Analytical Formulae of the Polyakov and the 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.

The Polyakov loop L P is one of the typical order parameters, and it relates to the single-quark free energy E q as L P ∝ e −Eq/T at temperature T . The Polyakov loop is also an order parameter of spontaneous breaking of the Z Nc center symmetry in QCD [5]. Also, its fluctuation is recently found to be important in the QCD phase transition [6].
In addition to the study of each nonperturbative phenomenon, to clarify the relation between confinement and chiral symmetry breaking is one of the challenging important subjects in theoretical physics [7][8][9][10][11][12][13][14][15][16], and their relation is not yet clarified directly from QCD.
A strong correlation between confinement and chiral symmetry breaking has been suggested by almost coincidence between deconfinement and chiral-restoration temperatures [5,17], although slight difference of about 25MeV between them is pointed out in recent lattice QCD studies [18]. Their correlation has been also suggested in terms of QCD-monopoles [7][8][9], which topologically appear in QCD in the maximally Abelian gauge. By removing the monopoles from the QCD vacuum, confinement and chiral symmetry breaking are simultaneously lost [7][8][9], which indicates an important role of QCDmonopoles to both phenomena, and thus these two phenomena seem to be related via the monopole.
As another type of pioneering study, Gattringer and Bruckmann et al. showed that the Polyakov loop can be analytically expressed with the Dirac eigenvalues under the temporally twisted boundary condition for temporal link-variables [10]. Although temporal (nontwisted) periodic boundary condition is physically required for linkvariables in real QCD at finite temperature, such an analytical relation would be useful to consider the relation between confinement and chiral symmetry breaking.
In a series of our recent studies [12][13][14], we have numerically investigated the Wilson loop and the Polyakov loop in terms of the "Dirac-mode expansion", and have found that quark confinement properties are almost kept even in the absence of low-lying Dirac modes. (Also, "hadrons" appear without low-lying Dirac modes [19], suggesting survival of confinement.) Note that the Diracmode expansion is just a mathematical expansion by eigenmodes |n of the Dirac operator D = γ µ D µ , using the completeness of n |n n| = 1. In general, instead of D, one can consider any (anti)hermitian operator, e.g., D 2 = D µ D µ , and the expansion in terms of its eigenmodes [20]. To investigate chiral symmetry breaking, however, it is appropriate to consider D and the expansion by its eigenmodes.
In this paper, we derive an analytical relation between the Polyakov loop and the Dirac modes using the lattice QCD formalism where the temporal lattice size is odd [15,16], and discuss the relation between confinement and chiral symmetry breaking.
The organization of this paper is as follows. In Sect. II, we briefly review the lattice QCD formalism for the Dirac operator, Dirac eigenvalues and Dirac modes. In Sect. III, we derive an analytical relation between the Polyakov loop and the Dirac modes in lattice QCD where the temporal size is odd-number. In Sect. IV, we investigate the properties of the obtained relation, and discuss the contribution from the low-lying Dirac modes to the Polyakov loop. In Sect. V, we consider the relation between the Wilson loop and Dirac modes on arbitrary square lattices, without restriction of odd-number size. Section VI will be devoted to the summary.

II. LATTICE QCD FORMALISM
To begin with, we state the setup condition of lattice QCD formalism adopted in this study. We use an ordinary square lattice with spacing a and size N 3 s × N t . The normal nontwisted periodic boundary condition is used for the link-variable U µ (s) = e iagAµ(s) in the temporal direction, with the gluon field A µ (s), the gauge coupling g and the site s. This temporal periodicity is physically required at finite temperature. In this paper, we take SU(N c ) with N c being the color number as the gauge group of the theory. However, arbitrary gauge group G can be taken for most arguments in the following.

A. Dirac operator, Dirac eigenvalues and Dirac modes in lattice QCD
In lattice QCD, the Dirac operator D = γ µ D µ is expressed with U µ (s) = e iagAµ(s) . In our study, we take the lattice Dirac operator of whereμ is the unit vector in µ-direction in the lattice unit, and U −µ (s) ≡ U † µ (s −μ). Adopting hermitian γ-matrices as γ † µ = γ µ , the Dirac operator D is antihermitian and satisfies D † s ′ ,s = − D s,s ′ . We introduce the normalized Dirac eigen-state |n as with the Dirac eigenvalue iλ n (λ n ∈ R). Because of {γ 5 , D} = 0, the state γ 5 |n is also an eigen-state of D with the eigenvalue −iλ n . Here, the Dirac eigen-state |n satisfies the completeness of n |n n| = 1.
For the Dirac eigenfunction ψ n (s) ≡ s|n , the explicit form of the Dirac eigenvalue equation Dψ n (s) = iλ n ψ n (s) in lattice QCD is written by The Dirac eigenfunction ψ n (s) can be numerically obtained in lattice QCD, besides a phase factor. By the gauge transformation of U µ (s) → V (s)U µ (s)V † (s +μ), ψ n (s) is gauge-transformed as which is the same as that of the quark field, although, to be strict, there can appear an irrelevant n-dependent global phase factor e iϕn[V ] , according to arbitrariness of the phase in the basis |n [13]. Note that the spectral density ρ(λ) of the Dirac operator D relates to chiral symmetry breaking. For example, from Banks-Casher's relation [4], the zero-eigenvalue density ρ(0) leads to qq as with space-time volume V phys . Thus, the low-lying Dirac modes can be regarded as the essential modes responsible to spontaneous chiral-symmetry breaking in QCD.
With the link-variable operator, the Dirac operator and covariant derivative are simply expressed as The Polyakov loop L P is also simply written as the functional trace ofÛ Nt 4 , with the four-dimensional lattice volume V ≡ N 3 s × N t andt =4. Here, "Tr c " denotes the functional trace of Tr c ≡ s tr c with the trace tr c over color index.
The Dirac-mode matrix element of the link-variable operatorÛ µ can be expressed with ψ n (s): Note that the matrix element is gauge invariant, apart from an irrelevant phase factor. Actually, using the gauge transformation (5), we find the gauge transformation of the matrix element as [13] To be strict, there appears an n-dependent global phase factor, corresponding to the arbitrariness of the phase in the basis |n . However, this phase factor cancels as e iϕn e −iϕn = 1 between |n and n|, and does not appear for physical quantities such as the Wilson loop and the Polyakov loop [13].

III. ANALYTICAL RELATION BETWEEN POLYAKOV LOOP AND DIRAC MODES IN LATTICE QCD WITH ODD TEMPORAL SIZE
Now, we consider lattice QCD with odd-number temporal lattice size N t , as shown in Fig.1. Here, we use an ordinary square lattice with the normal nontwisted periodic boundary condition for the link-variable in the temporal direction. (Of course, this temporal periodicity is physically required at finite temperature.) The spatial lattice size N s is taken to be larger than N t , i.e., N s > N t . Note that, in the continuum limit of a → 0 and N t → ∞, any number of large N t gives the same physical result. Then, in principle, it is no problem to use the odd-number lattice. In general, only gauge-invariant quantities such as closed loops and the Polyakov loop survive in QCD, according to the Elitzur theorem [5]. All the non-closed lines are gauge-variant and their expectation values are zero. Note here that any closed loop, except for the Polyakov loop, needs even-number link-variables on the square lattice, as shown in Fig.1.
Note also that, from the definition of the link-variable operatorÛ µ in Eq. (7), the functional trace of the product ofÛ µ along any non-closed trajectory is zero, i.e., for the non-closed trajectory with N k=1μ k = 0. In lattice QCD with odd-number temporal size N t , we consider the functional trace of where Tr c,γ ≡ s tr c tr γ includes tr c and the trace tr γ over spinor index. Its expectation value is obtained as the gauge-configuration average in lattice QCD. In the case of enough large volume V , one can expect O ≃ Tr O/Tr 1 for any operator O at each gauge configuration.
From Eq.(8),Û 4 D Nt−1 is expressed as a sum of products of N t link-variable operators, since the Dirac operator D includes one link-variable operator in each direction of ±µ. Then,Û 4 D Nt−1 includes many trajectories with the total length N t in the lattice unit on the square lattice, as shown in Fig.2. Note that all the trajectories with the odd-number length N t cannot form a closed loop on the square lattice, and thus give gauge-variant contribution, except for the Polyakov loop.  Actually, we can mathematically derive the following relation: We thus obtain the relation between I = Tr c,γ (Û 4ˆ D Nt−1 ) and the Polyakov loop L P , On the other hand, we calculate the functional trace in Eq. (14)  Combing Eqs. (16) and (17), we obtain the analytical relation between the Polyakov loop L P and the Dirac eigenvalues iλ n : for each gauge configuration.
Taking the gaugeconfiguration average, we obtain This is a direct relation between the Polyakov loop L P and the Dirac modes in QCD, and is mathematically valid in lattice QCD with odd-number temporal size in both confinement and deconfinement phases. The relation (18) is a Dirac spectral representation of the Polyakov loop, and we can investigate each Dirac-mode contribution to the Polyakov loop individually, based on Eq. (18). (For example, each contribution specified by n is numerically calculable in lattice QCD.) In particular, from Eq.(18), we can discuss the relation between confinement and chiral symmetry breaking in QCD. As a remarkable fact, because of the factor λ Nt−1 n , the contribution from low-lying Dirac-modes with |λ n | ≃ 0 is negligibly small in the Dirac spectral sum of RHS in Eq. (18), compared with the other Dirac-mode contribution. In fact, the low-lying Dirac modes have fairly small contribution to the Polyakov loop, regardless of confinement or deconfinement phase. This is consistent with the previous numerical lattice result that confinement properties are almost unchanged by removing low-lying Dirac modes from the QCD vacuum [12][13][14].

IV. DISCUSSIONS ON THE DIRAC SPECTRAL REPRESENTATION OF THE POLYAKOV LOOP
In this section, we consider the Dirac spectral representation of the Polyakov loop, i.e., the relation (18) between the Polyakov loop and Dirac modes, and discuss its physical meaning. In particular, we consider the contribution from low-lying Dirac modes to the Polyakov loop.

A. Properties of the relation between Polyakov loop and Dirac modes
First, we note that Eq. (18) is a manifestly gaugeinvariant relation. Actually, the matrix element n|Û 4 |n can be expressed with the Dirac eigenfunction ψ n (s) and the temporal link-variable U 4 (s) as and each term ψ † n (s)U 4 (s)ψ n (s +t) is manifestly gauge invariant, because of the gauge transformation property (5). Here, the irrelevant global phase factors also cancel exactly as e −iϕn e iϕn = 1 between n| and |n [12][13][14].
Third, Eq. (18) is correct for any odd number N t (> 1) and is applicable to both confinement and deconfinement phases. Then, Eq.(18) obtained on the odd-number lattice is expected to hold in the continuum limit of a → 0 and N t → ∞, since any number of large N t gives the same physical result.
Finally, we comment on generality and wide applicability of Eq. (18). In the argument to derive Eq. Note also that Eq. (18) is correct regardless of presence or absence of dynamical quarks. In fact, Eq.(18) can be derived also for the gauge configuration after integrating out quark degrees of freedom. Of course, the dynamical quark effect appears in the Polyakov loop L P , the Dirac eigenvalue distribution ρ(λ) and n|Û 4 |n . However, the relation (18) holds even in the presence of dynamical quarks. Therefore, the relation (18)  In this subsection, we consider the contribution from low-lying Dirac modes to the Polyakov loop based on Eq. (18). Due to the factor λ Nt−1 n , the contribution from low-lying Dirac-modes with |λ n | ≃ 0 is negligibly small in RHS in Eq. (18), compared with the other Dirac-mode contribution, so that the low-lying Dirac modes have small contribution to the Polyakov loop in both confinement and deconfinement phases.
If RHS in Eq. (18) were not a sum but a product, lowlying Dirac modes, or the small |λ n | region, should have given an important contribution to the Polyakov loop as a crucial reduction factor of λ Nt−1 n . In the sum, however, the contribution (∝ λ Nt−1 n ) from the small |λ n | region is negligible.
Even if n|Û 4 |n behaves as the δ-function δ(λ), the factor λ Nt−1 n is still crucial in RHS of Eq.(18), because of λδ(λ) = 0. In fact, without appearance of extra counter factor λ −(Nt−1) n from n|Û 4 |n , the crucial factor λ Nt−1 n inevitably leads to small contribution for low-lying Dirac modes. Note here that the explicit N t -dependence appears as the factor λ Nt−1 n in RHS of Eq. (18), and the matrix element n|Û 4 |n does not include N t -dependence in an explicit manner. Then, it seems rather difficult to consider the appearance of the counter factor λ −(Nt−1) n from the matrix element n|Û 4 |n .
One may suspect the necessity of renormalization for the Polyakov loop, although the Polyakov loop is at present one of the typical order parameters of confinement, and most arguments on the QCD phase transition have been done in terms of the simple Polyakov loop. Even in the presence of a possible multiplicative renormalization factor for the Polyakov loop, the contribution from the low-lying Dirac modes, or the small |λ n | region, is relatively negligible compared with other Dirac-mode contribution in the sum of RHS in Eq.(18).

C. Numerical confirmation with lattice QCD
It is notable that all the above arguments can be numerically confirmed by lattice QCD calculations. In this subsection, we briefly mention the numerical confirmation with lattice QCD Monte Carlo calculations [16].
Using actual lattice QCD calculations at the quenched level, we numerically confirm the analytical relation (18), non-zero finiteness of n|Û 4 |n for each Dirac mode, and the negligibly small contribution of low-lying Dirac modes to the Polyakov loop, in both confinement and deconfinement phases [16].
As for the matrix element n|Û 4 |n , its behavior drastically changes between confinement and deconfinement phases. In the confinement phase, we find a "positive/negative symmetry" on the distribution of the matrix element n|Û 4 |n [16], and this symmetry is one of the essence to realize the zero value of the Polyakov loop L P . In fact, due to this symmetry of n|Û 4 |n in the confinement phase, the contribution from partial Dirac modes in arbitrary region a ≤ λ n ≤ b leads to L P = 0, which is consistent with our previous studies [12][13][14]. In contrast, in the deconfinement phase, there is no such symmetry on the distribution of n|Û 4 |n , and this asymmetry leads to a non-zero value of the Polyakov loop [16].
In any case, regardless of the behavior of n|Û 4 |n , we numerically confirm that the contribution from low-lying Dirac modes to the Polyakov loop is negligibly small [16] in both confinement and deconfinement phases, owing to the factor λ Nt−1 n in Eq. (18). From the analytical relation (18) and the numerical confirmation, we conclude that low-lying Dirac-modes have small contribution to the Polyakov loop, and are not essential for confinement, while these modes are essential for chiral symmetry breaking. This conclusion indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.

V. SIMILAR RELATION FOR THE WILSON LOOP ON ARBITRARY SQUARE LATTICES
In this section, we attempt a similar consideration to the Wilson loop and the string tension on arbitrary square lattices (including anisotropic cases) with any number of N t , i.e., without restriction of odd-number size. We consider the ordinary Wilson loop on a R × T rectangle, where T and R are arbitrary positive integers. The Wilson loop is expressed by the functional trace [12,13] W ≡ Tr cÛ where we introduce the "staple operator"Û staple aŝ Here, the Wilson-loop operator is factorized as a product ofÛ staple andÛ T 4 , as shown in Fig.4. We note that W ∝ W gauge ave. for enough large volume lattice [12,13].
In the case of even number T , let us consider the functional trace of Again, owing to the factor λ T n , the contribution from lowlying Dirac modes is expected to be small also for the Wilson loop, although the matrix element n|Û staple |n includes explicit T -dependence and its behavior is not so clear, unlike the formula (18) for the Polyakov loop.
In the case of odd number T , the similar results can be obtained by considering instead of J. Actually, one finds so that one finds for odd T the similar relation of Finally, for even T case, we consider the inter-quark potential V (R) and the string tension σ. From the expression (26) for the Wilson loop W , we obtain the interquark potential V (R) and the string tension σ: Because of the factor λ T n in the sum, the low-lying Diracmode contribution is to be small for the Wilson loop W , the inter-quark potential V (R) and the string tension σ, unless the extra counter factor λ −T n appears from n|Û staple |n . Also for odd T case, similar arguments can be done with Eq.(30).
In this way, the string tension σ, or the confining force, is expected to be unchanged by the removal of the lowlying Dirac-mode contribution, which is consistent with our previous studies [12,13].

VI. SUMMARY AND CONCLUDING REMARKS
We have derived an analytical gauge-invariant relation between the Polyakov loop L P and the Dirac eigenvalues λ n in QCD, i.e., L P ∝ n λ Nt−1 n n|Û 4 |n , by considering Tr(Û 4ˆ D Nt−1 ) in lattice QCD formalism with odd-number temporal size N t . This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes |n . Here, we have used an ordinary square lattice with the normal nontwisted periodic boundary condition for link-variables U µ (s) in the temporal direction. From this relation, one can estimate each contribution of the Dirac eigenmode to the Polyakov loop.
Because of the factor λ Nt−1 n in the Dirac spectral sum, this analytical relation indicates quite small contribution of low-lying Dirac modes to the Polyakov loop in both confinement and deconfinement phases, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in confinement and deconfinement phases, we have numerically confirmed the analytical relation, non-zero finiteness of n|Û 4 |n for each Dirac mode, and negligibly small contribution of 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.
Finally, we have considered the relation between the Wilson loop and Dirac modes on arbitrary lattices, without restriction of odd-number size, and have found small contribution of low-lying Dirac modes to the string tension σ, or the confining force, i.e., σ is expected to be unchanged by the removal of the low-lying Dirac-mode contribution.
Thus, we conclude that low-lying Dirac-modes have quite small contribution to the Polyakov loop, and are not essential modes for confinement, while these modes are essential for chiral symmetry breaking. This conclusion indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.
As a next work, it is meaningful to perform the similar analysis for the Polyakov-loop fluctuation in terms of the Dirac modes, since such fluctuation gives a clear signal of the QCD phase transition at finite temperature [6]. It is also interesting to compare with other lattice QCD result on the important role of infrared gluons (below about 1GeV) for confinement in the Landau gauge [21], in contrast to the insensitivity of confinement against lowlying Dirac-modes.
This work indicates some independence between confinement and chiral symmetry breaking, and this may lead to richer phase structure of QCD in various environment.