Relevant gluonic energy scale of spontaneous chiral symmetry breaking from lattice QCD

We analyze which momentum component of the gluon field induces spontaneous chiral symmetry breaking in lattice QCD. After removing the high-momentum or low-momentum component of the gluon field, we calculate the chiral condensate and observe the roles of these momentum components. The chiral condensate is found to be drastically reduced by removing the zero-momentum gluon. The reduction is about 40% of the total in our calculation condition. The nonzero-momentum infrared gluon also has a sizable contribution to the chiral condensate. From the Banks-Casher relation, this result reflects the nontrivial relation between the infrared gluon and the zero-mode quark.

Figure 1

The schematic figure of momentum space. The shaded regions are the cut regions by the ultraviolet cutoff ${\Lambda }_{\mathrm{UV}}$ and the infrared cutoff ${\Lambda }_{\mathrm{IR}}$. The momentum-space lattice spacing is $a_p=2\pi /La$.

Figure 2

The chiral condensate $\Sigma \equiv -a^3⟨\overline{q}q⟩/N_f$ with the UV cutoff ${\Lambda }_{\mathrm{UV}}$ The quark mass is $ma=0.01$. The “renormalized” chiral condensate is multiplied by the renormalization factor $Z_S$.

Figure 3

The full QCD result of the chiral condensate $\Sigma \equiv -a^3⟨\overline{q}q⟩/N_f$ with the IR cutoff ${\Lambda }_{\mathrm{IR}}$. The lattice volume is ${16}^3\times 32$, and the quark mass is $ma=0.01$. PBC and APBC mean periodic and antiperiodic boundary conditions, respectively.

Figure 4

The quenched QCD result of the chiral condensate with the IR cutoff. The lattice volume is ${32}^4$, and the quark mass is $ma=0.01$. The notation is the same as Fig. 3.

Figure 5

The chiral extrapolation of the chiral condensate $\Sigma \equiv -a^3⟨\overline{q}q⟩/N_f$. ${\Lambda }_{\mathrm{IR}}\sim 0.1\mathrm{GeV}$ corresponds to the cutoff for the zero-momentum link variable.