The π3He interaction for the polarized target is studied in the fixed-centers approximation with all rescatterings included. Only the P33 wave is taken for the πN interaction. The nuclear wave function is taken either as a sum of Gaussians or as a Faddeev wave function in the s-wave approximation. The asymmetries for elastic π + 3He scattering at the lab energies Tπ =142, 180 and 256 MeV are calculated. The results are compared with experimental data.Razmatra se π3He interakcija s polariziranom metom u približenju nepomičnih središta, uključujući sva višestruka raspršenja. Za πN interakciju uzima se samo P33 val. Nuklearna se valna funkcija uzima bilo kao zbroj Gaussovih funkcija ili kao Faddeeva valna funkcija u aproksimaciji s-vala. Izračunali smo asimetrije za elastično π +3He raspršenje za laboratorijske energije Tπ =142, 180 i 256 MeV. Rezultati se uspoređuju s eksperimentalnim podacima

Braun, M. A.

Suslov, V. M.

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ISSN1330–0016CODEN FIZBE7LETTER TO THE EDITORELASTIC PION SCATTERING ON POLARIZED 3He IN THE ENERGYRANGE OF THE ∆-RESONANCEM. A. BRAUN and V. M. SUSLOVInstitute of Physics, S.-Petersburg State University, 198904 Ul’yanovskaya 1,Petrodvorets, S.-Petersburg, Russia, E-mail: suslov@mph.phys.spbu.ruReceived 24 December 1998; Accepted 7 June 1999The π-3He interaction for the polarized target is studied in the fixed-centers ap-proximation with all rescatterings included. Only the P33 wave is taken for the πNinteraction. The nuclear wave function is taken either as a sum of Gaussians or asa Faddeev wave function in the s-wave approximation. The asymmetries for elasticπ+ 3He scattering at the lab energies Tπ =142, 180 and 256 MeV are calculated.The results are compared with experimental data.PACS number: 25.80.Dj UDC 539.17Keywords: fixed-centers approximation, pion rescattering, asymmetryThe investigation of pion scattering off light nuclei requires reliable theoreticaldescription. The exact relativistic treatment of systems with four or more particlesis beyond present calculational possibilities. Therefore, simplifications are unavoid-able. Current studies of π-trinucleon reactions are mostly done in the frameworkof the optical potential model [1]. This may be a good approach at low energieswhen the pion wavelength λπ is larger than the internucleon distance R. However,at energies above 100 MeV, when λπ ≤ R, the optical potential model does notseem natural and alternative approaches deserve attention.From this point of view the energy region of the ∆-resonance exhibits two bigadvantages. First, at these energies the πN interaction can be well approximatedby the P33 wave, i.e., it is simple and essentially reduced to a separable form.Second, the pion energy is still considerably smaller than that of the nucleon.Therefore one can hope that a reasonable approximation might be to neglect thenucleon recoil.In this limit, the amplitude of the NN interaction vanishes and the problembecomes simplified to the fixed-scattering-centers approximation. It is well-knownFIZIKA B 8 (1999) 1, 107–112 107braun and suslov: elastic pion scattering on polarized . . .that this approximation gives an accuracy of about 10 - 20% for πd interaction [2].In Ref. 3, the πd breakup was studied including the summation of relativistic ladderdiagrams, and it has been shown that taking into account of the nucleon recoil,insignificant improvement was obtained as compared with the static approximation.Of course, the accuracy is decreasing with the increase of momentum transfer.However, any attempt to take the nucleon recoil into account inevitably requiresan ultraviolet cutoff, so that in principle a better accuracy can be achieved only atthe price of an extra parameter.In Ref. 4, a model based on the fixed-centers approximation for elastic π3H andπ3He reactions, with full account of multiple pion rescattering, was introduced.This paper reports on the application of the model to polarized targets.Neglecting the NN interaction and other waves in the πN interaction except theP33, the determination of the amplitude of the interaction with a nucleus is reducedto a summation of the diagrams shown in Fig. 1.• •• •• •+• • • •• •• •+ · · · +Fig. 1. Graphic representation of successive rescatterings of the π meson withproduction of the ∆33-resonance. The dotted lines correspond to the π meson,thick lines correspond to the ∆33-resonance, and thin lines correspond to nucleons.The exact treatment of these graphs requires the introduction of 216 amplitudes.In our numerical calculations, the spin-tensor interaction in the elementary blockof Fig. 2 was replaced by an averaged one. As a result, the number of independentamplitudes is reduced to twenty seven.••Fig. 2. The potential Wik. The notation is the same as in Fig. 1.The basic Faddeev-like amplitudes are Mik where i and k refer to the number ofthe initial and final nucleons (i, k = 1, 2, 3). Each Mik is a 3 ×3 matrix in both spinand isospin. In the fixed centers approximation, they are determined by a systemof linear equationsMik = R1P(i)δik + R1∑l /=iP (i)WilMlk . (1)Here R1(E) = (m∆ − mN + εb − k0 − i0)−1 and k0 is the pion energy. P(i) areprojectors onto states I = J = 3/2 for the system of pion plus the i-th nucleon.108 FIZIKA B 8 (1999) 1, 107–112braun and suslov: elastic pion scattering on polarized . . .The ”potentials” Wik describe intermediate pion propagation. They are com-plex:W (x) =λ23(2π)2∫d3qq2 exp(iqx)q2 + m2π − (k0 − εb)2 − i 0, λ2 =6πΓ∆k3res. (2)Here εb is the nucleus binding energy, Γ∆ is the resonance width and kres is the pionmomentum at the resonant energy. Once the system (1) is solved, the π 3He(3H)amplitudes with a given value of the total isospin T are found asA(T ) = A(T )1 + A(T )2 =3λ2√2k02k′0∑S=1/2,3/2(k′P (S)k)∫d3x1d3x2|ΨA(xik)|2×(exp(i(k − k′)x1)〈M11(T,S)〉 + 2 exp(i(kx1 − k′x2))〈M21(T,S)〉) .(3)Here A(T )1(2) refer to the 1st(2nd) term in the brackets; 〈Mik〉 are the amplitudesaveraged over spin and isospin of the 3He(3H) nuclei. The standard spin-non-flipand spin-flip amplitudes f and g are expressed via A(T )1(2) as followsf (T ) =13(A(T )1,S=1/2 + A(T )2,S=1/2 + 2(A(T )1,S=3/2 + A(T )2,S=3/2)g(T ) = 13(A(T )1,S=1/2 + A(T )2,S=1/2 − (A(T )1,S=3/2 + A(T )2,S=3/2)).(4)Here we used the approximation mN → ∞. However, both the kinematical correc-tion for the finite mN and the contribution from the s-wave πN interaction can inprinciple be taken into account.In our normalization, the differential cross-section on the unpolarized target isgiven bydσdθπ=k204π2(|f (T )|2 + |g(T )|2). (5)The asymmetry for the polarized target isAy =σ ↑ −σ ↓σ ↑ +σ ↓=2 Im(f (T )g∗(T ))|f (T )|2 + |g(T )|2. (6)To see the influence of the ground state structure of the nucleus, we performedcalculations with two different wave functions Ψ. The first choice was a sum of twoGaussians|Ψ|2 =∑j=1,2Nj exp(−αj∑i<kx2ik) (7)with the parameters αj chosen to give the best fit for the elastic cross-section atlarge angles (see Ref. 5).FIZIKA B 8 (1999) 1, 107–112 109braun and suslov: elastic pion scattering on polarized . . .Our second choice was the Faddeev wave function obtained as a solution of theFaddeev 3-nucleon equation with an NN potential taken as a sum of the Coulombpotential and a strong interaction potential in the s-wave approximation [6,7].Fig. 3. Asymmetry for elastic π+ scattering from polarized 3He. The solid linecorresponds to the Faddeev wave function, dashed line corresponds to a sum oftwo Gaussians with all rescatterings of pion taken into account. The dotted linecorresponds to a sum of two Gaussians for two rescatterings of pion taken intoaccount. The experimental data are taken from Ref. 8.Fig. 4. Asymmetry for elastic π+ scattering from polarized 3He at 180 MeV.The notation is the same as in Fig. 3.Fig. 5 (right). Asymmetry for elastic π+ scattering from polarized 3He at 256MeV. The notation is the same as in Fig. 3.In Figs. 3–5, the calculated asymmetries are shown for the reaction π+ 3He atthe pion lab. energy Tπ =142, 180 and 256 MeV together with the experimental110 FIZIKA B 8 (1999) 1, 107–112braun and suslov: elastic pion scattering on polarized . . .data [8]. To demonstrate the influence of multiple π rescatterings, also the resultswhich take into account only two rescatterings are shown (for the wave function asa sum of two Gaussians).The agreement with the experimental data for the asymmetry is found to beconsiderably worse than for the differential cross-sections. This is not astonishing,since, as stressed in all previous publications on the problem, the asymmetry isextremely sensitive to the details of the interaction.For energies 142 and 180 MeV, the calculated asymmetry is following the ex-perimental pattern, but it is shifted toward larger angles. This is not surprising. Inour approximation (pure P33 πN interaction, infinite mN ), the amplitude f vani-hes at θ = 90◦. Addition of the s-wave interaction and kinematical corrections forthe finite mN would both shift this zero towards smaller angles and thus wouldimprove the agreement with the experimental data. At 256 MeV, there seems tobe no agreement with the experimental data at all. However, as found in Ref. 1,the seemingly drastic change in the angular dependence of the asymmetry at thisenergy is in fact due to a quite small shift in the location of the zeros of the real andimaginary parts of the amplitude f . It was also found there that this shift could beeffectively achieved by the inclusion of a small angle-independent term to f . Thus,we expect that inclusion of the s-wave πN interaction will improve the situation at256 MeV considerably.From the figures, one can conclude that one observes a very strong effect ofmultiple rescatterings. For lower energies, the inclusion of multiple rescatteringssubstantially reduces the asymmetry, and at 256 MeV even completely changes itsbehaviour. Introduction of a very sophisticated Faddeev wave function does notseem to change the results in any significant manner. Application of our modelto the study of the asymmetry requires additional improvements, which includeintroduction of the s-wave πN interaction and finite mN kinematics. Work in thisdirection is now in progress.References1) C. Benhold, B. K. Jennings, S. S. Kamalov and L. Tiator, Nucl. Phys. A 540 (1992)621;2) M. A. Braun, W. Kallis and V. M. Suslov, Yad. Fiz. 27 (1978) 135;3) M. A. Braun and V. M. Suslov, Nucl. Phys. A 496 (1989) 687;4) M. A. Braun and V. M. Suslov, Yad. Fiz. 54 (1991) 404;5) M. A. Braun and V. M. Suslov, Yad. Fiz. 58 (1995) 1980;6) A. A. Kvitsinsky, Yu. A. Kuperin, S. P. Merkuriev, A. K. Motovilov and S. L. Yakovlev,Fiz. Elem. Chastis At. Yadra 17 (1986) 267;7) R. A. Malfliet and J. A. Tjon, Nucl. Phys. A 127 (1969) 161;8) O.Häusser, Few-Body Systems, Suppl. 9 (1995) 69.FIZIKA B 8 (1999) 1, 107–112 111braun and suslov: elastic pion scattering on polarized . . .ELASTIČNO RASPRŠENJE PIONA NA POLARIZIRANOM 3He UENERGIJSKOM PODRUČJU ∆-REZONANCIJERazmatra se π-3He interakcija s polariziranom metom u približenju nepomičnihsredǐsta, uključujući sva vǐsestruka raspršenja. Za πN interakciju uzima se samoP33 val. Nuklearna se valna funkcija uzima bilo kao zbroj Gaussovih funkcija ilikao Faddeeva valna funkcija u aproksimaciji s-vala. Izračunali smo asimetrije zaelastično π+3He raspršenje za laboratorijske energije Tπ =142, 180 i 256 MeV.Rezultati se uspored–uju s eksperimentalnim podacima.112 FIZIKA B 8 (1999) 1, 107–112

Elastično raspršenje piona na polariziranom 3He u energijskom području ∆-rezonancije

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