Lorentz-Lorenz Effect Research Articles

Letter to the Editor

The feature article by D.-O. Riska, "Nuclear Exchange Currents" in vol. 13/no. 3, p. 11-14, attributes the idea of axial quenching via virtual Δ(1232) resonant states to M. Rho (1974) (ref. 26) and to K. Ohta and M. Wakamatsu (1974) (ref. 27), with the quenching factor of gA given in eq. (6) (the Lorentz-Lorenz factor). These results were first given in M. Ericson, A. Figureau, and C. Thévenet, Phys. Lett. 45B, 19(1973). This priority is clearly acknowledged in the quoted papers. In ref. 27: "The purpose of this paper is to reinvestigate the model propsed by Ericson et al." In ref. 26: "The reduction of gA in nuclear matter by the Lorentz-Lorenz effect [was] first established by Ericson, Figureau and Thévenet." In addition, the presently favored value from data for the parameter g' Å 0.3 agrees with that of our original formulation.

A theoretical approach to pionic atoms and the problem of anomalies

We have constructed a theoretical optical potential for pionic atoms based on the first- and second-order terms of a many-body expansion on the number of ph excitations. The p-wave parts are very much influenced by the medium polarization. The quenching at large densities produced by this phenomenon, together with the one from the Lorentz-Lorenz effect, and modifications over standard potentials due to the isospin dependence, lead to a new potential which produces notoriously improved results in the problem of anomalies. A clearer definition of this problem is proposed as the inability of standard potentials to reproduce simultaneously the data for light and heavy nuclei. Finally a semiphenomenological potential is obtained by adding some small terms to the theoretical potential which are fitted to the data and by means of which a good overall fit to the data is obtained. This potential is then used to make predictions for energies and widths of deeply bound pionic states so far unobserved.

Lorentz-Lorenz effect and the Kemmer-Duffin-Petiau equation.

The introduction of scalar interactions in the Kemmer-Duffin-Petiau equation induces the appearance of Darwin terms in the effective Klein-Gordon equation that resemble the Ericson-Ericson Lorentz-Lorenz effect in pion-nucleus scattering. The ${\ensuremath{\xi}}_{\ensuremath{\pi}}$ parameter turns out to be ${\ensuremath{\xi}}_{\ensuremath{\pi}}$\ensuremath{\sim}2 in agreement with recent calculations of the effect.

New interpretation of the dirac approach to proton-nucleus scattering

We re-examine the Dirac approach to proton-nucleus elastic scattering at intermediate energies in the light conventional multiple scattering theory. We show that the main virtue of the relativistic treatment is to provide a cut-off for the ohterwise badly behaved spin-orbit potential at high momenta. Once this has been understood, there are many different and more plausible ways of restoring the agreement between the traditional theory and experiment. To illustrate this point, we show analytically that the Lorentz-Lorenz effect yields results which are identical to the Dirac approach. The only remnant of analytically that the Lorentz-Lorenz effect yields results which are identical to the Dirac approach. The only remnant of relativity in this case is kinematics.

Unified analysis of pionic atoms and low-energy pion-nuclear scattering: Hybrid analysis

Using the method of effective nuclear density, we apply a simple, $\ensuremath{\pi}$-nucleus optical potential (without ${\ensuremath{\rho}}^{2}$ terms and the Lorentz-Lorenz effect) to ${\ensuremath{\pi}}^{\ensuremath{-}}$ atoms and low-energy $\ensuremath{\pi}$-nucleus elastic scatterings. Data of both phenomena are analyzed in a unified, hybrid (phenomeno-logical and theoretical) manner: The ${\ensuremath{\pi}}^{\ensuremath{-}}$-atom data are analyzed first to determine phenomenologically the potential parameters at threshold. The parameters are then extrapolated successfully up to 50 MeV incident pion laboratory energy by a microscopic calculation in which the energy-dependence correction is made after including the Fermi-averaging and Pauli-blocking effects. In contrast to other work, our potential includes the minimum number of the parameters that describe the full information content of the data. We can thus conclude that these effects are the important microscopic corrections for the extrapolation, but neither the Lorentz-Lorenz effect nor some highly nonlocal effects are crucial ones. The potential we have used has angular transformation terms which are also found to be crucial in the unified treatment. During the course of this work we have found an interesting behavior of the terms. A short account of its discussion is also presented.NUCLEAR REACTIONS Unified analysis and calculation; strong-interaction shifts and widths in ${\ensuremath{\pi}}^{\ensuremath{-}}$ atoms and differential cross sections of elastic $\ensuremath{\pi}$-nucleus scattering up to 50 MeV; $\ensuremath{\pi}$-nucleus optical potential.

Pion coupling constants in nuclei and the isobar-hole model

We discuss the relevance of the on-shell pion-nucleus coupling constants to modern theories of pion and isobar degrees of freedom in nuclei. The excitation of specific transitions is first considered and the on-shell coupling constant is shown to carry other information than the familiar Gamow-Teller constant measured in β-decay; we present a critical discussion of the scarce existing determinations which are at considerable variance with our theoretical expectations. Next, the sum rule for the “effective” pion-nucleus coupling constant is calculated and predicted to be drastically reduced by δ-hole renormalization of the pion propagator besides the Lorentz-Lorenz effect. Dispersion relations strongly favour the validity of the model.

Quenching of the magnetic multipole strength distribution and of the anomalous magnetic moment in complex nuclei and mesonic renormalization of the nuclear spin current

The retardation of the magnetic multipole strength distribution (specifically M1 and M2) extracted from inelastic electron scattering to so-called pion-like states and the quenching of magnetic moments, both in heavy nuclei, is attributed to a mesonic renormalization of the nuclear spin current. The mechanism invoked is the same that leads to the renormalization of the axial current in nuclei and that also governs low energy p-wave pion-nucleus scattering (Lorentz-Lorenz effect).

The contribution of reflections to the Lorentz-Lorenz effect

We use the static limit and delta isobar dominance to calculate the contribution to the optical potential of pionic atoms of those terms in which a pion or a rho meson are exchanged any number of times between two nucleons, which constitute the ρ 2 term of the Lorentz-Lorenz effect. By suppressing the spin and isospin degrees of freedom, we are able to express the Lorentz-Lorenz parameter ξ in analytical form. In the general case when we include fully the spin and isospin variables, we obtain the Lorentz-Lorenz parameter ξ in numerical form. Our results show that ξ can not be larger than one and more likely is close to zero.

The quadrupole term of the Lorentz-Lorenz effect in pionic atoms

Two-nucleon contributions to the pion-nucleus p-wave optical potential at zero energy, not included in the standard Lorentz-Lorenz treatment, are considered. Such contributions involve coupling the external pion momenta k and k ′, which appear in the optical potential, to form a quadrupole term, and their neglect would appear to be justified only in a strict long-wavelength limit for the pion. The contribution of these quadrupole terms to the observed shifts of pionic atom energy levels is evaluated for 27Al and 40Ca, under the assumption of short-range repulsive nucleon-nucleon correlations, and found to constitute roughly between 10% and 15% of the standard Lorentz-Lorenz contribution, or 5% to 7% of the observed shifts.

Intermediate rho mesons in pi-nuclear scattering

The role of rho meson exchange between nucleons is examined by solving the Low equations which couple the pi-nuclear channels to the rho-nuclear channels. The result is in the form of a linear coupled-channel equation of the Lippmann-Schwinger type but including a propagator modification which strongly damps the effects of virtual rho mesons. In particular, the contribution of rho exchange to the Lorentz-Lorenz effect is found to be negligible.

Field theoretic treatment ofπ-nuclear scattering

The dynamical assumption that single mesons are emitted or absorbed by nucleons is applied to $\ensuremath{\pi}$-nucleus scattering and leads to a $\ensuremath{\pi}$-nucleus Low equation. With the neglect of multinucleon currents and true meson absorption, the $\ensuremath{\pi}$-nucleus Low equation is shown to be equivalent to a linear equation of the same form as in standard theories except for a striking propagator modification arising from the energy dependence of the driving term. The use of a corresponding linear equation for $\ensuremath{\pi}$-nucleon scattering enables us to derive a field-theoretic multiple-scattering series. It is found that the propagator modification eliminates the Kisslinger singularity, reduces the Lorentz-Lorenz effect and local field corrections. The effect of true meson absorption on $\ensuremath{\pi}$-nucleus elastic scattering is also examined and found to be somewhat important at low (\ensuremath{\sim}50 MeV) energies but negligible (due to strong first-order optical absorption) for energies greater than 100 MeV.NUCLEAR REACTIONS Scattering theory, $\ensuremath{\pi}$-nucleon scattering, $\ensuremath{\pi}\ensuremath{-}^{16}\mathrm{O}$ scattering, $E=50\ensuremath{-}275$ MeV.

Evidence for the Lorentz-Lorenz Effect in Low-Energy Pion-Nucleus Scattering

We point out the relatively clean geometrical aspect of the Lorentz-Lorenz effect and use it to extract the strength of the effect from an analysis of the elastic scattering of 50-MeV ..pi../sup +/ on /sup 16/O. We find that for a finite-range ..pi..-N interaction there is strong evidence for the effect with a value 0.2 < or = xi < or = 0.6.

Multiple Scattering Aspects of theπ-Nucleus Low Equation

The $\ensuremath{\pi}$-nucleus Low equation, under certain assumptions (implicit in conventional multiple-scattering theory), is equivalent to a linear equation. The propagator of this equation includes a factor of ${(\frac{z}{{H}_{0}})}^{2}$ which vastly reduces off-shell scattering. As a result, the Kisslinger singularity is eliminated, and the size of the Lorentz-Lorenz effect and local-field corrections is reduced.

Pauli principle and renormalization of the pion nucleon interaction in nuclei

We show that the exclusion principle provides an alternative mechanism to the Lorentz-Lorenz effect for the renormalization of the pion-nucleon coupling constant in nuclei. This mechanism proves to be less sensitive to the range of the π - N forces. The implications for pion-nucleus scattering are discussed.

Renormalisation of the πNN-vertex in a nuclear medium

Corrections to the πNN vertex in a nuclear medium (from mechanism other than the Lorentz-Lorenz effect) are estimated and found to be relatively small. At nuclear matter density the effective πN coupling constant (at a pion frequency ω = m π is reduced by less than 6% from its free space value.

Pion-nucleus interaction and the range of πN forces

We examine several definitions for the ranges r πN of the pion-nucleon ( πN) forces and conclude that 0.25 ≦ r πN ≦ 0.5 fm. We investigate two consequences of the finite range of the p-wave πN interaction: the additional spatial extension of the p-wave π-nucleus optical potential and the quenching of the Lorentz-Lorenz effect. For 0.25 ≦ r πN ≦ 0.5 fm (i) the additional extension of the p-wave optical potential gives negligible contributions to shifts and widths in π-mesic atoms and (ii) the Lorentz-Lorenz effect is largely quenched making nuclear correlations hard to detect by mesic atoms measurements.

The Lorentz-Lorenz effect in pion-nucleus interactions

The Lorentz-Lorenz effect in pion-nucleus interactions is derived from the Watson expansion of the optical potential. The use of a realistic, finite range πN interaction significantly reduces its importance for nuclear short-range correlations in pionic atoms.

Pions in nuclear matter—an approach to the pion-nucleus optical potential

The optical potential for pions propagating in nuclear matter is investigated from a many-body point of view. An exact expression for the optical potential is obtained in terms of the nuclear matter density-density correlation function. Explicit calculations of the pion self-energy, which is simply related to the optical potential, are performed via an expansion in the nuclear density n 0 . The familiar result of multiple scattering theory, 4π n 0f 0 where f 0 is the pion-nucleon scattering length, is recovered in lowest order. The terms quadratic in n 0 account for the long range Pauli correlation corrections to the pion-nucleon scattering process (quenching), as well as dispersion and absorption contributions coming from nucleon-nucleon collisions in the medium. The latter terms are proportional to off-shell nucleon-nucleon scattering amplitudes. Our treatment is compared and contrasted with that of Ericson and Ericson. In particular, we are able to establish what approxmations in the nuclear many-body problem lead to their form of s-wave potential that includes long-range correlation effects, as well as their p-wave potential that has a nonlinear density dependence (the Lorentz-Lorenz effect). Estimates of the dispersion and absorption terms are made using two simple models to describe interacting nuclear matter.

Optical properties of low-energy pions in nuclei

A simple nonlocal potential for low-energy pions in finite nuclei is calculated from the amplitudes for πN scattering and for π production in NN collisions. The potential includes absorption and has no free parameters. The appropriate multiple scattering equations are derived in the coordinate representation with nuclear pair correlations included. Owing to the large mass of the scatterers the pion field behaves nearly classically. It is shown that short-range pair correlations are important in the multiple scattering owing to the dominant dipole component in the πN scattering. To a good approximation this produces a Lorentz-Lorenz effect analogous to the one occurring in the scattering of electromagnetic waves in dense media. The contributions of Fermi motion to the potential are shown to be small. The isospin part of the potential is derived and is shown to have a tensor component in addition to the ordinary vector one. The former gives rise to direct double charge exchange of pions; its strength becomes important for high momentum pions. The potential is further found to have a term which gives a small but possibly observable hyperfine coupling. A comparison is made between predictions of the potential and experimental data on level shifts and widths for π mesic atoms. Satisfactory agreement is found within experimental uncertainties which is quantitative evidence for nuclear pair-absorption of pions. There is also some indication of short range anticorrelations between nucleons.