Clear correlation between monopoles and the chiral condensate in SU(3) QCD
YYITP-20-160
Clear correlation between monopoles and the chiral condensate in SU(3) QCD
Hiroki Ohata
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Hideo Suganuma
Department of Physics, Kyoto University, Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan (Dated: December 8, 2020)We study spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual supercon-ductor picture for quark confinement in the maximally Abelian (MA) gauge, using lattice QCDMonte Carlo simulations with 24 and β = 6 .
0, i.e., the spacing a (cid:39) INTRODUCTION
Since quantum chromodynamics (QCD) was estab-lished as the fundamental theory of strong interactionin 1970s, to understand its nonperturbative propertieshas been one of the most difficult central problems intheoretical physics for about a half century. In particu-lar, QCD exhibits two outstanding nonperturbative phe-nomena of quark confinement and spontaneous chiral-symmetry breaking in its low-energy region, many physi-cists have tried to clarify these phenomena and their re-lation directly from QCD, but this is still an unsolvedimportant issue in the particle physics.Chiral symmetry breaking in QCD is categorizedas well-known spontaneous symmetry breaking, whichwidely appears in various fields in physics, and is an im-portant phenomenon relating to dynamical quark-massgeneration [1, 2]. Indeed, apart from the dark matter,about 99% of the matter mass of our Universe originatesfrom chiral symmetry breaking, because the Higgs-originmass is just a small mass of u, d current quarks, electronsand neutrinos. The order parameter of chiral symmetrybreaking is the chiral condensate (cid:104) ¯ qq (cid:105) , and it is directlyrelated to low-lying Dirac modes, via the Banks-Casherrelation [3].In contrast, color confinement is a fairly unique phe-nomenon peculiar in QCD, and quark confinement ischaracterized by the linear inter-quark potential. Asfor the confinement mechanism, the dual superconduc-tor picture based on color-magnetic monopole condensa-tion was proposed by Nambu, ’t Hooft, and Mandelstamas a typical plausible physical scenario [4–6]. In latticeQCD, by taking the maximally Abelian (MA) gauge [7],this dual superconductor scenario has been investigatedin terms of Abelian dominance, i.e., dominant role of theAbelian sector [8–11], and the relevant role of monopoles[12–14]. The relation between confinement and chiral symmetrybreaking is not yet clarified directly from QCD. While astrong correlation between confinement and chiral sym-metry breaking has been suggested by almost coincidencebetween deconfinement and chiral-restoration tempera-tures [15], an lattice QCD analysis based on the Dirac-mode expansion indicates some independence of thesephenomena [16].Their correlation has been also suggested in termsof color-magnetic monopoles, which topologically ap-pear in QCD in the Abelian gauge [17]. In the dualGinzburg-Landau theory, the monopole condensate is re-sponsible to chiral symmetry breaking as well as quarkconfinement [18]. Also in SU(2) lattice QCD, Miya-mura and Woloshyn showed Abelian dominance [19, 20]and monopole dominance [19, 21] for chiral symmetrybreaking. In fact, by removing the monopoles from theQCD vacuum, confinement and chiral symmetry break-ing are simultaneously lost. In SU(3) lattice QCD witha 8 × a r X i v : . [ h e p - l a t ] D ec LATTICE SETUP AND ABELIAN PROJECTION
We perform SU(3) lattice QCD simulations at thequenched level with the standard plaquette action [15].On four-dimensional Euclidean lattices, the gauge vari-able is described as the SU(3) field U µ ( s ) ≡ e iagA µ ( s ) ∈ SU(3), with the gluon field A µ ( s ) ∈ su(3), the QCDgauge coupling g , and the lattice spacing a . In this work,we use the lattice size of 24 and β ≡ /g = 6 .
0, i.e., a (cid:39) a = 1 hereafter. Usingthe pseudo-heat-bath algorithm, we generate 100 gaugeconfigurations which are taken every 500 sweeps after athermalization of 5,000 sweeps. The jackknife method isused for the error estimate.Using the Cartan subalgebra (cid:126)H ≡ ( T , T ) of SU(3),the MA gauge fixing is defined so as to maximize R MA [ U µ ( s )] ≡ (cid:88) s (cid:88) µ =1 tr (cid:16) U † µ ( s ) (cid:126)HU µ ( s ) (cid:126)H (cid:17) = (cid:88) s (cid:88) µ =1 − (cid:88) i (cid:54) = j | U µ ( s ) ij | (1)under the SU(3) gauge transformation, and thus thisgauge fixing suppresses all the off-diagonal fluctuation ofthe SU(3) field U µ ( s ). In the MA gauge, the SU(3) gaugegroup is partially fixed remaining its maximal torus sub-group U(1) × U(1) , and QCD is reduced into an Abeliangauge theory like the non-Abelian Higgs theory.From the SU(3) field U MA µ ( s ) ∈ SU(3) in the MAgauge, the Abelian field is defined as u µ ( s ) = e i(cid:126)θ · (cid:126)H = diag (cid:16) e iθ µ ( s ) , e iθ µ ( s ) , e iθ µ ( s ) (cid:17) ∈ U(1) (2)with the constraint (cid:80) i =1 θ iµ ( s ) = 0 (mod 2 π ), by maxi-mizing the overlap R Abel ≡ Re tr (cid:8) U MA µ ( s ) u † µ ( s ) (cid:9) ∈ (cid:20) − , (cid:21) , (3)so that the distance between u µ ( s ) and U MA µ ( s ) becomesthe smallest in the SU(3) manifold.The Abelian projection is defined by the replacement ofSU(3) fields U µ ( s ) by Abelian fields u µ ( s ) for each gaugeconfiguration, i.e., O [ U µ ( s )] → O [ u µ ( s )] for QCD oper-ators. In this way, Abelian-projected QCD is extractedfrom SU(3) QCD. The case of (cid:104) O [ U µ ( s )] (cid:105) (cid:39) (cid:104) O [ u µ ( s )] (cid:105) iscalled “Abelian dominance” for the operator O . MONOPOLES IN QCD
Now, let us consider the Abelian plaquette variable, u µν ( s ) ≡ u µ ( s ) u ν ( s + ˆ µ ) u † µ ( s + ˆ ν ) u † ν ( s ) = e iθ µν ( s ) = diag( e iθ µν ( s ) , e iθ µν ( s ) , e iθ µν ( s ) ) ∈ U(1) . (4) The Abelian field strength θ iµν ( s ) ( i = 1 , ,
3) is the prin-cipal value of the exponent in u µν ( s ), and is defined as ∂ µ θ iν ( s ) − ∂ ν θ iµ ( s ) = θ iµν ( s ) − πn iµν ( s ) , − π ≤ θ iµν ( s ) < π, n iµν ( s ) ∈ Z , (5)with the forward derivative ∂ µ . Here, θ iµν ( s ) is U(1) gauge invariant and corresponds to the regular contin-uum Abelian field strength as a →
0, while n iµν ( s ) corre-sponds to the singular gauge-variant Dirac string [23].The electric current j iµ and the monopole current k iµ are defined from the Abelian field strength θ iµν , j iν ( s ) ≡ ∂ (cid:48) µ θ iµν ( s ) , (6) k iν ( s ) ≡ ∂ µ ˜ θ iµν ( s ) / π = ∂ µ ˜ n iµν ∈ Z , (7)where ∂ (cid:48) µ is the backward derivative. Both electric andmonopole currents are U(1) gauge invariant, accordingto U(1) gauge invariance of θ iµν ( s ). In the lattice formal-ism, k iµ ( s ) is located at the dual lattice L of s α + 1 / µ direction [14]. Hereafter, we will omit thecolor index i as appropriate.Abelian-projected QCD thus includes both electriccurrent j µ and monopole current k µ , and can be de-composed into the “photon part” which only includes j µ and the “monopole part” which only includes k µ approx-imately, as follows.First, we consider the photon part satisfying θ Ph µν ≡ ( ∂ ∧ θ Ph ) µν (mod 2 π ) , (8) ∂ (cid:48) µ θ Ph µν = j ν , ∂ µ ˜ θ Ph µν = 0 . (9)From ∂ µ ˜ θ Ph µν = 0, one can set θ Ph µν = ( ∂ ∧ θ Ph ) µν and then ∂ (cid:48) µ ( ∂ ∧ θ Ph ) µν = ∂ θ Ph ν − ∂ (cid:48) µ ∂ ν θ Ph µ = j ν . In the Landaugauge ∂ (cid:48) µ θ Ph µ = 0, the photon part θ Ph ν can be derivedfrom the electric current j ν , ∂ θ Ph ν = j ν , θ Ph ν = 1 ∂ j ν . (10)Therefore, we here define the photon part θ Ph ν by θ Ph ν ( s ) = (cid:88) s (cid:48) (cid:104) s | ∂ | s (cid:48) (cid:105) j ν ( s (cid:48) ) , (11)using the inverse d’Alembertian on the lattice [14]. Themonopole part θ Mo µ ( s ) is defined as θ Mo µ ( s ) ≡ θ µ ( s ) − θ Ph µ ( s ), and approximately satisfies θ Mo µν ≡ ( ∂ ∧ θ Mo ) µν (mod 2 π ) , (12) ∂ (cid:48) µ θ Mo µν (cid:39) , ∂ µ ˜ θ Mo µν (cid:39) k ν . (13)In this way, in Abelian-projected QCD, the contribu-tions from the electric current j µ and the magnetic cur-rent k µ can be well separated into the photon part θ Ph µ and the monopole part θ Mo µ , respectively. In Table I, weshow the monopole density ρ M and the electric-currentdensity ρ E defined as ρ M ≡ V (cid:88) i =1 (cid:88) s,µ (cid:12)(cid:12) k iµ ( s ) (cid:12)(cid:12) , (14) ρ E ≡ V (cid:88) i =1 (cid:88) s,µ (cid:12)(cid:12) j iµ ( s ) (cid:12)(cid:12) (15)for Abelian-projected QCD, monopole and photon parts,respectively. Field sector Monopole density Electric densityAbel 2 . × − . × − . . × − ρ M and the electric-currentdensity ρ E for Abelian-projected QCD, monopole and photonparts. Using the monopole and the photon link-variables, u Mo µ ( s ) ≡ e iθ Mo µ ( s ) ∈ U(1) , (16) u Ph µ ( s ) ≡ e iθ Ph µ ( s ) ∈ U(1) , (17)monopole and photon projection are defined by the re-placement of { u µ ( s ) } → { u Mo µ ( s ) } , { u Ph µ ( s ) } . The domi-nant role of the monopole part is called “monopole dom-inance,” and monopole dominance has been observed forquark confinement in lattice QCD [12]. CHIRAL CONDENSATE
First, we study Abelian dominance and monopoledominance for the chiral condensate in the chiral limit,using the Kogut-Susskind (KS) fermion [15] for quarksin SU(3) lattice QCD.Mathematically, the chiral condensate (cid:104) ¯ qq (cid:105) in the chirallimit is directly related to the low-lying Dirac eigenvaluedensity ρ (0) through the Banks-Casher relation [3], (cid:104) ¯ qq (cid:105) = − lim m → lim V →∞ πρ (0) . (18)The Dirac eigenvalue density ρ ( λ ) is defined as ρ ( λ ) ≡ V (cid:88) n (cid:104) δ ( λ − λ n ) (cid:105) , γ µ D µ | n (cid:105) = iλ n | n (cid:105) (19)with the space-time volume V .For the KS fermion, the Dirac operator γ µ D µ becomes η µ D µ with the staggered phase η µ ( s ) ≡ ( − s + ··· + s µ − ,and the Dirac eigenvalue λ n is obtained from12 (cid:88) µ =1 η µ ( s )[ U µ ( s ) χ n ( s + ˆ µ ) − U − µ ( s ) χ n ( s − ˆ µ )] = iλ n χ n ( s ) . (20)Here, the quark field q α ( x ) is described by a spinlessGrassmann variable χ ( x ), and the chiral condensate perflavor is given as (cid:104) ¯ qq (cid:105) = (cid:104) ¯ χχ (cid:105) /4 in the continuum limit.Figure 1 shows the Dirac eigenvalue densities ρ ( λ ) forSU(3) QCD, Abelian-projected QCD, monopole and pho-ton sectors, extracted from lattice QCD in the MA gauge. .
000 0 .
001 0 .
002 0 .
003 0 .
004 0 .
005 0 .
006 0 .
007 0 . λ . . . . . . . . π ρ ( λ ) V = 24 , β = 6 . SU(3)AbelMonopolePhoton
FIG. 1. The Dirac eigenvalue densities ρ ( λ ) for SU(3) QCD,Abelian-projected QCD, monopole and photon sectors, asfunctions of the Dirac eigenvalue λ . The line is the best fitwith a constant for the low-lying Dirac eigenvalue density. We find that the low-lying Dirac eigenvalue density ρ (0) in Abelian-projected QCD takes almost the samevalue in SU(3) QCD, which means Abelian dominance forthe chiral condensate in the chiral limit. For the photonsector, we find no eigenvalues below 0.13 in 10 configura-tions and conclude that ρ (0) in the photon sector is ex-actly zero. On the other hand, ρ (0) in the monopole partis close to that in SU(3) QCD, which means monopoledominance for the chiral condensate in the chiral limit.Next, we calculate the chiral condensate in a differentway using the quark propagator. Here, we adopt the KSfermion with the bare quark mass m , and consider thechiral extrapolation of m → U = { U µ ( s ) } , the Eu-clidean KS fermion propagator is given by the inversematrix, G ijU ( x, y ) ≡ (cid:104) χ i ( x ) ¯ χ j ( y ) (cid:105) U = (cid:104) x, i | (cid:18) η µ D µ [ U ] + m (cid:19) | y, j (cid:105) , (21)with the color index i and j . In this work, the propagatoris calculated by solving the large-scale linear equationwith a point source. Using the propagator for the gauge-field ensemble { U µ ( s ) } , { u µ ( s ) } , { u Mo µ ( s ) } , and { u Ph µ ( s ) } ,we calculate the local chiral condensate (cid:104) ¯ χ ( x ) χ ( x ) (cid:105) U = − Tr G U ( x, x ) (22)for SU(3) QCD, Abelian-projected QCD, monopole andphoton sectors, respectively. Here, we use 100 gauge con-figurations, and calculate the local chiral condensate at2 distant space-time points x for each gauge configu-ration. In fact, we perform 1,600 times calculations of (cid:104) ¯ χ ( x ) χ ( x ) (cid:105) U for each sector at each quark mass m . Here,we consider the net chiral condensate by subtracting thecontribution from the trivial vacuum U = 1 as (cid:104) ¯ χχ ( x ) (cid:105) U ≡ (cid:104) ¯ χ ( x ) χ ( x ) (cid:105) U − (cid:104) ¯ χχ (cid:105) U =1 , (23)where the subtraction term is exactly zero at the chirallimit m = 0. We eventually take its average over thespace-time x and the gauge ensembles U , U , ..., U N , (cid:104) ¯ χχ (cid:105) ≡ (cid:88) x,i (cid:104) ¯ χχ ( x ) (cid:105) U i / (cid:88) x,i . (24)Figure 2 shows the chiral condensates plotted againstthe bare quark mass m in the lattice unit, for SU(3),Abelian, monopole and photon sectors, extracted fromlattice QCD in the MA gauge. For each sector, m -dependence of the chiral condensate seems to be linearin this region, so that we evaluate the chiral condensatein the chiral limit, using the linear chiral extrapolation. . . . . . . . . . m . . . . . . | < ¯ χχ > | V = 24 , β = 6 . SU(3)AbelMonopolePhoton
FIG. 2. The chiral condensates for SU(3) QCD, Abelian-projected QCD, monopole and photon sectors, as functionsof the bare quark mass m in the lattice unit. The solid line isthe best fit with a linear function. Provided that the linear chiral extrapolation is valid,Abelian dominance and monopole dominance for the chi-ral condensate are realized in the chiral limit, whereas thephoton part has almost no chiral condensate in the chirallimit.These results are consistent with the above-mentionedconclusions using the Dirac eigenvalue density ρ ( λ ) andthe Banks-Casher relation. In Table II, we summarize thechiral condensate values in the chiral limit evaluated fromthe two different methods for SU(3), Abelian, monopoleand photon sectors.In the presence of bare quark masses of m = 0 . − . m = 0 . − . m dependence of the chiral condensate is fairly reduced inAbelian-projected QCD, and also in the monopole part. Field sector Banks-Casher PropagatorSU(3) 3 . × − . × − Abel 3 . × − . × − Monopole 2 . × − . × − Photon 0.00(0) 2 . × − TABLE II. The chiral condensate values in the chiral limitevaluated from the two different methods for SU(3) QCD,Abelian-projected QCD, monopole and photon sectors.
As an interesting possibility, the net chiral condensatein the Abelian/monopole sector is controlled by quark-mass independent object. This might be understood ifmonopoles are directly responsible for chiral symmetrybreaking, because monopoles have no bare quark massdependence in the quenched approximation. Then, inthe next section, we examine the correlation between thechiral condensate and monopoles in more direct manner.
LOCAL CORRELATION
Second, we study the local correlation between chiralcondensate and monopoles, by investigating the local chi-ral condensate around monopoles in Abelian-projectedQCD at each gauge configuration. Note that, at each lat-tice configuration, the monopoles topologically appear aslocal objects, so that they might locally influence the chi-ral condensate around them, although the translationalinvariance is recovered by the gauge ensemble average.For the visual demonstration, we show in Fig. 3 thelocal chiral condensate (cid:104) ¯ χχ ( x ) (cid:105) u and the monopole loca-tion at all three-dimensional space points at a time sliceof t = 12 in the 1st Abelian configuration. The barequark mass is taken as m = 0 .
02. Here, we show all themonopoles located at t = 11 . , . L of s α + 1 /
2. The value of the local chiral conden-sate |(cid:104) ¯ χχ ( x ) (cid:105) u | is visualized with the color graduation.(The same dark color is used for |(cid:104) ¯ χχ ( x ) (cid:105) u | > .
20, andno color is used for small |(cid:104) ¯ χχ ( x ) (cid:105) u | < . (cid:104) ¯ χχ ( x ) (cid:105) u and the localmonopole density ρ L ( s ) ≡ · (cid:88) i =1 (cid:88) s (cid:48) ∈ P ( s ) 4 (cid:88) µ =1 (cid:12)(cid:12) k iµ ( s (cid:48) ) (cid:12)(cid:12) , (25)where P ( s ) denotes the dual lattices in the vicinity of s ,i.e., P ( s ) = (cid:8) s (cid:48) ∈ L (cid:12)(cid:12) | s (cid:48) − s | = (cid:9) with the dual lat-tice L of s α +1 /
2. For this calculation, we use the lat-tice data of the local chiral condensate and the monopole x y z monopoles at t = 11.5monopoles at t = 12.5 .
04 0 .
06 0 .
08 0 .
10 0 .
12 0 .
14 0 .
16 0 .
18 0 . | < ¯ χχ ( x, y, z, t = 12) > u | at m = 0 . FIG. 3. The local chiral condensate at t = 12 and monopolesat t = 11 . , . m = 0 .
02. The value of the local chiralcondensate (cid:104) ¯ χχ ( x ) (cid:105) u is visualized with the color graduation.Monopoles at t = 11 . . current for 100 gauge configurations, which were used toobtain Fig.2 in the previous section.Figure 4 shows the correlation function C ( x − y ) be-tween the local chiral condensate (cid:104) ¯ χχ ( x ) (cid:105) u and the localmonopole density ρ L ( y ), C ( x − y ) ∝ (cid:104) ¯ χχ ( x ) ρ L ( y ) (cid:105) u − (cid:104) ¯ χχ (cid:105) u (cid:104) ρ L (cid:105) u , (26)as the function of | x − y | , for the bare quark masses of m = 0 . , . .
01. Here, the correlation function C ( x − y ) is normalized to be unity at | x − y | = 0 at each m .Within the error bar, the correlation function C ( x − y )seems to be a single-valued function of | x − y | , and nosignificant m -dependence of the correlation function isfound in this bare quark-mass region.It is likely that the correlation function C ( x − y ) mono-tonically decreases with the distance r ≡ | x − y | and al-most vanishes for large r such as r > (cid:39) . | x − y | − . . . . . . . . C ( x − y ) V = 24 , β = 6 . m = 0 . m = 0 . m = 0 . FIG. 4. The correlation function C ( x − y ) between the localchiral condensate (cid:104) ¯ χχ ( x ) (cid:105) u and the local monopole density ρ L ( y ) plotted against | x − y | for the bare quark masses of m = 0 . , . monopoles, which accompany a strong color-magneticfield around them. Therefore, as an interesting possi-bility, the strong magnetic field around the monopolesenhances chiral symmetry breaking also in this Abeliangauge theory. SUMMARY AND CONCLUSION
We have studied spontaneous chiral-symmetry break-ing in SU(3) QCD in terms of the dual superconductorpicture for quark confinement in the MA gauge, using lat-tice QCD Monte Carlo simulations with a large volumeof 24 .First, we have found Abelian dominance and monopoledominance for the chiral condensate in the chiral limit,using the two different methods of i) the Banks-Casherrelation with the Dirac eigenvalue spectral density andii) finite quark-mass calculations with the quark propa-gator and its chiral extrapolation. We have also foundthat bare-quark mass dependence of the chiral conden-sate is fairly reduced in Abelian-projected QCD and themonopole part.Second, we have investigated local correlation betweenthe chiral condensate and color-magnetic monopoles, andhave found that the chiral condensate takes a quite largevalue near the monopoles in Abelian-projected QCD.Thus, the color-magnetic monopoles topologically ap-pearing in the MA gauge significantly contribute to notonly quark confinement but also chiral symmetry break-ing in SU(3) QCD, that is, the dynamical origin of thematter mass in our Universe.H.S. is supported in part by the Grants-in-Aid for Sci-entific Research [19K03869] from Japan Society for thePromotion of Science. Most of numerical calculationshave been performed on NEC SX-ACE and OCTOPUSat Osaka University. We have used PETSc and SLEPcto solve linear equations and eigenvalue problems for theDirac operator, respectively [27–30]. [1] Y. Nambu and G. 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