Effects of Kaluza-Klein Theory and Potential on a Generalized Klein-Gordon Oscillator in the Cosmic String Space-Time

Hindawi Advances in High Energy Physics Volume 2020, Article ID 8107025, 15 pages https://doi.org/10.1155/2020/8107025

Research Article Effects of Kaluza-Klein Theory and Potential on a Generalized Klein-Gordon Oscillator in the Cosmic String Space-Time

Faizuddin Ahmed

Ajmal College of Arts and Science, Dhubri, 783324 Assam, India

Correspondence should be addressed to Faizuddin Ahmed; [email protected]

Received 19 April 2020; Revised 15 May 2020; Accepted 22 May 2020; Published 10 July 2020

Academic Editor: Shi-Hai Dong

Copyright © 2020 Faizuddin Ahmed. 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.

In this paper, we solve a generalized Klein-Gordon oscillator in the cosmic string space-time with a scalar potential of Cornell-type within the Kaluza-Klein theory and obtain the relativistic energy eigenvalues and eigenfunctions. We extend this analysis by replacing the Cornell-type with Coulomb-type potential in the magnetic cosmic string space-time and analyze a relativistic analogue of the Aharonov-Bohm eﬀect for bound states.

1. Introduction Bohm effect has been investigated in the context of Kaluza- Klein theory by several authors [18, 29–34], and the geomet- A unified formulation of Einstein’s theory of gravitation and ric quantum phase in graphene [35]. It is well-known in con- theory of electromagnetism in four-dimensional space-time densed matter [36–40] and in the relativistic quantum was first proposed by Kaluza [1] by assuming a pure gravita- systems [41, 42] that when there exists dependence of the tional theory in five-dimensional space-time. The so-called energy eigenvalues on geometric quantum phase [21], then, cylinder condition was later explained by Klein when the persistent current arises in the systems. The studies of persis- extra dimension was compactified on a circle S1 with a tent currents have explored systems that deal with the Berry microscopic radius [2], where the spatial dimension becomes phase [43, 44], the Aharonov-Anandan quantum phase [45, five-dimensional. The idea behind introducing additional 46], and the Aharonov-Casher geometric quantum phase space-time dimensions has found application in quantum [47–50]. Investigation of magnetization and persistent cur- field theory, for instance, in string theory [3]. There are rents of mass-less Dirac Fermions confined in a quantum studies on Kaluza-Klein theory with torsion [4, 5], in the dot in a graphene layer with topological defects was studied Grassmannian context [6–8], in Kähler fields [9], in the pres- in [51]. ence of fermions [10–12], and in the Lorentz-symmetry Klein-Gordon oscillator theory [52, 53] was inspired by violation (LSV) [13–15]. Also, there are investigations in the Dirac oscillator [54]. This oscillator field is used to study space-times with a topological defect in the context of the spectral distribution of energy eigenvalues and eigenfunc- Kaluza-Klein theory, for example, the magnetic cosmic string tions in 1 − d version of Minkowski space-times [55]. Klein- [16] (see also [17]), and magnetic chiral cosmic string [18] in Gordon oscillator was studied by several authors, such as in five-dimensions. the cosmic string space-time with external fields [56], with Aharonov-Bohm effect [19–21] is a quantum mechanical Coulomb-type potential by two ways: (i) modifying the mass phenomena that has been investigated in several branches of term m ⟶ m + SðrÞ [57], and (ii) via the minimal coupling physics, such as in, graphene [22], Newtonian theory [23], [58] in addition to a linear scalar potential, in the back- bound states of massive fermions [24], scattering of dislo- ground space-time generated by a cosmic string [59], in cated wave-fronts [25], torsion effect on a relativistic the Gödel-type space-times under the influence of gravita- position-dependent mass system [26, 27], and nonminimal tional fields produced by topological defects [60], in the Lorentz-violating coupling [28]. In addition, Aharonov- Som-Raychaudhuri space-time with a disclination parameter 2 Advances in High Energy Physics

[61], in noncommutative (NC) phase space [62], in ð1+2Þ Various potentials have been used to investigate the -dimensional Gürses space-time background [63], and in bound state solutions to the relativistic wave-equations. ð1+2Þ-dimensional Gürses space-time background subject Among them, much attention has given on Coulomb-type to a Coulomb-type potential [64]. The relativistic quantum potential. This kind of potential has widely used to study var- effects on oscillator field with a linear confining potential ious physical phenomena, such as in, the propagation of were investigated in [65]. gravitational waves [73], the confinement of quark models We consider a generalization of the oscillator as [74], molecular models [75], position-dependent mass sys- described in Refs. [34, 64] for the Klein-Gordon. This gener- tems [76–78], and relativistic quantum mechanics [57–59]. alization is introduced through a generalized momentum The Coulomb-type potential is given by operator where the radial coordinate r is replaced by a general function f ðrÞ. To author’s best knowledge, such a new η SrðÞ= c : ð5Þ coupling was first introduced by Bakke et al. in Ref. [41] r and led to a generalization of the Tan-Inkson model of a η fi two-dimensional quantum ring for systems whose energy where c is the Coulombic con ning parameter. levels depend on the coupling’s control parameters. Based Another potential that we are interested here is the on this, a generalized Dirac oscillator in the cosmic string Cornell-type potential. The Cornell potential, which consists space-time was studied by Deng et al. in Ref. [66] where the of a linear potential plus a Coulomb potential, is a particular four-momentum pμ is replaced with its alternative pμ + mω case of the quark-antiquark interaction, one more harmonic βf μðxμÞ. In the literature, f μðxμÞ has chosen similar to poten- type term [79]. The Coulomb potential is responsible by the interaction at small distances, and the linear potential leads tials encountered in quantum mechanics (Cornell-type, fi exponential-type, singular, Morse-type, Yukawa-like etc.). A to the con nement. Recently, the Cornell potential has been generalized Dirac oscillator in ð2+1Þ-dimensional world studied in the ground state of three quarks [80]. However, was studied in [67]. Very recently, the generalized K-G oscil- this type of potential is worked on spherical symmetry; in lator in the cosmic string space-time in [68] and noninertial cylindrical symmetry, which is our case, this type of potential effects on a generalized DKP oscillator in the cosmic string is known as Cornell-type potential [60]. This type of interac- space-time in [69] were studied. tion has been studied in [60, 65, 70, 81]. Given this, let us The relativistic quantum dynamics of a scalar particle of consider this type of potential mass m with a scalar potential SðrÞ [70, 71] is described by η the following Klein-Gordon equation: SrðÞ= η r + c , ð6Þ L r 1 pffiffiffiffiffiffi μν 2 pffiffiffiffiffiffi ∂μðÞ−gg ∂ν − ðÞm + S Ψ =0, ð1Þ where η , η are the confining potential parameters. −g L c The aim of the present work is to analyze a relativistic μν analogue of the Aharonov-Bohm effect for bound states with g is the determinant of metric tensor with g its – fi [19 21] for a relativistic scalar particle with potential in the inverse. To couple Klein-Gordon eld with oscillator [52, context of Kaluza-Klein theory. First, we study a relativistic 53], following change in the momentum operator is consid- scalar particle by solving the generalized Klein-Gordon oscil- ered as in [56, 72]: lator with a Cornell-type potential in the five-dimensional

! ! ! cosmic string space-time. Secondly, by using the Kaluza- p ⟶ p + imω r , ð2Þ Klein theory [1–3], a magnetic flux through the line- element of the cosmic string space-time is introduced and ! where ω is the oscillatory frequency of the particle and r = r ̂r thus wrote the generalized Klein-Gordon oscillator in the where r being distance from the particle to the string. To five-dimensional space-time. In the later case, a Coulomb- generalized the Klein-Gordon oscillator, we adopted the type potential by modifying the mass term m ⟶ m + SðrÞ idea considered in Refs. [34, 64, 66, 68, 69] by replacing is introduced which was not studied earlier. Then, we show r ⟶ f ðrÞ as that the relativistic bound states solutions can be achieved, where the relativistic energy eigenvalues depend on the geo- Xμ =0,ðÞfrðÞ,0,0,0 : ð3Þ metric quantum phase [21]. Due to this dependence of the relativistic energy eigenvalue on the geometric quantum ! ! So we can write p ⟶ p + imω f ðrÞ ̂r, and we have phase, we calculate the persistent currents [36, 37] that arise 2 ⟶ ! + ω ! ! − ω ! in the relativistic system. p ðp im f ðrÞ r Þðp im f ðrÞ r Þ. Therefore, the This paper comprises as follows: In section 2, we study a generalized Klein-Gordon oscillator equation: generalized Klein-Gordon oscillator in the cosmic string 1 ÀÁffiffiffiffiffiffi background within the Kaluza-Klein theory with a Cornell- ffiffiffiffiffiffi ∂ + ω p− μν ∂ − ω − + 2 Ψ =0, p− μ m Xμ fgg g ðÞν m Xν ðÞm S type scalar potential; in section 3, a generalized Klein- g Gordon oscillator in the magnetic cosmic string in the ð4Þ Kaluza-Klein theory subject to a Coulomb-type scalar potential and obtain the energy eigenvalues and eigenfunctions, where Xμ is given by Eq. (3). and the conclusion one in section 4. Advances in High Energy Physics 3

2. Generalized Klein-Gordon Oscillator in b frðÞ= ar+ , a, b >0: ð11Þ Cosmic String Space-Time with a Cornell- r Type Potential in Kaluza-Klein Theory Substituting the function (11) and Cornell potential (6) The purpose of this section is to study the Klein-Gordon into the Eq. (9), we obtain the following equation: equation in cosmic string space-time with the use of "# Kaluza-Klein theory with interactions. The ﬁrst study of the 2 1 2 2 η d + d + λ − Ω2 2 − j − m c − 2 η ψ =0, topological defects within the Kaluza-Klein theory was car- 2 r 2 m r ðÞr dr r dr r r L ried out in [16]. The metric corresponding to this geometry can be written as, ð12Þ

2 2 2 2 2 2 2 2 ds = −dt + dr + α r dϕ + dz + dx , ð7Þ where where t is the time coordinate, x is the coordinate associated λ = 2 − 2 − 2 − 2 − 2 ω − 2 2 ω2 − 2 η η , E k q m m a m ab L c with the fifth additional dimension, and ðr, ϕ, zÞ are cylindri- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cal coordinates. These coordinates assume the ranges −∞ < Ω = 2 ω2 2 + η2, m a L ðt, zÞ < ∞, 0 ≤ r < ∞, 0 ≤ ϕ ≤ 2 π, 0

− + ϕ+ + − 1/2 ρ+θ ρ ΨðÞt, r, ϕ, z, x = ei ðÞEt l kz qx ψðÞr , ð9Þ ψρðÞ= ρj e ðÞðÞHðÞρ : ð16Þ where E is the total energy, l = 0, ±1,±2::, and k, q are Substituting the solution Eq. (16) into the Eq. (14), we constants. obtain Substituting the above ansatz into the Eq. (8), we get the following equation for ψðrÞ: γ β ″ ρ + − θ − 2 ρ ′ ρ + − + Θ ρ =0, 17 " H ðÞ ρ H ðÞ ρ HðÞ ð Þ 2 2 d 1 d 2 2 2 l frðÞ + + E − k − q − − m ω f ′ðÞr + dr2 r dr α2 r2 r where # 2 2 2 2 − m ω f ðÞr − ðÞm + SrðÞ ψðÞr =0: γ =1+2j, θ2 10 Θ = ζ + − 21+ , ð Þ 4 ðÞj ð18Þ ð Þ θ We choose the function f r a Cornell-type given by [34, β = η + ðÞ1+2j : 64, 66, 69] 2 4 Advances in High Energy Physics

7 10

6 8 5 ) ) 6 c

4 L � � ( ( 1,1 1,1 E 3 E 4

2 2 1

0 0 0246810 0246810

�c �L

Figure ======η =1 α =0:5 ω =0:5: Figure ======η =1 α =0:5 ω =0:5 1: n l k M q a b L , , 2: n l k M q a b c , ,

15 Eq. (17) is the biconﬂuent Heun’sdiﬀerential equation [26, 27, 29–34, 58–61, 64, 65, 70, 83–87], and HðρÞ is the Heun polynomials.

The above Eq. (17) can be solved by the Frobenius 10 method. We consider the power series solution around the origin [88] (M) 1,1 E ∞ ρ = 〠 ρi 19 5 HðÞ ci ð Þ i=0

Substituting the above power series solution into the 0 Eq. (17), we obtain the following recurrence relation for the coefficients: 0 2 4 6 8 10 M 1 Figure 3: n = l = k = q = a = b = η = η =1, α =0:5, ω =0:5 c +2 = ½fgβ + θ ðÞn +1 c +1 − ðÞΘ − 2 n c : c L n ðÞn +2 ðÞn +2+2j n n ð20Þ By analyzing the condition Θ =2n, we get the expression of the energy eigenvalues E , : And the various coefficients are n l λ θ2 η θ 1 + − 21+ =2 ⇒ 2 = 2 + 2 + 2 +2Ω Ω 4 ðÞj n En,l k q m c1 = − c0, c2 = ½ðÞβ + θ c1 − Θ c0 : ð21Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! γ 2 41+ðÞj 2 +1+ l + 2 ω2 2 + η2 Á n α2 m b c The quantum theory requires that the wave function Ψ 2 2 must be normalized. The bound state solutions ψðρÞ can 2 2 m η +2m ω ab+2m ω a +2η η − L : be obtained because there is no divergence of the wave L c Ω2 ρ ⟶ 0 ρ ⟶ ∞ function at and . Since we have written ð23Þ the function HðρÞ as a power series expansion around the origin in Eq. (19). Thereby, bound state solutions We plot graphs of the above energy eigenvalues w. r. t. dif- can be achieved by imposing that the power series expan- ferent parameters. In Figure 1, the energy eigenvalues E1,1 sion (19) becomes a polynomial of degree n. Through the against the parameter η . In Figure 2, the energy eigenvalues recurrence relation (20), we can see that the power series c E1,1 against the parameter η . In Figure 3, the energy eigen- expansion (19) becomes a polynomial of degree n by L – – values E1,1 against the parameter M. In Figure 4, the energy imposing two conditions [26, 27, 29 34, 58 61, 64, 65, ω 70, 83–85]: eigenvalues E1,1 against the parameter . In Figure 5, the energy eigenvalues E1,1 against the parameter Ω. Now, we impose an additional condition c +1 =0to find Θ =2 =1,2,⋯ , =0 22 n n ðÞn cn+1 ð Þ the individual energy levels and corresponding wave functions Advances in High Energy Physics 5

12 It is noteworthy that a third-degree algebraic Eq. (24) has at least one real solution, and it is exactly this solution 10 that gives us the allowed values of the frequency for the lowest state of the system, which we do not write because its 8 expression is very long. We can note, from Eq. (24), that the possible values of the frequency depend on the quantum

( � ) 6 numbers and the potential parameter. In addition, for each 1,1

E relativistic energy level, we have a different relation of the 4 magnetic field associated to the Cornell-type potential and quantum numbers of the system fl, ng. For this reason, we 2 have labeled the parameters Ω and ω in Eqs. (24) and (25). Therefore, the ground state energy level and correspond- 0 ing wave-function for n =1are given by 0 1 2 3 4 5 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! � 2 2 2 2 2 l 2 2 2 2 E = k + q + m +2Ω1, 2+ + m ω b + η Figure ======η = η = =1 α =0:5 1,l l α2 c 4: n l k q a b c L M , 2 η2 2 2 m L +2m ω ab+2m ω1, a +2η η − , 1,l l L c Ω2 1,l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/2 l2/α2+m2 ω2 b2+η2 −ðÞ1/2 2 m η /Ω +ρ ρ ψ = ρ 1,l c ðÞL 1,l + ρ , 8 1,l e ðÞc0 c1 ð26Þ

6 where ( � ) 1,1

E 4 2 3 1 6 2 η η 7 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim c − m L : c1 1/2 4ÀÁ 5 c0 2 Ω 2 2 2 2 2 2 Ω 1,l 1+2 /α + ω + η 1,l l m 1,l b c 27 0 ð Þ 0 2 4 6 8 10 � Then, by substituting the real solution of Eq. (25) into the Eqs. (26)–(27), it is possible to obtain the allowed values Figure 5: n = l = k = q = a = b = η = η = M =1, α =0:5, ω =0:5 c L of the relativistic energy for the radial mode n =1of a position dependent mass system. We can see that the lowest =1 one by one as done in [89, 90]. As an example, for n ,we energy state defined by the real solution of the algebraic Θ =2 =0 have and c2 which implies equation given in Eq. (25) plus the expression given in Eq. (26) is defined by the radial mode n =1, instead of n =0. This 2 2 η θ 2 3 η 2 effect arises due to the presence of the Cornell-type potential c1 = c0 ⇒ − = Ω1, − Ω1, β + θ 1+2j 2 β + θ l 21+2ðÞj l in the system. 2 For α ⟶ 1, the relativistic energy eigenvalue (24) 1+j θ − ηθ Ω1, − ðÞ3+2j =0 becomes 1+2j l 8 ð24Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = 2 + 2 + 2 +2Ω +1+ 2 + 2 ω2 2 + η2 En,l k q m n l m b c Ω a constraint on the parameter 1,l. The relation given in 2 2 2 2 m η Eq. (24) gives the possible values of the parameter Ω1, that +2 ω +2 ω +2η η − L : l m ab m a L c 2 permit us to construct first degree polynomial to HðxÞ for Ω n =1. Note that its values changes for each quantum number ð28Þ Ω ⟶ Ω n and l, so we have labeled n,l. Besides, since this parameter is determined by the frequency; hence, the fre- Eq. (28) is the relativistic energy eigenvalue of scalar par- ω fi quency 1,l is so adjusted that the Eq. (24) can be satis ed, ticles via the generalized Klein-Gordon oscillator subject to a where we have simplified our notation by labeling: Cornell-type potential in the Minkowski space-time in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kaluza-Klein theory. 1 2 2 We discuss bellow a very special case of the above relativ- ω1, = Ω1, − η : ð25Þ l ma l L istic system. 6 Advances in High Energy Physics

( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Case A. Considering η =0, that is, only Coulomb-type 2 L 2 2 2 l 2 2 2 2 = η / E , =± k + q + m +2m ω an+2+ + m ω b + η potential SðrÞ c r. n l α2 c )1/2 We want to investigate the effect of Coulomb-type poten- +2 2 ω2 : tial on a scalar particle in the background of cosmic string m ab space-time in the Kaluza-Klein theory. In that case, the radial wave-equation Eq. (12) becomes ð36Þ "# 2 1 2 2 η The ground state energy levels and corresponding wave- d d 2 2 2 2 j m c =1 + + λ0 − m ω a r − − ψðÞr =0, function for n are given by dr2 r dr r2 r ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 29 2 ð Þ =± 2 + 2 + 2 +2 ω 3+ l + 2 ω2 2 + η2 E1,l k q m m 1,l a α2 m b c where )1/2 2 2 +2m ω ab , ψ1, ðÞρ 2 2 2 2 2 2 l λ0 = E − k − q − m − 2 m ω a − 2 m ω ab ð30Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l2/α2+m2 ω2 b2+η2 − ρ /2 = ρ 1,l c ðÞ + ρ , pffiffiffiffiffiffiffiffiffiffiffi e ðÞc0 c1 Transforming ρ = m ω a r into the Eq. (29), we get ð37Þ "# 2 1 λ 2 2 η 1 d d 0 2 j m where + + − ρ − − pffiffiffiffiffiffiffiffiffiffiffic ψρðÞ=0: dρ2 ρ dρ m ω a ρ2 m ω a ρ 2 η = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim c ð31Þ c1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ p ω 1+2 2/α2 + 2 ω2 2 + η2 m 1,l a l m 1,l b c 0 1 Suppose the possible solution to Eq. (31) is 1/2 B 2 C = @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 38 2 ÀÁ ð Þ ψρ = ρj −ðÞρ /2 1+2 2/α2 + 2 ω2 2 + η2 ðÞ E HðÞρ : ð32Þ l m 1,l b c 2 η2 , ω = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim c : Substituting the solution Eq. (32) into the Eq. (31), we c0 1,l ÀÁ 1+2 2/α2 + 2 ω2 2 + η2 obtain a l m 1,l b c 1+2j η~ λ0 ω H″ðÞρ + − 2 ρ H′ðÞρ + − + − 21+ðÞj HðÞρ , a constraint on the frequency parameter 1,l. ρ ρ m ω a ð33Þ Case B. We consider another case corresponds to a ⟶ 0, b ⟶ 0 and η =0, that is, a scalar quantum particle in ffiffiffiffiffiffiffiffiffiffiffi L ~η =2 η /p ω ’ ff the cosmic string background subject to a Coulomb-type where m c m a. Eq. (33) is the Heun sdi erential equation [26, 27, 29–34, 58–61, 64, 65, 70, 83–87] with scalar potential within the Kaluza-Klein theory. In that case, from Eq. (12) we obtain the following equation: HðρÞ is the Heun polynomial. Substituting the power series solution Eq. (19) into the "# ~2 Eq. (33), we obtain the following recurrence relation for coef- 1 ~ j 2 m η ψ″ðÞr + ψ′ðÞr + λ − − c ψðÞr =0: ð39Þ ficients r r2 r 1 λ0 c +2 = ~η c +1 − − 21+ðÞj − 2 n c Eq. (39) can be written as n ðÞn +2 ðÞn +2+2j n m ω a n 1 1 ÀÁ ð34Þ ″ ′ 2 ψ ðÞr + ψ ðÞr + −ξ1 r + ξ2 r − ξ3 ψðÞr =0, ð40Þ r r2 The power series solution becomes a polynomial of degree n provided [26, 27, 29–34, 58–61, 64, 65, 70, 83–85] where ÀÁ ξ = −λ~ = − 2 − 2 − 2 − 2 , λ0 1 E k q m − 21+ðÞj =2n ðÞn =1,2,⋯ c +1 =0: ð35Þ m ω a n ξ = −2 η , 2 m c ð41Þ 2 Using the first condition, one will get the following ~2 l 2 ξ3 = j = + η : energy eigenvalues of the relativistic system: α2 c Advances in High Energy Physics 7

Comparing the Eq (40) with Eq. (1) in the Appendix, Eq. (45) corresponds to the relativistic energy eigen- we get value of a scalar particle subject to a Coulomb-type scalar potential in the Minkowski space-time within the Kaluza- α1 =1, Klein theory. α =0, 2 3. Generalized Klein-Gordon Oscillator in the

α3 =0, Magnetic Cosmic String with a Coulomb- Type Potential in Kaluza-Klein Theory α4 =0,

α5 =0, Let us consider the quantum dynamics of a particle moving in the magnetic cosmic string background. In the Kaluza- α6 = ξ1, Klein theory [1, 2, 18], the corresponding metrics with fl Φ α = −ξ , 42 Aharonov-Bohm magnetic ux passing along the symme- 7 2 ð Þ try axis of the string assumes the following form α8 = ξ3, 2 2 2 2 2 2 2 2 Φ α9 = ξ1, ds = −dt + dr + α r dϕ + dz + dx + dϕ ð46Þ pffiffiffiffi 2 π α10 =1+2 ξ3, pffiffiffiffi with cylindrical coordinates are used. The quantum dynam- α11 =2 ξ1, pffiffiffiffi ics is described by the Eq. (4) with the following change in the inverse matrix tensor gμν, α12 = ξ3, pffiffiffiffi 0 1 α13 = − ξ1: −10 00 0 B C B C B 01 00 0 C The energy eigenvalues using Eqs. (41)–(42) into the B C B 1 Φ C Eq. (8) in the Appendix is given by μν B 00 0 − C g = B α2 2 2 πα2 2 C: ð47Þ B r r C vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 00 01 0 C u B C u η2 2 2 @ 2 A =± u1 − c + k + q , Φ Φ En,l m t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ 2 2 2 00− 01+ + 2/α2 + η2 + 1/2 m m 2 πα2 r2 4 π2 α2 r2 n l c ð43Þ By considering the line element (46) into the Eq. (4), we obtain the following differential equation: " 2 where n =0,1,2,:: is the quantum number associated with 2 2 1 1 Φ 2 2 −∂ + ∂ + ∂ + ∂ϕ − ∂ + ∂ + ∂ the radial modes, l = 0, ±1,±2,: are the quantum number t r r r α2 r2 2 π x z x associated with the angular momentum operator, k and q frðÞ 2 2 2 2 are arbitrary constants. Eq. (43) corresponds to the relativ- − m ω f ′ðÞr + − m ω f ðÞr − ðÞm + SrðÞ Ψ =0: istic energy eigenvalues of a free-scalar particle subject to a r Coulomb-type scalar potential in the background of cos- ð48Þ mic string within the Kaluza-Klein theory. The corresponding radial wave-function is given by Since the space-time is independent of t, ϕ, z, x, substituting the ansatz (9) into the Eq. (48), we get the fol- ~ ~ ψ = ∣ ∣ j/2 −ðÞr/2 ðÞj lowing equation: n,lðÞr N r e Ln ðÞr ÀÁpffiffiffiffiffiffiffiffiffiffiffiffi ð44Þ " pffiffiffiffiffiffiffiffiffiffiffiffi 2/α2+η2 2 1/2 2/α2+η2 − /2 l c 1 = ∣ ∣ l c ðÞr : 2 2 2 l frðÞ N r e Ln ðÞr ψ″ðÞr + ψ′ðÞr + E − k − q − ef f − m ω f ′ðÞr + r r2 r # Here, ∣N ∣ is the normalization constant and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2/α2 +η2 − ω − + ψ =0, ð ðl Þ c Þ m f ðÞr ðÞm SrðÞ ðÞr Ln ðrÞ is the generalized Laguerre polynomial. For α ⟶ 1, the relativistic energy eigenvalues Eq. (43) 49 becomes ð Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the effective angular quantum number u η2 2 2 u c k q E , =±m u1 − qffiffiffiffiffiffiffiffiffiffiffiffiffi + + : ð45Þ n l t 2 m2 m2 1 Φ + 2 + η2 + 1/2 = − q : 50 n l c lef f α l 2 π ð Þ 8 Advances in High Energy Physics

3.5 Substituting the power series solution Eq. (19) into the Eq. (55), we obtain the following recurrence relation for the 3.0 coeﬃcients: 2.5 2.0 1 λ0 )

c = ~η − − 2 − 2 χ − 2 : cn+2 cn+1 n cn ( � ðÞn +2 ðÞn +2+2χ m ω a 1,1 1.5 E ð56Þ 1.0

0.5 The power series becomes a polynomial of degree n by 0.0 imposing the following conditions [26, 27, 29–34, 58–61, 01234564, 65, 70, 83–85]

�c

Figure 6: n = l = k = q = a = b = M =1, α =0:5, ω =0:5, Φ = π λ0 c +1 =0, − 2 − 2 χ =2n ðÞn =1,2,⋯ ð57Þ n m ω a Substituting the function (11) into the Eq. (49) and using Coulomb-type potential (5), the radial wave-equation becomes By analyzing the second condition, we get the following energy eigenvalues E , : "#n l 2 1 χ2 2 η d + d + λ − 2 ω2 2 2 − − m c ψ =0, 2 0 m a r 2 ðÞr 2 2 2 2 dr r dr r r E , = k + q + m +2m ω a n l rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 ð51Þ ðÞl − q Φ/2 π 2 2 2 2 +2+ + ω + η ð58Þ Á n α2 m b c where 2 2 +2m ω ab: 2 2 2 2 2 2 λ0 = E − k − q − m − 2 m ω a − 2 m ω ab, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 52 − Φ/2 π 2 ð Þ Eq. (58) is the energy eigenvalues of a generalized Klein- χ = ðÞl q + 2 ω2 2 + η2: α2 m b c Gordon oscillator in the magnetic cosmic string with a Coulomb-type scalar potential in the Kaluza-Klein theory. pffiffiffiffiffiffiffiffiffiffiffi Observed that the relativistic energy eigenvalues Eq. (58) Transforming ρ = m ω a r into the Eq. (51), we get depend on the Aharonov-Bohm geometric quantum phase Φ + Φ = Φ Φ "#[21]. Thus, we have that En,lð 0Þ En,l∓τð Þ where 0 2 2 =± 2 π/ τ τ =0,1,2:: d 1 d λ0 2 χ ~η ð qÞ with . This dependence of the relativ- + + − ρ − − ψρðÞ=0, ð53Þ Φ dρ2 ρ dρ m ω a ρ2 ρ istic energy eigenvalue on the geometric quantum phase gives rise to a relativistic analogue of the Aharonov-Bohm effect for bound states [18–21, 26, 29]. pffiffiffiffiffiffiffiffiffiffiffi where ~η =2m η / m ω a. We plot graphs of the above energy eigenvalues w. r. t. c ff Suppose the possible solution to Eq. (53) is di erent parameters. In Figure 6, the energy eigenvalues η E1,1 against the parameter c. In Figure 7, the energy eigen- 2 values E1,1 against the parameter M. In Figure 8, the energy ψρ = ρχ −ðÞρ /2 ρ 54 ðÞ e HðÞ ð Þ eigenvalues E1,1 against the parameter ω. In Figure 9, the energy eigenvalues E1,1 against the parameter Φ. Substituting the solution Eq. (54) into the Eq. (53), we The ground state energy levels and corresponding wave- obtain function for n =1are given by 1+2χ ~η λ ″ ρ + − 2 ρ ′ ρ + − + 0 − 21+χ ρ : 2 2 2 2 H ðÞ H ðÞ ðÞHðÞ E = k + q + m +2m ω1, a ρ ρ m ω a 1,l rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil ! 55 − Φ/2 π 2 ð Þ 3+ ðÞl q + 2 ω2 2 + η2 Á α2 m b c ð59Þ +2 2 ω2 , Eq. (55) is the second-order Heun’sdifferential equation m 1,l ab – – – ρ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 [26, 27, 29 34, 58 61, 64, 65, 70, 83 87] with Hð Þ is the l−q Φ/2 π 2/α2+m2 ω2 b2+η2 − ρ /2 ψ ρ = ρ ðÞ 1,l c ðÞ + ρ , Heun polynomial. 1,lðÞ e ðÞc0 c1 Advances in High Energy Physics 9

3 6

2 (M) 4 ( Ф ) 1,1 1,1 E E

2 1

0 0 0 1 2 3 4 5 0 5 10 15 20 25 30 M Ф

Figure ======η =1 α =0:5 ω =0:5 Φ = π Figure ======η = =1 α =0:5 ω =0:5 7: n l k q a b c , , , 9: n l k q a b c M , ,

For α ⟶ 1, the energy eigenvalues (58) becomes

12 2 2 2 2 E = k + m + q +2m ω a n,l 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 Φ 2 @ +2+ − q + 2 ω2 2 + η2A 61 Á n l 2 π m b c ð Þ 8 2 2 +2m ω ab: 6 ( � ) 1,1 E Eq. (61) is the relativistic energy eigenvalue of the gener- 4 alized Klein-Gordon oscillator field with a Coulomb-type scalar potential with a magnetic flux in the Kaluza-Klein the- 2 ory. Observed that the relativistic energy eigenvalue Eq. (61) depend on the geometric quantum phase [21]. Thus, we have 0 Φ + Φ = Φ Φ =± 2 π/ τ that En,lð 0Þ En,l∓τð Þ where 0 ð qÞ with 0 1 2 3 4 5 τ =0,1,2::. This dependence of the relativistic energy eigen- � value on the geometric quantum phase gives rise to an ff ff Figure 8: n = l = k = q = a = b = η = M =1, α =0:5, Φ = π analogous e ect to the Aharonov-Bohm e ect for bound c states [18–21, 26, 29]. where Case A. We discuss below a special case corresponds to b ⟶ 0, a ⟶ 0, that is, a scalar quantum particle in a mag- 2 η netic cosmic string background subject to a Coulomb-type = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim c c1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ scalar potential in the Kaluza-Klein theory. In that case, from p ω 1+2 − Φ/2 π 2/α2 + 2 ω2 2 + η2 m 1,l a ðÞl q m 1,l b c Eq. (51), we obtain the following equation: 0 1 1/2 B 2 C 1 χ~2 2 η = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ψ″ + ψ′ + λ~ − − m c ψ =0, 62 @ ÀÁA c0 ðÞr ðÞr 2 ðÞr ð Þ 1+2 − Φ/2 π 2/α2 + 2 ω2 2 + η2 r r r ðÞl q m 1,l b c 2 m η2 where ω = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 1,l ÀÁ 1+2 − Φ/2 π 2/α2 + 2 ω2 2 + η2 a ðÞl q m 1,l b c rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − Φ/2 π 2 ð60Þ λ~ = 2 − 2 − 2 − 2, χ~ = ðÞl q + η2: 63 E k q m 0 α2 c ð Þ

ω a constraint on the physical parameter 1,l. The above Eq. (62) can be written as Eq. (59) is the ground states energy eigenvalues and corresponding eigenfunctions of a generalized Klein-Gordon oscil- 1 1 ÀÁ lator in the presence of Coulomb-type scalar potential in a ″ ′ 2 ψ ðÞr + ψ ðÞr + −ξ1 r + ξ2 r − ξ3 ψðÞr =0, ð64Þ magnetic cosmic string space-time in the Kaluza-Klein theory. r r2 10 Advances in High Energy Physics where For α ⟶ 1, the energy eigenvalues (66) becomes vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ u 2 2 2 ξ1 = −λ, u η k q =± u1 − c + + , En,l m t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 65 2 2 m m ξ2 = −2 m η , ð Þ + − Φ/2 π + + 1/2 c n ðÞl q kc ξ = χ~2: 3 0 ð68Þ Following the similar technique as done earlier, we get which is similar to the energy eigenvalue obtained in [30] (see the following energy eigenvalues E , : n l Eq. (12) in [30]). Thus, we can see that the cosmic string α modify the relativistic energy eigenvalue (66) in comparison vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u η2 k2 q2 to those results obtained in [30]. =± u1 − c + + , En,l m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Observe that the relativistic energy eigenvalues Eq. (66) t m2 m2 + 1/α2 − Φ/2 π 2 + η2 + 1/2 α n ðÞðÞl q c depend on the cosmic string parameter , the magnetic fl Φ η quantum ux , and potential parameter c. We can see that ð66Þ Φ + Φ = Φ Φ =± 2 π/ τ τ =0, En,lð 0Þ En,l∓τð Þ where 0 ð qÞ with 1, ::. This dependence of the relativistic energy eigenvalues where n =0,1,2,:: is the quantum number associated with on the geometric quantum phase gives rise to a relativistic ff radial modes, l = 0, ±1, ±2,: ⋯ are the quantum number analogue of the Aharonov-Bohm e ect for bound states – associated with the angular momentum, k and q are con- [18 21, 26, 29]. stants. Eq. (66) corresponds to the relativistic energy levels 3.1. Persistent Currents of the Relativistic System. By follow- for a free-scalar particle subject to Coulomb-type scalar ing [36–38], the expression for the total persistent currents potential in the background of magnetic cosmic string in a is given by Kaluza-Klein theory. The radial wave-function is given by = 〠 , 69 I In,l ð Þ ~ n,l χ~ /2 − /2 j ψ = ∣ ∣ 0 ðÞr ðÞ n,lðÞr N r e Ln ðÞr ÀÁ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − Φ/2 π 2/α2 +η2 where 1/2, − Φ/2 π 2/α2 +η2 − /2 ðÞðÞl q c ðÞðÞl q c ðÞr = ∣N∣ r e Ln ðÞr : ∂ ð67Þ = − En,l 70 In,l ∂Φ ð Þ ∣ ∣ pHere,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN is the normalization constant, and − Φ/2 π 2/α2 +η2 is called the Byers-Yang relation (36). ð ððl q Þ Þ c Þ Ln ðrÞ is the generalized Laguerre Therefore, the persistent current that arises in this relativ- polynomial. istic system using Eq. (58) is given by

∂ ω ∂ χ/∂ Φ = − En,l = ∓ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim a ðÞ , 71 In,l ∂Φ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ ð Þ 2 + 2 + 2 +2 2ω2 +2 ω +2+ − Φ/2 π 2/α2 + 2ω2 2 + η2 k q m m ab m an ðÞl q m b c

where Similarly, for the relativistic system discussed in Case A in this section, this current using Eq. (66) is given by ∂ χ qlðÞ− q Φ/2 π = − qÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð72Þ ∂ Φ 2 α2 π − Φ/2 π 2/α2 + 2 ω2 2 + η2 ðÞl q m b c

η2 − Φ/2 π =± mq c ðÞl q In,l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ3 qÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 πα2 + 1/2 + − Φ/2 π 2/α2 + η2 − Φ/2 π 2/α2 + η2 n ðÞl q c ðÞl q c 1 ð73Þ × sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ÀÁ 1 − η2/ + 1/α2 − Φ/2 π 2 + η2 + 1/2 + 2/ 2 + 2/ 2 c n ðÞðÞl q c k m ðÞq m Advances in High Energy Physics 11

For α ⟶ 1, the persistent currents expression given by energy eigenvalues obtained there depend on the global Eq. (73) reduces to the result obtained in Ref. [30]. Thus, parameters characterizing these space-times and the gravita- we can see that the presence of the cosmic string parameter tional analogue to the Aharonov-Bohm effect for bound modifies the persistent currents Eq. (73) in comparison to states [18–21, 26, 29] of a scalar particle was analyzed. those results in Ref. [30]. In this work, we have investigated the relativistic quan- By introducing a magnetic flux through the line element tum dynamics of a scalar particle interacting with gravita- of the cosmic string space-time in five dimensions, we see tional fields produced by topological defects via the Klein- that the relativistic energy eigenvalue Eq. (58) depend on Gordon oscillator of the Klein-Gordon equation in the pres- the geometric quantum phase [21] which gives rise to a rel- ence of cosmic string and magnetic cosmic string within the ativistic analogue of the Aharonov-Bohm effect for bound Kaluza-Klein theory with scalar potential. We have deter- states [18–21, 26, 29]. Moreover, this dependence of the rel- mined the manner in which the nontrivial topology due to ativistic energy eigenvalues on the geometric quantum the topological defects and a quantum magnetic flux modifies phase has yielded persistent currents in this relativistic the energy spectrum and wave-functions of a scalar particle. quantum system. We then have studied the quantum dynamics of a scalar particle interacting with fields by introducing a magnetic flux 4. Conclusions through the line element of a cosmic string space-time using the five-dimensional version of the general relativity. The In Ref. [30], the Aharonov-Bohm effects for bound states of a quantum dynamics in the usual as well as magnetic cosmic relativistic scalar particle by solving the Klein-Gordon equa- string cases allow us to obtain the energy eigenvalues and tion subject to a Coulomb-type potential in the Minkowski corresponding wave-functions that depend on the external space-time within the Kaluza-Klein theory were studied. parameters characterize the background space-time, a result They obtained the relativistic bound states solutions and cal- known by a gravitational analogue of the well studied culated the persistent currents. In Ref. [16], it is shown that Aharonov-Bohm effect. the cosmic string space-time and the magnetic cosmic string In section 2, we have chosen a Cornell-type function fi = + / = η + space-time can have analogue in ve dimensions. In Ref. f ðrÞ ar ðb rÞ and Cornell-type potential SðrÞ L r η / [18], quantum mechanics of a scalar particle in the back- ð c rÞ into the relativistic systems. We have solved the gen- ground of a chiral cosmic string using the Kaluza-Klein the- eralized Klein-Gordon oscillator in the cosmic string back- ory was studied. They have shown that the wave functions, ground within the Kaluza-Klein theory and obtained the the phase shifts, and scattering amplitudes associated with energy eigenvalues Eq. (23). We have plotted graphs of the particle depend on the global features of those space- the energy eigenvalues Eq. (23) w. r. t. different parameters times. This dependence represents the gravitational ana- by Figures 1–5. By imposing the additional recurrence con- ff =0 logues of the well-known Aharonov-Bohm e ect. In addi- dition cn+1 on the relativistic eigenvalue problem, for tion, they discussed the Landau levels in the presence of a example n =1, we have obtained the ground state energy cosmic string within the framework of Kaluza-Klein theory. levels and wave-functions by Eqs. (26)–(27). We have dis- η ⟶ 0 In Ref. [31], the Klein-Gordon oscillator on the curved back- cussed a special case corresponds to L and obtained ground within the Kaluza-Klein theory was studied. The the relativistic energy eigenvalues Eq. (36) of a generalized problem of the interaction between particles coupled har- Klein-Gordon oscillator in the cosmic string space-time monically with topological defects in the Kaluza-Klein theory within the Kaluza-Klein theory. We have also obtained the was studied. They considered a series of topological defects relativistic energy eigenvalues Eq. (43) of a free-scalar parti- and then treated the Klein-Gordon oscillator coupled to this cle by solving the Klein-Gordon equation with a Coulomb- background, and obtained the energy eigenvalue and corre- type scalar potential in the background of cosmic string sponding eigenfunctions in these cases. They have shown space-time in the Kaluza-Klein theory. that the energy eigenvalue depends on the global parameters In section 3, we have studied the relativistic quantum characterizing these space-times. In Ref. [32], a scalar particle dynamics of a scalar particle in the background of magnetic with position-dependent mass subject to a uniform magnetic cosmic string in the Kaluza-Klein theory with a scalar poten- field and a quantum magnetic flux, both coming from the tial. By choosing the same function f ðrÞ = ar+ ðb/rÞ and a = η / background which is governed by the Kaluza-Klein theory, Coulomb-type scalar potential SðrÞ c r, we have solved were investigated. They inserted a Cornell-type scalar poten- the radial wave-equation in the considered system and tial into this relativistic systems and determined the relativis- obtained the bound states energy eigenvalues Eq. (58). We tic energy eigenvalue of the system in this background of have plotted graphs of the energy eigenvalues Eq. (58) w. r. extra dimension. They analyzed particular cases of this sys- t. different parameters by Figures 6–9. Subsequently, the tem, and a quantum effect was observed: the dependence of ground state energy levels Eq. (59) and corresponding the magnetic field on the quantum numbers of the solutions. wave-functions Eq. (60) for the radial mode n =1by impos- =0 In Ref. [34], the relativistic quantum dynamics of a scalar ing the additional condition cn+1 on the eigenvalue prob- particle subject to linear potential on the curved background lem is obtained. Furthermore, a special case corresponds to within the Kaluza-Klein theory was studied. We have solved a ⟶ 0, b ⟶ 0 is discussed and obtained the relativistic the generalized Klein-Gordon oscillator in the cosmic string energy eigenvalues Eq. (66) of a scalar particle by solving and magnetic cosmic string space-time with a linear potential the Klein-Gordon equation with a Coulomb-type scalar within the Kaluza-Klein theory. We have shown that the potential in the magnetic cosmic string space-time in the 12 Advances in High Energy Physics

Kaluza-Klein theory. For α ⟶ 1, we have seen that the 2 α9 = α6 + α3 α7 + α3 α8, energy eigenvalues Eq. (66) reduces to the result obtained ffiffiffiffiffi in Ref. [30]. As there is an effective angular momentum α = α +2α +2pα , ⟶ = 1/α − Φ/2π 10 1 4 8 quantum number, l lef f ð Þðl q Þ; thus, the relativistic energy eigenvalues Eqs. (58) and (66) depend on pffiffiffiffiffi pffiffiffiffiffi α11 = α2 − 2 α5 +2ðÞα9 + α3 α8 , the geometric quantum phase [21]. Hence, we have that En,l Φ + Φ = Φ Φ =± 2 π/ τ τ =0,1,2,: ffiffiffiffiffi ð 0Þ En,l∓τð Þ where 0 ð qÞ with . p α12 = α4 + α8, This dependence of the relativistic energy eigenvalues on ffiffiffiffiffi ffiffiffiffiffi the geometric quantum phase gives rise to a relativistic ana- α = α − pα + α pα : logue of the Aharonov-Bohm effect for bound states [19– 13 5 ðÞ9 3 8 21, 29]. Finally, we have obtained the persistent currents by ðA:4Þ Eqs. (71)–(73) for this relativistic quantum system because of the dependence of the relativistic energy eigenvalues on A special case where α3 =0, as in our case, we find the geometric quantum phase. So in this paper, we have shown some results which are in α −1, α /α −α −1 α −1 lim ðÞ10 ðÞ11 3 10 1 − 2 α = 10 α , – Pn ðÞ3 s Ln ðÞ11 s addition to those results obtained in Refs. [18, 29 34] pre- α3→0 sents many interesting effects. ðA:5Þ −α12−ðÞα13/α3 α13 s lim 1 − α3 = : α →0ðÞs e Appendix 3 A. Brief Review of the Nikiforov-Uvarov Therefore, the wave-function from (2) becomes

(NU) Method α α α −1 ψ = 12 13 s 10 α , A:6 ðÞs s e Ln ðÞ11 s ð Þ The Nikiforov-Uvarov method is helpful in order to find eigenvalues and eigenfunctions of the Schrödinger like equa- where LðαÞðxÞ denotes the generalized Laguerre polynomial. ff n tion, as well as other second-order di erential equations of The energy eigenvalues equation reduces to physical interest. According to this method, the eigenfunctions of a second-order differential equation [94] pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi n α2 − ðÞ2 n +1 α5 +2ðÞn +1 α9 + α7 +2 α8 α9 =0: ÀÁ 2ψ α − α ψ −ξ 2 + ξ − ξ ðA:7Þ d ðÞs + ðÞ1 2 s d ðÞs + 1 s 2 s 3 ψ =0: 2 1 − α 2 2 ðÞs ds s ðÞ3 s ds s ðÞ1 − α3 s Noted that the simple Laguerre polynomial is the special ðA:1Þ case α =0of the generalized Laguerre polynomial: are given by ðÞ0 = : A:8 Ln ðÞx LnðÞx ð Þ α −α − α /α α −1, α /α −α −1 ψ = 12 1 − α 12 ðÞ13 3 ðÞ10 ðÞ11 3 10 1 − 2 α : ðÞs s ðÞ3 s Pn ðÞ3 s Data Availability ðA:2Þ No data were used to support this study. And that the energy eigenvalues equation Conflicts of Interest pffiffiffiffiffi pffiffiffiffiffi α2 n − ðÞ2 n +1 α5 +2ðÞn +1 ðÞα9 + α3 α8 ffiffiffiffiffiffiffiffiffiffi ðA:3Þ The author declares that there is no conflict of interest + − 1 α + α +2α α +2pα α =0: nnðÞ3 7 3 8 8 9 regarding publication of this paper.

The parameters α4, ⋯, α13 are obtained from the six Acknowledgments parameters α1, ⋯, α3 and ξ1, ⋯, ξ3 as follows: The author sincerely acknowledges the anonymous kind ref- 1 α = 1 − α , eree(s) for their valuable comments and suggestions and 4 2 ðÞ1 thanks the editor. 1 References α = α − 2 α , 5 2 ðÞ2 3 [1] T. H. Kaluza, “On the uniﬁcation problem in physics,” Inter- 2 α6 = α5 + ξ1, national Journal of Modern Physics D, vol. 14, no. 27, article 1870001, 2018. “ α7 =2α4 α5 − ξ2, [2] O. Klein, The atomicity of electricity as a quantum theory law,” Nature, vol. 118, no. 2971, p. 516, 1926. 2 [3] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, α8 = α4 + ξ3, vol. 1-2, Cambridge University Press, Cambridge, UK, 1987. Advances in High Energy Physics 13

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