Forward \bar{p}d elastic scattering and total spin-dependent \bar{p}d cross sections at intermediate energies

Forward elastic scattering and total spin-dependent cross sections at intermediate energies

Yu.N. Uzikov, J. Haidenbauer Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany

Spin-dependent total cross sections are considered using the optical theorem. For this aim the full spin dependence of the forward elastic scattering amplitude is considered in a model independent way. The single-scattering approximation is used to relate this amplitude to the elementary amplitudes of and scattering and the deuteron formfactor. A formalism allowing to take into account Coulomb-nuclear interference effects in polarized cross sections is developed. Numerical calculations for the polarized total cross sections are performed at beam energies 20-300 MeV using the interaction models developed by the Jülich group. Double-scattering effects are estimated within the Glauber approach and found to be in the order of 10 - 20 %. Existing experimental data on differential cross sections are in good agreement with the performed Glauber calculations. It is found that for the used models the total longitudinal and transversal cross sections are comparable in absolute value to those for scattering.

13.75Cs; 24.70.+s; 25.43.+t; 29.27.Hj

I Introduction

Recently the PAX collaboration was formed PAX () with the aim to measure the proton transversity in the interaction of polarized antiprotons with protons at the future FAIR facility in Darmstadt. In order to produce an intense beam of polarized antiprotons, the collaboration is going to use antiproton elastic scattering off a polarized hydrogen target (H) in a storage ring Rathmann (). The basic idea is connected to the result of the FILTEX experiment FILTEX (), where a sizeable effect of polarization buildup was achieved in a storage ring by scattering of unpolarized protons off a polarized hydrogen atoms at low beam energies of 23 MeV.

According to recent theoretical analyses MS (); NNNP (); NNNP1 (); NNNP2 () the polarization effect observed in Ref. FILTEX () has to be interpreted in such a way that solely the spin dependence of the hadronic (proton-proton) interaction provides the spin-filtering mechanism, i.e. is responsible for different rates of removal of protons from the ring for different initial polarization states. In other words MS (); NNNP (), proton scattering on the polarized electrons of hydrogen atoms cannot provide a sizeable effect of polarization buildup, as it was assumed before HOM (). Indeed, the maximal scattering angle in this process, mrad, is less than the beam acceptance angle , which is defined so that for scattering at smaller angle the projectile remains in the beam. For this case, the beam-into-beam scattering kinematics of this process in a storage ring allows the proton polarization buildup only due to spin-flip transitions between the initial and final spin states of the beam proton MS (). Furthermore, since the Coulomb interaction between the protons and electrons is spin-independent it cannot provide spin-flip transitions and, consequently, does not contribute to the polarization buildup. The same argument, obviously, is valid in case of antiproton scattering off a hydrogen target. Therefore, the authors of Ref. MS () concluded that only the hadronic interaction can be used to produce polarized antiprotons on the basis of the spin-filtering mechanism.

In contrast to the case, the spin-dependent part of the interaction is poorly known experimentally at present. Therefore, to investigate the polarization buildup mechanism in H scattering a new experiment is planned at CERN AD (). The stored antiprotons will be scattered off a polarized H target in that experiment AD () and the polarization of the antiproton beam will be measured at intermediate energies. Some theoretical estimations DmitrievMS () of the expected polarization effects were already performed employing a specific model of the interaction.

In this context, it is interesting and useful to explore other hadronic reactions as possible source for the antiproton polarization buildup too. Therefore, in the present work we study polarization effects in antiproton-deuteron () scattering for beam energies up to 300 MeV. Besides the issue of polarization buildup for antiprotons, scattering on a polarized deuteron, if it will be studied experimentally, can be also used as a test for our present knowledge of the and interactions. Our investigation is based on the Glauber-Sitenko theory for scattering and it utilizes the interaction models developed by the Jülich group Hippchen (); Hippchen1 (); Mull () as input for the elementary amplitudes. Since there are data on (unpolarized) total and differential cross sections in the considered energy range we can examine the reliability of the Glauber approach via a direct comparison of our results to experimental information. In addition we also present results for polarization effects for the system itself. With the potentials developed by the Jülich group we have conceptually rather different models at our disposal than the one used in Ref. DmitrievMS () and it will be instructive to see and compare the corresponding predictions. Moreover, we consider here the as well as the case.

Let us mention here for completeness that very recently new QED calculations for polarization transfer in proton-electron (and antiproton-positron) elastic scattering were performed Arenhoevel (). These calculations predict very large polarization transfer from polarized electrons (positrons) to unpolarized protons (antiprotons) in elastic proton-electron (antiproton-positron) scattering at low beam energies, less than 20 keV Arenhoevel (). On this basis new sources for polarized antiprotons are under discussion Walcher (). On the other hand, according to a calculation presented in Ref. MSS (), the effect of polarization buildup is negligible in this reaction, even at small relative velocites. A recent measurement performed at COSY Frankpc () intends to explore this method. Finally, another method for buildup of polarized antiprotons at high energies, based on production of and its subsequentional decay , has been proposed in Ref. Nurushev ().

The paper is structured in the following way. In the next section we consider the spin structure of the total cross section using the optical theorem. In Sect. III expressions for the forward elastic scattering amplitude are derived in the impulse approximation and the formalism for the polarized total cross sections is developed. The formalism for calculating the Coulomb-nuclear interference cross sections is presented in Sect. IV. The technical details for evaluating the Coulomb-hadronic interference cross sections for elastic scattering are summarized in Appendix A. The interaction model of the Jülich group is briefly reviewed in Sect. V. The amplitudes of this model are used as input for our calculations. We also provide and discuss results for spin-dependent cross sections in the as well as channels. Numerical results for the reaction are presented and discussed in section VI. A short summary is provided in the last section.

Ii Phenomenology of the spin dependence of the total cross section

Let us first consider only the purely hadronic part of the reaction amplitude. The modifications due to the presence of the Coulomb interaction will be discussed in Sect. IV. We use the optical theorem to derive the formalism for the total spin dependent cross sections. According to phillips () one has


where is the transition operator for elastic scattering at the angle , is the initial spin-density matrix, is the total cross section depending on the density matrix , and is the momentum in the center-of-mass system (cms).

The spin dependence of the amplitude of the elastic scattering is the same as for the elastic scattering. For collinear kinematics it contains four independent terms rekalo () and can be written as uzepan98 ()


where () are the Pauli spin matrices, is the fully antisymmetric tensor, are the Cartesian components of a unit vector pointing along the beam momentum, and are complex numbers determined by the dynamics of the reaction. Let us put the axis along the vector , so that . The elastic scattering amplitude is obtained by sandwiching the operator between antiproton and deuteron spin states,


where () is the Pauli spinor for the initial (final) antiproton with the spin projection () and () is the polarization vector of the initial (final) deuteron with the spin projection (.

When choosing different initial polarization states in Eq. (1), described by the product of the spin density matrices of the antiproton and the deuteron , , one can derive from Eqs. (1) and (2) different total spin-dependent cross sections in terms of the forward amplitudes . For example, for unpolarized antiprotons and unpolarized deuterons one has , and one finds from Eq. (1) the unpolarized total cross section to be


In general the spin density matrices are


for the antiproton and


for the deuteron. Here is the spin-1 operator, and are the vector and tensor polarizations of the deuteron, and is the spin-tensor operator.

Using Eqs. (2), (5) and (6) one can find from Eq. (1),


As seen from this formula, if the initial antiproton is polarized with the polarization vector and the deuteron has the polarization vector , then non-zero terms arise in the right-hand side of Eq. (1) for parallel (or antiparallel) orientation of the vectors and . For the tensor polarization of the deuteron only the diagonal components of the polarization tensor , and are connected with non-zero total cross section. The cross sections associated with and are the same and determined by , whereas the cross section connected with the component is given by . Taking into account the relation , one can find from Eq. (II) that the total tensor cross section can be connected only with the component.

All other combinations of polarizations of the antiproton and/or deuteron, namely, polarized antiproton, vector-polarized deuteron, and polarized antiproton – tensor-polarized deuteron, give zero contribution to the total cross section due to parity conservation.

Summarizing the above results, the total polarized section can be written as


where is given by Eq. (4), and the other coefficients are:


One can find from Eq. (8) that only the cross sections and are connected with the spin-filtering mechanism and, therefore, determine the rate of the polarization buildup in the scattering of unpolarized antiprotons off polarized deuterons. (More precisely, the rate of polarization buildup is determined by the ratio for transversal polarization and by for longitudinal polarization of target and beam in the storage ring.) The tensor cross section is not connected with the polarization of the beam and, therefore, is not relevant for the spin-filtering. However, this cross section, as well as the unpolarized cross section , determines the lifetime of the beam. When changing the sign of the tensor polarization , one may change the beam lifetime.

Iii elastic scattering at forward angles

iii.1 Glauber theory

Within the Glauber theory the amplitudes for the elastic () and breakup () reactions are given by the following matrix element


calculated between definite initial and final states of the two-nucleon system. Here the transition operator is


In Eqs. (10) and (11) is the transferred momentum, is the impact parameter, and () is the scattering amplitude. The amplitude of elastic scattering can be expressed via the elastic form factor of the deuteron, , and the elementary amplitudes of scattering. The differential scattering cross section for elastic () plus inelastic () scattering is calculated within the closure approximation FrancoGlauber (). That allows one to express the scattering cross section via , and the deuteron wave function. Since the D-wave component of the deuteron wave function becomes important only in the region of the first diffraction minimum of the differential cross section mahalabi (), we neglect its contribution in the present calculations and take into account only the S-wave component.

As seen from Eq. (11), in the Glauber theory of multiple scattering of hadrons off the deuteron only single-scattering (first two terms on the right-hand side) and double-scattering (third term on the right-hand side) mechanisms contribute to the transition amplitude. In forward direction the single-scattering mechanism dominates. The corrections related to double-scattering effects produce the so-called shadowing effect. As a result, the total unpolarized antiproton-deuteron cross section is not equal to the sum of the total and cross sections, and , but is given by


where stands for the corrections due to double-scattering effects.

iii.2 Impulse approximation

In the impulse approximation (IA) (or single-scattering approximation) one can present the forward elastic scattering amplitude in the following form


where is the scattering amplitude at zero degree, defined as in Ref. bystricky (), is the elastic form factor of the deuteron at zero transferred momentum , () is the invariant mass of the () system, and () is the mass of the deuteron (nucleon). The sum in Eq. (13) runs over the projections of the spin of the nucleon spectator , and of the initial () and recoil () nucleons inside the deuteron. The transition operator for the forward scattering amplitude has the form bystricky ()


where the matrix () acts on the spin state of the antiproton (nucleon) and , and () are complex amplitudes bystricky ().

The deuteron elastic form factor at can be written as


where is the relative weight of the S-wave () and D-wave components of the deuteron wave function with the normalization .

Using Eq. (2) one can find the invariant amplitudes from the following transition matrix elements of forward scattering rekalo ():


On the other hand, using Eqs. (13), (14) and (15), one can express the transition matrix elements in terms of the scattering amplitudes as


where , , and


From a comparison of Eqs. (III.2) with Eqs. (17) one can find that in the single-scattering approximation the invariant amplitudes can be written as


Here we used the standard relations between the amplitudes , , and and the helicity amplitudes of scattering bystricky () together with the notation . Utilizing Eqs. (4), (9) and (III.2) the total cross sections are thus


where we used the relations bystricky ()


and the fact that the cms momentum in the system, , is related to the cms momentum, , by


for equal () beam energies in the reactions and . The quantity in Eqs. (20) is defined by .

One can see from Eqs. (20) that in the impulse approximation all total cross sections are additive, i.e. given by the sum of the corresponding cross sections on the proton and neutron. While this result is obvious for the total unpolarized cross section , it was not expected for the spin-dependent cross sections, especially, in view of the opposite sign of the cross sections and with regard to and , respectively.

iii.3 Shadowing effects

As already mentioned above, the double-scattering mechanism dominates at large scattering angles, while its relative contribution decreases when approaching the scattering angle . The numerical calculations of the forward amplitude of elastic scattering, which will be presented below, demonstrate that the inclusion of the double-scattering mechanism reduces the total unpolarized cross section by around 15 % at energies 50-300 MeV as compared to the result obtained within the single-scattering approximation If one adopts the approximaton given in Ref. Alberi () then the effects of double-scattering for polarized cross sections should be likewise around 15-20 % in the energy region in question. Thus, we expect them to be significantly smaller than the variations in the predictions due to the uncertainties in the spin-dependence of the elementary interaction. Therefore, we do not consider the double-scattering effects on the spin-dependent cross sections in the present investigation, which anyway has an exploratory character. Note that an accurate evaluation of the contribution of double-scattering effects to polarized cross sections requires to consider the D-wave component of the deuteron as well as the angular dependence of all (ten) helicity amplitudes Alberi1 () and is therefore rather tedious.

Iv Coulomb-nuclear interference

The total polarized cross sections including the Coulomb interaction can be written as the sum of the purely hadronic contributions , the Coulomb-nuclear interference terms and the pure Coulomb contribution :


where is only present in case of the reaction. The hadronic contributions are those discussed in detail in the two preceeding sections. Note, however, that from now onwards we label the corresponding quantities with the superscript “h” for the sake of clarity.

As was found in Refs. MS (); HOM (), the interference between the Coulomb amplitude and the hadronic amplitudes in the total spin-dependent cross section of scattering plays an important role in the spin-filtering mechanism. When taken into account together with the purely hadronic total cross section this interference improves significantly the agreement between the theory of spin-filtering and the data of the FILTEX experiment MS (); NNNP (); HOM ().

Due to the singularity at , Coulomb effects in the total cross section cannot be taken into account by means of the optical theorem. Therefore, in order to obtain the Coulomb-nuclear interference cross section for elastic scattering one has to perform an integration over the scattering angle for terms like or , where is the Coulomb scattering amplitude, and is the amplitude of the purely hadronic interaction modified by the Coulomb interaction Landau (). As was shown in MS () and NNNP (), in such an integration the lower limit for the scattering angle has to be taken as (with ), because scattering events at lower angles, where the projectiles stay in the beam, do not lead to a beam polarization.

In the Glauber theory Coulomb effects can be taken into account by the method of Ref. Lesniak () in which the elementary eikonal phase is taken as sum of the purely strong and purely Coulomb phases. It is rather obvious that Coulomb effects appear in the total cross section only due to the presence of the pure Coulomb term in the elastic scattering amplitude (see Eq. (24)). However, there is also an interference effect between the Coulomb amplitude and the scattering amplitude, as can be seen immediately from the expressions for the single-scattering approximation given below.

iv.1 scattering

When calculating the Coulomb total cross section and the Coulomb-nuclear interference cross sections for the system we follow Ref. MS (), where these effects were considered for scattering. Here we take into account the difference in the electric charge between antiproton and proton, and we drop the exchange term , specific for scattering. The Coulomb scattering amplitude for is


where is the fine structure constant and () the velocity (momentum) of the antiproton in the cms. The Coulomb phase is


The cross sections were considered in Ref. MS () under the assumption that the beam acceptance satisfies the following condition: , where is the typical magnitude of the hadronic amplitude. Within this assumption the total Coulomb cross section was estimated in MS () to be


In contrast to scattering, for the interaction the spin-dependent Coulomb cross sections and are zero, because there is no antisymmetrization term in the elastic scattering amplitude. The interference terms and obtained in MS () in the logarithmic approximation (see Eq. (18) therein), have a fairly smooth dependence on , namely of the form . Adapting the formalism of MS () for the case we obtain the following expressions for the contribution of the Coulomb-nuclear interference terms to the spin-dependent cross sections:


where and are the hadronic helicity amplitudes, modified by the Coulomb interaction. One can see that in the limit Eqs. (IV.1) coincide with Eq. (18) of Ref. MS (). We should note that our definition for the cross section differs from that in Ref. MS (): our is equal to as definined in Eq. (2) of Ref. MS (). It is worth to note that at sufficiently high beam energies (above 50 MeV) one has and . As one can see from Eqs. (IV.1), in this case the total interference cross sections are determined by , and for , 1 and 2, respectively, whereas the purely hadronic total cross sections are given by the corresponding imaginary parts of those amplitude combinations, see Eqs. (21).

iv.2 elastic scattering

Also for the total Coulomb cross section contributes only for . In order to calculate the Coulomb-hadronic interference cross sections for one needs the elastic scattering amplitudes beyond the collinear kinematics, because it is necessary to perform an integration over the scattering angle. Therefore, the full spin structure of the scattering amplitude which consists of twelve independent terms, has to be considered. Details of the formalism for the general case are summarized in Appendix A. The final formulae for the polarized interference cross sections are Eqs. (Appendix: Coulomb-nuclear interference cross sections for elastic scattering).

In order to obtain within the impulse approximation one has to insert Eqs. (III.2) into Eqs. (Appendix: Coulomb-nuclear interference cross sections for elastic scattering) and after that use it in Eqs. (Appendix: Coulomb-nuclear interference cross sections for elastic scattering). The Coulomb amplitude for scattering in the impulse approximation at is , where is defined in Eq. (24). Thus, the Coulomb-nuclear interference contribution to the total cross sections, , () follows from Eqs. (IV.1) with the replacement . Furthermore, Eqs. (IV.1) should be multiplied by the factor and and will change their signs. Note that the cross sections and are equal to zero in the single-scattering approximation, because the hadronic amplitude vanishes in this approximation.

V Results for the system

In the present investigation we use two models developed by the Jülich group. Specifically, we use the models A(BOX) introduced in Ref. Hippchen () and D described in Ref. Mull (). Starting point for both models is the full Bonn potential MHE (); it includes not only traditional one-boson-exchange diagrams but also explicit - and -exchange processes as well as virtual -excitations. The G-parity transform of this meson-exchange model provides the elastic part of the considered interaction models. In case of model A(BOX) Hippchen () (in the following referred to as model A) a phenomenological spin-, isospin- and energy-independent complex potential of Gaussian form is added to account for the annihilation. It contains only three free parameters (the range and the strengths of the real and imaginary parts of the annihilation potential), fixed in a fit to the available total and integrated cross sections. In case of model D Mull (), the most complete model of the Jülich group, the annihilation into 2-meson decay channels is described microscopically, including all possible combinations of , , , , , , , , , , – see Ref. Mull () for details – and only the decay into multi-meson channels is simulated by a phenomenological optical potential.

Figure 1: Total (a) and integrated annihilation (b) cross sections versus beam momentum. The lines show predictions by the Jülich models A (dashed line) and D (solid line). Data are taken from Refs. Arm87 () (circles) and Iaz00 () (squares).

Results for the total and integrated elastic () and charge-exchange () cross sections and also for angular dependent observables for both models can be found in Refs. Hippchen (); Mull (). Evidently, with model A as well as with D a very good overall reproduction of the low- and intermediate energy data was achieved. Moreover, exclusive data on several 2-meson and even 3-meson decay channels are described with fair quality Hippchen1 (); Mull (); Betz (). Recently, it has been shown that the models of the Jülich group can also explain successfully the near-threshold enhancement seen in the mass spectrum of the reactions Sibi1 (), Sibi3 () and Sibi2 () and in the cross section Sibi4 ().

As already mentioned in the Introduction, the spin dependence of the interaction is not well known. There is a fair amount of data on analyzing powers, for elastic as well as for charge-exchange scattering, cf. Ref. Klempt () for a recent review. However, with regard to other spin-dependent observables there is only scant information on the depolarization and also on . Moreover, those data are of rather limited accuracy so that they do not really provide serious constraints on the interaction. The predictions of the Jülich models A and D are in reasonable agreement with the experimental polarizations up to beam momenta of MeV/c as can be seen in Ref. Mull (). In fact, model A gives a somewhat better account of the data and reproduces the measured polarizations even quantitatively up to MeV/c ( MeV). We consider both models here because it allows us to illustrate the influence of uncertainties in the spin-dependence of the interaction on the spin-dependent cross sections for the as well as the systems. In this context let us mention that a partial-wave analysis of scattering has been performed by the Nijmegen Group Timmermans () which, in principle, would allow to pin down the spin-dependence of the interaction. However, the uniqueness of the achieved solution was disputed in Ref. Richard (). Moreover, the actual amplitudes of the Nijmegen analysis are not readily available and, therefore, cannot be utilized for the present investigation.

For the computation of scattering we also need the amplitude. For this system, a purely isospin state, there is no experimental information. But there are data for the channel Arm87 (); Iaz00 (), which is identical to the former under the assumption of isospin symmetry. A comparison of our model results with those data is presented in Fig. 1. Obviously the predictions of the Jülich models are in nice agreement with the experimental information on the interaction too, despite the fact that those total and annihilation cross sections have not been included in the fitting procedure.

Figure 2: Total cross sections (a), (b), and (c) versus the antiproton laboratory energy . Results based on the purely hadronic amplitude, , (model D: solid line, model A: dashed line) and for the Coulomb-nuclear interference term, , (D: dash-dotted line, A: dotted line) are shown.

Let us now come to the spin-dependent and cross sections predicted by the Jülich interactions. Corresponding results are presented in Figs. 2 and 3. We display the cross sections based on the purely hadronic amplitude () and the Coulomb-nuclear interference term () separately so that one can see the magnitude of the latter. The total cross sections are then the sum of those two contributions. In the concrete calculations the acceptance angle was chosen to be mrad FILTEX ().

At low energies, i.e. around MeV, the interference terms are comparable to the corresponding purely hadronic cross sections and their magnitude increases further with decreasing energy due to the factor, cf. Eqs. (IV.1). With increasing energy the relevance of the Coulomb-nuclear interference terms diminishes more and more in case of the cross sections and . But for the term is still significant, as one can see from Fig. 2b. The large magnitude of as compared to is due to the fact that for 100 - 300 MeV, for both models. As already pointed out in the context of Eqs. (IV.1), is determined by the former quantity but by the latter. Note, that the three cross sections () themselves are all roughly of comparable magnitude for energies from around 50 MeV onwards.

While the predictions of the two models for are rather similar (cf. Figs. 2a and 3a), even for the Coulomb-nuclear interference cross section, this is not the case for the spin-dependent cross sections and . For energies below MeV there are drastic differences between the results based on the two models. Indeed, for at low energies even the sign differs in case of the channel. Obviously, here the variations in the hadronic amplitude are also reflected in large differences in the Coulomb-nuclear interference term.

For the total (including the hadronic and the Coulomb-nuclear interference terms) model A predicts a maximum of 12 mb at the beam energy MeV whereas model D yields a maximum of practically the same magnitude at MeV. In both cases becomes large and negative at very low energies due to the dominance of the Coulomb-nuclear interference term in this region. For comparison, in Ref. DmitrievMS (), where a version of the Paris model was employed, the largest value for was found to be -15 mb at MeV. In case of scattering both models exhibit a minimum in at MeV and reach values of around 20 mb (A) and 50 mb (D) close to threshold.

With regard to model A and D predict values around 10 mb for scattering at higher energies. Close to threshold large negative values are predicted for due to the Coulomb-nuclear interference term. One should note, however, that for beam energies below 5 MeV, say, the total Coulomb cross section becomes very large. In this case the beam lifetime turns out to be too short and the spin-filtering method cannot be used for polarization buildup in storage ring. The results for for scattering are comparable for both models, reaching a maximum of roughly 30 mb around MeV.

Finally, one should note that for both models A and D the polarized cross sections and exhibit a very different energy dependence in the and channels. Thus, the expected polarization buildup for scattering is likewise different from that of the reaction, as will be shown in the next section.

Figure 3: Total cross sections (a), (b), and (c) versus the antiproton laboratory energy . Results for model D (solid line) and model A (dashed line) are shown.

Vi Results and discussion of scattering

In this section we present numerical results for scattering employing the Jülich models A and D Hippchen (); Mull () for the elementary interactions. In order to estimate the role of the double-scattering mechanism, which will be not taken into account in our calculation of the polarized total cross sections, we first calculate the (unpolarized) total cross section and also differential cross sections for elastic as well as for elastic plus inelastic scattering events within the Glauber theory. This allows us also to compare our results directly with available experiments so that we can check the reliability of the approach. As was shown in detail in Refs. Kondratyuksb (); Dalkarov (); mahalabi (); Bendis1 (); Ma (); Bendis2 (); Robson (), in forward elastic scattering of antiprotons off nuclei the Glauber theory, though in principle a high-energy approach, works rather well even at fairly low antiproton beam energies like 50 MeV. This has to be compared with proton-nucleus scattering where the Glauber theory is known to give reliable results only for energies in the order of 1 GeV or above. The Glauber theory is applicable at such low energies, because of the presence of annihilation channels in the interaction. Due to strong annihilation effects, specifically in the -waves, higher partial waves start to play an important role already fairly close to threshold. As a consequence, the elastic differential cross section is peaked in forward direction already at rather low energies Mull (); Klempt () and, therefore, suitable for application of the eikonal approximation, which is the basis of the Glauber theory. In other words it can be seen from the optical theorem that the higher the annihilation cross section is, the larger is the modulus of the forward elastic scattering amplitude. Indeed, the elastic spin-averaged scattering amplitude can be very well parameterized by


where is the total unpolarized cross section, is the ratio of the real to imaginary part of the forward scattering amplitude , is the slope of the diffraction cone, is the transferred 3-momentum, and is the cms momentum.

MeV mb (GeV/c) mb (GeV/c)
10 A 409 46.24 -0.351 372 41.1 -0.372
D 441 59.1 -0.20 354. 51.4 -0.164
25.5 A 305 46.24 -0.176 267.2 36.0 -0.146
D 312 52.4 -0.130 260 48.8 -0.076
50 A 233.5 33 -0.095 209.6 26 -0.034
D 240 33.47 -0.121 219 33.47 -0.058
78 A 209.5 29.9 -0.03 192.6 26 0.0417
D 203 29.9 -0.03 192 29.2 0.0145
109 A 186 25.2 0.033 171.3 24 0.108
D 175.8 25.9 -0.029 165 25 0.0245
124 A 174.4 24.45 0.057 163. 24.3 0.133
D 168.4 24.45 -0.030 159.5 24.3 0.0245
138 A 174.0 23.24 -0.030 159.5 22.9 -0.03
D 162.5 23.24 -0.030 155 22.9 -0.03
149 A 164 22.58 0.009 156.2 22.58 0.166
D 159.26 22.58 0.002 149 22.58 0.056
179 A 156.3 23.4 0.110 148 20.7 0.187
D 150. 23.4 -0.0256 143 20.7 -0.0816
Table 1: Parameters of the and