Proca tubes with the flux of the longitudinal chromoelectric field and the energy flux/momentum density

We consider non-Abelian Proca theory with a Higgs scalar field included. Cylindrically symmetric solutions describing classical tubes either with the flux of the longitudinal electric field or with the energy flux (and hence with the nonzero momentum density) are obtained. It is shown that in quantum Proca theory there can exist tubes both with the flux of the longitudinal electric field and with the energy flux/momentum density simultaneously. An imaginary particle -- Proca proton~-- in which quarks are connected by tubes with a nonzero momentum density is considered. It is shown that this results in the appearance of the angular momentum related to the presence of the non-Abelian electric and magnetic fields in the tube, and this angular momentum is a part of the Proca proton spin. Because (i) the tube with the flux of the longitudinal electric field is required to exist in QCD to explain the phenomenon of confinement; and (ii) there is a contribution of the Proca gluon field to the Proca proton spin (which is similar to what is expected for an ordinary proton), we have concluded that non-Abelian Proca theories containing some extra fields may approximately describe QCD.


I. INTRODUCTION
In quantum chromodynamics (QCD), there are many deep and still unsolved problems, such, for example, as the confinement of quarks, the proton spin crisis, the mass gap problem, etc. All or almost all these problems are related to the fact that when it comes to quantizing strong interaction, it is necessary to employ a nonperturbative technique, which is still absent today. In order to describe nonperturbative effects in QCD, various approaches are used: dual superconductivity (for a review see Refs. [1,2]), the Abelian dominance, the Ads/CFT correspondence (for a review see Refs. [3,4]), etc. For example, in Ref. [5], the influence of the Abelian projection on quark confinement is studied using the dual superconductor picture. In Ref. [6], using the non-Abelian Stokes theorem, the proof of the Abelian dominance is given. In Ref. [7], the authors examine the properties of quasiparticles that presumably contribute a lot in the thermodynamic and statistical properties of a quark-gluon plasma.
In Refs. [8,9], we have assumed that, to study nonperturbative effects in QCD, one can use Proca theories. The latter appear when one explicitly introduces a mass term into gauge theories. At the present time, Proca fields are used in different aspects: in constructing models of black holes [10] and hypothetical Proca stars [11][12][13][14], in studying the generalized Proca theories [15] and solitons [16], in describing the massive spin-1 Z 0 and W ± bosons in the standard model [17], in considering various effects related to the possible presence of the rest mass of a photon [18], and within dark matter physics [19].
It is also known that in Proca theories involving some extra fields there can occur topologically trivial monopoles with an exponentially decaying radial magnetic field [8]. In the present paper we show that in non-Abelian Proca theories containing a Higgs scalar field there are cylindrically symmetric solutions describing tubes filled with color electric and magnetic fields. The distinctive feature of these solutions is that they can contain either the flux of the longitudinal electric field sourced by quarks located at ±∞ or the energy flux associated with the presence of the nonzero Poynting vector (and hence the nonzero momentum density).
Our main purpose here is to demonstrate that within non-Abelian Proca theories there are objects needed in QCD. For instance, this can be a flux tube containing a flux of longitudinal color electric field. The presence of such a tube is necessary to explain the phenomenon of confinement in QCD. According to many studies, the nonlinear nature of quantum SU(3) gauge field is such that the electric field between quarks, because of its nonlinearity, forms a tube, beyond which the field is practically absent. Since we will show below that such objects do exist in non-Abelian Proca theories containing Higgs scalar fields, this will imply that such theories share common features with QCD. We thereby continue our previous investigations started in Refs. [8,9].
In Ref. [8], the reason for the occurrence of the mass gap in the system consisting of a Proca field interacting with nonlinear scalar (Higgs) and spinor (Dirac) fields has been studied. It was shown there that this gap occurs due to the presence of the nonlinear spinor field. Since a mass gap must be present in QCD, one can assume that its occurrence there is caused by the same reasons as those in the Proca theory. This permits us to assume that in QCD the occurrence of the mass gap is caused by the interaction between sea quarks and gluons that can be approximately described using the nonlinear Dirac equation.
In Ref. [9], working within the Proca theory, we have studied statistical properties and thermodynamics of plasma, which is the analogue of quark-gluon plasma in QCD. We have shown that in such a plasma there is a phase transition associated with the destruction of quasiparticles filling the Proca plasma. Using the aforementioned analogy between Proca theories and QCD, we have concluded that similar processes may also take place in a quark-gluon plasma in QCD: the phase transitions are related to the destruction/appearence of some objects in the quark-gluon plasma (flux tubes, quasiparticles, etc.).
The paper is organized as follows. In Sec. II, we write down the general field equations for the non-Abelian-Proca-Higgs theory. In Sec. III, we obtain cylindrically symmetric solutions to the equations of Sec. II containing either the flux of the electric field or the energy flux (and hence the momentum density). In Sec. IV, we present some qualitative arguments in favor of the fact that in quantum Proca theory there can exist tubes containing simultaneously both the flux of the longitudinal electric field and the energy flux/momemtum density. In Sec. V, we show that tubes with a nonzero momentum density connecting three quarks in a Proca proton do create the angular momentum that is a part of the Proca proton spin. Finally, in Sec. VI, we give some arguments in favor of the fact that non-Abelian Proca theories containing some extra fields can approximately describe QCD and summarize the results obtained in the present paper.

II. NON-ABELIAN-PROCA-HIGGS THEORY
The Lagrangian describing a system consisting of a non-Abelian SU(3) Proca field A a µ interacting with nonlinear scalar field φ can be taken in the form (hereafter, we work in units such that c = = 1) Here F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν is the tensor of the Proca field in non-Abelian SU(3) Proca theory, where f abc are the SU(3) structure constants, g is the coupling constant, a = 1, 2, . . . , 8 are color indices, µ, ν = 0, 1, 2, 3 are spacetime indices. The Lagrangian (1) also contains the constants M, λ, Λ and the Proca field mass matrix µ 2 ab,µ ν . Using (1), the corresponding field equations can be written in the form and the energy density is where i = 1, 2, 3 and E a i and H a i are the components of the electric and magnetic field strengths, respectively.

III. PROCA FLUX TUBES WITH THE ENERGY FLUX (MOMENTUM DENSITY) AND THE FLUX OF THE ELECTRIC FIELD
The focus of this section is on obtaining and solving equations for two different types of flux tubes containing either the flux of the electric field or the energy flux (and hence the momentum density).
A. Proca tube with the flux of the longitudinal electric field To obtain a tube filled with a longitudinal color electric field, we choose the following Ansatz: where ρ, z, and ϕ are cylindrical coordinates. For such a choice of the SU(3) Proca field potentials, we have the following electric and magnetic field intensities: In this case the energy flux is absent, since the Poynting vector is zero, Here, ǫ ijk is the completely antisymmetric Levi-Civita symbol and γ is the determinant of the space metric. For such a tube, the energy density (4) yields with the following components of the Proca field mass matrix: µ 2 1 = µ 2 22,t t , µ 2 2 = µ 2 55,z z , and µ 2 3 = µ 2 77,ϕ ϕ . Substituting the potentials (5) in (2) and (3) and introducing the dimensionless is the central value of the scalar field], we get the following set of equations: Here, the prime denotes differentiation with respect to the dimensionless radius x. We seek a solution to Eqs.

FIG. 4: The profiles of the color magnetic fieldsH
in the vicinity of the origin of coordinates in the form v(x) =ṽ 0 +ṽ 2 where the expansion coefficientsh 0 ,ṽ 0 ,φ 0 , andw 1 are arbitrary. The derivation of solutions to the set of equations (10)-(13) is an eigenvalue problem for the parametersμ 1 ,μ 2 ,μ 3 , andM . The numerical solution describing the behavior of the Proca field potentials and of the corresponding electric and magnetic fields is given in Figs. 1-4. The asymptotic behavior of the functionsh,ṽ,w, andφ, which follows from Eqs. (10)- (13), is whereh ∞ ,ṽ ∞ ,w ∞ , andφ ∞ are integration constants. This flux tube solution has the following characteristics: • All physical quantities, such as the Proca field potentials, the electric and magnetic field intensities, and the energy density, are finite.
• The tube contains the finite flux of the longitudinal color electric field E 7 z , The existence of such a flux is necessary for understanding the essence of confinement in QCD. In our case we have obtained this flux from non-Abelian Proca theory containing a Higgs scalar field. This is one more evidence in favor of the assumption that there is some correspondence between non-Abelian Proca theories and QCD. It is possible that the non-Abelian-Proca-Higgs theory is some approximation for QCD.
• The tube possesses the finite linear energy density, Thus in this section we have demonstrated that there are cylindrically symmetric solutions within the non-Abelian-Proca-Higgs theory. These solutions describe a tube with the flux of the longitudinal color electric field, and this field is sourced by quarks/antiquarks located at z = ±∞.
B. Proca tube with the energy flux/momentum density Here, we choose the Ansatz which gives the following components of the electric and magnetic field intensities: In this case the Poynting vector (8) (2) and (3) and using the dimensionless variables given before Eq. (10), we derive the following equations:f This set of equations has a cylindrically symmetric solution describing a tube with a nonzero momentum density and energy flux (the Poynting vector). To demonstrate this, let us consider the simplest particular case wheref =ṽ andμ 1 =μ 2 =μ. In this case the set of equations (24)-(27) is split as follows. Eq. (24) takes the form of the Schrödinger equation where the effective potential for the "wave function"f is In order to ensure a regular solution of this equation, it is necessary that the effective potential would possess a well. In this case Eq. (28) must be solved as an eigenvalue problem for the parameterμ 2 with the eigenfunctionf . The remaining equations (26) and (27) for the functionsw andφ are then and they do not already contain the functionf . The effective potential which appears in (30) is Eq. (30) has the form of the Schrödinger equation with the "wave function"w and with the "energy"μ 2 3 . This means that it will have a regular solution only if the effective potential Uw ,eff possesses a well. Note also that in the limit x → 0 the effective potential Uw ,eff → 1/x 2 , i.e., it is repulsive and hence the "fall of a particle to the centre" is certainly absent.
We will seek regular solutions possessing a finite linear energy density. This means that asymptotically (as x → ∞) the functions behave asf (x),ṽ(x),w(x) → 0. Then, taking into account the positiveness of the effective potentials (29) and (32), one can conclude that the functionφ must go to a constant, and Eq. (31) implies that this constant is M , i.e.,φ →M as x → ∞.
We seek solutions of Eqs. (28), (30), and (31) in the vicinity of the origin of coordinates in the form where the expansion coefficientsf 0 ,φ 0 , andw 1 are arbitrary. In turn, the asymptotic behavior of the functions is where f ∞ ,w ∞ , andφ ∞ are integration constants. The results of numerical calculations are shown in Figs. 5-8, including the graph for the dimensionless energy densitỹ given in Fig. 6.
Note the presence of the gradient termf ′ṽ′ , which is the same as that in Maxwell's electrodynamics, and of the nonlinear termfṽw 2 /4, which appears because the Proca field is non-Abelian. The flux tube solution obtained in this section has the following characteristics: • All physical quantities (the Proca electric and magnetic field potentials and intensities, the energy density, the momentum density, and the energy flux) are finite.
• The tube contains a finite linear energy flux directed along the tube and computed as • Since the linear momentum density P is proportional to the linear energy flux, P = Π, the tube also contains the z-component of the linear momentum density P z = Π z .
• In the tube, there is the finite flux of the longitudinal color magnetic field H 7 z , • The tube possesses the finite linear energy density, Thus in this section we have shown that within the non-Abelian-Proca-Higgs theory there exist cylindrically symmetric solutions describing a tube possessing the energy flux (and hence the momentum density) between quark and antiquark located at z = ±∞.

IV. PROCA FLUX TUBES WITH A CLASSICAL (QUANTUM) FLUX OF THE ELECTRIC FIELD AND QUANTUM (CLASSICAL) ENERGY FLUX/MOMENTUM DENSITY
In Sec. III, we have shown that within the non-Abelian-Proca-Higgs theory there are two different types of tubes. The first one contains the flux of the longitudinal color electric field between quarks located at ±∞, but there is no energy flux (and hence the momentum density). The tubes of the second type contain the energy flux/momentum density, but a flux of the longitudinal color electric field is absent.
In this connection, the natural question arises as to whether a tube possessing simultaneously both the flux of the longitudinal electric field and the energy flux/momentum density does exist within classical non-Abelian-Proca-Higgs theory? It seems to us that solutions describing such tubes are absent in classical theory, but apparently they might exist in quantum non-Abelian-Proca-Higgs theory. In this section we give arguments in favor of such a point of view.
In Sec. III A, we have shown that, in having the Proca field potentials A 2 t , A 5 z , and A 7 ϕ given in the form (5), one can obtain the longitudinal electric field E 7 z (and hence the flux of this field between quarks which is needed to ensure confinement) that occurs due to the nonlinearity in the definition of the field intensity tensor. Similarly, in Sec. III B, we have shown that, in having the Proca field potentials A 5 t , A 5 z , and A 7 ϕ given in the form (21), one can obtain the energy flux/momentum density due to the appearance of electric and magnetic fields E 2 ϕ , H 2 ρ and E 5 ρ , H 5 ϕ which are perpendicular to each other. A comparison of these two Ansätze leads to the conclusion that for the simultaneous existence of the flux of the longitudinal electric field and of the energy flux/momentum density one needs the potentials A 2 t , A 5 t , A 5 z , and A 7 ϕ . We assume that this can be done in two different ways, but only in quantum Proca theory.
A. Tube with a classical flux of the longitudinal electric field and quantum energy flux/momentum density Here, the potentials are where . . . denotes the quantum average of the corresponding potentials. In this case, to describe the behavior of the quantum average of the potentialsÂ 2 t ,Â 5 z , andÂ 7 ϕ , one can approximately use Eqs. (10)- (13). Then, according to the definition of the Poynting vector (8), there is the following quantum average S z : Here, for brevity, we use the same designations as those given in (5) and (21): According to (45)-(47), the quantum average in (43) can be written in the form In classical Proca theory, according to Sec. III B and Eqs. (24)- (27), the functions f and v are interrelated. It is evident that in quantum theory this interrelation persists in the sense that the correlation of quantum fluctuations of the operatorsf and δv [the Green function G f,δv (x 1 , x 2 ) = f (x 1 ) δv(x 2 ) ] will be nonzero, Correspondingly, the Green function (48) will then also be nonzero. Using Eq. (43), the momentum density takes the form and it is nonzero. Notice that the energy flux (43) and the momentum density (51) are of a purely quantum nature. Thus in this section we have shown that in quantum Proca theory there exists a tube with the classical flux of the longitudinal electric field, sourced by quarks located at ±∞, and with the quantum energy flux/momentum density. For this case we need the following potentials: Then, to describe the behavior of the quantum average of the potentialsÂ 5 t ,Â 5 z , andÂ 7 ϕ , one can approximately use Eqs. (24)-(27).
In having these potentials, we can obtain the operator of longitudinal electric field needed for the existence of confinement in the formÊ Here, analogous to what was done in Sec. IV A, we employ the following designations: According to (54) and (55), the quantum average in (53) can be written in the form Arguments similar to those given in Sec. IV A concerning the relationship between the operators of the potentialŝ A 2 t =ĥ/g andÂ 5 z =v/g which follow from Eqs. (24)-(27) can be applied to obtain a Green function describing the quantum correlation between the operatorsÂ 2 t andÂ 5 z , Thus the expression for the quantum average of the flux of the longitudinal electric field (53) takes the form and it is nonzero. Note that this flux between quarks located at ±∞ is of a purely quantum nature. Thus in this section we have shown that • In quantum Proca theory, there can exist tubes possessing simultaneously both the energy flux/momentum density and the flux of the longitudinal electric field.
• There are two types of such tubes: -Tubes with (almost) classical flux of the longitudinal electric field and quantum energy flux/momentum density.
-Tubes with (almost) classical energy flux/momentum density and a quantum flux of the longitudinal electric field.
• According to our assumption about the relationship between the non-Abelian-Proca-Higgs theory and QCD, similar tubes must be present in QCD as well.
In the next section, we will show that the presence of tubes between quarks in a Proca proton results in the fact that the Proca proton spin includes a contribution from the momentum density of the Proca field containing inside such tubes. Note that by the Proca proton we mean a hadron in which the interaction between quarks is caused by a Proca gluon field. Actually, according to our assumption about the relationship between the Proca theory and QCD, the same mechanism will work for an ordinary proton as well.

V. ANGULAR MOMENTUM OF FLUX TUBES IN THE PROCA PROTON
Proca proton is an imaginary particle in which a SU(3) gauge field is replaced by a SU(3) Proca field. Analogously to an ordinary proton, we assume that there is a tube between quarks filled with the Proca field, which we have considered in Secs. IV A and IV B.
Consider a flux tube connecting two quarks (or quark and antiquark) in the Proca proton. We will model such a finite-size tube by cutting it from one of infinite tubes obtained in Sec. IV A or in Sec. IV B.
In each tube, there is the following linear momentum density (for clarity, in this section we resurrect c and in the equations): and this momentum is directed along the axis of each tube. The schematic sketch of the quarks, tubes, and directions of momenta is shown in Fig. 9. It is evident that in this case there is an angular momentum caused by the presence of the flux of the non-Abelian Proca field. Let us estimate this angular momentum as follows. Each tube has a momentum Π = P z l tb , where P z is the linear momentum density directed along the tube of the length l tb , and it is calculated either using the formula (38) for the tube with classical energy flux/momentum density and a quantum flux of the longitudinal electric field or using the formula (51) for the tube with a classical flux of the longitudinal electric field and quantum energy flux/momentum density. Let us consider the first case. Then the total angular momentum of the non-Abelian Proca field contained in the tubes can be estimated as where r 0 is an estimated value of the distance from the tube to the center of the Proca proton andl tb = l tb φ(0)/ √ c,r 0 = r 0 φ(0)/ √ c, and g ′ 2 = cg 2 are dimensionless quantities. Thus the fraction of the /2 Proca proton spin created by the non-Abelian Proca field contained in the tubes carrying the momentum is Here, we have employed the valueΠ ≈ 0.6 which has been computed using the solution obtained in Sec. III B and shown in Fig. 6. One can estimate the magnitude of L g / ( /2) using natural assumptions about the values of the quantitiesl tb andr 0 . In QCD, there is the dimensional parameter Λ QCD ≈ 1 fm −1 . Therefore it is natural to assume that φ(0) ≈ √ c Λ QCD . The length of the tube and the distance from the tube to the center can be estimated as l tb ≈ r 0 ≈ Λ −1 QCD . In QCD, the dimensionless coupling constant g ′ > 1; then, if one takes, say, g ′ = 8.65, the fraction of the angular momentum created by the Proca gluon field in the Proca proton spin is Thus in this section we have shown that the Proca theory permits one to explain how the Proca gluon field contributes to the Proca proton spin. According to our assumption about the relationship between the non-Abelian-Proca-Higgs theory on the one hand and QCD on the other, a similar mechanism must also occur for an ordinary proton: gluon fields in the proton create tubes with a nonzero momentum density, and this leads to the appearance of a nonzero angular momentum created by such gluon fields.

VI. DISCUSSION AND CONCLUSIONS
The main purpose of the present paper is further development of the idea suggested in Ref. [8], according to which non-Abelian Proca theories may approximately describe some phenomena in QCD. In [8], the non-Abelian Proca theory with a Higgs scalar field and a nonlinear spinor field has been considered, and the presence of the mass gap has been demonstrated (at least for fixed values of one of the parameters determining the properties of the particlelike solutions studied there). It was clarified that the occurrence of the mass gap is closely related to the nonlinear Dirac equation ensuring the presence of such a gap. Based on this, we proposed the mechanism of the appearance of the mass gap in QCD, according to which it occurs because of the fact that the interaction between sea quarks and gluons can be approximately described by the nonlinear Dirac equation.
The study of the idea about the interrelation between non-Abelian Proca theories and QCD was continued in Ref. [9] where we considered statistical physics and thermodynamics in a plasma consisting of quasiparticles supported by a nonlinear spinor field. We showed that this interrelation results in the appearance of a phase transition at which the quasiparticles decay, whereupon the spinor field in the plasma is believed to be described in a different way.
In the present work we have shown that there exist different types of tubes within non-Abelian Proca theory: (a) the tubes with the flux of the electric field between quarks located at ±∞; (b) the tubes with the nonzero momentum density directed along the tube; (c) the tubes with the classical flux of the electric field and the quantum momentum density; (d) the tubes with the classical momentum density and the quantum flux of the electric field. It is known that, to explain the phenomenon of confinement in QCD, the existence of tubes with a color longitudinal electric field between quarks is necessary. The appearance of such tubes in Proca theories gives us one more evidence in favor of the fact that non-Abelian Proca theories plus some extra fields (Higgs scalar fields, nonlinear spinor fields) may have some relevance to QCD. We believe that such fields can be used to get an approximate description of QCD.
We have also considered an imaginary particle -Proca proton -in which quarks are connected by the tubes obtained here. Using the fact that such tubes may contain a momentum density, we have shown that in this case the angular momentum created by these tubes contributes to the Proca proton spin. On the other hand, it is known that the proton spin contains a contribution from the angular momentum created by gluon fields. Thus, comparing our results concerning the contribution of the Proca gluon field to the Proca proton spin with the results regarding the presence of the contribution of gluon fields to the spin of ordinary proton, we have one more argument in favor of the presence of the relationship between Proca theories and QCD.
Thus in the present paper • We have obtained cylindrically symmetric solutions describing classical tubes possessing either the flux of the longitudinal electric field or the energy flux and the momentum density.
• At the qualitative level, we have shown that there can exist tubes filled simultaneously both with the flux of the longitudinal electric field and with the energy flux and the momentum density. But in this case either the flux of the longitudinal electric field or the energy flux/momentum density must be of a purely quantum nature.
• We have estimated the contribution of the Proca gluon field, contained in the tubes and creating there the momentum density, to the Proca proton spin.
• We have given arguments in favor of the fact that non-Abelian Proca theories are related to QCD: these theories can be used in an approximate description of QCD.
All this means that non-Abelian Proca theories contain many of the objects needed for solving some problems in QCD: Proca monopoles, tubes with a longitudinal electric field, tubes with a nonzero momentum density which leads to the appearance of angular momentum of a Proca gluon field in a Proca proton, and perhaps other objects. This enables us to argue that between QCD and non-Abelian Proca theories containing some extra fields there is a certain interrelation permitting one to use such theories in an approximate description of QCD.