Nonlinear Schrodinger solitons in massive Yang-Mills theory and partial localization of Dirac matter

We investigate the classical dynamics of the massive SU(2) Yang-Mills field in the framework of multiple scale perturbation theory. We show analytically that there exists a subset of solutions having the form of a kink soliton, modulated by a plane wave, in a linear subspace transverse to the direction of free propagation. Subsequently, we explore how these solutions affect the dynamics of a Dirac field possessing an SU(2) charge. We find that this class of Yang-Mills configurations, when regarded as an external field, leads to the localization of the fermion along a line in the transverse space. Our analysis reveals a mechanism for trapping SU(2) charged fermions in the presence of an external Yang-Mills field indicating the non-abelian analogue of Landau localization in electrodynamics.


I. INTRODUCTION
Over the last decades, the classical dynamics of Yang-Mills (YM) field theory has been thoroughly investigated in the literature, both in Minkowski and in Euclidean space (see, e.g., Ref. [1] and references therein). The motivation for this study has been mainly the effort to understand the vacuum structure of non-abelian gauge theories like Quantum Chromodynamics (QCD). In a spatially homogeneous description, one can show that the YM classical dynamics possesses a chaotic component attributed to the nonlinear form of the YM self-interaction [1,2]. Generalizing to the case of inhomogeneous solutions, the conformal structure of the YM Lagrangian and the associated absence of a characteristic scale does not permit the presence of localized solutions [3], and complicated patterns with fractal characteristics may appear [4]. Recently, it has been argued that classical Yang-Mills solutions may have impact on the properties of the quantum gauge fields. In particular, in Ref. [5], it was shown that periodic solutions of a special choice for the YM field configuration (Smilga's choice [1]) after quantization lead to a description of the gauge field propagator compatible with the calculations performed in lattice gauge theories.
On the other hand, localized inhomogeneous solutions could permit a particle interpretation of the YM-field, which may be relevant for several applications where quasi-particles are involved. Such a scenario appears, for example, when the YM-field is coupled to a condensate, breaking spontaneously the underlying gauge symmetry, or when the YM-field itself condensates at particular thermodynamic conditions. In these cases the gauge field can acquire a mass introducing a scale in the YM-theory and bypassing the restrictions of the Coleman theorem [3]. This allows for spatially inhomogeneous localized classical solutions -at least at the level of an effective theory.
In the present work, we follow this line of thoughts trying to explore the space of classical solutions in massive SU (2) Yang-Mills theory. Our primary interest is to display the capacity of the theory in terms of possible classical dynamical behavior, as well as the influence of the choice for the YM-field initial configuration on this dynamics. In particular we will show that at a given combination of scales the classical Yang-Mills theory contains the non-linear Schrödinger equation regime. We start our considerations with a Langrangian describing the interaction of the Yang-Mills field with a scalar field. Then we assume, at the level of the Langrangian, that the scalar field is constant and we remain with a massive Yang-Mills theory. The effect of the spatio-temporal fluctuations of the scalar field is considered in [6]. As a next step, making a choice similar to Smilga's [1], we are able to construct within the framework of a multi-scale perturbation theory a class of solutions which are localized along a line in the plane transverse to the momentum of the gauge field.
Furthermore, we study the dynamics of Dirac fields in the presence of such a gauge field configuration, considering the latter as an external classical field. We show that the Dirac field becomes bound in the subspace where the external gauge field is localized.
The paper is organized as follows: in section II we present the Lagrangian of the considered SU (2) YM field theory, we discuss the multiple scale approach used to solve the corresponding equations of motion and we obtain the associated solutions for the gauge field. We also give an interpretation of the involved parameters. In section III we use the solution found in section II as an external field for the Dirac dynamics of an SU (2)-charged matter field. Finally we end up, in section IV, with a summary and perspectives of our work.

II. SOLITON-LIKE SOLUTIONS IN THE MASSIVE YANG-MILLS DYNAMICS
We start our analysis by considering the Lagrangian describing the interaction of the SU (2) Yang-Mills field A a µ with a charged scalar field Φ = Φ 1 Φ 2 : where g is a dimensionless coupling and V [Φ † Φ] is the self-interaction potential of the scalar field which we need not to specify more. We only assume that the potential possesses at least one stable equilibrium point. As usual, we use greek letters to denote the space-time components and latin letters to denote the Lie group components of the YM fields. For the SU (2) case a, b, c take the values 1, 2, 3. Let us now further assume that the scalar field is constant (independent of space-time) and equal to a value corresponding to a stable equilibrium point of V . Then the Lagrangian (1), up to the constant term V [Φ † Φ] which can be neglected, becomes: In Eq. (2) M ab is the mass matrix of the YM field components which is diagonal in the group indices M 11 = M 22 = M 33 = m g = |Φ 1 | 2 + |Φ 2 | 2 . The corresponding evolution equations are given by: where δ ab and ε abc are the Kronecker delta and the full antisymmetric tensor in SU (2) space, respectively. We use the multiplescale perturbation theory (see, e.g., Ref. [7]) to solve the nonlinear Eqs. (3): first, we introduce the new space-time independent variables, X µn , as well as the partial derivatives thereof: and we assume that the corresponding field variables are expanded into an asymptotic series of the form: where is a formal small parameter (connected to the kink soliton amplitude and inverse width -see below). Substituting the above expressions into the equations of motion, and equating coefficients of the same powers of , we obtain a set of equations from which A a µ (k) (k = 0, 1, 2, · · · ) can be successively determined. Notice that each field A a µ (k) is to be determined so as to be bounded (nonsecular) at each stage of the perturbation.
In order to solve the evolution equations arising at various orders in , one can make an appropriate choice for the gauge field components, allowing for their decoupling -at least in the lowest orders in the perturbation expansion. Here, we will use the following configuration for the gauge fields: which allows us to decouple the corresponding equations of motion up to the order O( 3 ). This configuration is in fact a generalization of the Smilga's choice [1] for spatial non-homogeneous fields (see Appendix A). The resulting simplified equations for the component A k k (k = 1, 2) are given as follows: where we have used the notation: Here we should note that there is no summation over repeated latin indices in Eqs. (7)(8)(9). The equations of the remaining components are obtained in a similar way. Equation (9) still contains a coupling between A 1 1 and A 2 2 , due to the nonlinear term, which can be resolved using the further assumption: . Equations (7)(8)(9) can be solved self-consistently, leading to the following equations satisfied by the unknown A 1 1 (1) component: where k 2 0 − k 2 z = m 2 g . After some simple algebraic manipulations, the nonlinear evolution equation (12) takes the usual form of a nonlinear Schrödinger (NLS) equation with a repulsive (self-defocusing) nonlinearity (due to g 2 > 0 in the nonlinear term): which has been studied extensively in various branches of physics and, especially, in nonlinear optics [8] and atomic Bose-Einstein condensates [9]. The above NLS equation possesses a stationary kink-type (alias "dark") soliton solution [10], given by: where T 2 = 2 t and α = ko 2 F 0 . Details on the derivation of Eq. (13) are provided in Appendix A. In Fig. 1 we show a plot of the solution (14) using the parameter values: α = 53 MeV = 0.1, k 0 = 550 MeV, and F 0 = 3.2 MeV 1/2 . It can be seen that the obtained form is characterized by a free propagation in z-direction and a kink-soliton profile in the ξ-direction, with ξ = (x + y). It is obvious that Eq. (13), due to the presence of a first derivative in time, breaks the Lorentz invariance of the initial Lagrangian density; this is in accordance to the assumptions made to obtain the consistent solution (14) decomposing spacetime in two inequivalent subspaces ((x, y) and (z, t)). This property is inevitably expected to hold for gauge field solutions varying over a finite space interval. Additionally, gauge invariance is violated from the very beginning due to the presence of the gauge field mass term. However, the validity of the solution (14) is restricted to specific space-time scales and, therefore, there is no apparent contradiction with first principles.
After suitable rescaling in order to introduce dimensionless quantities, we have checked the validity of the solution (14) through numerical integration of eqs. (3). Adapting the choice (6) for the configuration of the gauge fields we concentrate on the equations of motion for the diagonal components A k k (k = 1, 2). The results of our numerical treatment in 1 + 1 dimensions for A 1 1 (Ξ, t) (Ξ = x + y) [13] is shown in the contour plot of Fig. 2. The solution (14) holds for more than 100 field oscillations indicating its remarkable stability and supporting the validity of our perturbative scheme.  14). The length scale is m −1 g . We have also used = 0.1.

III. PARTIAL LOCALIZATION OF DIRAC MATTER
In this section we will investigate the dynamics of an SU (2) charged Dirac field in the presence of an external gauge field which has the form found in Eq. (14). The corresponding Dirac equation is written as follows: where σ a (a = 1, 2, 3) are the Pauli spin matrices, γ µ (µ = 0, 1, 2, 3) are the Dirac matrices, and Ψ = Ψ 1 Ψ 2 is the SU (2) doublet for the fermionic field. For the fermionic mass matrix M we assume a diagonal form with m 11 = m 22 = m f . Due to the non-abelian character of the gauge group, the equations describing the dynamics of the two charged fields Ψ 1 and Ψ 2 , after expanding (15) and substituting Eq. (14) for the non-abelian gauge field, take the following coupled form: Taking into account that the expression (14) for the gauge field is non-covariant, it is consistent to consider the dynamics implied by Eqs. (16-17) in the non-relativistic limit. For that purpose, it is necessary to write the bispinors Ψ 1 and Ψ 2 in terms of their components. In that regard, we introduce the following notation: Applying the standard procedure (see, e.g., Ref. [11]) for obtaining the non-relativistic limit of Eqs. (16-17) (details of the calculations are given in Appendix B), we find the following set of coupled Schrödinger-type equations for the fermionic components χ ij (where χ ij = exp(−im f t) χ ij ): while φ ij are determined through χ ij as follows: Equations (19-22) can be consistently reduced, using χ 11 = χ 22 and χ 21 = i χ 12 , to the following two equations: where f (τ ) = cos(τ ) and A(ξ) = 2α √ 3g tanh(αξ). Without loss of generality we can choose k z = 0 (using the rest frame of the massive gauge field as reference frame) to further simplify the above expressions. Furthermore, in order to allow for non-trivial dynamics in the fermionic field, the corresponding mass m f has to be small (of order O( 2 )) as compared to the gauge field mass. In this case, writing m f = 2 m 0 , we obtain the following system of two equations: where m 0 is a mass scale of the order of m g . Let us now introduce the length scale ξ 0 and the time scale σ = m 0 ξ 2 0 to express Eqs. (29-30) in a dimensionless form. In these units, the dimensionless frequency of the oscillating YM-field becomes: ω 0 = m g m 0 ξ 2 0 . It also straightforward to define dimensionless variables ξ = ξ 0 ρ and t = σ τ . In these variables, we seek for solutions of the system (29-30) having the form: where F and G are slowly-varying functions of τ , while λ is the energy eigenvalue. In this limit, Eqs. (29-30) become: For ω 0 = m g m 0 ξ 2 0 1, Eqs. (32-33) can be integrated with respect to τ over a period T = 2π mgm0ξ 2 0 since in this time interval F and G are practically constant. Following this procedure, Eqs. (32-33) decouple and obtain the following form: allowing as a solution a fermionic state which is bound in the ξ direction and has the form [12]: where N is a normalization constant. The state (36) resembles the Landau levels of a particle in an external magnetic field in quantum electrodynamics. In the YM case under consideration, the magnetic field is generated by the term proportional to A 2 (ξ) in Eq. (30). The difference here is that we have a single level independently of the strength of the external Yang-Mills field. In addition, the Dirac particle is trapped only in the ξ-direction, where the external field is also localized. It should be noticed that the condition m g m 0 ξ 2 0 1, necessary for the existence of the solution (36), can be justified by either using a large ξ 0 value or a large m g value (or both).
It is illuminating to give an example of the energy and length scales involved in this solution. Assuming a gauge field mass of 500 MeV and a much smaller fermionic mass i.e., of order of O(5 MeV), we find that the SU (2) charged fermions are trapped in a region of radius of ≈ 150 f m in the (x, y)-plane with energy eigenvalue ≈ 1.5 M eV for an external field of amplitude 6 M eV . It must be noted that for this choice of parameter values the non-relativistic approximation is valid within an error of 15% estimated by the relative magnitude of the first relativistic correction term. In Fig. 3 we show the effective potential responsible for the trapping of the Dirac particle using the above mentioned parameter values. The dashed line indicates the energy λ = sF 2 0 of the associated bound state in the ξ-space. The fact that this state is very close to the continuum threshold explains the absence of a second bound state. In Fig. 4 we show the ξ-dependent wave function corresponding to the bound state displayed in Fig. 3. The broad spatial extension of this state is attributed to the small exponent in Eq. (36).

IV. CONCLUSIONS AND DISCUSSION
We have investigated classical solutions of the SU (2) massive Yang-Mills equations in the framework of multiple scale perturbation theory. Due to the presence of the mass term, conformal symmetry is explicitly broken and the Coleman theorem does not apply [3]. Therefore, the YM dynamics in this case admit soliton-like solutions localized in a subspace of the transverse space.
Such solutions of the Yang-Mills field break both Lorentz and gauge invariance in higher orders of the perturbation expansion, in consistency with the presence of a mass term as well as the appearance of partial localization. Dirac fermions with nonvanishing SU (2) charge, when exposed in an external YM field having the form of these soliton-like solutions, become trapped in a similar way as electrons in a transverse magnetic field (Landau levels). However, the trapping of the SU (2) colored fermions is a pure dynamical effect occurring in the non-adiabatic limit of very fast oscillations of the external YM field, and occurs only along the (x + y)-direction.
Our analysis reveals a mechanism for the occurrence of localized fermionic states with SU (2) charge based on the interaction with a massive Yang-Mills field. The simplifying assumptions made in our approach (two non-vanishing equal components of the gauge field at the leading order) may restrict the profile of the found solutions allowing, on the other hand, for an analytical treatment. Despite this restriction, the main ingredients of the present study could be used as a guide to obtain more general inhomogeneous classical solutions of the massive SU (2) field. However, such a task is a subject for future investigations.
where Ψ 1 and Ψ 2 are the two components of the bispinor Ψ defined in Eq. (18). In the following, we will apply the standard procedure [11] in order to obtain the non-relativistic limit of Eqs. (B1-B2). The necessity of this emerges by the violation of the covariance of the gauge field which we have imposed. Eventually, it is consistent to study the non-relativistic case.