Clear correlation between monopoles and the chiral condensate in SU(3) QCD

We study spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual superconductor picture for quark confinement in the maximally Abelian (MA) gauge, using lattice QCD Monte Carlo simulations with $24^4$ and $\beta=6.0$, i.e., the spacing $a \simeq$ 0.1 fm at the quenched level. First, in the MA gauge, we observe dominant roles of the Abelian part and monopoles for the chiral condensate in the chiral limit, using the two different methods of i) the Banks-Casher relation with the Dirac eigenvalue density and ii) finite quark-mass calculations with the quark propagator and its chiral extrapolation. Second, local correlation between the chiral condensate and monopoles are investigated. We find that the chiral condensate locally takes a quite large value near monopoles. Thus, the color-magnetic monopole topologically appearing in the MA gauge is responsible to chiral symmetry breaking in QCD, that is, dynamical origin of the matter mass in our Universe.


INTRODUCTION
Since quantum chromodynamics (QCD) was established as the fundamental theory of strong interaction in 1970s, to understand its nonperturbative properties has been one of the most difficult central problems in theoretical physics for about a half century. In particular, QCD exhibits two outstanding nonperturbative phenomena of quark confinement and spontaneous chiralsymmetry breaking in its low-energy region, many physicists have tried to clarify these phenomena and their relation directly from QCD, but this is still an unsolved important issue in the particle physics.
Chiral symmetry breaking in QCD is categorized as well-known spontaneous symmetry breaking, which widely appears in various fields in physics, and is an important phenomenon relating to dynamical quark-mass generation [1,2]. Indeed, apart from the dark matter, about 99% of the matter mass of our Universe originates from chiral symmetry breaking, because the Higgs-origin mass is just a small mass of u, d current quarks, electrons and neutrinos. The order parameter of chiral symmetry breaking is the chiral condensate qq , and it is directly related to low-lying Dirac modes, via the Banks-Casher relation [3].
In contrast, color confinement is a fairly unique phenomenon peculiar in QCD, and quark confinement is characterized by the linear inter-quark potential. As for the confinement mechanism, the dual superconductor picture based on color-magnetic monopole condensation was proposed by Nambu, 't Hooft, and Mandelstam as a typical plausible physical scenario [4][5][6]. In lattice QCD, by taking the maximally Abelian (MA) gauge [7], this dual superconductor scenario has been investigated in terms of Abelian dominance, i.e., dominant role of the Abelian sector [8][9][10][11], and the relevant role of monopoles [12][13][14].
The relation between confinement and chiral symmetry breaking is not yet clarified directly from QCD. While a strong correlation between confinement and chiral symmetry breaking has been suggested by almost coincidence between deconfinement and chiral-restoration temperatures [15], an lattice QCD analysis based on the Diracmode expansion indicates some independence of these phenomena [16].
Their correlation has been also suggested in terms of color-magnetic monopoles, which topologically appear in QCD in the Abelian gauge [17]. In the dual Ginzburg-Landau theory, the monopole condensate is responsible to chiral symmetry breaking as well as quark confinement [18]. Also in SU(2) lattice QCD, Miyamura and Woloshyn showed Abelian dominance [19,20] and monopole dominance [19,21] for chiral symmetry breaking. In fact, by removing the monopoles from the QCD vacuum, confinement and chiral symmetry breaking are simultaneously lost. In SU(3) lattice QCD with a 8 3 × 4 lattice, Thurner et al. showed a local correlation among monopoles, instantons, and the chiral condensate [22]. These studies indicate an important role of the monopoles to both phenomena, and thus these two phenomena might be related via the monopole. However, most of the pioneering lattice works were done in SU(2) lattice QCD or done on a small lattice [19][20][21][22].
In this Letter, we investigate correlation between chiral symmetry breaking and color-magnetic monopoles appearing in the MA gauge in SU(3) lattice QCD with a large-volume fine lattice at the quenched level. Using two different methods, we evaluate the chiral condensate in Abelianized QCD and the monopole system, extracted from lattice QCD. We also investigate correlation between the local chiral-condensate value and the monopole location. We perform SU(3) lattice QCD simulations at the quenched level with the standard plaquette action [15]. On four-dimensional Euclidean lattices, the gauge variable is described as the SU(3) field U µ (s) ≡ e iagAµ(s) ∈ SU(3), with the gluon field A µ (s) ∈ su(3), the QCD gauge coupling g, and the lattice spacing a. In this work, we use the lattice size of 24 4 and β ≡ 6/g 2 = 6.0, i.e., a 0.1 fm, and take the lattice unit a = 1 hereafter. Using the pseudo-heat-bath algorithm, we generate 100 gauge configurations which are taken every 500 sweeps after a thermalization of 5,000 sweeps. The jackknife method is used for the error estimate.
Using the Cartan subalgebra H ≡ (T 3 , T 8 ) of SU(3), the MA gauge fixing is defined so as to maximize under the SU(3) gauge transformation, and thus this gauge fixing suppresses all the off-diagonal fluctuation of the SU(3) field U µ (s). In the MA gauge, the SU(3) gauge group is partially fixed remaining its maximal torus subgroup U(1) 3 ×U(1) 8 , and QCD is reduced into an Abelian gauge theory like the non-Abelian Higgs theory. From the SU(3) field U MA µ (s) ∈ SU(3) in the MA gauge, the Abelian field is defined as u µ (s) = e i θ· H = diag e iθ 1 µ (s) , e iθ 2 µ (s) , e iθ 3 µ (s) ∈ U(1) 2 (2) with the constraint 3 i=1 θ i µ (s) = 0 (mod 2π), by maximizing the overlap so that the distance between u µ (s) and U MA µ (s) becomes the smallest in the SU(3) manifold.
The Abelian projection is defined by the replacement of O[u µ (s)] is called "Abelian dominance" for the operator O.

MONOPOLES IN QCD
Now, let us consider the Abelian plaquette variable, The Abelian field strength θ i µν (s) (i = 1, 2, 3) is the principal value of the exponent in u µν (s), and is defined as with the forward derivative ∂ µ . Here, θ i µν (s) is U(1) 2 gauge invariant and corresponds to the regular continuum Abelian field strength as a → 0, while n i µν (s) corresponds to the singular gauge-variant Dirac string [23].
The electric current j i µ and the monopole current k i µ are defined from the Abelian field strength θ i µν , where ∂ µ is the backward derivative. Both electric and monopole currents are U(1) 2 gauge invariant, according to U(1) 2 gauge invariance of θ i µν (s). In the lattice formalism, k i µ (s) is located at the dual lattice L 4 dual of s α + 1/2, flowing in µ direction [14]. Hereafter, we will omit the color index i as appropriate.
Abelian-projected QCD thus includes both electric current j µ and monopole current k µ , and can be decomposed into the "photon part" which only includes j µ and the "monopole part" which only includes k µ approximately, as follows.
First, we consider the photon part satisfying From ∂ µθ Ph µν = 0, one can set θ Ph µν = (∂ ∧ θ Ph ) µν and then In the Landau gauge ∂ µ θ Ph µ = 0, the photon part θ Ph ν can be derived from the electric current j ν , Therefore, we here define the photon part θ Ph ν by using the inverse d'Alembertian on the lattice [14]. The In this way, in Abelian-projected QCD, the contributions from the electric current j µ and the magnetic current k µ can be well separated into the photon part θ Ph µ and the monopole part θ Mo µ , respectively. In Table I, we show the monopole density ρ M and the electric-current density ρ E defined as for Abelian-projected QCD, monopole and photon parts, respectively.  Using the monopole and the photon link-variables, monopole and photon projection are defined by the re- The dominant role of the monopole part is called "monopole dominance," and monopole dominance has been observed for quark confinement in lattice QCD [12].

CHIRAL CONDENSATE
First, we study Abelian dominance and monopole dominance for the chiral condensate in the chiral limit, using the Kogut-Susskind (KS) fermion [15] for quarks in SU(3) lattice QCD.
Mathematically, the chiral condensate qq in the chiral limit is directly related to the low-lying Dirac eigenvalue density ρ(0) through the Banks-Casher relation [3], The Dirac eigenvalue density ρ(λ) is defined as 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) ≡ (−1) s1+···+sµ−1 , and the Dirac eigenvalue λ n is obtained from Here, the quark field q α (x) is described by a spinless Grassmann variable χ(x), and the chiral condensate per flavor is given as qq = χχ /4 in the continuum limit. Figure 1 shows the Dirac eigenvalue densities ρ(λ) for SU(3) QCD, Abelian-projected QCD, monopole and photon sectors, extracted from lattice QCD in the MA gauge. We find that the low-lying Dirac eigenvalue density ρ(0) in Abelian-projected QCD takes almost the same value in SU(3) QCD, which means Abelian dominance for the chiral condensate in the chiral limit. For the photon sector, we find no eigenvalues below 0.13 in 10 configurations and conclude that ρ(0) in the photon sector is exactly zero. On the other hand, ρ(0) in the monopole part is close to that in SU(3) QCD, which means monopole dominance for the chiral condensate in the chiral limit.
Next, we calculate the chiral condensate in a different way using the quark propagator. Here, we adopt the KS fermion with the bare quark mass m, and consider the chiral extrapolation of m → 0.
For the gauge-field ensemble of U = {U µ (s)}, the Euclidean KS fermion propagator is given by the inverse matrix, for SU(3) QCD, Abelian-projected QCD, monopole and photon sectors, respectively. Here, we use 100 gauge configurations, and calculate the local chiral condensate at 2 4 distant space-time points x for each gauge configuration. In fact, we perform 1,600 times calculations of χ(x)χ(x) U for each sector at each quark mass m. Here, we consider the net chiral condensate by subtracting the contribution from the trivial vacuum U = 1 as where the subtraction term is exactly zero at the chiral limit m = 0. We eventually take its average over the space-time x and the gauge ensembles U 1 , U 2 , ..., U N , Figure 2 shows the chiral condensates plotted against the bare quark mass m in the lattice unit, for SU(3), Abelian, monopole and photon sectors, extracted from lattice QCD in the MA gauge. For each sector, mdependence of the chiral condensate seems to be linear in this region, so that we evaluate the chiral condensate in the chiral limit, using the linear chiral extrapolation. Provided that the linear chiral extrapolation is valid, Abelian dominance and monopole dominance for the chiral condensate are realized in the chiral limit, whereas the photon part has almost no chiral condensate in the chiral limit.
These results are consistent with the above-mentioned conclusions using the Dirac eigenvalue density ρ(λ) and the Banks-Casher relation. In Table II, we summarize the chiral condensate values in the chiral limit evaluated from the two different methods for SU(3), Abelian, monopole and photon sectors.
In the presence of bare quark masses of m = 0.01−0.02, however, there appears a significant deviation of the chiral condensates between SU(3) and Abelian sectors, which quantitatively differs from SU(2) QCD, where Abelian dominance is observed at m = 0.05 − 0.3 [20]. In particular, compared with SU(3) QCD, bare-quark mass m dependence of the chiral condensate is fairly reduced in Abelian-projected QCD, and also in the monopole part. As an interesting possibility, the net chiral condensate in the Abelian/monopole sector is controlled by quarkmass independent object. This might be understood if monopoles are directly responsible for chiral symmetry breaking, because monopoles have no bare quark mass dependence in the quenched approximation. Then, in the next section, we examine the correlation between the chiral condensate and monopoles in more direct manner.

LOCAL CORRELATION
Second, we study the local correlation between chiral condensate and monopoles, by investigating the local chiral condensate around monopoles in Abelian-projected QCD at each gauge configuration. Note that, at each lattice configuration, the monopoles topologically appear as local objects, so that they might locally influence the chiral condensate around them, although the translational invariance is recovered by the gauge ensemble average.
For the visual demonstration, we show in Fig. 3 the local chiral condensate χχ(x) u and the monopole location at all three-dimensional space points at a time slice of t = 12 in the 1st Abelian configuration. The bare quark mass is taken as m = 0.02. Here, we show all the monopoles located at t = 11.5, 12.5 on the dual lattice L 4 dual of s α + 1/2. The value of the local chiral condensate | χχ(x) u | is visualized with the color graduation. (The same dark color is used for | χχ(x) u | > 0.20, and no color is used for small | χχ(x) u | < 0.04.) It is clearly observed that the local chiral condensate in a configuration has a large fluctuation and takes quite large values in the vicinity of the monopoles.
Finally, we calculate the correlation function between the local chiral condensate χχ(x) u and the local monopole density where P (s) denotes the dual lattices in the vicinity of s, i.e., P (s) = s ∈ L 4 dual |s − s| = 1 2 with the dual lattice L 4 dual of s α +1/2. For this calculation, we use the lattice data of the local chiral condensate and the monopole current for 100 gauge configurations, which were used to obtain Fig.2 in the previous section. Figure 4 shows the correlation function C(x − y) between the local chiral condensate χχ(x) u and the local monopole density ρ L (y), as the function of |x − y|, for the bare quark masses of m = 0.02, 0.015, and 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 no significant m-dependence of the correlation function is found in this bare quark-mass region. It is likely that the correlation function C(x−y) monotonically decreases with the distance r ≡ |x − y| and almost vanishes for large r such as r > 5 ( 0.5 fm), and thus a strong correlation between the local chiral condensate and the monopole density is quantitatively clarified.
From these lattice QCD results, we conclude that there exists a direct clear local correlation between monopoles and the chiral condensate.
Here, let us consider physical origin of the correlation between chiral symmetry breaking and monopoles in terms of the magnetic catalysis. In Abelian gauge theories, chiral symmetry breaking is generally enhanced in the presence of a strong magnetic field, which is called the magnetic catalysis [24][25][26]. In the MA gauge, infrared QCD resembles an Abelian gauge theory with monopoles, which accompany a strong color-magnetic field around them. Therefore, as an interesting possibility, the strong magnetic field around the monopoles enhances chiral symmetry breaking also in this Abelian gauge theory.

SUMMARY AND CONCLUSION
We have studied spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual superconductor picture for quark confinement in the MA gauge, using lattice QCD Monte Carlo simulations with a large volume of 24 4 .
First, we have found Abelian dominance and monopole dominance for the chiral condensate in the chiral limit, using the two different methods of i) the Banks-Casher relation with the Dirac eigenvalue spectral density and ii) finite quark-mass calculations with the quark propagator and its chiral extrapolation. We have also found that bare-quark mass dependence of the chiral condensate is fairly reduced in Abelian-projected QCD and the monopole part.
Second, we have investigated local correlation between the chiral condensate and color-magnetic monopoles, and have found that the chiral condensate takes a quite large value near the monopoles in Abelian-projected QCD.
Thus, the color-magnetic monopoles topologically appearing in the MA gauge significantly contribute to not only quark confinement but also chiral symmetry breaking in SU(3) QCD, that is, the dynamical origin of the matter mass in our Universe.
H.S. is supported in part by the Grants-in-Aid for Scientific Research [19K03869] from Japan Society for the Promotion of Science. Most of numerical calculations have been performed on NEC SX-ACE and OCTOPUS at Osaka University. We have used PETSc and SLEPc to solve linear equations and eigenvalue problems for the Dirac operator, respectively [27][28][29][30].