Non-Abelian Proca-Dirac-Higgs theory: particle-like solutions and their energy spectrum

We study a system consisting of a non-Abelian SU(2) Proca field interacting with nonlinear scalar (Higgs) and spinor fields. For such a system, it is shown that particle-like solutions with finite energy do exist. It is demonstrated that the solutions depend on three free parameters of the system, including the central value of the scalar field $\xi_0$. For some fixed values of $\xi_0$, we find energy spectra of the solutions. It is shown that for each of the cases under consideration there is a minimum value of the energy $\Delta=\Delta(\xi_0)$ (the mass gap $\Delta(\xi_0)$ for a fixed value of $\xi_0$). The behavior of the function $\Delta(\xi_0)$ is studied for some range of $\xi_0$.


I. INTRODUCTION
By Proca theories one means gauge theories (both Abelian and non-Abelian ones) where the gauge invariance is violated explicitly by introducing a mass term. The first use of a Proca field was by Yukawa to describe pions. Later on the Proca theory has found applications in various fields of modern theoretical physics. The use of a Proca field results in the following consequences: a photon may acquire a rest mass; Einstein-Proca gravity involves a graviton of nonzero rest mass; Einstein-Proca-Weyl theories can be applied to describe dark matter [1]; using a real Proca field with a negative mass square, one can describe tachyons -particles moving with the velocity greater than the velocity of light [2].
Additionally, the following researches involving a Proca field may be noted. Ref. [3] studies Einstein-non-Abelian-Proca theory in an asymptotically anti-de Sitter spacetime with gauge group SU (2). The results obtained describe a gravitating Proca monopole. Refs. [4] and [5] consider stars supported by Proca fields. In Ref. [6], theories of the generalized Proca field are under investigation: the generalized Proca action for a vector field with derivative self-interactions with only three propagating degrees of freedom is constructed. The paper [7] also deals with the generalized Proca action for an Abelian vector field. The work [8] considers relativistic quantum mechanics of a Proca particle in Riemannian spacetimes. In Ref. [9] the authors have placed observational constraints on a class of dark energy models within the framework of generalized Proca theories. The purpose of the paper [10] is to build the first-order terms of the generalized SU(2) Proca theory and to discuss a general form of the complete theory.
In the present paper we study a non-Abelian SU(2) Proca field interacting with nonlinear scalar and spinor fields. The scalar field is described by the Klein-Gordon equation with the Higgs potential. The spinor field ψ is described by the Dirac equation with a potential term of the form ψ ψ 2 . Our purpose here is to obtain particle-like spherically symmetric solutions and to study their energy spectra. We will show below that the energy spectrum depends on the parameters f 2 , E, and φ 0 . The parameter f 2 describes a behaviour of the SU(2) Proca field at the center of the system; E/ is a frequency of the stationary spinor field entering the factor e −iEt/ ; the parameter φ 0 is a central value of the Higgs field φ. We show that the energy spectrum has a minimum, at least for some values of φ 0 , and we argue that this will also take place for any value of φ 0 lying in the range 0 < φ 0 < ∞. The behaviour of this minimum as φ 0 → ∞ is of great interest: if in this limit the minimum is nonzero, one can say that there is a mass gap ∆ = 0 in non-Abelian Proca-Dirac-Higgs theory. If such a mass gap does exist, this would be of great significance. The reason is that in quantum field theory there is a problem to prove the existence of a mass gap in quantum chromodynamics. This problem is highly nontrivial and any examples of its existence (even within a classical theory) would be very useful for an understanding of the nature of existence of the mass gap.
Thus, the purpose of the present paper is to (i) obtain particle-like solutions within a theory with a non-Abelian SU(2) Proca field plus a Higgs scalar field plus a nonlinear Dirac field; (ii) study energy spectra of these solutions; and (iii) search for a minimum of the spectrum (a mass gap).
The paper is organized as follows. In Sec. II, we give the field equations describing a system consisting of a non-Abelian SU(2) Proca field interacting with nonlinear scalar and spinor fields. In Sec. III, we choose the ansatzs (stationary one for the spinor field and static ones for the Proca and scalar fields) to solve the equations of Sec. II, using which the corresponding complete set of equations is written down. For this set, in Sec. IV we numerically solve the equations and find regular solutions describing particle-like systems in non-Abelian Proca-Dirac-Higgs theory. In Sec. V, we construct the energy spectrum of the solutions obtained and show the existence of a mass gap for this spectrum. Finally, in Sec. VII, we summarize and discuss the results obtained.

II. NON-ABELIAN PROCA PLUS HIGGS SCALAR AND NONLINEAR DIRAC FIELDS THEORY
The Lagrangian describing a system consisting of a non-Abelian SU(2) Proca field A a µ interacting with nonlinear scalar, φ, and spinor, ψ, fields can be taken in the form Here m f is the mass of the spinor field; D µ = ∂ µ − i g 2 σ a A a µ is the gauge derivative, where g is the coupling constant and σ a are the SU(2) generators (the Pauli matrices); is the tensor of the Proca field in non-Abelian SU(2) Proca theory, where ǫ abc (the completely antisymmetric Levi-Civita symbol) are the SU(2) structure constants; µ, M, Λ, and λ 1,2 are constants; γ µ are the Dirac matrices in the standard representation.
The corresponding field equations are as follows: Let us note the following features of this set of equations: (i) the presence of the term λ 1 φ 2 − µ 2 in Eq.
(2) will lead to exponential damping of the SU(2) Proca field, and this is a distinctive feature of the Proca monopole compared with the 't Hooft-Polyakov monopole; (ii) the Higgs field φ is topologically trivial, and this is also the distinction of principle compared with the 't Hooft-Polyakov monopole; (iii) the nonlinear spinor field ψ allows the existence of particle-like solutions even in the absence of the fields A a µ and φ (see Refs. [11,12]); (iv) the system supported by the fields A a µ and φ also allows the existence of particle-like solutions -a Proca monopole; (v) particle-like solutions exist only for some special choices of the constants µ, M , and u 1 [for the definition of parameter u 1 , see Eq. (21)]; (vi) in the absence of the fields A a µ and φ, the nonlinear Dirac equation (4) describes a system with a mass gap (see Refs. [11,12]), in contrast to Eqs. (2) and (3) which yield a system whose energy spectrum has no mass gap.
To obtain particle-like solutions, the above equations will be solved numerically since apparently it is impossible to find their analytical solution. As will be shown below, these particle-like solutions have the following interesting feature: For each central value of the scalar field φ 0 the plane {f 2 , E} [where E/ is the spinor frequency, see Eq. (8), and f 2 is a free parameter associated with the Proca field, see Eq. (19)] is divided into two regions: in one of them, particle-like solutions do exist, and in the other one they are absent. This means that there is some curve γ dividing these two regions. Near this curve there is a complicated interaction between the fields A a µ , φ, and ψ, which possibly results in an infinite energy of the particle-like solution on the curve γ.

III. EQUATIONS FOR A PROCA MONOPOLE INTERACTING WITH NONLINEAR HIGGS AND DIRAC FIELDS
We seek particle-like solutions describing objects supported by a radial magnetic field (a Proca monopole or a Proca hedgehog) and a nonlinear spinor field. For this purpose, we employ the standard SU(2) monopole ansatz (5) and the ansatz (8) for the spinor field, each row of which describes a spinor for the ground state of an electron in a hydrogen atom; i.e., we seek a solution of Eqs. (2)-(4) in the following form: where E/ is the spinor frequency and the color index a = 1, 2, 3. The functions u and v depend on the radial coordinate r only. The ansatz (8) is taken from Refs. [13,14]. After substituting the expressions (5)-(8) into the Lagrangian (1), we havẽ Here, for convenience of performing numerical calculations, we have introduced the following dimensionless variables: g 2 = g 2 c is the dimensionless coupling constant for the SU(2) Proca gauge field; x = r/r 0 , where r 0 is a constant corresponding to the characteristic size of the system under consideration;ũ = √ r 0 u/g,ṽ = √ r 0 v/g,μ = r 0 µ,ξ = r 0 gφ/2m,M = gr 0 M/2m,λ 1 = 4λ 1 /g 2 ,λ = 4m 2 λ 2 /g 2 ,m f = r 0 m f c/ ,Ẽ = r 0 E/( c),Λ = 2Λ/(mr 3 0 ); m is a free parameter introduced for convenience; the prime denotes differentiation with respect to x.
Equations for the unknown functions f (x), ξ(x), u(x), and v(x) can be obtained either by substituting Eqs. (5)-(8) into the field equations (2)-(4) or by varying the Lagrangian (9) with respect to the corresponding functions f (x), ξ(x), u(x), and v(x). However, in the latter case one should take into account that the nonlinear term ψ ψ 2 = , where (ũ,ṽ)ψ are taken fromψ and (u, v) ψ -from ψ. Therefore the corresponding Dirac equation is obtained by varying (9) with respect to (ũ,ṽ)ψ, and then one must take (ũ,ṽ)ψ = (ũ,ṽ) ψ = (ũ,ṽ). Thus we obtain the following equations: For convenience of performing numerical calculations, we have takenλ 1 = 1. Eqs. (10) and (11) describe the Proca monopole with the sources appearing due to the presence of the Dirac and Higgs fields. We emphasize that these equations differ considerably from those describing the 't Hooft-Polyakov monopole. Eqs. (12) and (13) have been studied in a simplified form for the case of f = 1 andξ = const in Refs. [11,12], where it was shown that this set of equations has particle-like solutions whose energy spectrum possesses a mass gap (for brevity, we will refer to such solution as a spinball).
Next, by definition, the energy density of the spinor field is where the dot denotes differentiation with respect to time. The Lagrangian of the Dirac field L D,eff appearing here is given by the expression from (9), which is obtained using Eqs. (12) and (13). Then, using the ansatz (8), the energy density of the spinor field (14) can be found in the following dimensionless form: As a result, we get the following total energy density of the particle-like solution: whereǫ is the energy of the Proca monopole. The formula (17) is remarkable because the total energy of the particle-like solution is a sum of energies of the Proca monopole and of the spinball, despite the strong interaction between the fields f and ξ, which make up the Proca monopole, and the fields u and v supporting the spinball.

IV. PARTICLE-LIKE SOLUTIONS: A PROCA MONOPOLE PLUS A SPINBALL
This section is devoted to studying particle-like solutions of Eqs. (10)- (13). Since apparently there is no analytical solution of this set of equations, we seek a numerical solution. Because of the presence of terms containing x in the denominators of Eqs. (10)- (13), to perform numerical computations, we assign boundary conditions near the origin x = 0 where solutions are sought in the form of the Taylor series The expansion coefficients f 2 ,ξ 0 , andũ 1 appearing here are free parameters whose values cannot be found from Eqs. (10)- (13). The equations (10)- (13) are solved numerically as a nonlinear problem for the eigenvaluesμ,M , andũ 1 and for the eigenfunctions f,ξ,ṽ, andũ. The asymptotic behaviour of the solutions as x → ∞ is where f ∞ ,ξ ∞ ,ṽ ∞ , andũ ∞ are integration constants and the function δξ ≪M satisfies the asymptotic equation Its solution can be found in the form The dimensionless energy density of the system in question can be obtained from Eq. (17). Fig. 5 depicts distributions of the energy density for different f 2 (Fig. 5a),Ẽ (Fig. 5b), andξ 0 (Fig. 5c).
It is of great interest to follow the behaviour of the "magnetic" Proca field (we have been using the word "magnetic" in quotation marks because the concept of a magnetic field is not quite correct as applied to a Proca field). The radial color "magnetic" field is defined as follows: The presence of this field enables us to speak of the Proca monopole. Its asymptotic behaviour is It is seen from this expression that the Proca monopole differs in principle from the 't Hooft-Polyakov monopole by its asymptotic behaviour. The graphs for the radial "magnetic" Proca field H a r are given in Fig. 6.

V. ENERGY SPECTRUM OF THE PARTICLE-LIKE SOLUTIONS FOR FIXEDξ0
In this section we study the energy spectrum of the particle-like solutions as a function of the tree parameters f 2 ,Ẽ, andξ 0 . Each such solution describes a ball consisting of the Dirac, Proca, and Higgs fields. Its structure is described by Eqs. system in question is calculated asW Here, according to the decomposition (17), we have separated the Proca monopole and spinball energies. Solving the set of equations (10)-(13) numerically, we have computed this energy for different values of f 2 ,Ẽ, andξ 0 . Our strategy of studying the energy spectrum is as follows: (i) We obtain energy spectra for different values ofξ 0 . (ii) For the spectra obtained, we show the presence of minimum of the energy, i.e., of the mass gap ∆ for a givenξ 0 , ∆(ξ 0 ). (iii) We examine a behaviour of ∆ as a function ofξ 0 . Three-dimensional and contour plots for the energy (28) are given in Fig. 7. It is seen from Fig. 7b that there are closed lines characterising the presence of extremum of the energy. In the case under consideration, this is a minimum corresponding to the mass gap ∆(ξ 0 ) for a givenξ 0 . The approximate value of the dimensionless mass gap for the values of the parameters used here (Λ = 1/9,m f = 1,g = 1, and m = 3) is a) The dependenceM(f 2 ,Ẽ).
c) The dependenceũ 1 (f 2 ,Ẽ). (For simplicity, below we omit the tilde by ∆.) Similar graphs for the energy of the particle-like solutions can be obtained for other values ofξ 0 as well. In particular, Fig. 8 illustrates the behaviour of the energy (28) in a wide range of values of the parameters f 2 andẼ forξ 0 = 0.5. The numerical computations, as well as Fig. 7, indicate that for any fixedξ 0 (at least in the range 0.4 <ξ 0 < 4.5 under investigation) there is a minimum value of the energy, which can be called a mass gap ∆(ξ 0 ) for a fixed value ofξ 0 .
Notice here the following important features of the system under consideration: • The particle-like solutions obtained describe objects consisting of a Proca monopole and of a spinball supported by strongly interacting non-Abelian SU(2) Proca vector fields and nonlinear spinor and scalar fields.
• The numerical calculations enable us to speak with certainty that for any fixed value ofξ 0 there is a minimum value of the energy, which can be called a mass gap ∆(ξ 0 ) for fixedξ 0 , i.e., there is a dependence ∆(ξ 0 ). • The numerical analysis indicates that asẼ → 1, the total energyW t → +∞. In Sec. IV, we have shown that the set of equations describing the Proca-Dirac-Higgs system has particle-like Procamonopole-plus-spinball solutions. In Sec. V, we have studied the energy spectrum of the solutions for fixed values of the parameterξ 0 and have shown that the spectrum possesses a mass gap ∆(ξ 0 ) for some values ofξ 0 .
The next step is to study the behaviour of the mass gap ∆(ξ 0 ) in a whole range of values of the parameterξ 0 . The significance of this problem is due to the fact that if the quantity ∆(ξ 0 ) has a global minimum this means that the Proca-Dirac-Higgs theory has a mass gap. The problem of presence or absence of a mass gap is of great importance in modern quantum field theory. Its solving would practically mean solving the problem of a nonpertubative quantization, and this is necessary, for instance, in constructing a theory of strong interactions. In the present paper we study a classical field theory based on the coupled set of the Proca, nonlinear Klein-Gordon and Dirac equations. It seems to us that the presence of a mass gap, even in a classical field theory, would be of great interest since it would allow one to get insight into the reason for its appearance.
To study the question of the presence of the mass gap in a whole range of values of the parameters f 2 ,Ẽ, andξ 0 , it is necessary to calculate ∆(ξ 0 ) for 0 <ξ 0 < ∞ and to check whether or not there is a global minimum of this function. Fig. 10 shows this curve in the range 0.4 <ξ 0 < 4.5. Unfortunately, deriving solutions for small f 2 and largeξ 0 runs into great technical difficulty: it is impossible to find eigenvalues ofμ,M , and u 1 to the necessary accuracy.
The positions of minima of the energy (28) on the plane {f 2 ,Ẽ} are illustrated by Fig. 9 for different values ofξ 0 . Notice that numerical errors in calculating magnitudes and positions of minima of the energy ∆(ξ 0 ) are quite large, and perhaps this leads to shifting the points with ∆(ξ 0 = 0.6, 0.8) and ∆(ξ 0 = 1.1, 1.5).
Search for the mass gap ∆(ξ 0 ) forξ 0 > 4.5 runs into the aforementioned technical problems. For a qualitative understanding what can happen for largeξ 0 , we have investigated dependencies of the total energy,W t , and of the energies of the monopole, W t The results of calculations are given in Fig. 11. It is seen that there are local minima for all the energies for some values ofξ 0 . The behaviour of these quantities when the parameterξ 0 increases further is unclear, and it requires further investigations. From an analysis of the graphs, we can assume that W t s will tend to zero and W t Pm will tend to a constant value asξ 0 → ∞; this would mean thatW tξ

VII. DISCUSSION AND CONCLUSIONS
We have considered non-Abelian SU(2) Proca-Dirac-Higgs theory where a massive vector field interacts with nonlinear scalar and spinor fields. Within such theory, regular, nontrivial particle-like solutions describing finite-energy configurations -Proca-monopole-plus-spinball systems -have been constructed. A distinctive feature of the Proca monopole is that, unlike the 't Hooft-Polyakov monopole, the corresponding color field decays exponentially at infinity. In our opinion, the main reason why such solutions can exist is the presence of the mass of the non-Abelian Proca field, and also because of the structure of the Dirac equation leading to the existence of regular solutions of this equation.
Note that both a Proca monopole and a spinball can exist separately, i.e., one can get a Proca monopole as a solution of Eqs. (10) and (11) withũ =ṽ = 0. Also, there exists a particle-like solution (a spinball) of the nonlinear Dirac equation (12) and (13) with f = 1 and ξ = const (see Refs. [11,12]). The basic distinction between these two solutions is that the energy spectrum of the Proca monopole does not have a mass gap, while for the particle-like solution of the nonlinear Dirac equation such a gap does exist [11,12]. This means that if the mass gap exists for the complete set of equations (10)- (13), this is a consequence of the presence of the nonlinear Dirac field. Note in this connection that in the 1950's the mass gap has in fact been found in Refs. [11,12] in solving the nonlinear Dirac equation. However, the authors did not use such term, but said of "the lightest stable particle". Those papers were  (12) and (13) with replacing the functions f andξ by the asymptotic expressions (23) and (25).
devoted to study of the nonlinear Dirac equation, and W. Heisenberg offered to use it as a fundamental equation in describing the properties of an electron. To the best of our knowledge, the mass gap was first obtained in Refs. [11,12]. We have calculated the energy spectrum of the corresponding solutions for fixed values ofξ 0 and have shown that there is a minimum value of the energy (the mass gap ∆(ξ 0 ) for a fixedξ 0 ). Also, we have studied the dependence of ∆ onξ 0 in the range 0.4 <ξ 0 < 4.5. In order to understand whether the mass gap exists in a whole range of possible values ofξ 0 , we have examined the behaviour of the total energy of the particle-like solution, as well as the behaviour of the energies of the Proca monopole and of the spinball, which make up the particle-like configuration under consideration, for some fixed values of the parameters f 2 andẼ. As a result, we have shown that these energies have local minima, and the energy of the Proca monopole possibly tends to a constant value. This permits us to suggest that such a behaviour will occur for any values of f 2 andẼ; this eventually will lead to the appearance of a mass gap in a whole range of values of the parameters f 2 ,Ẽ, andξ 0 determining the characteristics of the particle-like solutions under investigation.
Thus, the following results have been obtained: • Particle-like solutions of the type Proca-monopole-plus-spinball have been found in some range of values of the system parameters f 2 ,Ẽ, andξ 0 .
• For such solutions, the energy spectra for some values of the parameterξ 0 have been constructed. It was shown that they possess a minimum ∆(ξ 0 ).
• The behaviour of ∆(ξ 0 ) as a function of the parameterξ 0 has been studied.
• It was shown that the solutions obtained give rise to a Meissner-like effect, which consists in the fact that a maximum value of the SU(2) gauge Proca field is located there where the Higgs field has a minimum.
• For the Proca monopole, it was shown that the Proca color "magnetic" field decreases asymptotically according to an exponential law.
• The non-Abelian Proca monopole obtained differs in principle from the 't Hooft-Polyakov monopole in the sense that the Proca monopole is topologically trivial.
• The main reason for the existence of the mass gap ∆(ξ 0 ) is due to the presence of the nonlinear Dirac field.
Note that when one tries to prove the presence (or absence) of the mass gap ∆, he/she encounters great technical difficulties related to the fact that with increasingξ 0 the mass gap ∆(ξ 0 ) shifts towards f 2 → 0 andẼ → m f . But for such values of f 2 andẼ the eigenvaluesμ,M , andũ 1 should be given to high accuracy, leading to such technical problems. In order to assume what can happen with the function ∆(ξ 0 ) with increasingξ 0 , we have studied the behaviour of the total energyW t , as well as the energies of the Proca monopole, W t Pm , and of the spinball, W t s , as functions ofξ 0 for some fixed values of the parameters f 2 ,Ẽ and have shown that in this case all the energies have a minimum (at least a local one). This allows us to hope that such a minimum will occur for any values of the parameters f 2 andẼ; this in turn assumes that there will be a minimum value of the energyW t in a whole range of values of the parameters f 2 ,Ẽ, andξ 0 .
Numerical study of the energy spectrum of a spinball [when one solves only Eqs. (12) and (13)] indicates that its energy goes to infinity whenẼ →m f and whenẼ → 0; this corresponds to the fact that somewhere inside this region there is a minimum value of the energy (a mass gap of the spinball). Our numerical calculations indicate that apparently when Proca and Higgs fields are applied the energy of our system also goes to infinity, but now wheñ E →m f or when f 2 tends to some critical value f 2 = (f 2 ) cr .
Taking into account all the above, the existence of a mass gap within a theory containing a SU(2) Proca field plus nonlinear Higgs and Dirac fields is not impossible. To confirm this, it is necessary to carry out further investigations in this direction which are connected with large technical problems associated with solving a nonlinear eigenvalue problem when eigenvalues must be determined to high accuracy.
Finally, note that the particle-like solutions obtained here can be considered as those describing a spinorial Proca glueball. In QCD, a glueball is a hypothetical particle consisting only of a gauge Yang-Mills field in the absence of quarks. In our case a spinor field is present, but this is a nonlinear field that does not describe quarks. Since the solutions obtained are supported by non-Abelian Proca and nonlinear Higgs and Dirac fields, we can refer to a configuration described by such solutions as a spinorial Proca glueball (or a Proca-monopole-plus-spinball system).