New Abelian-like monopoles and the dual Meissner effect
aa r X i v : . [ h e p - l a t ] N ov New Abelian-like monopoles and the dual Meissner effect
Atsuki Hiraguchi ∗ Graduate School of Integrated Arts and Sciences, Kochi University, Kochi 780-8520, Japan
Katsuya Ishiguro
Library and Information Technology, Kochi University, Kochi 780-8520, Japan
Tsuneo Suzuki
Research Center for Nuclear Physics (RCNP), Osaka University, Osaka 567-0047, Japan (Dated: December 1, 2020)Violation of non-Abelian Bianchi identity can be regarded as N − I. INTRODUCTION
The mechanism of color confinement is still unknownin quantum chromodynamics (QCD) [1].As a picture of color confinement, ’t Hooft [2] and Man-delstam [3] conjectured that the QCD vacuum is a dualsuperconducting state. An interesting idea to realize thisconjecture is to project QCD to the Abelian maximaltorus group by a partial (but singular) gauge fixing [4].After the Abelian projection, color magnetic monopolesappear as a topological current. The dual Meissner effectis caused by condensation of monopoles. Numerically,Abelian monopole dominance is observed clearly in themaximal Abelian gauge (MAG) fixing [5–7]. Similar re-sults are found also in various local unitary gauges [8].However, there are infinite ways of such a partial gaugefixing. It is not at all clear if the lattice results obtainedin a partial gauge fixing like MAG are gauge independent.In the works [9, 10], Abelian monopole dominance andthe dual Meissner effect are found to exist even withoutadopting any gauge fixing. By making use of a huge num-ber of thermalized vacua with additional random gaugetransformations, they found that the string tension fromthe monopole Polyakov loop correlations is identical tothat of the gauge-invariant non-Abelian static potential. ∗ e-mail:[email protected] There exists also the Abelian dual Meissner effect. Thevacuum type of pure SU(2) gauge theory was found tobe near the border between type 1 and type 2 dual su-perconductors. Although the results are interesting, thephysical meaning of such gauge-variant quantities with-out gauge fixing was not clear at all in the continuumlimit of QCD.Recently, it was shown in the continuum limit thatthe violation of the non-Abelian Bianchi identities (VN-ABI) J µ is equal to Abelian-like monopole currents k µ defined by the violation of the Abelian-like Bianchi iden-tities [11, 12]. Although VNABI is a gauge-variant ad-joint operator satisfying the covariant conservation rule,it gives us, at the same time, the Abelian-like currentconservation rules. There are N − ρ ( a ( β ) , n )is plotted versus the lattice spacing of the blocked lat-tice b = na ( β ), where a ( β ) is the lattice distance at thecoupling β . A single universal curve ρ ( b ) is found from n = 1 up to n = 12 for all four gauges adopted, whichsuggests that ρ ( a ( β ) , n ) is a function of b = na ( β ) aloneand gauge independent. Since the continuum limit isrealized as n → ∞ , scaling means that the lattice defi-nition of Abelian-like monopoles has a continuum limit.Afterwards, one of the present authors (T.S.) found thatcoupling constants of the effective monopole action de-rived from the inverse Monte Carlo method [14, 19] showalso a universal scaling behavior [20] for the above fourgauges.It is the purpose of this work first to investigatewhether the Abelian monopole dominance and theAbelian dual Meissner effect, which are observed in theMAG [7], are seen also in the above global color in-variant gauges (MCG, DLCG, and MAWL) with a rea-sonable number of field configurations. Since VNABIare gauge variant, finding various gauge-fixing methodsreducing lattice artifact monopoles without destroyingphysical monopole effects is very important for extractingany physical quantity concerning Abelian-like monopoles.Hereafter, the authors call such gauges as smooth in thiswork. Secondly, it is interesting to check global-colorindependence of the Abelian dual Meissner effect whenwe introduce a single color external source in the vac-uum. Such a study could not be done in practice atthe present stage without any gauge fixing as in Ref.[10].Hence, we adopt here the above global color invariantgauges smoothing the vacuum. Furtheremore, we usethe block-spin transformation of the monopole currentin comparing the monopole contribution to confinementin the MCG with that in the MAG. II. METHODA. The violation of non-Abelian Bianchi identities
If gauge fields have a line singularity in the continuumQCD, then the non-Abelian Bianchi identity is violated.The VNABI is found to be equivalent to that of Abelian-like Bianchi identity [11, 12]. Namely VNABI is regardedas eight Abelian-like monopoles in the continuum SU(3)QCD. Using a covariant derivative D µ = ∂ µ − igA µ , weget the following commutation relation:[ D µ , D ν ] = − igG µν + [ ∂ µ , ∂ ν ] , (1)where G µν is a non-Abelian field strength. The secondcommutator can not be discarded when a line singularity exists. The Jacobi identities, ǫ µνρσ [ D ν , [ D ρ , D σ ]] = 0 , (2)lead us to the following relation: D ν G ∗ µν = ∂ ν f ∗ µν = k µ , (3)where f µν is defined as f µν = ∂ µ A ν − ∂ ν A µ = ( ∂ µ A aν − ∂ ν A aµ ) λ a /
2. In the case of SU(3), λ a are the Gell-Mannmatrices. Relation (3) means that the VNABI is equiv-alent to eight Abelian-like magnetic monopole currentsin SU(3). In the case of SU(2), VNABI is equivalent tothree Abelian-like magnetic monopole currents. B. Abelian-like monopoles on a lattice
The direct definition of VNABI on lattice is very dif-ficult. Hence, we adopt defining Abelian-like monopoleson a lattice following Ref.[13] and study the continuumlimit of them since VNABI is equivalent to Abelian-likemonopoles in the continuum limit.We consider here also SU(2) gluodynamics for simplic-ity. SU(2) link variables are U µ ( s ) = U µ ( s ) + iσ a U aµ ( s ) , (4)where σ a are the Pauli matrices, a = 1 , , U µ ( s ) , U aµ ( s ) are real coefficients. We explainhow to define Abelian-like monopoles in SU(2) gauge the-ory below.First, Abelian-like gauge fields θ aµ ( s ) are derived to getthe maximum overlap with an original non-Abelian linkvariable, namely in such a way as the following quantityis maximized: R = X s,µ ReTr[ e iθ µ ( s ) σ U † µ ( s )] , (5)where only the case for color=1 is written as an example.Then we get θ aµ ( s ) = arctan (cid:18) U aµ ( s ) U µ ( s ) (cid:19) ( | θ aµ ( s ) | < π ) . (6)This is equal to the definition as adopted in previousworks [9, 10].We now define three monopole currents followingRef.[13]: k aν ( s ) = 14 π ǫ µνρσ ∂ µ ¯ θ aρσ ( s + ˆ ν ) , (7) θ aµν ( s ) = ∂ µ θ aν ( s ) − ∂ ν θ aµ ( s ) , ¯ θ aµν ( s ) = θ aµν ( s ) − πn aµν ( s ) , where θ aµν ( s ) is an Abelian-like field strength, ¯ θ aµν ∈ [ − π, π ], and n aµν ( s ) is an antisymmetric tensor. Note that n aµν ( s ) takes integer values { -2,-1,0,1,2 } . It can be inter-preted as the number of Dirac strings. We found thesemonopole currents have a continuum limit, studying themonopole density and the effective monopole action onthe lattice with the aid of a block-spin transformation ofmonopoles [12, 20]. C. Smooth gauge fixings
We adopt gauge-fixing techniques smoothing the vac-uum as in Ref.[12]. The gauge-fixing methods adoptedhere reduce lattice artifact monopoles well without de-stroying infrared long monopoles.1.
MCG . The first is the maximal center gauge[15, 16], which is usually discussed in the frame-work of the center vortex idea. We adopt the so-called direct maximal center gauge, which requiresmaximization of the quantity R = X s,µ (Tr U µ ( s )) , (8)with respect to local gauge transformations. Thecondition (8) fixes the gauge up to Z (2) gaugetransformation.2. DLCG . The second is the Laplacian center gauge[17], which is also discussed in connection with thecenter vortex idea.3.
MAWL . Another is the maximal Abelian Wilsonloop gauge, in which R = X s,µ = ν X a (cos( θ aµν ( s ))) , (9)is maximized [18]. Since cos( θ aµν ( s )) are 1 × MAU1 . The fourth is the combination of MAGand the U(1) Landau gauge [5, 6]. Namely, we firstperform maximal Abelian gauge fixing and then,with respect to what remains, U(1) symmetry Lan-dau gauge fixing is done. This case breaks theglobal SU(2) color symmetry contrary to the pre-vious three cases (MCG, DLCG, and MAWL), butwe consider this case since the vacuum is smoothedfairly well. The MAG is the gauge which maximizes R = X s,µ Tr (cid:16) σ U µ ( s ) σ U † µ ( s ) (cid:17) , (10)with respect to local gauge transformations. Thenthere remains a U(1) symmetry to which the Lan-dau gauge fixing is applied, i.e., P s,µ (cos θ µ ( s )) ismaximized [21]. D. Simulation details
In most cases we adopt the tadpole improved action inpure SU(2) gauge theory: S = β { X pl S pl − u X rt S rt } , (11) a V r/a β =3.9, V=48 non-AbelianAbelianMonopolePhoton FIG. 1:
The potential between quark and antiquark in theMCG. Only the data at β = 3 . latticeis shown as an example. where S pl and S rt denote plaquette and 1 × S pl,rt = 12 Tr(1 − U pl.rt ) , (12)the parameter u is the input tadpole improvement factortaken here equal to the fourth root of the average plaque-tte P = h trU pl i . In our simulations we do not includeone-loop corrections to the coefficients for the sake of sim-plicity. The lattices adopted are 48 for β = 3 . ∼ . for β = 3 . ∼ .
9. In the case of the tadpoleimproved action, we adopt the same vacuum ensemblesgenerated and used in the previous research [12].
III. RESULTSTABLE I:
The string tension at β = 3 . lattice √ σa σ A /σ NA σ mon /σ NA σ ph /σ NA MCG V NA . V A V mon V ph V NA . V A V mon V ph V NA . V A V mon V ph V NA . V A V mon V ph TABLE II:
The string tension in the MCG on the 48 lattice √ σa σ A /σ NA σ mon /σ NA σ ph /σ NA β = 3 . V NA . V A V mon V ph β = 3 . V NA . V A V mon V ph β = 3 . V NA . V A V mon V ph β = 3 . V NA . V A V mon V ph A. Abelian and monopole dominances
First, we check whether Abelian and monopole dom-inances observed in the MAG are seen in other smoothgauges like the MCG or not. We evaluate the potentialfrom Abelian Wilson loops and their monopole contribu-tions. Now, we take into account only a simple AbelianWilson loop, say, of size I × J . Then such an AbelianWilson loop operator is expressed as W aA = exp { i X J µ ( s ) θ aµ ( s ) } , (13)where J µ ( s ) is an external current taking ± J µ ( s ) is conserved, it is rewritten forsuch a simple Wilson loop in terms of an antisymmetricvariable M µν as J ν = ∂ ′ M µν ( s ) with a forward (back-ward) difference ∂ ν . ( ∂ ′ ν ). Note that M µν ( s ) take ± W aA = exp {− i X M µν ( s ) θ aµν ( s ) } . (14)We investigate the monopole contribution to the staticpotential in order to examine the role of monopoles forconfinement. The monopole part of the Abelian Wil-son loop operator is extracted as follows [22]. Usingthe lattice Coulomb propagator D ( s − s ′ ), which satisfies ∂ ν ∂ ′ ν D ( s − s ′ ) = − δ ss ′ , we get W aA = W amon W aph , (15) W amon = exp { πi X k aβ ( s ) × D ( s − s ′ ) 12 ǫ αβρσ ∂ α M ρσ ( s ′ ) } , (16) W aph = exp {− i X ∂ ′ µ ¯ θ aµν ( s ) D ( s − s ′ ) J ν ( s ′ ) } . (17) b V r/bn=1 in MCGn=2 in MCGn=3 in MCG FIG. 2:
The static-quark potentials from monopole Wilsonloops on a blocked reduced lattice with the spacing b = na in the MCG. The data at β = 3 . β = 3 . β = 3 .
6) are taken from n = 2 ( n = 3)blocked monopoles. We then compute the static potential from the AbelianWilson loops and the monopole Wilson loops in the MCGand MAU1 on the 48 lattices at β = 3 . , . , . , . at β = 3 .
5. Wefit the potential to the usual functional form V fit ( r ) = σr − c/r + µ, (18)where σ denotes the string tension, c the Coulombic co-efficient, and µ the constant. The static potential inthe MCG is shown in Fig.1. The results of the stringtensions in the above four smooth gauges on the 24 lattice are shown in Table I and on 48 in the MCGare summarized for various β in Table II. Here V NA , V A , V mon and V ph mean potentials from non-Abelian,Abelian, monopole, and photon Wilson loop, respec-tively. And σ NA , σ A , σ mon and σ ph are non-Abelian,Abelian, monopole, and photon string tensions. Fairlygood results of Abelian and monopole dominances areobtained also in the MCG in comparison with those inthe MAG. Both ratios σ A /σ NA and σ mon /σ NA approachmore to one as the coupling constant β becomes largeras expected from the previous data [9, 10]. TABLE III:
The string tension from the n blockedmonopole current in the MCG and the MAG.FR means the fitting range.n β √ σa FR(r/a) χ /N d.o.f σ mon /σ NA . . B. Monopole dominance after block-spintransformations of monopoles
Considering the previous data [9, 10] showing perfectmonopole dominance, insufficient monopole dominanceobtained here after smooth gauge fixings suggests thatthere still remain lattice artifact monopoles. Here, letus consider a block-spin transformation with respect tolattice monopoles. After the block-spin transformationof monopoles, we can study the monopole behaviors inthe long-range regions near to the continuum limit. InRef.[12], the scaling behavior is seen when the monopoledensity is plotted versus the lattice spacing of the blockedlattices b = na ( β ). This result suggests the contributionof the monopole on the blocked lattice must be largerthan that of the monopole on the original lattice. Weevaluate the monopole Wilson loop in the MCG and theMAG by using the monopole currents on the blocked24 (16 ) lattice after the n = 2 ( n = 3) block-spintransformation of monopoles on the original 48 lattice.Here the definition of the block-spin transformation ofthe monopole current is shown as k ( n ) µ ( s n ) = n − X i,j,l =0 k µ ( ns + ( n − µ + i ˆ ν + j ˆ ρ + l ˆ σ ) . (19)In the calculations of physics on a blocked lattice, itis important to adopt a corresponding improved oper-ator measuring physics correctly as well as the effectivemonopole action on the blocked lattice [23]. But in thecase of measuring the string tension, it is enough to con-sider flat Wilson loops on the blocked lattice as an im-proved operator. We evaluate the monopole contributionto the string tension at β = 3 . and at β = 3 . β = 3 .
6) on the n = 2 ( n = 3)blocked lattice 24 (16 ). These β points have similar b = na ( β ) values. As a result, the string tensions fromthe monopole Wilson loop on the blocked lattices in theMCG are larger than that on the original lattice as seenin Fig 2. The string tensions from monopoles in the MAGand the MCG are summarized in Table III. After n = 3blocking, the improvement in the MCG is bigger thanthat in MAG and the results in both gauges are almostthe same. This is consistent with the results showinggauge independence obtained in previous work [12]. C. The dual Meissner effect
Next, we show the results with respect to the Abeliandual Meissner effect. It is necessary to measure the cor-relation functions between an Abelian Wilson loop andvarious Abelian operators having the same, or differentcolors. But in the previous research [10], without anygauge fixing they could measure only the correlationsbetween a non-Abelian Wilson loop and Abelian oper-ators, which are connected by a Schwinger line, since the
FIG. 3: (a) is the schematic figure of the disconnectcorrelation between an Abelian Wilson loop and anAbelian operator. (b) is the definition of thecylindrical coordinate ( r, φ, z ) along the q -¯ q axis. -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 2 4 6 8 10 12 β =3.3, V=24 , W(5,5) a < E z > r/a Same colorDifferent colorDifferent color
FIG. 4:
The color distribution of electric fields E z on the24 lattice in the MCG. Only the β = 3 . -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0 2 4 6 8 10 12 β =3.5, V=24 , W(7,7) a < E > r/a EzErE φ FIG. 5:
The distribution of electric-field components E z , E r and E φ on the 24 lattice in the MCG. Onlythe β = 3 . -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 2 4 6 8 10 12 β =3.5, V=24 , W(7,7) a < E z > r/a MCGDLCGMAWLMAU1MAU1-off FIG. 6:
The profiles of the color electric field E z at β = 3 . lattice for four smooth gauge fixings -0.0001-5x10 -5 -5 β =3.5, V=24 , W(7,7) a < k φ > r/a Same colorDifferent colorDifferent color
FIG. 7:
The profiles of monopole current k φ distributionsat β = 3 . lattice in the MCG. There isno correlation between different colors. disconnected correlations are too small to get a reliableresult. The connected correlations, however, contain var-ious contaminations, and it is desirable to measure origi-nal disconnected correlations between an Abelian Wilsonloop and Abelian operators directly. Therefore, we hereadopt the above four gauge fixings smoothing the vacuumand evaluate such disconnected correlation functions: ρ ( W a , O b ) = h W a O b i − h W a i h O b ih W a i , (20)where W a is an Abelian Wilson loop, and O b is anAbelian operator. Here a and b denote color indices. Aschematic figure and the definition of coordinates are de-picted in Fig.3. In this simulation we adopt Wilson loopsof W(R=3,T=3) at β = 3 .
0, W(R=5,T=5) at β = 3 . β = 3 . lattice. Thephysical q − ¯ q distances are almost equal to 0.48 fm forthese Wilson loops. -0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 2 4 6 8 10 β =3.5, V=24 , W(7,7) a < O > r/a (rotE) φ ∂ t B φ π k φ FIG. 8:
The dual Amp`ere’s law at β = 3 . lattice in the MCG.
1. Color electric field distributions
To evaluate the profile of the Abelian color electricfield, we calculate the correlation between an AbelianWilson loop and an Abelian plaquette. In the naivecontinuum limit a →
0, the correlation becomes h E i i q ¯ q .From now on, only the MCG is discussed among globalcolor invariant gauges since the behaviors in the DLCGand the MAWL are much the same as those in the MCG.The results are as follows:1. When we put a static quark-antiquark producingan adjoint color flux, Abelian electric fields withthe same color alone exist around the quark pair asis naturally expected. It is shown in Fig.4.2. The Abelian electric fields are squeezed actually.Figure 5 shows the electric-field components atthe midpoint between the quark and the antiquarkpair. The electric field runs parallel to the directionbetween the quark and the antiquark static sources.3. Figure 6 shows h E z i q ¯ q in four smooth gauge fixings.In the case of MAU1, the global color symmetry isbroken. Hence, we evaluate both the diagonal com-ponent and the off-diagonal one separately. Thesedata are fitted to a function f ( r ) = c exp( − rλ ) + c . (21)The parameter λ corresponds to the penetrationdepth and the values for different gauge fixings aresummarized in Table IV. The difference appearsonly with respect to the coefficient c in the fittingfunction Eq.(21). These results show that there islittle gauge dependence with respect to the behav-ior of the squeezing of the Abelian color electricfield. TABLE IV:
The penetration length at β = 3 . λ [fm] c c MCG 0.189(16) 0.0330(12) -0.00045(44)DLCG 0.175(13) 0.0352(12) -0.00067(36)MAWL 0.189(16) 0.0336(13) -0.00043(45)MAU1 0.190(14) 0.0482(15) -0.00065(53)MAU1(off-diagonal) 0.175(17) 0.0175(8) -0.000(2)
D. The monopole-current distribution
We then evaluate the monopole-current k bi distribu-tions around the static quark and antiquark pair definedby the relation ρ ( W a , k bi ) = h W a k bi ih W a i , (22)where a and b are color indices.
1. The correlation between different color objects
It is interesting to see the color correlation between thecolor of the static quark source and that of monopoleskeeping the global color invariance. Here we evaluatethe correlations adopting the above three smooth gaugeskeeping the global color invariance at three different cou-plings β = 3 . , . , .
5. The example of the k φ distribu-tion in the MCG case is shown in Fig.7. We find the peakof the signal of the monopole current(VNABI) slightlyaway from the flux-tube. There are no correlations be-tween different colors. This result means that an Abeliancolor electric field is squeezed by the same color monopolecurrent alone. This is consistent with the Abelian con-finement picture.
2. The dual Amp ` e re’s law To see what squeezes the color-electric field, we inves-tigate the dual Amp`ere’s law derived from the definitionof the monopole current(rot E a ) φ = ∂ t B aφ + 2 πk aφ , (23)where index a is a color index with a = 1 , ,
3. We con-firm the dual Amp`ere’s law holds in four smooth gaugefixings. As a typical global-color invariant gauge, weshow graphs for the MCG alone in Fig.8. The electricfield is squeezed mainly due to Abelian monopole cur-rents as obtained in the MAG [24] and in the work with-out any gauge fixing [10]. In the case of MAU1, theglobal color invariance is broken. With respect to thediagonal component in MAU1 gauge, Abelian monopolecurrents are shown to squeeze the electric field [24]. But β =1.40, V=24 , W(7,7) a < E z > a < k > r/a Ezk FIG. 9:
The behaviors of the electric field squeezing andthe monopole density at β = 1 .
40 on the 24 lattice in the MCG. the behavior of the off-diagonal component looks differ-ent. In this case, the Abelian color magnetic displace-ment current ∂ t B seems to play the role of squeezing theoff-diagonal electric field instead of the Abelian monopolecurrent like in the Landau gauge [25]. But in the MAU1case, it is only apparent, since even the off-diagonal com-ponents contain monopoles if lattice artifacts are deletedenough as studied in Ref.[12], whereas in the Landaugauge, lattice monopoles `a la Degrand-Toussaint[13] donot exist. TABLE V:
The GL parameter in the MCG β λ [fm] ξ/ √ √ κ E. The vacuum type in the MCG
Finally, we evaluate the Ginzburg-Landau (GL) pa-rameter, which characterizes the type of the (dual) su-perconducting vacuum. In the previous result [10], theyfound that the vacuum type is near the border betweenthe type 1 and type 2 dual superconductors by using theSU(2) Iwasaki action without gauge fixing. The SU(2)Iwasaki action is adopted also to make a comparisonwith the previous result [10]. The Iwasaki improved ac-tion is essentially the same as (11) except the mixingparameter. Here, we evaluate the GL parameter in thecase of a smooth MCG. The lattices adopted are 24 for β = 1 . , . , . P µ k µ ( s ) k µ ( s ) and the Abelian Wilson loop by using thedisconnected correlation function. The typical data isshown in Fig.9. We fit the profile of h P µ k µ ( s ) k µ ( s ) i q ¯ q to the function g ( r ) = c ′ exp( − √ rξ ) + c ′ , (24)where the parameter ξ corresponds to the coherencelength. The number of gauge configurations is N conf =1000 to get the signal of the correlation. We show theresult of the GL parameter κ = λ/ξ in the Table V. TheGL parameter in MCG is close to the value of the pre-vious result [10]. These show that the vacuum after thesmooth MCG captures the essential property of the vac-uum in SU(2) gauge theory with a reasonable number offield configurations as opposed to the case of no gaugefixing. IV. CONCLUSION
In conclusion, we have studied monopole dominanceand the dual Meissner effect in three smooth gauge fixingswhich preserve global color symmetry as well as in theMAG. The summary is depicted as follows:1. The string tension of the static potential is re-produced fairly well by the monopole contribu-tion. When the string tension is evaluated afterthe block-spin transformation of monopoles, themonopole dominance is improved. The value of thestring tension in the MCG and the MAG becomeabout the same on the blocked lattice. These re-sults suggest that perfect monopole dominance andgauge independence are realized in the continuumlimit.2. In the study of the dual Meissner effect due toAbelian-like monopoles, the electric field havinga color is squeezed by the corresponding coloredmonopoles alone, as predicted by the Abelian pic-ture of confinement. We find the scaling behaviorof the dual Meissner effect in four gauge fixings.3. The vacuum type is determined to be at the borderbetween type 1 and type 2 in SU(2) gauge theorywith the smooth MCG gauge. This is consistentwith the previous data without gauge-fixing [10].4. The Abelian monopoles here correspond to VNABIin the continuum limit which are gauge variant.Hence, we have to adopt any method extracting the continuum gauge-invariant part on the lattice.One way is to adopt a very large number of vacuumensembles for an average as adopted in Ref.[10].Another method is to adopt a vacuum ensemblewhich is smooth enough to reduce the lattice ar-tifacts as much as possible. In this sense, adopt-ing a special gauge is important. The MAG is thesmoothest gauge known so far. Here, we show thatthe MCG is also a good gauge which can repro-duce roughly the essential monopole properties ofthe continuum SU(2) QCD with a reasonable num-ber of field configurations similarly as in the MAG.Moreover, contrary to the MAG, the MCG has theadvantage of preserving the global color invarianceand is so very interesting. To study the correlationbetween the Abelian monopoles and the center vor-tex in the MCG may also be interesting, since theMCG was first discussed in the framework of thecenter vortex model [15, 16].5. Since the Abelian-like monopoles studied in thiswork and the previous works[12, 20] have a gauge-invariant continuum limit, it is very important tostudy what quantity corresponds to the limit in theframework of continuum QCD.6. In the Abelian projection scenario of color confine-ment proposed by ’t Hooft [4], Abelian monopolesappear as topological objects corresponding to thehomotopy group by adopting a partial gauge fixing.There the singularity leading to Abelian monopolescomes from the partial gauge fixing. In our sce-nario, VNABI comes from a line singularity pos-sibly existing in original gauge fields. If this sce-nario is correct, we have to deal with a field theorycomposed of an operator having such a singularity.Such a singular operator is not considered in theframework of usual axiomatic field theory. It is in-teresting to extend a mathematical framework toaccommodate such a singular operator.
V. ACKNOWLEDGMENTS
The numerical simulations of this work were done us-ing HPC and NEC SX-ACE computer at RCNP of Os-aka University and partially by NEC SX-Aurora com-puter at KEK. The authors would like to thank RCNPand KEK for their support through computer facilities.T.S. was finacially supported by JSPS KAKENHI GrantNo. JP19K03848. A.H. was financially supported by theSasakawa Scientific Research Grant from The Japan Sci-ence Society. [1] K. Devlin,
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