Testing proposals for the Yang-Mills vacuum wavefunctional by measurement of the vacuum

We review a method, suggested many years ago, to numerically measure the relative amplitudes of the true Yang-Mills vacuum wavefunctional in a finite set of lattice-regulated field configurations. The technique is applied in $2+1$ dimensions to sets of Abelian plane wave configurations of varying amplitude and wavelength, and sets of non-Abelian constant configurations. The results are compared to the predictions of several proposed versions of the Yang-Mills vacuum wavefunctional that have appeared in the literature. These include (i) a suggestion in temporal gauge due to Greensite and Olejník; (ii) the “new variables” wavefunction put forward by Karabali, Kim, and Nair; (iii) a hybrid proposal combining features of the temporal gauge and new variables wavefunctionals; and (iv) Coulomb gauge wavefunctionals developed by Reinhardt and coworkers, and by Szczepaniak and coworkers. We find that wavefunctionals which simplify to a “dimensional-reduction” form at large scales, i.e., which have the form of a probability distribution for two-dimensional lattice gauge theory, when evaluated on long-wavelength configurations, have the optimal agreement with the data.

Figure 1

Typical plots of the data for $-\log (N_n/N_{\mathrm{tot}})$ vs the factor $2(\alpha +\gamma n)$ associated with the amplitude of the $n$th configuration. The straight line is a best linear fit, and the quantity ${\omega }_{\mathrm{MC}}({\tilde{k}}^2)$ is the slope of that line. The data shown is for ${\beta }_E=9$ and $L=24$, and two different ranges for $2(\alpha +\gamma n)$: (a) configurations generated with $\alpha =5$, $\gamma =0.5$, and the slope is ${\omega }_{\mathrm{MC}}=0.316(6)$; (b) configurations generated with $\alpha =80$, $\gamma =0.4$, and the slope is ${\omega }_{\mathrm{MC}}=0.309(2)$. The slope of the data is therefore essentially independent of the range of $2(\alpha +\gamma n)$.

Figure 2

Cumulative data for ${\omega }_{\mathrm{MC}}$ vs $p^2$ in physical units, on lattices of extensions $L=16$, 24, 32, 40, 48, and Euclidean lattice couplings ${\beta }_E=6$, 9, 12. The curves labeled “GO fit” and “KKN fit” (there are actually two curves, difficult to distinguish from one another), are the theoretical values for ${\omega }_{\mathrm{GO}}(p^2)$, and ${\omega }_{\mathrm{KKN}}(p^2)$, using the parameters of $m$ and $g^2$ in Table . The line labeled “Coulomb gauge” is obtained from the ansatz for the Coulomb gauge vacuum wavefunctional ${\Psi }_{\mathrm{CG}}[A]$ [Eq. (51)] as described in Sec. 4b.

Figure 3

The exponent $R$ from the variational approach Eq. (88) plotted against the lattice data for $-\ln {\Psi }^2$ for one set of non-Abelian constant configurations, choosing $\overline{\omega }(0)=c_1$ as fitting parameter ($c_1=0.1165$).

Figure 4

The exponent $R_{\mathrm{CG}}$ from the variational approach Eq. (90) plotted against the lattice data for $-\ln {\Psi }^2$ for the plane wave configurations with wave number $M\in \{1,2,4,8,12\}$. The lattice data was taken with lattice extension $L=24$ at $\beta =6.0$.

Figure 5

Dependence of $-\log (N_n/N_T)$ on the non-Abelianicity of the non-Abelian constant configurations, determined by $\sin ({\theta }_n)$.

Figure 6

Plot of $-\log (N_n/N_T)$ vs $R_{\mathrm{GO}}$ for the non-Abelian constant configurations with variable non-Abelianicity. The straight line fit has $\mathrm{\text{slope}}=1.02$.

Figure 7

Plot of $-\log (N_n/N_T)$ vs $R_{\mathrm{GO}}$ for non-Abelian constant configurations, maximal non-Abeliancity, at ${\beta }_E=6$, $L=32$, $\alpha =2$, $\gamma =0.15$. In this case, the straight line fit has a $\mathrm{\text{slope}}=0.98$.

Figure 8

Slopes for the GO wavefunctional vs $R$, at ${\beta }_E=6$, 9, 12 and $L=32$, using the values of $g^2$, $m$ derived from the Abelian plane wave fit.

Figure 9

${\beta }_E=12$ calculation, for both types of wavefunctionals.

Figure 10

The ghost propagator derived from standard Monte Carlo (MC) simulation at ${\beta }_E=9$, and the same quantity calculated by simulation of the GO and hybrid wavefunctionals, by the technique described in Ref. 8.

Figure 11

Data for the Coulomb potential at ${\beta }_E=9$ and $L=32$, derived from MC, GO, and hybrid simulations, with a cut on the data, discarding configurations for which $|V_0|$ is greater than 5, 10, 50, and 300, respectively.