Advances in High Energy Physics

Volume 2014 (2014), Article ID 784072, 8 pages

http://dx.doi.org/10.1155/2014/784072

Review Article

Remarks on the Spin-One Duffin-Kemmer-Petiau Equation in the Presence of Nonminimal Vector Interactions in Dimensions

L. B. Castro1,2 and L. P. de Oliveira³

¹Departamento de Física, CFM Universidade Federal de Santa Catarina (UFSC), CP. 476, 88040-900 Florianópolis, SC, Brazil

²Departamento de Física, Universidade Federal do Maranhão (UFMA), Campus Universitário do Bacanga, 65085-580 São Luís, MA, Brazil

³Instituto de Física, Universidade de São Paulo (USP), 05508-900 São Paulo, SP, Brazil

Received 24 March 2014; Revised 28 May 2014; Accepted 29 May 2014; Published 19 June 2014

Academic Editor: Elias C. Vagenas

Copyright © 2014 L. B. Castro and L. P. de Oliveira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We point out a misleading treatment in the recent literature regarding analytical solutions for nonminimal vector interaction for spin-one particles in the context of the Duffin-Kemmer-Petiau (DKP) formalism. In those papers, the authors use improperly the nonminimal vector interaction endangering in their main conclusions. We present a few properties of the nonminimal vector interactions and also present the correct equations to this problem. We show that the solution can be easily found by solving Schrödinger-like equations. As an application of this procedure, we consider spin-one particles in presence of a nonminimal vector linear potential.

1. Introduction

The Duffin-Kemmer-Petiau (DKP) formalism [1–4] describes spin-zero and spin-one particles and has been used to analyze relativistic interactions of spin-zero and spin-one hadrons with nuclei as an alternative to their conventional second-order Klein-Gordon (KG) and Proca counterparts. The DKP formalism proved to be better than the KG formalism in the analysis of decays, the decay-rate ratio , and level shifts and widths in pionic atoms [5–16]. The DKP formalism enjoys a richness of couplings not capable of being expressed in the KG and Proca theories [17, 18]. Although the formalisms are equivalent in the case of minimally coupled vector interactions [19–21], the DKP formalism opens new horizons as far as it allows other kinds of couplings which are not possible in the KG and Proca theories. The nonminimal vector interaction refers to a kind of charge conjugate invariant coupling that behaves like a vector under a Lorentz transformation. The invariance of the nonminimal vector potential under charge conjugation means that it does not distinguish particles from antiparticles. Hence, whether one considers spin-zero or spin-one bosons, this sort of interaction cannot exhibit Klein’s paradox [22]. Nonminimal vector potentials, added by other kinds of Lorentz structures, have already been used in a phenomenological context for describing the scattering of mesons by nuclei [23–32], but it should be mentioned that in [23–26, 28–30, 32] the nonminimal vector couplings have been used improperly. Nonminimal vector coupling with a quadratic potential [33] and with a linear potential [34] and mixed space and time components with a step potential [35, 36], double-step potential [37], a smooth step potential [38], a linear potential [22, 39, 40], and a linear plus inversely linear potential [41] have been explored in the literature. In a recent paper published in this journal, Hassanabadi et al. [42] analyze the DKP equation in the presence of nonminimal vectorial interactions (Coulomb and harmonic oscillator potentials) in dimensions for spin-one particles. In that paper, the authors used improperly the nonminimal vector interaction endangering in its main conclusions. The same mistake is found in recent works, in [43–51], for instance. Other misconception is found in [23–26], where the space component of the nonminimal vector potential is absorbed into the spinor. As it is shown in [22], there is no chance of dissociating from this term. Furthermore, the space component of the nonminimal vector potential could be irrelevant for the formation of bound states for potentials vanishing at infinity, but its presence is an essential ingredient for confinement.

In view of the misconceptions on the nonminimal vector interaction propagated in the literature, the purpose of this review paper is to review the DKP equation in the presence of a nonminimal vectorial interaction for spin-one particles in dimensions. We present a few properties of the nonminimal vector interactions and also present the correct equations to this problem. We show that the solution can be easily found by solving Schrödinger-like equations. As an application of this procedure, we consider spin-one particles in presence of a nonminimal vector linear potential. For this case in particular, the problem is mapped into the nonrelativistic three-dimensional harmonic oscillator.

2. The DKP Equation

The DKP equation for a free boson is given by [4] (with units in which ) as follows:

where the matrices satisfy the algebra

and the metric tensor is . The algebra expressed by (2) generates a set of 126 independent matrices whose irreducible representations are a trivial representation, a five-dimensional representation describing the spin-zero particles, and a ten-dimensional representation associated with spin-one particles. The DKP spinor has an excess of components and the theory has to be supplemented by an equation which allows us to eliminate the redundant components. That constraint equation is obtained by multiplying the DKP equation by ; namely,

This constraint equation expresses three (four) components of the spinor by the other two (six) components and their space derivatives in the scalar (vector) sector so that the superfluous components disappear and there only remain the physical components of the DKP theory. The second-order Klein-Gordon and Proca equations are obtained when one selects the spin-zero and spin-one sectors of the DKP theory.

A well-known conserved four-current is given by

where the adjoint spinor is given by with in such a way that (the matrices are Hermitian with respect to ). Despite the similarity to the Dirac equation, the DKP equation involves singular matrices, the time component of is not positive definite, and the case of massless bosons cannot be obtained by a limiting process [52]. Nevertheless, the matrices plus the unit operator generate a ring consistent with integer-spin algebra and may be interpreted as a charge density. The normalization condition can be expressed as

where the plus (minus) sign must be used for a positive (negative) charge.

3. Interactions in the DKP Equation

With the introduction of interactions, the DKP equation can be written as

where the more general potential matrix is written in terms of 25 (100) linearly independent matrices pertinent to five- (ten-) dimensional irreducible representation associated with the scalar (vector) sector. In the presence of interaction, satisfies the following equation:

Thus, if is Hermitian with respect to , then four-current will be conserved. The potential matrix can be written in terms of well-defined Lorentz structures. For the spin-zero sector there are two scalar, two vector, and two tensor terms [17], whereas for the spin-one sector there are two scalar, two vector, a pseudoscalar, two pseudovector, and eight tensor terms [18]. The tensor terms have been avoided in applications because they furnish noncausal effects [17, 18].

3.1. Nonminimal Vector Couplings in the DKP Equation

Considering only the nonminimal vector interaction, the DKP equation can be written as

where is a projection operator ( and ) in such a way that behaves like a vector under a Lorentz transformation as does . One very important point to note is that this potential leads to a conserved four-current, but the same does not happen if instead of one uses either or , as in [23–26, 28–30, 32, 42–51]. As a matter of fact, in [23] it is mentioned that and produce identical results. Considering explicitly the condition (7) for the potential (widely used in the literature), we obtain

The current is not conserved and it is proportional to . The fact that this current is not conserved has crucial consequences on the orthonormal condition of the DKP spinor [22, 35, 36, 38, 39].

The DKP equation is invariant under the parity operation, that is, when , if changes sign, whereas remains the same. This is because the parity operator is , where is a constant phase and changes into . Because this unitary operator anticommutes with and , they change the sign under a parity transformation, whereas and , which commute with , remain the same. Since or , the spinor components have definite parities. The charge conjugation operation can be accomplished by the transformation , where denotes the complex conjugation and is a unitary matrix such that . The matrix that satisfies these relations is . The phase factor is equal to , and thus . Note also that , as should be expected for a charge current. Meanwhile anticommutes with and the charge conjugation operation entails no change on . The invariance of the nonminimal vector potential under charge conjugation means that it does not couple to the charge of the boson. In other words, does not distinguish particles from antiparticles. Hence, whether one considers spin-zero or spin-one bosons, this sort of interaction cannot exhibit Klein’s paradox [22].

If the potential is time-independent, one can write , where is the energy of the boson, in such a way that the time-independent DKP equation becomes

In this case does not depend on time, so that the spinor describes a stationary state.

3.2. Vectorial Sector

For the case of spin-one (vectorial sector), the matrices are [53]

where are the 3 3 spin-one matrices , are the 1 3 matrices and , while and designate the 3 3 unit and zero matrices, respectively, while the superscript designates matrix transposition. In this representation ; that is, projects out the four upper components of the DKP spinor. The ten-component spinor can be written as and partitioned as (following the notation of [42])

The DKP equation in dimensions can be expressed in the following compact forms:

At this stage it is worthwhile to mention that (13)–(16) are completely different from those given in [42] and this fact is due to using improperly the nonminimal vector coupling. These facts should be enough to jeopardize the results presented in [42–51].

Using the standard procedure developed in [54], we put

where , , , and are radial wave functions, while are the usual spherical harmonics of order and are the vector spherical harmonics. Then, using the notation

and similar definitions for , , , and together with the properties of vector spherical harmonics (see the Appendix), we can get a set of first-order coupled differential radial equations. Substituting (18) and (20) in (13) and if we consider spherically symmetric potentials and , the radial differential equations obtained from (13) are

where and .

Similarly, substituting (17) and (19) in (14), we obtain

The radial equations obtained from (15) are

Finally, from (16) we get

Now, following the procedure used in [54], the ten coupled differential radial equations obtained above (equations (22)–(31)) can be decoupled into two classes of radial equations associated with a specific parity. For states of parity, the relevant differential equations are (22), (23), (28), and (29). The remaining six radial wave functions are zero. On the other hand, for states of parity, the relevant differential equations are (24), (25), (26), (27), (30), and (31). Similarly to the previous case, the other four radial wave functions are zero.

3.2.1. Parity States

Using (22), (23), and (29), the components , and can be eliminated in favor of then by inserting them in (28), the radial function obeys the second-order differential equation as follows:

where and because , unless the potentials contain a delta function at the origin, one must impose the homogeneous Dirichlet condition . At this stage it is worthwhile to mention that (32) is very similar to DKP equation for spin-zero particles in dimensions except for the term [40]. Therefore, for motion in a central field, the solution of the three-dimensional DKP equation with nonminimal vectorial interaction can be found by solving a Schrödinger-like equation for states of parity. The other components are obtained through (22), (23), and (29).

3.2.2. Parity States

Using (26) and (30), we obtain

where . Similarly, using (27) and (31), we get

where . In this general case, we are not able to obtain analytical solutions to this kind of parity states, because we cannot decouple the differential equations for the components and . An alternative to overcome this disadvantage is to restrict our analysis for . Considering , the differential equations are decoupled for the components and , but those differential equations are very complicated and do not furnish exact solutions.

For , (33) and (34) reduce to

where . In this case, we obtain that the radial functions and obey the second-order differential equations as follows:

Therefore, for the particular case , the solution of the three-dimensional DKP equation with nonminimal vectorial interaction can be found by solving two Schrödinger-like equations for states of parity. The other components are obtained through (35). It should not be forgotten, though, that the equations for and are not indeed independent because the energy appears in both equations. Therefore, one has to search for bound-state solutions for and with a common energy.

3.3. Nonminimal Vector Linear Potential

Having set up the spin-one equations for nonminimal vector interaction, we are now in a position to use the machinery developed above in order to solve the DKP equation for some specific form of the nonminimal interaction. As an application of this procedure, let us consider a nonminimal vector linear potential in the following form:

where and are dimensionless quantities.

3.3.1. Parity States

Substituting (37) in (32), one finds that obeys the second-order differential equation

where

Considering and , the solution for (38) with and real is the well-known solution of the Schrödinger equation for the three-dimensional harmonic oscillator. Note that the condition real implies that , meaning that the radial component of the nonminimal vectorial potential must be stronger than its time component in order for the effective potential to be a true confining potential. On the other hand, if or , we obtain , the effective potential in this case will be an inverted harmonic oscillator, and the energy spectrum will consist of a continuum corresponding to unbound states. Therefore, the presence of radial component of the nonminimal vector potential is an essential ingredient for confinement.

A detailed study of this effective potential is done in  [40]. Using the results of [40], the solution is expressed as

where is a normalization constant and with a nonnegative integer. Note that can take values when is an even number and when is an odd number and also that for each value of there are different values of . All the energy levels are degenerate with the exception of . The degeneracy of the level of energy for a given principal quantum number is given by as a consequence of the presence of essential and accidental degeneracy.

From (40), we can see that there is an infinite set of discrete energies (symmetrical about ) irrespective to sign of and although positive- and negative-energy levels do not touch, they can be very close to each other for moderately strong coupling constants without any danger of reaching the conditions for Klein’s paradox. The absence of Klein’s paradox for this kind of interaction is attributed to fact that the nonminimal vectorial interaction does not distinguish particles from antiparticles [36].

For the case of (), the solution is expressed as

where .

3.3.2. Parity States

As mentioned in Section 3.2.2, we cannot consider the general case (37), because we are not able to obtain analytical solutions to this kind of parity states.

Otherwise, considering (37) with () and using the notations, and , (36) reduces to

where

The solution for (43) with and real is the solution of the Schrödinger equation for the three-dimensional harmonic oscillator, as in the case of states of parity.

The energy can be obtained from the relation

Now we move on to match a common energy to spin-one particles problem for states of parity. The compatibility of the solutions for and demands that the quantum number and must satisfy the relation

4. Final Remarks

In this review paper, we showed the correct use and also presented a few properties of the nonminimal vector interactions in the Duffin-Kemmer-Petiau (DKP) formalism. A relativistic wave equation must carry a conserved four-current to exhibit symmetries in physical problems. In this spirit, we showed that the four-current is not conserved when one uses either the matrix potential or (widely used in the literature), even though the linear forms constructed from those matrices potentials behave as true Lorentz vectors. Also, we presented the correct equations for the problem addressed in [42]. In this case, we found an equation very similar to DKP equation for spin-zero particles in dimensions, except for some additional terms. Therefore, the solution of the three-dimensional DKP equation with nonminimal vectorial interactions can be found by solving Schrödinger-like equations. As an application of the procedure developed, we considered the problem of spin-one particles in the presence of a nonminimal linear vector potential and discussed the necessary conditions in order for the effective potential to be true confining potential. The absence of Klein’s paradox is attributed to the fact that the nonminimal vectorial interaction does not distinguish particles from antiparticles [36].

Our results are definitely useful because they shed some light on the understanding of the nonminimal vector interactions. Furthermore, the correct use of the nonminimal vectorial interaction may be useful due to wide applications in the description of elastic meson-nucleus scattering.

Appendix

The Vector Spherical Harmonics

The properties of the vector spherical harmonics used in this work are obtained from [55]. The list of properties is the following:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are indebted to the anonymous referee for an excellent and constructive review. This work was supported in part by means of funds provided by CAPES. This work was partially done during a visit (L. B. Castro) to UNESP, Campus de Guaratinguetá.

References

G. Petiau, University of Paris thesis, Published in Académie Royale de Médecine de Belgique, Classe des Sciences, Mémoires in 8, 16, 2, 1936

N. Kemmer, “Quantum theory of Einstein-Bose particles and nuclear interaction,” Proceedings of Royal Society A, vol. 166, pp. 127–153, 1938.

R. J. Duffin, “On the characteristic matrices of covariant systems,” Physical Review, vol. 54, no. 12, p. 1114, 1938.

N. Kemmer, “The particle aspect of meson theory,” Proceedings of the Royal Society A, vol. 173, pp. 91–116, 1939.

E. Fischbach, F. Iachello, A. Lande, M. M. Nieto, and C. K. Scott, “Kl3 form factors and the scaling behavior of spin-0 fields,” Physical Review Letters, vol. 26, no. 19, pp. 1200–1203, 1971.

E. Fischbach, M. M. Nieto, H. Primakoff, C. K. Scott, and J. Smith, “Resolution of the discrepancy in the determination of the cabibbo angle from Kl3 and nuclear β decays,” Physical Review Letters, vol. 27, no. 20, pp. 1403–1406, 1971.

E. Fischblach, M. M. Nieto, H. Primakoff, et al., “Resolution of the puzzle: ,” Physical Review Letters, vol. 29, no. 15, pp. 1046–1049, 1972.

N. G. Deshpande and P. C. McNamee, “Some difficulties with the use of the Kemmer-Duffin formalism for pseudoscalar mesons,” Physical Review D, vol. 5, no. 4, pp. 1012–1014, 1972.

A. O. Barut and Z. Z. Aydin, “Value of the ξ parameter in Kl3 decay from the pion-gauge condition,” Physical Review D, vol. 6, no. 11, pp. 3340–3342, 1972.

E. Fischbach, M. M. Nieto, and C. K. Scott, “Spin-0 phenomenology of Kl3 decays in the K-pole model,” Physical Review D, vol. 7, no. 1, pp. 207–214, 1973.

Z. Z. Aydin and A. O. Barut, “Kl3 and π e3 decays in terms of the Kemmer β current,” Physical Review D, vol. 7, no. 11, pp. 3522–3525, 1973.

M. D. Scadron and R. L. Thews, “Inconsistency of a Duffin-Kemmer-Petiau model for meson decay rates,” Physical Review D, vol. 9, no. 7, pp. 2180–2182, 1974.

E. Fischbach, M. M. Nieto, H. Primakoff, and C. K. Scott, “Meson decays and the DKP equation,” Physical Review D, vol. 9, no. 7, pp. 2183–2186, 1974.

E. Fischblach, M. M. Nieto, and C. K. Scott, “Structure of the Duffin-Kemmer-Petiau matrix element for Kl3 decay,” Progress of Theorical Physics, vol. 51, pp. 1585–1597, 1974.

F. T. Meiere, E. Fischbach, A. McDonald, M. M. Nieto, and C. K. Scott, “-symmetric coupling of mesons to baryons and the strong D/F ratio,” Physical Review D, vol. 8, no. 11, pp. 4209–4212, 1973.

E. Friedman, G. Kälbermann, and C. J. Batty, “Kemmer-Duffin-Petiau equation for pionic atoms and anomalous strong interaction effects,” Physical Review C, vol. 34, no. 6, pp. 2244–2247, 1986.

R. F. Guertin and T. L. Wilson, “Noncausal propagation in spin-0 theories with external field interactions,” Physical Review D, vol. 15, no. 6, pp. 1518–1531, 1977.

B. Vijayalakshmi, M. Seetharaman, and P. M. Mathews, “Consistency of spin-1 theories in external electromagnetic fields,” Journal of Physics A: Mathematical and General, vol. 12, no. 5, pp. 665–677, 1979.

M. Riedel, Relativistische Gleichungen Fuer Spin-1-Teilchen, Diplomarbeit, Institute for Theoretical Physics, Johann Wolfgang Goethe-University, Frankfurt, Germany, 1979.

M. Nowakowski, “The electromagnetic coupling in Kemmer-Duffin-Petiau theory,” Physics Letters A, vol. 244, no. 5, pp. 329–337, 1998.

J. T. Lunardi, B. M. Pimentel, R. G. Teixeira, and J. S. Valverde, “Remarks on Duffin-Kemmer-Petiau theory and gauge invariance,” Physics Letters A, vol. 268, no. 3, pp. 165–173, 2000.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “On the nonminimal vector coupling in the Duffin-Kemmer-Petiau theory and the confinement of massive bosons by a linear potential,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 5, Article ID 055306, 2010.

B. C. Clark, S. Hama, G. R. Kälbermann, R. L. Mercer, and L. Ray, “Relativistic impulse approximation for meson-nucleus scattering in the kemmer-duffin-petiau formalism,” Physical Review Letters, vol. 55, no. 6, pp. 592–595, 1985.

G. Kälbermann, “Kemmer-Duffin-Petiau equation approach to pionic atoms,” Physical Review C, vol. 34, no. 6, pp. 2240–2243, 1986.

R. E. Kozack, B. C. Clark, S. Hama et al., “Relativistic deuteron-nucleus scattering in the Kemmer-Duffin-Petiau formalism,” Physical Review C, vol. 37, no. 6, pp. 2898–2901, 1988.

R. E. Kozack, B. C. Clark, S. Hama, V. K. Mishra, R. L. Mercer, and L. Ray, “Spin-one Kemmer-Duffin-Petiau equations and intermediate-energy deuteron-nucleus scattering,” Physical Review C, vol. 40, no. 5, pp. 2181–2194, 1989.

V. K. Mishra, S. Hama, B. C. Clark, R. E. Kozack, R. L. Mercer, and L. Ray, “Implications of various spin-one relativistic wave equations for intermediate-energy deuteron-nucleus scattering,” Physical Review C, vol. 43, no. 2, pp. 801–811, 1991.

L. J. Kurth, B. C. Clark, E. D. Cooper et al., “Relativistic impulse approximation treatment of the elastic scattering of 400 MeV on Si²⁸,” Physical Review C, vol. 50, no. 5, pp. 2624–2626, 1994.

R. C. Barrett and Y. Nedjadi, “Meson-nuclear interactions in the Duffin-Kemmer-Petiau formalism,” Nuclear Physics A, vol. 585, no. 1-2, pp. 311–312, 1995.

L. J. Kurth, B. C. Clark, E. D. Cooper et al., “Meson-nucleus scattering in the KDP formalism,” Nuclear Physics A, vol. 585, no. 1-2, pp. 335–336, 1995.

S. Ait-Tahar, J. S. Al-Khalili, and Y. Nedjadi, “A relativistic model for α-nucleus elastic scattering,” Nuclear Physics A, vol. 589, no. 2, pp. 307–319, 1995.

B. C. Clark, R. J. Furnstahl, L. Kurth Kerr, J. Rusnak, and S. Hama, “Pion-nucleus scattering at medium energies with densities from chiral effective field theories,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 427, no. 3-4, pp. 231–234, 1998.

Y. Nedjadi, S. Ait-Tahar, and R. C. Barrett, “An extended relativistic quantum oscillator for particles,” Journal of Physics A: Mathematical and General, vol. 31, no. 16, pp. 3867–3874, 1998.

D. A. Kulikov, R. S. Tutik, and A. P. Yaroshenko, “An alternative model for the Duffin-Kemmer-Petiau oscillator,” Modern Physics Letters A: Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 20, no. 1, pp. 43–49, 2005.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Comment on solutions of the duffin-kemmer- petiau equation for a pseudoscalar potential step in (1 + 1) dimensions,” Canadian Journal of Physics, vol. 87, no. 8, pp. 857–859, 2009.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Absence of Klein's paradox for massive bosons coupled by nonminimal vector interactions,” Canadian Journal of Physics, vol. 87, no. 11, pp. 1185–1189, 2009.

L. P. de Oliveira and A. S. de Castro, “Scattering and bound states of spin-0 particles in a nonminimal vector double-step potential,” Canadian Journal of Physics, vol. 90, no. 5, pp. 481–486, 2012.

L. B. Castro, T. R. Cardoso, and A. S. de Castro, “On the scattering of massive spinless bosons by a nonminimal vector smooth step potential,” Nuclear Physics B, vol. 199, no. 1, pp. 207–210, 2010.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Confining solutions of massive spin-0 bosons by a linear nonminimal vector coupling in the Duffin-Kemmer-Petiau theory,” Nuclear Physics B, vol. 199, no. 1, pp. 203–206, 2010.

L. B. Castro and A. S. de Castro, “Spinless bosons embedded in a vector Duffin-Kemmer-Petiau oscillator,” Physics Letters A, vol. 375, no. 27, pp. 2596–2600, 2011.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “An effective singular oscillator for Duffin-Kemmer-Petiau particles with a nonminimal vector coupling: a two-fold degeneracy,” Journal of Physics A: Mathematical and Theoretical, vol. 45, no. 7, Article ID 075302, 2012.

H. Hassanabadi, Z. Molaee, M. Ghominejad, and S. Zarrinkamar, “Spin-one DKP equation in the presence of Coulomb and harmonic oscillator interactions in -dimension,” Advances in High Energy Physics, vol. 2012, Article ID 489641, 10 pages, 2012.

Z. Molaee, M. Ghominejad, H. Hassanabadi, et al., “S-wave solutions of spin-oneDKP equation for a deformed Hulthén potential in (1 + 3) dimensions,” European Physical Journal Plus, vol. 127, p. 116, 2012.

H. Hassanabadi, Z. Molaee, and S. Zrrinkamar, “Relativistic vector bosons under pöschl-teller double-ring-shaped coulomb potential,” Modern Physics Letters A, vol. 27, no. 39, Article ID 1250228, 2012.

H. Hassanabadi, Z. Molaee, M. Ghominejad, and S. Zarrinkamar, “Duffin-Kemmer-Petiau equation with a hyperbolical potential in (2 + 1) dimensions for spin-one particles,” Few-Body Systems, vol. 54, pp. 1765–1772, 2012.

H. Hassanabadi, Z. Molaee, and A. Boumali, “Exact solutions of the Duffin-Kemmer-Petiau equation with a pseudoharmonic potential in the presence of a magnetic field in (1 + 2) dimensions,” Foundations of Physics, vol. 43, no. 2, pp. 225–235, 2013.

Z. Molaee, M. K. Bahar, F. Yasuk, and H. Hassanabadi, “Solutions of the Duffin-Kemmer-Petiau equation in the presence of Hulthén potential in (1 + 2) dimensions for unity spin particles using the asymptotic iteration method,” Chinese Physics B, vol. 22, no. 6, Article ID 060306, 2013.

S. Hassanabadi, A. A. Rajabi, S. Zarrinkamar, and H. Hassanabadi, “DKP equation under a vector Yukawa-type potential,” Physics of Particles and Nuclei Letters, vol. 10, no. 1, pp. 28–32, 2013.

H. Hassanabadi, M. Kamali, and B. H. Yazarloo, “Spin-one Duffin-Kemmer-Petiau equation in the presence of Manning-Rosen potential plus a ring-shaped-like potential,” Canadian Journal of Physics, vol. 92, no. 3, pp. 1–7, 2014.

A. Moradzadeh and H. Hassanabadi, “Quasi-Maxwell equation for spin-1 particles,” International Journal of Modern Physics E, vol. 23, Article ID 1450007, 2014.

M. K. Bahar and F. Yasuk, “Relativistic solutions for the spin-1 particles in the two-dimensional Smorodinsky–Winternitz potential,” Annals of Physics, vol. 344, pp. 105–117, 2014.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Effects due to a scalar coupling on the particle-antiparticle production in the Duffin-Kemmer-Petiau theory,” International Journal of Theoretical Physics, vol. 49, no. 1, pp. 10–17, 2010.

T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Inconsistencies of a purported probability current in the Duffin-Kemmer-Petiau theory,” Physics Letters A, vol. 372, no. 38, pp. 5964–5967, 2008.

Y. Nedjadi and R. C. Barrett, “Solution of the central field problem for a Duffin-Kemmer-Petiau vector boson,” Journal of Mathematical Physics, vol. 35, no. 9, pp. 4517–4533, 1994.

E. L. Hill, “The theory of vector spherical harmonics,” American Journal of Physics, vol. 22, no. 6, pp. 211–214, 1954.