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Motion of Colored Particle in a Chromomagnetic Field

The Dirac equation in a chromomagnetic field is solved for colored particle moving in a limited space volume. Quantized energy levels and the corresponding wave functions are found for backgrounds both directed along third axes and having spherical symmetry. It was shown interrelation with the case of motion in an infinite space volume.


Introduction
Quantum and classical mechanics of non-abelian charged particles has been studied by various authors. Problems in this theory serve to find the solution of QCD problems and are connected with description of quark's motion inside hadrons or any other limited space volume. Since QCD has confinement property, presumably it will be better to consider QCD problems in a finite volume as well [1]. As is known, there are color background fields in the QCD vacuum [2] and so, it makes sense to study the problem of motion of a colored particle in a color background in limited space volume. The simplest case, a constant color background could be given by two different types of vector potentials [3,4]. Problems in an external color field given by abelian vector potentials are solved analoguosly to the ones in abelian theory. The second type of vector potentials are non-commuting constant vector potentials, which are not gauge equivalent to first type [4]. These potentials are used to solve the ground state problem of QCD [4,5] and other different problems connected with the QCD vacuum [5][6][7][8][9] as well and sometimes give physically different results from that ones in which first type potensials was used. Here we shall solve Dirac equation in a finite space volume in a constant background chromomagnetic field given by the second type vector potentials. The solutions and spectra found are more suited to the physical situation and by use of them could be constructed quark's Green's function by the help of exact solution method [10], which could be used in solving QCD problems.

Constant homogenous background
Let us define an external chromomagnetic field by constant vector potentials. Within the SU(3) color symmetry group they look like where a = 1, 8 is a color index, τ is a constant and δ µa is the Kroneker symbol. The field (1.1) is directed along the third axes of ordinary and color spaces : Here g is the color interaction constant. The Dirac equation for a colored particle in the external color field has the form where P µ = p µ + gA µ = p µ + gA a µ λ a 2 , λ a are Gell-Mann matrices describing particle's color spin. Equation (1.3) written for Maiorana spinors φ and χ has the well-known form 4) where the Pauli matrices σ i describe a particle's spin. Here and afterwards ψ means φ or χ. The spinors φ and χ have two components, corresponding to the two spin states of a par-  (3) and has three color components describing color states of a particle corresponding to three eigenvalues of color spin λ 3 Since the background field (1.1) is time independent, the equation (1.4) has got the form (1.5) we get two independent systems of differential equations From (1.7) we get the same equation for all states ψ (i) where p 2 ⊥ = p 2 1 + p 2 2 . This equation possesses rotational invariance around the z axis. If we consider free motion of the particle in all − → x space, then we should not impose any boundary condition on solution of (1.8) and shall get continous energy spectra found in [11,12] for plane wave solutions ψ This spectrum is used in solving of various problems [5][6][7][8][9]. Since free motion of the colored particle in nature could be take place in limited space, for example inside a hadron, let us consider motion of particle in limited space bounded by a cylinder with radius r 0 and hight z 0 and solve (1.8) on boundary conditions ψ (i) . Let us rewrite (1.8) in the form ± it has got the following form (1.11) Acting on equation ( we get an equation for η (i) ± that has the same form as for ψ ( 1.12) Equation (1.12) is equivalent to (1.10). In the appendix is shown another way to reduce (1.10) to the form (1.12). In a cylindrical coordinate system the solution of (1.12) can be looked for using the separation ansatz ψ The solutions of last two equations with the boundary condition v (z 0 ) = 0 are u (ϕ) = 1 √ 2π e imϕ (m = 0, ±1, ±2, ... and v (z) = sin p 3 z λ 2 = p 2 3 , p 3 = πn z 0 . For the p 3 = 0 case we can choose v (z) = 1. It is worth to remark that (1.12) and (1.10) become equivalent after taking the last equation in them into account and there is no such problem for the case p 3 = 0. Taking into account explicit expression of v (z) and (1.12) in (1.10) we get the following equation for the constant k 2 from which one finds the following expression for the constant k 2 (1.14) The first equation in (1.13) is the Bessel equation. Under the finitness condition on r → 0 it has a solution determined by the Bessel function If we do not impose any boundary condition on the states (1.15) with given m, then they have an energy from continous spectrum (1.9), which does not depend on m. Imposing the boundary condition ψ 1,2 (r 0 ) = 0 we find two branchs of quantized energy levels of the colored particle in a chromomagnetic field (1.1) Here α (N ) m are the zeros of Bessel function and the quantity N labels the sequence of zeros N = 1, 2, 3.... From (1.16) one sees that these energy branchs, as branches of the continous spectrum (1.9), are defined by ± sign of the field strength, which is not connected with direction of field. The energy levels E (N ) m are determined by the m -chromomagnetic quantum number, i.e. by the projection of the chromomagmetic moment of the particle onto the chromomagnetic field. Another quantum number is N, determining the energy level E (N ) m , a result of quantization due to the finitness of the motion. In contrast to the discrete spectum (1.16) the continous spectum (1.9) is not determined by the quantum number m. Thus, finitness of space volume turns continous spectra (1.9) for a given m into a discrete series determined by the quantum number N. Both spectra (1.9) and (1.16) are infinitely degenerate. Since (1.12) is universal for the spin ± and color (i) indices, any of these states may have any energy from spectra (1.16) or (1.9). If in the energy branchs (1.16) one makes the replacement α (N) m r 0 = p ⊥ we get continous spectra (1.9). This is suitable to quantization of momentum in standing waves. Taking into account spectrum (1.16) in (1.14) we find the value of constant k 2 , which is the same for both energy branchs and so, the solution (1.15) is the same for these branchs. Thus, the wave functions ψ Thus, the motion of colored particle in the field (1.1) along z axis is a standing wave and in the (x, y) plane is a rotation. Writing (1.13) for the radius a of the cirle in which particle rotates 1 a ∂ ∂r a ∂ψ (a) ∂r we can find discrete values for these orbits In the continous spectrum case the constant k 2 and radius a are , a = m p ⊥ and the solution (1.15) becomes The states ψ (3) ± ( − → r ) correspond to states of freely moving colorless scalar particle.

Spherically symmetric configuration of background
Let us consider the case of spherical symmetry in the ordinary space with constant chromomagmetic field defined by non-commuting potentials The non-zero components of the field strength tensor are In the external field (2.1) the explicit form of (1.4) is Let us denote components of the spinor Ψ by ψ 1 and ψ 2 : Ψ = ψ 1 ψ 2 . Since the external field has three components in both spaces, in this case, different from the previous one, the components ψ (a) 1,2 do not describe states with definite value of the spin and color spin projection onto the field. Equation (2.2) can be written in terms of a system of differential equations for the components ψ where operator A denotes A = − → p 2 + M 2 + 3g 2 τ 4 − E 2 . From the system (2.3) we get the same differential equation for the all states ψ which possesses rotational invariance. Since the operator in the first square brackets commutes with the second one, (2.4) could be divided into two equations 1,2 = 0.
(2.5) From (2.5) we get four branchs of the continous energy spectrum found in [11] for plane wave solutions ψ (2.6) In the same manner as used in the previous section from (2.5) we get the equivalent equations If we take (2.7) into account corresponding equations (2.5) we found values of k 2 (2.8) The solution of (2.7) can be found in textbooks of quantum mechanics ( [13] ). In a spherical coordinate system for the wave functions ψ (i) j one used separation ansatz, i.e. it is chosen as a multiplication of the radial R(r) and angular Y m l (θ, ϕ) parts Here r = √ x 2 + y 2 + z 2 , l and m are the orbital angular momentum and chromomagnetic quantum numbers, θ, ϕ are the polar and azimuthal angles, respectively. The spherical functions Y m l (θ, ϕ) are expressed by means of Legendre polynomials P |m| l (cos θ) The equations for the radial parts of wave functions are the same as for many quantum mechanical problems possessing rotational invariance where ν = 1, 2, 3, 4; k 2 1,2 = k 2 , k 2 3,4 = k 2 − g 2 τ. With the notations Q (r) = √ rR(r) (2.10) turns into Bessel's equations for Q (r) The function R(r) must be finite on r → 0. This means that solution of (2.11) are first type Bessel functions R ν l = C ν l k ν √ r J l+1/2 (k ν r) . (2.12) Imposing boundary conditions R ν l (r 0 ) = 0 we enclose motion of the particle by a sphere with a radius r 0 , where r 0 agrees with the hadron's size. Since, the external field (2.1) is not depend on r, we have got the same equation for R(r) as for a freely moving particle enclosed in a sphere, differing only from the expression by k 2 constant. From these boundary conditions one finds quantized energy levels of spectrum branchs (2.13) In this case the boundary condition on R ν l (r) with fixed value of the angular momentum quantum number l chooses from the continous energy spectrum (2.6) the discrete series (2.13) of the radial quantum number N for this value of l, and the energy degeneracy for this state remains infinite. The replacement connecting discrete and continous spectra is α (N) l r 0 =| − → p |. By means of spectra (2.13) one finds the same values for k ν The radius a of the particle could be found from (2.10) using the maximum condition R ′ (a) = 0 and is equal to As the energy spectrum is described by (2.13), the motion orbits of the particle are quantized and defined by the orbital and radial quantum numbers l and N as well. Thus, general solution of (2.7) is (2.14) In the case of this field configuration the particle moves on "s", "p", "d", "f ".... orbitals known from orbital angular momentum eigenfuctions. For the continous spectrum the constant k and radius a are k = | − → p | , a l (p) = l(l + 1) | − → p | .
The plane wave could be decomposed by means of spherical waves and therefore, plane wave solution obey initial equation as well [13]. Using the energy spectra (1.16) and (2.13), estimates for field strength of gluon condensate [2] and hadron's size the energy of a photon emitted by excited hadron may be calculated.
Using the wave functions (1.17) and (2.14) the explicit expressions of Green's functions for these particles in finite temperature can be found [10], which gives a possibility to investigate different radiation corrections in the chromomagnetic field and the quark condensate (2.15) The spectra found, (1.16) and (2.13) and the quark condensate (2.15) open new possibilties to investigate chromomagnetic catalysis of color superconductivity as well [6].