We study vector meson photoproduction from the deuteron at high momentum transfer, accompanied by break-up of the deuteron into a proton and neutron. The large $-t$ involved allows one of the nucleons to be identified as struck, and the other as a spectator to the $γN\rightarrow VN$ subprocess. Corrections to the plane wave impulse approximation involve final state interactions (FSIs) between the struck nucleon or the vector meson, either of which is energetic, with the slow spectator nucleon. In this regime, the eikonal approximation is valid, so is employed to calculate the cross-section for the reaction. Due to the high-energy nature of the FSIs, the maxima of the rescatterings are located at nearly transverse directions of the fast hadrons. This results in two peaks in the angular distribution of the spectator nucleon, each corresponding to either the $V$-$N$ or the $p$-$n$ rescattering. The $V$-$N$ peak provides a new means of probing the $V$-$N$ interaction. This is checked for near-threshold $ϕ$ and $J/Ψ$ photoproduction reactions which demonstrate that the $V$-$N$ peak can be used to extract the largely unknown amplitudes of $ϕ$-$N$ and $J/Ψ$-$N$ interactions.
Two additional phenomena are observed when extending the calculation of $J/Ψ$ photoproduction to the sub-threshold and high-energy domains. In the first case we observe overall suppression of FSI effects due to a restricted phase space for sub-threshold production in the rescattering amplitude. In the second, we observe cancellation of the $V$-$N$ rescattering amplitudes for vector mesons produced off of different nucleons in the deuteron.
Transversity generalized parton distributions (GPDs) can be probed in diffractive electro- and photoproduction of two vector mesons on a hadron in kinematics where the two vector mesons are separated by a large rapidity gap. We report on calculations for this process in the case of coherent $ρ^0-ω$ meson production on a deuteron target. Our cross section results show that an electron-ion collider with deuteron beams and forward detectors could probe deuteron transversity GPDs.
The production of pions by antiproton-deuteron annihilation at rest is analyzed. Assuming the possible existence of two delta-isobars in a deuteron some enhancement in the distribution over the invariant mass of two negative charged pions is predicted.
We demonstrate that measurement of tagged structure functions of the deuteron in (e,e'N) semi-inclusive reactions can discriminate between different hypotheses on the origin of the nuclear EMC effect. By choosing extreme backward kinematics for the spectator nucleon to minimize effects from the deuteron wave function and final state interactions, one can isolate the modifications in the structure of the bound nucleon within the impulse approximation. The same reaction can be used to extract the large-x neutron to proton structure function ratio.
The fragmentation of deuterons into pions emitted forward in the kinematic region forbidden for free nucleon-nucleon collisions is analyzed. It is shown that the inclusion of the non-nucleonic degrees of freedom in a deuteron results in a satisfactory description of the data for the inclusive pion spectrum and improves the description of the data about $T_{20}$. According to the data, $T_{20}$ has very small positive values, less than 0.2, which contradicts the theoretical calculations ignoring these degrees of freedom.
We present an analysis of the electric dipole moment (EDM) of the deuteron as induced by CP-violating operators of dimension 4, 5 and 6 including theta QCD, the EDMs and color EDMs of quarks, four-quark interactions and the Weinberg operator. We demonstrate that the precision goal of the EDM Collaboration's proposal to search for the deuteron EDM, (1-3)\times 10^{-27} e cm, will provide an improvement in sensitivity to these sources of one-two orders of magnitude relative to the existing bounds. We consider in detail the level to which CP-odd phases can be probed within the MSSM.
We discuss deuteron $A(d,p)X$ breakup and cumulative pion production $A(d,π)X$ in the framework of constituent quark model of the deuteron. We demonstrate that consideration of the Pauli principle at the quark level, as well as multiple scattering, affect drastically cross section and polarization observables of these reactions and provide good description of the experimental data.
The d(e,e'p) cross section at a momentum transfer of 3.5 (GeV/c)^2 was measured over a kinematical range that made it possible to study this reaction for a set of fixed missing momenta as a function of the neutron recoil angle theta_nq and to extract missing momentum distributions for fixed values of theta_nq up to 0.55 GeV/c. In the region of 35 (deg) <= theta_nq <= 45 (deg) recent calculations, which predict that final state interactions are small, agree reasonably well with the experimental data. Therefore these experimental reduced cross sections provide direct access to the high momentum …
I provide a brief, and therefore necessarily biased, review of the application of effective field theory to electromagnetic processes on the deuteron. Electron-deuteron scattering is discussed for virtual photon momenta up to 700 MeV, and photon-deuteron scattering is examined for incident photon energies of order the pion mass.
A calculation of the deuteron polarization observables $A^d_y$, $A_{yy}$, $A_{xx}$, $A_{xz}$ and the differential cross-section for elastic nucleon-deuteron scattering at incident deuteron energies 270 and 880 MeV in lab is presented. A comparison of the calculations with two different deuteron wave-functions derived from the Bonn-CD $NN$-potential model and the dressed bag quark model is carried out. A model-independent approach, based on an optical potential framework, is used in which a nucleon-nucleon $T$-matrix is assumed to be local and taken on the energy shell, but still depends on the…
Understanding the role of Quantum Chromodynamics in generating nuclear forces is important for uncovering the mechanism of short-ranged nuclear interactions and their manifestation in short range correlations (SRC). The future Electron-Ion-Collider (EIC) at Brookhaven National Laboratory in the US will provide an unprecedented opportunity to systematically investigate the underlying physics of SRC for energies and kinematic regions that are otherwise impossible to reach. We study SRCs in electron-deuteron scattering events using the Monte Carlo event generator BeAGLE. Specifically, we investig…
Employing the polarization degrees of freedom in the deuteron allows to isolate smaller than average inter-nucleon distances in the deuteron. As a result one can identify set of high $Q^2$ reactions off polarized deuteron which are particularly sensitive to the short range dynamics of strong interaction. We concentrate on the studies of several aspects of the short range phenomena. These are the relativistic dynamics of electron--bound-nucleon scattering, color coherence in high $Q^2$ electro-production as well as the formation of the vector mesons in coherent electro-production from the deute…
We present results for the three-cluster breakup in deuteron-deuteron collisions at 130 and 270 MeV deuteron beam energy. The breakup amplitude is calculated using the first term in the Neumann series expansion of the corresponding exact four-nucleon equations. In analogy with nucleon-deuteron breakup where an equivalent approximation is compared with exact calculations, we expect this single-scattering approximation to provide a rough estimation of three-body breakup observables in quasifree configurations. We predict the nucleon-deuteron and deuteron-deuteron three-cluster breakup cross sect…
Deuteron-deuteron elastic scattering and transfer reactions in the energy regime above four-nucleon breakup threshold are described by solving exact four-particle equations for transition operators. Several realistic nuclear interaction models are used, including the one with effective many-nucleon forces generated by the explicit $Δ$-isobar excitation; the Coulomb force between protons is taken into account as well. Differential cross sections, deuteron analyzing powers, outgoing nucleon polarization, and deuteron-to-neutron polarization transfer coefficients are calculated at 10 MeV deuteron…
We analyze the momentum distributions of constituents in 3He, as well as the spin-dependent observables for (3He,d), (3He,p), and (d,p) breakup reactions. Special attention is paid to the region of small relative momenta of the helium-3 and the deuteron constituents, where a single parameter, D_2, has determining role for the spin-dependent observables. We extract also this parameter for the deuteron, basing on existing data for the tensor analyzing power of this reaction.
An analysis of the $^{231}$Pa$(d,3n)$$^{230}$U reaction excitation function at energies around the Coulomb barrier has taken into account the pre-equilibrium and compound-nucleus cross sections corrected for the deuteron-breakup decrease of the total reaction cross section, as well as the inelastic breakup enhancement. The analysis reveals the dominance of the deuteron breakup mechanism unlike a former assessment in this respect of the deuteron-induced fission process.
In recent years, high-accuracy data for pionic hydrogen and deuterium have become the primary source of information on the pion-nucleon scattering lengths. Matching the experimental precision requires, in particular, the study of isospin-breaking corrections both in pion-nucleon and pion-deuteron scattering. We review the mechanisms that lead to the cancellation of potentially enhanced virtual-photon corrections in the pion-deuteron system, and discuss the subtleties regarding the definition of the pion-nucleon scattering lengths in the presence of electromagnetic interactions by comparing to …
The exclusive deuteron break-up reaction is analyzed within a covariant approach based on the Bethe-Salpeter equation with realistic meson-exchange interaction. Relativistic effects in the cross section, tensor analyzing power and polarization transfer are investigated in explicit form. Results of numerical calculations are presented for kinematical conditions in forthcoming p + D reactions at COSY.
The break-up of the deuteron during deeply-virtual Compton scattering, gamma* d --> gamma(*) n p, is explored. In the effective field theory describing nucleon dynamics at momenta below the pion mass, the EMC effect results from four-nucleon interactions with the twist-2 operators, appropriate for describing forward, and near-forward, matrix elements in the two-nucleon system. We point out that the break-up of the deuteron to low-energy final states during deeply-virtual Compton scattering is a process with which to explore strong-interaction physics closely related to that responsible for the…
The complex Kohn variational principle and the (correlated) hyperspherical harmonics method are applied to study the proton-deuteron elastic scattering at energies above the deuteron breakup threshold. Results for the elastic cross section and various elastic polarization observables have been obtained by fully taking into account the long-range effect of the Coulomb interaction and using a realistic nucleon-nucleon interaction model. Detailed comparison between the theoretical predictions and the accurate and abundant proton-deuteron experimental data can now be performed.
We discuss how to reduce theoretical uncertainties in the neutrino-deuteron breakup cross-sections crucial to the Sudbury Neutrino Observatory's efforts to measure the solar neutrino flux. In effective field theory, the dominant uncertainties in all neutrino-deuteron reactions can be expressed through a single, common, isovector axial two-body current parameterized by $L_{1,A}$. After briefly reviewing the status of fixing $L_{1,A}$ experimentally, we present a constraint on $L_{1,A}$ imposed by existing reactor antineutrino-deuteron breakup data. This constraint alone leads to an uncertainty …
Model dependence of the reaction rates for the weak breakup of deuterons by low energy neutrinos is studied starting from the cross sections derived from potential models and also from pionless effective field theory. Choosing the spread of the reaction yields, caused basically by the different ways the two-body currents are treated, as a measure of the model dependent uncertainty, we conclude that the breakup reactions are $\sim$ 2 - 3 % uncertain, and that even the ratio of the charged to neutral current reaction rates is also $\sim$ 2 % uncertain.
The advantages to study charge--symmetry breaking and especially the phenomenon of f0-a0 mixing in nucleon--nucleon and deuteron--deuteron induced reactions are discussed.
The available data on the forward charge exchange of nucleons on the deuteron up to 2GeV per nucleon are reviewed. The value of the inclusive nd->pnn/np->pn cross section ratio is sensitive to the fraction of spin-independent neutron-proton backward scattering. The measurements of the polarisation transfer in d(\vec{n},\vec{p})nn or the deuteron analysing power in p(\vec{d},pp)n in high resolution experiments, where the final nn or pp pair emerge at low excitation energy, depend upon the longitudinal and transverse spin-spin np amplitudes. The relation between these types of experiments is dis…
Theoretical predictions for the nucleon induced deuteron breakup process based on solutions of the three-nucleon Faddeev equation including such relativistic features as the relativistic kinematics and boost effects are presented. Large changes of the breakup cross section in some complete configurations are found at higher energies. The predicted relativistic effects, which are mostly of dynamical origin, seem to be supported by existing data.
A microscopic formalism for describing observed cross sections for deuteron breakup by three-nucleon nuclei was developed on the basis of the diffraction nuclear model. A general formula that describes the amplitude for the reaction d+3He->3He+p+n and which involves only one adjustable parameter was obtained by using expansions of the integrands involved in terms of a Gaussian basis. This formula was used to analyze experimental data on the exclusive cross sections for deuteron breakup by He-3 nuclei at the projectile energy of 89.4 MeV. The importance of employing, in calculations, a deuteron…
A new computational method for solving the nucleon-deuteron breakup scattering problem has been applied to study the inelastic neutron-deuteron scattering on the basis of the configuration-space Faddeev equations. This method is based on the spline-decomposition in the angular variable and on a generalization of the Numerov method for the hyperradius. The Merkuriev-Gignoux-Laverne approach has been generalized for arbitrary nucleon-nucleon potentials and with an arbitrary number of partial waves. Neutron-deuteron observables at the incident nucleon energy 14.1 MeV have been calculated using th…
Use of deuteron-induced spallation reactions at intermediate energies has recently been proposed for transmutation of several long-lived fission products (LLFPs). In the design study of a transmutation system using a deuteron primary beam, accurate cross section data of deuteron-induced reactions on the LLFPs are indispensable. Reliable model predictions play an important role in completing the necessary cross section data since currently available experimental data are very limited. Under the circumstances, we have been developing a code system dedicated for deuteron-induced reactions, which …
We examine a description of available cross section data for symmetric space star (SST) configurations in the neutron-deuteron (nd) and proton-deuteron (pd) breakup reaction using numerically exact solutions of the three-nucleon (3N) Faddeev equation based on two- and three-nucleon (semi)phenomenological and chiral forces. The predicted SST cross sections are very stable with respect to the underlying dynamics for incoming nucleon laboratory energies below $\approx 25$ MeV. We discuss possible origins of the surprising discrepancies between theory and data found in low-energy nd and pd SST bre…
We study the electrodisintegration of deuteron at quasi-elastic kinematics and high transferred momentum as a probe for the short distance structure in nuclei. In this reaction, an electron hits a nucleus of deuterium, which breaks up into a proton-neutron pair. We focus our attention on events where fast nucleons emerge, corresponding to nuclear configurations where the bound nucleons have a high relative momentum (exceeding 700 MeV/c). The present research is relevant to physical systems where high-density nuclear matter is present. This condition covers a wide range of physics, from neutron…
Fully converged calculations of deuteron-deuteron elastic scattering observables are performed at energies above three- and four-body breakup threshold. Differential cross sections and analyzing powers are obtained using realistic nucleon-nucleon force models together with the Coulomb repulsion between protons. For all observables we find a very reasonable agreement with the available experimental data limited to deuteron beam energies up to 25.3 MeV.
An isoscalar $pn$ pair is expected to emerge in nuclei that have similar proton and neutron numbers and it may be a candidate for a deuteron ``cluster.'' There is, however, no experimental evidence for it. The purpose of this paper is to construct a new reaction model for the ($p,pd$) reaction including the deuteron breakup in the elementary process and the deuteron reformation by the final-state interactions (FSIs). How these processes contribute to the observables of the reaction is investigated. The distorted wave impulse approximation is extended in twofold. The elementary processes of the…
The reactions of nucleon and polarized deuteron scattered off a heavy target at large impact parameter with intermediate energies have been investigated by using the improved quantum molecular dynamics model. It is found that, due to the difference effect of isovector potential on proton and neutron, there is a significant difference between the angle distribution of elastic scattering protons and neutrons. To overcome the lack of monochromatic neutron beam, the reaction of polarized deuteron peripherally scattered off the heavy target is used to replace the reaction of individual proton and n…
We analyzed contribution of quark degrees of freedom in the deuteron to the longitudinal, P_z', and transverse, P_x', polarizations of the proton in the d(\vec e,e'\vec p)n break-up. It is demonstrated that such effects work in correct direction to explain experimental data. We predict that the polarizations should change qualitatively behavior at p_m >200 MeV/c, (i) the polarizations become strongly dependent on the out-of-plain angle and (ii) there appears a structure in the both polarizations at p_m ~200 MeV/c.
The consequences for direct reactions of the dynamical non-locality generated by the excitation of the target and projectile are much less studied than the effects of non-locality arising from exchange processes. Here we are concerned with the dynamical non-locality due to projectile excitation in deuteron induced reactions. The consequences of this non-locality can be studied by the comparison of deuteron induced direct reactions calculated with alternative representations of the elastic channel wave functions: (i) the elastic channel wave functions from coupled channel (CC) calculations invo…
The problem of the calculation of inclusive breakup cross sections in nuclear reactions is reexamined. For that purpose, the post-form theory proposed by Ichimura, Austern and Vincent [Phys. Rev. C32, 431 (1985) is revisited, and an alternative derivation of the non-elastic breakup part of the inclusive breakup is presented, making use of the coupled-channels optical theorem. Using the DWBA version of this model, several applications to deuteron and $^6$Li reactions are presented and compared with available data. The validity of the zero-range approximation of the DWBA formula is also investig…
Quasi-binary reaction of the deuteron breakup $p+d\to (pp)+n$ with the final proton-proton pair $(pp)$ in the $^1S_0$ state is analyzed at initial energies $0.5 - 2$ GeV in the kinematics of backward elastic pd-scattering $pd\to dp$. On the basis of the main mechanisms of the $pd\to dp$ process, including initial and final state interaction, we show that unpolarized cross section and spin observables of this reaction exhibit important properties of the half-off-shell $pp(^1S_0)$-scattering amplitude, which are relevant to the nucleon-nucleon interaction at short distances.
Neutrino emissions from electron/positron capture on the deuteron and the nucleon-nucleon fusion processes in the surface region of a supernova core are studied. These weak processes are evaluated in the standard nuclear physics approach, which consists of one-nucleon and two-nucleon-exchange currents and nuclear wave functions generated by a high precision nucleon-nucleon potential. In addition to the cross sections for these processes involving the deuteron, we present neutrino emissivities due to these processes calculated for typical profiles of core-collapsed supernovae. These novel neutr…
We extended the study of relativistic effects in neutron-deuteron scattering to the exclusive breakup. To this aim we solved the three-nucleon Faddeev equation including such relativistic features as relativistic kinematics and boost effects at incoming neutron lab. energies E_n^{lab}=65 MeV, 156 MeV and 200 MeV. As dynamical input a relativistic nucleon-nucleon interaction exactly on-shell equivalent to the CD Bonn potential has been used. We found that the magnitude of relativistic effects increases with the incoming neutron energy and, depending on the phase-space region, relativity can inc…
An extended analysis of the key role of direct interactions, i.e., breakup, stripping and pick-up processes, has been carried out for deuteron-induced reactions. Particular comments concern the deuteron breakup which is the dominant mechanism involved in surrogate reactions on heavy nuclei, around the Coulomb barrier.
The Continuum Discretized Coupled Channels (CDCC) method is a well established theory for direct nuclear reactions which includes breakup to all orders. Alternatively, the 3-body problem can be solved exactly within the Faddeev formalism which explicitly includes breakup and transfer channels to all orders. With the aim to understand how CDCC compares with the exact 3-body Faddeev formulation, we study deuteron induced reactions on: i) $^{10}$Be at $E_{\rm d}= 21.4, 40.9 \; {\rm and} \; 71$ MeV; ii) $^{12}$C at $E_{\rm d} = 12 \; {\rm and} \; 56$ MeV; and iii) $^{48}$Ca at $E_{\rm d} = 56$ MeV…
We discuss the quasi-adiabatic approximations to the three-body wavefunction in breakup processes, clarifying the assumptions underlying the model. This suggests alternative approximation schemes. Using different theoretical three-body models, calculated differential cross section angular distributions for the Be-11(p,d) reaction,for which new preliminary data have been reported at 35 MeV, are presented. We show that calculations are sensitive to the inclusion of deuteron breakup and to the breakup model used, particularly if used to deduce absolute spectroscopic information on the 0{+} and 2{…
An extensive study of three-nucleon force effects in the entire phase space of the nucleon-deuteron breakup process, for energies from above the deuteron breakup threshold up to 200 MeV, has been performed. 3N Faddeev equations have been solved rigorously using the modern high precision nucleon-nucleon potentials AV18, CD Bonn, Nijm I, II and Nijm 93, and also adding 3N forces. We compare predictions for cross sections and various polarization observables when NN forces are used alone or when the two pion-exchange Tucson-Melbourne 3NF was combined with each of them. In addition AV18 was combin…
Recent data on the reaction pD -> (pp) n with a fast forward pp pair with very small excitation energy is analyzed within a covariant approach based on the Bethe-Salpeter formalism. It is demonstrated that the minimum non-relativistic amplitude is completely masked by relativistic effects, such as Lorentz boost and the negative-energy P components in the 1S_0 Bethe-Salpeter amplitude of the pp pair.
The deuteron fragmentation by fast protons has been studied both near the kinematics of quasi-free proton - proton scattering and far away from it. We have concentrated on the interplay between different reaction mechanisms associated with the antisymmetrization of the initial and final states and rescattering contributions. A multiple-scattering-expansion technique has been applied to evaluate the reaction amplitude. An essential element of this approach in the momentum representation is the use of the effective nucleon- nucleon interaction constructed by Love and Franey as a two-body t-matri…
We investigate the sensitivity of the non-exclusive nucleon induced deuteron breakup reaction to the three-nucleon interaction and distributions of three-nucleon force effects in inclusive spectra. To this end we solve the three-nucleon Faddeev equation at a number of incoming nucleon laboratory energies using the CD Bonn nucleon-nucleon interaction alone or combined with the 2π-exchange Tucson-Melbourne three-nucleon force. Based on these solutions energy spectra of an outgoing nucleon, at a specified detection angle as well as spectra integrated over that angle, are calculated. By integratin…
In view of recent neutron-deuteron (nd) breakup data for neutron-neutron (nn) and neutron-proton (np) quasi-free-scattering (QFS) arrangements and the large discrepancy found between theoretical predictions and measured nn QFS cross sections, we analyze the sensitivity of the QFS cross sections to different partial wave components of the nucleon-nucleon (NN) interaction. We found that the QFS cross section is strongly dominated by the 1S0 and 3S1-3D1 contributions. Because the standard three-nucleon force (3NF) only weakly influence the QFS region, we conjecture, that it must be the nn 1S0 for…
Gluon density and its distributions inside nuclei and the parton modification of bounded nucleons inside a nucleus, are some of the main standing problems in nuclear and particle physics. In recent years, ultra-peripheral collisions (UPC) of heavy ions have provided a new way of probing the gluon density, which is based on coherent diffractive vector-meson productions, e.g., $J/ψ$ meson. For heavy ions, e.g., Pb, the gluon density is found to be significantly suppressed through the UPC $J/ψ$ measurement, suggesting a strong gluon shadowing effect in heavy nuclei. In this analysis, we aim to lo…
The break up of the deuteron is studied at high $Q^2$ kinematics, with main motivation to probe the deuteron at small internucleon distances. For this, two main issues are studied: electromagnetic interaction of the virtual photon with the bound nucleon and the strong interactions of the produced baryons in the final state of the reaction. Within virtual nucleon approximation we developed a new prescription to account for the bound nucleon effects in the electromagnetic interaction. The final state interaction at high $Q^2$ kinematics is calculated within generalized eikonal approximation ~ (G…
We present in this paper a formalism for deuteron-induced inclusive reactions. We disentangle direct elastic breakup contributions from other processes (which we generically call non-elastic breakup) implying a capture of the neutron both above and below the neutron emission threshold. The reaction is described as a two step process, namely the breakup of the deuteron followed by the propagation of the neutron-target system driven by an optical potential. The final state interaction between the neutron and the target can eventually form an excited compound nucleus. Within this context, the dir…
The effect of core excitations in transfer reactions of the form A(d, p)B has been found to affect significantly the calculated cross sections and to depend strongly and non-linearly on the incident deuteron energy in Faddeev/AGS calculations. Our goal is to investigate these effects within a coupled-channels formulation of the scattering problem which, in addition of being computationally less demanding than the Faddeev counterpart, may help shed some light into the physical interpretation of the cited effects. We use an extended version of the continuum-discretized coupled-channels (CDCC) me…
Selected Nd breakup data over a wide energy range are compared to solutions of Faddeev equations based on modern high precision NN interactions alone and adding current three-nucleon force models. Unfortunately currently available data probe phase space regions for the final three nucleon momenta which are rather insensitive to 3NF effects as predicted by current models. Overall there is good to fair agreement between present day theory and experiment but also some cases exist with striking discrepancies. Regions in the phase space are suggested where large 3NF effects can be expected.
Three-nucleon scattering equations with irreducible three-nucleon force are solved in momentum-space. The Coulomb interaction between the two protons is included using the method of screening and renormalization. The need for the renormalization of the scattering amplitudes is demonstrated numerically. The Coulomb and Urbana IX three-nucleon force effects on the observables of elastic proton-deuteron scattering and breakup are studied.
We derive the cross-section for exclusive vector meson production in high energy deeply inelastic scattering off a deuteron target that disintegrates into a proton and a neutron carrying large relative momentum in the final state. This cross-section can be expressed in terms of a novel gluon Transition Generalized Parton Distribution (T-GPD); the hard scale in the final state makes the T-GPD sensitive to the short distance nucleon-nucleon interaction. We perform a toy model computation of this process in a perturbative framework and discuss the time scales that allow the separation of initial …
For deuteron photo- and electrodisintegration the selection of complete sets of polarization observables is discussed in detail by applying a recently developed new criterion for the check of completeness of a chosen set of observables. The question of ambiguities and their resolution by considering additional observables is discussed for a numerical example, for which the role of experimental uncertainties is also investigated. Furthermore, by inversion of the expressions of the observables as hermitean forms in the $t$-matrix elements a bilinear term of the form $t_{j'}^*t_j$ can be given as…
