Complete-experiment reconstruction of the nucleon-nucleon elastic scattering matrix at fixed energy and angle, and the discrete ambiguities that survive complete sets of polarization observables

Elastic scattering of two spin-1/2 nucleons is described, at a fixed energy and center-of-mass angle, by five complex amplitudes, that is, by nine real functions up to an unobservable overall phase. Because every polarization observable is a bilinear form in these amplitudes, a 'complete experiment' - a finite set of measurements from which the amplitudes can be reconstructed directly, without phase-shift analysis or other dynamical input - is a nontrivial nonlinear inverse problem with discrete branches. Schumacher and Bethe (1961) proposed a reconstruction at a single energy and angle from fourteen bilinear observables that combine the spin-correlation tensor with the polarization-transfer tensor, to be made experimentally feasible by polarized targets, and left open the treatment of realistic experimental errors, the minimum sufficient observable set, and the extension to neutron-proton scattering with its isospin T=0 / T=1 separation. Later work established the general theory of complete and sufficient sets (a minimum of 2N-1 error-free measurements for N amplitudes), proved that discrete 'quadrant' ambiguities survive even complete sets, and executed the program with polarized beams and targets: the pp matrix was reconstructed model-independently at nineteen energies between 447 MeV and 5135 MeV, and the np amplitudes between 0.80 and 1.1 GeV, but the phase ambiguities persisted at higher energies, the np reconstructions produced several statistically independent solutions, and the isoscalar amplitudes below about 800 MeV were never reconstructed from the later complete np data set. Unique, model-independent determination of the nucleon-nucleon amplitudes from complete observable sets in the presence of realistic errors, in both isospin channels, remains open and is currently pursued as the model-free pp amplitude-reconstruction program of the SPASCHARM facility at U-70.

Background

Elastic nucleon-nucleon (NN) scattering is the cleanest probe of the nuclear force, and between a few hundred MeV and several GeV of beam energy its spin structure carries much of the dynamical information. In the center-of-mass (c.m.) frame, the most general scattering matrix for two spin-1/2 particles consistent with rotational invariance, parity conservation, and time reversal can be built from the invariance analysis of Wolfenstein and Ashkin and parametrized by five complex invariant amplitudes $a,b,c,d,e$ (functions of the energy and the c.m. scattering angle $\theta$); for identical protons the Pauli principle restricts the problem to $0\le\theta\le\pi/2$, and the Coulomb-nuclear interference at small angles - interference between the electromagnetic and the strong-interaction amplitudes, whose electromagnetic phase is known - can fix the otherwise arbitrary overall phase. Since only relative phases are observable, nine real functions must be determined at each energy and angle. Every measurable quantity is a bilinear form in these amplitudes: the differential cross section, the analyzing power $P$ (the left-right asymmetry, historically called polarization, produced with an unpolarized beam), the two-spin correlation coefficients $C_{ij}$ (spin correlations between beam and target, or between the two outgoing nucleons), the polarization-transfer coefficients $K_{ij}$ (how the polarization of an initial nucleon is transferred to a final one, obtained from double- and triple-scattering measurements), the depolarization tensor $D_{ij}$ (a companion two-spin quantity relating transverse polarization components of an incoming and an outgoing nucleon), and three-spin parameters that combine beam or target polarization with polarization analysis of an outgoing nucleon. Inverting these bilinear expressions is a system of quadratic equations with discrete branches, not a linear fit.

Puzikov, Ryndin, and Smorodinsky introduced the 'complete experiment' - a set of measurements sufficient to reconstruct the amplitudes without dynamical assumptions - and showed how elastic unitarity (which relates the amplitudes at one angle to integrals of their products over all angles, and holds only below the pion-production threshold) can compensate for incomplete single-angle information. Schumacher and Bethe (1961) took the complementary route: reconstruction from data at a single energy and angle, without unitarity or phase-shift analysis, exploiting the complementary algebraic structure of the correlation tensor and the polarization-transfer tensor, with fourteen bilinear equations for nine unknowns. Their appendix solves the system exactly when the data are exact: the moduli follow from sums and differences of the measured quantities, one amplitude follows from a positive-definite expression, and the individual signs of the real parts $E_r$ and $F_r$ ($E$ and $F$ denoting two of the five amplitudes in the notation of that paper) are fixed only by the last four of the fourteen observables. They argued that polarized targets (first demonstrated at Saclay shortly before) would make the transfer measurements practical: they replace difficult triple-scattering arrangements with double scattering and move the polarimetry away from recoil nucleons of only 20-100 MeV laboratory energy, where polarization measurement is hard; and they sketched the n-p extension, in which the observed amplitudes mix the isospin channels $T=0$ and $T=1$ (isospin being the internal quantum number under which the proton and the neutron form a doublet: p-p scattering is purely $T=1$, while n-p superposes $T=0$ and $T=1$). The alternative working method of the era was the phase-shift (partial-wave) analysis, a unitary fit across all angles and energies whose solutions came in discrete near-degenerate sets - precisely the situation a direct reconstruction could discriminate.

The theory was subsequently sharpened. For $N$ amplitudes, $2N-1$ well-chosen error-free measurements are necessary and sufficient (nine for p-p), with the quadratic relations among the twenty-five linearly independent 'pure' observables (measurable quantities that are bilinear in the amplitudes) supplying consistency checks; Barker, Randa, and Storrow derived necessary and sufficient conditions for completing a standard p-p observable set and, decisively, showed that discrete 'quadrant' ambiguities - distinct amplitude sets, related by sign flips, that reproduce the same observables while placing amplitudes in different quadrants of the complex plane - survive even complete sets. In experiments the system must therefore be overdetermined and the reconstruction performed statistically. Below about 350 MeV the field settled on energy-dependent partial-wave analyses of the complete database (for example the Nijmegen analysis of Stoks et al.), which is precise but model-dependent in parametrization; the direct, model-independent reconstruction remained the alternative, executed at the SIN (Schweizerisches Institut fuer Nuklearforschung, Villigen), LAMPF (Los Alamos Meson Physics Facility), and SATURNE II (Saclay) accelerators, and continued today as the model-free amplitude-reconstruction program of the SPASCHARM experiment (a polarized-beam and polarized-target experiment at the U-70 proton synchrotron in Protvino).

Problem Statement

The problem, arising from the 1961 analysis of Schumacher and Bethe and still not closed, is the direct, model-independent determination of the nucleon-nucleon elastic-scattering amplitudes - all five complex functions, up to the unobservable overall phase, for the p-p ($T=1$) case and for the mixed-isospin n-p case - at fixed energy and center-of-mass angle, from measured polarization observables with realistic experimental errors. Its components serve one coherent objective: (1) characterize which finite sets of observables are complete or sufficient for this determination, identify the discrete ambiguities that survive even complete sets (the sign ambiguities of the type affecting the real parts $E_r$ and $F_r$ in the 1961 table, and the quadrant and phase ambiguities found in practice), and determine what additional observables or external input remove them, establishing this characterization in a stated amplitude representation and observable convention and for a stated observable class; (2) establish the optimal reconstruction procedure under realistic error propagation, deciding when the multi-solution landscape of the bilinear system collapses to a single statistically acceptable solution; (3) realize the measurements, using polarized beams and polarized targets where they replace difficult triple-scattering arrangements and recoil polarimetry; and (4) extend the determination to n-p, separating the $T=0$ and $T=1$ amplitudes. Parts (3) and the p-p part of (2) have been demonstrated at selected energies; the open instances are the uniqueness question under realistic errors (the n-p reconstructions produced several statistically independent solutions, and it is not known whether reduced error bars alone yield uniqueness), the reconstruction of the isoscalar $T=0$ amplitudes from the complete n-p observable set measured between 200 and 590 MeV, and the completion of the model-free reconstruction at energies where no complete set has been measured, as planned at the SPASCHARM facility at U-70.

References

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