Spontaneous chiral-symmetry breaking in four-dimensional QCD with $A_x=A_y=0$ projection in dimensional reduction gauge

We investigate spontaneous chiral-symmetry breaking in four-dimensional (4D) QCD in the dimensional reduction (DR) gauge, defined so as to minimize $R_{\mathrm{D}\mathrm{R}}\equiv \int d^{4}s\mathrm{T}\mathrm{r}[A_x^2(s)+A_y^2(s)]$. We consider $tz$-projection" of $A_{x,y}(s) \to 0$ and $xy$-projection" of $A_{t,z}(s)\to 0$, respectively, for the gauge configuration generated in the DR gauge, in a similar sense to Abelian, monopole and photon projections in the maximally Abelian gauge. The chiral condensate is estimated from the low-lying Dirac-eigenvalue density $\rho (\lambda )$ using the Banks-Casher relation. By the $tz$-projection in the DR gauge, the chiral condensate is largely reserved. In contrast, the chiral condensate becomes zero by the $xy$-projection in the DR gauge. This indicates that $A_t(s)$ and $A_z(s)$ play a dominant role, while $A_x(s)$ and $A_y(s)$ play no role, in spontaneous chiral-symmetry breaking in the DR gauge.