Effects of Lorentz Symmetry Violation on a Relativistic Scalar Particle In

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clrptnili omcsrn pc-ieRf [32], Ref. space-time correlations string (Einstein–Podolsky–Rosen) cosmic EPR to in relativistic geometric subject potential particles relativistic [27], scalar scalar the Ref. a [29–31], [28], Refs. oscillator phases Ref. parti- Klein–Gordon quantum oscillator scalar the Dirac the Lan- [26], limit, par- [25], Ref. relativistic Dirac Ref. ticles oscillator the and oscilla- quantum quantization in harmonic the dau–He–McKellar–Wilkens [21–24], the ; Ref. and [20] cles quantum 19], Ref. 18, geometric 13, tor [12, [17], [14, Refs. Ref. Refs. phases system couplings Dirac Landau-type Rashba-like the neutral 16], hydro- [12–15], [10], [11], Refs. the Ref. conductivity Ref. particles of Hall effect in spectrum quantum Aharonov-Bohm-Casher [9], example, the atom for limit, gen investigated, nonrelativistic been the have (LSV) olation the of obtained. solutions be bound-state can the equation that Klein-Gordon see modified also Landau We standard the the that to the levels. comparison see on in and modified effects function gets wave result the the analyze and eigenvalue We energy (SME). Extension Model ainlAaeyGuiu,Asm 831 India 783331, Assam, Gauripur, Academy National Scal Relativistic AHMED FAIZUDDIN a on Systems Violation Quantum in Symmetry Particle Lorentz of Effects arXiv:2106.02458v2 symmetry Lorentz of ( effect tensor a the motion by under defined quantum breaking particle the scalar investigate therein). a we references of work, related present and the [1–8] In Refs. quantum (see, in systems investigated various been in have types background different spacetime of potential Klein-Gordon (-out) the with solving by equation bosons like particles, scalar of [gr-qc] 24 Jun 2021 (a) nqatmsses ffc fLrnzsmer vi- symmetry Lorentz of effect systems, quantum In nrdcin – Introduction. E-mail: Abstract PACS qaincnb band n h pcrmo nryadtew the and energy of spectrum the solu parameters. breaking and bound-state symmetry obtained, the Lorentz that be see can We equation investigated. is Extension fLrnzsmer ilto eemndb esr( tensor a by determined violation symmetry Lorentz of PACS PACS [email protected] 11.30.Qc 03.65.Pm 11.30.Cp nti ae,terltvsi unu yaiso scala a of dynamics quantum relativistic the paper, this In – h eaiitcqatmdynamics quantum relativistic The pnaeu n aitv ymtybreaking symmetry radiative and Spontaneous – eaiitcwv equations wave Relativistic – oet n Poincaré invariance and Lorentz – (a) K F ) µναβ u fteStandard the of out p-1 oie li-odnoclao ne iercentral linear a [35]. a and Ref. under of potential [34], oscillator influence Ref. Klein-Gordon the modified potential under a central field linear scalar and massive Coulomb-type a [33], Ref. inetbihdb esrfil sitoue nothe viola- coupling into symmetry introduced nonminimal is a Lorentz CPT-even field through The tensor equation the a Klein–Gordon and by called established 38] [42–48]. [37, are tion classes Refs. Refs. two sector sector These electromag- CPT-odd of spacetime. the properties SME in propagation waves the netic of the sector modify electromagnetic that the in contributions [41]. symmetry Ref. Lorentz in detail the in of found coefficients be cur- can The the violation to Model. on added Standard limits are the rent of scalars density Lorentz Lagrangian observer op- the field these Model and Standard the erators fields with contracted background suitably tensor-valued interac- are framework, fundamental this of In Model tions. an Standard [37–40], the Refs. known of CPT- (SME) is extension Extension the Model breaking Standard symmetry of framework the Lorentz as manifestations general the A low-energy and symmetry the fields. testing tensor ex- for vacuum of nonvanishing values mechanism violated of is pectation breaking appearance symmetry the symmetry Lorentz by the triggered spontaneous that a shown Ref. through is symmetry Lorentz It the fields, of [36]. violation tensor the a implies the through which for mechanism breaking the symmetry of extension spontaneous the is Model Standard hr r w ye rcasso Lorentz-violating of classes or types two are There the beyond scenario a with dealing of way possible A K F in ftemdfidKlein-Gordon modified the of tions ) µναβ v ucindpnso the on depends function ave u fteSadr Model Standard the of out atceudrteeffect the under particle r ar F. Ahmed et al.

g µν αβ given by 4 (KF )µναβ F (x) F (x), where g is the cou- Let us consider a possible scenario of Lorentz sym- pling constant whose mass dimension is 2, Fµν (x) is the metry violation determined by (κDE )rr = const = κ , − 1 electromagnetic (EM) field tensor, and (KF )µναβ corre- (κHB )zz = const = κ2, (κDB)rz = const = κ3 with the sponds to the CPT-even tensor of the electromagnetic sec- configuration of electric and magnetic fields given by Refs. tor of the SME Refs. [37–40] and it is a dimensionless ten- [13, 24, 27, 28]: sor. The tensor (K ) has the symmetries of the Rie- F µναβ λ mann tensor R and zero on double trace (K )µν = 0, B~ = B zˆ , E~ = r,ˆ (5) µναβ F µν r so it contains 19 independent real components (see, Refs. [37, 38, 44–48] for details). where B > 0,z ˆ is a unit vector in the z-direction, λ is the Therefore, the relativistic quantum dynamics of a scalar linear charge density of the electric charge distribution, particle under the effects of LSV Refs. [21–25, 27, 29, 37, andr ˆ is the unit vector in the radial direction. 38, 45–47] is given by Hence, equation (4) using the configuration (5) becomes 2 2 2 2 2 g ∂ ∂ 1 ∂ 1 ∂ ∂ gκ1 λ pµ p + (K ) F µν (x) F αβ (x) Ψ= M 2 Ψ, (1) + + + + Ψ µ F µναβ −∂t2 ∂r2 r ∂r r2 ∂φ2 ∂z2 − 2 r2 − 4 h i gκ B2 gκ λB where Ψ is the scalar wave function and M is the rest mass + 2 3 M 2 Ψ=0. (6) 1 2 − r − of the particle . The structure of this paper is as follows: in section 2, we Since the metric (2) is independent of time and symmet- establish the background of Lorentz symmetry violation ric with respect to translations along the z-axis, as well as defined by the tensor (KF )µναβ out of the SME. Then, we with respect to rotations, it is reasonable to write the to- analyze the behaviour of a relativistic scalar particle by tal wave function Ψ(t,r,φ,z) in terms of the radial wave solving the modified Klein-Gordon equation; in section 3, function ψ(r) as follows: we present our conclusions. − Ψ(t,r,φ,z)= ei ( ε t+mφ+kz) ψ(r), (7) Lorentz Symmetry Breaking Effects on a Rel- ativistic Scalar Particle. – We consider Minkowski where ε is the energy of the scalar particle, m = spacetime in cylindrical coordinates (t,r,φ,z) 0, 1, 2, .... are the eigenvalues of the z-component of the± angular± momentum operator, and k is a constant. ds2 = dt2 + dr2 + r2 dφ2 + dz2, (2) Substituting the solution (7) into the Eq. (6), we obtain − the following radial wave equation for ψ(r): where the ranges of the coordinates are < (t,z) < , 2 −∞ ∞ ′′ 1 ′ j δ r 0 and 0 φ 2 π. ψ (r)+ ψ (r)+ Λ2 ψ(r)=0, (8) ≥For the geometry≤ ≤ (2), the modified Klein-Gordon equa- r − − r2 − r tion (1) becomes where 2 2 2 2 ∂ ∂ 1 ∂ ∂ 1 ∂ 2 2 2 2 1 2 + + + + Ψ Λ = M + k ε gB κ2, −∂t2 ∂r2 r ∂r ∂z2 r2 ∂φ2 − − 2 g µν αβ 2 2 1 2 + (KF )µναβ F (x) F (x)Ψ= M Ψ. (3) j = m + g λ κ1, 4 r 2 δ = gλBκ3. (9) Based on Refs. [44–48], the tensor (KF )µναβ can be written in terms of 3 3 matrices that represent Let us now perform a change of variables via x =2Λ r. × the parity-even sector: (κDE)jk = 2 (KF )0j0k and Then, from Eq. (8) we have κ 1 ǫ ǫ K pqlm − ( HB )jk = 2 jpq klm ( F ) , and the parity-odd sec- 2 0jpq ′′ 1 ′ 1 j δ tor: (κDB )jk = (κHE )kj = ǫkpq (KF ) . The matri- ψ (x)+ ψ (x)+ ψ(x)=0, (10) − x −4 − x2 − 2Λ x ces (κDE)jk and (κHB )jk are symmetric and the matrices (κDB)jk and (κHE )kj have no symmetry. In this way, we By analysing the asymptotic behaviour of Eq. (10) for can rewrite (3) in the form : x 0 and x , we can write a possible solution to the Eq.→ (10) that→ can ∞ be expressed in terms of an unknown ∂2 ∂2 1 ∂ 1 ∂2 ∂2 + + + + Ψ function F (x) as −∂t2 ∂r2 r ∂r r2 ∂φ2 ∂z2 − x j 2 g i j g i j ψ(x)= x e F (x). (11) + (κDE)ij E E + (κHB )ij B B Ψ − 2 2 h i j 2 i Thereby, substituting the solution (11) into the Eq. (10), g (κDB)ij E B Ψ= M Ψ. (4) − we obtain the following differential equation : 1In the sign convention (−, +, +, +), momentum four-vector is µ ′′ ′ δ 1 defined p = (E, ~p). So, its covariant form is given by pµ = (−E, ~p). x F (x)+(1+2 j x) F (x)+ j F (x)=0, − µ − − 2 2 2 − −2Λ − − 2 Thus, p pµ = ( E +~p )= M using the mass-energy relation. Here, we have chosen the units c =1=¯h (12)

p-2 Scalar Particle under LSV which is called the confluent hyper-geometric equation breaking terms are no longer constant breaking momen- Refs. [49, 50]. The function F (x) is the confluent tum conservation. Studying the effects related to momen- 1 hyper-geometric function, that is, F (x) = 1F1(j + 2 + tum nonconservation is beyond the scope of the present δ article. After solving the radial wave equation, we have 2 Λ , 2 j +1, x). It is well-known that the confluent hyper- geometric series becomes a polynomial of degree n when obtained the energy eigenvalues Eq. (13) and the wave j + 1 + δ = n Refs. [49, 50] where, n =0, 1, 2, 3, ..... function Eq. (14) of a relativistic scalar particle. We can 2 2 Λ − After simplification of j + 1 + δ = n, we have ob- see that the presence of the Lorentz symmetry breaking 2 2 Λ − parameters (g,B,λ,κ ,κ ,κ ) implies the modified energy tained the energy eigenvalue εn,m as follows: 1 2 3 spectrum by Eq. (13) and the wave function by Eq. (14) in comparison to the results obtained in Ref. [24]. 2 2 1 2 εn,m = M + k gB ∆, (13) ± r − 2 ∗∗∗ 2 2 g λ κ3 κ 2 where ∆ = 2 + 1 2 1 2 . I sincerely acknowledge the kind referee(s) for his/her 2 n+ 2 +√m + 2 g λ κ1 ! valuable comments and suggestions. The radial wave function is given by

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