Scattering and bound states of Dirac Equation in presence of cosmic string for Hulthén potential
Mansoureh Hosseinpour, Fabiano M. Andrade, Edilberto O. Silva, Hassan Hassanabadi
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Scattering and bound states for the Hulthén potential in a cosmic stringbackground
Mansoureh Hosseinpour a,1 , Fabiano M. Andrade b,2 , Edilberto O. Silva c,3 , HassanHassanabadi d,1 Physics Department, Shahrood University of Technology, P. O. Box: 3619995161-316, Shahrood, Iran Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil Departamento de Física, Universidade Federal do Maranhão, 65085-580 São Luís, MA, BrazilReceived: date / Accepted: date
Abstract
In this work we study the Dirac equation withvector and scalar potentials in the spacetime generated bya cosmic string. Using an approximation for the centrifu-gal term, a solution for the radial differential equation is ob-tained. We consider the scattering states under the Hulthénpotential and obtain the phase shifts. From the poles of thescattering S -matrix the states energies are determined as well. To study the relativistic quantum dynamics of particles withspin under the influence of electromagnetic fields in curvedspacetime we must consider the modified covariant form ofthe Dirac equation (¯ h = c =
1) [1, 2] (cid:8) i γ µ ( x ) (cid:2) ∂ µ + Γ µ ( x ) + ieA µ ( x ) (cid:3) − M (cid:9) ψ ( x ) = , (1)where A µ denotes the vector potential associated with theelectromagnetic field, Γ µ ( x ) is the spinor affine connectionand γ µ ( x ) are the Dirac matrices in the curved spacetime.The γ µ ( x ) matrices are constructed from the standard Diracmatrices in Minkowski spacetime, which are written in termsof local coordinates, and write them in terms of global co-ordinates using the inverse vierbeins e µ a ( x ) through the rela-tion γ µ ( x ) = e µ a ( x ) γ a , ( µ , a = , , , ) , (2)with γ a = (cid:0) γ , γ i (cid:1) being the standard gamma matrices. The γ µ ( x ) matrices also define the covariant Clifford algebra, { γ µ ( x ) , γ ν ( x ) } = g µν ( x ) . (3) a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] If we want the particle to interact with scalar potentials itis more convenient to write the Dirac equation (1) in thefollowing form [3]: (cid:8) α i [ p i − i Γ i − eA i ] + eA − i Γ + β M (cid:9) ψ ( x ) = E ψ ( x ) , (4)for i = , ,
3, and then introduce the vector and scalar cou-plings, E → E − V ( r ) , (5) M → M + S ( r ) , (6)respectively.It is important to note that these couplings differ in themanner how they are inserted into the Dirac equation. InRef. [4] it was shown how the vector coupling in (5) acts onelectron and positron states. As a result, the potential cou-ples to the charge and a great number of physical phenomenacan be studied through the Dirac equation (1). For instance,it is used to study the relativistic quantum motion of chargedspin-0 and spin-1/2 particles in the presence of a uniformmagnetic field and scalar potentials [5], to study the influ-ence of topological defects on the spin current as well as thespin Hall effect [6] and to investigate the role of the cosmicstring on spin current and Hall electric field [7]. On the otherhand, contrary to the coupling (5), the scalar coupling (6)acts equally on particles and antiparticles. Namely, we canadd it directly to the mass term of the Dirac equation. Themost common interpretation known in the literature for thiscoupling is that of a position-dependent effective mass [8].This coupling has been used, for example, to study the prob-lem of a relativistic particle with position-dependent mass inthe presence of a Coulomb and a scalar potential in the back-ground spacetime generated by a cosmic string [9].These couplings are also used to study various phys-ical models by including interactions and then to investi-gate their possible physical implications on system dynam-ics. For example, the Aharonov–Bohm effect [10, 11], the a r X i v : . [ h e p - t h ] A p r Dirac particle in a Morse potential [12, 13], the Dirac equa-tion for an attractive Coulomb potential in supersymmet-ric quantum mechanics [14], the Dirac equation with scalarand vector potentials under the exact spin and pseudospinsymmetries limits [15–18] and particles interacting with theHulthén [19] and Rosen–Morse [20] potentials.Among the most important problems that we can studyusing the couplings (5) and (6), we refer to the scatteringproblems. The scattering problems of quantum systems, i.e.,the prediction of reaction probabilities when two objects col-lide, provide us with a reliable understanding of the physicalsystem under investigation and they belong the most effec-tive ways to study the structure of matter. Great parts of ourcurrent theories are built on the basis of scattering experi-ments. The best-known example of this claim is the atomicnucleus. Scattering problems are among the most techni-cally demanding problems in quantum physics. The under-lying difficulty lies in the unbounded nature of the wavefunction in these problems. The process of scattering of par-ticles by potentials changes their wave function by introduc-ing phase shifts. The study of these phase shifts allow usto predict experimental observations from the fundamentalinteractions postulated by the theory. These studies have al-ready been done in nonrelativistic and relativistic quantummechanics [4, 21–26]. Nevertheless, the dominant part ofthese scattering studies is limited to flat spacetime and thecases regarding the curved spacetime are actually less fre-quent. On the other hand, the dynamics of quantum mechan-ics on curved spaces in the presence of topological defectshas attracted much attention in recent years [27–35].In the present work, we are going to consider the scatter-ing problem of the Dirac equation produced by the Huthénpotential in a cosmic string background. As will be showed,due to the presence of the centrifugal term in the radial dif-ferential equation, it cannot be solved in an exact manner.An approximation is used in order to obtain an approximatedsolution for the problem. The cosmic string is a linear defectthat changes the topology of the medium. This field has beenan appealing research field in the past years as not much isknown in comparison with the ordinary Dirac equation inflat space [36–41]. From the field theory point of view, thecosmic string can be viewed as a consequence of symmetrybreaking phase transition in the early Universe [42]. Untilnow some problems have been investigated in curved space-time including the one-electron atom problem [43–45]. Thedynamics of non-relativistic particles in curved spacetime isconsidered in [29, 30, 46–50] as well.This paper is organized as follows. In Sect. 2, we studythe covariant Dirac equation in the spacetimes generated bya cosmic string in the presence of vector and scalar poten-tials of electromagnetic field. We then find special cases ofthe equation for equal and opposite scalar and vector poten-tials. In Sect. 3, we consider the Dirac equation with the Hulthén potential in the context of spin and pseudo-spinsymmetries and obtain the scattering solutions as well as thephase shifts. In Sect. 4, we derive the scattering S -matrixand from its poles we determine the bound state energies.Finally, in Sect. 5, we present our conclusions. The line element corresponding to the cosmic string space-time [51, 52] is given, in spherical coordinates, by [53, 54] ds = − dt + r d θ + α r sin θ d ϕ , (7)where t ∈ ( − ∞ , ∞ ) , r ∈ [ , ∞ ) , θ ∈ [ , π / ] and φ ∈ [ , π ] .The α parameter in Eq. 7 is related to the linear mass ¯ µ ofthe string by α = − µ and it is defined in the range ( , ] .Now, in order to write the Dirac equation (4) in the cos-mic string spacetime, we must rewrite the Dirac matrices interms of global coordinates. Additionally, we have to cal-culate the affine spinorial connection ( Γ , Γ i ) . The details ofthis calculation can be found in [3, 24]. By using the wavefunction decomposition in the form ψ ( x ) = e − iEt χ ( r , θ , ϕ ) , (8)the Dirac equation in (4) can be written as (cid:26) i α r ∂ r + i α θ r ∂ θ + i α r sin θ α ϕ ∂ ϕ + i r (cid:18) − α (cid:19) (cid:16) α r + cot θ α θ (cid:17) − γ [ M + S ( r )] + E − V ( r ) (cid:27) χ ( r , θ , ϕ ) = , (9)where we have included a scalar M → M + S ( r ) and a vector E → E − V ( r ) coupling. Now, assuming that the solutionsof Eq. (9) are of the form [45] χ ( r , θ , ϕ ) = r − ( α − ) / α ( sin θ ) − ( α − ) / α F ( r ) Θ ( θ ) Φ ( ϕ ) , (10)we find the radial equation (cid:26) ˜ α r p r + ir ˜ α r σ z κ ( α ) + V ( r ) + [ M + S ( r )] σ z (cid:27) F n , κ ( α ) ( r )= EF n , κ ( α ) ( r ) , (11)with [45]˜ α r = (cid:18) − ii (cid:19) , σ z = (cid:18) − (cid:19) . (12) In Eq. (11), κ ( α ) represents the generalized spin–orbit oper-ator in the spacetime of a cosmic string whose eigenvaluesare given by κ ( α ) = ± (cid:20) j ( α ) + (cid:21) = ± (cid:20) j + m (cid:18) α − (cid:19) + (cid:21) , (13)with j ( α ) representing the eigenvalues of the generalized to-tal angular momentum operator. The operator κ ( α ) is givenby σ z κ ( α ) = ˜ α · L ( α ) + , (14)where L ( α ) is the generalized angular momentum operatorin the spacetime of the cosmic string, L ( α ) Y m ( α ) (cid:96) ( α ) ( θ , ϕ ) = (cid:96) ( α ) (cid:0) (cid:96) ( α ) + (cid:1) Y m ( α ) (cid:96) ( α ) ( θ , ϕ ) , (15)with Y m ( α ) (cid:96) ( α ) ( θ , ϕ ) being the generalized spherical harmonicsand m ( α ) and (cid:96) ( α ) not necessarily being integers. In partic-ular m ( α ) = m / α , where m = , ± , ± , . . . , α ∈ ( , ] and (cid:96) ( α ) = n + m ( α ) = (cid:96) + m ( / α − ) , (cid:96) = , , , . . . , n −
1. Here (cid:96) and m are, respectively, the orbital angular momentumquantum number and the magnetic quantum number in theflat space (i.e., for α = n is the principal quantumnumber.By choosing the radial wave function as [45] F ( r ) = r (cid:18) − i f ( r ) g ( r ) (cid:19) , (16)we obtain the coupled equations − i [ E − M − ( S + V )] f ( r ) + dg ( r ) dr + κ ( α ) r g ( r ) = , (17) − i [ E + M + S − V ] g ( r ) + d f ( r ) dr + κ ( α ) r f ( r ) = . (18)Let us now consider the special case of S ( r ) = V ( r ) (exactspin symmetry limit) and S ( r ) = − V ( r ) (exact pseudo-spinsymmetry limit). After elimination of one component in fa-vor of the other, for S ( r ) = V ( r ) , we have − d f ( r ) dr + (cid:20) κ ( α ) (cid:0) κ ( α ) − (cid:1) r − ( E − M − V ( r )) ( E + M ) (cid:21) f ( r ) = . (19)Additionally, for the case S ( r ) = − V ( r ) , we obtain − d g ( r ) dr + (cid:20) κ ( α ) (cid:0) κ ( α ) + (cid:1) r − ( E + M − V ( r )) ( E − M ) (cid:21) g ( r ) = . (20)Thus comparing Eqs. (19) and (20), we can see that thesolution for the case S ( r ) = − V ( r ) , can be obtained from the solution for the case S ( r ) = V ( r ) with the replacements κ ( α ) − → κ ( α ) + M → − M . Therefore we shall onlydeal with the latter because the results for the former canbe obtained in a straightforward manner by using the abovereplacements. In this work, we are interested in considering the Hulthénpotential, which has remarkable applicabilities because ofits short-range nature. It should be noted that this potentialis a special case of the Eckert potential [55]. Therefore, weare going to investigate scattering state solutions of the Diracequation in the presence of the Hulthén potential, V ( r ) = − ξ ω e ω r − , (21)where ω is the screening parameter and ξ is a positive con-stant. When V ( r ) is used to describe atomic phenomena, ξ is interpreted as Ze , with Z the atomic number. In this step,we want to evaluate phase shifts and normalization factorfor the pseudo-spin symmetry limit (i.e., S ( r ) = V ( r )) . Thusinserting Eq. (21) into (19), we obtain (cid:20) − d dr + κ ( α ) ( κ ( α ) − ) r − ( E + M ) ξ ω e − ω r − e − ω r (cid:21) f ( r )= k f ( r ) , (22)where k = E − M . It is worthwhile to note here that,for small values of ω , the Hulthén potential behaves likethe Coulomb potential, consequently Eq. (22) turns into theBessel equation [56–59]. In contrast with the Coulomb po-tential, unfortunately, the Dirac equation for the Hulthén po-tential cannot be solved analytically due to the presence ofthe centrifugal term. In this manner, it is necessary to usesome approximation. Considering small values of ω , a com-mon approximation for the centrifugal term is1 r ≈ ω e − ω r ( − e − ω r ) . (23)The above approximation, as we shall see, leads us to a solv-able differential equation [55, 60]. Therefore, by using theapproximation in Eq. (23), followed by the change of vari-able y = − e − ω r , we can write the Eq. (22) in the form (cid:20) d dy − − y ddy − κ ( α ) ( κ ( α ) − ) y ( − y ) + η y ( − y )+ ε ( − y ) (cid:21) f ( y ) = , (24)in which η = ( E + M ) ξ / ω and ε = k / ω . Equation(24) can be turned into a well-known differential equationif we choose f ( y ) = y γ ( − y ) ν h ( y ) , (25) as the solutions, where ν and γ are arbitrary constants to bedetermined. Therefore, substituting Eq. (25) into Eq. (24)we obtain (cid:20) d dy + γ − ( + ν + γ ) yy ( − y ) ddy + ν + ε ( − y ) + η − γν − γ y ( − y ) + γ ( γ − ) − κ ( α ) ( κ ( α ) − ) y ( − y ) (cid:21) h ( y ) = . (26)We determine the parameters ν and γ by imposing that thecoefficients of the terms 1 / ( − y ) and 1 / [ y ( − y ) ] vanishidentically. In this manner, we have ν = ± ik ω , γ = κ ( α ) or γ = − κ ( α ) . (27)This set of parameters leads us to the same solution for Eq.(26) and we are free to choose one set. Thus choosing ν = ik / ω and γ = κ ( α ) this leads to (cid:26) y ( − y ) d dy + [ η − ( + η + η y ] ddy − η η (cid:27) h ( y ) = , (28)where η = γ + ν + (cid:112) η + ν = κ ( α ) + ik ω + (cid:114) ( E + M ) ξω − k ω , η = γ + ν − (cid:112) η + ν = κ ( α ) + ik ω − (cid:114) ( E + M ) ξω − k ω , η = γ = κ ( α ) . (29)Equation (28), has the form of a hypergeometric differentialequation [44], h ( y ) = F ( η , η , η ; y ) . (30)Therefore, the radial wave functions can be written as f ( r ) = N ( − e − ω r ) (cid:96) e ikr F (cid:0) η , η , η ; 1 − e − ω r (cid:1) . (31)Now, in order to obtain the scattering phase shift and thenormalization factor we write the asymptotic form of theabove wave function. For this purpose we use the propertiesof the hypergeometric functions [61] and the asymptotic be-havior of (31) to r → ∞ [24] f ( r ) ∼ N [ Γ ( η )] (cid:12)(cid:12)(cid:12)(cid:12) Γ ( η − η − η ) Γ ( η − η ) Γ ( η − η ) (cid:12)(cid:12)(cid:12)(cid:12) (32) × sin (cid:16) kr + π + δ (cid:17) , (33)where N is a normalization constant. Recalling the boundarycondition for r → ∞ imposed in Ref. [44] as f ( r ) ∼ (cid:18) kr − (cid:96) π + δ (cid:96) (cid:19) , (34) and comparing with Eq. (33), the phase shift and the nor-malization factor can be found. The result is δ (cid:96) = π ( (cid:96) + ) + arg (cid:20) Γ ( η − η − η ) Γ ( η − η ) Γ ( η − η ) (cid:21) , (35)and N = [ Γ ( η )] (cid:12)(cid:12)(cid:12)(cid:12) Γ ( η − η − η ) Γ ( η − η ) Γ ( η − η ) (cid:12)(cid:12)(cid:12)(cid:12) . (36)It can be seen that when α = κ = ± (cid:18) j + (cid:19) , (37)where γ = κ . Therefore we can obtain the phase shifts andthe normalization factor in flat spacetime in this case if werewrite the Dirac equation in the flat spacetime by standardDirac matrices. As expected our result is the limit of Eqs.(35) and (36) when α = The Hulthén potential also admits bound state solutions. Inorder to find the bound states energies, let us analyze thescattering S -matrix. It is well known that poles of the S -matrix in the upper half of the complex plane are associatedwith the bound states. Using Eq. (35), the S -matrix can bewritten as S (cid:96) = e i δ (cid:96) = e i π ( (cid:96) + ) e i arg [ Γ ( η − η − η )] × e − i arg [ Γ ( η − η )] e − i arg [ Γ ( η − η )] . (38)Therefore, the poles of the S -matrix are given by the polesof the gamma functions Γ ( η − η ) and Γ ( η − η ) . In thismanner, based on the relations η − η = η ∗ , η − η = η ∗ and η − η − η = ( η + η − η ) ∗ = ik / ω , we are lookingfor the poles of Γ (cid:32) κ ( α ) − ik ω ± (cid:114) ( E + M ) ξω − k ω (cid:33) . (39)The gamma function Γ ( z ) has poles at z = − n , where n isa non-negative integer. Then the bound state energies aregiven by k ≡ E − M = − (cid:2) ( n + κ ( α ) ) ω − ( E + M ) ξ (cid:3) ( n + κ ( α ) ) , (40)whit n = , , , . . . . In this work we considered the Dirac equation in curvedspacetime and the topology of spacetime in order to de-scribe physics of the system in the presence of the gravi-tational fields of a cosmic string. We obtained the solutionof the Dirac equation in the curved spacetime by consider-ing vector and scalar potentials. We considered the scatter-ing states of the Dirac equation under the Hulthén potentialand obtained as scattering phase shifts. From the poles of thescattering S -matrix we determined the bound state energies.When α =
1, we recover the general solution of the Diracequation in usual spherical coordinates, as we should.
Acknowledgments
With great pleasure, the authors thank the kind referee forhelpful comments. FMA acknowledges funding from theConselho Nacional de Desenvolvimento Científico e Tec-nológico (CNPq), Grants No. 460404/2014-8 and No. 311699/2014-6. EOS acknowledges funding from CNPq, Grants No. 482015/2013-6, No. 306068/2013-3, No. 476267/2013-7, FAPEMA GrantNo. 01852/14 (PRONEM) and FAPES.
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