On contributions to the pion-nucleus optical potential non-linear in nuclear density: The Ericson-Ericson Lorentz-Lorentz correction

A well-defined set of higher order corrections to the lowest order pion-nucleus optical potential is examined within the framework of many-body theory for an infinite nuclear medium. The connection between the pion self-energy operators and multiple scattering formalisms is thereby indicated and a deriviation of the EELL effect given.     

Low-energy pion-nucleus optical potentials

We discuss the local and non-local pion-nucleus optical potentials. We find that the local potential becomes non-local when two-nucleon correlations are included. The two potentials (including correlations) can be made local through a transformation on the wave function. The new local potentials agree up to quadratic terms when expanded in powers of the density. The influence of finite-range correlations and off-shell pion-nucleon form factors is also investigated.     

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.     

Correlations and the pion-nucleus optical potential

We investigate the influence of nucleon-nucleon correlations on the parameters of the pion-nucleus optical potential. In the (3, 3) resonance region the corrections can be as large as 30% and their nature is intermediate between well-known high- and low-energy limits. Binding and correlation effects add constructively below the resonance and destructively above.     

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.     

Impulse-approximation correction in pion-nucleus optical potential

The local density approximation is used in this paper to calculate the first-order pion-nucleus optical potential. TheπN scattering matrix in nuclear medium is computed by employing separableπN scattering matrix. This nuclear-mediumπN scattering matrix, which includes impulse-approximation correction is then used to construct the pion-nucleus optical potential.π-¹²C elastic scattering results obtained by using this potential are compared with the impulse-approximation potential results.     

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.     

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.     

The reactive two-body part of the pion-nucleus optical potential

We calculate the reactive component of the two-body contribution to the pion-nucleus optical potential from a two-nucleon pion absorption mechanism that predicts the total cross section and angular distributions for π⁺d → pp very well. At threshold the calculated absorptive parts explain most of the values obtained from pionic atom level widths, whereas the dispersive parts, which are very sensitive to wave function correlations are considerably more attractive than what the conventional phenomenological parameters would suggest.     

Off-shell sensitivity,repulsive correlations and the pion-nucleus optical potential

Repulsive nucleon-nucleon correlations tend to reduce the dependence of pion-nucleus elastic scattering upon the off-shell pion-nucleon dynamics. However, optical potential calculations can in practice be quite sensitive to the particular choice of off-shell model parameters. It is argued that this sensitivity results from the nature of the optical potential as a one-body operator which introduces extra off-shell dependence not found in the physical many-body process itself. Thus, one must be very careful in any attempt to extract correlation or off-shell information, or to predict pion-nucleus phase shifts, by means of an optical potential theory. Results of model calculations are presented for purposes of illustration.     

Investigations of the pion-nucleus optical potential from pionic atoms

The explicit dependence of the zero-energy π^?-nucleus optical potential on its various parameters, as well as its implicit dependence on neutron radii, is investigated for a selected set of particularly accurate level shift and width data. Among the various effects studied are some possible variations in the form of the Lorentz-Lorenz term, terms induced by the so-called “angle-transformation”, and the smearing of the π 2N terms which allows for a finite range in the absorption process. The pionic atom data are equally well fitted by any of the several forms used for the optical potential, with the persistent failure to fit the recently reported widths of 3d levels in ¹⁸¹Ta, Re and ²⁰⁹Bi and the width of the 1s level in ²³Na.     

A pion-nucleus optical potential containing effects of genuine pion absorption and emission

We derive in two ways a lowest-order optical potential $\text{V}$_πA using the projectile and a pair of target nucleons as the basic interacting unit. The first derivation—a heuristic one—uses a multiple-scattering theory, while the second employs a field theory where pions can be absorbed and emitted as well as scattered. The lowest-order terms of $\text{V}$_πA contain the two-particle ground state density and the πNN → πNN scattering matrix. In the same field-theoretical model one analyzes the latter in standard multiple-scattering contributions and genuine absorption corrections. The form of these corrections and their relation to often ad hoc assumed forms is discussed.     

The necessity for an improved optical potential in low energy pion-nucleus scattering

Earlier findings that there is a considerable disagreement between theory and experiment in low energy pion-nucleus elastic scattering are confirmed. By explicit calculation, it is shown that this may be largely due to poor πN input. Nevertheless, a residual factor of two difference (in the first peak) could easily be corrected by simple (kinematic and binding) corrections to the first order optical potential.     

The contribution of pion absorption with two-nucleon emission to the pion-nucleus optical potential

Four terms in the pion-nucleus optical potential which arise from pion absorption with two nucleon emission are calculated using a model in which the nucleon is treated as a bound state of a pion and a nucleon. The contribution of these terms to the reaction cross section of carbon is considered. Two of the terms give rise to a negative reaction cross section, but it is shown that the sum of the four is positive definite. We find that the absorption reaction cross sections are very sensitive to the off-shell behavior of the free pion-nucleon T-matrix.     

A density expansion of the pion-nucleus optical potential. II. Second order terms

The second order terms in a density expansion of the pion optical potential V_opt are evaluated quantitatively. The coefficients of these terms are proportional to various combinations of on- and off-shell nucleon-nucleon T-matrices, averaged over the distribution of two nucleon relative momenta in the Fermi sea. The on-shell contributions can be obtained directly from experimental phase shifts, but the calculation of the off-shell-parts requires a model for the nucleon-nucleon potential. We consider a number of realistic local and nonlocal, separable potentials which fit nucleon-nucleon phase shifts, in order to study the variations in V_opt which arise from differences in the off-shell behavior of T. We find absorptive (imaginary) and dispersive (real) contributions to V_opt which are of comparable magnitude. The dispersive part, which leads to a real energy shift associated with the two nucleon absorptive process, has not been previously estimated quantitatively. We compare our results to empirical potentials obtained by fitting energy level shifts and widths in pi-mesic atoms, as well as theoretical estimates based on threshold cross sections for the processes π + N + N ? N + N.     

Vibrational mechanics in an optical lattice: controlling transport via potential renormalization

We demonstrate theoretically and experimentally the phenomenon of vibrational resonance in a periodic potential, using cold atoms in an optical lattice as a model system. A high-frequency (HF) drive, with a frequency much larger than any characteristic frequency of the system, is applied by phase modulating one of the lattice beams. We show that the HF drive leads to the renormalization of the potential. We used transport measurements as a probe of the potential renormalization. The very same experiments also demonstrate that transport can be controlled by the HF drive via potential renormalization.     

A survey of selected pion-nucleus reactions using phase-shift-equivalent potentials

Since pion-nucleus elastic scattering data can only constrain the asymptotic behavior of the scattered wave, it remains for inelastic scattering and reaction data to test the wave function in the nuclear interior. To this end, distorted-wave impulse approximation calculations have been performed for a variety of low-energy pion reactions using phase-shift-equivalent pion-nucleus potentials: inelastic excitation, photoproduction, and the (π, p) reaction. Sensitivity of the reaction cross sections to variations in the equivalent potentials is examined, with an eye toward understanding what can be learned about pion-nucleus dynamics from a given set of reaction data.