#### Abstract

The approach in which the electron and positron are treated as ordinary, different particles, each being characterized by the complete set of the Dirac plane waves, is examined. This completely symmetric representation that is beyond the standard QED makes it necessary to choose another solution of the Dirac equation for the free particle propagator as compared to that used currently. The Bethe-Salpeter equation with these particle propagators is solved in the ladder approximation. A new solution has been found represented by the massless composite bosons formed by the coupled electron-positron pairs with the coupling equal to the fine structure constant. It has been demonstrated that the massless boson states have normalizable complex wave functions which are transversely compressed plane waves; the transverse radius of the wave functions diverges as the boson energy goes to zero; that is, the composite bosons cannot be at rest; increasing the boson energy results in an extension of the transverse wave function in the momentum space and a corresponding contraction of the real space coordinate wave function. The new reaction is investigated with the products composed of the massless composite boson and two photons. The cross-section of this reaction is derived for nonrelativistic colliding beams of spin-polarized electrons and positrons. In this case the angular correlation spectrum is characterized by a narrow peak with the full-width-at-half-maximum not exceeding 0.2 mrad. It is shown that in order to distinguish between the conventional annihilation of the singlet electron-positron pair with the two-photon emission and the new examined reaction yielding the three particles, experiments are proposed with the extremely nonrelativistic colliding beams.

#### 1. Introduction

During the development of the QED theory, Feynman considered essentially two possible ways [1, 2]. In the first way, assuming that the fermion-antifermion symmetry must exist in nature, Feynman derived the free fermion propagator (Equation in [1]). In this propagator the negative-energy states are assumed to be not available to the electrons; the upper continuum is assumed to be not available to the positrons which are recognized as particles traveling backwards in time [2]. The total number of degrees of freedom that is determined by the complete basis of the Dirac plane waves [3] is divided into half. One half of the degrees of freedom is assigned to the electron and the other to the positron. The modern description of the electron-positron field is based on this propagator.

Note that, in many situations, the filled electronic states with negative energies cannot be ignored. So, these states play a prominent role in the behavior of an electron in external fields, for example, in the Coulomb field [4, 5]. The filled lower continuum is important in the analysis of the electronic structure in super-heavy nuclei, for which at a certain nuclear charge the electron lower level 1S merges with the bottom of the lower continuum [6, 7]. A similar situation arises in discussion of the value of the cosmological constant. Apart from the positive contribution from the zero-point energy of boson quantized fields, another energy source is derived from the Dirac theory of the electron because the filled levels lead inevitably to negative contribution to the vacuum energy [8]. In a study of the radiation scattering by free electrons it was concluded that radiation-induced electron quantum jumps in the intermediate states of negative energies are crucial for the scattering [9].

Taking into account the contradictory situation presented above, in the present paper we have tried to find out what one can expect from the second way that was also discussed by Feynman [1], but has not been studied so far. In this second way Feynman also proposed the free electron propagator, in which the electron is characterized by the complete set of the Dirac plane waves [3]. In this case there is the only possibility of treating the electron and the positron as independent particles and of using a similar propagator for the positron. Then both the electron and the positron are characterized by the number of degrees of freedom that exactly corresponds to the Dirac theory. There is no reason to doubt that the complete spectrum of states for any system of interacting particles can be deduced only when the full basis of states is taken into account for each particle of the system. According to our considerations the division of the complete plane-wave basis into two parts as discussed above leads to the following fact: neither electron states nor positron states separately form the complete system of the wave functions. Therefore, the full coupled electron-positron system may possess additional states as compared to the partial electron and positron subsystems.

The electron-positron field theory derived from the hole theory of positrons [1] leads to a clear picture of the annihilation process of electron-positron pairs. In this process nothing remains of the electron and positron, and the reaction products are just a few photons [1, 3, 5, 10]. The singlet pair of free particles with the center of mass at rest is converted with the greatest probability into two photons which, due to the momentum conservation, should be emitted exactly in the two opposite directions, at the angle of to each other.

Below we consider only reactions of low-energy electron-positron pairs. This article does not take into account any high-energy electron-positron reactions, when the reaction products can be either charged leptons (electrons, muons, taus) or hadrons [11â€“18]. In the case of the existence of the predicted massless composite bosons there should be a process that is, in a sense, similar to the conventional process of electron and positron annihilation, but it has a fundamental difference from the latter. The reaction products in this new process involve, together with emitted photons, the massless boson which is formed by the strongly coupled electron-positron pair. This annihilation-like process with emission of two photons can be represented as follows:where denotes the massless composite boson. It is crucial for this reaction that the two-photon angular correlation spectra must have finite angular widths even for the electron-positron pairs with the center of mass at rest. This is due to the fact that there are three particles in the reaction products. To prove the existence of the third body in the reaction outcome it is of fundamental importance to find the angular width of the correlation spectra.

The article consists of two parts. The first part, presented in Section 2, is devoted to derivation of the massless boson states formed by the coupled electron-positron system with the actual coupling equal to the fine structure constant. Results obtained for the massless boson wave functions are presented. The goal of the second part presented in Section 3 of the paper is to study the cross-section of the reaction (1) for the extremely nonrelativistic colliding beams of the electrons and positrons. The minimal angular width of the correlation spectrum is obtained numerically. In addition, in Section 4 experiments are suggested to establish whether there is the conventional annihilation of singlet electron-positron pair with the two-photon emission or the proposed new reaction with the three-particle outcome. Note that such experiments constitute a relatively simple possibility of testing the central particle-antiparticle concept of the Standard Model.

Natural units () will be used throughout.

#### 2. The Massless Composite Bosons

##### 2.1. The Free Fermion Propagator

At present in QED the free propagator for the Dirac equation is used in the following form [1, 4, 5]:Here is the Dirac plane wave representing the state of the free particle with energy , respectively, and the Dirac conjugate wave function. In (2) the contribution to at is due to the electron terms and at to the positron terms.

The Bethe-Salpeter equation [19, 20] with the propagator (2) was studied in many works, as a rule, in the ladder approximation [21]. After the work [22], considerable attention is given to the problem of strongly coupled states for fermion-antifermion systems. In the most commonly used approach to the problem the Bethe-Salpeter equation is regarded as eigenvalues task for the coupling constant [23â€“30]. That is, an eigenvalue is considered as the necessary strength of the attractive potential to make a massless bound state.

The Dirac equation, as well as any differential equation, has several solutions for the Green function [31]. The Green function for the Dirac equation can also be presented in the following form:Both the propagators (2) and (3) were discussed in [1]. In the positron hole theory that is the foundation of QED, the negative-energy states are assumed to be not available to the electron. Therefore from this point of view the choice of (3), for which the negative-energy states are available for the electron, is unsatisfactory, as noted in [1].

On the other hand, the situation changes radically when the consideration is based on the Dirac theory. Because the electron must be characterized by the total number of degrees of freedom, the propagator (2) is not suitable, and the propagator (3), in which all of the spectrum of the Dirac plane waves is taken into account, should be applied. There is no doubt that the positron can be described by the Dirac equation as well. Considering (3) as the electron propagator, the only opportunity to use a free propagator similar to (3) for the positron is to assume that the electron and positron are independent particles.

In this approach, in the vacuum state, the lower continua for each of these particles are completely filled and the upper continua are not occupied. Then this vacuum state is charge-neutral and stable. The latter is due to the fact that the annihilation of the electrons and the positrons in the negative-energy states is forbidden by the energy conservation law since when both the electron energy and the positron energy are negative.

The interaction between electron and positron is attractive. In the ladder approximation the retarded interaction function can be written as follows [32]:Here are the velocity operators of the electron (âˆ’) and positron (+), are the Pauli matrices, and is the invariant distance between the particles.

This kind of the interaction function (4) is convenient for the subsequent development in this paper but at the same time we must exclude the matrices from the definition of the free fermion propagator (3). By doing this, in the approach of the absolutely symmetric representation of the particles, the electron propagator should be written asand, similarly, for the positron propagatorHere and are the Dirac plane waves for the free electrons and positrons and and are the Hermitian conjugate matrices with respect to and . The latter are given by and , whereand is the unit vector .

Considering (4)â€“(7), in the ladder approximation the bound-state Bethe-Salpeter equation for the electron-positron system iswhere .

Below we do not consider the positronium states. Note only that in the nonrelativistic limit in which one neglects the interaction retardation and the interaction through the vector potential and assumes that the characteristic velocity of particles in the bound pair is much smaller than the speed of light, (8) with the propagators (5)-(6) is reduced to the Schrodinger equation for the Ps states.

##### 2.2. The Boson Wave Function

We search for a solution of (8) in the form of a stationary wave with the phase velocity equal to the speed of light. Let ( and are the momentum of the electron and positron, respectively), and the momentum of the pair, , is directed along the -axis. It is the strongly coupled state with the momentum dependence of the boson energy that is valid only for massless particles. Due to the symmetry of the problem, we have to put and for this massless boson state that allows us to introduce a two-dimensional relative vector between the particles, . Then the wave function is

One can imagine (9) as transversely compressed plane wave. The wave cross-section is determined by the wave function of the transverse motion of the coupled pair , which should be normalizable:For the state (9) the particle distribution is stationary and depends only on .

The function in (4) was given in [32]. In our case it can be written aswhere . It was taken into account that since the phase velocity of the wave (9) is equal to the velocity of light, the interaction between the electron and positron can only occur in the same layers (), which are perpendicular to the wave vector . Considering that (here and are the* z*-components of the velocity operators for the electron and positron, respectively), the factor of in (4) should be replaced by . The latter means that in this bound state the electron and positron do not interact through the Coulomb potential and their retarded interaction occurs through the vector potential which is due to the particles transverse motion defined by the function .

As a result, for the massless composite boson state (9), (8) is reduced toHere and are the electron operators, and are the positron ones, is the electron mass, is the electron energy, and is the positron energy. The matrices and are given in the standard representation.

At first, analyzing only the *-* dependent functions in (12), we integrate over and :

Since the function depends only on , in (12), we replace the integration variables: . Thereafter the integral over on the right side of (12) is easily calculated and gives us . Consequently (12) takes the form:where , , , and was replaced by the fine structure constant, .

Now integrating over and , (14) is rewritten aswhere the functions are given by Here defines the rule for bypassing simple poles.

All the three integrals over on the right side of (15) are easily calculated. We obtainwhere ,with andHere and are the integral sine and cosine functions.

Now we treat the bispinors of the function . It is apparent from (15) that the action of the following operators , , , and on these bispinors must be reduced only to the multiplication of these spin functions on some scalars. For the boson state each particle of the bound pair can be characterized by the projection of the particle spin on the wave vector or, in other words, the particle helicity. There are eight bispinor functions for which the helicities of both the electron and positron are simultaneously either positive, or negative, Because , one can verify that

It implies that the interaction function vanishes, and the massless boson state cannot be formed from these electron and positron states.

Analyzing the remaining possibilities, the required functions in (9) can be written as , where the bispinor functions are given byand are the coordinate wave functions of the transverse motion of the strongly coupled electron-positron pair.

In the states (23) the helicities of the particles are opposite. For these functions we have

As a result, (15) is transformed to the following integral equation for the coordinate function (since , the lower index 1, 2 can be omitted):where

Equation (25) with the notations (26) is a rather complicated integral equation for the transverse wave function . It is important to show that this equation has solutions for the normalizable eigenfunctions. Below it will be demonstrated for the case of the* S*-state of the bound pair and small momentum of the boson, . Then, can be used in (26).

For the* S*-state the angular momentum of the relative motion of the bound pair is zero, and the transverse wave function depends only on the modulus of the relative vector, that is, . In this case after integration over the azimuthal angle of the vector and integration over the azimuthal angle of the vector , (25) is reduced toHere is the Bessel function of the first kind. In the case from (26) we find that andSubstituting (28) to (27), the latter equation is reduced to the homogeneous Fredholm integral equation of the second kind:with the kernel

One can make sure that the kernel (30) is a non-Fredholm one. According to the asymptotic property of the Bessel function , the integral in the right-hand side of (30) can only be defined as the principal value integral. That is, (29) with the kernel (30) is still difficult to solve.

In the momentum space, the equation for the transverse wave function can be reduced to a simpler form convenient for a numerical solution. To this end, in (25), we use the Fourier transform of the -functions, . Then, for the wave function in the momentum space we obtain:For the* S*-state of the bound pair, (31) can be written asIn the case the function can be replaced by the expression (28). Then (32) is reduced to the homogeneous Fredholm integral equation of the second kind:with the kernel

Now the integral in the right-hand side of (34) that has a direct relationship with the discontinuous Weber-Schafheitlin integral is an absolutely convergent integral which, however, is expressed through a discontinuous function, as will be shown below.

It is convenient to use the dimensionless variables: , , , and . Here is the Compton wavelength of the electron. Then (33) remains unchanged, and (34) is rewritten as

The integral on the right-hand side of (35) was previously calculated [33]. Thus the complex kernel is given byHere is the complete elliptic integral of the first kind. Since this kernel, having the weak logarithmic singularity, is a complex function, the boson wave function is complex too. Partition of into the imaginary and real parts is, in a sense, arbitrary since (33) is invariant under the phase transformation with a constant .

It is easy to see that the kernel (36)-(37) is a non-Fredholm one as well. Therefore we can expect that for this kernel the spectra of both the characteristic eigenvalues and the eigenfunctions should be continuous.

##### 2.3. Numerical Procedure

The studies of the Fredholm equation with non-Fredholm kernels are extremely rare in the literature because the search for its solutions is very difficult. In our case, the kernel eigenfunctions must satisfy the normalization condition, . Hence, ; that simplifies the problem. Assume that goes to zero fast enough so that (33) can be reduced to the form:where the value of in units of depends on the boson kinetic energy and .

The kernel given by (36)-(37) is replaced by the two matrices: which is denoted as . Here the number is the partition of the interval . The wave function is replaced by the two -dimensional vectors . Then (38) is reduced to two related linear equations:Here is the unit matrix.

From (40)-(41) we obtain the homogeneous system of linear -equations for the vector :where the matrix isHere means the inverse of .

In contrast to the Fredholm procedure, to solve (42) with the definition (43), it is required to introduce a boundary condition. It is sufficient to putHere is small. Typically this value is assumed to be equal to . This selection does not matter because of the subsequent normalization of the wave function.

Using the boundary condition (44), from (42) we found the vector . Then, from (41) rewritten asthe vector was obtained. The function was normalized to unity. Having thus determined the first approximation to the boson wave function , (40) was represented asand the system of two interrelated equations (45) and (46) was solved by the iterative method. The number of iterations necessary to provide a convergent solution appeared to be about a few hundred.

Then we found the average momentum of the transverse motion of the strongly coupled electron-positron pair:

The transverse wave function in the coordinate representation was derived ():and the average transverse radius of the massless boson wave function was calculated:In other words, is the average relative distance between the electron and the positron in the massless boson state.

For all results presented below, was used. To obtain reproducible results, the upper limit of integration (in units of ) has a bottom restriction which depends on the boson kinetic energy. Therefore, was chosen separately for each energy .

##### 2.4. Numerical Results for the Transverse Wave Functions

Figure 1 shows the kinetic energy dependencies of the integral characteristics (47) and (49) for the composite massless boson. With increasing energy increases, and decreases. In the low-energy region â€‰eVâ€‰eV, with decreasing energy the average momentum approaches monotonically to , and the average transverse radius increases sharply, as shown in the inset in Figure 1. In a narrower region â€‰eVâ€‰eV the dependence of is close to logarithmic, . There is a reason to suppose that this logarithmic divergence persists to the limit because the massless particles cannot be at rest.

Because of computational constraints we were unable to carry out calculations for boson energies lower than 130â€‰eV. Figure 2 demonstrates the boson wave function for the energy eV (the dimensionless value of ) that is slightly larger than the boson energy restriction. Above we pointed out the arbitrariness of the choice between the imaginary and real parts of . Therefore we do not introduce the corresponding notations for the presented curves. Note that Figure 2 shows one of the possible representations of obtained in our calculations, since the phase transformation of the wave function will change these curves.

**(a)**

**(b)**

The momentum wave function is maximal at and then abruptly decays with increasing . This feature for small boson energies does not allow us to calculate accurately the coordinate wave function at , since the integrand function in (48) vanishes at . However, this does not affect the average transverse radius of the massless boson wave function. It was found that and for the energy â€‰eV.

Computational noise on the curves presented in Figure 2(a) correlates with the step of the finite difference grid . With decreasing energy eV the region of this noise becomes more extended. After the transformation (48) the noise features vanish, as shown in Figure 2(b).

According to Figure 1, with increasing boson energy, there is the extension of the transverse wave function in the momentum space, and the contraction of the real space coordinate wave function. This contraction means that the electron and the positron become closer to each other in the -space. The region 1â€‰keV2.3â€‰keV can be considered as a transition region, in which . Outside this region decreases monotonically with increasing boson energy, and the dependence of becomes close to linear. Figure 3 shows both the momentum and coordinate wave functions for keV.

**(a)**

**(b)**

Comparing the data in Figures 2 and 3, one can conclude that, with increasing of the boson energy, oscillations of the wave functions in both the momentum and coordinate spaces are enhanced. The wave function in the momentum space presented in Figure 3(a) is more extended as compared with that for eV (see Figure 2(a)). Consequently, the average transverse momentum increases to the value of . The probability of zero relative momentum of the particles decreased significantly in comparison to that presented in Figure 2(a). As a consequence of the contraction of the transverse wave function, the average transverse radius of the boson wave function is changed from at eV (Figure 2(b)) to the value at keV (Figure 3(b)).

With a further increase in the boson energy the transverse contraction of the wave function tends to slow down. At the same time the probability of finding the electron and positron with nearly zero distance between them increases abruptly. This is clearly demonstrated by Figure 4 where the boson state corresponding to the kinetic energy keV is presented. The transverse momentum wave function (Figure 4(a)) is very extended, and the average relative transverse momentum . The wave function in coordinate space is presented in Figure 4(b). The average transverse radius between the electron and positron , but with the highest probability density the relative distance between the particles .

**(a)**

**(b)**

Thus, the consideration of the electron and positron as independent particles leads to the appearance of the branch of the massless composite bosons formed by the coupled electron-positron pairs with the coupling equal to the fine structure constant. The results obtained above for the normalizable wave functions of the massless bosons are used in the next section, in which reaction (1) is investigated theoretically.

#### 3. The Angular Correlation Spectrum

##### 3.1. The Initial and Final States of the Reaction (1)

We will need the massless boson state for an arbitrary direction of the boson momentum. When is directed along the -axis, the boson wave function is given by (9) with , where the bispinor parts have the forms (23). The bispinors define only the -projection of the total spin . That is, the helicities of the particles are opposite.

When is directed along the -axis, the bispinors take the forms: For (50) we have . The form of these functions is obvious when is parallel to the -axis.

Now consider an arbitrary direction of the momentum boson. Let be an arbitrary radius vector lying in the plane perpendicular to the vector (). Then the wave function (9) takes the form:where is the component of the radius vector which is collinear to and are the bispinor parts which can depend only on the angles of the unit vector . For the wave function (51) one should understand that , where and , and that and with the condition .

As it is well known, the three-dimensional spinors corresponding to definite helicities have the following form:where and are the polar and azimuthal angles of the vector . Taking into account (52), one can see that (23) and (50) are the special cases of the following bispinor functions:

Now we turn to the initial state of the free electron and positron. In principle one can consider colliding non-spin-polarized particles with equal energy. Currently, however, the production of spin-polarized low-energy positron beams with the kinetic energy of a few electron volts became possible [34â€“38]. It is therefore of particular interest to study the annihilation-like process (1) for the polarized beams.

Suppose that the electron and positron collide with the center of mass at rest (). For definiteness let the electron momentum, , be directed along the -axis; then the positron momentum, , is against the -axis. The wave function of the free pair can be represented aswhere and are the four vectors for the electron, and are the four vectors of the positron, and . The spin function in (54) can be defined by the -projections of the spin for the particles in the rest system. Then, for the total spin projection , the function can be represented as

The function (55) can be considered as the two-particle spin state with opposite helicities. One can also say that (55) corresponds to the opposite longitudinal polarization of the particles.

Free pairs of the transversely polarized particles relative to the vector can be prepared as well, and it is easy to write, for example, for the polarization along the -axis. Besides, the colliding electron-positron pair with or can be experimentally obtained, and it is not difficult to write the corresponding bispinor parts for these cases.

Interaction for the reaction (1), which will be discussed in Section 3.2, predetermines the choice of the photon wave function. The photon plane wave can be represented aswhere is the photon polarization that can be chosen real, , and are the energy and wave vector of the photon, respectively. For the transverse photons with .

##### 3.2. Interaction for the Reaction (1)

The products of reaction (1) include two photons and the massless boson. This composite boson is formed by the strongly coupled electron-positron pair. Therefore, the radiative transition of any particle from the initial free pair (the left-hand side of (1)) to any intermediate state does not lead to the formation of the massless boson. The simultaneous emission of one photon by the electron and other photon by the positron and, accordingly, the simultaneous transition of both the electron and positron into the massless boson state are the only processes for which reaction (1) occurs. For this reason it is necessary to determine the interaction operator for the simultaneous emission of photons by each particle of the pair.

The additive energy of any free pair from the beams is defined asHere is the pair mass. Note that in (57) the sum of the particle momenta is presented.

The interaction operator for single-photon emission by the electron and positron at the same time can be obtained from (57) by introducing the canonical momenta in the presence of electromagnetic fields, and ( are the operators of the vector potentials generated by the electron and positron). Assuming for the nonrelativistic particles, from (57), we obtain the expression for this operator:

A similar approach can be applied to positronium. The kinetic energy of the unperturbed Hamiltonian is where is the momentum operator of the relative motion, corresponding to the relative radius vector between the particles. Substituting into this energy and making the transition to the canonical momenta of the particles, we get, up to the sign, the operator (58) for simultaneous emission of two photons by the two particles.

The electromagnetic interaction operator (58) is used below in the study of the reaction (1).

##### 3.3. Possible Reaction Channels

Now, taking into account the spin conservation, we discuss the possible reaction channels with initially free electron-positron pairs. Let the colliding pairs be in the triplet state, . First of all, the reaction with single-photon emission is* impossible in principle*. Here denotes the initial free pair, is the massless boson, and is the photon emitted. For this reaction, the single photon is only emitted either by the electron or by the positron. Therefore, the simultaneous transition of these particles into the composite massless boson state cannot occur.

The reaction with emission of three photons,can take place. For (61), at first one of the particles (electron or positron) emits one photon (say ) and passes to an intermediate state, and then, in a subsequent point in time, simultaneous emission of two other photons () is accompanied by the simultaneous transition of these particles into the boson state .

In the case of prepared pairs with the reaction with single-photon emission is also impossible in principle as was discussed above.

The reaction with the emission of two photons:can occur due to the interaction (58). As will be shown below, for (62), the emission of composite bosons with relatively low energies is possible and preferable. Their energy is several orders of magnitude smaller than the electron mass. With the increasing of the boson energy the cross-section of this reaction decreases sharply.

Below, we derive the cross-section of the reaction (62) and study numerically the angular correlation spectra.

##### 3.4. Cross-Section of the Reaction (62)

In the center of mass system and . The cross-section of the reaction (62) has the following form:Here , is the relative velocity of the particles, , and is the matrix element of the operator (58) for the transition from the initial state (*i*) of the free pair (54)-(55) into the final state (*f*). The latter include the composite massless boson (51)â€“(53) and the two photons (56), one of which has the energy , the momentum , and the polarization () and the second photon, , , and (), respectively. This matrix element is written asHere satisfies (29), is the two-dimensional relative vector between the electron and positron, , and . As noted above, when the boson momentum is directed along -axis, the -components of the radius-vectors of particles coincide, and . Now we need to take into account the fact that for the composite boson. On the right side of (64), the last factor in round brackets is the sum of two terms. The first contribution corresponds to the emission of photon with the wave vector by the electron, and in the second term the electron emits photon with the wave vector . The second of two photons is emitted simultaneously by the positron.

To calculate the spatial integrals in (64), it is convenient to use new variables: and . As a result we obtainmultiplied by . We do not write the latter, because it is already included in (63). In (65) the Fourier transform of the wave function of the transverse motion of the coupled pair in the massless composite boson state is used: Here is the component of the photon wave vector which is perpendicular to ,and is the component of the free electron momentum perpendicular to the boson momentum,According to the momentum conservation, , where is the corresponding component of the wave vector of the second emitted photon.

Note that has the dimension of length.

Substituting (64)-(65) in (63), the cross-section is given bywhere is the electromagnetic radius of the electron and , where is the Compton wavelength of the electron.

##### 3.5. Transformation of (69)

In (69) we have the summation over photon polarization: . Because for and , the sum is reduced to For the unit polarization vectors we can use . Then, taking into account , finally we obtain .

In (69) the factor can be considered as the overlap of bispinor functions. Using (53) and (55), we find that the factor is equal to .

Considering these results, in the nonrelativistic limit, the cross-section (69) is reduced to