SU(2) gluodynamics and ${\mathrm{HP}}^1$ $\sigma$-model embedding: Scaling, topology, and confinement

We investigate recently proposed ${\mathrm{HP}}^1$ $\sigma$-model embedding method aimed to study the topology of SU(2) gauge fields. The ${\mathrm{HP}}^1$ based topological charge is shown to be fairly compatible with various known definitions. We study the corresponding topological susceptibility and estimate its value in the continuum limit. The geometrical clarity of ${\mathrm{HP}}^1$ approach allows to investigate nonperturbative aspects of SU(2) gauge theory on a qualitatively new level. In particular, we obtain a numerically precise estimation of gluon condensate and its leading quadratic correction. Furthermore, we present clear evidences that the string tension is to be associated with global (percolating) regions of sign-coherent topological charge. As a by-product of our analysis we estimate the continuum value of quenched chiral condensate and the dimensionality of regions, which localize the lowest eigenmodes of overlap Dirac operator.

Figure 1

Cumulative distribution of $Q_{{\mathrm{HP}}^1}$, Eq. (19) and $Q_{\mathrm{clover}}$, Eq. (21), on 300 cooled configurations initially thermalized at $\beta =2.400$ on ${16}^4$ lattice.

Figure 2

Cumulative distribution of $Q_{{\mathrm{HP}}^1}$, Eq. (19) and $Q_{\mathrm{overlap}}$, Eq. (24), on 200 fixed volume configurations at $\beta =2.500$, Table .

Figure 3

Projection of the cumulative distribution of $Q_{{\mathrm{HP}}^1}$ and $Q_{\mathrm{overlap}}$ at $\beta =2.500$ onto the lines $Q_{{\mathrm{HP}}^1}=\pm Q_{\mathrm{overlap}}$, along which $Q_{\mathrm{proj}}\sqrt{2}$ is the corresponding integer valued coordinate. Note that both histograms are normalized separately.

Figure 4

The topological susceptibility as a function of lattice volume both being expressed in physical units (fixed spacing configurations at $\beta =2.4180$, Table ).

Figure 5

Topological charge distribution obtained at $\beta =2.475$ on ${16}^4$ lattices (380 configurations). Curve represents the best fit to Eq. (27).

Figure 6

Scaling of the ${\mathrm{HP}}^1$-based topological susceptibility with lattice spacing; line is the best fit to Eq. (28).

Figure 7

Scaling of averaged plaquette $P=⟨1/2\mathrm{Tr}{U}_p^{{\mathrm{HP}}^1}⟩$ constructed from ${\mathrm{HP}}^1$ projected gauge fields. Line is the best fit according to Eq. (31).

Figure 8

$-\ln ⟨W(T,R)⟩$ as a function of $T$ at various $R$ measured at $\beta =2.600$ on ${28}^4$ lattice.

Figure 9

Heavy quark potential $V^{{\mathrm{HP}}^1}(R)$ extracted from Wilson loops measured at $\beta =2.600$ on ${28}^4$ lattice. Solid curve is the fit to Eq. (35).

Figure 10

Scaling of ${\mathrm{HP}}^1$ projected string tension ${\sigma }^{{\mathrm{HP}}^1}$ compared to that of full SU(2) string tension ${\sigma }^{\mathrm{SU}(2)}$. Lines are the best fits according to Eq. (36) in the whole range of available spacings and in the region $a<0.12\mathrm{fm}$.

Figure 11

${\Lambda }_q$ dependence of the ${\mathrm{HP}}^1$ string tension (squares) and of the topological susceptibility (circles) both normalized to their respective values at ${\Lambda }_q=0$ ($\beta =2.475$, ${16}^4$). Shaded box stands for the approximate location of the lumps percolation transition (see text).

Figure 12

The spectral density of lowest Dirac eigenmodes calculated on fixed volume configurations, Table , at different spacings. Line represents linear interpolation towards $\lambda =0$.

Figure 13

IPRs $I_{\lambda }$ for lowest eigenmodes at various lattice spacings. Note the explicitness of the ’mobility edge’ around $\lambda \approx 200\mathrm{MeV}$.

Figure 14

IPR for lowest eigenmodes, Eq. (49), as a function of lattice spacing. The fitting curves (50) are represented by various lines (see text).

Figure 15

Distribution of trace of the plaquette matrix constructed for a particular configuration ($\beta =2.475$, ${16}^4$) from $U_{x,\mu }$ and $V_{x,\mu }$ fields.

Figure 16

$-\ln ⟨W(T,R)⟩$ as a function of $T$ at various $R$ measured on $V$ configurations generated on $\beta =2.475$, ${16}^4$ lattice.

Figure 17

The heavy quark potential extracted from $V$ fields on $\beta =2.475$, ${16}^4$ lattice. The curve is the fit to analytical dependence (54).

Figure 18

The spectral density of overlap Dirac operator measured on $V$ fields. Note the complete absence of eigenmodes in $\lambda \lesssim 200\mathrm{MeV}$ interval.