Improved theoretical pion-nucleus optical potentials

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC¹² scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π^?C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.     

Analysis of elastic scattering of pi-mesons by nuclei within the microscopic optical potential

The microscopic pion-nucleus optical potential defined by the pion-nucleon scattering amplitude and the density distribution function of the target nucleus is applied to obtain the differential cross section of the elastic pion-nucleus scattering based on the solution to the relativistic wave equation. This allow one to account for effects of the relativization and distortion by the potential field. Data on π^±-meson scattering on ²⁸Si, ⁵⁸Ni, and ²⁰⁸Pb nuclei at T _lab = 291 MeV are analyzed and the parameters of the in-medium πN-amplitude are obtained. The parameters are compared with similar parameters for scattering on free nucleons.     

The DWIA analysis of the pion photoproduction on carbon

The inclusive pion spectra in the¹²C(γ,π) reaction are calculated in the distorted wave impulse approximation (DWIA). The final state pion-nucleus interaction was constructed within the framework of the multiple scattering theory. The sensitivity of the cross section to the choice of different optical potential variants is investigated.     

Quenching in the basic πN amplitude for pion-nucleus scattering

An upward shift of the 3,3 resonance energy in pion-nucleus scattering is discussed, which arises from the absorption of the transferred pion in the Chew-Low πN amplitude, when the πN system is embedded in the nuclear medium.     

Isobar-hole doorway states and π-16O scattering

We describe and apply the isobar-hole approach to intermediate energy pion-nucleus reactions. Pion propagation, nucleon and isobar binding, Pauli restrictions and Δ propagation are calculated explicitly within a shell model framework. Intermediate coupling to multihole channels, for example through pion absorption, is treated phenomenologically through an isobar spreading potential. We find strongly collective Δ-hole states, leading to a reformulation of the approach in an extended schematic model. This entails systematic construction of a Δ-hole doorway state basis within which the Δ-hole propagator is evaluated. We find that this doorway space can be truncated at very low dimensionality while preserving accuracy, thereby simplifying the calculations appreciably. We make a detailed comparison between the theoretical results and recent data for π⁺ — ¹⁶O scattering in the pion energy range 50–340 MeV. Nonresonant πN interactions and the π-nucleus Coulomb interaction are included in the calculations. The data is reproduced quite well both below and in the resonance region, and we discuss in detail the role of various dynamical mechanisms. Above the resonance, the calculations are far less successful. We discuss possible shortcomings, stressing the role of inclusive pion-nucleus reactions for revealing the important dynamics. As a test of the Δ spreading potential used for describing elastic scattering, we calculate the total cross section for pion absorption. The result agrees reasonably well with the available data.     

The absorptive pion-nucleus optical potential

We calculate s-wave and p-wave absorptive pion-nucleus optical potentials assuming that a pion is absorbed by a pair of nucleons. Employing a model which takes into account both a single nucleon absorption with nucleon-nucleon correlations and rescattering, we obtain simple analytic expressions for Im B₀ sid Im C₀ of the pion-nucleus optical potential. The off-shell effect on the s-wave pion absorption is examined and shown to be strongly modified by short range correlations. The result for the p-wave absorptive part Im C₀ clearly shows the importance of the tensor correlations. The enhanced nn emission after π^? absorption is shown to be related with a large p-wave πN scattering length a₃₃ via the tensor correlations.     

Coherent meson production in antinucleon annihilation on nuclei

A phenomenological optical potential for antinucleon-nucleon (N?N) interaction in an energy range of 1 to 15 GeV is presented in an eikonal model. In addition to the discussion of the strength and energy dependence, the potential is used to describe initial state interactions for coherent meson production on nuclei. To describe nuclear matter effects, Hartree-Fock Bogoliobuv (HFB) densities are used. As a first step we calculate the production of two mesons, namely pions. To cover the large energy range, the final state pion-nucleus ( π A) interaction is improved by adding resonances beyond the Δ resonance to a Kisslinger-type ansatz. Results for N?N and π A cross sections are presented.     

Pion-nucleus inelastic scattering and charge exchange

Using a separable fixed scatterer pion-nucleon interaction and the distorted wave impulse approximation we have made predictions for medium energy pion inelastic scattering from ¹⁶O and (π^?, π⁰) charge exchange from ⁴⁸Ca. An optical potential based on the pion-nucleon interaction adopted in this work has been shown previously to provide good fits to pion-nucleus elastic scattering. After a discussion of the basic formulae, we present results of calculations for pion inelastic scattering from ¹⁶O for initial pion lab kinetic energies of 70 and 180 MeV. Because of the strong energy dependence of the pion-nucleon interaction there are qualitative differences between the predictions for the nuclear response in the momentum transfer, energy loss plane for T_π^lab = 70 and 180 MeV. At these energies, states not prominently excited by other probes are predicted to be observable. In particular, J^π = 3^? and 4^?, T = 0 states appear prominently in the excitation spectrum region at large momentum transfer. A comparison of π^? and π⁺ scattering showing the effects of the Coulomb interaction is presented. The predictions for pion single-charge exchange on ⁴⁸Ca indicate that this interaction would be useful for studying the location of the T_> states arising from the splitting of the giant dipole resonance in T ≠ 0 nuclei.     

The pion propagator in relativistic quantum field theories of the nuclear many-body problem

Pion interactions in the nuclear medium are studied using renormalizable relativistic quantum field theories. Previous studies using pseudoscalar πN coupling encountered difficulties due to the large strength of the πNN vertex. We therefore formulate renormalizable field theories with pseudovector πN coupling using techniques introduced by Weinberg and Schwinger. Calculations are performed for two specific models: the scalar-vector theory of Walecka, extended to include π and ρ mesons in a non-chiral fashion, and the linear σ-model with an additional neutral vector meson. Both models qualitatively reproduce low-energy πN phenomenology and lead to nuclear matter saturation in the relativistic Hartree formalism, which includes baryon vacuum fluctuations. The pion propagator is evaluated in the onenucleon-loop approximation, which corresponds to a relativistic random-phase approximation built on the Hartree ground state. Virtual $\text{N}\text{N}$ loops are included, and suitable renormalization techniques are illustrated. The local-density approximation is used to compare the threshold pion self-energy to the s-wave pion-nucleus optical potential. In the non-chiral model, s-wave pion-nucleus scattering is too large in both pseudoscalar and pseudovector calculations, indicating that additional constraints must be imposed on the lagrangian. In the chiral model, the threshold self-energy vanishes automatically in the pseudovector case, but does so for pseudoscalar coupling only if the baryon effective mass is chosen self-consistently. Since extrapolation from free space to nuclear density can lead to large effects, pion propagation in the medium can determine which πN coupling is more suitable for the relativistic nuclear many-body problem. Conversely, pion interactions constrain the model lagrangian and the nuclear matter equation of state. An approximately chiral model with pseudovector coupling is favored. The techniques developed here allow for a consistent treatment of these models using renormalizable relativistic quantum field theores.     

Pion scattering from a bound nucleon at medium energies

In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable πN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for ⁴He ～ ¹⁶O. The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 ～ 70%) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.     

The pion optical potential and its application to reaction processes

The gross features of the pion-nucleus interaction are studied within the framework of an optical potential approach. The optical potential is determined by a systematic analysis of the data on pionic atoms and elastic scattering. It is then decomposed into quasi-free and absorption terms through theoretical consideration of the reactive content. A method of calculating the absorption cross section using the optical potential with the above decomposition is proposed. It takes into account not only direct absorption from the elastic channel but also the absorption after the quasi-free processes. The calculated absorption cross sections are in reasonable agreement with the observed ones. The cross sections for inelastic, single-charge-exchange and double-charge-exchange processes are calculated as an extended application of the method. The results are consistent with the available data. Inclusive spectra of pions and protons are also considered in this approach. An application to the (γ, π) reaction is briefly discussed.     

On the pion-nucleus dynamics at low energies

Low-energy pion scattering (up to 50 MeV) on ¹²C and ¹⁶O is described quantitatively in the framework of the UST approach. It is shown that at T_π ∼ 50 MeV the differential cross sections arise as a result of a strong interference between the pure potential and absorption channels. The importance of nuclear structure effects in the pion-nucleus dynamics at low energies is discussed.     

The absorptive P-wave pion-nucleus optical potential

We derive the absorptive part of the P-wave pion-nucleus optical potential from a two-body model for the absorption mechanism which involves rescattering of a pion and ?-meson through a Δ₃₃ resonant state. The model gives an adequate explanation for the fundamental π⁺d → pp reaction cross section and leads to values for the optical potential parameter which are in fair agreement with those obtained from pionic atom level widths.     

A three-body calculation of πd scattering

We present a theoretical treatment of the pion-deuteron system, meant specifically for the energy region below 100 MeV, and based on the Faddeev method for three-body scattering. This includes all orders of multiple scattering, two- and three-body unitarity (to a good approximation), nucleon recoil, deuteron d-state and a correct treatment of spin and isospin. For consistency with nuclear physics we treat the nucleons non-relativistically. However, relativistic kinematics are used for the pion. In order to obtain one-dimensional integral equations in the three-body system, we have constructed a set of separable πN t-matrices (with analytic form factors), which fit selected data up to 300 MeV. A comparison is made with existing π⁺d data at 48 MeV. This data tends to favour the Faddeev type of energy dependence for the πN t-matrix in the πd system. This could also be important in low-energy pion-nucleus scattering.     

Optical model studies of the 13C (π+, πo) 13N reaction

The pion single-charge-exchange reaction ¹³C(π⁺, π^o) ¹³N is calculated using the pion-nucleus optical-model approach. The two versions of the optical potential, local and non-local, are investigated and the results are compared with the recently measured cross sections in the energy range of 30 to 90 MeV.     

Final state interaction in pion electroproduction from nuclei near threshold

We have calculated cross sections for pion electroproduction from nuclei near threshold, including the distortion of the pion wave function by the pion-nucleus optical potential. Distortion effects near threshold are found to be significant, changing the ratio σ_π-/σ_π+ for production of 10 MeV pions from ¹⁶O by nearly fifty percent.     

Elastic π± scattering on 4He, 16C, 16O and 40Ca

A calculation of π⁺ and π^? elastic scattering in the resonance region by four spin-zero nuclei is presented. A non-static first order optical potential which includes binding corrections and short range correlations is used. The comparison with the experimental data shows rather good agreement at and above the resonance but elastic scattering at lower energies is largely overestimated. The calculated angular distributions are more sensitive to the variations of the rms radius than to variations of the surface thickness of the nucleus.     

Remarks on calculating the pion-nucleus first-order optical potential

It is shown that the factorization approximation should not be used in any serious theoretical effort to calculate the first order optical potential for low energy π-nucleus scattering. In contrast to earlier work, the required off-shell πN t matrix for non-zero total momentum is consistently evaluated using relativistic particle quantum mechanics.