The next obvious task to formulate QCD on a lattice is to rewrite the QCD La-grangian in discretised form such that the gauge and other symmetries of QCD are preserved. This can be divided in two separate problems, the discretisation of the gauge action SG and the fermionic action SF. In order to discretise those a four
x x+aµ x+aµ+aν x+aν
Figure 3.1: The plaquette termUµν of the Wilson gauge action
dimensional grid of space-time points is introduced. The introduction of a hyper-cubic lattice breaks Lorentz invariance which should be restored in the continuum limit. However the remaining hypercubic symmetry is still present in the theory [99, 100]. To translate the QCD Lagrangian to the lattice we need some clever fi-nite difference techniques, in particular for the Dirac operator. The discretisation of the gauge part is quite straight forward and therefore we will start with it. As second we will address the fermionic part which is more demanding and is still problematic in ourdays lattice calculations.
3.4.1 Gauge Action
In the continuum gauge transformations involve space time derivatives of group-valued functions. On the lattice we do not have infinitesimally close points and replace the derivative by finite differences. However, we need to be more care-fully as we want to keep the gauge invariance in the discretised theory. Therefore we need the concept of so called parallel transporter introduced independently by Wegner and Wilson, as well as Smit. On discretised spacetime the parallel trans-porter U from point x to the neighbour point in µ direction x+ aˆµ is defined as
U(x, x+aµ)ˆ ≡Ux,µ =Pexp
iag0
1
Z
0
dξ Gµ(x+ (1−ξ)aˆµ)
, (3.10) a straight line path between x andx+aµˆ withµˆ the unit vector in µdirection.
The local gauge transformation of the parallel transporter has then the form:
U(x, x+aµ)ˆ →Ω(x)U(x, x+aˆµ)Ω(x+aˆµ)† (3.11) where Ω are elements of SU(3)C, the colour group. Connecting two separated space-time points the parallel transporters naturally lives on the links of the lattice
(Figure 3.1). These link variables are used to construct the gauge part of the action. The colour trace of any closed loop (’Wilson loop’) of link variables is gauge invariant. The smallest possible loop, the plaquette, one can obtain on the lattice is the usual building block of the gauge action
UP(x;µν) = Ux,µUx+ˆµ,νUx+ˆ† ν,µUx,ν† . (3.12) Then the standard Wilson action for the gauge sector is the sum over all plaquettes
SG=−β X withβthe lattice coupling constant. In the limit of vanishing lattice spacinga→0 it reduces to the usual continuum Yang-Mills action
SG = β 4Nc
Z
dxtr (FµνFµν) +O(a2). (3.14)
3.4.2 Fermion Action
The discretisation of the fermion action is much more troublesome. Naively dis-cretising the fermionic part of the QCD action leads in the continuum to the fa-mous ’fermion doubling’ problem [101, 102]. The fermion action can be generi-cally written as
SF =XψMψ¯ (3.15)
where Mis the fermion operator, a lattice approximation to the Dirac operator.
In this work we use the so called Wilson-Clover action which avoid the fermion doublers by adding the Wilson term to the naive discretised fermion action. Thus on the lattice we will use in our calculations the lattice fermion action of the form
SF =X where in the first line we have a simple discretisation of the fermion action. The covariant derivative
D↔is constructed from forward and backward derivatives as D→µψ(x) =
and setting
D↔ = 1 2
→
D−D←
. (3.19)
Higher order derivatives can be obtained by appropriately combining the forward and backward derivative operators.
Taking the limit of the (free) naive fermion propagator however gives poles in the massless fermion propagator not only at ap = (0,0,0,0)but also whenever apµ =π causing the already mentioned fermion doubling. Therefore the Wilson term is introduced in the second line of eq. (3.16) to avoid this problem. This term gives a contribution∝P
µ = (1−cos(apµ))/ato the propagator and so the doubler masses are∝1/aand vanish as the continuum limita →0is taken. But we have to pay a three-fold price:
1. Even for massless fermions we loose completely the chiral symmetry, which is present in the continuum (at least approximately).
2. Without chiral symmetry the fermion mass is not prevented anymore from running up to the lattice scale1/a. For about two decades lattice QCD suf-fered from this hierarchy problem in the fermion sector. Recovering chiral symmetry in the continuum limit then requires additive mass renormalisa-tion, involving a delicate fine-tuning of the bare fermion mass
am=am0−amcont0 (3.20)
where one has to determineamcont0 .
3. In the continuum limit the fermion part is now worse than the gauge part as we have nowO(a)discretisation errors.
Point 1 and 2 are a fundamental problem in lattice QCD calculations. The solutions are provided by more sophisticated actions like overlap [103, 104] or domain wall [105, 106]. These actions are numerically more expensive and will become available with reasonably high statistics only in the near future. However for point three we can achieve some progress by adding a further term and so restore the O(a2) behaviour. Adding further irrelevant operators (i.e. higher di-mensional) to the action [107] and tuning them to remove the discretisation errors we can restore theO(a2)behaviour. At the lowest order, i.e. O(a)improvement (for review cf. [108, 109]), there are 5 possible gauge invariant irrelevant opera-tors [110] (including the Wilson term). Restricting ourselves to the improvement of on-shell quantities, when the correlation functions are evaluated at non-zero physical distances and no contact terms between operators appear, the equations
Figure 3.2: The clover-leaf of the discretised field strength tensor.
of motions can be used and we are left with only one remaining operator, the
"clover" term. This is the last term in eq. (3.16) with Fµν(x) = 1
8ig X
±µ,±ν
UµνP (x)−UµνP (x)†
, (3.21)
where we have extended the definition of the plaquette to the Clover leave (Fig-ure (3.2))
Now if we can achieve that one physical quantity (e.g. physical mass ratio) does not have aO(a)term then
• this fixescsw(g)
• all other physical quantities are automatically improved toO(a2).
However the practical realisation is difficult and numerically demanding as one has to perform simulations at different csw(g) values for one β. The ALPHA Collaboration were able to find [111] numericallycsw(g)
csw = 1−0.454g2−0.175g4+ 0.012g6+ 0.045g8
1−0.720g2 (3.22)
and using this values it is possible to estimatecsw(g)for theβ values used in our simulations.