SCALING BEHAVIOUR

In this note we focus on modelling the gluon propagator calculated in [1]. We use the same conventions and definitions here as in [1].

The lattice gluon propagator is related to the renormalised continuum propagator D_R(q;$\mu$) via a renormalisation constant,

 

$$\mu \mu$$ (1)

The asymptotic behaviour of the renormalised gluon propagator in the continuum is given to one-loop level by

 

$${\frac{1}{q^2}} \left(\vphantom{\frac{1}{2}\ln(q^2/\Lambda^2)}\right. {\textstyle\frac{1}{2}} \Lambda^{2}_{} \left.\vphantom{\frac{1}{2}\ln(q^2/\Lambda^2)}\right)^{-d_D}_{}$$ (2)

with d_D = 13/44 in Landau gauge for quenched QCD.

Since the renormalised propagator D_R(q;$\mu$) is independent of the lattice spacing, we can use (1) to derive a simple, q-independent expression for the ratio of the unrenormalised lattice gluon propagators at the same value of q:

 

$${\frac{D_c(qa_c)}{D_f(qa_f)}} {\frac{Z_3(\mu,a_c)D_R(q;\mu)/a_c^2}{Z_3(\mu,a_f)D_R(q;\mu)/a_f^2}} {\frac{Z_c}{Z_f}} {\frac{a_f^2}{a_c^2}}$$ (3)

where the subscript f denotes the finer lattice ($\beta$ = 6.2 in this study) and the subscript c denotes the coarser lattice ($\beta$ = 6.0). We can use this relation to study directly the scaling properties of the lattice gluon propagator by matching the data for the two values of $\beta$. This matching can be performed by adjusting the values for the ratios R_Z = Z_f/Z_c and R_a = a_f/a_c until the two sets of data lie on the same curve. We have implemented this by making a linear interpolation of the logarithm of the data plotted against the logarithm of the momentum for both data sets. In this way the scaling of the momentum is accounted for by shifting the fine lattice data to the right by an amount $\Delta_{a}^{}$ as follows

 

$$\Delta_{a}^{} \Delta_{Z}^{}$$ (4)

Here $\Delta_{Z}^{}$ is the amount by which the fine lattice data must be shifted up to provide the optimal overlap between the two data sets. The matching of the two data sets has been performed for values of $\Delta_{a}^{}$ separated by a step size of 0.001. $\Delta_{Z}^{}$ is determined for each value of $\Delta_{a}^{}$ considered, and the optimal combination of shifts is identified by searching for the global minimum of the $\chi^{2}_{}$/dof. The ratios R_a and R_Z are related to $\Delta_{a}^{}$ and $\Delta_{Z}^{}$ by R_a = exp(- $\Delta_{a}^{}$); R_Z = R_a²exp(- $\Delta_{Z}^{}$).

We considered matching the lattice data using $\hat{q}$ = 2$\pi$n/L as the momentum variable. The minimum value for $\chi^{2}_{}$/dof of about 1.7 was obtained for R_a $\sim$ 0.815. This value for R_a is considerably higher than the value of 0.716 $\pm$ 0.040obtained from an analysis of the static quark potential in [2]. From this discrepancy, as well as the relatively high value for $\chi^{2}_{}$/dof, we may conclude that the gluon propagator, taken as a function of $\hat{q}$, does not exhibit scaling behaviour for the values of $\beta$ considered here.

Fig. 1 shows the result of the matching using q = 2sin($\hat{q}$/2) as the momentum variable. This gives much more satisfactory values both for $\chi^{2}_{}$/dof and for R_a. The minimum value for $\chi^{2}_{}$/dof of 0.6 is obtained for R_a = 0.745. Taking a confidence interval where $\chi^{2}_{}$/dof < $\chi^{2}_{min}$ + 1 gives us an estimate of R_a = 0.745$$\;\stackrel{\scriptstyle +32}{\scriptstyle -37}\;$$, which is fully compatible with the value [2] of 0.716 $\pm$ 0.040.

  

Figure: $\chi^{2}_{}$ per degree of freedom as a function of the ratio of lattice spacings for matching the small and fine lattice data, using q as the momentum variable. The dashed line indicates the ratio R_Z of the renormalisation constants.