Spin- 0 scalar particle interacts with scalar potential in the presence of magnetic field and quantum flux under the effects of KKT in 5D cosmic string space-time
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Effects of Kaluza-Klein theory on a position dependentmass system with uniform magnetic field in a magneticcosmic string space-time
Faizuddin Ahmed Ajmal College of Arts and Science, Dhubri-783324, Assam, India
Abstract
In this paper, we study the relativistic quantum dynamics of aposition mass system on curved background within the Kaluza-Kleintheory (KKT) with Cornell-type potential. We solve the Klein-Gordonequation in the magnetic cosmic string space-time background subjectto a uniform magnetic field with a Cornell-type scalar potential andobserve a relativistic analogue of Aharonov-Bohm effect for boundstates. We show the energy levels get modify due to the presenceof global parameters characterizing the space-time and break theirdegeneracy. keywords: cosmic string, Relativistic wave equation, elecromagnetic inter-actions, potential, energy spectrum, wave-functions, Aharonov-Bohm effect,special functions.
PACS Number:
The relativistic wave-equations are of current research interest for theoreticalphysicists [1, 2] including in nuclear and high energy physics [3, 4]. In recentyears, many studies have carried out to explore the relativistic energy eigen-values and eigenfunctions on the curved background with the cosmic string(see, [5, 6, 7, 8, 9] and references their in). [email protected] ; faiz4U.enter@rediffmail.com Position dependent mass system in a mag-netic cosmic string space-time
In the context of Kaluza-Klein theory [18, 19], the metric with a magneticquantum flux (Φ) passing along the symmetry axis of the string assumes thefollowing form ds = − dt + dr + α r dφ + dz + [ dx + K A µ ( x ) dx µ ] , (1)where t is the time-coordinate, x is the coordinate associated with fifth ad-ditional dimension having ranges 0 < x < π a where, a is the radius ofthe compact dimension of x , ( r, φ, z ) are the cylindrical coordinates with theusual ranges, and K is the Kaluza constant [46]. The parameter α = (1 − µ )[11] characterizing the wedge parameter where, µ is the linear mass densityof the string.Based on [46, 48, 53, 52], we introduce a uniform magnetic field B andmagnetic quantum flux Φ through the line-element of the cosmic string space-time (1) in the following form ds = − dt + dr + α r dφ + dz + (cid:20) dx + (cid:18) − α B r + Φ2 π (cid:19) dφ (cid:21) , (2)where the gauge field given by A φ = K − (cid:18) − α B r + Φ2 π (cid:19) (3)gives rise to a uniform magnetic field ~B = ~ ∇ × ~A = − K − B ˆ z [68], ˆ z is theunitary vector in the z -direction. Here Φ = const. is the magnetic quantumflux [30, 68] through the core of the topological defects [69].The relativistic quantum dynamics of a position dependent mass systemis described by [34, 35, 52]: (cid:20) √− g ∂ µ ( √− g g µν ∂ ν ) − ( m + S ) (cid:21) Ψ = 0 , (4)3ith g is the determinant of metric tensor with g µν its inverse. For the metric(2) g µν = − α r − K A φ α r − K A φ α r K A φ α r . (5)By considering the line-element (2) into the Eq. (4), we obtain the fol-lowing differential equation :[ − ∂ ∂t + ∂ ∂r + 1 r ∂∂r + 1 α r (cid:18) ∂∂φ − K A φ ∂∂x (cid:19) + ∂ ∂z + ∂ ∂x − ( m + S ) ] Ψ( t, r, φ, z ) = 0 . (6)Since the line-element (2) is independent of t, φ, z, x . One can choose thefollowing ansatz for the function Ψ as:Ψ( t, r, φ, z, x ) = e i ( − E t + l φ + k z + q x ) ψ ( r ) , (7)where E is the total energy of the particle, l = 0 , ± , ± , .. ∈ Z , and k, q are constants.Substituting the ansatz (7) into the Eq. (6), we obtain the followingequation: (cid:20) d dr + 1 r ddr + E − k − q − ( l − K q A φ ) α r − ( m + S ) (cid:21) ψ ( r ) = 0 . (8) Case A : Interactions with Cornell-type potential
Cornell-type potential consists of linear plus Coulomb-like term is a par-ticular case of the quark-antiquark interaction [70, 71]. The Coulomb poten-tial is responsible at small distances or short range interactions and linearpotential leads to confinement of quark. This type of potential is given by[72, 35, 35] S ( r ) = η c r + η L r (9)4here η c , η L are the potential parameters.Sunstituting the (3) and (9) into the Eq. (8), we obtain the followingequation: (cid:20) d dr + 1 r ddr + λ − j r − Ω r − ar − b r (cid:21) ψ ( r ) = 0 , (10)where λ = E − k − q − m − η c η L − m ω ( l − q Φ2 π ) α , Ω = q m ω + η L ,j = s ( l − q Φ2 π ) α + η c ,ω = q B m ,a = 2 m η c ,b = 2 m η L . (11)Introducing a new variable ρ = √ Ω r , Eq. (10) becomes (cid:20) d dρ + 1 ρ ddρ + ζ − j ρ − ρ − ηρ − θ ρ (cid:21) ψ ( ρ ) = 0 , (12)where ζ = λ Ω , η = a √ Ω , θ = b Ω . (13)Suppose the possible solution to Eq. (12) is ψ ( ρ ) = ρ j e − ( ρ + θ ) ρ H ( ρ ) . (14)Substituting the solution Eq. (14) into the Eq. (12), we obtain H ′′ ( ρ ) + (cid:20) γρ − θ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − βρ + Θ (cid:21) H ( ρ ) = 0 , (15)5here γ = 1 + 2 j, Θ = ζ + θ − j ) ,β = η + θ j ) . (16)Equation (15) is the biconfluent Heun’s differential equation [34, 35, 41, 51,52, 73, 74] and H ( ρ ) is the Heun polynomials.The above equation (15) can be solved by the Frobenius method. Weconsider the power series solution [75] H ( ρ ) = ∞ X i =0 c i ρ i (17)Substituting the above power series solution into the Eq. (15), we obtain thefollowing recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) [ { β + θ ( n + 1) } c n +1 − (Θ − n ) c n ] . (18)And the various coefficients are c = (cid:18) ηγ + θ (cid:19) c ,c = 14 (1 + j ) [( β + θ ) c − Θ c ] . (19)We must truncate the power series by imposing the following two condi-tions [34, 35, 41, 48, 49, 50, 51, 52]:Θ = 2 n, ( n = 1 , , ... ) c n +1 = 0 . (20)By analyzing the condition Θ = 2 n , we get the following second degree6xpression of the energy eigenvalues E n,l : λ Ω + θ − j ) = 2 n ⇒ E n,l = ± { k + q + m + 2 Ω n + 1 + s ( l − q Φ2 π ) α + η c +2 η c η L + 2 m ω ( l − q Φ2 π ) α − m η L Ω } . (21)For α →
1, the relativistic energy eigenvalue (21) is consistent with thoseresult in [49].Now, we impose additional recurrence condition c n +1 = 0 to find theindividual energy levels and wave-functions one by one as done in [34, 35, 51,52, 41]. For n = 1, we have Θ = 2 and c = 0 which implies from Eq. (19) c = 2 β + θ c ⇒ (cid:18) η j + θ (cid:19) = 2 β + θ Ω ,l − a j ) Ω ,l − a b ( 1 + j j ) Ω ,l − b j ) = 0 (22)a constraint on the parameter Ω ,l . The magnetic field B ,l is so adjustedthat Eq. (22) can be satisfied and we have simplifed by labelling: ω ,l = 1 m q Ω ,l − η L ↔ B ,l = 2 q q Ω ,l − η L . (23)Therefore, the ground state energy level for n = 1 is given by E ,l = ± { k + q + m + 2 Ω ,l s ( l − q Φ2 π ) α + η c +2 η c η L + 2 m ω ,l ( l − q Φ2 π ) α − m η L Ω ,l } . (24)And the radial wave-functions is ψ ,l = ρ r ( l − q Φ2 π )2 α + η c e − m ηL Ω 321 ,l + ρ ρ ( c + c ρ ) , (25)7here c = m η c p Ω ,l (1 + 2 q ( l − q Φ2 π ) α + η c ) + m η L Ω ,l c . (26) Case B : Interactions with Coulomb-type potential
We consider η L → S . Thus the Coulomb potentialis given by S ( r ) = η c r , (27)This kind of potential has used to study position-dependent mass systems[52, 49, 78, 79] in the relativistic quantum mechanics.The radial wave-equations Eq. (10) becomes (cid:20) d dr + 1 r ddr + ˜ λ − j r − m ω r − ar (cid:21) ψ ( r ) = 0 , (28)where ˜ λ = E − k − q − m − m ω ( l − q Φ2 π ) α .Introduce a new variable ρ = √ m ω r , Eq. (28) becomes " d dρ + 1 ρ ddρ + ˜ λm ω − j ρ − ρ − ˜ ηρ ψ ( ρ ) = 0 , (29)where ˜ η = a √ m ω .Suppose the possible solution to Eq. (29) is ψ ( ρ ) = ρ j e − ρ H ( ρ ) . (30)Substituting the solution Eq. (14) into the Eq. (12), we obtain H ′′ ( ρ ) + (cid:20) jρ − ρ (cid:21) H ′ ( ρ ) + (cid:20) − ˜ ηρ + ˜Θ (cid:21) H ( ρ ) = 0 , (31)where ˜Θ = ˜ λm ω − j ). 8quation (31) is the biconfluent Heun’s differential equation [34, 35, 41,51, 52, 73, 74] and H ( ρ ) is the Heun polynomials.Substituting the power series solution (17) into the Eq. (31), we obtainthe following recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 j ) h ˜ η c n +1 − ( ˜Θ − n ) c n i . (32)And the various coefficients are c = ˜ η j c , c = 14 (1 + j ) [˜ η c − Θ c ] . (33)The power series expansion (17) becomes a polynomial of degree n by im-posing two conditions [34, 35, 41, 48, 49, 50, 51, 52]: c n +1 = 0 , ˜Θ = 2 n ( n = 1 , , .... ) (34)By analyzing the condition ˜Θ = 2 n , we get the following energy eigen-values E n,l : E n,l = ± vuuut k + q + m + 2 m ω n + 1 + s ( l − q Φ2 π ) α + η c + ( l − q Φ2 π ) α . (35)For the radial mode n = 1, we have ˜Θ = 2 and c = 0 which implies ω ,l = 2 m η c (cid:18) q ( l − q Φ2 π ) α + η c (cid:19) ↔ B ,l = 4 m η c q (cid:18) q ( l − q Φ2 π ) α + η c (cid:19) . (36)a constraint on the parameter ω ,l or the magnetic field B ,l .The ground state energy eigenvalues for n = 1 is E ,l = ± vuuut k + q + m + 2 m ω ,l s ( l − q Φ2 π ) α + η c + ( l − q Φ2 π ) α , (37)9here ω ,l is given by Eq. (36).Equation (37) with (36) corresponds to the allowed values of energy levelsfor the radial mode n = 1 of a position-dependent mass particle subject toa Coulomb-type scalar potential in the context of Kaluza-Klein theory. For α →
1, the energy eigenvalues is consistent with those result in [49].
Special case
In this special case, we choose zero magnetic field, B →
0. The radialwave-equations from Eq. (28) becomes (cid:20) d dr + 1 r ddr + E − k − q − m − j r − ar (cid:21) ψ ( r ) = 0 . (38)The above can now be expressed as [41, 43, 81] ψ ′′ ( r ) + 1 r ψ ′ ( r ) + 1 r ( − ξ r + ξ r − ξ ) ψ ( r ) = 0 . (39)where ξ = k + q + m − E , ξ = − a , ξ = j . (40)The energy eigenvalues is given by E n,l = ± m vuuut − η c (cid:18) n + + q ( l − q Φ2 π ) α + η c (cid:19) + k m + q m , (41)where n = 0 , , , .... .Equation (41) is the relativistic energy eigenvalues of a scalar chargedparticles in the magnetic cosmic string background in the Kaluza-Klein the-ory with a Coulomb-type scalar potential. For α →
1, the energy eigenvaluesEq. (41) is consistent with those result in [48].The corresponding radial wave functions is given by ψ n,l ( r ) = | N | n,l r j e − √ k + q + m − E n,l r L (2 j ) n ( r ) , (42)10here | N | n,l = 2 j (cid:0) k + q + m − E n,l (cid:1) j + (cid:16) n !( n +2 j )! (cid:17) is the normalizationconstant and L (2 j ) n ( r ) is the generalized Laguerre polynomials. The poly-nomilas L ( j ) n ( r ) are orthogonal over [0 , ∞ ) with respect to the measure withweighting function r j e − r as Z ∞ r j e − r L ( j ) n L ( j ) m ′ dr = ( n + j )! n ! δ n m ′ . (43) Case C : Interactions with Linear potential
We consider η c →
0. Thus the linear scalar potential is given by S ( r ) = η L r, (44)The linear potential have studied by many authors in the relativistic quantummechanics [9, 78, 79, 80, 82, 83].The radial wave-equations from Eq. (10) becomes (cid:20) d dr + 1 r ddr + ˜ λ − l r − Ω r − b r (cid:21) ψ ( r ) = 0 . (45)Introduce a new variable ρ = √ Ω r , then the Eq. (45) becomes " d dρ + 1 ρ ddρ + ˜ λ Ω − l ρ − ρ − θ ρ ψ ( ρ ) = 0 . (46)Let the possible solution to Ee. (46) is ψ = ρ | l | e − ( θ + ρ ) ρ H ( ρ ) (47)Substituting Eq. (47) into the Eq. (46), we obtain H ′′ ( ρ ) + (cid:20) (1 + 2 | l | ) ρ − θ − ρ (cid:21) H ′ ( ρ ) + " − θ (1 + 2 | l | ) ρ + Θ H ( ρ ) = 0 , (48)11here Θ = ˜ λ Ω − | l | ) + θ .Equation (48) is the biconfluent Heun’s differential equation [34, 35, 41,51, 52, 73, 74] and H ( ρ ) is the Heun polynomials.Substituting the power series solution (17) into the Eq. (48), we obtainthe following recurrence relation for the coefficients: c n +2 = 1( n + 2)( n + 2 + 2 l ) (cid:20) θ n + 3 + 2 | l | ) c n +1 − (Θ − n ) c n (cid:21) . (49)And the various coefficients are c = θ c ,c = 14 (1 + j ) (cid:20) θ | l | ) c − Θ c (cid:21) . (50). The power series expansion (17) becomes a polynomial of degree n byimposing two conditions [34, 35, 41, 48, 49, 50, 51, 52]: c n +1 = 0 , Θ = 2 n ( n = 1 , , , , .... ) (51)By analyzing the condition Θ = 2 n , we get the following energy eigen-values E n,l : E n,l = ± vuut k + q + m + 2 m ω l + 2 Ω n + 1 + | l − q Φ2 π | α ! − m η L Ω , (52)where l = α ( l − q Φ2 π ).Equation (52) is the relativistic energy eigenvalues of a scalar chargedparticles in the magnetic cosmic string background in the Kaluza-Klein the-ory with a linear confining potential. For α →
1, the energy eigenvalues Eq.(52) is consistent with those result in [49].For the radial mode n = 1, c = 0 which impliesΩ ,l = (cid:20) m η L | l | ) (cid:21) . (53)12 constraint on the parameter Ω ,l . Therefore, the magnetic field is given by ω ,l = 1 m q Ω ,l − η L ⇒ B ,l = 2 q q Ω ,l − η L ⇒ = 2 q s(cid:20) m η L | l | ) (cid:21) − η L . (54)Therefore the ground state energy levels E ,l = ± { k + q + m + 2 m ω ,l ( l − q Φ2 π ) α +2 Ω ,l n + 1 + | l − q Φ2 π | α ! − m η L Ω ,l } . (55)Equation (55) with (54) corresponds to the allowed values of relativistic en-ergy levels for the radial mode n = 1 of a position-dependent mass particlesubject to a linear confining potential in a possible scenario described by aKKT. Special case
In this special case, we choose zero magnetic field, B →
0. The radialwave-equations from Eq. (45) becomes (cid:20) d dr + 1 r ddr + ˜ λ − l r − η L r − b r (cid:21) ψ ( r ) = 0 . (56)Transforming ρ = √ η L r into the Eq. (45), we have " d dρ + 1 ρ ddρ + ˜ λη L − l ρ − ρ − bη L ρ ψ ( r ) = 0 . (57)Let us now discuss the asymptotic behavior of the possible solutions toEq. (57), that is, we hope that ψ ( ρ ) → ρ → ρ → ∞ . Suppose the13ossible solution to Eq. (57) is ψ ( ρ ) = ρ | l | e − ( ρ + bη L ) ρ H ( ρ ) (58)Substituting the solution Eq. (58) into the Eq. (57), we obtain H ′′ ( ρ ) + " | l | ρ − ρ − bη L H ′ ( ρ )+ ˜ λη L + b η L − | l | ) − b η L (1 + 2 | l | ) ρ H ( ρ ) = 0 . (59)Substituting the power series solution Eq. (14) into the above equation,we get c n +2 = 1( n + 2)( n + 2 + 2 | l | ) [ bη L ( n + 32 + | l | ) c n +1 − ( ˜ λη L + b η L − | l | ) − n ) c n ] , (60)where few coefficients are c = b η L c ,c = 14 (1 + | l | ) " b η L (3 + 2 | l | ) c − ˜ λη L + b η L − − | l | ! c . (61)The power series solution becomes a polynomial of degree n . for this, wemust have [34, 35, 41, 48, 49, 50, 51, 52]˜ λη L + b η L − | l | ) = 2 n ( n = 1 , , ... ) , c n +1 = 0 . (62)For n = 1, we have c = 0 which implies from (61) η ,l L = m | l | ) . (63)14 constraint on the potential parameter η ,l .By analysing the condition ˜ λη L + b η L − | l | ) = 2 n , we get E n,l = ± p k + q + 2 η L ( n + 1 + | l | ) . (64)Therefore, the ground state energy eigenvalue is given by E ,l = ± q k + q + 2 η ,l L (2 + | l | )= ± m s k m + q m + (3 + 2 | l − q Φ2 π | α ) (2 + | l − q Φ2 π | α ) . (65)Equation (65) represents energy levels associated with the radial mode n = 1of a Klein-Gordon particle subject to a linear central potential in a back-ground governed by the Kaluza-Klein theory. For α →
1, the energy eigen-value is consistent with those result in [50].
Case D : Interactions without potential
In this case, we consider zero scalar potential, S = 0. Therefore, theradial wave-equations Eq. (8) becomes (cid:20) d dr + 1 r ddr + λ − l r − m ω r (cid:21) ψ ( r ) = 0 , (66)where λ = E − k − q − m − m ω ( l − q Φ2 π ) α ,ω = q B m . (67)Transforming to a new variable ρ = m ω r into the Eq. (66), we obtain[41, 43, 81] ψ ′′ ( ρ ) + 1 ρ ψ ′ ( ρ ) + 1 ρ (cid:0) − ξ ρ + ξ ρ − ξ (cid:1) ψ ( ρ ) = 0 , (68)15here ξ = 14 , ξ = λ m ω , ξ = l . (69)Therefore, the energy eigenvalues is given by E n,l = ± vuut k + q + m + q B n + 1 + ( l − q Φ2 π ) α + | l − q Φ2 π | α ! , (70)where n = 0 , , , ... .Equation (70) is the relativistic energy eigenvalues of a scalar chargedparticle subject to a uniform magnetic field including a magnetic quantumflux in cosmic string space-time within the Kaluza-Klein theory. For zeromagnetic quantum flux, Φ →
0, the energy eigenvalues Eq. (70) is consistentwith those result obtained in [46]. Thus we can see that the energy eigenval-ues Eq. (70) get modify in comparison to those in [46] due to the presenceof a magnetic quantum flux Φ.The wave-functions is given by ψ n,l ( ρ ) = | N | n,l ρ | l − q Φ2 π | α e − ρ L ( l − q Φ2 πα ) n ( ρ ) , (71)where | N | n,l = n !2 ( n + | l − q Φ2 π | α )! ! is the normalization constant and L ( l − q Φ2 πα ) n ( ρ )is the generalized Laguerre polynomials.We have observed in all cases that the angular momentum number l isshifted, l → l = α ( l − q Φ2 π ), an effective angular quantum number. Therefore,all the relativistic energy eiganvalues obtained here depend on the geometricquantum phase [30, 68]. Thus, we have that, E n,l (Φ+Φ ) = E n,l ∓ τ (Φ), whereΦ = ± πq τ with τ = 0 , , , ... . This dependence of the relativistic energylevels on the geometric quantum phase gives rise to a relativistic analogueof the Aharonov-Bohm effect for bound states [34, 35, 39, 40, 41, 42, 43, 44,45, 46, 47, 48, 49, 50, 51, 52]. 16 Conclusions
In this work, we have investigated the quantum dynamics of massive chargedparticles with a uniform magnetic field in the cosmic string space-time inthe context of Kaluza-Klein theory with various potential form. In
Case A ,we have considered a Cornell-type scalar potential and obtained the energyeigenvalues (21). In
Case B , we have considered a Coulomb-type scalarpotential and obtained the energy eigenvalues (35). Furthermore, we havediscussed a special case corresponds to zero external magnetic field, B → Case C , we have consideredlinear confining potential and obtained the energy eigenvalues (52). Further-more, we have discussed a special case corresponds to zero external magneticfield, B →
0, and obtained the energy eigenvalues (64). We have observedthat for α →
1, the energy eigenvalues reduces to those results obtained in[48, 49, 50]. Thus the presence of the topological defect parameter, α , modifythe energy specrtum of the quantum system and shifted the energy levels.In Case D , we have solved the Klein-Gordon equation subject to a uniformmagnetic field including a magnetic quantum flux in cosmic string space-timein the context of Kaluza-Klein theory without potential. We have obtainedthe energy eigenvalues (70) and seen that for zero magnetic quantum flux,Φ →
0, this energy eigenvalues is consistent in [46]. Thus the energy eigenval-ues (70) get modify in comparison to those result in [46] due to the presenceof magnetic quantum flux Φ in the quantum systems.
Data Availability
No data has used to prepare this paper.17 onflict of Interest
Author declares that there is no conflict of interest regarding publication thispaper.
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