NONPOTENTIAL ELASTIC SCATTERING OF SPINNING PARTICLES

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1 NONPOTENTIAL ELASTIC SCATTERING OF SPINNING PARTICLES A.K.Aringazin 1,, D.A.Kirukhin 1 and R.M.Santilli 1 Department of Theoretical Physics Karaganda State University Karaganda 4774, Kazakhstan and Istituto per la Ricerca di Base Castello Principe Pignatelli I-8675 Monteroduni IS Molise, Italy Received February 8, 15 Hadronic J Abstract In this paper, we consider nonpotential elastic scattering of two spinning particles, within the framework of hadronic mechanics, which is used to account for nonpotential effects. Permanent address.

2 1 Introduction In this paper, we consider nonpotentital elastic scattering of spinning particles. When particles have spins their interactions are in general of noncentral type, i.e., the potential of the interaction depends not only on the relative distance but also on the mutual orientation of spins and on vector of the relative distance, and the same holds for the nonpotential contribution. We consider nonpotetial scattering of particles with spins s 1 and s. Within the framework of non-potential scattering theory[1, ], we use the isotopically deformed Schrödinger equation, the iso-schrödinger equation[3], characterized by isotopic operator T is assumed to be responsible for the non-potential part of the scattering. In the limit T 1, the usual potential scattering theory is recovered. The isotopic lifting of the spin space giving rise to the isotopic spin space[3] is made by the Lie-Santilli theory[4] with the use of operator R acting on spin parts of the wave functions. In the limit R 1, the usual spin theory is recovered. In the previous paper[5], we calculated the free iso-propagator in the coordinate and momenta representations, to find the general solution of the iso- Schrödinger equation, the nonpotential scattering amplitude, and represent the solution and amplitude for particular choices of T. The iso-lippmann- Schwinger equation for the scattering matrix, its general on-shell solution for a separable potential, and non-potential scattering length have been investigated. We presented computations of the nonpotential scattering length for the Yamaguchi potential, and particular cases of the isotopic operator T k. The present paper relies on the considerations and results of Ref.[5]. The paper is organized as follows. In Section we develop a formalism to describe T -isotopic nonpotential scattering of spinning particle off spinless particle in vacuum. In Section 3 we generalize the results of Sec. to the case of two spinning particles. In Section 4 we briefly review the eigenvalue problem of spin operators. The generalization of ordinary spin theory is reached via a spin matrix R, which is chosen diagonal. Section 5 is devoted to T -isotopic scattering problem of spin-half particles. 1

3 Scattering of spinning particle off spinless particle Let us consider the scattering of spinning particle off spinless particle. The potential V entering the Hamiltonian of the system, H = H + V, depends both on the relative distance between the particles and on the spin, i.e., V is an operator matrix in the spin space see for example [6]. Let us take the initial wave function of the system, ˆϕ +, in the form of T - isotopic product of the iso-free wave, characterizing relative motion with momentum k, and spin function, ˆϕ + kµ = exp{ikt kr} ˆχ µ, 1 where ˆχ µ is the eigenfunction of the spin operators in isospace[3], ŝ and ŝ z, with µ being 3-projection of spin see Section 4 for a review. Then the wave function ˆψ kµ + satisfying the iso-schrödinger equation and describing the scattering can be written as follows: ˆψ kµ + r = ˆψ kµ + r + dr Ĝ + r r ˆV r ˆψ kµ + r, or, in more details, ˆψ kµ + r, σ = ˆψ kµ + r, σ + σ,σ dr Ĝ + σσ r r ˆV σ,σ r ˆψ + kµ r, σ, 3 where Ĝ+ σσ r r is the iso-propagator, which in the coordinate representation has the form Ĝ + σσ r r = m π exp{ikt k r r } T k ˆχ r r µ σr ˆχ + µ σ, 4 µ ˆV = V T and R is s + 1 s + 1 matrix. We assume that ˆχ µ s form a complete set of spin functions, ˆχ µ σr ˆχ + µ σ = ˆδ σσ = Îδ σσ. 5 µ According to this, the propagator can be taken in a diagonal form, Ĝ + σσ r r = m π exp{ikt k r r } T kˆδ r r σσ. 6

4 Below, we follow [6] to derive the scattering amplitude and scattering length. The operator Ĵ satisfies the equation ˆV T k ˆψ + kµ = ĴT k ˆϕ+ kµ. 7 It can be used in equation to write down the initial wave function in the form } ˆψ kµ {exp[ikt + r = r] + dr Ĝ + r r T ĴT exp[ikt r ] T ˆχ µ. 8 Expression in curly brackets is an operator in spin space. asymptotics of the wave function 8 can be presented as Using 6 the { ˆψ kµ + r r exp[ikt kr] + exp[ikt } kr ] ˆf µ k, k T kˆχ µ, r where ˆf µ k, k is the nonpotential scattering amplitude; k and k are income and outcome momenta respectively, in iso-euclidean space with isometric ˆδ = T δ, δ = diag1, 1, 1. The solution of the associated iso-lippman-schwinger equation for the scattering matrix, Ĵ, in this case is similar to that of the spinless case [6]. We do not repeat the derivation here and present the result. Namely, the nonpotential scattering amplitude for the spinning particle is or, taking m = 1/, ˆf µ k, k = m π k T ĴT k = m π k Ĵ k 1 The scattering length, â µ = ˆf µ,, is then ˆf µ k, k = k T ĴT k. 11 4π â µ = T ĴT. 1 4π Note that the nonpotential scattering amplitude 11 as well as Ĵ are operators matrices in the isotopic space. The function obtained as the action of the operator ˆf µ on the initial spin function, ˆχ = ˆf µ ˆχ µ. 13 3

5 can be considered as the spin function of the particle after the scattering, i.e. the operator ˆf µ transforms the spin state ˆχ µ to the final spin state ˆχ. The coefficients of the expansion of ˆχ in terms of the complete set of ˆχ µ s, ˆχ = µ ˆfµµ ˆχ µ. 14 are the amplitudes of elastic scattering accompanied by changing of the projection of spin, ˆf µµ = kµ Ĵ k µ. 15 4π Here, µ and µ are projections of spin in the initial and final states respectively. It should be noted that when the interactions between the particles depend on spin there is no an azimutal symmetry of the scattering. Namely, the potential or nonpotential amplitude ˆf µµ depends not only on the scattering angle ˆθ but also on the azimutal angle ˆϕ, which is the angle between the plane defining the quantization axis and the plane of the scattering. Using equation 7, we can define the scattering amplitude in terms of the potential see Ref.[6] for details, ˆf µµ ˆθ, ˆϕ = 1 4π dr ˆχ + T k r µ T kv rt k ˆψ + Re ik kµ r, 16 when ˆθ and ˆϕ are isoangles[3]. The differential cross section at fixed values of the projections of spin can then be written i.e., ˆσµ µ = σµ µ Î = ˆf µµ ˆθ, ˆϕ ˆ ˆf µµ ˆθ, ˆϕ T ˆf µµ ˆθ, ˆϕ Î, 17 or, in terms of the scattering length, σµ µ = ˆf µµ ˆθ, ˆϕ, 18 ˆσµ µ = 4π â µµ ˆθ, ˆϕ ˆ. 1 If the projections are not fixed then the cross section 17 should be averaged on the initial projections, and summed over the final projections, 4

6 namely, ˆσˆθ = 1 s + 1 R µµ ˆf µ µˆθ, ˆϕ ˆ. Note that since the initial spin projections are averaged, the cross section depends only on the scattering isoangle ˆθ. 3 Scattering of spinning particles In this section, we generalize the results of the previous section to the case when both particles have non-zero spins. Let ŝ 1 and ŝ denote spins of the incoming and target particles, respectively. The initial wave function can be then written as ˆϕ + kµ 1 µ = exp{ikt kr}t kˆχ s1 µ 1 R ˆχ s µ, 1 where µ 1 and µ are projections of spins s 1 and s respectively. Introduce the wave functions of the channel spin ˆχ sµ with the help of T -isotopically lifted sum rules, ˆχ sµ = µ 1 µ s 1 µ 1 s µ sµrˆχ s1 µ 1 R ˆχ s µ. Inverting this relation, we have ˆχ s1 µ 1 R ˆχ s µ = µ 1 µ s 1 µ 1 s µ sµr ˆχ sµ. 3 Inserting 3 into and noting that the spin function of distinct channels are orthogonal to each other, one can see that the scatterings for the distinct incoming channels are independent of each other. Thus, the scattering of spinning particles can be described in terms of the scattering of spinning particle off spinless particle which has been considered in the previous section. Namely, the wave function for a fixed incoming channel can be written as cf. eqs. 8,7,4 ˆψ + ksµ r = eikt r R ˆχ sµ m π s µ ˆχ s µ T k dr exp[ikt r r ] R 4 r r 5

7 ˆχ s µ, ˆV rt ˆψ ksµ + r. It should be emphasized that the channel spin is not conserved so that in general s s. Starting with expression 4, we obtain after tedious but straightforward calculations cf. eq.16, ˆf s µ sµˆθ, ˆϕ = m π T k dr ˆχ + T k r s µ T k ˆV rt k ˆψ + Re ik ksµ r. 5 Since the projections of spins are observables while the channel spins are not, the scattering amplitude 5 has no direct physical meaning. However, the observable scattering amplitude, ˆfµ 1 µ µ 1µ, can be readily expressed in terms of 5 with the use of sum rules 3, ˆf µ 1 µ µ 1µ ˆθ, ˆϕ = sµs µ s 1 µ 1 s µ sµrs 1µ 1s µ s µ ˆf s µ sµˆθ, ˆϕ. 6 In equation 6 the sum is over all allowed values of the channel spins s and s. The generalization of the cross section 1 then reads or, in terms of the scattering length, σµ 1 µ µ 1µ = ˆf µ 1 µ µ 1µ ˆθ, ˆϕ, 7 σµ 1 µ µ 1µ = 4π â µ 1 µ µ 1µ ˆθ, ˆϕ. Similarly, the generalization of cross section reads σˆθ = 1 s 1 + 1s + 1 R sµs µ ˆf sµs µ ˆθ, ˆϕ. 8 4 Isoeigenfunctions of spin operators The wave function ˆψ + r, σ of particle with spin s has s + 1 components, while spin operators are s + 1 s + 1 matrices. These dimensionalities are preserved under isotopy[3]. Let us denote ˆχ sµ the eigenfunctions of the operators ŝ = srs and ŝ z, see Chapter 6 of Ref.[3] ŝ R ˆχ sµ = ss + 1 R ˆχ sµ, ŝ z Rˆχ sµ = µ R ˆχ sµ. 6

8 The matrix R is assumed to be invertible and detr >. Note that when R is the usual unit matrix the ordinary eigenvalue problem and notion of spin is recovered. The functions ˆχ sµ are orthogonal and normalized due to ˆχ sµ σr ˆχ + sµσ = ˆδ σσ ÎRδ σσ, Î R = R 1. 3 s This is the isotopic generalization of the standard condition of completeness of the set of eigenfunctions. The representation for which the spin projection has a certain value we have ˆχ sµ σ = ˆδ µ,σ. 31 This means that ˆχ sµ σ can be presented as s+1 column with all elements zero except for one, with σ = µ, ˆχ sµ σ =. Î Ṛ. s s 1. µ. s, 3 where ÎR = R 1 is the isounit operator; ÎRR ˆχ = ˆχ. Due to this representaion it is easy to check that the ˆχ sµ σ form a complete set, i.e. ˆχ sµ σrˆχ + sµσ = ˆδ σσ, 33 µ and can be considered as a basis in the s + 1-dimensional spin space. Therefore, any spin state ˆχ can be presented as the linear superposition, ˆχσ = where the â µ s satisfy the expresion s µ= s â µ Rˆχ µ σ, 34 â µ Râ µ = ÎR. 35 mu 7

9 5 Isoscattering of spin-half particles In this section we specify the considerations made in the previous sections to the spin-half case, s 1 = s = 1/ in isospace[3] From equation 6 we see that the scattering amplitude for two spinning particles can be expressed in terms of the scattering amplitude when one of the particles is spinless. Indeed, ˆf µ 1 µ µ 1µ ˆθ, ˆϕ = s 1 µ 1 s µ sµrs 1µ 1s µ s µ T ˆf ˆθ, ˆϕR ˆχ + s µ. 36 sµs µ Introducing the notation Ĉ = equation 36 can be rewritten sµs µ s 1 µ 1 s µ sµrs 1µ 1s µ s µ, 37 ˆf µ 1 µ µ 1µ ˆθ, ˆϕ = Ĉ ˆf ˆθ, ˆϕR ˆχ + s µ. 38 where ˆf ˆθ, ˆϕ is the nonpotential scattering amplitude for spinless particles. To determing ˆf one should find Ĉ and ˆχ+ s µ while ˆf has been found in [5]. 5.1 Isotopy of Clebch-Gordan coefficients Introducing the notation Ĉ 1 = s 1 µ 1 s µ ˆ sµ = Ĉ s1 s sµ; µ 1 µ, 3 Ĉ = s 1µ 1s µ ˆ s µ = Ĉs 1 s s µ ; µ 1µ, we rewrite 36 in the more convenient form Ĉ = sµs µ Ĉ 1 Ĉ sµs µ Ĉ 1 RĈ. 4 Using the isotopy[3] of Clebch-Gordan coefficients[6] we have Ĉ 1 = s 1 µ 1 s µ ˆ sµ = s1 µ 1 s µ ˆ sµ 41 8

10 and where s 1 j 1, s j ; s 1 µ 1 s µ R s 1 j 1, s j ; sµ Ĉ = s 1µ 1s µ ˆ s µ = s 1µ 1s µ ˆ s µ s 1j 1, s j ; s 1µ 1s µ R s 1j 1, s j ; s µ R = diagg kk = ±b k b k, g kk >, b k = b k t, r, ṙ,..., k = 1,, 3. 4 In equation 4, the sum runs over four indeces, s, µ, s, and µ. However, we note that since the sum over the primed indeces is the same as the sum over the unprimed indeces, we can perform the sum over s and µ, Ĉ = sµs µ Ĉ 1 Ĉ sµ Ĉ 1 RĈ. 43 To rewrite 4 in an explicit form, one needs an explicit form of the orthohonality condition of the iso-clebch-gordan coefficients[3, 6], Ĉ s1 s sµ; µ µ 1 µ 1 g µs1 s µ 1 µ µ s Ĉ 1 s µ 1 µ s 1 s s µ ; µ µ 1 µ 1 = δ ss g kk, µ 1 µ 44 Ĉ s1 s sµ; µ µ 1 µ 1 g sµ1 µ s 1 s µ 1 s 1 s µ s Ĉ s 1 s s µ ; µ µ 1 µ 1 = δ µµ g kk, 45 s 1 Ĉ s1 s sµ; µ µ 1 µ 1 g s1 s µ µsµ s s 1 s µ Ĉ s 1 s s µ ; µ µ 1 µ 1 = δ µ1 µ g 1 kk, s so that s 1 s Ĉ s1 s sµ; µ µ 1 µ 1 RĈs 1 s s µ ; µ µ 1 µ 1 = δ µµ δ µ1 µ 1 g kk. 46 Combining equations 45 and 46 we finally have sµs µ Ĉ 1 RĈ = δ ssδ µµ δ µ 1 µ 1 g5 kk. 47

11 5. Spin-half case Let us define spin isoeigenfunctions for the case of spin-half particles. For triplet state S 1 there are three symmetric triplet spin functions, ˆχ 1 11, = ˆα1 ˆα, ˆχ 11, = 1 [ˆα1 ˆβ + ˆα ˆβ1], 48 ˆχ 1 1 1, = ˆβ1 ˆβ. For the singlet state S, we have the antisymmetric spin function where ˆχ 1, = 1 [ˆα1 ˆβ ˆα ˆβ1], 4 ˆα = ˆχ 1/ 1/ g 1/ = 11 1/, ˆβ = ˆχ 1/ = g 1/ With 5 we have for 48 and 4 explicitly ˆχ 11, = 1 1/ g 11 ˆχ 1 11, = 1 ˆχ 1 1 1, = ˆχ 1, = 1 1/ g / 1/ g 11 g 11, g 1/ g 1/ g 1/ + 1 g 1/ 1 g 1/ 1 g 1/, g 1/ g 1/, 51 5 Straightforward calculations lead to the following final form of the spin functions: ˆχ 1 1, = g 1 11, ˆχ 1 1 1, =, 53. 1

12 ˆχ 11, = 1 ˆχ 1, = Scattering amplitudes g 11 g 1/ g 11 g 1/,. 54 By using the results obtained for the spin functions above we can then easily write the triplet and singlet non-potential amplitudes, ˆfˆθ, ˆϕtrpl and ˆfˆθ, ˆϕ sngl, respectively Triplet and singlet states The triplet non-potential amplitude S = 1 is ˆfˆθ, ˆϕ trpl = 1 g 11 /g 1/ Singlet non-potential amplitude S = is ˆfˆθ, ˆϕ sngl = 1 g 11 /g 1/ δ ss δ µµ δ µ 1 µ 1 δ ss δ µµ δ µ 1 µ 1 ˆf ˆθ, ˆϕ. 55 ˆf ˆθ, ˆϕ. 56 Comparing the amplitude 56 with 55 we arrive at the conclusion that the triplet and singlet isoscattering amplitudes coincide in isospace, i.e., ˆfˆθ, ˆϕ trpl = ˆfˆθ, ˆϕ sngl. 57 We see that the non-potential scattering amplitudes for the triplet and singlet states differ only by the overal minus sign, as it is in the usual potential theory. However, we note that this result follows from the choice of diagonal form 4 of the matrix R. Off-diagonal terms in R may spoil such a correspondence between the amplitudes for the triplet and singlet states. Using the definition of the non-potential scattering length, â = ˆf,, we find from 55-57, â trpl = 1 g 11 /g 1/ 11 δ ss δ µµ δ µ 1 µ 1â, 58

13 and â sngl = 1 g 11 /g 1/ δ ss δ µµ δ µ 1 µ 1â, 5 â trpl = â sngl Elastic np-scattering We now present the application of preceding analyzis to the triplet and singlet scattering amplitudes for the elastic np-scattering, with three specific choices of the isotopic element T k. The basic formulas are 58-6 while the forms of the scattering length â for a specific T k have been calculated in the previous paper[5]. a T k = T = const. â trpl = 1 g 11 /g 1/ b T k = 1 + α k, α = const. â trpl == 1 g 11 /g 1/ c T k = 1 + cosαk n, α = const, n. â trpl == 1 g 11 /g 1/ 1 + κ β1 + κ T, 61 â sngl = â trpl κ β1 + κ 1 + α n, 63 â sngl = â trpl κ β1 + κ siniα n, 65 â sngl = â trpl. 66 References [1] R.Mignani, Lett. Nuovo Cim. 3, ; 43, ; Hadronic J., [] R.M.Santilli, Hadronic J. Suppl.4B, issue no

14 [3] R.M.Santilli, Elements of hadronic mechanics, Vol.1, Naukova Dumka, Kiev 14. [4] G.T.Tsagas and A.S.Sourlas, Mathematical foundations of the Lie-Santilli theory Academy of Sciences of Ukraine, Kiev 13. [5] A.K.Aringazin, D.A.Kirukhin, R.M.Santilli, Nonpotential two-body elastic scattering problem, Karaganda Univ. preprint KSU-DTP-4/1; Hadronic J [6] A.G.Sitenko, Scattering theory Naukova Dumka, Kiev