Gauge-invariant formalism with a Dirac-mode expansion for confinement and chiral symmetry breaking

Using the eigenmode of the Dirac operator $\mathrm{D̸}={\gamma }^{\mu }{D}^{\mu }$ in quantum chromodynamics (QCD), we develop a manifestly gauge-covariant expansion and projection of the QCD operators such as the Wilson loop and the Polyakov loop. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD Monte Carlo calculation on $6^4$ at $\beta =5.6$. Even after removing the low-lying Dirac modes, which are responsible for chiral symmetry breaking, we find that the Wilson loop obeys the area law, and the slope parameter corresponding to the string tension, or confinement force, is almost unchanged. We find also that the Polyakov loop remains to be almost zero even without the low-lying Dirac modes, which indicates the $Z_3$-unbroken confinement phase. These results indicate that one-to-one correspondence does not hold between confinement and chiral symmetry breaking in QCD.

Figure 1

(a) The spectral density $\rho (\lambda )$ of the Dirac operator in lattice QCD at $\beta =5.6$ and $6^4$. The negative-$\lambda$ region is omitted, because of $\rho (-\lambda )=\rho (\lambda )$. The volume $V$ is multiplied. (b) The IR-cut Dirac spectral density ${\rho }_{\mathrm{IR}}(\lambda )\equiv \rho (\lambda )\theta (|\lambda |-{\Lambda }_{\mathrm{IR}})$ with the IR cutoff ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}\simeq 0.4\mathrm{GeV}$.

Figure 2

The lattice QCD result of the quark condensate $⟨\overline{q}q{⟩}_{{\Lambda }_{\mathrm{IR}}}$ as a function of the current quark mass $m$ in the presence of IR cut ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}$. The vertical axis is normalized by the original value of $⟨\overline{q}q⟩$ without cut. A large reduction is found as $⟨\overline{q}q{⟩}_{{\Lambda }_{\mathrm{IR}}}/⟨\overline{q}q⟩\simeq 0.02$ for ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}\simeq 0.4\mathrm{GeV}$ around the physical region of $m\simeq 0.006a^{-1}\simeq 5\mathrm{MeV}$.

Figure 3

The lattice QCD result of the Dirac-mode projected Wilson loop $⟨W^P(R,T)⟩\equiv \mathrm{Tr}{\widehat{W}}^P(R,T)$ after removing low-lying Dirac modes, plotted against the area $R\times T$. Circles denote the Wilson loop obtained with the IR cut of ${\rho }_{\mathrm{IR}}(\lambda )\equiv \rho (\lambda )\theta (|\lambda |-{\Lambda }_{\mathrm{IR}})$ with the IR cutoff ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}$. Squares denote the original Wilson loop $⟨W(R,T)⟩$. $⟨W^P(R,T)⟩$ seems to obey the area law with the same slope parameter, ${\sigma }^P\simeq \sigma$.

Figure 4

The effective mass plot of the interquark potential $V_{\mathrm{eff}}(R,T)\equiv \ln [⟨W^P(R,T)⟩/⟨W^P(R,T+1)⟩]$ after removing the low-lying Dirac modes. $V_{\mathrm{eff}}(R,T)$ is plotted against $T$ for each $R$. The horizontal lines denote the best-fit value in the exponential fit of Eq (39) at each $R$.

Figure 5

The lattice QCD result (circles) of the interquark potential $V^P(R)$ after removing low-lying Dirac modes below the IR cutoff ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}$. Squares denote the original interquark potential. The potential is almost unchanged even after removing the low-lying Dirac modes, apart from an irrelevant constant.

Figure 6

The scatter plots of the Polyakov loop. The left panel shows the original Polyakov loop $⟨L_P⟩$. The right panel shows the Polyakov loop $⟨L_P{⟩}_{\mathrm{IR}}$ after cutting off the low-lying Dirac modes below the IR cutoff ${\Lambda }_{\mathrm{IR}}=0.5a^{-1}$.

Figure 7

(a) The UV-cut Dirac spectral density ${\rho }_{\mathrm{UV}}(\lambda )\equiv \rho (\lambda )\theta ({\Lambda }_{\mathrm{UV}}-|\lambda |)$ with the UV cutoff ${\Lambda }_{\mathrm{UV}}=2a^{-1}\simeq 1.6\mathrm{GeV}$. (b) The UV-cut Wilson loop $\mathrm{Tr}{W}^P(R,T)$ (circles) after removing the UV Dirac modes, plotted against $R\times T$. The slope parameter ${\sigma }^P$ is almost the same as that of the original Wilson loop (squares). (c) The corresponding UV-cut interquark potential (circles), which is almost unchanged from the original one (squares), apart from an irrelevant constant.

Figure 8

The left panels show the intermediate (IM)-cut Dirac spectral density ${\rho }_{\mathrm{IM}}(\lambda )$: The IM Dirac modes of $0.5–0.8[a^{-1}]$ (top panel), $0.8–1.0[a^{-1}]$ (middle panel), and $1.0–1.2[a^{-1}]$ (bottom panel) are cut. The central panels show the IM-cut Wilson loop $\mathrm{Tr}{W}^P(R,T)$ (circles) after removing the IM Dirac modes, plotted against $R\times T$. For each case, the slope parameter ${\sigma }^P$ is almost the same as that of the original Wilson loop (squares). The right panels show the corresponding IM-cut interquark potential (circles), which is almost unchanged from the original one (squares), apart from an irrelevant constant.