Generalized Scalar Potential in the Dirac Equation

Chiang Mai J. Sci. 2012; 39(4)

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Chiang Mai J. Sci. 2012; 39(4) : 723-730 http://it.science.cmu.ac.th/ejournal/ Contributed Paper

Generalized Scalar Potential in the Dirac Equation Lalit K. Sharma*, Moletlanyi Tshipa, Samuel Chimidza and Dominique P. Winkoun Department of Physics, University of Botswana, Gaborone, Botswana. *Author for correspondence; e-mail: [email protected] Received: 10 July 2011 Accepted: 9 November 2011

ABSTRACT A superposed scalar potential which is the combination of a general even power, Coulomb and linear potentials is solved in the Dirac equation. The eigenvalues are found to be real for this potential indicating that quarks could be confined in this set up. Keywords: superposed Scalar potential, dirac equation and quark confinement 1. INTRODUCTION

The quark model has been very successful in predicting the hadron spectroscopy, high-energy and large transverse- momentum behavior of cross-sections, and current algebra. In any consistent theory of hadrons based on the quark model the question of quark confinement is of crucial importance [1]. The possibility that infrared catastrophe may provide confinement in an asymptotically free gauge theory of quarks and gluons with exact SU (3) color symmetry has been a most exciting field of investigation [2]. On a more direct level the dynamic models proposed by [3] and SLAC groups[4], in which the quarks are confined to a region of space called a “bag”, provide a new insight into the question of confinement as well as into the properties of the hadronic states. Therefore, the crucial point is to assume that quarks are confined as taken in bag models [5],where the Lorentz invariance is assumed. As a first approximation the quarks may be considered in a central potential, behaving as independent particles [6].In various QCD motivated models, the

confining potentials are taken as a mixture of Lorentz scalar and vector parts. This is because a Lorentz vector potential in the Dirac equation cannot confine the particles, and the confining component which is a consequence of multi-gluon exchange must be scalar [7]. Due to the relativistic motion of at least the light quarks, a relativistic treatment is necessary. However, the Dirac equation with both negative and positive energy solutions, may lead to the Klein paradox. This difficulty may be resolved by the dominance of scalar component in the Dirac equation [8].Precisely for this reason and for studying quark confining potentials, interest has been shown in solving the Dirac equation with scalar potential functions. In these “bag” models, the quarks satisfy Dirac’s equation with either covariant boundary condition at the surface of the bag or a self- consistent confining “potential” through their interaction with a scalar field, [5, 9]. Much interest has therefore been shown in solving the Dirac equation with scalar

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potential functions. The use of scalar potential in the Dirac equation is equivalent to a dependence of the rest mass upon position. Of particular importance is the case in which the potential is a function of radius about a fixed centre. Such potentials appear in the “quasi-independent models” [10-12] and “bag models” [4,13] of hadrons. Critchfield [6] has solved the Dirac equation with a linear scalar potential and has obtained real eigenvalues from which he constructed a relativistic zero-mass quark model, describing the dynamics of bound quark states. From these solutions, values of some of the static properties of proton viz., its magnetic moment and root-mean square radius were evaluated. It may also be noted that Shahnas et al [14] have applied a saddle point variational method to the Dirac equation with both positive and negative energy solutions. They calculated the total energy per unit mass, relativistic magnetic moment and the electromagnetic energy for a unit charge and magnetic moment of quarks. Their results were found to be in good agreement with their corresponding exact solutions. Critchfield [5] further generalized his work by considering a potential of the form: , 2 where μ is the bare mass, λ and K characterize the scalar potential. In this case the solution with finite values of the bare mass μ remain unchanged and led to real eigenvalues, indicative of pure bound states. Dirac equation with a quadratic scalar potential has been solved by Ram et al [15]. They have shown that for zero bare mass the energy eigenvalues are not pure real but are complex for zero and for small bare mass of the quark. On the basis of these observations, Ram et al [15] have concluded that for this potential it is not possible to confine light quarks and that they leak out. Recently, Sharma [16] has considered a


potential which is in the form of equal admixture of scalar and vector parts in the Dirac equation and applied this potential to obtain the magnetic moments of octet baryons, axial vector coupling of neutron for β-decay and finally to the computation of mean square charge radius of the proton. Further, Sharma et al [17] have derived the asymptotic expressions for the wave functions and s-wave phase shifts for a linear potential in the Dirac equation. It may also be interesting to note that the present authors have recently worked on a scalar potential which is the combination of a linear and a quadratic potentials [18] and observed that for zero and lower values of bare quark mass, the energy eigenvalues were real, indicating thereby that light quarks could be confined in such a set up. Recently, these authors have also worked on a logarithmic scalar potential and found that their calculated eigenvalues are real. This work has now been submitted for publication elsewhere. In this paper, we have solved the Dirac equation for a scalar potential which is proportional to r-1, r and r2. For zero and lower values of the bare mass the eigenenergies are found to be real, indicating that for such an interaction, confinement is possible. In fact authors have used a general even power potential in combination with a linear and a Coulomb potential to work on this problem. It would be worth mentioning that the general evenpower potential studied by us here is of particular interest in the potential theory. This is due to the fact that well-known potentials such as harmonic oscillator, the Gauss, and the anharmonic oscillator (with even anharmonicities) may be derived from this, as particular cases. Thus the results of this problem provide a unified treatment of these potentials. Precisely for this reason,


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one of the authors of this paper, has extensively worked on this general potential and have solved it in the Schr dinger, Klein-Gordon and Dirac equation, set ups [19-21]. SOLUTION OF THE DIRAC EQUATION The scalar potential for which the Dirac equation is solved has the form 2.

(1) where μ is the bare mass, N 2i , g1 , g 2 and g3 are constants which characterize the scalar potential. The Dirac equation for potential (1) [with units = c = 1], on separation in spherical coordinates can be expressed as two coupled equation in ψa and ψb, the large and the small components of the Dirac wave functionψ [22]:

(4) In expanding the series, it is convenient to define the new variables as P = (f + g), Q = (f - g) (5) and also substituting (6) Then equations (2) lead to the following recursion relation for Pk AkPk+1+BkPk+CkPk-1+DkPk-2i+1 = 0

(7)

where Ak = (k+g3+s)(k+1)(k+2s+1) Bk = [EK-2μ(k+g3+s)(k+g3+s+1)] Ck = [E2-2g2(k+g3+s)](k+g3+s+1) (8) Since Pk vanishes for k < 0, one sees from equation (7) A0P1+B0P0 = 0 (9) Further, defining

(2a) (2b) In equations (2), K = l+1 when the total angular momentum j = l + ,and K = -l when j = l - . Following the procedure adopted in our previous paper [23], we solve equations (2) by setting (3a) and (3b) where f(r) and g(r) are the series in non-negative powers of r with leading terms fo and go respectively. The boundary conditions at r = 0 and r = ∞ yield

(10) The recursion relation (7) is then transformed to (11) For i = 0 to 1, and N0 = 0 , (12) For square-integrable functions(ψa & ψb), following Critchfield [5, 6] and Sharma et al [18], we assume that for large k =N′ → ∞ the ratio of successive Pk obeys the relation (13)

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With this condition, equation (12) reduces to the following cubic equation HN′b3N′+b2N′+MN′bN′+LN′ = 0, (14) which has at least one real root smaller than zero leading to real normalizable wave functions. Note that similar expressions can also be written, choosing higher values of i. These higher values of i will yield r4, r6, r8 terms and so forth, availing an opportunity of studying their influence on the eigen energies in combination with the linear and the Coulomb terms. For calculating the real eigenenergies, we take the values of N′ equal to 26 and 25 and a trial input value of Ein in equation (14) to obtain the corresponding real

negative roots b26, b25. Using equation (12), the values of bk for k = 24, k = 23,…,k = 1 are calculated. The value of b1 thus calculated is substituted in equation (9) to obtain Ein = Eout. If Eout is not equal to Ein, we use 1 the average Eave =2 (Ein + Eout) as input in equation (14). This process is repeated until we obtain Ein = Eout within an accuracy of 10-2 3. RESULTS AND DISCUSSIONS

(A) It should be noted that our results derived in equations (1) to (13) are applicable to any general value of i = 0, 1, 2,-----. In this paper, however, we have calculated the ground-state quark energies, as a function of bare mass for our potential (1), for the

Table 1. The ground state quark energy values as a function of the bare mass for the potential of the present paper {a}compared with that of Sharma et al [18] {b}, and Ram et al [15] {c}, for the coupling constants, g1 = -0.5 GeV, g2 = 0.1 GeV g3 = 0.5 GeV, N0 = 0.0, N2 = 0.4 and i= 1. Quark Mass

Energy (GeV)

(GeV)

{a}

{b}

{c}

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.78739 2.27100 3.91107 5.35123 6.94384 8.49991 10.20995

1.63475 2.17253 2.79241 3.27954 3.84698 4.48167 5.18971

0.88576 1.35852 1.82645 2.30925 2.76758 3.24268 3.709904

Figure 1. The comparison of quark ground state energies as a function of bare mass for potential (1) {a} for N0 =0.0, i = 1; with that of Sharma et al, [18] {b}, and Ram et al [15] {c}.


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particular value i =1. Our potential thus effectively is a combination of a quadratic, linear and Coulomb potentials. These values {a} have been shown in Table I. For

comparison, similar values for the same bare mass, calculated by Sharma et al [18]{b} and by Ram et al [15] {c}are also given in the same Table. Graphical representation of all the three

Table 2. The ground state and excited states quark energy eigenvalues for g1 = -0.5 GeV, g2 = 0.1 GeV, g3 = 0.5 GeV, N0 = 0.0, i = 1 and N2 = 0.4. Quark Mass

Energy (GeV)

(GeV)

l=0

l=1

l=2

l=3

0.0 0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0

1.78739 2.27100 3.91107 5.35123 6.94384 8.49991 16.98030 25.42815 33.89771

2.09467 3.09082 4.32130 5.83713 7.52184 9.26168 18.28213 27.38520 36.50136

2.462448 3.63995 5.05946 6.83992 8.81742 10.88002 21.48392 32.18342 42.89742

2.82281 4.12188 5.83082 7.95238 10.28202 12.69890 25.10490 37.61233 50.13516

Figure 2. The dependence of the quark ground state energies on the mass for different values of l with N0 = 0.0 and i = 1. calculations is shown in Figure 1. (B) For the present potential, the energy eigenvalues for different angular momenta (different l values) have been shown in Table 2. The dependence of the quark energies on the quark bare mass for different angular momenta is depicted in Figure 2.

( C ) In Table 3, using equation (14) along with equations (12) and (9), Regge trajectories for different bare masses and different values of coupling constant g1, keeping g2 = 0.1GeV have been calculated. These trajectories have been shown graphically in Figure 3.

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Table 3. The Regge trajectories for different bare masses and different values of the coupling constant g1 for g2 = 0.1 GeV, N0 = 0.0, i =1 and N2 = 0.4. Energy (GeV) μ = 0.00 GeV

l(E) 0.0 0.1 0.2 0.3 0.7 1.0 2.0 3.0 4.0

g1=-0.5 GeV 1.78739 1.80762 1.83198 1.85948 1.98823 2.09467 2.46229 2.82271 3.16815

g1=-5.0 GeV 3.701301 3.74308 3.80541 3.85153 4.12217 4.34641 5.12228 5.88402 6.61524

μ = 1.00 GeV

g1=-0.5 GeV 8.57609 8.52171 8.51251 8.54311 8.87921 9.26432 10.88002 12.69916 14.59515

μ = 2.00 GeV

g1=-5.0 GeV 9.35045 9.32848 9.35104 9.40669 9.83201 10.26757 12.01626 13.92444 15.88568

g1=-0.5 GeV 16.97067 16.85183 16.82906 16.87847 17.52567 18.28214 21.48387 25.10490 28.88401

g1=-5.0 GeV 17.20153 17.09464 17.08264 17.14178 17.82008 18.59404 21.83381 25.47660 29.26896

Figure 3. The Regge trajectories for the indicated bare masses for N2 = 0.4, i = 1 and g1 = -0.5 GeV. The dash dotted curves are for g1 = -5.0 GeV. 4. CONCLUSIONS

In conclusion, we state the following as principal observations of our investigations: (I)From Figure 1, we find that : (a) For the same value of bare mass μ, the quark energy increases more rapidly for the present potential than the potentials considered by Ram et al [ 15]

and Sharma et al [18]. (b) The addition of a Coulomb term to the potential of Sharma et al [18] and the addition of a Coulomb-cum- linear term to the potential of Ram et al [15] gives a linear variation of eigenenergies with bare mass μ for heavier quarks, and a quasiparabolic variation for very light quarks.


(II)In Figure 2: (a)We have studied the quark energy values of the ground state and a few excited states for several values of μ for our superposed scalar potential. (b)We observe from this figure that for lower values of the bare mass μ, the energy values E for the ground state, which were closer to the higher values of l, diverge more and more from the ground state values, as the values of bare mass μ increases. (c)Quark energies have been seen to increase with increasing quark masses, the gradient or the rate of change increases with increasing angular momentum values. (III)In Figure 3: (a)We have plotted Regge trajectories. These trajectories are linear and rising for the range considered. (b)These trajectories shift to higher energy values as the coupling strength g1 becomes more negative. (c)The degree of this shift, however, decreases as the bare mass μ increases. For lower bare masses the Regge trajectories are almost linear as a function of quark energies. This observation is in agreement with that of Duviryak [24] and Chu et al [25]. Chu et al [25] studied the properties of potentials that may provide linear Regge trajectories. Duviryak [24] in particular, obtained these trajectories by solving the two-body Dirac equation involving matrices, which was reduced to the pair of ordinary second-order differential equations for radial components of a wave function.

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(d)It is also found that larger bare mass and higher values of l yield higher energies at an ever increasing rate. (e) Finally it may be stated in conclusion that since for our superposed potential (1), the calculated energies are real even for zero bare mass, hence one can infer that light quarks may be confined in this interaction. Our observation differs from that of Ram et al [15], where they have considered a quadratic potential. Since their set up provides complex values of eigenenergies, it prohibits the confinement of even light quarks. REFERENCES [1] Grinstein B., A modern introduction to Quarkonium theory, Int. J. Mod. Phys. A., 2000 15: 461-495. [2] Weinberg S., Non-Abelian Gauge theories of the strong interaction, Phys. Rev. Lett., 1973; 33: 494-497. [3] Chodos A., Jaffe R.L., Johnson K., Thorn C.B. and Weisskopf V.F. New extended model of hadrons, Phys. Rev. D., 1974; 9: 3471-3495. [4] Bardeen W.A., Chanowitz M.S., Drell S.D., Weinstein M. and Yan T.-M. Heavy quarks and strong binding: A field theory of hadron structure, Phys. Rev. D., 1975; 11: 1094-1136. [5] Critchfield C.L., Scalar potentials in the Dirac equation, J. Math. Phys., 1976; 17: 261-266. [6] Critchfield C.L., Scalar binding of quarks, Phys. Rev. D., 1975; 12: 923925. [7] Palladino B.E. and Ferreira Leal P., Mass differences of light hadron isomultiplets, Phys. Rev. D., 1989; 40: 3024-3034. [8] Fishbane P.M., Gasorowicz S.G., Johnnesen D.C. and Kaus P. Vector

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