^{1, 2}

^{3}

^{1}

^{2}

^{3}

Constructing the effective action for dyonic field in Abelian projection of QCD, it has been demonstrated that any charge (electrical or magnetic) of dyon screens its own direct potential to which it minimally couples and antiscreens the dual potential leading to dual superconductivity in accordance with generalized Meissner effect. Taking the Abelian projection of QCD, an Abelian Higgs model, incorporating dual superconductivity and confinement, has been constructed and its representation has been obtained in terms of average of Wilson loop.

Quantum chromodynamics (QCD) is the most favored color gauge theory of strong interaction whereas superconductivity is a remarkable manifestation of quantum mechanics on a truly macroscopic scale. In the process of current understanding of superconductivity, Rajput et al. [

The first model of QCD vacuum, in which the non-Abelian dyons are responsible for confinement, was given by Simonov [

Evaluating Wilson loops under the influence of the Abelian field due to all monopole currents, monopole dominance has been demonstrated [

Starting with generalized field equations and the corresponding Lagrangian of the field associated with Abelian dyons in this paper, it has been demonstrated that topologically, a non-Abelian gauge theory is equivalent to a set of Abelian gauge theories supplemented by dyons which undergo condensation leading to confinement and consequently to superconducting model of QCD vacuum, where the Higgs fields play the role of a regulator only. It has also been demonstrated that for the self-dual fields, the Abelian monopoles become the Abelian dyons, and in low energy QCD the dyon interactions are saturated by duality when Abelian projection is described by the Abelian Higgs model where dyons are condensed leading to confinement and the state of dual superconductivity. Constructing the effective action for dyonic field in Abelian projection of QCD in terms of electric and magnetic constituents,

The quantum average of Wilson loop has been obtained in the dyonic theory specified by a partition function in terms of dyon Lagrangian in Abelian Higgs model, and the effective electric and magnetic charges and four-currents of dyons have been determined from Wilson loop given in terms of electromagnetic field tensor satisfying field equations identical to those for usual electrodynamics.

A gauge invariant and Lorentz covariant quantum field theory of fields associated with dyons has been developed [

generalized charge

and generalized four-potential

In the compact form, these equations may be written as

The Lagrangian density for spin-1 generalized charge (i.e., bosonic dyon) of rest mass

An Abelian dyon moving in the generalized field of another dyon carries a residual angular momentum [

With the development of non-Abelian gauge theories, Dirac monopole has mutated in another way as we have to take into account not only electromagnetic U(1) gauge group but also the color gauge group SU(3)c describing strong interaction. In QCD, because SU(3) is compact, the color electric charges defined with respect to any maximal Abelian subgroup are quantized. It implies that we can write down gauge field configurations that asymptotically look like magnetic monopole of any chosen Abelian direction. The confinement of color electric charge corresponds to the screening of color magnetic charge. There are monopole field configurations in any non-Abelian gauge theory. The phase structure of any such theory can be probed by adding a scalar field (i.e., Higgs field) in the adjoint representation so long as it does not change the nature of flow of the coupling constant with energy. For asymptotically free theories, the low energy behavior is dominated by the Abelian monopoles of almost zero mass which are almost point-like. The interaction of these point-like monopoles with gluons and charged particles can be studied as a dual analogue of point-like charged particle interactions. It leads to condensation of monopole. Thus topologically, a non-Abelian gauge theory is equivalent to a set of Abelian gauge theories supplemented by monopoles which undergo condensation leading to confinement. Thus the non-Abelian confinement of dyonic charge is related to linear Abelian theory in a dyonic superconductor.

Let us first consider the effective action for dyonic field in this Abelian projection of QCD in the following manner [

Using (

Let us apply (

Let us consider electric and magnetic charges on different particles (i.e., not dyons). Then field equations (

These relations show that charged particles

The non-Abelian nature of gauge group [SU(3) or SU(2)] is quite crucial to dyon condensation as mechanism of confinement. The method of Abelian projection is one of the popular approaches to the confinement problem in the non-Abelian gauge theories. A general non-Abelian theory of dyons consists of usual four-space (external) and n-dimensional internal group space, where the field associated with dyons has

A suitable Lagrangian density of a spontaneously broken non-Abelian gauge theory SU(2), yielding the classical dyonic solutions, may be constructed as

The gauge-dependent part of Lagrangian, that is, first term of rhs in (

In the dyon theory, specified by partition function (

Let us apply the transformation (

Equations (

The gauge depended part of the Lagrangian density, given by (

It is generally suspected that the dyonic theory is CP-violating contrary to QCD in the sense that dyon