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International Journal of Modern Physics A Vol 31, No 23 (2016) 1650126 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0217751X16501268 High energy scattering of Dirac particles on smooth potentials Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only Nguyen Suan Han,∗,‡ Le Anh Dung,∗ Nguyen Nhu Xuan†,§ and Vu Toan Thang∗ ∗ Department of Theoretical Physics, Hanoi University of Science, Hanoi, Vietnam † Department of Physics, Le Qui Don University, Hanoi, Vietnam ‡ lienbat76@gmail.com § xuannn@mta.edu.vn Received 26 March 2016 Revised June 2016 Accepted 15 July 2016 Published 19 August 2016 The derivation of the Glauber type representation for the high energy scattering amplitude of particles of spin 1/2 is given within the framework of the Dirac equation in the Foldy–Wouthuysen (FW) representation and two-component formalism The differential cross-sections on the Yukawa and Gaussian potentials are also considered and discussed Keywords: Foldy–Wouthuysen representation; eikonal scattering theory PACS numbers: 11.80.−m, 04.60.−m Introduction Eikonal representation, or Glauber type representation, for the scattering amplitude of high energy particles with small scattering angles was first obtained in Quantum Mechanics1 and, then, in Quantum Field Theory based on the Logunov– Tavkhelidze quasipotential equation.2,3 The assumption of the smoothness of local pseudo-potential4–6 allows us to explain successfully physical characteristics of high energy scattering of hadrons More generally, it leads to a simple qualitative model of interactions between particles in the asymptotic region of high energy Eikonal representation for high energy scattering amplitude has been studied by other authors.7–20 However, these investigations not take into account the spin structure of the scattered particles Some authors included spin effects in their studies,19,20 but their methods were not complete, or could not be applied for various potentials On the other hand, experiments showed that spin effects, for example, the non-negligibility of the ratio of spin-flip to spin-nonflip amplitudes and Coulomb-hadron interference,21–23 play an important role in many physical processes, such as in the recent RHIC and LHC experiments.24,25 Moreover, present 1650126-1 Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only N S Han et al investigations did not utilize the Foldy–Wouthuysen (FW) transformation which is very convenient for passing to the quasiclassical approximation Consequently, this paper aims to generalize the eikonal representation for the scattering amplitude of spinor particles, in particular, to establish the Glauber type representation for the scattering amplitude of spin 1/2 particles on smooth potentials at high energies within the framework of the Dirac equation in an external field after using the FW transformation.26–32 The paper is organized as follows In Sec 2, we obtain the Dirac equation in an external field in the FW representation This representation has a special place in the field of relativistic quantum mechanics due to the following properties First, Hamiltonian and quantum mechanical operators for relativistic particles in external fields in the FW representation are similar to those in the nonrelativistic quantum mechanics Second, the quasiclassical approximation and the classical limit of relativistic quantum mechanics can be obtained by a replacement of operators in quantum-mechanical Hamiltonian and equations of motion with the corresponding classical quantities.28–31 This property is significant since most quantum effects are measured using classical apparatuses Moreover, the FW representation is perfect for a description of spin effects which will be discussed later We need also to mention that the relativistic FW transformation is widely and successfully used in quantum chemistry (see Refs 35–41) In Sec 3, employing the smoothness of external potentials and the Dirac equation in the FW representation, we end up with the Glauber type representation for the high energy scattering amplitude of spin −1/2 particles In Sec 4, the analytical expressions of the differential crosssection in the Yukawa and Gaussian potentials are derived The contribution of terms in the FW Hamiltonian to the scattering processes discussed The results and possible generalizations of this approach are also discussed Foldy Wouthuysen Transformation for the Dirac Equation in External Field In general, there are two regular ways to perform the FW representation of the Dirac Hamiltonian: one approach gives a series of relativistic corrections to the nonrelativistic Schrăodinger Hamiltonian;26,27,32 the other approach allows one to obtain a compact expression for the relativistic FW Hamiltonian (see Refs 28–31, 33, 34 and references therein) In this section, we utilize the method in Ref 34 The Dirac equation for a particle with charge e = q in an external electromagnetic field Vµ (V, A ) is given by ∂Ψ(r, t) = HD Ψ(r , t) dt with the Dirac Hamiltonian HD and bi-spinor Ψ defined as follows: i (2.1) HD = βm + O + E , (2.2) ψ , η (2.3) Ψ= 1650126-2 High energy scattering of Dirac particles on smooth potentials Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only where O = α(p − qA ), E = qV , and ψ, η are two-component spinors One can see that, the Dirac Hamiltonian (2.3) contains both the odd operator O and the even operator E The odd operator leads to the nondiagonal form of the Hamiltonian As a result, the spinor Ψ with positive energy is “mixed” with the negative-energy one η However, it is necessary to isolate the positive-energy (particle) spinor, which will be employed in the next section to derive the scattering amplitude Let us consider the following unitary transformation34 √ + + X + βX O ΨFW = U Ψ , U = , X= (2.4) √ √ m + X2 + + X2 with (2.4), the Dirac Hamiltonian is transformed as HFW = i∂t + U (HD − i∂t )U −1 (2.5) The explicit expression for the FW Hamiltonian is34 HFW = βε + E − 1 , [O, [O, F ]] ε(ε + m) , (2.6) + where m2 + O , ε= F = E − i∂t (2.7) In Eq (2.6), commutators of the third and higher orders as well as degrees of commutators of the third and higher orders are disregarded In the specific case, when the external field is scalar (A = 0), Eq (2.6) becomes HFW = βε + qV + q , i Σ · (p × ∇V ) + Σ · (∇V × p) + ∇2 V ε(ε + m) + (2.8) In this study, we obtain explicit relations for the scattering of a nonrelativistic particle, while we plan to consider the corresponding relativistic problem in the future p2 Using the nonrelativistic approximation, ε = m2 + p ≈ m + 2m , the FW Hamiltonian (2.8) to the order m12 can be written as HFW = β m + + p2 2m + qV + iq Σ · (p × ∇V ) 8m2 q q Σ · ( ∇V × p) + ∇2 V 4m 8m2 (2.9) In our study, the external field is a scalar central potential V = V (r) The FW Hamiltonian, therefore, becomes: HFW = β m + p2 2m + qV + q dV q Σ·L+ ∇2 V 4m r dr 8m2 1650126-3 (2.10) N S Han et al Due to the β-matrix, the FW Hamiltonian (2.10) contains relativistic corrections for both particle and antiparticle which include the spin-orbit coupling and the Darwin term The Darwin term is added to direct interaction of charged particles as point charges, and it characterizes the Zitterbewegung motion of Dirac particles It is related to particles in the FW representation being not concentrated at one 27 point but rather spreading out over a volume with radius of about m Since the only particle is considered in our scattering problem, one needs to deal with the positive-energy component of the Hamiltonian (2.10) Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only + HFW =m+ p2 q dV q + qV + σ·L+ ∇2 V 2m 4m r dr 8m2 (2.11) It is also important to note that, the relativistic correction terms guarantee that the wave function in the FW representation agrees with the nonrelativistic Pauli wave function for spin −1/2 particles.27 This Hamiltonian (2.11) include the contribution of the Darwin term in the scattering amplitude As shown in the next section, this term leads to different result compared with that obtained in Ref 18 Glauber Type Representation for Scattering Amplitude + With Hamiltonian HFW , the equation for two-component wave function ψ(r , t) is given by m+ p2 e dV e ∂ψ(r , t) + eV + σ·L+ ∇2 V ψ(r , t) = i 2 2m 4m r dr 8m ∂t (3.1) By the variable separation ψ(r , t) = e−iEt ϕ(r ) (3.2) Equation (3.1) can be reduced to m− ∇2 ie dV e + eV − (σ × r ) ∇ + ∇2 V − E ϕ(r) = 2 2m 4m r dr 8m2 (3.3) The solution to (3.3) will be sought in the form ϕ(r) = eipz ϕ(+) (r) + e−ipz ϕ(−) (r) = eipz ϕ(+) (b, z) + e−ipz ϕ(−) (b, z) , (3.4) where r = ( b, z), and the z-axis is chosen to be coincident with the direction of incident momentum p Two-component spinors ϕ(+) (r) and ϕ(−) (r) satisfy the following boundary conditions ϕ(+) ( b, z)|z→−∞ = ϕ0 , ϕ(−) ( b, z)|z→−∞ = 1650126-4 (3.5) High energy scattering of Dirac particles on smooth potentials Two terms in (3.4) describe the propagation of incident and reflected waves along the z-axis, respectively Substitution of (3.4) into (3.3) yields eipz 2m U+ 1 dU ∂ ∇2 U + p(σ × r )z − 2ip ϕ(+) 2 8m 4m r dr ∂z − ∇2 ϕ(+) − Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only + e−ipz 2m i dU (σ × r )∇ϕ(+) 4m2 r dr U+ − ∇2 ϕ(−) − 1 dU ∂ ∇2 U − p(σ × r )z + 2ip ϕ(−) 2 8m 4m r dr ∂z i dU (σ × r )∇ϕ(−) = , 4m2 r dr (3.6) p (in the nonrelativistic approximation) where U (r ) = 2meV (r ) and E ≈ m + 2m Due to the smoothness of the potential, the quasiclassical condition of scattering is satisfied18,35 U˙ V˙ U 2meV = ≪ 1, = ≪ (3.7) Up Vp p p2 With the condition (3.7), the spinors ϕ± (b, z) are slowly varying functions and approximately satisfy the equations 1 dU ∂ϕ(+)( b, z) (+) U+ ϕ ( b, z) = 2ip ∇ U + p(σ × r ) , z 8m2 4m2 r dr ∂z (3.8) 1 dU ∂ϕ(−)( b, z) (−) U+ ∇ U− p(σ × r )z ϕ ( b, z) = −2ip 8m2 4m2 r dr ∂z Note that (σ × r )z = (σ × b)z + (σ × z )z = (σ × b)z = −b(n × σ )z , (3.9) here n = bb = (sin φ, cos φ) where φ is the azimuthal angle in the (x, y)-plane Equations (3.8) can be rewritten as pb dU ∂ϕ(+) ( b, z) (+) ∇ U − (n × σ ) ϕ ( b, z) = 2ip , U + z 8m2 4m2 r dr ∂z (3.10) pb dU ∂ϕ(−)( b, z) (−) U + 8m2 ∇ U + 4m2 r dr (n × σ )z ϕ ( b, z) = −2ip ∂z The solutions of Eqs (3.10) with the boundary conditions (3.5) can be written in the form z 1 pb dU ϕ(+) ( b, z) = ϕ0 exp U (r) + (∇2 U (r)) − (n × σ )z dz ′ , 2ip −∞ 8m2 4m2 r dr (3.11) ϕ(−) ( b, z) = 1650126-5 (3.12) N S Han et al From the boundary condition (3.5), one can see that the reflected wave equals to zero From (3.4), the wave function for scattering particle is ϕ(r) = eipz ϕ0 · exp[χ0 (b, z) + i(n × σ )z χ1 (b, z)] , (3.13) where functions χ0 (b, z) and χ1 (b, z) are defined as χ0 ( b, z) = Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only χ1 ( b, z) = 2ip z U (r) + −∞ z b 8m2 −∞ (∇2 U (r)) dz ′ , 8m2 dU ′ dz r dr (3.14) (3.15) For the scattering amplitude, we obtain respectively f (θ) = − = 4π p 2iπ ′ dr e−ip r ϕ∗0 (p′ ) U + ∇2 U pb dU − (n × σ )z ϕ(r) 8m2 4m2 r dr d2 b e−ib∆ ϕ∗0 (p′ ) eχ0 +i(n×σ)z χ1 − ϕ0 (p) (3.16) One can rewrite this formula as f (θ) = ϕ∗0 (p ′ )[A(θ) + σy B(θ)]ϕ0 (p) (3.17) here42 ϕ0 = or ϕ0 = ∆ = p ′ − p = 2p sin A(θ) = −ip B(θ) = −ip θ ; ∞ ∞ for λ = χ0 = χ0 ( b, ∞) , or λ = − , (3.18) χ1 = χ1 ( b, ∞) , (3.19) b db J0 (b∆) eχ0 cos χ1 − , (3.20) b db J1 (b∆)eχ0 sin χ1 , (3.21) where p′ and θ are the momentum after scattering and the scattering angle; J0 (b∆) and J1 (b∆) are the Bessel functions of the zeroth- and the first-orders The presence of quantities A(θ) and B(θ) determined by formulas (3.20) and (3.21) in the highenergy limit shows that there are both spin-flip and nonspin-flip parts contributing to the scattering amplitude Differential Scattering Cross-Section In this section, using the obtained expression for the scattering amplitude shown above, we derive the differential cross-sections for the scattering in Yukawa and 1650126-6 High energy scattering of Dirac particles on smooth potentials Gaussian potentials for cases in which the Darwin term is included or excluded, respectively In fact, the Gaussian potential is a smooth and nonsingular function that ensures the constancy of the total hadron cross-section.43 Those will then be used to evaluate the contribution of the Darwin term in different regions of momentum and to study the behavior of the Coulomb-nuclear interference in our in-process studies.51 4.1 Yukawa potential Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only Let us consider the Yukawa potential44 given by g −µr g r e = e− R , r r U (r) = (4.1) where g is a magnitude scaling constant whose dimension is of energy, µ is another scaling constant which is related to R — the effective size where the potential is nonzero as µ = R1 From (3.14) and (3.15), one gets (see App A) χ0 (b) = µ2 πg 1+ K0 (µb) , ip 8m2 χ1 (b) = − (4.2) µπg K1 (µb) , 4m2 (4.3) where K0 (µb) and K1 (µb) are the MacDonald function of zeroth-order52 and firstorder, respectively Substitution of (4.2) and (4.3) into (3.20) and (3.21) yields πg + A(θ) = − B(θ) = µ2 8m2 , (4.4) iπgp ∆ · 4m2 µ2 + ∆2 (4.5) ∆2 + µ2 The differential cross-section is then dσ dΩ YD = |A(θ)|2 + |B(θ)|2 = π2 g 4p2 sin2 (θ/2) + µ2 1+ µ2 8m2 + p4 sin2 (θ/2) 4m4 (4.6) This expression of differential cross-section is for the case in which the Darwin term is included If we ignore this term, the differential cross-section is dσ dΩ = Yo π2 g µ2 + 4p2 sin2 (θ/2) 1650126-7 1+ p4 sin2 (θ/2) 4m4 (4.7) N S Han et al p 45 µ, With a dimensionless q defined as q = (4.7), respectively, as dσ dΩ YD = Yo π2 g 2 µ4 4q sin (θ/2) + π2 g + 4q sin2 µ4 1+ 8q 2 θ 1+ + µ4 q sin2 4m4 µ4 q sin2 4m4 θ θ , (4.8) (4.9) The dependence of the differential cross-section on q (or, in other words, on the incident momentum) and the scattering angle θ in both two cases is graphically illustrated in Figs and (constants are set to unit) In Fig 2, the differential cross-section has a peak at a small value of scattering angle Also, the behavior of the differential cross-section in those figures is similar to one obtained formerly in Refs 44–46 − Differential Cross Section (*10 3) 1568 with Darwin term without Darwin term 1566 1564 1562 1560 1558 100 200 300 400 500 600 p − momentum 700 800 900 1000 (a) 200 Differential Cross Section Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only dσ dΩ = one can rewrite expressions (4.6) and with Darwin term without Darwin term 150 100 50 0.1 0.2 0.3 0.4 0.5 0.6 p− momentum 0.7 0.8 0.9 (b) Fig Dependence of the differential cross-section on the momentum of incident particle (with a specific small value of the scattering angle, θ = 0.1 rad), (a) for large p-momentum, (b) for small p-momentum 1650126-8 High energy scattering of Dirac particles on smooth potentials Differential Cross Section 200 with Darwin term without Darwin term 150 100 50 0.02 0.04 0.06 0.08 0.1 Theta (rad) 0.12 0.14 0.16 (a) 700 Differential cross section Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only with q = 100 with q = 200 600 500 400 300 200 100 0 0.02 0.04 0.06 0.08 0.1 Theta (rad) 0.12 0.14 0.16 0.18 (b) Fig Dependence of the differential cross-section on the scattering angle: (a) differential crosssection with and without the Darwin term with q = 100, (b) differential cross-section with the Darwin term with q = 100 and q = 200 4.2 Gaussian potential Now, we consider the Gaussian potential of the following form44 U (r) = λe−αr = λ exp − r2 2R2 , (4.10) where λ is a magnitude scaling constant, R is the effective size where the potential is nonzero and α is another scaling constant, α = 2R1 To get the differential cross-section, we performed some calculations similar to the calculation of the differential cross-section with Yukawa potential in Subsec 4.1 (see App B for detail) As a result, we obtain dσ dΩ = GD πλ2 2p2 sin2 (θ/2) exp − 16α α × 1− p2 sin2 2m2 θ 1650126-9 + p4 sin2 4m4 θ (4.11) N S Han et al Differential cross section 1.4 0.8 0.6 0.4 0.2 0 20 40 60 80 100 120 p− momentum 140 160 180 200 (a) 1.005 Differential cross section Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only with Darwin term without Darwin term 1.2 with Darwin term without Darwin term 0.995 0.99 0.985 0.98 0.975 0.97 0.1 0.2 0.3 0.4 0.5 0.6 p− momentum 0.7 0.8 0.9 (b) Fig Dependence of the differential cross-section on the momentum of incident particle (at a particular small value of the scattering angle), (a) with large p-momentum, (b) with small p-momentum Now, if the Darwin term is ignored, the differential cross-section is dσ dΩ = Go πλ2 2p2 sin2 (θ/2) exp − 16α3 α 1+ p4 sin2 (θ/2) 4m4 (4.12) Figures and graphically describe the dependence of the differential cross-section on the momentum of incident particle and the scattering angle (constants are set to unit) Unlike the case of Yukawa potential considered above, in the case of Gaussian potential the Darwin term causes non-negligible contributions to the differential cross-section as shown in Figs and In the region of small values of momentum and very small scattering angles, the contribution of the Darwin term is significant 1650126-10 High energy scattering of Dirac particles on smooth potentials Differential cross section 0.6 0.4 0.2 0 0.02 0.04 0.06 0.08 0.1 Theta (rad) 0.12 0.14 0.16 0.18 (a) 100 Differential cross section Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only with Darwin term without Darwin term 0.8 with Darwin term without Darwin term 80 60 40 20 0 0.02 0.04 0.06 0.08 0.1 Theta (rad) 0.12 0.14 0.16 0.18 (b) Fig Dependence of the differential cross-section on the scattering angle: (a) at p = 100, (b) at p = 100 Conclusion By employing the step-by-step FW transformation which consists of two unitary transformation to the order m12 , we obtained the nonrelativistic expression for Dirac Hamiltonian in the FW representation, which describes the interaction of particles and antiparticles having spin 1/2 with an electromagnetic field With the assumption of smooth potential, we ended up with the Glauber type representation for the high energy scattering amplitude of Dirac particles with small scattering angles The resultant scattering amplitude includes the contribution of the Darwin term This term guarantees that the wave function in the FW representation agrees with the nonrelativistic Pauli wave function for spin 1/2 particles The expressions for the differential cross-section with and without the Darwin term in the Yukawa and Gaussian potentials, which are two different forms of nuclear potential serving for the problem of Coulomb-nuclear interference,51 are derived, respectively We showed that the Darwin term has relatively significant contribution at some finite 1650126-11 N S Han et al ranges of incident particle’s momentum p However, this contribution is very small for large particle’s momenta For the problem of scattering on the gravitational field, due to the relation to some basic problems such as strong gravitational forces near black-holes, string modification of theory of gravity and some other effects of quantum gravity,14,47–51 the Darwin term derived in Refs 53 and 54 might be important; and, therefore, in our point of view, the application and generalization of the method proposed in this paper are necessary Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only Acknowledgments We are grateful to Profs B M Barbashov, A V Efremov, M K Volkov, O V Selyugin and V V Nesterenko for useful discussions N S Han is indebted to Profs V N Pervushin, A J Silenko for reading the manuscript and making useful remarks for improvements, N S Han is also indebted to Prof A B Arbuzov for support during his stay at JINR in Dubna and warm hospitality This research is funded by NAFOSTED under grant number 103.03-2012.02, by the Joint Institute for Nuclear Research Dubna Appendix A Yukawa Potential For the Yukawa potential (4.1), since e−µr r dU g d = r dr r dr = −g(1 + µr)e−µr r3 (A.1) We can rewrite χ1 (b) as χ1 (b) = − ∞ gb 8m2 −∞ (1 + µr)e−µr ′ dz r3 (A.2) On the other hand, if we employ the MacDonald function of zeroth-order52 K0 (µb) = 2π ∞ e−µr ′ dz = r 2π ∞ √ ′2 e−µ b +z √ dz ′ b2 + z ′ (A.3) with the following property d d (K0 (µb)) = db 2π db = 2π =− 2π =− b 2π ∞ e−µr ′ dz r d dr e−µr r ∞ dr ′ dz db ∞ e−µr + µr e−µr r2 ∞ + µr −µr ′ e dz r3 0 · 1650126-12 b · dz ′ r (A.4) High energy scattering of Dirac particles on smooth potentials one gets χ1 (b) = πg d µπg (K0 (µb)) = − K1 (µb) , 4m2 db 4m2 (A.5) d where K1 (µb) is the MacDonald function of first-order, K1 (µb) = − µ1 db (K0 (µb)) Now, we turn to the calculation of χ0 (b) ∞ 2ip χ0 (b) = U (r) + −∞ [∇2 U (r)] dz ′ 8m2 (A.6) Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only For the Yukawa potential (4.1), we get ∇2 U (r) = ∂ ∂U r2 r2 ∂r ∂r g ∂ µ2 ge−µr [(1 + µr)e−µr ] = r ∂r r (A.7) e−µr ′ πg µ2 dz = 1+ K0 (µb) r ip 8m2 (A.8) =− Thus χ0 (b) = ∞ g µ2 1+ 2ip 8m2 Substitution of (A.5) and (A.8) into (3.20) and (3.21) one gets A(θ) = −ip ∞ b dbJ0 (b∆) πg µ2 µπg 1+ K0 (µb) · cos − K1 (µb) − ip 8m2 8m2 × exp ≃ −πg + = −πg + µ2 8m2 ∆2 + µ2 B(θ) = −ip ∞ b dbJ0 (b∆)K0 (µb) , (A.9) b dbJ1 (b∆) × exp ≃ ip ∞ µ2 8m2 ∞ πg µ2 µπg 1+ K0 (µb) sin − K1 (µb) ip 8m 4m2 b dbJ1 (b∆) + ∞ ≃ iµπgp 4m2 = iπgp ∆ · 4m2 µ2 + ∆2 πg µ2 1+ K0 (µb) ip 8m2 µπg K1 (µb) 4m2 b dbJ1 (b∆)K1 (µb) (A.10) 1650126-13 N S Han et al The differential cross-section is then dσ dΩ YD = |A(θ)|2 + |B(θ)|2 π2 g = ∆2 + µ2 4p2 sin2 θ + µ2 µ2 8m2 π2 g = Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only 1+ + µ2 8m2 1+ p2 ∆2 16m4 + p4 sin2 4m4 θ (A.11) This expression of differential cross-section is for the case in which the Darwin term is included If we ignore this term in (4.6), the differential cross-section is dσ dΩ = Yo = π2 g (µ2 + 1+ ∆2 ) p2 ∆2 16m4 π2 g µ2 + 4p2 sin2 (θ/2) 1+ p4 sin2 (θ/2) 4m4 (A.12) Appendix B Gauss Potential For this case, we similar to the case of Yukawa potential In particular, we have ∇2 U = ∂ ∂U r2 r2 ∂r ∂r U (r) + =− 2λα ∂ −αr2 r e = −2λα(3 − 2αr2 )e−αr , r ∂r ∇2 U (r) 8m2 =λ 1− 3α λα2 −αr2 −αr e + r e 4m2 2m2 (B.1) (B.2) Substitution of (B.2) into (3.14) and (3.15) yields the final expressions for χ0 (b, ∞) and χ1 (b, ∞) χ0 (b) = = 2ip −∞ λ 1− λ 3α 1− 2ip 4m2 + = ∞ λα2 4ipm2 ∞ 3α λα2 −αr2 −αr e + r e dz ′ 4m2 2m2 ∞ e−α(b +z ′ ) dz ′ −∞ b2 + z ′ e−α(b +z ′ ) dz ′ −∞ λ 3α α2 b2 1− + 2ip 4m 2m2 + λα exp(−αb2 ) 8ipm2 exp(−αb2 ) π α 1650126-14 π α High energy scattering of Dirac particles on smooth potentials λ α α2 b2 1− + 2ip 2m2 2m2 = exp(−αb2 ) π α = C1 exp(−αb2 ) + C2 b2 exp(−αb2 ) , (B.3) dU ′ αbλ +∞ −α(b2 +z′ ) ′ dz = − e dz 4m2 −∞ −∞ r dr √ 2 π απλbe−αb αbλe−αb = − = C3 be−αb , =− 2 4m α 4m ∞ b 8m2 χ1 (b) = (B.4) Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only where C1 = π λ α 1− α 2ip 2m2 , π λα2 , α 4ipm2 C2 = √ απλ C3 = − 4m2 (B.5) Substituting the expressions (B.3) and (B.4) into (3.20) and (3.21), performing the Taylor’s approximation, and keeping only the terms up to the first-order, one gets A(θ) = −ip ∞ b db J0 (b∆) × exp C1 exp(−αb2 ) + C2 b2 exp(−αb2 ) cos C3 e−αb ≃ −ip ∞ −1 b db J0 (b∆) C1 + C2 b2 e−αb 2 exp − ∆ 8α ∆2 √ = −iC1 p M 12 ,0 ∆ α 4α − iC2 p exp − ∆ 8α ∆α M 23 ,0 ∆2 4α (B.6) and ∞ B(θ) = ip b db J1 (b∆) × exp C1 exp(−αb2 ) + C2 b2 exp(−αb2 ) sin C3 be−αb ≃ iC3 p ∞ b2 dbJ1 (b∆)e−αb = iC3 p exp − ∆ 8α α∆ M1, 12 ∆2 4α (B.7) In (B.6) and (B.7), the following integral identity for Bessel functions has been used ∞ xµ e−αx Jv (βx)dx = 1 2µ + 2ν + 1 βα µ Γ(ν + 1) Γ 1650126-15 exp − β2 β2 M 12 µ, 12 ν 8α 4α , (B.8) N S Han et al where Re(α) > 0; Re(µ + ν) > −1 and Mµ,ν (z) is the Whittaker function With the notice that Mµ,ν ∆2 4α ∆2 = exp − 8α ∆2 8α = exp − ν+ 12 · ∆2 4α · ∆2 4α ν+ 12 ∞ ∆2 · F1 ν − µ + ; + 2ν; 4α a(n) ∆2 b(n) n! 4α n=0 n , (B.9) where Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only a= ν −µ+ a (0) = 1, a (n) , b = + 2ν , (B.10) = a(a + 1)(a + 2) + · · · + (a + n − 1) From (B.9), one gets ∆2 4α = exp − ∆2 8α ∆2 4α M 21 ,0 ∆2 = exp − 8α ∆2 4α M 32 ,0 ∆2 4α M1, 12 ∆2 4α = exp − ∆2 8α ∆2 4α , 1− ∆2 , 4α (B.11) Substituting (B.11) into (B.6) and (B.7) we obtain the following expressions for A(θ) and B(θ) A(θ) = λ(∆2 − 8m2 ) 32m2 α B(θ) = −i λp∆ 16αm2 π ∆2 exp − α 4α π ∆2 exp − α 4α , (B.12) (B.13) The differential cross-section is then dσ dΩ GD = |A(θ)|2 + |B(θ)|2 = πλ2 2p2 sin2 (θ/2) exp − 16α α × 1− p2 θ sin2 2m 2 + p4 θ sin2 4m (B.14) Now, if the Darwin term is ignored, the differential cross-section is dσ dΩ = Go πλ2 2p2 sin2 (θ/2) exp − 16α3 α 1650126-16 1+ p4 sin2 (θ/2) 4m4 (B.15) High energy scattering of Dirac particles on smooth potentials Int J Mod Phys A 2016.31 Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 08/28/16 For personal use only References R J Glauber, Lectures in Theoretical Physics (New York, 1959) V R Garsevanishvili, V A Matveev, L A Slepchenko and A N Tavkhelidze, Phys Lett B 29, 191 (1969) A A Logunov and A N Tavkhelidze, Nuovo Cimento 29, 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the high energy scattering amplitude of Dirac particles with small scattering angles The resultant scattering