$⟨A^2⟩$ condensate, Bianchi identities, and chromomagnetic field degeneracy in SU(2) Yang-Mills theory

We consider the non-Abelian Bianchi identities in SU(2) pure Yang-Mills theory in $D=3,4$ focusing on the possibility of their violation and the significance of the chromomagnetic fields degeneracy points. We show that the recently proposed non-Abelian Stokes theorem allows one to formulate the Bianchi identities in terms of the physical fluxes and their relative color orientations. Then the violation of Bianchi identities becomes a well defined concept ultimately related to the degeneracy points. The locality and gauge invariance of our approach allows one to study the problem numerically. We present evidence that in $D=4$ the suppression of the Bianchi identities violation is likely to destroy confinement, while the removal of the degeneracy points drives the theory to the topologically nontrivial sector. However, confronting the results obtained in three and four dimensions, we argue that it is the mass dimension two condensate $⟨A_{\min }^2⟩$ which probably explains our findings.

Figure 1

Graphical representation of lattice Bianchi identities.

Figure 2

Segment of the Wilson loop $W(C)$ in the original and ribbonlike representation. The operator in between solid blobs is $U_{k,k+1}$, Eq. (23).

Figure 3

${\Omega }_x$ (left) and ${\gamma }_x$ (right) terms in Eq. (27). Arrows correspond to the orientation induced by $C$.

Figure 4

Left: The non-Abelian Stokes theorem in application to the closed contour $C$ which bounds two distinct surfaces $S_C$, $S_C^{\prime }$, $\delta S_C=\delta S_C^{\prime }=C$. Arrows indicate the order of plaquette vertices induced by the orientation of $C$. Right: The non-Abelian Bianchi identities for a single lattice cube (see the text).

Figure 5

Left: The flux directions in the plane $(\mu ,\nu )$ around point $x$ in the weak coupling limit, Eq. (44). Right: The only nontrivial 3-cell in $D=3$ (see the text).

Figure 6

The densities (47, 48) versus $\beta$ coupling. The lines are drawn to guide the eye.

Figure 7

The densities (47, 48) and the mean plaquette $⟨1/2\mathrm{Tr}{U}_p⟩$ on the line $\tilde{\gamma }=0$ as functions of $\gamma$.

Figure 8

The correlation function of Polyakov lines and the heavy quark potential in $D=3$ at $\tilde{\gamma }=0$ and various $\gamma$.

Figure 9

The Polyakov lines correlator in $D=4$ at $\tilde{\gamma }=0$ and various $\gamma$.

Figure 10

The heavy quark potential $V(R)$ and the topological susceptibility $\chi$ in $D=4$ at $\tilde{\gamma }=0$ and various $\gamma$.

Figure 11

The oscillations of Wilson loops $⟨W(R,T)⟩$ at $\gamma =0$, $\tilde{\gamma }=4$ and the Monte Carlo history of the topological charge at $\gamma =0$ in $D=4$.

Figure 12

The quantity $⟨A^2⟩$ in the Landau gauge at $\tilde{\gamma }=0$ as function of $\gamma$ in three and four dimensions.

Figure 13

${\mathcal{D}}^{(2)}$ 3-cell in $D=4$ (see the text).