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Multiple scattering theory with proper wave function symmetry applied to piondeuteron scttering at threshold Bendix, Peter Bernard


The scattering amplitude for pions on deuterons is calculated in the threshold limit using Watson's multiple scattering theory. Care is taken to use wave functions with proper symmetry throughout and it is shown that the results are identical with those obtained using unsymmetrized wave functions in intermediate states. Terms up to second order in the multiple scattering series are calculated, gradually increasing the complexity of the assumptions until all quantitatively relevant features are taken into account. Specifically, we treat single scattering, double elastic scattering, double charge-exchange scattering, and second order binding correction terms. New quantitative results are obtained which account for non-zero binding energy of the deuteron and nucleon excitation in the propagators, Lorentz-invariant and inelastic scattering kinematic factors in the two-body scattering amplitudes, phase-shift fitted pion-nucleon scattering amplitudes up to P-waves, an S-wave Gartenhaus deuteron wave function, and relativistic effects in high-momentum intermediate states. In addition, a general method utilizing graphs analogous to Feynman diagrams is presented which easily reproduces each order contribution of the multiple scattering series (for constant two-body T matrices) and allows ene to sum the whole series in closed form. In particular, we find the sum of the whole series for π⁻-deuteron scattering at threshold, including all isospin-flipping terms, a result incorrectly obtained in previous literature. We also find the series sum for π⁻ scattering on an arbitrary nucleus of neutrons and protons,including charge-exchange scattering. (This result does not appear in the literature,). From the series sum we then calculate the higher-order contribution with a Hulthen and then a Gartenhaus S-wave deuteron wave function first neglecting charge-exchange and then including it. We find the higher-order contribution to be roughly twenty per cent of the first and second order terms combined (at threshold). Our best estimate of the pion-deuteron scatter ing amplitude at threshold (the pi-d scattering length) is F[sub πd],= -.0273 fermis. Because pion-deuteron scattering is a three-body problem and because of the similarities with multiple scattering theory, we have included a short discussion of the Faddeev equations. We give particular emphasis to wave function symmetry in the Faddeev approach.

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