TENSOR STRUCTURE

By studying the tensor structure of the gluon propagator, we may be able to determine how well the Landau gauge condition is satisfied, and also discover violations of continuum rotational invariance.

The continuum tensor structure (3) follows from the condition q_$\mu$A_$\mu$ = 0. This translates directly to the lattice provided we use the symmetric definition of the gluon field. If we use the asymmetric definition, we will instead obtain the condition $\sum_{\mu}^{}$(isin$\hat{q}_{\mu}^{}$ + cos$\hat{q}_{\mu}^{}$ - 1)A'_$\mu$($\hat{q}$) = 0.

The tensor structure may be measured directly by taking the ratios of different components of D_$\mu$$\nu$(q) for the same value of q. The results for the small lattice are summarised in table 2, and compared to what one would expect from (3), and to what one would obtain by replacing qwith $\hat{q}$ in (3). The results are similar for the two other lattices. It is clear from table 2 that our numerical data are consistent with the expectation from (3). We can also see that in general, the asymmetric definition A' of the gluon field gives results which are inconsistent with this form.

  

Table: Tensor structure for the small lattice. $\hat{q}$ is in units of 2$\pi$/L_s, where L_s is the spatial length of the lattice. The theoretical predictions are the values for the ratios one obtains from (3), and from (3) with q$\to$$\hat{q}$. The numbers in brackets are the statistical uncertainties in the last digit(s). Where no error is quoted, the statistical uncertainty is less than 10^-6.