Vol. 10 (2017)

Acta Physica Polonica B Proceedings Supplement

No 3

INTERPLAY BETWEEN DECONFINEMENT
AND CHIRAL PROPERTIES∗
Hideo Suganuma, Takahiro M. Doi
Department of Physics, Kyoto University, Kyoto, Japan

Krzysztof Redlich, Chihiro Sasaki
Institute of Theoretical Physics, University of Wrocław, Wrocław, Poland(Received February 16, 2017)
We study interplay between confinement/deconfinement and chiralproperties. We derive some analytical relations of the Dirac modes withthe confinement quantities, such as the Polyakov loop, its susceptibility
and the string tension. For the confinement quantities, the low-lying Diraceigenmodes 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 eigenmodesof the Wilson, the clover and the domain-wall fermion kernels, respectively.
DOI:10.5506/APhysPolBSupp.10.741

1. Introduction
The relation between quark confinement and spontaneous chiral-symmetry breaking has been a longstanding difficult problem remaining in QCD
physics. In this paper, considering the essential role of low-lying Dirac modes
to chiral symmetry breaking [1], we derive analytical relations between theDirac modes and the confinement quantities, e.g., the Polyakov loop [2], its
fluctuations [3] and the string tension [4]. We mainly use the lattice unit,
a = 1.
2. Dirac operator, Dirac eigenvalues and Dirac modes
We use an ordinary square lattice with spacing a and size V ≡ Ns3 × Nt ,and impose the temporal periodicity/antiperiodicity for gluons/quarks. In∗Presented at the “Critical Point and Onset of Deconfinement” Conference, Wrocław,Poland, May 30–June 4, 2016.

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H. Suganuma et al.

lattice QCD, the gauge variable is expressed as the link-variable Uµ (s) =eiagAµ (s) , and the simple Dirac operator is given as4D̂
6 =1 X 
γµ Ûµ − Û−µ ,2a(1)µ=1where the link-variable operator Û±µ is defined by [2–4]hs|Û±µ |s0 i = U±µ (s)δs±µ̂,s0 ,(2)with U−µ (s) ≡ Uµ† (s− µ̂). For the anti-hermitian Dirac operator D
6 ˆ satisfying†D
6 ˆ 0 =−D6 ˆ 0 , we define the Dirac mode |ni and the Dirac eigenvalue λns ,ss,sD
6 ˆ |ni = iλn |ni(λn ∈ R) ,X

hm|ni = δmn ,|nihn| = 1 .

(3)n

3. Polyakov loop and Dirac modes on odd-number lattice
We use here a temporally odd-number lattice [2–4], where the temporal
lattice size Nt (< Ns ) is odd. In general, only gauge-invariant quantities
such as closed loops and the Polyakov loop survive in QCD, according tothe Elitzur theorem [1]. All the non-closed lines are gauge-variant and their
expectation values are zero. Now, we consider the functional trace [2, 4]
 X
X
Nt −1
I ≡ Trc,γ Û4D=6ˆhn|Û4 D̂
6 Nt −1 |ni = iNt −1
λnNt −1 hn|Û4 |ni , (4)nwhere Trc,γ ≡nPs trc trγ , and we use the completeness of the Dirac mode.
Nt −1
From Eq. (1), Û4D
6ˆis expressed as a sum of products of Nt linkNt −1
variable operators. Then, Û4D6ˆincludes many trajectories with the total

length Nt , as shown in Fig. 1. Note that all the trajectories with the oddnumber length Nt cannot form a closed loop on the square lattice, andgive gauge-variant contribution, except for the Polyakov loop. Thus, in
Nt −1
hIi = hTrc,γ (Û4D6ˆ)i, only the Polyakov-loop can survive as the gaugeinvariant component, and hIi is proportional to the Polyakov loop hLP i.
Actually, we can mathematically derive the following relation [2, 4]: Nt −1 DE Nt −1hIi = Trc,γ Û4D= Trc,γ Û4 γ4 D̂46ˆ4DnoE4Nc VNtTrÛ
= − Nt −1 hLP i , (5)c42Nt −12
where the last minus reflects the temporal antiperiodicity of D
6 ˆ [4].
E
D= 4 Trc Û4 D̂4Nt −1=

Interplay Between Deconfinement and Chiral Properties

743

Nt −1Fig. 1. Examples of the trajectories stemming from I = Trc,γ (Û4D6ˆ). For
each trajectory, the total length is Nt , and the “first step” is positive temporal
direction, Û4 . All the trajectories with the odd length Nt cannot form a closed
loop on the square lattice, so that they are gauge-variant and give no contribution,
except for the Polyakov loop. Thus, only the Polyakov-loop component survives
in hIi.

Thus, we obtain the analytical relation between the Polyakov loop hLP i
and the Dirac modes in QCD on the temporally odd-number lattice [2, 4]*+(2i)Nt −1 X Nt −1
λn hn|Û4 |ni,(6)
hLP i = −4Nc Vngauge ave

which is mathematically valid in both confined and deconfined phases. FromEq. (6), we can investigate each Dirac-mode contribution to the Polyakovt −1 in Eq. (6), low-lying Dirac modesloop. Remarkably, due to the factor λNngive negligible contribution to the Polyakov loop [2, 4]. In lattice QCD simulations, we have numerically confirmed relation (6) and scarce contributionof low-lying Dirac modes to the Polyakov loop in both confined and deconfined phases [2].
4. Polyakov-loop fluctuations and Dirac eigenmodes
Next, we consider the Polyakov-loop fluctuations, which can be a good
indicator of the QCD transition [5]. On the temporally odd lattice, we derive
Dirac-mode expansion formula for the Polyakov-loop fluctuations [3], e.g., D
P Nt −1 nn E2P Nt −1 nn 2λn Û4
λn Û4−RA = DPE2 ,P Nt −1  2πki/3 nn 2t −12πki/3 Û nn
λn Re eÛ4−λNReen4(7)where Û4nn ≡ hn|Û4 |ni, and k is chosen such that the transformed Polyakov
loop lies in its real sector [3, 5]. The damping factor λnNt −1 appears in theDirac-mode sum. By removing low-lying Dirac modes, the quark condensate
rapidly reduces, but the Polyakov-loop fluctuation is almost unchanged [3].

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H. Suganuma et al.

5. The Wilson loop and Dirac modes on arbitrary square lattices
In this section, we investigate the string tension and the Dirac modes,
using the Wilson loop on R × T rectangle on arbitrary square lattices withany number of Nt [4]. The Wilson loop is expressed by the functional trace
TRÛ−1Û4T = Trc Ûstaple Û4T ,W ≡ Trc Û1R Û−4TRÛstaple ≡ Û1R Û−4Û−1.

(8)For even T (odd T case is similar [4]), we consider the functional trace
TJ ≡ Trc,γ ÛstapleD
6ˆ =X
T X
hn|ÛstapleD
6 T |ni = (−) 2λTn hn|Ûstaple |ni . (9)
nn

T
2 T P
Similarly in Sec. 3, one can derive hW i = (−) 4 2 h n λTn hn|Ûstaple |nii [4].
Then, the string tension σ is expressed as*+X
1
1σ = − limlnhW i = − limln(2λn )T hn|Ûstaple |ni . (10)
R,T →∞ RT
R,T →∞ RTnBecause of the factor λTn in the sum, the string tension σ (the confining force)is to be unchanged by the removal of the low-lying Dirac-mode contribution.
6. The Polyakov loop and Wilson/clover/domain-wall fermionsFinally, we express the Polyakov loop with the eigenmodes of the Wilson,
the clover (O(a)-improved Wilson) and the domain-wall fermion kernels,
where light doublers are absent [1]. The clover fermion kernel is given asK̂ =4±4 r Xarg
1 X γµ Ûµ − Û−µ +γµ Ûµ −1 + m +σµν Gµν , (11)
2a2a2
µ=1

µ=±1

which becomes the Wilson fermion kernel without the last term in RHS. We
define eigenmodes and eigenvalues of K̂ as K̂|nii = iλ̃n |nii with λ̃n ∈ C.
We adopt the lattice with Nt = 4l + 1, and consider the functional trace
 XX 2l
J ≡ Tr Û42l+1 K̂ 2l = hhn|Û42l+1 K̂ 2l |nii =iλ̃n hhn|Û42l+1 |nii . (12)nnSince the kernel K̂ in Eq. (11) includes many terms, J ≡ Tr(Û42l+1 K̂ 2l )consists of products of link-variable operators. In each product, the total
number of Û does not exceed Nt . Each product gives a trajectory as inFig. 2. Among the trajectories, however, only the Polyakov loop can form aInterplay Between Deconfinement and Chiral Properties

745Fig. 2. Some examples of the trajectories in J ≡ Tr(Û42l+1 K̂ 2l ). The length does
not exceed Nt . Only the Polyakov loop can form a closed loop and survives in hJi.

closed loop and survives in hJi, and we derive+*EDX
2l+12lhLP i ∝
λ̃n hn|Û4 |nin.

(13)

gauge ave

Due to λ̃2ln , one finds small contribution from low-lying modes of K̂ to thePolyakov loop. We also derive a similar formula for the domain-wall fermion.
7. Summary
We have derived relations between the Dirac modes and the confinement quantities (the Polyakov loop, its fluctuations and the string tension)and have found scarce contribution from the low-lying Dirac modes. This
indicates some independence of confinement from chiral properties in QCD.
H.S. and T.M.D. are supported by the Grants-in-Aid for Scientific Research [Grant No. 15K05076, 15J02108] from Japan Society for the Promotion of Science. The work of K.R. and C.S. is partly supported by theNational Science Centre, Poland (NCN) under Maestro Grant No. DEC2013/10/A/ST2/0010.

REFERENCES[1] H.-J. Rothe, Lattice Gauge Theories, 4th edition, World Scientific, 2012.
[2] T.M. Doi, H. Suganuma, T. Iritani, Phys. Rev. D 90, 094505 (2014).
[3] T.M. Doi, K. Redlich, C. Sasaki, H. Suganuma, Phys. Rev. D 92, 094004(2015).
[4] H. Suganuma, T.M. Doi, T. Iritani, PTEP 2016, 013B06 (2016).
[5] P.M. Lo et al., Phys. Rev. D 88, 014506 (2013); 88, 074502 (2013).