5 QCD on the Lattice
153
The last fermionic discretization we wish to mention here was originally
constructed to address another problem of Wilson’s discretization, namely the fact
that they are not protected against the occurrence of zero modes for any non-zero
value of the bare quark mass. These unphysical zero modes manifest themselves
as “exceptional” configurations, which occur with a certain frequency in numerical
simulations with Wilson quarks and which can lead to strong statistical fluctuations.
The problem can be cured by introducing a so-called “chirally twisted” mass term,
after which the fermionic part of the QCD action in the continuum assumes the form
S
tm; cont
F
=
d
4 x ¯
ψ(x)(γ μ D μ + m + iμ q γ 5 τ
3 )ψ(x).
(5.50)
Here, μ q is the twisted mass parameter, and τ 3 is a Pauli matrix. The standard action
in the continuum can be recovered via a global chiral field rotation:
ψ
(x) = e
iαγ 5 τ 3 /2 ψ(x),
¯
ψ
(x) = ¯
ψ(x)e
iαγ 5 τ 3 /2 .
(5.51)
Fixing the twist angle α by requiring that tan α = μ q /m one finds
S
F =
d
4 x ¯
ψ
(x)(γ μ D μ + M)ψ
(x),
M =
m 2 + μ 2
q ,
(5.52)
which demonstrates the complete equivalence of the twisted formulation with
“ordinary” QCD. The lattice action of twisted mass QCD for N f = 2 flavours is
defined as [26]
S
tm
F [U, ¯
ψ, ψ] = a
4
x∈ E
¯
ψ(x)(D w + m 0 + iμ q γ 5 τ
3 )ψ(x).
(5.53)
Although this formulation breaks physical parity and flavour symmetries, is has a
number of advantages over standard Wilson fermions. In particular, the presence of
the twisted mass parameter μ q protects the discretized theory against unphysical
zero modes. Another attractive feature of twisted mass lattice QCD is the fact
that the leading lattice artefacts are of order a 2 without the need to add the
Sheikholeslami-Wohlert term [27], even though the Wilson-Dirac operator is used
in Eq. (5.53). Although the problem of explicit chiral symmetry breaking remains,
the twisted formulation is particularly useful to circumvent some of the problems
that are encountered in connection with the renormalization of local operators on
the lattice. Recent review of twisted mass lattice QCD can be found in [28, 29].
We wish to end this part with a few general remarks. Although we have
discussed discretizations of the QCD action in some detail, including the most
recent developments, many more variants of the basic types of action—including
several different combinations of fermionic and pure gauge parts—can be found in
the literature. This reflects the fact that the discretization is not unique. The actual
choice of lattice action in a particular simulation will influence the convergence rate
153
The last fermionic discretization we wish to mention here was originally
constructed to address another problem of Wilson’s discretization, namely the fact
that they are not protected against the occurrence of zero modes for any non-zero
value of the bare quark mass. These unphysical zero modes manifest themselves
as “exceptional” configurations, which occur with a certain frequency in numerical
simulations with Wilson quarks and which can lead to strong statistical fluctuations.
The problem can be cured by introducing a so-called “chirally twisted” mass term,
after which the fermionic part of the QCD action in the continuum assumes the form
S
tm; cont
F
=
d
4 x ¯
ψ(x)(γ μ D μ + m + iμ q γ 5 τ
3 )ψ(x).
(5.50)
Here, μ q is the twisted mass parameter, and τ 3 is a Pauli matrix. The standard action
in the continuum can be recovered via a global chiral field rotation:
ψ
(x) = e
iαγ 5 τ 3 /2 ψ(x),
¯
ψ
(x) = ¯
ψ(x)e
iαγ 5 τ 3 /2 .
(5.51)
Fixing the twist angle α by requiring that tan α = μ q /m one finds
S
F =
d
4 x ¯
ψ
(x)(γ μ D μ + M)ψ
(x),
M =
m 2 + μ 2
q ,
(5.52)
which demonstrates the complete equivalence of the twisted formulation with
“ordinary” QCD. The lattice action of twisted mass QCD for N f = 2 flavours is
defined as [26]
S
tm
F [U, ¯
ψ, ψ] = a
4
x∈ E
¯
ψ(x)(D w + m 0 + iμ q γ 5 τ
3 )ψ(x).
(5.53)
Although this formulation breaks physical parity and flavour symmetries, is has a
number of advantages over standard Wilson fermions. In particular, the presence of
the twisted mass parameter μ q protects the discretized theory against unphysical
zero modes. Another attractive feature of twisted mass lattice QCD is the fact
that the leading lattice artefacts are of order a 2 without the need to add the
Sheikholeslami-Wohlert term [27], even though the Wilson-Dirac operator is used
in Eq. (5.53). Although the problem of explicit chiral symmetry breaking remains,
the twisted formulation is particularly useful to circumvent some of the problems
that are encountered in connection with the renormalization of local operators on
the lattice. Recent review of twisted mass lattice QCD can be found in [28, 29].
We wish to end this part with a few general remarks. Although we have
discussed discretizations of the QCD action in some detail, including the most
recent developments, many more variants of the basic types of action—including
several different combinations of fermionic and pure gauge parts—can be found in
the literature. This reflects the fact that the discretization is not unique. The actual
choice of lattice action in a particular simulation will influence the convergence rate