Figure 2.2: Picture of the simplest formulation of the field strength tensor on the lattice. The clover-like shapes are responsible for the name of the clover improved Wilson action.
Within perturbation theory the coefficient of the clover term is given by
cSW = 1 +O(g02). (2.52)
It can, however, also be calculated non-perturbatively, e.g., within the framework of the Schr¨odinger functional [34, 35, 36, 37].
A very general introduction to non-perturbative improvement can be found in Ref. [38].
2.3 Gauge fields on the lattice
2.3.1 Coupling to the gauge fields
In Section 2.1, we have derived the interaction between fermions and gauge fields by requiring the fermion Lagrangian and thus the action to be invariant under a local gauge transformation of the fermion fields. We follow the same procedure on the lattice. This means that the fermion action
S =a4X
x,y
ψ(x)D(x, y)ψ(y)¯ (2.53)
again has to be invariant under the transformation
ψ(x)−→ψ0(x) = Λ(x)ψ(x), (2.54)
ψ(x)¯ −→ψ¯0(x) = ¯ψ(x)Λ†(x). (2.55) Therefore, the Dirac operator must transform as
D(x, y)−→D0(x, y) = Λ(x)D(x, y)Λ†(y). (2.56)
18 Chapter 2: QCD on the lattice
This means that the Dirac operator has to connect points x and y in a gauge covariant way. For that purpose, the concept of parallel transporters [7, 39], usually called Wilson lines, is used. In the continuum, they are given by
U(x, y) =Pexp
i Z y
x
dzµAµ(z)
, (2.57)
where P denotes path ordering. After discretizing this expression on a lattice with lattice spacing a we obtain
Uµ(x) :=U(x, x+aµ)ˆ ≈exp(iaAµ(x))∈SU(3), (2.58) whereAµ(x) are the continuum gauge fields. One immediately sees that the object Uµ(x) lives on the link connecting the site x to its nearest neighbor in direction µ. We therefore refer to them in the following as link variables, or simply just links. They represent the gauge field degrees of freedom on the lattice.
Using these results the covariant Wilson Dirac operator, for example, then reads DW(x, y) = (m+ 4/a)1δx,y− 1
2a
±4
X
µ=±1
(1+γµ)Uµ(x)δx+aˆµ,y, (2.59) where we have used the convention
U−µ(x) = Uµ(x−aˆµ)† (2.60) Another important object we want to define here is theplaquette
Uµνpl(x) =Uµ(x)Uν(x+aµ)Uˆ −µ(x+aˆµ+aˆν)U−ν(x+aˆν). (2.61) It represents the closed loop around a square with side length a.
2.3.2 Wilson gauge action
We can use the plaquette to construct the Wilson gauge action [7]
Sgauge =a4βX
x
X
1≤µ<ν≤4
1− 1
NcRe[Tr(Uµνpl(x))]
, (2.62)
where Re[X] denotes the real part of X and Nc = 3 in the case of QCD. The invariance of this action under a local gauge transformation Λ(x) is shown very easily. The link variables are defined to transform as
Uµ0(x) = Λ(x)Uµ(x)Λ−1(x+aˆµ). (2.63)
2.3 Gauge fields on the lattice 19
By using Equation (2.60) and the invariance of the trace under cyclic permuta-tions, one can immediately see thatSgauge is invariant.
Now we want to show that Equation (2.62) has the right continuum limit. There-fore, we expand the links to first order in a. This results in
Sgauge=− β 4Nc
Z
d4xTr[Fµν(x)Fµν(x)] +O(a2). (2.64) We compare this to (2.7)and find
β = 2Nc
g2 . (2.65)
Another thing we see is that Equation (2.62) still has discretization errors of order O(a2). To reduce these errors, we have to include more complicated objects than simple plaquettes in the construction of the gauge action. Two examples of improved gauge actions are described in the following section.
2.3.3 Improved gauge actions
We have seen that the Wilson gauge action still has quite large discretization errors. In the following we want to discuss two improvement schemes to reduce these errors.
The first one is based on the transformation behavior of lattice actions under the renormalization group. By an iterative blocking of the lattice action until a fixed point of the renormalization group flow is reached, we obtain an improved gauge action with reduced discretization errors.
A different improvement scheme was proposed by Symanzik [40, 41]. He has shown that a Lagrangian field theory on a lattice is equivalent order by order in g and a to a local effective Lagrangian in the continuum. This allows one to im-prove lattice actions order by order by adding higher dimensional operators with appropriately chosen coefficients. As an example for this improvement scheme we want to discuss the L¨uscher-Weisz gauge action. As we have seen in Section 2.2.4, this improvement scheme can be also applied to lattice fermion actions.
Iwasaki gauge action
The Iwasaki gauge action is a renormalization group improvedSU(3) gauge action SGRG= β
6 c0
X
x,µ<ν
Uµνpl(x) +c1
X
x,µ,ν
Uµνrt(x)
!
, (2.66)
20 Chapter 2: QCD on the lattice
with
Uµνrt(x) = Uµ(x)Uν(x+aˆµ)Uν(x+aˆµ+aˆν)
U−µ(x+aµˆ+ 2aˆν)U−ν(n+ 2aˆν)U−ν(n+aˆν). (2.67) The coefficient c1 = −0.331 of the rectangular loops Uµνrt is fixed by an approx-imate renormalization analysis [8], while c0 = 1−8c1 = 3.648 is determined by the normalization condition, which defines the bare coupling β = 6/g2. From the point of view of Symanzik improvement, which we want to discuss next, the leading discretization error of this action areO(a2), the same as for Wilson gauge action. However, in a number of calculations, some improvement has been found.
L¨uscher-Weisz gauge action
Following Symanzik’s improvement program, L¨uscher and Weisz [9, 42] have been able to derive an improved lattice gauge action. This is achieved by adding rectangular and “parallelogram” terms with appropriately chosen coefficients to the Wilson gauge action. The resulting action reads
SGLW =β0X
S
1− 1
NReTrUµνpl
+β1X
R
1− 1
NReTrUµνrt
(2.68) +β2X
P
1− 1
NReTrUµνρpg
,
where the additional “parallelogram” terms are given by Uµνρpg (x) = Uµ(x)Uν(x+aˆµ)Uρ(x+aµˆ+aν)ˆ
U−µ(x+aµˆ+aˆν+aρ)Uˆ −ν(x+aνˆ+aρ)Uˆ −ρ(x+aˆρ). (2.69) A comparison with perturbative calculated quantities however has shown that the bare coupling which is incorporated in β and β0 for standard Wilson and improved L¨uscher-Weisz action, respectively, is not a meaningful expansion pa-rameter. A more useful parameter can be obtained after a redefinition of the coupling constant within the framework of tadpole improved perturbation theory [43]. When combining these considerations with the L¨uscher-Weisz gauge ac-tion the coefficients are given in terms of the expectaac-tion value of the plaquette