G.C. Oades

Electromagnetic corrections to the s-wave scattering lengths in pionic hydrogen

Abstract Motivated by the recently performed X-ray precision experiments on pionic hydrogen (preceding paper), we reconsider the problem of electromagnetic corrections to the π - N scattering lengths. Based on a relativistic two-channel approach, we find corrections to the π − p elastic and single-charge-exchange (SCE) scatering lengths due to the point-Coulomb interaction, the finite-size Coulomb interaction (including the pion size), the first-order vacuum polarization (Uehling potential) and the ( π − p ) - ( π 0 n ) mass difference (mass difference effect). We also estimate the contribution due to the ( γ , n ) decay channel. The total corrections to the elastic and the SCE scattering lengths are found to be δ e = −(2.1 ± 0.5) × 10 −2 and δ Γ = −(1.3 ± 0.5) × 10 −2 . Previously published values for the corrections are critically compared with our results.

Phase-shift analysis of low-energy π±p elastic-scattering data

Abstract Using electromagnetic corrections previously calculated by means of a potential model, we have made a phase-shift analysis of the π ± p elastic-scattering data up to a pion laboratory kinetic energy of 100 MeV . The hadronic interaction was assumed to be isospin invariant. We found that it was possible to obtain self-consistent databases by removing very few measurements. A pion–nucleon model, based on s- and u-channel diagrams with N and Δ in the intermediate states, and σ and ρ t-channel exchanges, was fitted to the elastic-scattering database obtained after the removal of the outliers. The model-parameter values showed an impressive stability when the database was subjected to different criteria for the rejection of experiments. Our result for the pseudovector πNN coupling constant (in the standard form) is 0.0733 ± 0.0014 . The six hadronic phase shifts up to 100 MeV are given in tabulated form. We also give the values of the s-wave scattering lengths and the p-wave scattering volumes. Big differences in the s-wave part of the interaction were observed when comparing our hadronic phase shifts with those of the current GWU solution. We demonstrate that the hadronic phase shifts obtained from the analysis of the elastic-scattering data cannot reproduce the measurements of the π − p charge-exchange reaction, thus corroborating past evidence that the hadronic interaction violates isospin invariance. Assuming the validity of the result obtained within the framework of chiral perturbation theory, that the mass difference between the u- and the d-quark has only a very small effect on the isospin invariance of the purely hadronic interaction, the isospin-invariance violation revealed by the data must arise from the fact that we are dealing with a hadronic interaction which still contains residual effects of electromagnetic origin.

THE EXTRACTION OF HADRONIC PARAMETERS FROM EXPERIMENTS ON PIONIUM

Abstract We show how experimental values of the lifetime of the 1 s level of pionium and of the difference between the energies of the 2 s and 2 p levels yield values of the elements a oc and a cc respectively of the s -wave scattering matrix for the two-channel ( π + π − , π 0 π 0 ) system at the π + π − threshold. We then develop a method, using energy independent hadronic potentials which reproduce the best available pion-pion phase shifts up to 500 MeV total energy in the c.m. frame, for obtaining the values of the isospin invariant quantities a 0 2 − a 0 0 and 2 a 0 0 + a 0 2 from a oc and a cc respectively. We emphasize that the isospin invariant scattering lengths a 0 2 and a 0 2 universally used in the literature cannot be considered to be purely hadronic quantities.

Protonium annihilation into two pseudoscalar mesons

Abstract The annihilation of a p p atomic bound state into a pseudoscalar meson and its antiparticle is analyzed in terms of threshold values of the helicity amplitudes for the basic annihilation process p p → m m . Parity and angular momentum conservation restrict such decays to the 3 L J states having J = L ± 1. In addition, apart from annihilation from the 3 P 0 state, the threshold behaviour of the helicity amplitudes heavily suppresses this decay mode in the case of the 3 L J states having J = L − 1. Simple expressions for the partial widths, total widths and branching ratios are derived. The branching ratios are independent of the principal quantum number but depend on the angular momentum quantum numbers. Measurement of the angular distribution of one of the mesons with respect to the X-ray emitted in the transition leading to the decaying 2P atomic state enables separation of the amplitudes for the decays of the 3 P 0 and 3 P 2 states. Polarization of the initially captured proton may provide otherwise inaccessible information about the dynamical evolution of the protonium atomic system during the cascade down from the state into which the antiproton is initially captured.

Electromagnetic corrections for the analysis of low energy π−p scattering data

Abstract We calculate the electromagnetic corrections to the isospin invariant mixing angle and to the two eigenphases for the s -, p 1/2 - and p 3/2 -partial waves for π − p elastic and charge exchange scattering. These corrections have to be applied to the nuclear quantities in order to obtain the two hadronic phase shifts for each partial wave. The calculation uses relativised Schrodinger equations containing the sum of an electromagnetic potential and an effective hadronic potential. The mass differences between π − and π 0 and between p and n are taken into account. We compare our results with those of previous calculations and estimate the uncertainties in the corrections.

Microscopic calculation of antiproton atomic-like bound states in light nuclei

Abstract The antiproton-nucleon K -matrices corresponding to the central, spin-orbit and tensor parts of the Dover-Richard interaction are parametrized by linear combinations of Yukawa functions. The antiproton-nucleus optical potential is then constructed from the two-body K -matrix or a series of light nuclei. Basically simple folding with harmonic oscillator nucleon wave functions is used. However, the angular momentum couplings are treated with special care to allow for the inclusion of the spin-orbit and tensor parts of the basic interaction. Energy level shifts in antiprotonic atoms are then calculated for 4 He, 6 Li, 7 Li, 12 C, 16 O, 17 O, and 18 O. The numerical results show that both the spin-orbit and tensor parts contribute terms significantly larger than the present experimental uncertainties. Hyperfine and fine splitting are of the same order of magnitude and analyses of experimental data should include such splittings. Considering the crudeness of the calculations the results are in surprisingly good agreement with experimental data.

Determination of the s-wave pion-nucleon threshold scattering parameters from the results of experiments on pionic hydrogen

Abstract We give the conversion equations which lead from experimental values of the 3 p → 1 s transition energy in pionic hydrogen and the total width of the 1s level to values of the s-wave threshold scattering parameters for the processes π − p → π − p and π − p → π 0 n respectively. Using a three-channel potential model, we then calculate the electromagnetic corrections to these quantities, which remove the effects of the Coulomb interaction, the external mass differences and the presence of the γn channel. We give the s-wave scattering parameters obtained from the present experimental data and these electromagnetic corrections. Finally we discuss the implications for isospin invariance.

Electromagnetic corrections to the hadronic phase shifts in low energy π + p elastic scattering

Abstract We calculate for the s -, p 1/2 - and p 3/2 -waves the electromagnetic corrections which must be subtracted from the nuclear phase shifts obtained from the analysis of low-energy π + p elastic scattering data, in order to obtain hadronic phase shifts. The calculation uses relativised Schrodinger equations containing the sum of an electromagnetic potential and an effective hadronic potential. We compare our results with those of previous calculations and estimate the uncertainties in the corrections.

NN → ππ In the resonance region: (I) Invariant amplitudes

Abstract An analysis has been made of data for the reactions p p → π − π + and p p → π 0 π 0 in the mass range W ⩽ 2.5GeV using a method exploiting analyticity and crossing symmetry via hyperbolic dispersion relations. Results are presented for the invariant amplitudes of the process.

Microscopic calculation of antiproton-nucleus elastic scattering

Abstract The antiproton-nucleon T -matrix of the Dover-Richard interaction is fitted by a linear combination of simple functions. The one-pion exchange potential is included explicitly, and the remainder is approximated by either Gauss or Yukawa functions. Including a linear energy dependence improves the fit significantly. The optical potential is then obtained by folding with the nuclear density, taking into account the size of the antiproton. When an energy or relative momentum-squared dependence of the central part of the T -matrix is included, the corresponding gradient operators lead to an optical potential containing unusual terms. This corresponds to a generalization of Skyrme-like forces to finite range and to the inclusion of an imaginary effective mass. For scattering the latter term is equivalent to a definite energy-dependent optical potential. Elastic scattering cross sections for 300 and 600 MeV/ c antiprotons on 12 C, 40 Ca and 208 Pb are computed using the KMT procedure for the optical potentials resulting from the various T -matrix approximations. Comparisons with available experimental values give solχ 2 N values typically around 4. The different choices of functions give rather similar results.