Decomposition of the SU(2) gauge field in the Maximal Abelian gauge
aa r X i v : . [ h e p - l a t ] J a n Decomposition of the SU(2) gauge field in the Maximal Abelian gauge
V. G. Bornyakov
NRC “Kurchatov Institute” - IHEP, Protvino, 142281 Russia,NRC “Kurchatov Institute” - ITEP, Moscow, 117218 Russia
I. Kudrov
NRC “Kurchatov Institute” - ITEP, Moscow, 117218 Russia
R. N. Rogalyov
NRC ”Kurchatov Institute” - IHEP, 142281 Protvino, Russia
We study decomposition of SU (2) gauge field into monopole and monopoleless components. Afterfixing the Maximal Abelian gauge in SU (2) lattice gauge theory with Wilson action we decomposethe nonabelian gauge field into the Abelian field created by monopoles and the modified nonabelianfield with monopoles removed. We then calculate respective static potentials in the fundamentaland adjoint representations and confirm earlier findings that the sum of these potentials approx-imates the nonabelian static potential with good precision at all distances considered. Repeatingthese computations at three lattice spacings we find that in both representations the approximationbecomes better with decreasing lattice spacing. Our results thus suggest that this approximationbecomes exact in the continuum limit. We further find the same relation (for one lattice spacing)to be valid also in the cases of improved lattice action and in the theory with quarks. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.AwKeywords: gauge field theory, confinement, monopoles, maximal Abelian gauge
I. INTRODUCTION
We study numerically the lattice SU (2) gluodynamics in the Maximal Abelian gauge (MAG) and consider decom-position of the lattice gauge field U µ ( x ) U µ ( x ) = U modµ ( x ) U monµ ( x ) (1)where U monµ ( x ) is the monopole component and U modµ ( x ) is respectively the monopoleless component which we alsowill call a modified gauge field. By modification we understand removal of monopoles.It is well known [2–6] that after performing the Abelian projection in the MAG [7, 8], the Abelian string tensioncalculated from the Abelian static potential is very close to the nonabelian string tension and the correspondingcoefficient of the Coulomb term is about 1/3 of that in the nonabelian static potential. The former observation,like many others, supports the concept of Abelian dominance (for a review see e.g. [11]). It was further discovered[4, 9, 10] that the monopole static potential also has string tension close to the nonabelian one and small coefficientof the Coulomb term. These observations are in agreement with conjecture that monopole degrees of freedom areresponsible for confinement [12]. It is then interesting to see what kind of static potential one obtains if the monopolecontribution into the gauge field switches off, that is, if only off-diagonal gluons and the so called photon part of theAbelian gluon field are left interacting with static quarks.Previously computations of this kind were made in [13, 14], where it was shown that the topological charge, chiralcondensate and effects of chiral symmetry breaking in quenched light hadron spectrum disappear after removal ofthe monopole contribution from the relevant operators. Similar computations were made within the scope of the Z projection studies [15]. It was shown that modified gauge field with removed projected center vortices (P-vortices)produces Wilson loops without area law, i.e. devoid of the confinement property. We do a similar removal withmonopoles. We consider three types of the static potential: V mod ( r ) obtained from the Wilson loops of the modifiedgauge field U modµ ( x ), V mon ( r ) obtained from the Wilson loops of the monopole gauge field U monµ ( x ) and the sum ofthese two static potentials.The decomposition (1) was first considered in [1]. It was demonstrated for one value of the lattice spacing that V mod ( r ) could be well fitted by purely Coulomb fit function and the sum V mod ( r ) + V mon ( r ) was a good approximationof the original nonabelian static potential, V ( r ), at all distances.Here we study this phenomenon at three lattice spacings using the Wilson lattice gauge field action and thus wecan make conclusions about the continuum limit. We also present the results for one lattice spacing obtained withthe improved lattice field action thus checking the universality. Furthermore, we present results for the SU (2) theorywith dynamical quarks, i.e. for QC D.The paper is organized as follows. In the next section we introduce relevant definitions and describe details ofour computations. In section 3 results for the static potential are presented. Section 4 is devoted to discussion andconclusions.
II. DEFINITIONS AND SIMULATION DETAILS
We consider the SU (2) lattice gauge theory after fixing MAG. The Abelian projection means coset decompositionof the nonabelian lattice gauge field U µ ( x ) into the Abelian field u µ ( x ) and the coset field C µ ( x ) [22]: U µ ( x ) = C µ ( x ) u µ ( x ) , (2)The Abelian gauge field can be further decomposed into the monopole (singular) part u monµ ( x ) and the photon(regular) part u phµ ( x ) [16]: u µ ( x ) = u monµ ( x ) u phµ ( x ) . (3)In terms of the corresponding angles it has the form θ µ ( x ) = θ monµ ( x ) + θ phµ ( x ) , (4)where θ µ ( x ) ∈ ( − π, π ] is defined by u µ ( x ) = e iθ µ ( x ) , and θ mon,phµ ( x ) are defined analogously. θ monµ ( x ) can be presentedas follows: θ monµ ( x ) = − π X y D ( x − y ) ∂ ′ ν m νµ ( y ) , (5)where D ( x ) is lattice inverse Laplacian, ∂ ′ ν is lattice backward derivative, m νµ ( x ) are Dirac plaquettes. This solutionsatisfies the Lorenz gauge condition ∂ ′ µ θ mon ( s, µ ) = 0. We calculate the usual Wilson loops W ( C ) = 12 Tr W ( C ) , W ( C ) = Y l ∈ C U ( l ) ! , (6)the monopole Wilson loops W mon ( C ) = 12 Tr Y l ∈ C u mon ( l ) ! , (7)and the nonabelian Wilson loops with removed monopole contribution W mod ( C ) = 12 Tr W mod ( C ) , W mod ( C ) = Y l ∈ C ˜ U ( l ) ! , (8)where the modified nonabelian gauge field ˜ U µ ( x ) is defined as˜ U µ ( x ) = C µ ( x ) u phµ ( x ) . (9)Note that u ph ( x ) is the Abelian projection of ˜ U µ ( x ) and involves no monopoles.It is known that MAG fixing leaves U (1) gauge symmetry unbroken. The general form of the U (1) gauge transfor-mation is given by θ ′ µ ( x ) = θ µ ( x ) + ∂ µ ω ( x ) + 2 πn µ ( x ) , (10)where θ ′ µ ( x ) , ω ( x ) ∈ ( − π, π ], n µ ( x ) = 0 , ±
1. Thus there are ’small’ gauge transformations with n µ ( x ) = 0 and’large’ gauge transformations with n ( s, µ ) = ±
1. The monopole Wilson loop W mon ( C ) is invariant under these gaugetransformations. This is not true for W mod ( C ). It was shown in [1] that W mod ( C ) is invariant only under ’small’gauge transformations and it is necessary to remove ’large’ gauge transformations. To this end we fix the Landau U (1) gauge using the gauge condition max ω X x,µ cos ( θ ′ µ ( x ) . (11)Up to Gribov copies this conditions fixes configuration of Dirac plaquettes m µν ( x ) completely. Fixing U (1) Landaugauge is excessive for our purposes but is eligible for calculations of W mod .We calculated r × t rectangular Wilson loops W ( r, t ), W mon ( r, t ) and W mod ( r, t ). To extract respective staticpotentials the APE smearing [18] has been employed. Computations were done with the Wilson lattice action at β = 2 . , . lattices and at β = 2 . lattices using 100 statistically independent configurations. With thetadpole improved action the simulation were made at β = 3 . lattices. The simulations in QC D were madeon 32 lattice with small lattice spacing [20]. To fix MAG, the simulated annealing algorithm [4] with one gauge copywas used. III. STATIC POTENTIAL IN FUNDAMENTAL AND ADJOINT REPRESENTATIONS
We present our results for the sum V mon ( r ) + V mod ( r ) and compare it with the nonabelian potential V ( r ) in Fig.1for lattice Wilson action and three lattice spacings. One can see that the nonabelian static potential V ( r ) is wellapproximated by this sum, i.e. V ( r ) ≈ V mon ( r ) + V mod ( r ) . (12)This observation can be formulated in the following way: potential V ( r ) between static sources interacting withthe nonabelian gauge field U µ ( x ) can be approximated by the sum of the potential V mon ( r ) between the sourcesinteracting only with the monopole field U monµ ( x ) and the potential V mod ( r ) between the sources interacting only withthe modified (monopoleless) field U modµ ( x ).We fitted all static potentials to the fit function V ( r ) = V + α/r + σr. (13)The results for the string tension σa and the Coulomb coefficient α are presented in Table I, irrelevant parameter V is not shown.One can see that the agreement between V mon ( r ) + V mod ( r ) and V ( r ) improves with decreasing lattice spacing.This is the main result of this paper. To make it more explicit we show in Fig. 2 the relative deviation determined asfollows: ∆( r ) = V ( r ) − ( V mon ( r ) + V mod ( r )) V ( r ) . (14)More extended study with increased precision and enlarged set of lattices is needed to make final conclusion aboutthe continuum limit.In Fig.1 we also show the monopole V mon ( r ) and the modified field V mod ( r ) potentials separately. We find that V mon ( r ) is linear at large distances and has small curvature at small distances, which can be well fitted by the Coulombbehavior with small positive coefficient. The slope of V mon ( r ) at large distances agrees better and better with thatof V ( r ) with decreasing lattice spacing. We shall note that increasing of the ratio σ mon /σ with decreasing latticespacing was reported before in [21].It can be seen that V mod ( r ) is of Coulombic form. Indeed it can be very well fitted by the fitting function V mod − α mod /r with α mod = 0 . β = 2 . β = 2 . , .
6. One can see from Fig.1 that V mod ( r ) is ina very good agreement with the Coulombic part of V ( r ). Thus removing the monopole contribution from the Wilsonloop operator leaves Wilson loop which has no area law, i.e., the confinement property is lost. This result is similarto that obtained in [15] after removing P-vortices.Apart from approach to the continuum limit we studied the question of universality of the decomposition eq. (12).The simulations were made with the tadpole improved action at β = 3 .
4. The lattice spacing at this coupling isapproximately equal to that of the Wilson action at β = 2 .
5. The result is presented in the Fig. 3 (left). One can seethat agreement between V ( r ) and V mod ( r ) + V mon ( r ) is nearly as good as in Fig. 1 for β = 2 . D on 32 lattice with small lattice spacing (for details of simulationssee, e.g. [20]). The result is presented in Fig. 3 (right). One can see clearly that approximate decomposition is fulfilledwith rather high precision in this case as well. V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum 0123456 0 0.5 1 1.5 2 2.5 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum01234567 0 0.5 1 1.5 2 2.5 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum FIG. 1: Comparison of the nonabelian potential V ( r ) (filled squares) with the sum V mod ( r ) + V mon ( r ) (filled circles) for β = 2 . β = 2 . β = 2 . V mod ( r ) (empty squares) and V mon ( r ) (empty circles) arealso depicted. The solid curve shows the fit to eq. (13). Two dashed curves show its Coulomb and linear terms with adjustedconstants. TABLE I: Parameters of the potentials obtained by fits to the function (13)Potential β = 2 . β = 2 . β = 2 . σa α σa α σa αV V mon + V mod V mon V mod - -0.25(1) - -0.27(1) 0.002(1) -0.27(1) Next we come to the static potential in the adjoint representation. In this case we check the validity of the relation V adj ( r ) ≈ V adj,mod ( r ) + V mon,q ( r ) (15)Our numerical results for three lattice spacing for the Wilson action and for one lattice spacing for the improvedaction are presented in Fig. 4. In this case the precision of our results is lower still it is seen that the relation (15)is satisfied quite well. The signature of improving the agreement between lhs and rhs in (15) with decreasing latticespacing is also seen although this should be checked in more precise measurements. -0.15-0.1-0.0500.050.10.15 0 0.5 1 1.5 2 2.5 ∆ r*sqrt( σ )2.42.52.6 FIG. 2: The relative deviation ∆( r ) defined in eq. (14) vs. distance r for three values of the coupling constant β . V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum 012345678 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)mon+modmodmon FIG. 3: Comparison of the nonabelian potential V ( r ) (filled squares) with the sum V mod ( r )+ V mon ( r ) (filled circles) for improvedaction at β = 3 . D (right). V mod ( r ) (empty squares) and V mon ( r ) (empty circles) are also shown. The solidcurve and dashed curves carry same meaning as in Fig. 1. IV. CONCLUSIONS
We studied the decomposition of the static potential in the fundamental and adjoint representations into the linearterm produced by the monopole (Abelian) gauge field U mon ( x ) and the Coulomb term produced by the monopolelessnonabelian gauge field U mod ( x ). We confirm the results of Ref. [1] and improve them in a few respects. First, we madecomputations with varying lattice spacing and found that in both representations the agreement becomes better withdecreasing lattice spacing. Our results suggest that the relations (12) and (15) become exact in the continuum limit.Further work is needed to provide more evidence for this conclusion. Second, we checked that the decompositionis valid also in the case of improved lattice action and in the theory with quarks. These results make it even moreinteresting to check this decomposition in the case of SU (3) gauge group.There are few conclusions to be drawn from the decomposition (12). It suggests that the monopole part U mon ( x )is responsible for the classical part of the hadronic string energy while the monopoleless part U mod ( x ) produces thefluctuating part of that energy, i.e. while at small distances U mod ( x ) should reproduce the perturbative results atlarge distances it contributes to the nonperturbative physics. Acknowledgments
Computer simulations were performed on the Central Linux Cluster of the NRC “Kurchatov Institute” - IHEP(Protvino) and Linux Cluster of the NRC “Kurchatov Institute” - ITEP (Moscow). This work was supported by the V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 V (r) / s q r t ( σ ) r*sqrt( σ )SU(2)monmodifsum FIG. 4: Comparison of the adjoint nonabelian potential V adj ( r ) (filled squares) with the sum V mod,adj ( r ) + V mon,q ( r ) (filledcircles) for Wilson action at β = 2 . β = 2 . β = 2 . β = 3 . V mod,adj ( r ) (empty squares) and V mon,q ( r ) (empty circles) are also shown.The solid curve and dashed curves carry same meaning as in Fig. 1. Russian Foundation for Basic Research, grant no.20-02-00737 A. The authors are grateful to G. Schierholz, T. Suzuki,S. Syritsyn, V. Braguta, A. Nikolaev for participation at the early stages of this work and for useful discussions. [1] V. G. Bornyakov, M. I. Polikarpov, G. Schierholz, T. Suzuki and S. N. Syritsyn, Nucl. Phys. B Proc. Suppl. , 25-32(2006) [arXiv:hep-lat/0512003 [hep-lat]].[2] T. Suzuki and I. Yotsuyanagi, Phys. Rev. D (1990) 4257.[3] S. Hioki, S. Kitahara, S. Kiura, Y. Matsubara, O. Miyamura, S. Ohno and T. Suzuki, Phys. Lett. B , 326 (1991)[Erratum-ibid. B , 416 (1992)].[4] G. S. Bali, V. Bornyakov, M. Muller-Preussker and K. Schilling, Phys. Rev. D (1996) 2863.[5] V. Bornyakov and M. Muller-Preussker, Nucl. Phys. Proc. Suppl. , 646 (2002).[6] N. Sakumichi and H. Suganuma, Phys. Rev. D (2014) no.11, 111501 doi:10.1103/PhysRevD.90.111501 [arXiv:1406.2215[hep-lat]].[7] A. S. Kronfeld, M. L. Laursen, G. Schierholz and U. J. Wiese, Phys. Lett. B , 516 (1987).[8] G. ’t Hooft, Nucl. Phys. B (1981) 455.[9] H. Shiba and T. Suzuki, Phys. Lett. B , 461 (1994).[10] J. D. Stack, S. D. Neiman and R. J. Wensley, Phys. Rev. D , 3399 (1994).[11] M. N. Chernodub and M. I. Polikarpov, in ”Confinement, Duality and Non-perturbative Aspects of QCD” , p.387, PlenumPress, 1998, hep-th/9710205; R. W. Haymaker, Phys. Rept. (1999) 153.[12] G. ’t Hooft, in High Energy Physics , ed. A. Zichichi, EPS International Conference, Palermo (1975); S. Mandelstam,
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