reliably. In the ideal situation one expects that the dependence of
Z MOM
(g 0 , aμ) on the virtuality μ inside the “window” is well described by the
perturbative RG function.
The RI/MOM prescription is a flexible method to introduce an intermediate
renormalization scheme and can easily be adapted to a range of operators and
lattice actions. In particular, the extension to discretizations of the quark action
based on the Ginsparg-Wilson relation is straightforward. This contrasts with the
situation encountered in the Schrödinger functional, where extra care must be taken
to ensure that imposing Schrödinger functional boundary conditions is compatible
with the Ginsparg-Wilson relation [79–81]. On the other hand, the non-perturbative
scale evolution, for which the Schrödinger functional is tailored, is not so easy to
incorporate into the RI/MOM framework. Hence, the matching between RI/MOM
and MS schemes is usually performed at fairly low scales, i.e. ¯
μ = μ 0 in the
notation of Fig. 5.12. Furthermore, the accessible momentum scales in the matching
of hadronic and RI/MOM schemes are typically quite narrow, i.e. aμ 0 ≈ 1. Special
care must also be taken when one considers operators that couple to the pion, such as
the pseudoscalar density. In this case the vertex function receives a contribution from
the Goldstone pole, which for p ≡ μ = 0 diverges in the limit of vanishing quark
mass. The fact that the chiral limit is ill-defined may spoil a reliable determination
of the renormalization factor, in particular when the accessible “window” is narrow
such that μ cannot be set to large values.
5.5.4 Mean-Field Improved Perturbation Theory
Another widely used strategy is to avoid the introduction of an intermediate
renormalization scheme altogether and attempt the direct, perturbative matching
between hadronic and MS schemes via an effective resummation of higher orders in
the expansion. In this sense one regards the bare coupling and masses as parameters
that run with the cutoff scale a −1 .
The bad convergence properties of perturbative expansions such as Eq. (5.118)
has been attributed to the presence of large gluonic tadpole contributions in the
relation between the link variable U μ (x) and the continuum gauge potential A μ (x).
It was already suggested by Parisi [82] that the convergence of lattice perturbation
theory could be accelerated by replacing the bare coupling g 2
0 by an “improved”
191
which is not afflicted with the bad convergence properties encountered in the direct
matching of hadronic and MS-schemes. Finally, for the whole method to work, one
must be able to fix the virtualities μ of the external fields such that
QCD μ 1/a.
(5.134)
In other words, the method relies on the existence of a “window” of scales in which
lattice artefacts in the numerical evaluation are controlled, μ 1/a, and where
μ is also large enough such that the perturbative matching to the MS scheme can
be performed reliably. In the ideal situation one expects that the dependence of
Z MOM
(g 0 , aμ) on the virtuality μ inside the “window” is well described by the
perturbative RG function.
The RI/MOM prescription is a flexible method to introduce an intermediate
renormalization scheme and can easily be adapted to a range of operators and
lattice actions. In particular, the extension to discretizations of the quark action
based on the Ginsparg-Wilson relation is straightforward. This contrasts with the
situation encountered in the Schrödinger functional, where extra care must be taken
to ensure that imposing Schrödinger functional boundary conditions is compatible
with the Ginsparg-Wilson relation [79–81]. On the other hand, the non-perturbative
scale evolution, for which the Schrödinger functional is tailored, is not so easy to
incorporate into the RI/MOM framework. Hence, the matching between RI/MOM
and MS schemes is usually performed at fairly low scales, i.e. ¯
μ = μ 0 in the
notation of Fig. 5.12. Furthermore, the accessible momentum scales in the matching
of hadronic and RI/MOM schemes are typically quite narrow, i.e. aμ 0 ≈ 1. Special
care must also be taken when one considers operators that couple to the pion, such as
the pseudoscalar density. In this case the vertex function receives a contribution from
the Goldstone pole, which for p ≡ μ = 0 diverges in the limit of vanishing quark
mass. The fact that the chiral limit is ill-defined may spoil a reliable determination
of the renormalization factor, in particular when the accessible “window” is narrow
such that μ cannot be set to large values.
5.5.4 Mean-Field Improved Perturbation Theory
Another widely used strategy is to avoid the introduction of an intermediate
renormalization scheme altogether and attempt the direct, perturbative matching
between hadronic and MS schemes via an effective resummation of higher orders in
the expansion. In this sense one regards the bare coupling and masses as parameters
that run with the cutoff scale a −1 .
The bad convergence properties of perturbative expansions such as Eq. (5.118)
has been attributed to the presence of large gluonic tadpole contributions in the
relation between the link variable U μ (x) and the continuum gauge potential A μ (x).
It was already suggested by Parisi [82] that the convergence of lattice perturbation
theory could be accelerated by replacing the bare coupling g 2
0 by an “improved”