5 QCD on the Lattice
145
unique, and hence one is free to add further gauge invariant terms to the plaquette
action which formally vanish as a → 0, but which produce a discretization with
an accelerated rate of convergence to the continuum limit. The most widely chosen
alternatives are the Symanzik [7] and Iwasaki [8] actions.
Quark and antiquark fields, ψ(x) and ¯
ψ(x), are associated with the lattice sites
and transform under the gauge group as
ψ(x) → g(x)ψ(x),
¯
ψ(x) → ¯
ψ(x)g(x)
−1 .
(5.27)
Using the transformation property of the link variables, it is straightforward to write
down a discretized version of the covariant derivative, i.e.
∇ μ ψ(x) :=
1
a
U μ (x)ψ(x + a ˆ
μ) − ψ(x)
∇
∗
μ ψ(x) :=
1
a
ψ(x) − U μ (x − a ˆ
μ)
−1 ψ(x − a ˆ
μ)
,
(5.28)
where ∇ μ and ∇ ∗
μ denote the “forward” and “backward” derivatives, respectively.
Finally, we note that in Euclidean space-time, the Dirac matrices can be defined to
satisfy
γ μ , γ ν
= 2δ μν .
Before we attempt to construct the fermionic part of the action of lattice QCD, it
is useful to identify the basic properties that the discretized, massless Dirac operator,
D, should satisfy:
(a) D is local;
(b)
D(p) = iγ μ p μ + O(ap 2 );
(c)
D(p) is invertible for p = 0;
(d) γ 5 D + D γ 5 = 0.
Locality, i.e. the absence of long-ranged interactions, is a basic property of any
quantum field theory describing elementary particles. Property (b) implies that
the correct continuum behaviour of the quark-gluon interaction is reproduced.
Furthermore, condition (c) ensures that the correct fermion spectrum is obtained:
fermion masses are associated with poles of {
D(p)} −1 , which, in the continuum
theory, only occur at vanishing four-momentum. Finally, property (d) ensures that
the massless theory respects chiral symmetry.
Using the definition of the covariant derivative and the conventions for the Dirac
matrices in Euclidean space-time, we can now write down the simplest discretized
version of the massless lattice Dirac operator:
D disc =
1
2 γ μ (∇ μ + ∇ ∗
μ ).
(5.29)
It turns out, however, that this “naïve” discretization violates condition (c) and
therefore produces spurious fermionic degrees of freedom. This is the so-called
fermion doubling problem, which is most easily explained by considering D disc in
145
unique, and hence one is free to add further gauge invariant terms to the plaquette
action which formally vanish as a → 0, but which produce a discretization with
an accelerated rate of convergence to the continuum limit. The most widely chosen
alternatives are the Symanzik [7] and Iwasaki [8] actions.
Quark and antiquark fields, ψ(x) and ¯
ψ(x), are associated with the lattice sites
and transform under the gauge group as
ψ(x) → g(x)ψ(x),
¯
ψ(x) → ¯
ψ(x)g(x)
−1 .
(5.27)
Using the transformation property of the link variables, it is straightforward to write
down a discretized version of the covariant derivative, i.e.
∇ μ ψ(x) :=
1
a
U μ (x)ψ(x + a ˆ
μ) − ψ(x)
∇
∗
μ ψ(x) :=
1
a
ψ(x) − U μ (x − a ˆ
μ)
−1 ψ(x − a ˆ
μ)
,
(5.28)
where ∇ μ and ∇ ∗
μ denote the “forward” and “backward” derivatives, respectively.
Finally, we note that in Euclidean space-time, the Dirac matrices can be defined to
satisfy
γ μ , γ ν
= 2δ μν .
Before we attempt to construct the fermionic part of the action of lattice QCD, it
is useful to identify the basic properties that the discretized, massless Dirac operator,
D, should satisfy:
(a) D is local;
(b)
D(p) = iγ μ p μ + O(ap 2 );
(c)
D(p) is invertible for p = 0;
(d) γ 5 D + D γ 5 = 0.
Locality, i.e. the absence of long-ranged interactions, is a basic property of any
quantum field theory describing elementary particles. Property (b) implies that
the correct continuum behaviour of the quark-gluon interaction is reproduced.
Furthermore, condition (c) ensures that the correct fermion spectrum is obtained:
fermion masses are associated with poles of {
D(p)} −1 , which, in the continuum
theory, only occur at vanishing four-momentum. Finally, property (d) ensures that
the massless theory respects chiral symmetry.
Using the definition of the covariant derivative and the conventions for the Dirac
matrices in Euclidean space-time, we can now write down the simplest discretized
version of the massless lattice Dirac operator:
D disc =
1
2 γ μ (∇ μ + ∇ ∗
μ ).
(5.29)
It turns out, however, that this “naïve” discretization violates condition (c) and
therefore produces spurious fermionic degrees of freedom. This is the so-called
fermion doubling problem, which is most easily explained by considering D disc in