New scheme for color confinement and violation of the non-Abelian Bianchi identities

A new scheme for color confinement in QCD due to violation of the non-Abelian Bianchi identities is proposed. The violation of the non-Abelian Bianchi identities (VNABI) $J_{\mu }$ is equal to Abelian-like monopole currents $k_{\mu }$ defined by the violation of the Abelian-like Bianchi identities. Although VNABI is an adjoint operator satisfying the covariant conservation law $D_{\mu }{J}_{\mu }=0$, it satisfies, at the same time, the Abelian-like conservation law ${\partial }_{\mu }{J}_{\mu }=0$. The Abelian-like conservation law ${\partial }_{\mu }{J}_{\mu }=0$ is also gauge-covariant. There are $N^2-1$ conserved magnetic charges in the case of color $SU(N)$. The charge of each component of VNABI is quantized à la Dirac. The color-invariant eigenvalues ${\lambda }_{\mu }$ of VNABI also satisfy the Abelian conservation law ${\partial }_{\mu }{\lambda }_{\mu }=0$ and the magnetic charges of the eigenvalues are also quantized à la Dirac. If the color invariant eigenvalues condense in the QCD vacuum, each color component of the non-Abelian electric field $E^a$ is squeezed by the corresponding color component of the solenoidal current $J_{\mu }^a$. Then only the color singlets alone can survive as a physical state and non-Abelian color confinement is realized. This confinement picture is completely new in comparison with the previously studied monopole confinement scenario based on an Abelian projection after some partial gauge-fixing, where Abelian neutral states can survive as physical. To check if the scenario is realized in nature, numerical studies are done in the framework of lattice field theory by adopting pure $SU(2)$ gauge theory for simplicity. Considering $J_{\mu }(x)=k_{\mu }(x)$ in the continuum formulation, we adopt an Abelian-like definition of a monopole following DeGrand-Toussaint as a lattice version of VNABI, since the Dirac quantization condition of the magnetic charge is satisfied on lattice partially. To reduce severe lattice artifacts, we introduce various techniques of smoothing the thermalized vacuum. Smooth gauge fixings such as the maximal center gauge (MCG), block-spin transformations of Abelian-like monopoles and extraction of physically important infrared long monopole loops are adopted. We also employ the tree-level tadpole improved gauge action of $SU(2)$ gluodynamics. With these various improvements, we measure the density of lattice VNABI: $\rho (a(\beta ),n)=\sum _{\mu ,s_n}\sqrt{\sum _a(k_{\mu }^a(s_n){)}^2}/(4\sqrt{3}{V}_n{b}^3)$, where $k_{\mu }^a(s_n)$ is an $n$ blocked monopole in the color direction $a$, $n$ is the number of blocking steps, $V_n=V/n^4$ ($b=na(\beta )$) is the lattice volume (spacing) of the blocked lattice. Beautiful and convincing scaling behaviors are seen when we plot the density $\rho (a(\beta ),n)$ versus $b=na(\beta )$. A single universal curve $\rho (b)$ is found from $n=1$ to $n=12$, which suggests that $\rho (a(\beta ),n)$ is a function of $b=na(\beta )$ alone. The universal curve seems independent of a gauge fixing procedure used to smooth the lattice vacuum since the scaling is obtained in all gauges adopted. The scaling, if it exists also for $n\to \infty$, shows that the lattice definition of VNABI has the continuum limit and the new confinement scenario is realized.

Figure 1

$b=na(\beta )$ in unit of $1/\sqrt{\sigma }$ versus $\beta$.

Figure 2

The VNABI (Abelian-like monopoles) density versus $a(\beta )$ in MCG on $48^4$. Top: total density; bottom: infrared density. $n^3$ in the legend means $n$-step blocked monopoles.

Figure 3

The VNABI (Abelian-like monopoles) density versus $b=na(\beta )$ in MCG on $48^4$. Top: total density; bottom: infrared density.

Figure 4

The fit of the infrared VNABI (Abelian-like monopoles) density data in MCG on $48^4$ lattice to Eq. (43).

Figure 5

The VNABI (Abelian-like monopole) density at $b=0.5$, 1.0, 1.5, 2.0 for different $n$ in MCG on $48^4$. The data used are derived by a linear interpolation of two nearest data below and above for the corresponding $b$ and $n$. As an example, see the original data at $b=1.0$ in Table 4.

Figure 6

The VNABI (Abelian-like monopoles) density versus $b=na(\beta )$ in AWL on $48^4$. Top: total density; bottom: infrared density.

Figure 7

The VNABI (Abelian-like monopoles) density versus $b=na(\beta )$ in DLCG on $24^4$.

Figure 8

The VNABI (Abelian-like monopoles) density versus $b=na(\beta )$ for $k^2$ and $k^3$ components in MAU1 on $48^4$. Top: total density; bottom: infrared density.

Figure 9

The VNABI (Abelian-like monopoles) density (42) versus $b=na(\beta )$ in MAU1 on $48^4$. Top: total density; bottom: infrared density.

Figure 10

Comparison of the VNABI (Abelian-like monopoles) densities versus $b=na(\beta )$ in MCG, AWL, DLCG and MAU1 cases. DLCG data only are on $24^4$ lattice. Here $\rho (b)$ is a scaling function (43) determined from the Chi-Square fit to the IF monopole density data in MCG. Top: total density; bottom: infrared density.

Figure 11

Volume dependence of VNABI (Abelian-like monopole) density in the case of MCG in $48^4$ and $24^4$ tadpole improved gauge action. The data for $3.0\le \beta \le 3.6$ and $1\le n\le 6$ alone are plotted for comparison.

Figure 12

Gauge action dependence of VNABI (Abelian-like monopole) densities in the case of DLCG in $24^4$ tadpole improved and Wilson gauge actions, The data for $3.3\le \beta \le 3.7$ and $1\le n\le 6$ alone are plotted.