Modeling the Landau-gauge ghost propagator in 2, 3, and 4 spacetime dimensions

We present an analytic description of numerical results for the ghost propagator $G(p^2)$ in the minimal Landau gauge on the lattice. The data were produced in the SU(2) case using the largest lattice volumes to date, for $d=2$, 3 and 4 spacetime dimensions. Our proposed form for $G(p^2)$ is derived from the one-loop relation between ghost and gluon propagators, considering a tree-level ghost-gluon vertex and our previously obtained gluon-propagator results [A. Cucchieri et al., Phys. Rev. D 85, 094513 (2012). Although this one-loop expression is not a good description of the data, it leads to a one-parameter fit of our ghost-propagator data with a generally good value of ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$, comparable to other fitting forms used in the literature. At the same time, we present a simple parametrization of the difference between the lattice data and the one-loop predictions.

Figure 1

Plot of the average reduced chi-squared statistic ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ as a function of the parameter $r$ [see Eq. (47) and explanation in the text]. The average is taken over all lattice volumes $V$ and $\beta$ values considered.

Figure 2

Plot of the Gribov ghost form factor $\sigma (p^2)$ [see Eq. (2)] for the lattice volume $V={128}^4$ at $\beta =2.2$, as a function of unimproved momenta [see Eq. (48)] (left plot) and of improved momenta [see Eq. (47)] with $r=1/12$ (right plot). In both plots, red data points correspond to momenta along the diagonal direction ($p_{\mu }=\overline{p}$ for $\mu =1,\dots ,4$), while green data points correspond to off-diagonal momenta. All momenta are in physical units. Error bars have been estimated using propagation of errors.

Figure 3

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volumes $V={140}^3$ (symbol $*$ in blue), $V={240}^3$ (symbol $\times$ in green) and $V={320}^3$ (symbol $+$ in red) at $\beta =3.0$. Here we show the data corresponding to momenta with only one component different from zero. The data are (multiplicatively) normalized to 1 for $p=1.0\mathrm{GeV}$. Notice the logarithmic scale on the $y$ axis.

Figure 4

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volumes $V={48}^4$ (symbol $*$ in blue), $V={64}^4$ (symbol $\times$ in green) and $V={128}^4$ (symbol $+$ in red) at $\beta =2.2$. Here we show the data corresponding to momenta with only one component different from zero. The data are (multiplicatively) normalized to 1 for $p=1.0\mathrm{GeV}$. Notice the logarithmic scale on the $y$ axis.

Figure 5

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volumes $V={80}^2$ (symbol $*$ in blue), $V={200}^2$ (symbol $\times$ in green) and $V={320}^2$ (symbol $+$ in red) at $\beta =10.0$. Here we show the data corresponding to momenta with only one component different from zero. The data are (multiplicatively) normalized to 1 for $p=1.0\mathrm{GeV}$. Notice the logarithmic scale on the $y$ axis.

Figure 6

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={320}^3$ at $\beta =3.0$. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. We also show the function $F_1(p^2)$ [see Eq. (49)] (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (12)–(22); the corresponding parameters are reported in Eq. (51) and we set $g^2=1.313\mathrm{GeV}$. On the other hand, in the right plot, we fix the analytic form to match the numerical result at $p=p_{\min }=14\mathrm{MeV}$, the smallest nonzero (lattice) momentum for the pair $(V,\beta )$ considered, yielding $3.38353F_1(p^2)$. Notice the logarithmic scale on both axes.

Figure 7

Plot of $F_1(p^2)$ [see Eq. (49), blue curve] and $F_2(p^2)$ [see Eq. (50), magenta curve] as functions of the momentum $p$ for the 3D case, with $\sigma (p^2)$ given by ${\sigma }_{1L}(p^2)$ [see Eqs. (12)–(22) and (51)] with $g^2=1.313\mathrm{GeV}$. For both curves we consider $\overline{\mu }=2.5\mathrm{GeV}$. Notice the logarithmic scale on both axes.

Figure 8

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={320}^3$ at $\beta =3.0$ together with the fitting forms discussed in the text. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. In the left plot, we show the function ${\tilde{F}}_1(p^2)$ (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (12)–(22); the corresponding parameters are reported in Eq. (51) and we use the fitted value 10.0831 GeV for $g^2$. On the other hand, in the right plot, we show the fitting function $F_3(p^2)$ [see Eq. (54)] with the parameters given in Eqs. (55)–(57). Notice the logarithmic scale on both axes.

Figure 9

Plot of the term $W(p^2)$ [see Eqs. (58) and (59)] as a function of the (unimproved) momenta $p$ (in physical units) for the lattice volume $V={320}^3$ at $\beta =3.0$ (data points). We also show (in red) the fitting function $\tilde{W}(p^2)$ [see Eq. (60)] with the fitting parameters reported in Eq. (61).

Figure 10

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={128}^4$ at $\beta =2.2$. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. We also show the function $F_2(p^2)$ [see Eq. (50)] (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (23)–(34); the corresponding parameters are reported in Eq. (62) and we set $g^2=7.794$. On the other hand, in the right plot, we fix the analytic form to match the numerical result at $p=p_{\min }=46\mathrm{MeV}$, the smallest nonzero (lattice) momentum for the pair $(V,\beta )$ considered, yielding $2.01654F_2(p^2)$. Notice the logarithmic scale on both axes.

Figure 11

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={128}^4$ at $\beta =2.2$ together with the fitting forms discussed in the text. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. In the left plot we show the function ${\tilde{F}}_2(p^2)$ (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (23)–(34); the corresponding parameters are reported in Eq. (62) and we use the fitted value 14.6165 for $g^2$. On the other hand, in the right plot, we show the fitting function $F_3(p^2)$ [see Eq. (54)] with the parameters given in Eqs. (66)–(68). Notice the logarithmic scale on both axes.

Figure 12

Plot of the term $W(p^2)$ [see Eqs. (58) and (59)] as a function of the (unimproved) momenta $p$ (in physical units) for the lattice volume $V={128}^4$ at $\beta =2.2$ (data points). We also show (in red) the fitting function $\tilde{W}(p^2)$ [see Eq. (60)] with the fitting parameters reported in Eq. (69).

Figure 13

Plot of the Gribov ghost form factor $\sigma (p^2)$ [see Eq. (2)] for the lattice volumes $V={320}^2$ at $\beta =10.0$ (symbol $+$ in red), $V={320}^3$ at $\beta =3.0$ (symbol $\times$ in green) and $V={128}^4$ at $\beta =2.2$ (symbol $*$ in blue), as a function of the unimproved momenta [see Eq. (48)], for momenta with only one component different from zero. All momenta are in physical units and we show the data only in the IR limit, i.e. for $p\le 0.25\mathrm{GeV}$. Error bars have been estimated using propagation of errors. One clearly sees that in the 3D and 4D cases $\sigma (p^2)$ becomes almost constant at small momenta, with a value $\sigma (0)<1$, implying a freelike behavior for the ghost propagator in the IR limit. On the contrary, in the 2D case $\sigma (p^2)$ is still clearly increasing for momenta of the order of 20 MeV.

Figure 14

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={320}^2$ at $\beta =10.0$. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. We also show the function $F_1(p^2)$ [see Eq. (49)] (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (38)–(44); the corresponding parameters are reported in Eq. (70) and we set $g^2=0.516{\mathrm{GeV}}^2$. On the other hand, in the right plot, we fix the analytic form to match the numerical result at $p=p_{\min }=18\mathrm{MeV}$, the smallest nonzero (lattice) momentum for the pair $(V,\beta )$ considered, yielding $4.41862F_1(p^2)$. Notice the logarithmic scale on both axes.

Figure 15

Plot of the ghost propagator $G(p^2)$ as a function of the magnitude of the (unimproved) momenta $p$ (both in physical units) for the lattice volume $V={320}^2$ at $\beta =10.0$ together with the fitting forms discussed in the text. The data are (multiplicatively) normalized to $1/{\overline{\mu }}^2$ for $p=\overline{\mu }=2.5\mathrm{GeV}$. In the left plot we show the function ${\tilde{F}}_1(p^2)$ (normalized in the same way) with the Gribov ghost form factor $\sigma (p^2)$ given by the one-loop results (38)–(44); the corresponding parameters are reported in Eq. (70) and we use the fitted value $13.4556{\mathrm{GeV}}^2$ for $g^2$. On the other hand, in the right plot, we show the fitting function $F_{2\mathrm{D}}(p^2)$ [see Eq. (73)] with the parameters given in Eqs. (74)–(76). Notice the logarithmic scale on both axes.

Figure 16

Plot of the term $W(p^2)$ [see Eqs. (58) and (59)] as a function of the (unimproved) momenta $p$ (in physical units) for the lattice volume $V={320}^2$ at $\beta =10.0$ (data points). We also show (in red) the fitting function $\tilde{W}(p^2)$ [see Eq. (60)] with the fitting parameters reported in Eq. (77).