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Scalar and spinor solutions in the spacetimes of vacuumless defects

We obtain solutions of the Klein-Gordon and Dirac equations in the gravitational fields of vacuumless defects. We calculate the energy levels and the current, respectively, in the scalar and spinor cases. In all these situations we emphasize the role played by the defects on the solutions, energy and current.


Introduction
The study concerning the interaction of a quantum mechanical system with a gravitational field has been an exiting research field [1]- [5]. One of the conclusions of these investigations is that the energy levels of an atom placed in a gravitational field are shifted as a result of the its interaction with the spacetime curvature(local aspects) [2] as well as with the spacetime topology (global aspects) [6]. Therefore, it is also important to investigate the role played by the topological features of a given gravitational field interacting with a quantum system. More recently, other investigations on this subject have been carried out [6,7] motivated by the fact that these can have considerable as well as possible observational interest.
One of the fascinating aspects of the gauge theories is that they can create topological objects called topological defects, that appear due to the spontaneous symmetry breaking. Among these defects we can have domain walls, cosmic strings, global monopoles [8] and their hybrid [9]. These defects have been studied in models where the scalar potential , V (Φ), that appears in the Lagrangian that describes the system, possesses an absolute minimum. However, topological defects can also be formed when the potential V (Φ) does not possess a minimum, but, is a monotonically decreasing function of the scalar field. These defects are called vacuumless defects [10].
The aim of this paper is to find the solution of the Klein-Gordon equation and study the energy shifts associated with scalar particles in the spacetimes generated by a gauge vacuumless string, a global vacuumless string and a vacuumless global monopole. In the spinor case, we solve the Dirac equation and find the currents associated with relativistic particles in these backgrounds.
In order to do these investigations, we will consider a scalar and spinor quantum particles embedded in a classical background gravitational field. In the scalar case, the behavior of the particle is described by the covariant Klein-Gordon equation where µ is the mass of the particle andh = c = 1 units are chosen.
In the case of a spinor particle, we consider the generally covariant form of the Dirac equation which is given by where Γ µ (x) are the spinor affine connections which can be expressed in terms of the set of tetrad fields e µ (a) (x) and the standard flat spacetime γ (a) Dirac matrices as The generalized Dirac matrices γ µ (x) satisfies the anticommutation relations and are defined by where e µ (a) (x) obeys the relation η ab e µ (a) (x) e ν (b) (x) = g µν ; µ, ν = 0, 1, 2, 3 are tensor indices and a, b = 0, 1, 2, 3 are tetrad indices. This paper is organized as follows. In section II, we solve the Klein-Gordon and Dirac equations, determine the energy shifts and calculate the current associated with relativistic particles in the spacetime of a gauge vacuumless string. In section III, we perform similar calculations in the background spacetime generated by a global vacuumless string. In section IV, once again, we do similar calculations, but now , in the spacetime of a vacuumless global monopole. Finally, in section V, we end up with ours conclusions.
2 Scalar and spinor particles in the spacetime of a gauge vacuumless string Firstly, let us consider the metric corresponding to a gauge vacuumless string which is given by [10] where Φ(ρ) = 2πa 2 ln ρ δ , is the Newtonian potential, in which δ = λ 1/2 M −1 is the size of the defect core and λ, M and a are positive constants. Initially, we will consider a scalar quantum particle in the spacetime generated by a gauge vacuumless string, whose line element is given by eq. (5). In this spacetime, the covariant Klein-Gordon equation turns into The solution of eq.(6) can be obtained writing Ψ(t, ρ, ϕ, z) in the following form where E , m and k are constants. Substituting eq.(7) into (6), we get the equation The solution of eq. (8) can be obtained if we assume that the particle is restricted to move in a cylindrical narrow region of width ρ 2 − ρ 1 , with ρ 2 ∼ = ρ 1 > 0. This implies that the potential Φ(ρ) is approximately constant and equal to Φ 0 . With this hypothesis, the solution of (8) is where J |ν| ( kρ) and N |ν| ( kρ) are Bessel functions of first and second kind, respectively, with If we consider that the boundaries ρ = ρ 1 > 0 and ρ = ρ 2 > ρ 1 are impenetrable, we can demand that Ψ(t, ρ, ϕ, z) vanishes on these boundaries and outside the cylindrical region. In this case, we can determine the energy spectrum of the particle, from the boundary conditions The condition expressed in eq.(10) furnishes us the following equation In order to obtain the energy spectrum explicitly, we will consider the situation in the which kρ 1 >> 1 and kρ 2 >> 1. With these conditions, we can use the Hankel ′ s asymptotic expansions [11] to obtain the following results and and similarly for J |ν| ( kρ 2 ) and N |ν| ( kρ 2 ) , by interchanging ρ 1 by ρ 2 . Substituting eqs. (12) and (13) and its similar equations for J |ν| ( kρ 2 ) and N |ν| ( kρ 2 ) into eq.(11) , we obtain the following result Thus, remembering that (14), that the energy spectrum is given by Note that the energy depends upon the value Φ 0 of the Newtonian potential in the narrow cylindrical region. Now, we will consider a spinor particle in this spacetime. To do this let us solve the Dirac equation where γ (ρ) = γ (ρ) cos ϕ + γ (ϕ) sin ϕ and γ (ϕ) = −γ (ρ) sin ϕ + γ (ϕ) cos ϕ , Φ 0 is the Newtonian potential in the narrow cylindrical region defined in the scalar case and we have used the following set of tetrads In order to take into account the symmetry let us adopt the following representation for the Dirac matrices where σ 1 , σ 2 and σ 3 are the Pauli matrices.
Let us write the spinor Ψ in terms of the bi-spinors Ψ A and Ψ B , in the form Ψ = Ψ A Ψ B and choose the following Ansatz Thus, we find the following equations for u 1 (ρ) and u 2 (ρ) and whose solutions are given in the general form by where i,m and C i,m are constant bi-spinors. Now, let us calculate the current, j µ , which can be written as where σ µλ = i 2 γ µ , γ λ . In the spacetime of a gauge vacuumless string, we have the following results and Therefore, using the expression for the current given by the eq.(22) and the results given in (23) and (24), we obtain the following results for the components of the current where the components of the convective part of the current are denoted by the subindex ′′ conv. ′′ and are derived from the term i is the gradient operator in the gauge vacuumless string. The components of the polarization and magnetization densities are given by and In the presence of an external electromagnetic field, M represents the density of the magnetization current.
The obtained results indicate us the dependence of the current on the Newtonian potential, and therefore, with the presence of the gauge vacuumless string.
3 Scalar and spinor particles in the spacetime of a global vacuumless string As a second example of a string, let us consider the global vacuumless string whose metric is written as where K ∼ 1 is a numerical coefficient and Φ(ρ) is the Newtonian potential given by Φ(ρ) = −κM 2 ρ δ 4 n+2 with κ being a coefficient equal to 15.5, for n = 2. As in the case of a gauge vacuumless string, the solution of the Klein-Gordon equation, Ψ(t, ρ, ϕ, z), can be separated as where R(ρ) obeys the following equation whose solution is where ν = (1 + KΦ 0 ) m 2 and we have assumed that the particle is restricted to move in a cylindrical narrow region in which the Newtonian potential, Φ 0 , is approximately constant. In this case, the energy spectrum is given by which shows up the dependence on the Newtonian potential Φ 0 . Now, we will consider a spinor particle in the spacetime generated by a global vacuumless string. To solve the Dirac equation in this background, we will choose the following tetrads Thus, the Dirac equation in this background is given by where we have assumed, once again, that the particle is restricted to move in a narrow cylindrical region, around the string, of radii ρ 1 > 0 and ρ 2 , with ρ 1 ∼ = ρ 2 > ρ 1 .
Choosing the following Ansatz considering the same representation of the Dirac matrices of the previous section and taking into account that the present problem can be treated as bi-dimensional, we find the following equations for u 1 (ρ) and u 2 (ρ) and Solutions of eqs.(43) and (44) are given in the general form by where i = 1, 2 , ς was defined in the previous section and v i = m and C (2) i,m are constant bi-spinors. The current can be calculated following the same steps of the previous section and taking into account the appropriate modifications. Doing this we get the following results for the components of the current The operator ∇ Φ 0 is given by is the gradient operator in the global vacuumless string. The components of the polarization and magnetization densities are the same as given by Eqs. (30)-(32) and (33)-(35), respectively. The obtained results indicate us the dependence of the current on the Newtonian potential, and therefore, with the presence of the global vacuumless string.
4 Scalar and spinor particles in the spacetime of a vacuumless global monopole Now, let us consider the solution that corresponds to a vacuumless global monopole, which is described by a static and spherically symmetric metric given by [10] where K is a coefficient which depends upon the value of n and it is equal to 9.4, for n = 2, in the case of a vacuumless global monopole. The Newtonian potential, Φ(r), is given by , with κ being a numerical coefficient which is approximately equal to 1. Now, we will obtain the solution of the Klein-Gordon equation for a particle in this spacetime, which can be written as The solution of this equation can be separated as where Y lm (θ, ϕ) are the spherical harmonics. Substituting (53) into (52), we obtain the following radial equation whose solution is given by in terms of Bessel functions as where k 2 ≡ −µ 2 (1 − KΦ 0 ) + E 2 [1 + (2 + K)Φ 0 ], v 2 = l(l + 1)(1 + KΦ 0 ) and Φ 0 a constant value for the Newtonian potential. Note that we are assuming that the particle is restricted to move in a spherical region of radii r 1 > 0 and r 2 and such that r 1 ∼ = r 2 > r 1 .
Analogously to the case of the vacuumless strings, we can determine the energy spectrum which is given by This result shows explicitly the dependence of the energy on the presence of the vacuumless global monopole. Now, we will obtain the solution of the Dirac equation in the vacuumless global monopole spacetime and determine the current associated with the particle of mass µ. In order to solve the Dirac equation in this spacetime we will choose the following tetrads [12] where we also assume that the particle is restricted to move inside a very narrow spherical region. Thus, the Dirac equation turns into where γ (r) = sin θ cos ϕγ (1) + sin θ sin ϕγ (2) , γ (θ) = cos θ cos ϕγ (1) + cos θ sin ϕγ (2) − sin θγ (3) , γ (ϕ) = cos θγ (2) − sin ϕγ (1) .
In this case, it is convenient to use the following representation of the Dirac matrices in Minkowski spacetime The set of solutions of eq.(58) can be written in the form where ψ (1) j,m and ψ (2) j,m are bi-spinors Ψ (1) j,m = f (r)ψ (1) j,m g(r)ψ (2) j,m and Ψ (2) j,m = f (r)ψ

Conclusions
In this paper we solved Klein-Gordon and Dirac equations in the gauge and global vacuumless strings and in the vacuumless global monopole spacetimes, and showed that the solutions of these equations depend on the Newtonian potentials associated each one of these spacetimes. We also calculated the energy spectra and the currents, for the scalar and spinor cases, respectively, and showed their dependence on the respective Newtonian potential. In the obtained results we have made the hypotheses that the particles are restricted to move in very narrow regions, so that the Newtonian potentials are approximately constant. These hypotheses are reasonable, from the physical point of view, since we are treating with gravitational fields in the weak field approximation.