Quark confinement due to nonAbelian magnetic monopoles in SU(3) YangMills theory
Abstract
We present recent results on quark confinement: in SU(3) YangMills theory, confinement of fundamental quarks is obtained due to the dual Meissner effect originated from nonAbelian magnetic monopoles defined in a gaugeinvariant way, which is distinct from the wellknown Abelian projection scenario. This is achieved by using a nonAbelian Stokes theorem for the Wilson loop operator and a new reformulation of the YangMills theory.
address=Department of Physics, Graduate School of Science, Chiba University, Chiba 2638522, Japan , address=Computing Research Center, High Energy Accelerator Research Organization, Tsukuba 3050801, Japan address=Department of Physics, Graduate School of Science, Chiba University, Chiba 2638522, Japan address=Fukui National College of Technology, Sabae 9168507, Japan
Quark confinement due to nonAbelian magnetic monopoles in SU(3) YangMills theory
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Keywords: quark confinement, dual Meissner effect, dual superconductor, magnetic monopole
PACS: 12.38.Aw, 12.38.Gc, 21.65.Qr
Introduction. – The dual superconductor picture proposed long ago dualsuper () is believed to be a promising mechanics for quark confinement. For this mechanism to work, however, magnetic monopoles and their condensation are indispensable to cause the dual Meissner effect leading to the linear potential between quark and antiquark, namely, area law of the Wilson loop average. The Abelian projection method proposed by ’t Hooft tHooft81 () can be used to introduce such magnetic monopoles into the pure YangMills theory even without matter fields. Indeed, numerical evidences supporting the dual superconductor picture resulting from such magnetic monopoles have been accumulated since 1990 in pure SU(2) YangMills theory SY90 (); SNW94 (); AS99 (). However, the Abelian projection method explicitly breaks both the local gauge symmetry and the global color symmetry by partial gauge fixing from an original nonAbelian gauge group to the maximal torus subgroup, . Moreover, the Abelian dominance SY90 () and magnetic monopole dominance SNW94 () were observed only in a special class of gauges, e.g., the maximally Abelian (MA) gauge and Laplacian Abelian (LA) gauge, realizing the idea of Abelian projection.
For , we have already succeeded to settle the issue of gauge (in)dependence by introducing a gaugeinvariant magnetic monopole in a gauge independent way, based on another method: a nonAbelian Stokes theorem for the Wilson loop operator DP89 (); Kondo98b () and a new reformulation of YangMills theory rewritten in terms of new field variables KMS06 (); KMS05 (); Kondo06 () and KKMSSI05 (); IKKMSS06 (); SKKMSI07 (), elaborating the technique proposed by Cho Cho80 () and Duan and Ge DG79 () independently, and later readdressed by Faddeev and Niemi FN99 ().
For , , there are no inevitable reasons why degrees of freedom associated with the maximal torus subgroup should be most dominant for quark confinement. In this case, the problem is not settled yet. In this talk, we give a theoretical framework for describing nonAbelian dual superconductivity in dimensional YangMills theory, which should be compared with the conventional Abelian dual superconductivity in YangMills theory, hypothesized by Abelian projection. We demonstrate that an effective lowenergy description for quarks in the fundamental representation (abbreviated to rep. hereafter) can be given by a set of nonAbelian restricted field variables and that nonAbelian magnetic monopoles in the sense of Goddard–Nuyts–Olive–Weinberg nAmm () are the most dominant topological configurations for quark confinement as conjectured in KT99 (); Kondo99Lattice99 (). This is the nonAbelian dual superconductor scenario for quark confinement for YangMills theory proposed in KSSK10 ().
Reformulation of YangMills theory using new variables – By using new variables, we have reformulated the SU(3) YangMills theory in the continuum KSM08 () and on a lattice latticef (). For , there exist two possible options: maximal one with the maximal stability subgroup Cho80c (); FN99a () and the minimal one with the maximal stability subgroup KSM08 (). The minimal option we have found in KSM08 () is a new formulation. In our reformulation, all the new variables are obtained by the change of variables from the original variable, once the color field is determined by solving the reduction condition for a given set of the original variables. In the continuum, for the change of variables from to , and :
(1) 
the reduction condition is given by
(2) 
Numerical simulations on a lattice – On a fourdimensional Euclidean lattice, the gauge field configurations (link variables) are generated by using the standard Wilson action and pseudo heatbath method. For a given , color field are determined by imposing a lattice version of reduction condition. Then new variables are introduced by using the lattice version of change of variables latticef ().
In Fig. 1, we give the result KSSK10 (): the full quarkantiquark potential obtained from the Wilson loop average , the restricted part obtained from the Wilson loop average of the restricted variable , and magnetic–monopole part obtained from following from the nonAbelian Stokes theorem. They are gauge invariant by construction. These results exhibit infrared dominance in the string tension (85–90%) and nonAbelian magnetic monopole dominance in the string tension (75%) in the gauge independent way.
In Fig.2, we give the color flux produced by a quarkantiquark pair obtained in Shibata12 (). In order to explore the color flux in the gauge invariant way, we use the connected correlator of the Wilson line (see the right panel of Fig.2):
(3) 
In the naive continuum limit, reduces to the field strength:
(4) 
Thus, the color filed strength produced by a pair is given by .
These are numerical evidences supporting the “nonAbelian” dual superconductivity due to nonAbelian magnetic monopoles as a mechanism for quark confinement in SU(3) YangMills theory.
Summary. – We have shown for the YangMills theory in dimensions:

We have defined a gaugeinvariant magnetic monopole inherent in the Wilson loop operator by using a nonAbelian Stokes theorem for the Wilson loop operator, even in YangMills theory without adjoint scalar fields. The Wilson loop operator can be rewritten Kondo08 () in terms of a pair of the gaugeinvariant magneticmonopole current (form) and the associated geometric object defined from the Wilson surface bounding the Wilson loop , and another pair of an electric current (oneform independently of ) and the associated topological object, due to a nonAbelian Stokes theorem for the Wilson loop operator Kondo08 ().
For quarks in the fundamental representation, the stability group is given by for .
G=SU(2) Abelian magnetic monopole SU(2)/U(1)
G=SU(3) nonAbelian magnetic monopole SU(3)/U(2) 
We have constructed a new reformulation KSM08 () of the YangMills theory in terms of new field variables obtained by change of variables from the original YangMills gauge field , so that it gives an optimal description of the nonAbelian magnetic monopole defined from the Wilson loop operator in the fundamental rep. of quarks. The reformulation allows options discriminated by the maximal stability group of the gauge group .
For , two options are possible:
The minimal option gives an optimized description of quark confinement through the Wilson loop in the fundamental representation.
The maximal option, , the new theory reduces to a manifestly gaugeindependent reformulation of the conventional Abelian projection in the maximal Abelian gauge.
The idea of using new variables is originally due to Cho Cho80c () and Faddeev and Niemi FN99a (), where color fields () are introduced. However, our reformulation in the minimal option is new for : we introduce only a single color field for any , which is enough for reformulating the quantum YangMills theory to describe confinement of the fundamental quark.

We have constructed a lattice version latticef () of the reformulation of the YangMills theory and performed numerical simulations for the case on a lattice, Numerical simulations of the lattice YangMills theory give numerical evidences that the restricted field variables become dominant in the infrared for the string tension and correlation functions (infrared dominance of the restricted nonAbelian variables) and that the magnetic monopole gives a dominant contribution to the string tension obtained from Wilson loop average (nonAbelian magnetic monopole dominance for for quark confinement (in the string tension)). This should be compared with the infrared Abelian dominance and magnetic monopole dominance in MA gauge.

We have shown the numerical evidence of the dual Meissner effect caused by nonAbelian magnetic monopoles in YangMills theory: the tubeshaped flux of the chromoelectric field originating from the restricted field including the nonAbelian magnetic monopoles.
To confirm the nonAbelian dual superconductivity picture proposed KSSK10 () for YangMills theory, we plan to do further checks, e.g., determination of the type of dual superconductor, measurement of the penetrating depth, induced magnetic current around color flux due to magnetic monopole condensations, and so on.
Acknowledgments. – This work is supported by GrantinAid for Scientific Research (C) 24540252 from Japan Society for the Promotion Science (JSPS), and also in part by JSPS GrantinAid for Scientific Research (S) 22224003. The numerical calculations are supported by the Large Scale Simulation Program No.09/1019 (FY20092010) and No.1013 (FY2010) of High Energy Accelerator Research Organization (KEK).
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