Quantitative extraction of two-nucleon correlation information from low-energy pion double charge exchange

The 1987 analysis by Bleszynski and Glauber of 50 MeV pion single and double charge exchange on $^{14}$ C concluded that double charge exchange (DCX) "furnishes a direct means of detecting" the position correlation of the two valence neutrons, and reported sensitivity to the short-distance behavior of the pion-nucleon ( $\pi N$ ) charge-exchange amplitude, expressed through a phenomenological off-shell form factor with fitted range $\Lambda=3.5~\mathrm{fm}^{-1}$ (rms interaction radius about 0.7 fm). The claim rested on a second-order impulse approximation with closure over intermediate nuclear states, harmonic-oscillator wave functions, and approximate optical-potential distortions, and the paper itself flagged the resulting model dependence. The problem is to place this connection on a quantitatively controlled microscopic footing: treat pion distortion and absorption, higher-order multiple scattering, and two-body (meson-exchange) currents microscopically, go beyond harmonic-oscillator/closure nuclear structure, and determine whether the existing low-energy pion DCX data can discriminate between models of two-nucleon short-range correlations and constrain the short-range $\pi N$ amplitude. Later work (1988-2000) executed much of this agenda: microscopic coupled-channel theories described the $^{14}$ C excitation functions for the first time, and correlation and two-body-current effects were quantified at the factor-of-two level, but no work demonstrated that the data discriminate between correlation models, and the experimental program ended around 2000. The surviving open core is the quantitative extraction, or demonstrated exclusion, of two-nucleon correlation information and short-range $\pi N$ constraints from low-energy pion DCX.

Background

Pions ( $\pi^{+}$ , $\pi^{-}$ , $\pi^{0}$ ) are the lightest strongly interacting mesons, and when a charged pion scatters from a nucleus it can exchange charge with individual nucleons. Single charge exchange (SCX), $\pi^{+}\to\pi^{0}$ , converts a neutron into a proton; double charge exchange (DCX), the $(\pi^{+},\pi^{-})$ reaction, converts two neutrons into two protons and changes the nuclear charge by two units. At incident pion kinetic energies of a few tens of MeV, well below the $\Delta(1232)$ resonance (the prominent pion-nucleon excitation near 180 MeV), DCX proceeds mainly through two successive SCX scatterings on a nucleon pair, so the reaction amplitude is a second-order quantity that weights the spatial and spin structure of that pair. Transitions to isobaric analog states (IAS; states related to the target ground state by an isospin rotation, i.e. by replacing neutrons with protons) and to double isobaric analog states (DIAS) provide the cleanest exclusive channels, because the structure of the initial and final nuclear states is then fixed by symmetry. The meson factories LAMPF (the Los Alamos pion accelerator) and TRIUMF (the Vancouver cyclotron laboratory) measured $\pi^{\pm}$ on $^{14}\mathrm{C}$ at $T_{\pi}=50$ MeV: SCX to the $^{14}\mathrm{N}$ analog and DCX to the $^{14}\mathrm{O}$ double-analog ground state (Navon et al. 1984; Leitch et al. 1985), with a forward-peaked DCX angular distribution of order $\mu$ b/sr.

Bleszynski and Glauber (1987) analyzed both reactions in the impulse approximation, in which the pion scatters from individual bound nucleons with the free pion-nucleon ( $\pi N$ ) amplitude, to first order for SCX and to second order for DCX. Their DCX amplitude summed over all intermediate nuclear states by closure (using completeness of the nuclear spectrum instead of resolving individual intermediate states, thereby neglecting intermediate-state excitation energies), used harmonic-oscillator $p$ -shell wave functions for the two valence neutrons of $^{14}\mathrm{C}$ coupled to ${}^{1}S_{0}$ or ${}^{3}P_{0}$ two-nucleon configurations, and parametrized the unknown half-off-shell behavior of the $\pi N$ amplitude (the amplitude for a pion propagating between the two scatterings away from its mass shell, which elastic $\pi N$ phase shifts do not determine) with a monopole form factor

h(p,k)=\frac{\Lambda^{2}+k^{2}}{\Lambda^{2}+p^{2}}.

Pion distortion and absorption in the nucleus were included approximately through optical potentials (complex effective potentials fitted to pion-nucleus elastic scattering). With the Cohen-Kurath shell-model mixing of the valence $p_{1/2}$ and $p_{3/2}$ orbitals (the standard effective-interaction description of $1p$ -shell nuclei; Cohen and Kurath 1965) and a fitted $\pi N$ range parameter $\Lambda=3.5~\mathrm{fm}^{-1}$ , corresponding to an rms interaction radius $\sqrt{6}/\Lambda\approx0.7$ fm, the calculation reproduced the measured SCX and DIAS angular distributions simultaneously. Its central physical conclusion was that DCX is dominated by the position correlation of the two valence neutrons: the ${}^{1}S_{0}$ pair density is large at small internucleon separation, where the intermediate- $\pi^{0}$ propagator weights the two-step amplitude most strongly, while the ${}^{3}P_{0}$ density vanishes at zero separation, so that DCX furnishes a direct means of detecting this correlation. Distortion corrections modified the plane-wave results only at the 10-30% level for SCX and about 10% for DCX.

The same paper identified the model dependence of these conclusions, and those items constitute the problem: (i) the off-shell form factor's shape is a phenomenological choice, unconstrained by direct experiment, so the extracted $\Lambda$ and the nuclear mixing parameters inherit an uncontrolled uncertainty, as does the DCX sensitivity to the short-distance $\pi N$ interaction; (ii) the optical-potential parameters are not uniquely fixed by elastic-scattering fits, leaving residual model dependence; (iii) harmonic-oscillator wave functions and closure are schematic stand-ins for realistic two-body densities and intermediate-state dynamics, in particular short-range correlations (the modification of the two-nucleon relative wave function at separations below about 1 fm caused by the repulsive core and tensor forces of the nucleon-nucleon interaction), which are exactly the physics DCX is claimed to probe and which are absent from the harmonic-oscillator ${}^{1}S_{0}$ density; and (iv) higher-order multiple scattering and explicit many-body dynamics beyond the second-order impulse approximation were neglected. The unresolved target is whether the claimed correlation sensitivity survives a quantitatively controlled microscopic treatment of all these effects, and whether DCX data then discriminate between models of two-nucleon correlations.

Problem Statement

Within a quantitatively controlled microscopic theory of exclusive low-energy pion double charge exchange (the regime, roughly $T_{\pi}=20$ -80 MeV, in which the 1987 $^{14}\mathrm{C}$ analysis and its successors were made), determine whether DCX observables carry quantitative information about two-nucleon correlations in nuclei, and how much. The reaction theory must treat pion distortion and absorption microscopically (not through phenomenological optical potentials with free parameters), include higher-order multiple scattering and two-body meson-exchange currents, use realistic nuclear wave functions and an intermediate-state treatment beyond harmonic-oscillator closure, and state the $\pi N$ half-off-shell input in a principled way. On that basis, establish: (a) how the DCX amplitudes and cross sections depend on the two-nucleon correlations of the target (the valence-neutron position correlations of the source analysis, with realistic configuration mixing, and the dynamical short-range correlation models: repulsive-core correlation functions, $\rho$ -meson exchange, $\Delta N$ interactions), and whether the existing low-energy DCX data base ( $^{14}\mathrm{C}(\pi^{+},\pi^{-})^{14}\mathrm{O}$ from 19 to 80 MeV, the calcium-isotope DIAS data, and the heavier-target systematics) discriminates between these correlation models; (b) whether the data constrain the short-distance behavior and effective range of the $\pi N$ charge-exchange amplitude, as suggested by the 1987 extraction $\Lambda=3.5~\mathrm{fm}^{-1}$ (rms radius $\sqrt{6}/\Lambda\approx0.7$ fm), given that half-off-shell amplitudes are not observables; and (c) what residual model dependence the reaction mechanism (distortion, higher-order scattering, two-body currents, intermediate-state treatment) leaves on any extracted correlation or $\pi N$ information. These are facets of one objective: establishing, or excluding, DCX as a quantitative probe of two-nucleon correlations. The open question is whether the low-energy exclusive DCX data yield a controlled discrimination or constraint on the two-nucleon correlation models and the short-distance $\pi N$ amplitude, or are provably insensitive, within stated bounds, to the correlation-model differences.

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