Eur. Phys. J. C (2020) 80:1111 https://doi.org/10.1140/epjc/s10052-020-08681-6 Regular Article - Theoretical Physics Scattering of charged fermion to two-dimensional wormhole with constant axial magnetic flux Kulapant Pimsamarn1,a, Piyabut Burikham2,b, Trithos Rojjanason2,c 1 Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand 2 High Energy Physics Theory Group, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand Received: 14 April 2020 / Accepted: 18 November 2020 / Published online: 3 December 2020 © The Author(s) 2020 Abstract Scattering of charged fermion with (1 + 2)- boundary phenomena that have no bulk analogs emerge, e.g. dimensional wormhole in the presence of constant axial mag- quantum (spin) hall effects, topological matters, and more netic flux is explored. By extending the class of fermionic recently graphene physics [1]. Two dimensional surface can solutions of the Dirac equation in the curved space of worm- be manipulated in the laboratory so that it can be curved, hole surface to include normal modes with real energy and strained, and twisted. A cage structure of graphene worm- momentum, we found a quantum selection rule for the scat- holes, the schwarzite, can even be produced with promising tering of fermion waves to the wormhole. The newly found properties [2,3]. The gauge fields such as electric [4] and momentum–angular momentum relation implies√ that only magnetic fields [5] can be applied to the system, leading to fermion with the quantized momentum k = m/a q can be unique intriguing 2D phenomena, notably the well-known transmitted through the hole. The allowed momentum is pro- Landau quantization of fermionic states on a plane. It has portional to an effective angular momentum quantum number been shown that effects of strain and gauge fields could be m and inversely proportional to the radius of the throat of similar in 2D [6,7]. Behaviour of quantum particle on the the wormhole a. Flux dependence of the effective angular curved surface in the presence of gauge fields can be con- momentum quantum number permits us to select fermions siderably different from the flat situation [8,9]. Similar to that can pass through according to their momenta. A conser- the strain, curvature effects [10–12] can mimic gauge fields vation law is also naturally enforced in terms of the unitar- [13,14], specifically curvature and gauge connection appears ity condition among the incident, reflected, and transmitted in the equation of motion with equal role. For example, waves. The scattering involving quasinormal modes (QNMs) fermions on two-dimensional sphere and wormhole experi- of fermionic states in the wormhole is subsequently explored. ence spin-orbit coupling induced from the surface curvature It is found that the transmitted waves through the wormhole [15–18] even in the absence of the gauge fields. Addition of for all scenarios involving QNMs are mostly suppressed and axial magnetic field generates Landau quantization distinc- decaying in time. In the case of QNMs scattering, the uni- tive from the planar case. For the wormhole, the fermionic tarity condition is violated but a more generic relation of states can be in quasinormal modes with complex energies, the scattering coefficients is established. When the magnetic its quantum statistics can be altered by the magnetic flux flux φ = mhc/e, i.e., quantized in units of the magnetic flux through the hole [18,19]. quantum hc/e, the fermion will tunnel through the wormhole Dirac fermion in graphene wormhole without gauge field with zero reflection. has been discussed in Refs. [20,21]. In Ref. [18], we investi- gate the effects of axial magnetic field on a charged fermion in a (1 + 2)-dimensional wormhole. This two-dimensional 1 Introduction wormhole is fundamentally different from the (1 + 3)- dimensional wormhole in general relativity (see e.g. [22]), The physics of fermion on two-dimensional surface has been there is no time dilation in the two-dimensional wormhole one of the major research topics in recent years. Interesting under consideration. For the constant magnetic flux scenario, the system can be solved analytically and exact solutions are a e-mail: [email protected] found to contain “normal” (real energy but complex momen- b e-mail: [email protected] (corresponding author) tum) and quasinormal modes (QNMs). c e-mail: [email protected] 123 1111 Page 2 of 13 Eur. Phys. J. C (2020) 80 :1111 where = (t, u,v) represents the Dirac spinor field on the wormhole and M represents the rest mass of the particle, c is the speed of light, e is electric charge, and Aμ is the electromagnetic four-potential. The γ a are the Dirac matrices given by σ k γ 0 = i 0 ,γk = 0 i , 0 −i −iσ k 0 where σ k are the Pauli matrices defined by − σ 1 = 01 ,σ2 = 0 i ,σ3 = 10 . 10 i 0 0 −1 They obey the Clifford algebra Fig. 1 Geometric structure of a wormhole surface where a is a radius {γ a,γb}=2ηab. (2.3) at the radius function R(u = u0 = ln q/2).Andr is the radius of curvature of the wormhole surface along u direction The Pauli matrices have a useful identity that we will use later In this work the scattering of fermions to the worm- σ i σ j = δij + i ijkσ k, (2.4) hole is explored. The more generic solutions to the equation of motion of the fermion in the magnetized wormhole are where ijk is Levi-Civita symbol. constructed in terms of hypergeometric functions. The nor- The covariant derivative of the spinor interaction with mal modes are actually found when the (effective) angular gauge field in the curved space is given by quantum number m (see definition below) is related to the ∇ ≡ ∂ − , m μ μ μ (2.5) momentum of the fermion by k = √ ≡ km . Such quan- a q where the spin connection μ [23]is tized momentum–angular momentum relation is unique to the 2D wormhole under consideration. =−1γ aγ b ν ∂ β − β , μ ea μ gνβ eb eb βμν (2.6) In Sect. 2, the mathematical formulation of the wormhole 4 and fermion in curved space is established. In Sect. 3 as a where β,μ,ν ∈{t, u,v} and the Christoffel symbols βμν review of the main results of Ref. [18], the Dirac equation in are defined by the magnetized wormhole is written and solved analytically, 1 βμν = (∂μgβν + ∂ν gβμ − ∂β gμν). then the general solutions in the upper and lower plane con- 2 nected to the wormhole is discussed. Matching conditions of Then for the metric (2.1), the scattering of fermionic waves to the wormhole is con- sidered in Sect. 4. Section 5 discusses the use of Wronskian 1 2 − uvv = vuv = vvu = ∂u R = RR , (2.7) to derive a general relation between scattering coefficients. 2 The scattering scenarios are categorized into normal-modes and zero otherwise. It is then straightforward to show that and QNMs scattering and some of the results are presented the spin connections are in Sect. 6. Section 7 concludes our work. 1 1 2 t = 0, u = 0, v = γ γ R . (2.8) 2 2 Geometric and gauge setup of the wormhole We will apply an external magnetic field such that the z- component Bz = B(z) is uniform with respect to the ( , ) In the (1 + 2)-dimensional curved spacetime, the line ele- plane x y and the magnetic flux through the circular area ment on the surface of wormhole (Fig. 1) can be cast in the enclosed by the wormhole at a fixed z is constant, namely ∼ / 2 following form Bz 1 R . Due to the axial symmetry, the electromagnetic four-potential can be expressed in the axial gauge as 2 μ υ 2 2 2 2 2 ds = gμυ dx dx =−c dt + du + R (u)dv . (2.1) 1 1 Aμ (t, x, y, z) = 0, − By, Bx, 0 , The fermion will experience the effective curvature that can 2 2 be addressed by considering the Dirac equation in curved and in the wormhole coordinates as spacetime ∂xν 1 a μ e ( , ,v)= ( , , , ) = , , 2 . γ e −h¯ ∇μ + i Aμ − Mc = 0, (2.2) Aμ t u μ Aν t x y z 0 0 BR (2.9) a c ∂x 2 123 Eur. Phys. J. C (2020) 80 :1111 Page 3 of 13 1111 The magnetic field is then given by of the effective potential arising from the wormhole geo- x y metrical structure. In the presence of external magnetic field B = − ∂z B, − ∂z B, B . (2.10) =× 2 2 B A along the z direction, the charged fermion mov- ing in v direction is expected to form a stationary state with We will now consider the Dirac equation of fermion in the quantized angular momentum and energy, i.e. the Landau wormhole in the presence of constant magnetic flux. levels in the curved space with hole. To show this, we need to solve for the stationary states of the system. We can start by considering −iD×Eq. (3.5)- E + Mc2 /hc¯ ×Eq. (3.6)to 3 The Dirac equation in magnetized wormhole obtain Utilizing the results from above equations, the Dirac equation E2 − M2c4 D2 + ϕ(u) = 0. (3.7) Eq. (2.2) can be written in the form then h¯ 2c2 ⎛ ⎞ ∂ + Mc For constant magnetic flux, the operator D2 in the equation i ct h¯ iD ⎝ ⎠ = 0, (3.1) of motion now takes the form −iD −i∂ + Mc ct h¯ 2 φ 2 2 2 R R R 1 D = ∂ + ∂ − + + ∂v − i u u 2 where D is a differential operator R 2R 2R R φ0 φ 3 R 1 R 2 1 ie −iσ ∂v − i .