Spin-dependent part of $p\bar{p}$ interaction cross section and Nijmegen potential

Low energy $p\bar{p}$ interaction is considered taking into account the polarization of both particles. The corresponding cross sections are calculated using the Nijmegen nucleon-antinucleon optical potential. Then they are applied to the analysis of the polarization buildup which is due to the interaction of stored antiprotons with polarized protons of a hydrogen target. It is shown that, at realistic parameters of a storage ring and a target, the filtering mechanism may provide a noticeable polarization in a time comparable with the beam lifetime.


I. INTRODUCTION
An extensive research program with polarized antiprotons has been proposed recently by the PAX Collaboration [1]. This program has initiated a discussion of various methods to polarize stored antiprotons. One of the methods is to use multiple scattering on a polarized hydrogen target. If all particles remain in the beam (scattering angle is smaller than acceptance angle θ acc ), only spin flip can lead to polarization buildup, as was shown in Refs. [2,3].
However, spin-flip cross section is negligibly small in both cases of proton-antiproton [2] and electron-antiproton [4] scattering. Hence the most realistic method is spin filtering [5].
This method implements the dependence of scattering cross section on orientation of particles spins. Therefore number of antiprotons scattered out of the beam after interaction with a polarized target depends on their spins, which results in the polarization buildup.
Interaction with atomic electrons can't provide noticeable polarization because in this case antiprotons will scatter only in small angles and all antiprotons remain in the beam [2].
Thus it is necessary to study proton-antiproton scattering.
At present, quantum chromodynamics can't give reliable predictions for pp cross section below 1 GeV and different phenomenological models are usually used for numerical estimations. As a result, the cross sections obtained are model-dependent. All models are based on fitting of experimental data for scattering of unpolarized particles. These models give similar predictions for spin-independent part of the scattering cross sections, but predictions for spin-dependent parts may differ drastically. Different nucleon-antinucleon potentials have similar behavior at large distance (r 1 fm) because long-range potentials are obtained by applying G-parity transformation to well-known nucleon-nucleon potential. The most important difference between nucleonantinucleon and nucleon-nucleon scattering is existence of annihilation channels. A phenomenological description of annihilation is usually based on an optical potential of the Imaginary part of this potential describes annihilation into mesons and is important at small distance. The process of annihilation has no uniform description, and short-range potentials in various models are different.
Spin-dependent part of the cross section of pp interaction was previously calculated in Ref. [6] using the Paris potential. In Ref. [6] a possibility to obtain a noticeable beam polarization in a reasonable time was also investigated. Similar calculations were performed in Ref. [7] where various forms of Julich potentials were explored. Note that the contribution of interference between the Coulomb and strong amplitudes to the scattering cross section has been omitted in Ref. [7]. In the present paper, we calculate the spin-dependent part of the cross section of pp scattering using the Nijmegen model and analyze the polarization buildup which is due to the interaction of stored antiprotons with polarized protons.

II. CROSS SECTIONS
It is convenient to calculate the cross section in the center-of-mass frame, where antiproton and proton have momenta p and −p, respectively. In the nonrelativistic approximation (p ≪ M, where M is the nucleon mass), the antiproton momentum in the lab frame is p lab = 2p. Therefore the acceptance angle (maximum scattering angle when antiprotons remain in the beam) in the lab frame is connected with the acceptance angle in the centerof-mass frame by the relation θ acc = 2θ acc . The pp scattering process has several channels: elastic scattering (pp → pp), charge exchange (pp → nn), and annihilation into mesons (pp → mesons). As explained above, noticeable polarization can be obtained only if some antiprotons are dropped out of the beam.
The corresponding cross section can be written in the form where σ cex is the charge exchange cross section, σ ann is the annihilation cross section, and σ el is the elastic cross section integrated over scattering angle from θ acc to π. All cross sections are summed up over final spin states. The cross section σ el includes pure Coulomb cross section, hadronic cross section, and interference term, which can't be omitted.

Spin-dependent cross section can be written in the form
where ζ 1 and ζ 2 are unit vectors collinear to the particles spins, and v = p/p is unit momentum vector. Here σ 0 is the spin-independent cross section, σ 1 describes spin effects in the case when both vectors of polarization are perpendicular to v, and σ 2 describes spin effects in the case when all three vectors are collinear. We direct quantization axis along the vector v and express the cross sections (3) via cross sections Σ Sµ calculated for states with total spin S and projection of total angular momentum µ: Here we use the relation The potential of proton-antiproton interaction is a sum of the Coulomb potential and optical Nijmegen potential [8]. Therefore the amplitude of elastic scattering can be written as a sum of the Coulomb amplitude and strong amplitude, which doesn't coincide with the amplitude calculated in the absence of the Coulomb field. Strong amplitude is not singular at small scattering angles, so that we can integrate hadronic cross section over the whole range from 0 to π. However, finite θ acc should be taken into account at calculation of the Coulomb cross section and the interference between the Coulomb and strong amplitudes. In the nonrelativistic limit, the Coulomb amplitude is spin-independent and has the form where χ L = arg Γ (L + 1 + iη) are the Coulomb phases, η = − α v lab is the Sommerfeld parameter, v lab = p lab /M, and α is the fine structure constant.
For the strong elastic triplet scattering amplitude, we have The sum over L, L ′ is performed under conditions L, L ′ = J, J ± 1 and |L − L ′ | = 0, 2.
Strong singlet amplitude reads Here S J LL ′ and S L are partial elastic triplet and singlet scattering amplitudes, respectively, Y Lm (θ, ϕ) are the spherical functions and C Jµ Lm,1µ−m are the Clebsch-Gordan coefficients. Charge exchange scattering amplitudes have the form and Here S J LL ′ and S L are partial charge exchange triplet and singlet scattering amplitudes, respectively.
The cross sections Σ 1µ and Σ 00 can be represented as a sum of pure Coulomb cross sections Σ C 1µ and Σ C 00 , hadronic contributions Σ h 1µ and Σ h 00 , and interference terms Σ int 1µ and Σ int 00 . For the Coulomb contribution we have where smallness of θ acc is taken into account. The total hadronic cross section can be calculated using the optical theorem The interference contributions can be calculated in the logarithmic approximation, The hadronic contributions to the elastic cross sections have the form The charge exchange cross sections are given by (14)

III. NUMERICAL RESULTS
We follow the method of calculations of scattering amplitudes S described in Refs. [8,9].
To calculate S-matrix and cross sections (10 -14), we use formula (15) of Ref. [8]. The partial cross sections obtained are in agreement with the values from Table V of Ref. [8].
Let us analyze the kinetics of polarization. Let P T be the target polarization vector and ζ T = P T /P T . Antiproton beam polarization vector P B is collinear to ζ T in both cases ζ T ⊥ v and ζ T v. General solution of the kinetic equation is given in Refs. [2,3]. In our case, when only spin-filtering mechanism is important, we have where Here n is the areal density of the target and f is the beam revolving frequency. It follows from our calculations that Ω out − − Ω out which allows us to simplify Eq. (15).
The beam lifetime due to the interaction with a target is For the polarization degree at t 0 , we have Positive (negative) sign of P B (t 0 ) means that the beam polarization is parallel (antiparallel) to ζ T .
The dependence of σ 1 and σ 2 on T lab for different acceptance angles is shown in Fig. 2.
These cross sections depend on the acceptance angle completely due the interference contribution. It was necessary to take interference into account in the case of pp scattering [2].
In the case of pp scattering the interference contribution is also important. Corresponding contributions σ int 1,2 are also shown in Fig. 2. The dependence of polarization degree P B (t 0 ) for P T = 1 on T lab (MeV) for ζ T · v = 0 (P ⊥ ) and |ζ T · v| = 1 (P ) is shown in Fig. 3. In the case ζ T · v = 0, the polarization degree becomes independent of antiproton energy at T lab about 100 MeV. With increasing acceptance angle the polarization degree raises faster. In the case |ζ T · v| = 1, the polarization degree increases slower but amounts to 40% at energy about 200 MeV. Let us compare our results with the previous calculations. In Ref. [6], spin-dependent part of the cross section of pp scattering has been calculated using Paris potential in energy range 20 ÷ 100 MeV. Authors predict positive P ⊥ with maximum 8% at energy about 60 MeV and negative P , raising up to 12%. Analogous calculations have been performed in Ref. [7] using several modifications of the Julich model. Note that the contribution of interference between the Coulomb and strong amplitudes has been omitted. Qualitatively, the dependence of polarization degree on the antiproton energy, obtained in our paper, is similar to that in Ref. [7], but quantitative disagreement is rather big.
In conclusion, using the Nijmegen nucleon-antinucleon optical potential, we have calculated the spin-dependent part of the cross section of pp interaction and the corresponding degree of the beam polarization. Our results indicate that a filtering mechanism can provide a noticeable beam polarization in a reasonable time. However, we state that today it is impossible to predict the beam polarization with high accuracy because different models give essentially different predictions. Only experimental investigation of the spin-dependent part of the cross section of pp scattering can prove the applicability of potential models. Nevertheless, since polarization degree in all models are rather big, we can hope that filtering mechanism can be used to get polarized antiproton beam.