5 QCD on the Lattice
143
5.2.2 Lattice Actions for QCD
Our goal now is to find a lattice transcription of the Euclidean QCD action in the
continuum, i.e.
S QCD =
d
4 x
−
1
2g 2
0
Tr (F μν F μν ) +
f =u,d,s...
¯
ψ f
γ μ D μ + m f
ψ f
, (5.16)
where g 0 denotes the gauge coupling, and our conventions are chosen such that the
covariant derivative is defined through
D μ = ∂ μ + A μ ,
(5.17)
while the field tensor reads
F μν = ∂ μ A ν − ∂ ν A μ + [A μ , A ν ],
A
†
μ = −A μ .
(5.18)
Before attempting to write down a discretized version, we must first elucidate the
notion of a lattice gauge field in a non-Abelian theory. In fact, in this case it turns
out that the gauge potential A μ must be abandoned when the theory is discretized.
The reason is that the familiar non-Abelian transformation law, i.e.
A μ (x) → g(x)A μ (x)g(x)
−1
+ g(x)∂ μ (x)g(x)
−1 , g(x) ∈ SU(3),
(5.19)
no longer holds exactly when ∂ μ is replaced by its discrete counterpart d μ of
Eq. (5.5). Strict gauge invariance at the level of the regularized theory cannot be
maintained in this fashion.
The definition of a lattice gauge field relies on the concept of the parallel
transporter. If a quark moves in the presence of a background gauge field from y
to x, it picks up a non-Abelian phase factor, given by
U (x, y) = P.O. exp
−
x
y
dz μ A μ (z)
,
(5.20)
where “P.O.” denotes path ordering, as a consequence of the non-Abelian nature of
the gauge field. By contrast to the gauge potential A μ , which is an element of the Lie
algebra of SU(3), the parallel transporter U (x, y) is an element of the gauge group
itself. On the lattice, the parallel transporter between neighbouring lattice sites x
and x + a ˆ
μ is called link variable:
U(x, x + a ˆ
μ) ≡ U μ (x),
U (x + a ˆ
μ, x) = U(x, x + a ˆ
μ)
−1
= U μ (x)
−1 .
(5.21)
143
5.2.2 Lattice Actions for QCD
Our goal now is to find a lattice transcription of the Euclidean QCD action in the
continuum, i.e.
S QCD =
d
4 x
−
1
2g 2
0
Tr (F μν F μν ) +
f =u,d,s...
¯
ψ f
γ μ D μ + m f
ψ f
, (5.16)
where g 0 denotes the gauge coupling, and our conventions are chosen such that the
covariant derivative is defined through
D μ = ∂ μ + A μ ,
(5.17)
while the field tensor reads
F μν = ∂ μ A ν − ∂ ν A μ + [A μ , A ν ],
A
†
μ = −A μ .
(5.18)
Before attempting to write down a discretized version, we must first elucidate the
notion of a lattice gauge field in a non-Abelian theory. In fact, in this case it turns
out that the gauge potential A μ must be abandoned when the theory is discretized.
The reason is that the familiar non-Abelian transformation law, i.e.
A μ (x) → g(x)A μ (x)g(x)
−1
+ g(x)∂ μ (x)g(x)
−1 , g(x) ∈ SU(3),
(5.19)
no longer holds exactly when ∂ μ is replaced by its discrete counterpart d μ of
Eq. (5.5). Strict gauge invariance at the level of the regularized theory cannot be
maintained in this fashion.
The definition of a lattice gauge field relies on the concept of the parallel
transporter. If a quark moves in the presence of a background gauge field from y
to x, it picks up a non-Abelian phase factor, given by
U (x, y) = P.O. exp
−
x
y
dz μ A μ (z)
,
(5.20)
where “P.O.” denotes path ordering, as a consequence of the non-Abelian nature of
the gauge field. By contrast to the gauge potential A μ , which is an element of the Lie
algebra of SU(3), the parallel transporter U (x, y) is an element of the gauge group
itself. On the lattice, the parallel transporter between neighbouring lattice sites x
and x + a ˆ
μ is called link variable:
U(x, x + a ˆ
μ) ≡ U μ (x),
U (x + a ˆ
μ, x) = U(x, x + a ˆ
μ)
−1
= U μ (x)
−1 .
(5.21)