The Scattering of Dirac Spinors in Rotating Spheroids
aa r X i v : . [ a s t r o - ph . H E ] J u l Noname manuscript No. (will be inserted by the editor)
The Scattering of Dirac Spinors in Rotating Spheroids
Zhi Fu, Gao
Ci Xing, Chen † a,3 Na, Wang Xinjiang Astronomical Observatory, CAS, 150, Science 1-Street, Urumqi, Xinjiang, 830011, China. Key Laboratory of Radio Astronomy, CAS, Nanjing, Jiangshu, 210008, China Department of Astronomy, University of Sciences and Technology of China, CAS, Hefei 230026, China.Received: date / Accepted: date
Abstract
There are many stars that are rotating spheroidsin the Universe, and studying them is of very impor-tant significance. Since the times of Newton, many as-tronomers and physicists have researched gravitationalproperties of stars by considering the moment equationsderived from Eulerian hydrodynamic equations. In thispaper we study the scattering of spinors of the Diracequation, and in particular investigate the scatteringissue in the limit case of rotating Maclaurin spheroids.Firstly we give the metric of a rotating ellipsoid star,then write the Dirac equation under this metric, andfinally derive the scattering solution to the Dirac equa-tion and establish a relation between differential scat-tering cross-section, σ , and stellar matter density, µ . Itis found that the sensitivity of σ to the change in µ isproportional to the density µ . Because of weak gravita-tional field and constant mass density, our results arereasonable. The results can be applied to white dwarfs,main sequence stars, red giants, supergiant stars and soon, as long as their gravitational fields are so weak thatthey can be treated in the Newtonan approximations,and the fluid is assumed to be incompressible. Noticethat we take the star’s matter density to be its averagedensity and the star is not taken to be compact. Obvi-ously our results cannot be used to study neutron starsand black holes. In particular, our results are suitablefor white dwarfs, which have average densities of about10 − g cm − , corresponding to a range of mass ofabout 0 . − . M J and a range of radius of about6000 − a e-mail: [email protected] As we know, a physical model usually contains a set ofmost basic physical quantities, some of which can beused to describe the structure and evolution of mat-ter, while others can be used to describe the motionstate of an object. A mathematical model of the mo-tion of matter is usually composed of fundamental equa-tions, fundamental physical quantities and definite so-lution conditions such as initial conditions, boundaryconditions and joint conditions. The research method ofphysics is to abstract the objective world into a physicalmodel, and then convert it into a mathematical model.By solving the mathematical model, we can obtain thefunctional relationship between physical quantities andspace-time coordinates, then shall examine the physicalmodel by observation or experiment, and improve thephysical model based on comparisons.Assuming that three semi-axes of an elliposoid are a, b and c , respectively, we define a spheroid as an el-liposoid with a = b > c . As the most general physicalmodel, a rotating spheroid is composed of fluids (mag-netohydrodynamic, turbulence and perfect fluid) withthe pressure, density and temperature in a gravitationalfield, particle groups in the non-thermal equilibrium(distribution function), and boson and fermion fields.Especially for scalar fields, the fundamental equationsare the Klein-Gordon equations in flat space-time andcurved space-time, respectively [1],[2],[3],[4]. A theory ofgravity will be tested by the gravitational observationeffect in different celestial bodies. For the same theoryof gravity, different coupling coefficients give differentsolutions to the basic equations. There may exist sig-nificant differences between different theories of gravity,e.g., the theory of gravity with torsion, superstring the-ory and supergravity [5] [6] [7],[8]. In the nearly 80 years since the Dirac spinors weredefined, their roles in describing fermion fields havebeen well established and widely accepted. For conve-nience, let us review the previous research work con-ducted by other authors related to spinors: Villalbaand de Fisica [9] solved the massless Dirac equationin a non-stationary rotating causal Godel-type cosmo-logical universe using the method of separation of vari-ables; Dolan et al. [10] studied the scattering of massivespin-half waves by a Schwarzschild black hole using an-alytical and numerical methods; Deleo & Rotelli [11]developed the potential scattering of a spinor withinthe context of perturbation field theory; In 2011, Ste-fano et al. [12] studied Dirac spinors in Bianchi type-Icosmological models within the framework of torsional f ( R )-gravity, while Bord´e et al. [13] presented a second-quantized field theory of massive particles with a spinof a half or antiparticle in the presence of a weak gravi-tational field treated as spin two external field in a flatMinkowski background. In 2012, Poplawski [14] madeuse of Einstein-Cartan-Sciama-Kibble (ECSK) theoryof gravity to discuss non-singular, big-bounce cosmol-ogy from spinor-torsion coupling, and Daude and Kam-ran [15]considered massive Dirac fields evolving in theexterior region of a 5-D Myers-Perry black hole andstudied their propagation properties; In 2014, Brihayeet al.[16] studied Dirac equation for spherically sym-metric space-time and application to a boson star inEGB gravity, and Ambrus & Winstanley [17] discussedDirac fermions on an anti-desitter background; In 2015,Bini et al. [18] discussed massless Dirac particles inthe vacuum C -metric; In 2017, R¨oken [19] showed theseparability of the massive Dirac equation in the non-extreme Kerr geometry in the horizon-penetrating ad-vanced Eddington-Finkelstein coordinates; Dzhunushaliev& Folomeev [20] investigated Dirac star in the presenceof Maxwell and Proca fields, and Oliveira [21] investi-gated the influence of non-inertial and spin effects onthe 2 − D Dirac oscillator interacting with a uniformmagnetic field and with Aharonov-Bohm effects in thecosmic string space-time.As we know, there are two kinds of methods to studyDirac equation describing fermions in the vicinity ofdifferent types black holes. One is the Newman-Penrose(NP) formalism, and another is direct methods. In 2016,Batic et al. [22] first gave the Dirac equation in theSchwarzschild black hole metric to study the problemof embedded eigenvalues adopting the NP formalisms; In 2018, Kraniotis [3] [23] mmade use of the samemethods to derive the Dirac equation in the KerrC-NewmanCde Sitter (KNdS) black hole background us-ing a generalized Kinnersley null tetrad, and the sameyear Blazquez-Salcedo and Knoll & Knoll [24] used the direct methods to study the massive Dirac equation inthe near-horizon metric of the extremal five dimensionalMyers-Perry black hole with equal angular momenta.Moreover, Dariescu et al. [25] used the direct meth-ods to obtain the Dirac equation in the background ofthe Garfinkle-Horowitz-Strominger black hole. In 2020,Ahmad et al. [26] took advantage of the NP formalismto study the Dirac equation around a regular Bardeenblack hole surrounded by quintessence. The above-mentionedarticles are related to the Dirac equation in curvedspace-time.The remainder of this paper is organized as follows.In Sec. 2, we present the calculation method. In Sec-tion 3, we define connections and give the derivationsof related quantities. In Section 4, we derive the Diracequation in the weak gravitational field approximation.In Sec. 5, we give the scattering solutions to the Diracequation and discuss the scattering issue in a limit caseof rotating Maclaurin spheroids, and summarize thewhole paper and look forward to the future work inSec. 6. In an appendix we give concrete expressions ofthe Dirac equation in the weak gravitational potentialand write the matrix M and the scattering solutions ofthe Dirac equation. Let us emphasize that there are two approaches to studygravitational properties: (1) moment method, and (2)Diracfield scattering, particle geodesic motion and scalar fieldscattering. We will use the second approach, becausethe advantage of this approach is that the gravitationalproperties are closely related to the observations, and aphysical model will be constrained by observations. Inthis paper, we will discuss the physical effect of a ro-tating spheroid: the scattering of Dirac spinors, whichhas never been done before. A specific method is first togive the metric of a rotating spheroid, then write out theDirac equation under the metric,and find the scatter-ing solution, and finally give a relationship between thescattering cross-section and the stellar density. Accord-ing to the observed scattering amplitude, the density isexpected to be determined to study the gravitationalcharacteristics of spheroids, and the sensitivity of thescattering cross-section to the change of the density willbe discussed.In this paper, we will discuss the physical effectof a rotation spheroid: the scattering of Dirac spinors,which has never been done before. The specific methodis to first give the metric of the spheroid, then writethe Dirac equation under the metric, find the scatter-ing solution, and finally give the relation between the scattering cross-section and the star density. Accord-ing to the observed scattering amplitude, the gravita-tional properties of a rotating spheroid will be studiedby determining the density, and the sensitivity of thescattering cross-section to the density change will bediscussed.Here, employing the quantum field theory of curvedspace-time, we will study the scattering of spinors ofthe Dirac equation by taking the following three steps:(1)In the first step, we get the metric g µν , g µν ⇒ e ( a ) µ , e ν ( b ) , (1) Γ λµν ⇒ ω ( a ) µ ( b ) = − e ν ( b ) ( ∂ µ e ( a ) ν − Γ λµν e ( a ) λ ) . (2)(2)In the second step, we will study the Dirac equationin a curved space-time (weak gravitational field) iγ ( c ) e µ ( c ) D µ Ψ ( a ) − mΨ ( a ) = 0 . (3)where a = 0 , , , D µ Ψ ( a ) isgiven as D µ Ψ ( a ) = ∂ µ Ψ ( a ) + ω ( a ) µ ( b ) Ψ ( b ) (4)with Ψ ( a ) = ( φ ( a ) , χ ( a ) ) T . For convenience, we replace γ ( c ) with γ k ( D ) , which is dependent on the Pauli matrix σ k . Also we get the second term on the right side ofEq. (4), namely Π ( a ) µ ≡ ω ( a ) µ ( b ) Ψ ( b ) , (5)where the summation convention is used. At the sametime, the term of γ ( c ) e µ ( c ) will be calculated, then weobtain the term of γ ( c ) e µ ( c ) D µ Ψ ( b ) . Therefore, we caninvestigate the Dirac equation in curved space-time.Rearranging the above Dirac equation leads to ∂ Π − σ k · ▽ Π + P Π = 0 , (6)For the physical meanings of quantities in Eq (6), tosee the following sections. (3)In the last step, we willobtain the scattering solution of the Dirac equation ina curved space-time i.e., Ψ ( x ) = ϕ ( x ) − lim ε → ∞ Z −∞ ∞ Z −∞ ∞ Z −∞ d ´ xG ± ε ( x , ´ x ) V (´ x ) Ψ (´ x ) , (7) where the Green function G ± ε ( x , ´ x ) = H − E ± iε δ ( x − ´ x ),with the first-order Hamiltonian H = − σ k · ▽ + P ± iε. (8)Using the first-order weak gravitational field approxi-mation, the Ψ (´ x ) on the right side of Eq. (4) is replacedby the solution of free particle equation, and the scat-tering solution is obtained. The Newtonian limit corresponds to the following lineelement [30] ds = (1 + 2 Φ )( dx ) − (1 − Φ )[( dx ) + ( dx ) +( dx ) ] ≡ η ( a )( b ) ω ( a ) ω ( b ) , ω ( a ) = e ( a ) µ dx µ . (9) From Eq. (9), it is obtained that e ( a ) µ = diag (1 + Φ, − Φ, − Φ, − Φ ) ,e ν ( b ) = diag (1 − Φ, Φ, Φ, Φ ) . (10)According to the metric above, we get nonvanishingconnections of Γ λµν ( µ, ν, λ = 0 , , ,
3, and there is nosummation over repeated indices)(1)Case of Γ µν Γ i = Γ i = (1 − Φ ) ∂Φ∂x i , ( i = 1 , , (2)Case of Γ jµν ( j = 1 , , Γ jij = Γ jji = − (1 + 2 Φ ) ∂Φ∂x i ( i, j = 1 , , ,Γ jii = (1 + 2 Φ ) ∂Φ∂x j ( i, j = 1 , , , i = j ); (12) Based on Eq. (2), the quantities ω ( a ) µ ( b ) can be calculatedas follows. In the following cases of (1-4), we set a, b, c =1 , ,
3, and there is no summation over the indices(1)Case of ω (0)0(0) , ω (0)0( b ) , ω ( a )0(0) and ω ( a )0( b ) : ω (0)0( b ) = (1 + Φ ) (1 − Φ ) ∂Φ∂x b ,ω ( a )0(0) = (1 − Φ ) (1 + 2 Φ ) ∂Φ∂x a , ω (0)0(0) = ω ( a )0( b ) = 0; (13) (2)Case of ω (0) c (0) ; ω (0) c (0) = − (1 − Φ ) ∂Φ∂x c + (1 − Φ )(1 − Φ ) ∂Φ∂x c ; (14) (3)Case of ω (0) c ( b ) and ω ( a ) c (0) : ω (0) c ( b ) = − (1 + Φ ) ∂Φ∂x c , ω ( a ) c (0) = (1 − Φ ) ∂Φ∂x c ; (15) (4)Case of ω ( a ) c ( b ) : ω ( a ) c ( a ) = (1 + Φ ) ∂Φ∂x c − (1 − Φ )(1 + 2 Φ ) ∂Φ∂x c ,ω ( a ) c ( c ) = (1 + Φ ) ∂Φ∂x c + (1 − Φ )(1 + 2 Φ ) ∂Φ∂x a , ( a = c ) ,ω ( a ) a ( b ) = (1 + Φ ) ∂Φ∂x a − (1 − Φ )(1 + 2 Φ ) ∂Φ∂x b , ( a = b ) ω ( a ) c ( b ) = ω ( b ) c ( a ) = (1 + Φ ) ∂Φ∂x c , ( a = b = c ) . (16) As shown in Section 2, the Dirac equation in curvedspace-time in a weak gravitational field approximationcan be described by Eq. (3) and Eq. (4), with the wavefunction Ψ ( a ) = ( φ ( a ) , χ ( a ) ) T , φ ( a ) = ( φ ( a )1 , φ ( a )2 ) T and χ ( a ) = ( χ ( a )1 , χ ( a )2 ) T , where a = 0 , , ,
3. There are alsomounting concerns that mΨ ( a ) = m (cid:18) φ ( a ) χ ( a ) (cid:19) and D µ Ψ ( a ) = D µ (cid:18) ϕ ( a ) χ ( a ) (cid:19) = (cid:18) ∂ µ φ ( a ) ∂ µ χ ( a ) (cid:19) + ω ( a ) µ ( b ) (cid:18) φ ( b ) χ ( b ) (cid:19) . (17) For convenience, by making the following substitution γ ( c ) → γ k ( D ) , γ D ) = (cid:18) I − I (cid:19) , γ k ( D ) = (cid:18) σ k − σ k (cid:19) , (18) it is easy to obtain γ ( c ) e c ) = (cid:18) I − I (cid:19) and γ ( c ) e k ( c ) = (cid:18) σ k − σ k (cid:19) , here k = 1 , ,
3, and σ k is the Pauli matrix Notice that we reconsider Eq. (17), and the second termon its right side is denoted as Π ( a ) µ = ω ( a ) µ ( b ) (cid:18) ϕ ( b ) χ ( b ) (cid:19) , µ =0 , , , . The concrete values of Π ( a ) µ are written in amore compact way as follows Π (0)0 = X b =1 ∂Φ∂x b Ψ ( b ) , Π ( a )0 = ∂Φ∂x a Ψ (0) ,Π (0) c = − Φ ∂Φ∂x c Ψ (0) − X b =1 (1 + Φ ) ∂Φ∂x c Ψ ( b ) ,Π ( a ) a = (1 − Φ ) ∂Φ∂x a Ψ (0) + X b =1 [(1 + Φ ) ∂Φ∂x a − (1 + 2 Φ ) ∂Φ∂x b ] Ψ ( b ) ( b = a ) − Φ ∂Φ∂x a Ψ ( a ) ,Π ( a ) c ( c = a ) = (1 − Φ ) ∂Φ∂x c Ψ (0) + [(1 + Φ ) ∂Φ∂x c + (1 + 2 Φ ) ∂Φ∂x a ] Ψ ( c ) + (1 + Φ ) ∂Φ∂x c Ψ ( b ) ( b = a = c ) − Φ ∂Φ∂x c Ψ ( a ) . (19) (Here, we don’t adopt Einstein convention in order toavoid confusion.)In order to study the Dirac equation in curved space-time, we also calculate the quantity of iγ ( c ) e µ ( c ) D µ Ψ ( a ) iγ ( c ) e µ ( c ) D µ Ψ ( b ) = i (cid:18) I − I (cid:19) (1 − Φ )[ ∂ (cid:18) φ ( b ) χ ( b ) (cid:19) + Π ( b )0 ]+ iσ i (cid:18) I − I (cid:19) (1 + Φ )[ ∂ i (cid:18) ϕ ( b ) χ ( b ) (cid:19) + Π ( b ) i ] , (20) In quantum mechanism the three components of the Paulimatrix under the 2 × σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , and σ = (cid:18) − (cid:19) . where b = 0 , , , ∂ Π − σ k · ▽ Π + P Π = 0 , (21)where Π = ( φ (0) , χ (0) , φ (1) , χ (1) , φ (2) , χ (2) , φ (3) , χ (3) ) T , σ k is the Pauli matrix and P is a [8 ×
8] matrix, which isrelated to gravitational potential, the matrix elementsare given as following:1. For the first row, P = im, P = 0 , P = ∂Φ∂x ,P = P = P = − ∂Φ∂x i σ i ( i = 1 , , ,P = ∂Φ∂y , P = ∂Φ∂z ; (22)
2. For the second row, P = 0 , P = − im,P = P = P = − ∂Φ∂x i σ i ( i = 1 , , ,P = − ∂Φ∂x , P = ∂Φ∂y , P = ∂Φ∂z ; (23)
3. For the third row, P = ∂Φ∂x , P = ∂Φ∂x i σ i ( i = 1 , , ,P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = im, P = P = P = 0; (24)
4. For the fourth row, P = ∂Φ∂x , P = ∂Φ∂x i σ i ( i = 1 , , ,P = P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = P = P = 0 , P = − im ; (25)
5. For the fifth row, P = ∂Φ∂y , P = ∂Φ∂x i σ i ( i = 1 , , ,P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = im, P = P = P = 0; (26)
6. For the sixth row, P = ∂Φ∂y , P = ∂Φ∂x i σ i ( i = 1 , , ,P = P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = P = P = 0 , P = − im ; (27)
7. For the seventh row, P = ∂Φ∂z , P = ∂Φ∂x i σ i ( i = 1 , , ,P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = P = P = 0 , P = im ; (28)
8. For the eighth row, P = ∂Φ∂z , P = ∂Φ∂x i σ i ( i = 1 , , ,P = P = ∂Φ∂x i σ i + ε ij ( ∂Φ∂x i σ j ) ,P = P = ∂Φ∂x i σ i − ε ij ( ∂Φ∂x i σ j ) ,P = P = P = 0 , P = − im, (29) where ε kij is the Levi-Civita symbol.By defining the matrix of P ∗ ≡ P − P that is asmall quantity, we get ∂ Π + HΠ = 0 , and H = H + P ∗ = ( − σ k · ∇ + P ) + P ∗ , (30)where P =diag( im, − im, im, − im, im, − im, im, − im ).In order to get the scattering solution to the aboveDirac equation using a perturbation theory, for conve-nience, we first set Φ = V e iwt , ( H − E ) V = − P ∗ V and make the substitutions of V → | Ψ > , P ∗ → V and iw → E , then we obtain ( H − E ) | Ψ > = − V | Ψ > and H | ϕ > = E (0) | ϕ > . The main research objects of quantum mechanics aredivided into two types: bound states and scatteringstates. Theoretically, the scattering state is a non-boundstate, which involves the continuous region of the en-ergy spectrum of a system. One can freely control theenergy of incident particles, which is different from deal-ing with particles in the bound state. The bound statetheory mainly involves the eigenvalues and eigenstatesof discrete, quantized energies of the system [31], [32], [33]..The scattering theory mainly deals with the redistribu-tion of scattering particles and their properties (such aspolarization, correlation, etc.) in the scattering process.By analyzing the scattering results, one can find thestructures inside particles, which promotes the devel-opment of basic theories. From trapped atoms to liber-ated quarks, a better understanding of the structure ofmatter depends largely on the study of scattering [34]. 5.1 The scattering solution to the Dirac equation in aweak gravitational field of a rotating spheroidIn this subsection, at first, we focus on a solution to theequation of Dirac spinors of free particles, | ϕ > = | ϕ >e − i k · r , which satisfies the following equation( σ k · i k + P ) | ϕ > = E (0) | ϕ > . (31)From the above expression, it is obvious that the secularequation is written as [det( σ k · i k + im − E (0) )det( σ k · i k − im − E (0) )] = 0 , (32) where σ k · i k = σ ik x + σ ik y + σ ik z = (cid:18) ik z ik x + k y ik x − k y − ik z (cid:19) . (33) There are four different solutions to Eq. (32): E (0) = im ± ik and E (0) = − im ± ik . As the space is lim-ited, we only choose the solution of E (0) = im + ik . Itshould be noted that the choice of E depends on theobservation of the scattering cross section (see the endof this section). Supposing k x = k y = 0 and k = k z andsolving the above eigenvalues, we obtain φ (0) = φ (1) = φ (2) = φ (3) = e ik z z (cid:18) (cid:19) ,χ (0) = χ (1) = χ (2) = χ (3) = 0 . (34) According to the scattering formula in quantum me-chanics, we have Ψ ( x ) ≈ ϕ ( x ) − lim ε → Z d x ′ G ± ε ( x , x ′ ) V ( x ′ ) ϕ ( x ′ ) ,G ± ε ( x , x ′ ) = 1 H − E ± iε δ ( x , x ′ )= 1 − σ k · ∇ + P − E ± iε δ ( x , x ′ ) , (35) Based on the expressions of σ k ·∇ = (cid:18) ∂ z ∂ x − i∂ y ∂ x + i∂ y − ∂ z (cid:19) and δ ( x − x ′ ) = π ) R d pe − i p · ( x − x ′ ) , we first set ε →
0, and define the matrices of H , H , and H as fol-lows: H = − σ k · ∇ + P − E ± iε,H = − σ k · ∇ + im − E ± iε,H = − σ k · ∇ − im − E ± iε. (36)Then we have H = (cid:18) ip z + im − E ± iε ip x + p y ip x − p y − ip z + im − E ± iε (cid:19) H = ip z − im − E ± iε ip x + p y ip x − p y − ip z − im − E ± iε . (37) Accordingly, their inverse matrices are written as H − = 1( im − E ± iε ) + p × (cid:18) − ip z + im − E ± iε − ip x + p y − ip x − p y ip z + im − E ± iε (cid:19) H − = 1( im − E ± iε ) + p × − ip z − im − E ± iε − ip x + p y − ip x − p y ip z − im − E ± iε . (38) Thus, the Green’s function is given as G ± ε ( x , x ′ ) = H − δ ( x − x ′ )= H − π ) Z d pe − i p · ( x − x ′ ) = 1(2 π ) Z d p diag ( H − , H − , H − ,H − , H − , H − , H − , H − ) · e − i p · ( x − x ′ ) , (39) and the scattering formula of Eq. (35) becomes Ψ ( x , t ) ≈ ϕ ( x , t ) − π ) lim ε → Z d x ′ Z d p diag ( H − , H − , H − , H − , H − ,H − , H − , H − ) e − i p · ( x − x ′ ) V ( x ′ ) ϕ ( x ′ , t ) . (40) where ϕ ( x , t ) = e iwt ϕ ( x ) is the eigenfunction men-tioned above. Letting S ≡ (cid:18) (cid:19) , we obtain ϕ ( x ) = e ik z z S (1 , , , , , , , T . (41)Next, we’re going to derive V ( x ) ϕ ( x ) ≡ P ∗ ( x ) ϕ ( x ). Itshould be noticed that P ∗ is a [8 ×
8] matrix, while ϕ ( x )is a [8 ×
1] matrix. If both of S and σ are assumed tobe numbers, the matrix elements of [ P ∗ ( x ) ϕ ( x )] wouldbe given as follows.1. For the first row, [ P ∗ ( x ) ϕ ( x )] = ( ∂Φ∂x + ∂Φ∂y + ∂Φ∂z ) e − ik z z S = e − ik z z ∂Φ∂x + ∂Φ∂y + ∂Φ∂z ! , (42)
2. For the second row, [ P ∗ ( x ) ϕ ( x )] = − ∂Φ∂x σ + ∂Φ∂y σ + ∂Φ∂z σ ) × e − ik z z S = − e − ik z z ∂Φ∂z∂Φ∂x + i ∂Φ∂y ! , (43)
3. For the third row [ P ∗ ( x ) ϕ ( x )] = ∂Φ∂x e − ik z z S = e − ik z z (cid:18) ∂Φ∂x (cid:19) , (44)
4. For the forth row [ P ∗ ( x ) ϕ ( x )] = [(3 ∂Φ∂x − ∂Φ∂y − ∂Φ∂z ) σ + (3 ∂Φ∂y + ∂Φ∂x ) σ + (3 ∂Φ∂z + ∂Φ∂x ) σ ] e − ik z z S = e − ik z z ∂Φ∂z + ∂Φ∂x (3 + i ) ∂Φ∂x + ( − i ) ∂Φ∂y − ∂Φ∂z ! , (45)
5. For the fifth row, [ P ∗ ( x ) ϕ ( x )] = ∂Φ∂y e − ik z z S = e − ik z z ∂Φ∂y ! , (46)
6. For the sixth row [ P ∗ ( x ) ϕ ( x )] = [(3 ∂Φ∂x + ∂Φ∂y ) σ + (3 ∂Φ∂y − ∂Φ∂x − ∂Φ∂z ) σ + (3 ∂Φ∂z + ∂Φ∂y ) σ ] e − ik z z S = e − ik z z ∂Φ∂z + ∂Φ∂y (3 − i ) ∂Φ∂x + (1 + 3 i ) ∂Φ∂y − i ∂Φ∂z ! (47)
7. For the seventh row, [ P ∗ ( x ) ϕ ( x )] = ∂Φ∂z e − ik z z S = e − ik z z (cid:18) ∂Φ∂z (cid:19) , (48)
8. For the eighth row, [ P ∗ ( x ) ϕ ( x )] = [(3 ∂Φ∂x + ∂Φ∂z ) σ + (3 ∂Φ∂y + ∂Φ∂z ) σ + (3 ∂Φ∂z − ∂Φ∂x − ∂Φ∂y ) σ ] e − ik z z S = e − ik z z ∂Φ∂z − ∂Φ∂x − ∂Φ∂y ∂Φ∂x + 3 i ∂Φ∂y + (1 + i ) ∂Φ∂z . ! (49) We write the scattering solution of the Dirac equation Ψ ( x , t ) ≈ e iωt ϕ ( x ) − e iωt lim ε → π ) Z d p diag ( H − , H − , H − , H − , H − , H − ,H − , H − ) e − i p · x Z d x ′ e i p · x ′ P ∗ ( x ′ ) ϕ ( x ′ ) . (50) Defining M ≡ R d x ′ e i p · x ′ P ∗ ( x ′ ) ϕ ( x ′ ), which is a[8 ×
1] matrix, the eight matrix entries are given inAppendix B.Let us continue exploring the matrix entries M ∼ M . Firstly, the relations between Cartesian coordi-nates x, y , and z and oblate elliptic coordinates ξ, η and ϕ are given as follows x = ρ p (1 + ξ )(1 − η ) cos ϕ,y = ρ p (1 + ξ )(1 − η ) sin ϕ,z = ρ ξη, (0 ≤ ξ < ∞ , − ≤ η ≤
1) (51)where ρ is the focal length of a rotating spheroid, andthe azimuth ϕ ∈ (0 , π ). By making a coordinate trans-formation ( x, y, z ) → ( ξ, η, ϕ ), we get d x ≡ dxdydz = ρ ( ξ + η ) dξdηdϕ . When ξ > ξ , the gravitational po-tential Φ of a rotating spheroid with constant density µ is given by Φ = − Mρ { arccot ξ + 34 [ ξ − ( ξ + 13 )arccot ξ ](1 − η ) } , (52) where M is the mass of an ellipsoid star, which canbe estimated as M = 4 / πa cµ = 4 / × πρ µ (1 + ξ ) ξ ,( a = b = ρ p ξ is the semi-axis length in theequatorial plane, and c = ρ ξ is the semi-axis length in the axis of rotation). When 0 ≤ ξ ≤ ξ , the gravita-tional potential Φ becomes Φ = V + 12 Ω ρ (1 + ξ )(1 − η ) − C × (cid:20) − ρ (1 + ξ )(1 − η ) a + C ρ ξ η c (cid:21) , (53) where V = A ( ξ ), Ω = p B ( ξ ) and A = − M [( ξ + 1)arccot ξ − ξ ]2 ρ , B = 3 M × ρ (1 + ξ )[(3 ξ + 1)arccot ξ − ξ ] , C = 3 Mξ (1 − ξ arccot ξ )2 ρ . (54) For details about the gravitational potential Φ , seeChandrasekhar [32]. From Eq. (B6), we get the matrixentry M = − Z ∞ Z − Z π dξdηdϕρ ( ξ + η ) × exp[ ip x ρ q (1 + ξ )(1 − η ) cos ϕ + ip y ρ q (1 + ξ )(1 − η ) sin ϕ + i ( p z − k z ) ρ ξη ] × (cid:18) ip x + ip y + i ( p z − k z )0 (cid:19) Φ. (55) For convenience, we introduce the notation T to denotethe following integral, T = − Z ∞ Z − Z π dξdηdϕρ ( ξ + η ) × exp[ ip x ρ q (1 + ξ )(1 − η ) sin ϕ + ip y ρ q (1 + ξ )(1 − η ) cos ϕ + i ( p z − k z ) ρ ξη ] Φ. (56) Thus, the matrix entry of M becomes M ≡ (cid:18) ip x + ip y + i ( p z − k z )0 (cid:19) T. (57) In the same way, we get the other seven matrix entries M ≡ (cid:18) − i ( p z − k z ) − ip x + 3 p y (cid:19) T, M ≡ (cid:18) ip x (cid:19) T, (58) M ≡ (cid:18) i ( p z − k z ) + ip x (3 + i ) ip x + ( − i + 3 i ) ip y − i ( p z − k z ) (cid:19) T, (59) M ≡ (cid:18) ip y (cid:19) T, M ≡ (cid:18) i ( p z − k z )0 (cid:19) T, (60) M ≡ (cid:18) i ( p z − k z ) + ip y (3 − i ) ip x + (1 + 3 i ) ip y + ( p z − k z ) (cid:19) T, (61) and M ≡ (cid:18) i ( p z − k z ) − ip x − ip y ip x − p y + (1 + i ) i ( p z − k z ) (cid:19) T. (62) Since the system we are studying is a rotating ellip-soid star, its gravitational potential Φ does not involveazimuth ϕ . Seeing from Eq. (55)-(62), the matrix en-tries of M and the symbol T are functions of p x , p y and p z . In the following, the scattering solution to theDirac equation will be reexpressed with M or T . Thenwe have Ψ ( x , t ) = e iωt ϕ ( x ) − e iωt lim ε → π ) Z d p diag ( H − ,H − , H − , H − , H − , H − , H − , H − ) e − i p · x M. (63) where the expression of Ψ ( x , t )= ( φ (0) , χ (0) , φ (1) , χ (1) ,φ (2) , χ (2) , φ (3) , χ (3) ) is used. 5.2 A special case: Maclaurin spheroidsIn this part, we will study the scattering of Dirac spinorsin a special limiting case of Maclaurin spheroids. Tounderstand the properties of Maclaurin spheroids, atfirst, let us discuss the motion of scalar particles in arotating Maclaurin spheroid belonging to a special classof boson stars. When the distance between the centerof the spheroid and a scalar particle, r , is larger thanthe stellar radius R , the exterior potential is given by Φ ex = − M/r , corresponding to its partial derivative( −∇ Φ ext ) r = M/r and the acceleration of gravity, a r = − M/r [30]. From the Kepler’s law, we get theangular velocity of the particle ω = M/r (or M/r = rω from the Newton Second Law of Motion). Simi-larly, when ( r ≤ R , the interior potential is given by Φ in = − πµ ( R − r / ∇ Φ in ) r =4 / × πµr and ω = 4 / × πµ .According to statistical mechanics, the scatteringcross section, also known as the collision section, is aphysical quantity describing the scattering probabilityof microscopic particles. The dimension of the scatter-ing cross-section is the same as that of the area. Onecore issue of scattering theory is to solve the scatter-ing amplitude by studying the probability of the parti-cles being scattered to the unit solid angle in the direc-tion of ( p, θ, ϕ ). This probability can be expressed bythe scattering differential cross-section σ ( p, θ, ϕ ), deter-mined by the amplitude of the spherical scattered wave f ( p, θ, ϕ ), i.e. σ ( p, θ, ϕ ) = | f ( p, θ, ϕ ) | .In order to evaluate T in a rotating Maclaurin spheroid,we define another new quantity as p = − p x e x − p y e y − ( p z − k z ) e z here e x , e y and e z are the unit vectors inthe x , y and z directions, respectively, then we have T = Z + ∞−∞ d x e i p · x − ip z z Φ = Z R dr Z π dθ Z π dϕr sin θe − ip r cos θ × [ − πµ ( R − r Z + ∞ R dr Z π dθ Z π dϕ × r sin θe − ip r cos θ ( − Mr )= 8 π µRP ( R p ) cos( p R )+ 4 πMp (cos ∞ − cos( p R )) ≈ π µRP ( R p ) cos( p R ) − πMp cos( p R )= ( − π µR p + 8 π µRp ) cos( p R ) . (64) Since the scattering field is anisotropic, we can givethe values of the scattering amplitude, f i ( p, θ, ϕ ) ( i =0 , , , , , , , i = 0, i.e. ϕ (0) . The scattering amplitude f ( p, θ, ϕ ) is determined by the following expression: f ( p, θ, ϕ ) = lim ε → ik z ∓ iε ) + p × (cid:18) − ip z − ik z ± iε − ip x + p y − ip x − p y ip z − ik z ± iε (cid:19) × (cid:18) ip x + ip y + i ( p z − k z )0 (cid:19) T = lim ε → ik z ∓ iε ) + p × (cid:18) − ip cos θ − ik z ± iε − ip cos ϕ sin θ + p sin ϕ sin θ − ip cos ϕ sin θ − p sin ϕ sin θ ip cos θ − ik z ± iε (cid:19) × (cid:18) ip cos ϕ sin θ + ip sin ϕ sin θ + i ( p cos θ − k z )0 (cid:19) T. (65) We also obtain the average value of f ( p, θ, ϕ ), ¯ f ( p, θ ) = 12 π Z π f ( p, θ, ϕ ) dϕ = (cid:18) cos θ (1 − i ) sin θ (cid:19) T. (66) The probability of scattering particle in the interval of p ∼ p + dp and θ ∼ θ + dθ is determined by dW i d p = dW i p sin θdpdθ = | ¯ f i ( p, θ ) | . (67)Taking the long-wavelength approximation k z →
0, andpaying attention to p = − p x e x − p y e y − ( p z − k z ) e z ,we get the first component of ¯ f ( p, θ ),¯ f +0 ( p, θ ) = cos θ [ − π µR p + 8 π µRp ] cos( pR ) , (68)and its corresponding scattering cross-section, ¯ σ +0 ( p, θ ) = | f +0 ( p, θ ) | = cos θ [ − π µR p + 8 π µRp ] cos ( pR ) . (69) In the same way, the second component of the scatter-ing amplitude is given as ¯ f − ( p, θ ) = 12 (1 − i ) sin θ [ − π µR p + 8 π µRp ] cos( pR ) , (70) corresponding to the scattering cross-section ¯ σ − ( p, θ ) = 12 sin θ [ − π µR p + 8 π µRp ] cos ( pR ) . (71) Substituting Eq. (64) into the scattering solution, wefind that the scattering cross-section is proportional to µ , and depends on the radius of the rotating star.In addition, we can determine the constant density µ from observations, ¯ σ ± ( p, θ ). From the above expression,it is obvious that the higher the density µ , the largerthe sensitivity of scattering amplitude ¯ σ ± ( p, θ ) with re-gard to µ . Notice that above we have chosen the long-wavelength approximation E = − im ± ik ≈ − im , andthe star is not too compact. The properties of whitedwarfs (WDs) in Einstein- ∧ gravity were investigatedby Liu and L¨u [35]. Considering the temperature ef-fects [36], the massive WD, with a mass of about 0.61 M J and a radius of about 6000 km, has an average density of about 10 g cm − , while the lower-mass WD,with a mass of about 0.21 M J and a radius of about10,000 km, has an average density of about 10 g cm − ,where M J is the mass of the sun. Thus, our results arereasonable for WDs.Similar to φ (0) , the scattering amplitudes for otherseven quantities χ (0) , φ (1) , χ (1) , φ (2) , χ (2) , φ (3) and χ (3) also can be obtained. However, due to limited space,the expressions of these seven scattering amplitudes willnot be given in detail. Next, we shall continue to takethe long-wavelength approximation. After a long butstraightforward calculation, the scattering amplitudesare given as follows:¯ f ∗ ( p, θ ) = 12 π Z π f ∗ ( p, θ, ϕ ) dϕ = 1 p − m (cid:18) p cos θ + 2 imp cos θ (1 − i ) p sin θ (cid:19) T ≡ (cid:18) ¯ f +0 ∗ ( p, θ )¯ f − ∗ ( p, θ ) (cid:19) (72)From the above expression, we obtain the two compo-nents of ¯ σ ∗ ( p, θ ),¯ σ +0 ∗ ( p, θ ) = | ¯ f +0 ∗ ( p, θ ) | = 1( p − m ) ( p cos θ + 4 m p cos θ ) × [ − π µR p + 8 π µRp ] cos ( pR ) , (73)and¯ σ − ∗ ( p, θ ) = | ¯ f − ∗ ( p, θ ) | = 12( p − m ) × p sin θ × [ − π µR p + 8 π µRp ] cos ( pR ) . (74)From the above, we find that scattering cross sections¯ σ ± ( E = im ± ik ≈ im ) are independent of the massof particles, m , while other scattering cross sections¯ σ ± ∗ ( E = − im ± ik ≈ − im ) depend on m . Thereforethe observations of scattering cross sections can deter-mine which form of energy ( E = ± im ± ik ≈ ± im )should be adopted in our physical systems. In this work, we have studied the scattering of spinors inthe Dirac equation in detail, and discussed the scatter-ing issue in the limit case of rotating Maclaurin spheroids,and established a relationship between the scatteringcross section σ and the density of matter µ . We alsofound: the higher the density µ , the higher the sensi-tivity of the scattering cross section σ to the change of µ . The results can be applied to all stars that canbe treated with the Newtonian approximation approxi-mately. In the future we’ll determine the constant den-sity of the star from the observed values of σ +0 ( p, θ, ϕ ).Here we provide the prospect of the follow-up work,and study the direction continuing being advanced. First,we’ll find out the other seven scattering cross sections¯ σ ± i ( p, θ )( E = im ± ik ≈ im ), and ¯ σ ± i ∗ ( p, θ )( E = − im ± ik ≈ − im ), ( i = 1 , , , , , , h νυ away from flat space-time, defined as g µν = η µν + h µν , and explore the gravitational properties of rotat-ing spheroids, as well as the scattering solution undersupergravity. Acknowledgements
We are grateful the referee for helpfulcomments. This work was supported by National Key Re-search and Development Program of China under grant num-ber 2018YFA0404602, and Chinese National Science Founda-tion through grants No. 11673056 and 11173042, and XinjiangNatural Science Foundation No.2018D01A24 and Guizhouprovincial Science and Technology Project ([2017]7349, [2019]1241).This work was also supported by CAS Light of West ChinaProgram (No.2016-QNXZ-B-25), Xiaofeng Yangs Xinjiang TianchiBairen project and CAS Pioneer Hundred Talents Program.
Appendix A: The concrete expressions of theDirac equation in the weak gravitationalpotential
Here iγ ( c ) e µ ( c ) D µ Ψ ( b ) − mΨ ( b ) = 0 ( b = 0 , , ,
3) arewritten as follows Q b ( φ, χ ) (cid:18) (cid:19) + Q b ( φ, χ : φ ↔ χ ) (cid:18) − (cid:19) − m (cid:18) Φ ( b ) χ ( b ) (cid:19) = 0 , (A.1) where Q = i [ ∂ φ (0) + ∂Φ∂x φ (1) + ∂Φ∂y φ (2) + ∂Φ∂z φ (3) ]+ iσ [ ∂ χ (0) − ∂Φ∂x χ (1) − ∂Φ∂x χ (2) − ∂Φ∂x χ (3) ]+ iσ [ ∂ χ (0) − ∂Φ∂y χ (1) − ∂Φ∂y χ (2) − ∂Φ∂y χ (3) ]+ iσ [ ∂ χ (0) − ∂Φ∂z χ (1) − ∂Φ∂z χ (2) − ∂Φ∂z χ (3) ] , (A.2) Q = i [ ∂ φ (1) + ∂Φ∂x φ (0) ] + iσ [ ∂ χ (1) + ∂Φ∂x χ (0) + ( ∂Φ∂x − ∂Φ∂y ) χ (2) + ( ∂Φ∂x − ∂Φ∂z ) χ (3) ] + iσ [ ∂ χ (1) + ∂Φ∂y χ (0) + ( ∂Φ∂y + ∂Φ∂x ) χ (2) + ∂Φ∂y χ (3) ] + iσ [ ∂ χ (1) + ∂Φ∂z χ (0) + ∂Φ∂z χ (2) + ( ∂Φ∂z + ∂Φ∂x ) χ (3) ] , (A.3) Q = i [ ∂ φ (2) + ∂Φ∂y φ (0) ] + iσ [ ∂ χ (2) + ∂Φ∂x χ (0) + ( ∂Φ∂x + ∂Φ∂y ) χ (1) + ∂Φ∂x χ (3) ] + iσ [ ∂ χ (2) + ∂Φ∂y χ (0) + ( ∂Φ∂y − ∂Φ∂x ) χ (1) + ( ∂Φ∂y − ∂Φ∂z ) χ (3) ]+ iσ [ ∂ χ (2) + ∂Φ∂z χ (0) + ∂Φ∂z χ (1) + ( ∂Φ∂z + ∂Φ∂y ) χ (3) ] , (A.4) and Q = i [ ∂ φ (3) + ∂Φ∂z φ (0) ] + iσ [ ∂ χ (3) + ∂Φ∂x χ (0) + ( ∂Φ∂x + ∂Φ∂z ) χ (1) + ∂Φ∂x χ (2) ]+ iσ [ ∂ χ (3) + ∂Φ∂y χ (0) + ∂Φ∂y χ (1) + ( ∂Φ∂x + ∂Φ∂z ) χ (2) ]+ iσ [ ∂ χ (3) + ∂Φ∂z χ (0) + ( ∂Φ∂z − ∂Φ∂x ) χ (1) + ( ∂Φ∂z − ∂Φ∂y ) χ (2) ] . (A.5) Appendix B: The matrix M and thescattering solutions to the Dirac equation The eight matric entries are given as following: M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ ∂Φ∂x ′ + ∂Φ∂y ′ + ∂Φ∂z ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ × (cid:18) ip x + ip y + i ( p z − k z )0 (cid:19) Φ, (B.6) M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ ∂Φ∂z ′ ∂Φ∂x ′ + 3 i ∂Φ∂y ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ × (cid:18) i ( p z − k z )3 ip x + 3 p y (cid:19) Φ, (B.7)0 M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ (cid:18) ∂Φ∂x ′ (cid:19) = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ (cid:18) ip x (cid:19) Φ, (B.8) M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ × ∂Φ∂z ′ + ∂Φ∂x ′ (3 + i ) ∂Φ∂x ′ + ( − i ) ∂Φ∂y ′ − ∂Φ∂z ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ × (cid:18) i ( p z − k z ) + ip x (3 + i ) ip x + ( − i ) ip y − i ( p z + k z ) (cid:19) Φ, (B.9) M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ ∂Φ∂y ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ (cid:18) ip y (cid:19) Φ, (B.10) M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ × ∂Φ∂z ′ + ∂Φ∂y ′ (3 − i ) ∂Φ∂x ′ + (1 + 3 i ) ∂Φ∂y ′ − i ∂Φ∂z ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ × (cid:18) i ( p z − k z ) + ip y (3 − i ) ip x + (1 + 3 i ) ip y + i ( p z − k z ) (cid:19) Φ, (B.11) M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ (cid:18) ∂Φ∂z ′ (cid:19) = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ (cid:18) i ( p z − k z )0 (cid:19) Φ, (B.12) and M ≡ Z d x ′ e i p · x ′ [ P ∗ ( x ′ ) ϕ ( x ′ )] = Z d x ′ e i p · x ′ e − ik z z ′ × ∂Φ∂z ′ − ∂Φ∂x ′ − ∂Φ∂y ′ ∂Φ∂x ′ + 3 i ∂Φ∂y ′ + (1 + i ) ∂Φ∂z ′ ! = − Z d x ′ e ip x x ′ + ip y y ′ + i ( p z − k z ) z ′ × (cid:18) i ( p z − k z ) − ip x − ip y ip x − p y + (1 + i ) i ( p z − k z ) (cid:19) Φ. (B.13) The scattering solutions to the Dirac equation aregiven as φ (0) ≈ e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x H − M = e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x ik z ∓ iε ) + p (cid:18) − ip z − ik z ± iε − ip x + p y − ip x − p y ip z − ik z ± iε (cid:19) × (cid:18) ip x + ip y + i ( p z − k z )0 (cid:19) T, (B.14) χ (0) ≈ − e iωt lim ε → π ) Z d pe − i p · x H − M = − e iωt lim ε → π ) Z d pe − i p · x × − im − ik z ± iε ) + p × (cid:18) − ip z − im − ik z ± iε − ip x + p y − ip x − p y ip z − im − ik z ± iε (cid:19) × (cid:18) − i ( p z − k z ) − ip x + 3 p y (cid:19) T, (B.15) φ (1) ≈ e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x H − M = e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) × Z d pe − i p · x ik z ∓ iε ) + p × (cid:18) − ip z − ik z ± iε − ip x + p y − ip x − p y ip z − ik z ± iε (cid:19) (cid:18) ip x (cid:19) T, (B.16) χ (1) ≈ − e iωt lim ε → π ) Z d pe − i p · x H − M = − e iωt lim ε → π ) Z d pe − i p · x × − im − ik z ± iε ) + p × (cid:18) − ip z − im − ik z ± iε − ip x + p y − ip x − p y ip z − im − ik z ± iε (cid:19) × (cid:18) i ( p z − k z ) + ip x (3 + i ) ip x + ( − i ) ip y − i ( p z − k z ) (cid:19) T, (B.17) φ (2) ≈ e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x ik z ∓ iε ) + p × (cid:18) − ip z − ik z ± iε − ip x + p y − ip x − p y ip z − ik z ± iε (cid:19) (cid:18) ip y (cid:19) T, (B.18)1 χ (2) ≈ − e iωt lim ε → π ) Z d pe − i p · x H − M = − e iωt lim ε → π ) Z d pe − i p · x × − im − ik z ± iε ) + p × (cid:18) − ip z − im − ik z ± iε − ip x + p y − ip x − p y ip z − im − ik z ± iε (cid:19) × (cid:18) i ( p z − k z ) + ip y (3 − i ) ip x + (1 + 3 i ) ip y + ( p z − k z ) (cid:19) T, (B.19) φ (3) ≈ e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x H − M = e iωt − ik z z (cid:18) (cid:19) − e iωt lim ε → π ) Z d pe − i p · x ik z ∓ iε ) + p (cid:18) − ip z − ik z ± iε − ip x + p y − ip x − p y ip z − ik z ± iε (cid:19) × (cid:18) i ( p z − k z )0 (cid:19) T, (B.20) and χ (3) ≈ − e iωt lim ε → π ) Z d pe − i p · x H − M = − e iωt lim ε → π ) Z d pe − i p · x × − im − ik z ± iε ) + p × (cid:18) − ip z − im − ik z ± iε − ip x + p y − ip x − p y ip z − im − ik z ± iε (cid:19) × (cid:18) i ( p z − k z ) − ip x − ip y ip x − p y + (1 + i ) i ( p z − k z ) (cid:19) T. 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