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Advances in High Energy Physics
Volume 2010, Article ID 713659, 18 pages
http://dx.doi.org/10.1155/2010/713659
Research Article

Dyon Condensation and Dual Superconductivity in Abelian Higgs Model of QCD

1I-11, Gamma-2, Greater Noida (UP), India
2Department of Physics, Kumaon University, Nainital, Uttrakhand, India
3Department of Physics, D.S. College, Aligarh, Uttar Pradesh, India

Received 5 September 2010; Accepted 9 November 2010

Academic Editor: A. Petrov

Copyright © 2010 B. S. Rajput and Sandeep Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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. [13] and Kumar [4] have conceived its hopeful analogy with QCD and demonstrated that the essential features of superconductivity, that is, the Meissner effect and flux quantization, provided the vivid models [59] for actual confinement mechanism in QCD. Mandelstam [1012] propounded that the color confinement properties may result from the condensation of magnetic monopoles in QCD vacuum. In a series of papers [1316], Ezawa and Iwazaki made an attempt to analyze a mechanism of quark confinement by demonstrating that the Yang-Mills vacuum is a magnetic superconductor and such a superconducting state is considered to be a condensed state of magnetic monopole. The condensation of magnetic monopole incorporates the state of magnetic superconductivity [17] and the notion of chromomagnetic superconductor where the Meissner effect confining magnetic field in ordinary superconductivity would be replaced by the chromoelectric Meissner effect (i.e., the dual Meissner effect), which would confine the color electric flux. As such one conceives the idea of correspondence between quantum chromodynamic situation and chromomagnetic superconductor. However, the crucial ingredient for condensation in a chromomagnetic superconductor would be the non-Abelian force in contrast to the Abelian ones in ordinary superconductivity. Topologically, a non-Abelian gauge theory is equivalent to a set of Abelian gauge theories supplemented by monopoles [18]. The method of Abelian projection is one of the popular approaches to confinement problem, together with dual superconductivity [19, 20] picture, in non-Abelian gauge theories. Monopole condensation mechanism of confinement (together with dual superconductivity) implies that long-range physics is dominated by Abelian degrees of freedom [21] (Abelian dominance).

The first model of QCD vacuum, in which the non-Abelian dyons are responsible for confinement, was given by Simonov [19], and almost simultaneously it was demonstrated by Bornyakov and Schierholz [22] that for (anti-) self-dual fields [23, 24] the Abelian monopoles, participating in confinement, become dyons. The non-Abelian dyons give rise to Abelian dyons in the Abelian projection [22]. The mere notions of dyons and of the Kraan-van Baal instantons [25, 26] (i.e., calorons) imply that the Y-M field is periodic in the Euclidean time direction and thus leads to confinement. It has recently been demonstrated by Diakonov and Petrov [27] that the ensemble of dyons can be described mathematically by an exactly solvable field theory in three dimensions and that the resulting vacuum built of dyons has certain features expected for the confining pure Y-M theory. It has also been shown by them that the dyons ensemble induces the area law for spatial Wilson loops which fulfill confinement in three dimensions. To investigate the possible physical implications of the topological structure of non-Abelian dyons in connection with the issue of quark confinement in QCD, extended gauge theory has been formulated [8, 9] in SU(2) and SU(3) groups from the corresponding restricted chromodynamics (RCD), and it has been shown that in this extended QCD the confinement mechanism of the corresponding RCD remains intact, and physical spectrum contains color-singlet generalized electric glue balls made of valence gluon pairs as well as the generalized magnetic glue balls as massive collective modes of condensed vacuum. Recently, it has been shown [3, 4] that a perfect confinement can be achieved with pure dyonic states participating in actual dyonic condensation of RCD vacuum as the result of magnetic symmetry breaking in strong coupling limit.

Evaluating Wilson loops under the influence of the Abelian field due to all monopole currents, monopole dominance has been demonstrated [21, 28]. In the Abelian projection the quarks are the electrically charged particles and, if monopoles are condensed, the dual Abrikosov-string carrying electric flux is formed between quark and antiquark. Due to nonzero tension in this string, the quarks are confined by the linear potential. The conjecture that the dual Meissner effect is the color confinement mechanism is realized if we perform Abelian projection in the maximal gauge where the Abelian component of gluon field and Abelian monopoles are found to be dominant [29, 30]. Then the Abelian electric field is squeezed by solenoidal monopole current [31]. The vacuum of gluodynamics behaves as a dual superconductor, and the key role in dual superconductor model of QCD is played by Abelian monopole. The infrared properties of QCD in the Abelian projection can be described by the Abelian Higgs Model (AHM) in which dyons are condensed. There exists the model [27, 3234] of QCD vacuum in which the non-Abelian dyons are responsible for the confinement and the non-Abelian dyons give rise to Abelian dyons in the Abelian projection. Therefore, an important problem, before studying the vacuum properties of non-Abelian theories, is to Abelianize them so as to make contribution of the topological magnetic degrees of freedom to the partition function explicit. Such a construction for non-Abelian gauge theories and its relevance to topological magnetic charge and hence to confinement are still lacking in spite of large amount of recent literature [3540] on the subject.

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, 𝐴𝜇 and 𝐵𝜇, of the generalized four-potential 𝑉𝜇, the dyonic current correlators have been derived and it has been demonstrated that the dyonic electric charge produces screening effect for 𝐴𝜇-propagator and antiscreening effect for 𝐵𝜇-propagator while the dyonic magnetic charge produces screening effect for 𝐵𝜇-propagator and antiscreening effect for 𝐴𝜇-propagator. These antiscreening effects have been shown to lead to dyonic condensation and dual superconductivity and also to maintain the asymptotic freedom of non-Abelian gauge theory (QCD) in its Abelian version. A non-Abelian SU(2) gauge has been obtained in terms of Lagrangian density describing the fields associated with non-Abelian dyons and it has been shown that these non-Abelian dyons give rise to Abelian dyons in the Abelian projection of QCD. The infrared properties of QCD in this Abelian projection have been described by constructing an Abelian Higgs model in which dyons are condensed and the relevant degrees of freedom are two massive gluons, a U(1) gluon and a dyon. This AHM has been shown to incorporate dual superconductivity and confinement as the consequence of dyonic condensation.

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.

2. Electromagnetic Duality and Dyonic Interaction

A gauge invariant and Lorentz covariant quantum field theory of fields associated with dyons has been developed [4144] in purely group theoretical manner by using two four-potentials and assuming the generalized charge, generalized current, and generalized four-potential as complex quantities with their real and imaginary parts as electric and magnetic constituents, that is, generalized charge𝑞=𝑒𝑖𝑔,(2.1a) generalized four-current 𝐽𝜇=𝑗𝜇𝑖𝑘𝜇,(2.1b) and generalized four-potential 𝑉𝜇=𝐴𝜇𝑖𝐵𝜇,(2.1c) where 𝑒 and 𝑔 are electric and magnetic charges on dyon, 𝑗𝜇 and 𝑘𝜇 are electric and magnetic four-current densities, and 𝐴𝜇 and 𝐵𝜇 are the electric and magnetic four-potentials associated with dyons. Taking the wave function associated with generalized field as 𝜓=𝐸𝑖𝐻,(2.1d)the generalized field equations of these fields may be written as Ψ=𝐽0,𝜕Ψ×Ψ=𝑖𝐽𝑖,𝜕𝑡(2.2) where 𝐽0 and 𝐽 are the temporal and spatial parts of 𝐽𝜇 defined by (2.1b).

In the compact form, these equations may be written as𝐺𝜇𝜈,𝜈=𝐽𝜇,𝐺𝑑𝜇𝜈,𝜈=0,(2.3) where 𝐺𝜇𝜈 the generalized field tensor, is given as𝐺𝜇𝜈=𝜕𝜇𝑉𝜈𝜕𝜈𝑉𝜇(2.4) and 𝐺𝑑𝜇𝜈 is its dual given as𝐺𝑑𝜇𝜈=12𝜀𝜇𝜈𝛼𝛽𝐺𝛼𝛽.(2.5) Equation (2.4) may also be written as𝐺𝜇𝜈=𝐹𝜇𝜈𝑖𝐻𝜇𝜈,(2.6a) where 𝐹𝜇𝜈=𝜕𝜇𝐴𝜈𝜕𝜈𝐴𝜇,𝐻(2.6b)𝜇𝜈=𝜕𝜇𝐵𝜈𝜕𝜈𝐵𝜇.(2.6c)Then (2.3) reduces to the following form:𝐹𝜇𝜈,𝜈=𝑗𝜇,𝐻(2.7a)𝜇𝜈,𝜈=𝑘𝜇.(2.7b)These equations are symmetrical under the duality transformations𝐹𝜇𝜈𝐻𝜇𝜈,𝐻𝜇𝜈𝐹𝜇𝜈,𝑗𝜇𝑘𝜇,𝑘𝜇𝑗𝜇.(2.8)

The Lagrangian density for spin-1 generalized charge (i.e., bosonic dyon) of rest mass 𝑚0 may be written as follows [41] in the Abelian theory: 𝐿=𝑚014𝛼𝐴𝜈,𝜇𝐴𝜇,𝜈2𝐵𝜈,𝜇𝐵𝜇,𝜈2𝐴2𝛽𝜈,𝜇𝐴𝜇,𝜈𝐵𝜈,𝜇𝐵𝜇,𝜈+𝛼𝐴𝜇𝛽𝐵𝜇𝑗𝜇𝛼𝐵𝜇+𝛽𝐴𝜇𝑘𝜇=𝐿𝑃+𝐿𝐹+𝐿𝐼,(2.9) where 𝛼 and 𝛽 are real positive unimodular parameters, that is,|𝛼|2+||𝛽||2=1.(2.10)𝐿𝑃,𝐿𝐹, and 𝐿𝐼 are free particle, field, and interaction Lagrangians, respectively. The action integral may be written as𝑆=𝑡2𝑡1𝐿𝑑𝑡=𝑆𝑃+𝑆𝐹+𝑆𝐼.(2.11) Varying the trajectory of particle without changing the field, we get the following equation of motion:𝑚̈𝑥𝜇=Re𝑞𝐺𝜇𝜈𝑢𝜈,(2.12) where Re denotes the real part and 𝑢𝜈 is the 𝜈th component of four-velocity of dyon.

An Abelian dyon moving in the generalized field of another dyon carries a residual angular momentum [45] (field contribution) besides its orbital and spin-angular momenta. If we consider 𝑖th Abelian dyon moving in the field of 𝑗th dyon (assumed at rest), its gauge invariant rotationally symmetric orbital angular momentum may be written as [45] 𝐽=𝑟×𝑝𝜇𝑖𝑗𝑉𝑇+𝜇𝑖𝑗𝑟𝑟,(2.13) where 𝑟 is the position vector and 𝑝 is the linear momentum of 𝑖th dyon, 𝑉𝑇 is the transverse generalized vector potential of the field associated with 𝑗th dyon, and 𝜇𝑖𝑗 is the magnetic coupling parameter defined as𝜇𝑖𝑗=𝑒𝑖𝑔𝑗𝑒𝑗𝑔𝑖.(2.14) The last term in (2.14) is the residual angular momentum carried by 𝑖th dyon besides its usual orbital angular momentum and spin-angular momentum𝐽res=𝜇𝑖𝑗𝑟𝑟.(2.15) For each pair of dyons, this residual angular momentum generates a one-dimensional representation of the pair of four-momentum associated with these particles. This is the subgroup of the Lorentz group which leaves both four-momenta invariant. This residual angular momentum leads to chirality-dependent multiplicity in the eigenvalues of angular momentum of an Abelian dyon.

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 [17]:1𝑆=4𝐺𝜇𝜈(𝑥)(𝑥𝑦)𝐺𝜇𝜈𝑑4𝑥𝑑4𝑦+𝐽𝜇𝑉𝜇,(2.16) where (𝑥𝑦) is the generalized dielectric constant defined as (𝑥𝑦)=(𝑥𝑦)𝑖𝜇(𝑥𝑦)(2.17) with (𝑥𝑦) as ordinary dielectric constant and 𝜇(𝑥𝑦) as magnetic permeability such that (𝑥𝑦)𝜇(𝑥𝑧)𝑑4𝑦=𝛿(𝑥𝑧),(2.18) where 𝛿(𝑥) is Dirac-Delta function. The generalized field tensor 𝐺𝜇𝜈(𝑥) of (2.16) satisfies field equations (2.3) or equivalently the field equation (2.7a). The generalized four-current of field equation (2.3) couples to 𝑉𝜇, with the current-correlators given by𝐽𝜇=𝛿𝑆𝛿𝑉𝜇,𝑗(2.19)𝜇=𝛿(𝑥)𝐽(𝑦)2𝑆𝛿𝑉𝜈(𝑥)𝛿𝑉𝜇(𝑦).(2.20)

Using (2.17) and (2.20), we have𝐽𝜇(𝑥)𝐽𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝑒𝑖𝑘(𝑥𝑦)𝑘2𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝑘2,(2.21) where (𝑘2) is Fourier transform of (𝑥𝑦). For free fields in vacuum, (𝑘2)=1. In the perturbation theory, the deviation of (𝑘2) from 1 can be interpreted as the vacuum polarization due to dyon loops. For perturbatively small 𝜒(𝑘2), we have 𝑘2𝑘=1+𝜒2,(2.22) where𝜒𝑘2=𝜒𝑒𝑘2𝑖𝜒𝑔𝑘2(2.23) with 𝜒𝑒(𝑘2) as perturbation related with electric charge loop and 𝜒𝑔(𝑘2) as the perturbation related with magnetic charge loop.

Let us apply (2.21) to the case of dual superconductivity where includes fully nonperturbative effects. This rigidly excludes generalized electromagnetic field in side dual superconductor in conformity with the generalized Meissner effect with its real and imaginary constituents as the strict Meissner effect and dual Meissner effect, respectively. Then the generalized field 𝑉𝜇 can penetrate into a generalized superconductor up to the generalized London penetration depth𝜆𝐿=𝜆𝑒𝑖𝜆𝑔,(2.24) where 𝜆𝑒 is strict penetration depth due to Meissner effect and 𝜆𝑔 is the dual penetration depth due to dual Meissner effect. For small values of 𝑘2, we have𝑘2=𝑚2𝐿𝑘2𝑖𝑘2𝑚2𝐿,(2.25) where𝑚𝐿=1𝜆𝐿=𝑚𝑙𝑒𝑖𝑚𝐿𝑔(2.26a) or𝑚𝐿=1𝜆𝑒𝑖𝜆𝑔=𝜆𝑒+𝑖𝜆𝑔||𝜆𝐿||2.(2.26b)It gives𝑚𝐿𝑒=𝜆𝑒||𝜆𝐿||2,𝑚𝐿𝑔𝜆=𝑔||𝜆𝐿||2.(2.27) Equations (2.21) and (2.1b) then give𝑗𝜇(𝑥)𝑗𝜈(𝑦)+𝑘𝜇(𝑥)𝑘𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝑒𝑖𝑘(𝑥𝑦)𝑘2𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝑘2,𝑗𝜇(𝑥)𝑘𝜇(𝑦)𝑗𝜈(𝑥)𝑘𝜇𝑑(𝑦)=4𝑘(2𝜋)4𝑒𝑖𝑘(𝑥𝑦)𝑘4𝛿𝜇𝜈𝑘2𝑘𝜇𝑘𝜈𝜇𝑘2/𝑚2𝐿.(2.28) These equations give the generalized propagator associated with generalized field 𝑉𝜇.

3. Dual Superconductivity through Generalized Meissner Effect

Let us consider electric and magnetic charges on different particles (i.e., not dyons). Then field equations (2.3) reduce to the following form:𝐹𝜇𝜈,𝜈=𝑗𝜇,𝐹𝑑𝜇𝜈,𝜈𝐻=0,𝜇𝜈,𝜈=𝑘𝜇,𝐻𝑑𝜇𝜈,𝜈=0(3.1a) or equivalently 𝐴𝜇=𝑗𝜇,𝐵𝜇=𝑘𝜇,(3.1b)and equation of motion (2.12) becomes𝑚̈𝑥𝜇=𝑒𝐹𝜇𝜈+𝑔𝐻𝜇𝜈𝑢𝜈.(3.2) All these equations are dual invariant under the transformations (2.8). The effective action in this Abelian projection of QCD may be written as follows from (2.16):1𝑆=4𝐹𝜇𝜈(𝑥)(𝑥𝑦)𝐹𝜇𝜈(𝑦)𝑑4𝑥𝑑41𝑦4𝐻𝜇𝜈(𝑥)𝜇(𝑥𝑦)𝐻𝜇𝜈(𝑦)𝑑4𝑥𝑑4𝑦+𝑗𝜇𝐴𝜇+𝑘𝜇𝐵𝜇.(3.3) The current-correlations (2.20) may then be written as follows:𝑗𝜇=𝛿𝑆𝛿𝐴𝜇,𝑘𝜇=𝛿𝑆𝛿𝐵𝜇,𝑗𝜇(𝑥)𝑗𝜈=𝛿(𝑦)2𝑆𝛿𝐴𝜈(𝑦)𝛿𝐴𝜇(,𝑘𝑥)𝜇(𝑥)𝑘𝜈=𝛿(𝑦)2𝑆𝛿𝐵𝜈(𝑦)𝛿𝐵𝜇.(𝑥)(3.4) For the given action in the present case, these relations lead to 𝑗𝜇(𝑥)𝑗𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝑘2𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝑘2,𝑘𝜇(𝑥)𝑘𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝑘2𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝜇𝑘2.(3.5) For small perturbations, we have𝑘2=1±𝜒𝑒𝑘2,𝜇𝑘2=1𝜒𝑔𝑘2,(3.6) where the upper signs in the right-hand sides correspond to vacuum polarization due to charged particle-loops and the lower signs correspond to that due to monopole-loops. Relations (3.5) may also be written as𝑗𝜇(𝑥)𝑗𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝑘2𝑚2𝐿𝑒,𝑘𝜇(𝑥)𝑘𝜈𝑑(𝑦)=4𝑘(2𝜋)4𝛿𝜇𝜈𝑘𝜇𝑘𝜈𝑘2𝑚2𝐿𝑔.(3.7)

These relations show that charged particles [𝜒𝑒(𝑘2)1] produce screening effect for the 𝐴𝜇-propagator, with the corresponding photon acquiring the mass 𝑚𝐿𝑒 and antiscreening effect for the 𝐵𝜇-propagator. On the other hand, the monopole loops produce screening effect for 𝐵𝜇-propagator, with corresponding photon acquiring the mass 𝑚𝐿𝑔 and antiscreening effect for 𝐴𝜇-propagator. Thus any particle (electrically charged or a monopole) screens its own direct potential to which it minimally couples and antiscreens the dual potential (𝐵𝜇 for electric charge and 𝐴𝜇 for monopole). This dual antiscreening effect leads to dual superconductivity in accordance with generalized Miessner effect.

4. Dyon Condensation and Confinement

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 𝑛-fold internal multiplicity and the multiplets of gauge field transform as the basis of adjoint representation of n-dimensional non-Abelian gauge symmetry group. Choosing the internal gauge group as SU(2), the generalized dyonic field tensor may be constructed as𝐺𝜇𝜈=𝐺𝑎𝜇𝜈𝑇𝑎(4.1) with the generalized four-potential defined as𝑉𝜇=𝑉𝑎𝜇𝜈𝑇𝑎,(4.2) where repeated indices are summed over 1, 2, and 3 (internal degrees of freedom), vector sign is denoted in the internal group space, and the matrices 𝑇𝑎 are infinitesimal generators of group SU(2), satisfying the commutation relation𝑇𝑎,𝑇𝑏=𝑖𝜀𝑎𝑏𝑐𝑇𝑐(4.3) with 𝜀𝑎𝑏𝑐 as a structure constant of internal group. We may connect 𝐺𝜇𝜈 and 𝑉𝜇 through the following non-Abelian version of (2.4):𝐺𝑎𝜇𝜈=𝜕𝜈𝑉𝑎𝜇𝜕𝜇𝑉𝑎𝜈+||𝑞||𝜀𝑎𝑏𝑐𝑉𝜇𝑏𝑉𝜈𝑐,(4.4) where the dyonic generalized charge 𝑞 is given by (2.1a).

A suitable Lagrangian density of a spontaneously broken non-Abelian gauge theory SU(2), yielding the classical dyonic solutions, may be constructed as1𝐿=4𝐺𝑎𝜇𝜈𝐺𝑎𝜇𝜈+12𝐷𝜇𝜙𝑎(𝐷𝜇𝜙)𝑎𝑉(𝜙)=𝐿dyon𝐴𝜇,𝐵𝜇,𝜙,where𝐷𝜇𝜙=𝜕𝜇𝜙𝑖Re𝑞𝑉𝜇𝜕𝜙=𝜇𝑖𝑒𝐴𝜇𝑖𝑔𝐵𝜇𝜙,(4.5) with Re denoting the real part and1𝑉(𝜙)=4𝜙𝑎𝜙𝑎212𝑣2𝜙𝑎𝜙𝑎0||𝜙||0with𝑣=𝜙=(4.6) determining the vacuum expectation value of Higgs field. In simplest manner, this equation may be written as||𝜙||𝑉(𝜙)=𝜂2𝑣22(4.7) with 𝜂 as a constant.

The gauge-dependent part of Lagrangian, that is, first term of rhs in (4.5) is invariant under the following transformations of the fields 𝐴𝜇 and 𝐵𝜇:𝑉𝜇=𝐴𝜇𝐵𝜇𝐴𝜇𝐵𝜇=𝑉𝜇𝐴=𝑅(𝛿)𝜇𝐵𝜇=𝑅(𝛿)𝑉𝜇,where𝑅(𝛿)=cos𝛿sin𝛿sin𝛿cos𝛿(4.8) with𝛿=tan1𝑔𝑒.(4.9) Using the Lagrangian density given by (4.5), the electric and magnetic fields of dyons may be calculated by imposing the following ansatz [46]:𝑉𝑖𝑎=𝜀𝑎𝑖𝑗𝑟𝑗𝐾(𝑟)1||𝑞||𝑟2,𝑉0𝑎=𝑟𝑎𝐽(𝑟)||𝑞||𝑟2,𝜙𝑎=𝑟𝑎𝐻(𝑟)||𝑞||𝑟2,(4.10) where the functions 𝐾(𝑟), 𝐽(𝑟), and 𝐻(𝑟) satisfy the following equations:𝑟2𝐻(𝑟)=2𝐻𝐾2,𝑟2𝐽(𝑟)=2𝐽𝐾2,𝑟2𝐾𝐾(𝑟)=𝐾2𝐻1+𝐾2𝐽2.(4.11) A solution of these equations may be written as follows:̃𝐽(𝑟)=𝛼𝜙(𝑟),𝐻(𝑟)=𝛽𝜙(𝑟),𝐾(𝑟)=𝐶𝑟,̃𝛽sinh𝐶𝑟where2𝛼2=1,𝜙(𝑟)=𝐶(𝑟)coth𝐶𝑟1.(4.12) In the Prasad-Sommerfield limit [47], 𝑉(𝜙)=0;but𝑣=𝜙0.(4.13) In this limit, the dyons have lowest possible energy for given electric and magnetic charges 𝑒 and 𝑔, respectively. Thus we get the following expression for dyonic mass:𝑒𝑀=𝑣2+𝑔21/2||𝑞||=𝑣,(4.14) where the electric and magnetic fields associated with dyons obey the first-order equations𝐸𝑎𝑖=𝐺𝑎0𝑖=𝜕𝑖𝑉a0+||𝑞||𝜀𝑎𝑏𝑐𝑉𝑖𝑏𝑉0𝑐=𝐷𝑖𝜙𝑎𝐵sin𝛼,𝑎𝑖=𝜀𝑖𝑗𝑘𝐺𝑗𝑘𝑎=𝐷𝑖𝜙𝑎𝐷cos𝛼,0(𝜙)𝑎=0,where𝛼=tan1𝑒𝑔.(4.15) In these equations, 𝑖 and 0 indicate space and time directions and a is an SU(2) vector index. These electric and magnetic fields associated with dyons are non-Abelian in nature having external as well as internal components. In the Abelian projection, obtained by setting𝐾(𝑟)0,𝐽(𝑟)𝑏+𝑐𝑟,(4.16) where 𝑏 and 𝑐 are positive constants having the dimensions of charge and mass, respectively, these fields reduce to the following form in the asymptotic limit:𝐸𝑎𝑗=3𝑏||𝑞||𝑟4𝑟𝑎𝑟𝑗2𝑐||𝑞||𝑟3(𝑟)𝑎𝑟𝑗,𝐵𝑎𝑗=𝑟𝑗𝑟𝑎||𝑞||𝑟4.(4.17) For vanishing 𝑐 (i.e., vanishing mass), these fields correspond to point-like mass-less dyons with electric charge 3𝑏/|𝑞| and magnetic charge 1/|𝑞|. Thus non-Abelian dyons give rise to the Abelian dyons in the Abelian projection. The infrared properties of QCD in the Abelian projection can be described in the Abelian Higgs Model (AHM) in which dyons are condensed. In this model, the relevant degrees of freedom are two massive gluons 𝑊±𝜇, a U(1) gluon (associated with generalized field 𝑉𝜇), and a dyon which we take to be scalar represented by complex field 𝜙. Then the Lagrangian (4.5) reduces to𝐿dyon𝐴𝜇,𝐵𝜇1,𝜙=4𝐺𝜇𝜈𝐺𝜇𝜈+12||𝜕𝜇𝑖𝑒𝐴𝜇𝑖𝑔𝐵𝜇𝜙||2||𝜙||+𝜂2𝑣22.(4.18) In terms of this Lagrangian, the partition function in the Euclidean space-time may be written as𝑍dyon=𝐷𝐴𝜇𝐷𝐵𝜇𝑑𝐷𝜙exp4𝑥𝐿dyon𝐴𝜇,𝐵𝜇,𝜙.(4.19) Applying the transformation (4.8) and integrating over the field 𝐴𝜇, this partition function reduces to the following form in AHM:𝑍dyon=𝐷𝐵𝜇𝑑𝐷𝜙exp4𝑥𝐿AHM𝐵𝜇,,𝜙with𝐿AHM𝐵𝜇1,𝜙=4𝐻𝜇𝜈𝐻𝜇𝜈+12||𝜕𝜇𝑖𝑔𝐵𝜇𝜙||2||𝜙||+𝜂2𝑣22,(4.20) where the Higgs field 𝜙 has the magnetic charge||𝑞||𝑔=,𝐻𝜇𝑣=𝜕𝜇𝐵𝑣𝜕𝑣𝐵𝜇.(4.21) This model (AHM) incorporates dual superconductivity and hence confinement as the consequence of dyonic condensation since the Higgs-type mechanism arises here.

5. Dyonic Loop in Abelian Higgs Model

In the dyon theory, specified by partition function (4.19), the quantum average of the Wilson loop is [48]𝑊𝑐𝑙dyon=1𝑍dyon𝐷𝐴𝜇𝐷𝐵𝜇𝑑𝐷𝜙exp4𝑥𝐿dyon𝐴𝜇,𝐵𝜇𝑊,𝜙𝑙𝑐𝐴𝜇,(5.1a) where 𝑊𝑙𝑐𝐴𝜇=exp𝑖𝑒0𝑑4𝑥𝜂𝜇𝐴𝜇(5.1b) with 𝜂𝜇(𝑥)=𝐶𝑑̆𝑥𝜇𝛿(4)(𝑥̆𝑥(𝜏)),(5.1c)which creates the particle with electric charge 𝑒0 on the world trajectory 𝐶.

Let us apply the transformation (4.8) to the quantum average (5.1a) and then integrate over the field 𝐴𝜇. Thus we get𝑊𝑐𝑙dyon=𝐾𝑐(𝑞𝑙,𝑞𝑚)𝐵𝜇AHM(5.2) with the operator 𝐾𝑐(𝑞𝑒,𝑞𝑚) as the product of t Hooft loop and the Wilson loop 𝑊𝑐:𝐾𝑐(𝑞𝑒,𝑞𝑚)𝐵𝜇=𝐻𝑐𝑞𝑒𝐵𝜇𝑊𝑐𝑞𝑚𝐵𝜇,where𝑞𝑒=𝑒0𝑔||𝑞||,𝑞𝑚=𝑒0𝑒||𝑞||.(5.3) Then the effective electric and magnetic four-current density may be written as follows: 𝑗𝜇=𝑞𝑒𝜂𝜇,𝑘𝜇=𝑞𝑚𝜂𝜇.(5.4) In (5.3), the operator 𝐻𝑐𝑞𝑒(𝐵𝜇) is𝐻𝑐𝑞𝑒𝐵𝜇1=exp4𝑑4𝑥𝐻𝜇𝑣12𝜀𝜇𝑣𝛼𝛽𝐹𝛼𝛽2𝐻𝜇𝑣𝐻𝜇𝑣,where𝐻𝜇𝑣=𝜕𝜇𝐵𝑣𝜕𝑣𝐵𝜇(5.5) and 𝐹𝛼𝛽 is the dual field tensor satisfying𝐹𝜇𝜈,𝜈=𝑗𝜇(5.6) which is identical to (2.7a) for the usual electrodynamic field tensor of the field associated with Abelian dyons.

6. Discussion

Equations (2.21), (2.22), and (2.23) for dyonic current correlations show that dyonic electric charge produces the screening effect for 𝐴𝜇-propagator and antiscreening effect for 𝐵𝜇-propagator, while the dyonic magnetic charge produces screening effect for 𝐵𝜇-propagator and antiscreening effect for 𝐴𝜇-propagator. This antiscreening effect maintains the asymptotic freedom of non-Abelian gauge theory (QCD) in its Abelian version. In QCD, because of asymptotic freedom, the Landau singularity (led by charged particles in ordinary electrodynamics) is in the infrared regime and hence the most convenient microscopic theory of low energy QCD is produced by the chromodynamic dyons. The correlations (2.28) give the generalized propagator associated with generalized field 𝑉𝜇 of dyons. In the Abelian projection of QCD with the simultaneous existence of electric charges and monopoles (but not dyons), the effective action is given by (3.3) and the current correlations are given by (3.5), (3.6), and (3.7) which demonstrate that any particle screens its own direct potential to which it minimally couples and antiscreens the dual potential (𝐵𝜇 for electric charges and 𝐴𝜇 for monopoles). This dual antiscreening effect leads to dual superconductivity in accordance with generalized Miessner effect. This dual superconductivity is the Higgs phase of QCD in its Abelian projection. The antiscreening, described by (3.7), provides the prescription that the magnetic photon (𝐵𝜇)-charge particle vertex is identical to the 𝐴𝜇-charge particle vertex with the constant 𝑒 replaced by 𝑖𝑒. Such prescription of coupling of a gauge particle to its dual charge must be used only when all dual charges appear in loops. The duality prescribed by these equations may be a strong guide to the description of confinement, and interactions of chromomagnetic monopoles should be saturated by this duality, at least for low energy.

The gauge depended part of the Lagrangian density, given by (4.5) for the fields associated with the non-Abelian dyons in the minimal gauge theory, is invariant under the linear transformation (4.8). Equations (4.15) and (4.17) demonstrate that the non-Abelian dyons give rise to Abelian dyons in the Abelian projection obtained by setting up conditions given by (4.16). The infrared properties of QCD in this Abelian projection can be described by the Abelian Higgs model with Lagrangian density given by (4.20) in which dyons are condensed. In this model, the partition function in the Euclidean space-time is given by the first part of (4.20). This model incorporates dual superconductivity and confinement as the consequence of dyonic condensation. In the dyon theory, specified by the partition function given by (4.19) in terms of dyon Lagrangian (4.18), the quantum average of Wilson loop given by (5.1a) corresponds to quark Wilson loop if we consider this partition function as an effective theory of QCD. In (5.2) this average is given in AHM with the effective electric and magnetic charges and the effective electric and magnetic four-current densities given by (5.3) and (5.4), respectively. t Hooft loop is precisely given by (5.5) in terms of electromagnetic field tensor 𝐻𝜇𝜈 and the dual field tensor satisfies field equation (5.6) which is identical to (2.7a) for the usual electromagnetic field tensor of field associated with Abelian dyons. It is what we expect in the Abelian projection of QCD in the present Abelian Higgs Model of Abelian dyons in the Abelian version of QCD.

It is generally suspected that the dyonic theory is CP-violating contrary to QCD in the sense that dyon (𝑒,𝑔) and anti-dyon (𝑒,𝑔) with the same density, when combined in one vacuum in equal amount, may violate CP-invariance. We have carried out [49] the study of behavior of dyonium in non-Abelian gauge theory and also the study of dyon-dyon bound states [50] and showed that the Bohr radius of dyonium is much smaller than the atomic Bohr radius. The study of bound state of a dyon and an anti-dyon has also been carried out [51], and it has been demonstrated that this state is very short lived and decays in to four or six photons depending on the spin-statistics relationship of the dyons involved. Furthermore, CP-invariance of the vacuum requires that there must be equal number of self-dual and anti-self-dual configurations of dyons up to the thermodynamic fluctuations 𝑉 [5254]. Recently, it has been demonstrated [27] that the integration measure over dyons has the drastic effect on the ensemble of dyons when determinant over nonzero modes is ignored and only the salient features like the renormalization of the coupling constant and the perturbative potential energy [55] are taken into consideration. The question of CP-invariance for dyonic system has also been addressed by Diakonov [56]. It has recently been shown [57] that though the dyon-antidyon pair appears to violate CP-invariance, the CPT-invariance is an exact symmetry for generalized dyon-antidyon system.

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