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.


Introduction
Color confinement and spontaneous chiral-symmetry breaking [1] are two outstanding nonperturbative phenomena in QCD, and they and their relation [2][3][4] have been studied as one of the important difficult problems in theoretical particle physics. For quark confinement, the Polyakov loop L P is one of the typical order parameters, which relates to the single-quark free energy E q as L P ∝ e −E q /T at temperature T . For chiral symmetry breaking, the standard order parameter is the chiral condensate qq , and low-lying Dirac modes are known to play the essential role [5]. A strong correlation between confinement and chiral symmetry breaking has been suggested by approximate coincidence between deconfinement and chiral-restoration temperatures [6]. Their correlation has been also suggested in terms of QCD-monopoles [7,8], which topologically appear in QCD in the maximally Abelian (MA) gauge. By removing the monopoles from the QCD vacuum, confinement and chiral symmetry breaking are simultaneously lost in lattice QCD [7,8]. (See Fig. 1.) This indicates an important role of QCD-monopoles to both confinement and chiral symmetry breaking, and thus these two phenomena seem to be related via the monopole. However, the direct relation of confinement and chiral symmetry breaking is still unclear.
Actually, an accurate lattice QCD study [9] shows about 25MeV difference between the deconfinement and the chiral-restoration temperatures, i.e., T deconf ≃ 176MeV and T chiral ≃ 151MeV. We also note that some QCD-like theories exhibit a large difference between confinement and chiral symmetry breaking. For example, in an SU(3) gauge theory with adjoint-color fermions, the chiral transition occurs at much higher temperature, T chiral ≃ 8T deconf [10]. In 1+1 QCD with N f ≥ 2, confinement is realized, but spontaneous chiral symmetry breaking does not occur, because of the Coleman-Mermin-Wagner theorem. Also in N = 1 SUSY 1+3 QCD with N f = N c + 1, while confinement is realized, chiral symmetry breaking does not occur. A recent lattice study of SU(2)-color QCD with N f = 2 shows that a confined but chiral-restored phase is realized at a large baryon density [11].  Figure 1. In the MA gauge, monopoles topologically appear. By removing the monopole from the QCD vacuum, confinement and chiral symmetry breaking are simultaneously lost [7,8]. This means crucial role of monopoles to both confinement and chiral symmetry breaking, but does not mean the direct correspondence between them.
In this paper, considering the essential role of low-lying Dirac modes to chiral symmetry breaking [6], we derive analytical relations between the Dirac modes and the confinement quantities, e.g., the Polyakov loop [2], its fluctuations [3] and the string tension [4], in the lattice QCD formalism.

Dirac operator, Dirac eigenvalues and Dirac modes in lattice QCD
In this paper, we take an ordinary square lattice with spacing a and the size V ≡ N 3 s ×N t , and impose the standard temporal periodicity/anti-periodicity for gluons/quarks. In lattice QCD, the gauge variable is expressed as the link-variable U µ (s) ≡ e iagA µ (s) with the gauge coupling g and the gluon field A µ (x), and the simple Dirac operator and the covariant derivative operator are given as where the link-variable operatorÛ ±µ [2][3][4]12] is defined by with U −µ (s) ≡ U † µ (s −μ). This simple Dirac operatorˆ D is anti-hermite and satisfieŝ We define the normalized Dirac eigenmode |n and the Dirac eigenvalue λ n , D|n = iλ n |n (λ n ∈ R), m|n = δ mn .
Because of anti-hermiticity of D, the Dirac eigenmode |n satisfies the complete-set relation, n |n n| = 1.
In lattice QCD, any functional trace becomes a sum over all the space-time site, i.e., Tr = s tr, which is defined for each lattice gauge configuration. For enough large volume lattice, e.g., N s → ∞, the functional trace is proportional to the gauge ensemble average, TrÔ = s trÔ ∝ Ô gauge ave. , for any operatorÔ. Note also that, because of the definition ofÛ ±µ in Eq.(2), the functional trace Tr(Û µ 1Û µ 2 · · ·Û µ N ) of any product of link-variable operators corresponding to "non-closed line" is exactly zero [2][3][4] at each lattice gauge configuration, before taking the gauge ensemble average.
In this paper, we mainly use the lattice unit, a = 1, for the simple notation.

Polyakov loop and Dirac modes in temporally odd-number lattice QCD
To begin with, we study the Polyakov loop and Dirac modes in temporally odd-number lattice QCD [2][3][4], where the temporal lattice size N t (< N s ) is odd. In general, only gauge-invariant quantities such as closed loops and the Polyakov loop survive in QCD, according to the Elitzur theorem [6]. All the non-closed lines are gauge-variant and their expectation values are zero. Now, we consider the functional trace [2][3][4], where Tr c,γ ≡ s tr c tr γ includes the sum over all the four-dimensional site s and the traces over color and spinor indices. In Eq.(6), we have used the completeness of the Dirac mode, n |n n| = 1. From Eq.(1),Û 4ˆ D N t −1 is expressed as a sum of products of N t link-variable operators. Then, U 4ˆ D N t −1 includes many trajectories with the total length N t , as shown in Fig. 2.
where the last minus reflects the temporal anti-periodicity of D [4]. (Because of Eq.(2), the functional trace of "non-closed line" is exactly zero [2][3][4] at each lattice gauge configuration.) In this way, we obtain the analytical relation between the Polyakov loop L P and the Dirac modes in QCD on the temporally odd-number lattice [2][3][4],

Polyakov-loop fluctuations and Dirac eigenmodes
Next, we investigate Polyakov-loop fluctuations. As shown in Fig. 3(a), the Polyakov loop L P has fluctuations in longitudinal and transverse directions. We define its longitudinal and transverse components, L L ≡ ReL P and L T ≡ ImL P , withL P ≡ L P e 2πik/3 where k ∈ {0, ±1} is chosen such that the Z 3 -transformed Polyakov loop lies in its real sector [3,13]. We introduce the Polyakov-loop fluctuations as Some ratios of them largely change around the transition temperature, and can be good indicators of the QCD transition [13]. The lattice QCD result for the infrared Dirac-mode cut quantities of R conf (Λ IRcut ) ≡ R A (Λ IRcut )/R A and R chiral (Λ IRcut ) ≡ qq Λ IRcut / qq plotted against the infrared cutoff Λ IRcut introduced on Dirac eigenvalues [3]. This figure is taken from Ref. [3]. In contrast to the sensitivity of the chiral condensate R chiral , the Polyakov-loop fluctuation ratio R conf is almost unchanged against the infrared cutoff Λ IRcut of the Dirac mode.
In temporally odd-number lattice QCD, we derive Dirac-mode expansion formulae for Polyakovloop fluctuations [3]. For example, the Dirac spectral representation of the ratio R A ≡ χ A /χ L is  .
Because of the reduction factor λ N t −1 n in the Dirac-mode sum, all the Polyakov-loop fluctuations are almost unchanged by removing low-lying Dirac modes [3].
As the demonstration, we show in Fig. 3(b) the lattice QCD result of R conf (Λ IRcut ) ≡ R A (Λ IRcut )/R A and R chiral (Λ IRcut ) ≡ qq Λ IRcut / qq in the presence of the infrared Dirac-mode cutoff Λ IRcut introduced on Dirac eigenvalues [3]. As the Dirac-mode cutoff Λ IRcut increases, the chiral condensate R chiral rapidly reduces, but the Polyakov-loop fluctuation ratio R conf is almost unchanged [3].

The Wilson loop and Dirac modes on arbitrary square lattices
In this section, we investigate the Wilson loop and the string tension in terms of the Dirac modes, on arbitrary square lattices with any number of N t [4]. We consider the ordinary Wilson loop of the R×T rectangle. The Wilson loop on the x i -t (i = 1, 2, 3) plane is expressed by the functional trace, where we introduce the "staple operator"Û staple defined bŷ In fact, the Wilson-loop operator is factorized as a product ofÛ staple andÛ T 4 , as shown in Fig. 4.  [4]. Here, T , R and the lattice size N 3 s × N t are arbitrary.

Case of even T
In the case of even number T , we consider the functional trace, where the completeness of the Dirac mode, n |n n| = 1, is used. Similarly in Sec. 3, one finds at each lattice gauge configuration. In fact, U 4 must be selected in all the D ∝ µ γ µ (U µ − U −µ ) in D , to form a loop in the functional trace. All other terms correspond to non-closed lines and give exactly zero, because of the definition ofÛ ±µ in Eq. (2). Thus, we obtain [4] Then, the inter-quark potential V(R) is written as and the string tension σ is expressed as Owing to the reduction factor λ T n in the sum, the string tension σ, i.e., the confining force, is to be unchanged by the removal of the low-lying Dirac-mode contribution.

Case of odd T
In the case of odd number T , the similar results can be obtained by considering Actually, one finds and obtains for odd T the similar formula of [4] Then, the inter-quark potential V(R) and the sting tension σ are written as where σ is unchanged by removing the low-lying Dirac-mode contribution due to λ T −1 n in the sum.

The Polyakov loop v.s. Wilson, clover and domain wall fermions
All the above formulae are mathematically correct, because we have just used the Elitzur theorem (or precisely Eq.(2) forÛ ±µ ) and the completeness n |n n| = 1 on the Dirac operator (1). However, one may wonder the doublers [6] in the use of the simple lattice Dirac operator (1). In this section, we express the Polyakov loop with the eigenmodes of the kernel of the Wilson fermion [6], the clover (O(a)-improved Wilson) fermion [14,15] and the domain wall (DW) fermion [16,17], respectively [18]. In these fermionic kernel, light doublers are absent.

The Wilson fermion
The Wilson fermion kernel [6] can be described with the link-variable operatorÛ ±µ as [18] where each term ofK includes oneÛ ±µ at most, and connects only the neighboring site or acts on the same site. Near the continuum, a ≃ 0, Eq.(21) becomesK ≃ (ˆ D + m) + arD 2 and the Wilson term For the Wilson fermion kernelK, we define its eigenmode |n and eigenvalueλ n aŝ Note that, without the Wilson term, the eigenmode ofK =ˆ D + m is the simple Dirac eigenmode |n , i.e.,K|n = (iλ n + m)|n , and satisfies the completeness of n |n n| = 1. In the presence of the O(a) Wilson term,K is neither hermite nor anti-hermite, and the completeness may include an O(a) error, Now, on the lattice with N t = 4l + 1, we consider the functional trace, Using the quasi-completeness of Eq.(23) for the eigenmode |n , one finds, apart from an O(a) error, We note that the kernelK in Eq. (21) includes many terms, and J ≡ Tr(Û 2l+1 4K 2l ) consists of products of link-variable operators, accompanying with c-number factors. In each product, the total number of U does not exceed N t , because of Eq. (21). Each product gives a trajectory as shown in Fig. 5. Among the trajectories, however, only the Polyakov loop L P can form a closed loop and survives in J, so that one gets J ∝ L P . Thus, apart from an O(a) error, we obtain [18] L P ∝ nλ 2l n n|Û 2l+1 4 |n .
Due to the suppression factor ofλ 2l n in the sum, one finds again small contribution from low-lying modes ofK to the Polyakov loop L P .

The clover (O(a)-improved Wilson) fermion
The clover fermion is an O(a)-improved Wilson fermion [14], and its kernel is expressed as [18] with σ µν ≡ i 2 [γ µ , γ ν ]. Here, G µν is the clover-type lattice field strength defined by with The sum of the Wilson and the clover terms is O(a 2 ) near the continuum, and the clover fermion gives accurate lattice results [15]. Since G µν acts on the same site, each term ofK in Eq.(27) connects only the neighboring site or acts on the same site, so that one can use almost the same technique as the Wilson fermion case. For the clover fermion kernelK, we define its eigenmode |n and eigenvalueλ n aŝ K|n = iλ n |n ,λ n ∈ C, Again, on the lattice with N t = 4l + 1, we consider the functional trace, where we have used the quasi-completeness for |n in Eq.(30) within an O(a 2 ) error. J ≡ Tr(Û 2l+1 4K 2l ) consists of products of link-variable operators, accompanying with c-number factors, and each product gives a trajectory as shown in Fig. 5. Among the trajectories, only the Polyakov loop L P can form a closed loop and survives in J, i.e., J ∝ L P . Thus, apart from an O(a 2 ) error, we obtain [18] and find small contribution from low-lying modes ofK to the Polyakov loop L P , due toλ 2l n in the sum.

The domain wall (DW) fermion
Finally, we consider the domain wall (DW) fermion [16,17], where the "exact" chiral symmetry is realized in the lattice formalism by introducing an extra spatial coordinate x 5 . The DW fermion is formulated in the five-dimensional space-time, and its (five-dimensional) kernel is expressed aŝ where the last two terms in the RHS are the kinetic and the mass terms in the fifth dimension. Here, x 5 -dependent mass M(x 5 ) is introduced as shown in Fig.6, where M 0 = |M(x 5 )| = O(a −1 ) is taken to be large. As for the extra coordinate x 5 , there are only kinetic and mass terms inK 5 , so that the eigenvalue problem is solved in the fifth direction, and chiral zero modes are found to appear [16,17]. For the five-dimensional DW fermion kernelK 5 , we define its eigenmode |ν and eigenvalue Λ ν aŝ Note that each term ofK 5 in Eq.(33) connects only the neighboring site or acts on the same site in the five-dimensional space-time, and hence one can use almost the same technique as the Wilson fermion case. On the lattice with N t = 4l + 1, we consider the functional trace, where the quasi-completeness for |ν in Eq.(34) is used. J ≡ Tr(U 2l+1 4 K 2l 5 ) consists of products of link-variable operators with other factors, and each product gives a trajectory as shown in Fig. 5 in the projected four-dimensional space-time. Among the trajectories, only the Polyakov loop L P can form a closed loop and survives in J, i.e., J ∝ L P . Thus, apart from an O(a) error, we obtain Because of the simple x 5 -dependence inK 5 in Eq.(33), the extra degrees of freedom in the fifth dimension can be integrated out in the generating functional, and one obtains the four-dimensional physical-fermion kernelK 4 [17]. The physical fermion mode is given by the eigenmode |n ofK 4 , K 4 |n = iλ n |n ,λ n ∈ C. (37) We find that the four-dimensional physical fermion eigenvalueλ n ofK 4 can be approximated by the eigenvalue Λ ν of the five-dimensional DW kernelK 5 as where M 0 = |M(x 5 )| = O(a −1 ) is taken to be large. Combining with Eq.(36), apart from an O(a) error, we obtain and find small contribution from low-lying physical-fermion modes ofK 4 to the Polyakov loop L P , because of the suppression factorλ 2l n ν in the sum.

Summary and Concluding Remarks
In QCD, we have derived analytical relations between the Dirac modes and the confinement quantities such as the Polyakov loop, its fluctuations and the string tension in the lattice formalism, and have found negligible contribution from the low-lying Dirac modes to the confinement quantities. We have also investigated the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain wall fermion kernels, respectively, and have obtained the similar results.
These relations indicate no direct one-to-one correspondence between confinement and chiral symmetry breaking. In other words, there is some independence of confinement from chiral properties in QCD. This seems natural because confinement is realized independently of the quark mass.