Polyakov Loop In Terms Research Articles

Relating quark confinement and chiral symmetry breaking in QCD

We study the relation between quark confinement and chiral symmetry breaking in QCD. Using lattice QCD formalism, we analytically express the various ‘confinement indicators’, such as the Polyakov loop, its fluctuations, the Wilson loop, the inter-quark potential and the string tension, in terms of the Dirac eigenmodes. In the Dirac spectral representation, there appears a power of the Dirac eigenvalue such as , which behaves as a reduction factor for small . Consequently, since this reduction factor cannot be cancelled, the low-lying Dirac eigenmodes give negligibly small contribution to the confinement quantities, while they are essential for chiral symmetry breaking. These relations indicate that there is no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD. In other words, there is some independence of quark confinement from chiral symmetry breaking, which can generally lead to different transition temperatures/densities for deconfinement and chiral restoration. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain-wall fermion kernels, and find similar results. The independence of quark confinement from chiral symmetry breaking seems to be natural, because confinement is realized independently of quark masses and heavy quarks are also confined even without the chiral symmetry.

Interplay Between Deconfinement and Chiral Properties

We study interplay between confinement/deconfinement and chiral properties. We derive some analytical relations of the Dirac modes with the confinement quantities, such as the Polyakov loop, its susceptibility and the string tension. For the confinement quantities, the low-lying Dirac eigenmodes are found to give negligible contribution, while they are essential for chiral symmetry breaking. This indicates no direct, one-to-one correspondence between confinement/deconfinement and chiral properties in QCD. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain-wall fermion kernels, respectively.

Some Relations for Quark Confinement and Chiral Symmetry Breaking in QCD

We analytically study the relation between quark confinement and spontaneous chiral-symmetry breaking in QCD. In terms of the Dirac eigenmodes, we derive some formulae for the Polyakov loop, its fluctuations, and the string tension from the Wilson loop. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain wall fermion kernels, respectively. For the confinement quantities, the low-lying Dirac/fermion eigenmodes are found to give negligible contribution, while they are essential for chiral symmetry breaking. These relations indicate no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD, which seems to be natural because confinement is realized independently of the quark mass.

Analytical formulae of the Polyakov and Wilson loops with Dirac eigenmodes in lattice QCD

We derive an analytical gauge-invariant formula between the Polyakov loop $L_P$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $L_P \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, in ordinary periodic square lattice QCD with odd-number temporal size $N_t$. Here, $|n\rangle$ denotes the Dirac eigenstate, and $\hat U_4$ temporal link-variable operator. This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. Because of the factor $\lambda_n^{N_t -1}$ in the Dirac spectral sum, this formula indicates negligibly small contribution of low-lying Dirac modes to the Polyakov loop in both confinement and deconfinement phases, while these modes are essential for chiral symmetry breaking. Next, we find a similar formula between the Wilson loop and Dirac modes on arbitrary square lattices, without restriction of odd-number size. This formula suggests a small contribution of low-lying Dirac modes to the string tension $\sigma$, or the confining force. These findings support no crucial role of low-lying Dirac modes for confinement, i.e., no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD, which seems to be natural because heavy quarks are also confined even without light quarks or the chiral symmetry.

Lattice QCD analysis of the Polyakov loop in terms of Dirac eigenmodes

Using the Dirac-mode expansion method, which keeps the gauge invariance, we analyze the Polyakov loop in terms of the Dirac modes in SU(3) quenched lattice QCD in both confined and deconfined phases. First, to investigate the direct correspondence between confinement and chiral symmetry breaking, we remove low-lying Dirac-modes from the confined vacuum generated by lattice QCD. In this system without low-lying Dirac modes, while the chiral condensate $\langle \bar{q} q\rangle$ is extremely reduced, we find that the Polyakov loop is almost zero and $Z_3$-center symmetry is unbroken, which indicates quark confinement. We also investigate the removal of ultraviolet (UV) Dirac-modes, and find that the Polyakov loop is almost zero. Second, we deal with the deconfined phase above $T_c$, and find that the behaviors of the Polyakov loop and $Z_3$-symmetry are not changed without low-lying or UV Dirac-modes. Finally, we develop a new method to remove low-lying Dirac modes from the Polyakov loop for a larger lattice of $12^3 \times 4$ at finite temperature, and find almost the same results. These results suggest that each eigenmode has the information of confinement, i.e., the "seed" of confinement is distributed in a wider region of the Dirac eigenmodes unlike chiral symmetry breaking, and there is no direct correspondence between confinement and chiral symmetry breaking through Dirac-eigenmodes.

Analytical relation between quark confinement and chiral symmetry breaking in odd-number lattice QCD

To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size $N_t$ being odd. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. By considering ${\rm Tr} (\hat{U}_4\hat{\not D}^{N_t-1})$, we analytically derive a gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. Owing to the factor $\lambda_n^{N_t -1}$ in the Dirac spectral sum, this relation generally indicates fairly small contribution of low-lying Dirac modes to the Polyakov loop, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in both confined and deconfined phases, we numerically confirm the analytical relation, non-zero finiteness of $\langle n|\hat U_4|n \rangle$ for each Dirac mode, and negligibly small contribution from low-lying Dirac modes to the Polyakov loop, i.e., the Polyakov loop is almost unchanged even by removing low-lying Dirac-mode contribution from the QCD vacuum generated by lattice QCD simulations. We thus conclude that low-lying Dirac modes are not essential modes for confinement, which indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.

Critical exponents and abelian dominance in SU(2) QCD

The critical properties of the abelian Polyakov loop and the Polyakov loop in terms of Dirac string are studied in finite temperature abelian projected SU(2) QCD. We evaluate the critical point and the critical exponents from each Polyakov loop in the maximally abelian gauge using the finite-size scaling analysis. Abelian dominance in this case is proved quantitatively. The critical point of each abelian Polyakov loop is equal to that of the non-abelian Polyakov loop within the statistical errors. Also, the critical exponents are in good agreement with those from non-abelian Polyakov loops.

Critical exponents in abelian projected SU(2) QCD

The critical properties of the abelian Polyakov loop and the Polyakov loop in terms of Dirac string are studied in finite temperature abelian projected SU(2) QCD. The critical point and the critical exponents are determined from each Polyakov loop in the maximally abelian gauge using the finite-size scaling analyses. Those critical points and exponents are in good agreement with those from non-abelian Polyakov loops.