LATTICE FORMALISM

We use the `symmetric' definition of the gluon field, given by U_$\mu$(x) = exp(ig₀aA_$\mu$(x + $\hat{\mu}$/2)). This gives the gluon field in momentum space,

$$\mu \hat{q} {\frac{e^{-i\hat{q}_{\mu}a/2}}{2ig_0a}} \left[\vphantom{B_\mu(\hat{q}) -\frac{1}{3}{\rm Tr}\,B_\mu(\hat{q})}\right. \mu \hat{q} {\textstyle\frac{1}{3}} \mu \hat{q} \left.\vphantom{B_\mu(\hat{q}) -\frac{1}{3}{\rm Tr}\,B_\mu(\hat{q})}\right]$$ (1)

where $\hat{q}$ denotes the discrete momenta $\hat{q}_{\mu}^{}$ = 2$\pi$n_$\mu$/(aL_$\mu$), B_$\mu$($\hat{q}$) $\equiv$ U_$\mu$($\hat{q}$) - U^$\dagger$_$\mu$(- $\hat{q}$), and U_$\mu$($\hat{q}$) $\equiv$ $\sum_{x}^{}$e^-i$\hat{q}$xU_$\mu$(x). An alternative, `asymmetric' definition of the gluon field can be provided by U_$\mu$(x) = exp(ig₀aA'_$\mu$(x)). In momentum space, this differs from A_$\mu$(x) by a factor exp(i$\hat{q}_{\mu}^{}$a/2).

The gluon propagator D^ab_$\mu$$\nu$($\hat{q}$) is defined as

$$\mu \nu \hat{q} \langle \mu \hat{q} \nu \hat{q} \rangle$$ (2)

where A_$\mu$($\hat{q}$) $\equiv$ t^aA_$\mu$^a($\hat{q}$). In the continuum Landau gauge, the propagator has the structure

 

$$\mu \nu \delta^{ab}_{} \delta_{\mu\nu}^{} {\frac{q_{\mu}q_{\nu}}{q^2}}$$ (3)

At tree level, D(q²) will have the form D⁽⁰⁾(q²) = 1/q². On the lattice, this becomes D⁽⁰⁾($\hat{q}$) = a²/(4$\sum_{\mu}^{}$sin²($\hat{q}_{\mu}^{}$a/2)). Since QCD is asymptotically free, we expect that up to logarithmic corrections, q²D(q²)$\to$1 in the ultraviolet. Hence we define the new momentum variable q by q_$\mu$ $\equiv$ (2/a)sin($\hat{q}_{\mu}^{}$a/2), and use this throughout

 

 

Table: Simulation parameters. The lattice spacing is taken from the string tension [6].

Name	$\beta$	a-1 (GeV)	Volume	Nconf
Small	6.0	1.885	163 x 48	125
Large	6.0	1.885	323 x 64	75
Fine	6.2	2.63	243 x 48	223

We have analysed three lattices, with different values for the volume and lattice spacing. The details are given in table 1. All the configurations have been fixed to Landau gauge with an accuracy $\langle$($\partial_{\mu}^{}$A_$\mu$)²$\rangle$ < 10^-12.