Dynamics of Quarks in a 2D Flux Tube

. On the basis of a compactiﬁcation of the (3 + 1) into (1 + 1) dimensional space-time [1] , the quark states inside the 2D ﬂux tube are studied for the case of a linear transverse conﬁning potential. The derived states are classiﬁed by both the projections of the orbital momentum and the spin along the tube direction. The spectrum of the fermion states is evaluated. It is found that the energy eigenvalues of the quarks appear to be approximately related to the square root of the eigenvalues of the two-dimensional harmonic oscillator.


Introduction
The three-dimensional QCD flux tube between a quark and an antiquark and its idealization as a one-dimensional string are useful concepts (for a review see [2]). The flux tube enters into the flux-tube model of hadrons [3], and the QCD string provides the arena for the fragmentation model of particle production [4,5]. The idealization of the flux-tube as a string has the limitation that it pertains mainly to the dynamics along the longitudinal direction and cannot describe the dynamics in the transverse directions. To describe the transverse dynamics, it is necessary to examine the structure and the states of particles in the flux tube. The knowledge of these states furnishes useful information for the investigation of the transverse momentum distribution of the produced particles, for which a wealth of information have been collected recently [6][7][8][9][10][11][12].
In the flux tube model, the q andq partons in the flux tube are assumed to be confined and are subject to transverse confinement. We can describe transverse confinement in terms of a confining scalar interaction S (r ⊥ ) in transverse coordinates r ⊥ , with the quark mass function described by m(r ⊥ ) = m 0 + S (r ⊥ ) where m 0 is the quark rest mass.
Previously, we show from action integral that under the assumption of transverse confinement and longitudinal dominance, QCD 4 in (3+1) dimensional space-time can be approximately compactified into QCD 2 in (1+1) dimensional space-time [1]. In such a process, we find the equations of motion for quarks in the flux tube, and the relation between the coupling constant g 2D in QCD 2 and the coupling constant g 4D = g in QCD 4 . We show how the quark and the gluon in QCD 2 acquire masses arising their being confined within the tube, and how all these quantities depend on the transverse wave functions of the quarks in the tube. The variation of the gluon fields leads to an a e-mail: A_Kosh@internets.ru b e-mail: wongc@ornl.gov additional mass that depends on the coupling constant g 2D . The compactification facilitates the investigation of some problems in QCD 4 in the simpler dynamics of QCD 2 and brings us to a better understanding of the different degrees of freedom of the produced particles.

Equations of Motion for Transverse Dynamics of Quarks
We write the fermion field Ψ(x) in terms of the following spinor with transverse functions G ± (r ⊥ ) and x 0 -x 3 functions f ± (x 0 , x 3 ) [1,13], where we use the notations of [1], and r ⊥ =(x 1 , x 2 ) is a vector in the plane perpendicular to the x 3 axis. To obtain the equations of motion for the functions G + (r ⊥ ) and G − (r ⊥ ), we vary the action integral A(4D) in Eq. (11) of [1], under the constraint of the normalization condition. To do this we construct a new functional A ′ : where λ 2 is the Lagrange multiplier; ψ(2D, X) and m qT are the 2D spinor and the fermion effective mass in the (1+1) space-time [1]. The last term in Eq. (2) takes into account the unitarity of a fermion field in the 4D spacetime. Varying A ′ with respect to the functions G + (r ⊥ ) and G − (r ⊥ ), we derive Combining the above two equations, we obtain ( We have also The general equations coupling G + (r ⊥ ) and G − (r ⊥ ) for a scalar mass function m(r ⊥ ) in transverse confinement are ( The above equations show that λ 2 plays the role of the fermion eigenenergy.

Axisymmetric Potential
In this manuscript, we consider an axisymmetric linearly confining potential where the parameter κ can be interpreted as the string tension in the radial direction of the flux tube. Then, the set of coupled equations become ( In matrix form they are or where ) .
The Hamiltonian H 0 is given by Each fermion eigenstate is characterized by the zcomponent of the total angular momentum Ω. To separation out the φ and r ⊥ dependencies, we introduce the radial wave function R Ωσ (r ⊥ ) to write wave function G ± as where σ = ±1. Then, we obtain from Eq.(9) .

Eigenvalue equation
For fermions, Ω are positive and negative half-integers. Eqs. (13)-(15) suggests the usefulness of expanding the radial wave function R Ωσ in terms of a complete set of Harmonic oscillator radial basis R nΩσ . Accordingly, we expand |G(Ω)⟩ in terms of a complete set of Harmonic oscillator states |nΩσ⟩ with specific phase factors as where Here, the radial basis state wave functions R nΩσ (ξ) are where ξ = κr 2 ⊥ , and the normalization constant is .
The basis states |nΩσ⟩=ψ nΩσ χ σ are orthonormal, Substitute the expansion (16) into Eq. (10), we can separate out the φ degrees of freedom and get where the subscripts "(±)" refer to the σ = ±1 quantum numbers and H 0 (Ω ∓ 1/2) is given by Eq. (15). So, the eigenvalue equation becomes the following set of coupled equations: We note that we have chosen R nΩσ to be the radial eignestates of H 0 (Ω ∓ σ/2). It satisfies the eigenvalue equation where we have introduced dimensionless ϵ 2 as

01004-p.2
Upon taking the scalar product with R nΩ+ for the first equation in Eq. (21) and with R Ωn− for the second equation, we get where the scalar product ⟨nΩσ ′ , Ωmσ⟩ involves only the radial wave functions, The set of coupled equations in Eq. (24) for C nΩσ can be written as a matrix eigenvalue equation in the form where G(Ω) ≡ |G(Ω)⟩ is a column vector of coefficients C Ωnσ with known basis states |nΩσ⟩, and H is the matrix The matrix equation (26) can be diagonalized numerically.
After an eigenvalue λ 2 /κ is obtained, the corresponding eigenstate |G(Ω)⟩ can be normalized by choosing a normalization factor for the C nΩσ coefficients such that We note that the set of equations in (24) is invariant when we interchange Ω↔(−Ω) and σ↔(−σ). Thus, the ±Ω are doubly degenerate with and the wave functions are related to each other, Therefore, without losses of generality, we can focus our attention on positive Ω = 1/2, 3/2, 5/2, ... As an illustration, it is illuminating to obtain the approximate eigenvalues and eigenfunctions when we consider the following simplifying approximation then the original full Hamiltonian will be truncated and we obtain the approximate eigenvalue equation, ( ϵ 2 (nΩ+) ⟨nΩ+, nΩ−⟩ ⟨nΩ−, nΩ+⟩ ϵ 2 (nΩ−) This eigenvalue equation is of the form with eigenvalues and (unnormalized) eigenvectors When V is zero, the state C 1 will be a purely |nΩ(σ=1/2)⟩ state and the state C 2 will be a purely |nΩ(σ=−1/2⟩ state.

Numerical Solutions
The Hamiltonian for the eigenvalue equation (26) can be constructed and the eigenvalues and eigenvectors obtained by numerical Gauss-Laguerre integration and by matrix diagonalization. The analysis of the Hamiltonian in the matrix representation shows that the matrix element H(i j) ≪ H(ii) for i j and furthermore, H(i j) decreases as the difference between i and j increases. The eignevalues λ 2 /κ is therefore dominated by the diagonal matrix elements.
The eigenstate energy E is the square root of λ 2 and we have  Fig.1. Each state is doubly degenerate and is labelled by the quantum numbers (n, Ω, σ) of its dominant component. An eigenstate wave function is therefore given approximately by where R nΩσ is given by Eq. (19). It is reasonable to approximate the corresponding energy eigenvalue as In Fig. 1, the long horizontal lines are the positive energy levels given by the above approximate eigenvalues, which are close to the eigenvalues obtained by numerical diagonalization. Thus, the spectrum of E is related approximately to the square root of the eigenvalues of the twodimensional harmonic oscillator, where the radial quantum number n have become approximately a "good" quantum number.

Effective Masses of Quarks and Gluons and 2D coupling constant
It was shown in [1] that the quark effective mass m qT in the 2D flux tube considerations coincides with the eigenvalue λ. For the approximate eigenstate (37) with the dominant component of |nΩσ⟩, we therefore have As for the effective gluon mass m gT generated by QCD 4 → QCD 2 compactification, it is proportional to the kinetic energy term of the Hamiltonian H 0 given by Eq. (10). Since the mean kinetic energy of an oscillator is one half of its total energy, we have In the flux tube consideration the g 2D coupling constant is [1]: Substituting the wave functions given by Eq. (37) into the last formula, we obtain for n = 0: ( 2 |Ω−σ/2| Γ 2/3 (3/2|Ω − σ/2| + 1) 3 |Ω−σ/2| Γ(|Ω − σ/2| + 1) The total effective gluon mass M tT which is induced by the QCD 4 → QCD 2 compactification and interaction between 2D fermions and gluons is given by the formula [1]: where N f is the number of flavors.

Conclusion
On the basis of the QCD 4 →QCD 2 compactification [1], we study the quark states in an axial symmetric radiallyconfining potential. The eigenvalues and eigenvectors of these state are evaluated and the spectrum of the eigenstates is obtained by diagonalizing the exact Hamiltonian.
The results of the calculations show that the energy spectrum is related approximately to the square root of the eigenvalues of the two-dimensional harmonic oscillator. The obtained fermion eigenstates allow us to calculate the effective masses of quarks and gluons in the 2D flux tube, as well as the coupling constant in the compactified QCD 2 [1]. Following the string fragmentation concept [2,4], they will enable us to derive the p T distribution of the observable hadrons arising from the flux tubes formed in the collisions of high energy proton and nuclear collisions.