The effective dipole-type field approach and the dual Higgs model

We consider the dual picture of the Yang-Mills theory at large distances. The dual Higgs model is reformulated in terms of two-point Wightman functions with the equations of motion involving higher derivatives.


Introduction
The nature of confining forces in gauge theories has attracted a lot of attention in recent years. We consider a model (in four-dimensional space-time (4d)) based on the dual description of a long-distance Yang-Mills (LDY-M) theory which can provide quark confinement in a system of static test charges. This article pursues the idea that the vacuum of quantum Yang-Mills (Y-M) theory is realized by a condensate of monopole-antimonopole pairs [1]- [4]. Since there are no monopoles as classical solutions with finite energy in a pure Y-M theory, it has been suggested by 't Hooft [5] that one might go to the Abelian projection where the gauge group SU(2) is broken by a suitable gauge condition to its Abelian subgroup U (1). Now there is the well-known assertion that the interplay between a quark and an antiquark is analogous to the interaction between a monopole and an antimonopole in a superconductor.
It is known that the topology of the Y-M SU(N ) manifold and that of its Abelian subgroup [U (1)] N −1 are different, and new topological objects can appear if the local gauge transformation of some gauge function is introduced in our model, e.g., the field strength tensor for the gauge field A µ (x) in quantum chromodynamics (QCD) with D µ (x) = ∂ µ + ieA µ (x):

16.2
which transforms with the gauge function Ω(x) as The last term in (1) reflects the singular character of the above-mentioned gauge transformation. One can identify the Abelian projection by making the replacement where A Ω µ → A α µ , while the label α reflects the Abelian world. This leads to the Dirac string and magnetic current in the Abelian gauge sector. Formally, a gauge group element which transforms a generic SU(3) connection onto the gauge-fixing surface in the space of connections is not regular everywhere in space-time. The projected (or transformed) connections contain topological singularities (or defects). Such a singularity may form the worldline(s) of magnetic monopoles. Hence, this singularity leads to the monopole current J mon µ . This is a natural way of performing the transformation from the Y-M theory to a model dealing with Abelian fields.
Analytical models of the dual QCD with monopoles have been intensively investigated [6]- [8]. The monopole confinement mechanism is confirmed by many lattice calculations (see reviews [9] and references therein). We study the Lagrangian model where the fundamental variables are an octet of dual potentials coupled minimally to three octets of monopole (Higgs) fields. The dual gauge model is studied at the lowest order of the perturbative series using canonical quantization. The basic manifestation of the model is that it generates the equations of motion, where one of them for the scalar (Higgs) field is the equation involving higher derivatives. Our aim is to apply the previous results [10]- [16] on Wightman functions in 4d models, with equations of motion (for the scalar fields) involving the dipole-type structure at least, to the model with monopoles. The monopole fields obeying such equations are classified by their two-point Wightman functions (TPWF). In the scheme presented in this paper, the flux distribution in the tubes formed between two heavy colour charges is understood via the following statement: the Abelian monopoles are excluded from the string region while the Abelian electric flux is squeezed into the string region. In our model, we use the dual gauge fieldĈ a µ (x) and the monopole field , which are relevant modes for infrared behaviour. The local coupling of theB a i -field to theĈ a µ -field provides the mass of the dual field and, hence, a dual Meissner effect. The commutation relations, TPWF and Green functions, as well-defined distributions in the space S(R d ) of complex Schwartz test functions on R d , will be defined in the following text.

The Lagrangian density
Let us consider the Lagrangian density (LD) L of the U (1) × U (1) dual Higgs model corresponding to the LDY-M theory [8]: (3)).F µν becomes the Abelian tensor with C µ being the dual to an ordinary vector potential. The Higgs fields develop their vacuum expectation values (v.e.v.)B 0 i and the Higgs potential W (B i ) has a minimum atB 0 i . The v.e.v.B 0 i produce a colour monopole-generating current confining the electric colour flux. The interaction between a quark Q and antiquarkQ is provided byĈ µ via a Dirac string tensor [17]Ĝ µν having the same colour structure asĈ µ , Following [8], we choose the colour structure forB i -fieldsB The effective potential looks as follows (see also [8]): ; the generating current of (3) is nothing but the monopole current confining the electric colour flux J mon The formal consequence of the J mon µ -conservation, ∂ µ J mon µ = 0, means that monopole currents form closed loops.
Since φ 1 and φ 2 couple to C µ in the same way [8], we choose (3) is divided into two parts: L = L 1 + L 2 where L 1 is of lowest order in g and λ, and with the minimal weak interaction looks as follows: Here, m ≡ gB 0 and µ ≡ √ 2λB 0 . The equations of motion for the fields b,b, b 3 and C µ are where µ 2 1 = (50/3)µ 2 , µ 2 2 = 12µ 2 , m 2 1 = 3m 2 . The formal solution of equation (4) looks as follows: We obtain that the dual gauge field is defined via the divergence of the Dirac string tensorG µν (x) shifted by the divergence of the scalar fieldb(x). For large enough 16.4 x, the monopole field is going to its v.e.v. while C µ ( x → ∞) → 0 and J mon µ ( x → ∞) → 8m 2 C µ . This implies that in the d = 2h dimensions theb(x) field obeys the equation but ∆ 2b (x) = 0. Here, the solutions of equation (5) obey locality, Poincaré covariance and spectral conditions, and look like the dipole 'ghosts' at h = 2.
The general solution of (6) should be Lorentz invariant and is given in the following form [10] at h = 2: where a i (i = 1, 2, 3) are the coefficients, l is an arbitrary length scale. The coefficients a 1 and a 2 in (7) can be fixed using the canonical commutation relations (CCR) The standard commutator for the scalar fieldb is are taken into account. The direct calculation leads to a 1 = (m 2 /48π 2 ) and a 2 = −(1/24π 2 ).
The propagator of theb-field in S (R 4 ) iŝ τ 2 (p) = weak lim Here, κ is a parameter of the representation and not an analogue of the infrared mass;κ 2 ≡ κ 2 /p 2 .
The free dual gauge field propagator in S (R 4 ) in any local covariant gauge is given bŷ where The gauge parameter ζ in (11) is a real number. The following requirement: (∆ 2 ) 2τ µν (x) = iδ 4 (x) on the Green function τ µν (x) leads to the constant c having to be equal to zero and τ µν (x) = τ µν (x)/(ξm 2 1 ).

An approximate solution
To obtain an approximate topological solution for this dual model, we fix the equations of motion: using the new scalar variables χ(x) and f (x). The equation of motion (12) transforms into the following one: which means that theb(x) field is nothing but a mathematical realization of the 'massive' phase B 0 f (x) at large enough x, i.e.,b(x) Integrating out G µν over the 2D surface element σ µν in the flux Π = G µν (x) dσ µν , we conclude that the phase f (x) is varied by 2πn for any integer number n associated with the topological charge [18] inside the flux tube.
Using the cylindrical symmetry we arrive at the field equation ( C → (C(r)/r) e θ ): with asymptotic transverse behaviour of its solution: The field equation (13) is given by whereχ(r) = χ(r) + B 0 . The profile of the colour electric field in the flux tube at large r looks as follows: 16.6

The confinement potential in analytic form
Now, our aim is to obtain the confinement potential in an analytic form for the system of interacting static test charges of a quark and an antiquark. According to the distribution (8), the static potential in R 3 is a rising function with r = | x| [14,15]: and the Fourier transformation of the analytic function We define the static potential as P stat (r) = lim T →∞ [T −1 · A(r)] and the action A(r) is given by the colour source current part of the LD Here, Q α = e ρ α is the Abelian colour electric charge of a quark while ρ α is the weight vector of the SU(3) algebra: ; x 1 and x 2 are the position vectors of a quark and an antiquark, respectively.
As a consequence of the dual field propagator (11), and using the following representation in the sense of generalized functions [11]: weak lim Hence, the string tension a in P stat (r) ∼ ar emerges as where the logarithmic correction in (14) has been ignored at large distances r ∼μ −1 . We obtain a 0.18 GeV 2 for the mass of the dual C µ -field m = 0.8 GeV, and α s = π/g 2 = e 2 /(4π) = 0.37, fixed from fitting the heavy quark-antiquark spectrum [19]. The string tension a f ixed = 1.0 GeV fm −1 extracted from the Regge slope of the hadrons [20] then determines the fixed value of the mass m f ixed = 0.85 GeV. Performing a formal comparison, let us recall that the string tensions in papers [21] and [22] are equal to a = (430 MeV) 2 and (440 MeV) 2 , respectively. We found that for a sufficiently long string the ∼r behaviour of the static potential is dominant; for a short string the singular interaction provided by the second term in (14) becomes important if the average size of the monopole is even smaller. 16.7

Conclusions
Finally, some conclusions can be stated: (a) We have actually derived the analytic expressions for the propagators of both theb-field (8) and the dual gauge boson field (11) in S (R 4 ). Our result should be regarded as the distributions (8) and (11) in a weak sense. (b) The dual gauge bosons become massive due to their interaction with scalar field(s).
But not every scalar species becomes massive, since the symmetry-breaking pattern is SU(3) → U (1) × U (1) and one scalar field remains massless. We see that the fields b(x) and b 3 (x) receive their masses, and theb(x) field-in combination with ∂ νG µν (x)-forms the vector field C µ (x) obeying the equation of motion for the massive vector field with the mass m = gB 0 . The solution for theb(x) field can be identified as a 'ghost'-like particle in a similar manner. (c) The form of the potential (14) grows linearly with the distance r apart from a logarithmic correction. (d) The string tension a obtained is quite close to a phenomenological value. (e) Since no real physics can depend on the choice of the gauge group (where the Abelian group appears as a subgroup), there seems to be a new mechanism of confinement [23].