### 6.5.1. Main features

The most intriguing aspects of the PS185 measurements of the $\bar{p}p \to \bar{\Lambda}\Lambda$ reactions are:

1. the energy dependence of the cross-section immediately after the threshold. It cannot be accom-
modated by a dominant S wave. This is an indication for P-waves and even higher $\ell$ contributing
very early. This is confirmed by the shape of the angular distribution and the content of their
Legendre analysis. It was even speculated (on the basis of early results which were not confirmed)
that narrow resonances are responsible for this behaviour. Even if the resonance interpretation is
questionable, it remains that there is a specific dynamical mechanism to be identified, giving rise
to the early onset of high partial waves, in spite of the centrifugal barrier. This mechanism was
searched for either in K, $K^*$ exchange process or in a specific topology of quark diagrams. Is is
difficult to distinguish between the two approaches, as pointed out, e.g., in Ref. [381], where the
low-energy behaviour is examined.

2. the $\bar{\Lambda}\Lambda$ production occurs practically always in a triplet state, the singlet contribution being almost completely suppressed. This property was noticed in earlier experiments [382,383]. The debate on how to reproduce the observed spin correlations focused again on the issue of kaon-exchange versus quark dynamics.

It was hoped that measurement of spin transfer from the proton target will be more decisive to
discriminate among the models. The preliminary results are somewhat disappointing, with values in
between the predictions of simple quark models and those of simple kaon-exchange models. The
final analysis of the last runs of PS185 are, however, not yet published.

### 6.5.2. Kaon-exchange models

This is the nuclear-physics type of approach to $\bar{p}p \to \bar{\Lambda}\Lambda$. As the exchange of charged mesons such as $\pi^+$ or $\rho^+$ mediates the charge-exchange process $\bar{p}p \to \bar{n}n$, exchanging a strange meson induces a transition from $\bar{N}N$ to $\bar{Y}Y$, where Y denotes a hyperon. Several groups have studied this mechanism, sometimes in parallel with quark-model pictures, to be discussed later.

*Lebedev Institute study of $\bar{p}p \to \bar{\Lambda}\Lambda$:* In the reviews summarising the work of the Lebedev group on $\bar{N}N$ dynamics with the coupled-channel model (CCM) [242,240], the last part is dedicated to the modifications done to describe the reaction $\bar{p}p \to \bar{\Lambda}\Lambda$ near threshold with this CCM: $\bar{\Lambda}\Lambda$ is introduced as a new channel, with an explicit diagonal interaction, and a transition potential from $\bar{N}N$ to $\bar{\Lambda}\Lambda$.

For instance, Carbonell et al. [384] have analysed the very-low-energy data on the $\bar{p}p \to \bar{\Lambda}\Lambda$ cross-section, at the time where a near-threshold resonance was suggested by the data. Their results are reproduced in Fig. 6.27. A resonance is produced with $J^{PC} = 1^{--}$ in the $^3\text{SD}_1$ coupled waves. It is remarkable that D-wave contributions are necessary to understand the data at such low energy above the threshold. This is confirmed in the model-independent analysis of Ref. [381].

$\bar{p}p \to \bar{\Lambda}\Lambda$ analysis by Tabakin et al.: Tabakin and Eisenstein were among the first to develop a formalism to analyse the $\bar{p}p \to \bar{\Lambda}\Lambda$ reaction. Their paper [385] contains a comprehensive formalism that was used by other authors.
<figure><img src="image_123.png" /><figcaption>Fig. 6.27. Cross-section for $\bar{p}p \to \bar{\Lambda}\Lambda$ as measured in early PS185 runs, and analysed by Carbonell et al. within the Lebedev model.</figcaption></figure>

They studied in some detail the meson-exchange mechanism, including K, $K^*$ and $K^{**}$. They found that the two latter are particularly important, in contrast to the conclusions by Kohno and Weise [386], or by LaFrance et al. [387].

They produce predictions for several spin observables, not yet measured at the time of the paper. Their polarisation is obviously too small. However, the large and positive values of $C_{nn}$ are remarkably anticipated.

In a second paper [388], Tabakin et al. concentrated on the region very near threshold. Assuming only S and P wave (plus $^3D_1$ which is coupled to $^3S_1$), they were able to determine the contribution of these complex amplitudes (up to an overall phase). They found large S–D mixing, and large differences between the different P waves, confirming what was implicitly found by other authors.

$\bar{p}p \to \bar{\Lambda}\Lambda$ analysis by the Helsinki group: The Helsinki group has made extensive studies of $\bar{N}N$ dynamics, in particular of annihilation mechanisms. In Ref. [389], Niskanen fitted the data with a CCM, using a variant of the Dover–Richard model for $\bar{N}N$ and K and $K^*$ exchange for the transition. See also the review articles by Green and Niskanen [390].

Note that $\Delta-\bar{\Delta}$ and $\Sigma-\bar{\Sigma}$ channels are introduced explicitly. The cross-sections are fitted, and then predictions are produced for the spin observables. In particular, this model easily accommodates a very small spin-singlet fraction. Also the large and positive values of the spin-correlation coefficient $C_{nn}$ were predicted. For the other spin observables, the predictions were not confirmed by the data.

$\bar{p}p \to \bar{\Lambda}\Lambda$ analysis by LaFrance et al.: In his first paper [391], LaFrance et al. sketched a meson-exchange picture of the $\bar{p}p \to \bar{\Lambda}\Lambda$ reaction, and underlined the role of initial- and final-state interaction. In Ref. [387], LaFrance and Loiseau proposed a more detailed study: they use the G-parity transformed Paris potential in the entrance channel, and a similar meson-exchange potential for $\bar{\Lambda}\Lambda$. The transition is mainly due to K exchange, as for Kohno and Weise. The $K^*$ contribution becomes appreciable only when energy increases.
An interesting result is displayed in Table 1 of this paper: they give predictions for $\bar{\Lambda}\Lambda \rightarrow \bar{\Lambda}\Lambda$ cross-section, which is heavily dominated by its inelastic part.

**$\bar{p}p \rightarrow \bar{\Lambda}\Lambda$ analysis by the Regensburg group:** The Regensburg group has studied several aspects of $\bar{N}N$ physics. We mentioned several times the Kohno-Weise optical model of $\bar{N}N$, with applications to $\bar{N}N$ scattering and specific two-meson annihilation channels. Concerning $\bar{p}p \rightarrow \bar{\Lambda}\Lambda$, Ref. [386] stressed the importance of initial and final state distortion for obtaining realistic transition rates. In Ref. [392], Kohno and Weise found a good agreement with the data using K exchange only, without need for $K^*$ or higher resonances. Again, it seems difficult to distinguish K exchange from simple quark dynamics on the basis of $\bar{p}p \rightarrow \bar{\Lambda}\Lambda$ only. It is argued, however, that a systematic measurement of strangeness-exchange reactions should allow one to make a choice. See also, Ref. [393].

**$\bar{p}p \rightarrow \bar{\Lambda}\Lambda$ analysis by the Bonn group:** The Bonn group has published several interesting papers on strangeness-exchange reactions.

In Ref. [394], Haidenbauer et al. developed a meson-exchange model, a natural extension of their work for $\bar{N}N$. They stressed the role of the coherent superposition of K and $K^*$ exchanges, building a strong tensor force, that naturally leads to a dominance of triplet states. Their conclusion is thus somewhat intermediate between Kohno et al., LaFrance et al., who are satisfied with K exchange, and Tabakin et al. who call for even higher kaon resonances ($K^{**}$) in the *t*-channel.

The articles [341,342] were rather influential for the last measurements at LEAR. It is pointed out that the depolarisation parameter $D_{nn}$ is predicted to be positive in simple quark model and negative in their meson-exchange picture, so that its measurement, made possible by scattering antiprotons off a polarised target, could distinguish between the two mechanisms. Their figure is reproduced in Fig. 6.28. More details are provided in Ref. [342], where all measured or measurable spin observables are considered at several energies.

In Refs. [395,396], the study is extended to other hyperon-antihyperon channels. The data on $\bar{p}p \rightarrow \bar{\Sigma}^0\Lambda + \text{c.c.}$ are rather well reproduced, as illustrated in Fig. 6.29.

An interesting result is that the $\bar{p}p \rightarrow \bar{\Sigma}^+\Sigma^-$ reaction, which requires exotic states in its *t*-channel, is not very much suppressed as compared to $\bar{p}p \rightarrow \bar{\Sigma}^-\Sigma^+$ which can proceed via a single kaon exchange. But this later mechanism is not too much effective in their model, because the pK$\Sigma$ coupling is small. Then both $\bar{p}p \rightarrow \bar{\Sigma}^+\Sigma^-$ and $\bar{p}p \rightarrow \bar{\Sigma}^-\Sigma^+$ get dominant contributions from $\bar{\Lambda}\Lambda$ intermediate states, and thus tend to be of the same order of magnitude.

A further extension to $\bar{p}p \rightarrow \Xi\Xi$ is proposed in Ref. [397]. This, of course, would require energies which were not accessible at LEAR.

**$\bar{p}p \rightarrow \bar{\Lambda}\Lambda$ analysis by the Nijmegen group:** The Nijmegen group has extended its study of $\bar{N}N$ to strangeness exchange, again insisting on the low value of their $\chi^2$, which reflects, indeed, a good agreement with the data [398–400].

The transition is described in terms of kaon exchange. It is found that the data allow one to recover the mass of the kaon, $m(K) = 480 \pm 60$ MeV/c$^2$, to be compared to the experimental value 493.7. The $\Lambda$NK coupling constant at the pole is found $f_{\Lambda$NK}^2 = 0.071 \pm 0.007$. This value is in agreement with the value $f_{\Lambda$NK}^2 = 0.0734$ used in the recent soft-core Nijmegen hyperon-nucleon potential [344]. Note, however, that these hyperon-nucleon potentials are regularly updated; see, for instance, Ref. [345].
<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /><figcaption>Fig. 6.28. Polarisation and depolarisation parameter for $\bar{p}p \to \bar{\Lambda}\Lambda$ as predicted by the Bonn group [341]. The solid curve corresponds to a meson-exchange picture (the dashed one to a variant), while the dashed-dotted and dotted curves are obtained from simple quark-gluon models.</figcaption></figure>

<figure><img src="image_126.png" /></figure>

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /><figcaption>Fig. 6.29. Some observables for the $\bar{p}p \to \Sigma^0\Lambda + \text{c.c.}$ reaction at $p_{\text{lab}} = 1695 \text{ MeV/c}$, as calculated by the Bonn group [395].</figcaption></figure>

When presenting their $\bar{p}p \to \bar{\Lambda}\Lambda$ results at a Conference [400], the Nijmegen group provided a number of comments:

* It is underlined that accurate data for the reactions $\bar{p}p \to \bar{\Sigma}^0\Lambda + \text{c.c.}$ could give access to the $\Sigma\text{NK}$ coupling constant, and thus to the SU(3) ratio $\alpha = F/(F+D)$.
* They obviously favour meson exchange, with a well-defined formalism and well-determined coupling, as compared to quark models, whose low-energy limit is not very much under control.

* They again overestimate the power of their fitting technology, and do not encourage performing further experiments. To quote Ref. [400]: "It is absolutely not necessary to measure the spin transfer to distinguish between the K(494)- and K*(892)-exchange picture and a simple quark-gluon-exchange picture. This distinction has already been made using our PWA as a tool and using just the differential cross-sections and polarisations."

We do not share this opinion: two observables are not sufficient to fix six complex amplitudes.

* The dominance of triplet states is explained by the crucial role of tensor forces, which act in triplet and vanish for singlet states.

### 6.5.3. Quark models for $\bar{p}p \to \bar{Y}Y$

In simple constituent models, the ud pair in a $\Lambda$ hyperon has spin 0 and isospin 0, and the quantum numbers of the $\Lambda$ are carried by the strange quark s. Thus, measuring how a $\bar{\Lambda}\Lambda$ pair is produced indicates in which state an s$\bar{s}$ pair of quark is created out of the available energy. The $^3P_0$ model has been developed to describe the strong decay of meson and baryon resonances. In this model, which is reviewed in [401], a q$\bar{q}$ pair is created with vacuum quantum numbers. On the other hand, a $^3S_1$ configuration would correspond to the quantum numbers of the gluon. The question of the quantum numbers of the q$\bar{q}$ pair also arises in baryon and meson decay, multiparticle production, hadronisation, etc.

This is perhaps too naive a point of view to believe that q$\bar{q}$-creation contributes in a single partial wave. However, given the intricate and non-perturbative nature of QCD, one should look at opportunities to extract some simple signals from the background [402].

*Rubinstein and Snellman:* This paper [402] contains considerations on the relative rates for the various proton-antiproton to baryon-antibaryon reactions, as a function of the assumed $2^{s+1}\ell_J$ partial wave for the annihilating q$\bar{q}$ pair and the created s$\bar{s}$ pair.

*Kroll and Schweiger:* In Ref. [403], the $\bar{p}p \to \bar{Y}Y$ reactions are analysed in a diquark model already used by the authors, or others, in different contexts. A diquark-antidiquark mechanism supplements the quark-antiquark annihilation of Fig. 3.12 of Section 3. This provides in particular transitions from $\bar{p}p$ to $\bar{\Sigma}^+\bar{\Sigma}^-$ and to $\bar{\Xi}\Xi$. High-energy data are well reproduced in this model. To our knowledge, it has not been adapted to the PS185 energy-range.

$\bar{p}p \to \bar{\Lambda}\Lambda$ analysis by the Genz and Tatur: In Refs. [404,405], Genz and collaborators analysed various reactions where $\bar{p}p$ is transformed into an antibaryon-baryon pair. Reduced cross-sections (with phase-space factors removed) can be compared with various assumptions for the quantum numbers of the $q\bar{q} \to s\bar{s}$ transition. Assuming gluon quantum numbers ($^3S_1$) leads to a good agreement with most data.

$\bar{p}p \to \bar{\Lambda}\Lambda$ analysis by the CERN-Seattle group: Mary Alberg, with varying but always eminent collaborators, showed a persistent interest in these strangeness-exchange reactions. She also gave several enlightening talks at Conferences, in particular at LEAP98 [47] and LEAP2000 [48].

In Ref. [406], in collaboration with W. Weise, the problem of the spin-singlet fraction is addressed. A rather natural cancellation of pseudo-scalar intermediate states ($\eta, \eta', \dots$) is found in a generalisation
of the Nambu–Jona–Lasinio model. It remains to extend this idea to spin-singlet states with higher
$L$ ($^1P_1$, etc.).

In an interesting note [338], Alberg et al. discuss the respective merits of the $^3P_0$ and $^3S_1$ models of quark-pair creation or annihilation, following a debate initiated by Burkardt and Dillig [337]. They underline the role of initial and final state interaction, and suggest a superposition of both mechanisms to fit the data.

In an often-cited paper [215], Alberg et al. suggested an alternative to kaon exchange or s¯s creation following p̄p annihilation: strangeness exchange is achieved by “extracting” an s¯s pair out of the sea of the proton (or antiproton). See Figs. 3.12 and 3.13 in Section 3. As this sea is likely to be polarised, from the analysis of the proton structure functions, very specific effects on the spin-transfer observables are predicted, in particular on $D_{nn}$.

This investigatory paper did not account for the corrections due to initial- or final-state interaction, and also ignored that the allowed range for $D_{nn}$ is already bound from the existing data on the correlations coefficients [343,183]. It had, however, the great merit to trigger the interest on this development of the PS185 experiment.

Their model predicted $D_{nn} < 0$, while conventional quark models tend to predict $D_{nn} > 0$. Meson-exchange models predict also $D_{nn} < 0$. Preliminary experimental results, displayed in Fig. 6.32, give $D_{nn} \simeq 0$. Remember that $-1 \le D_{nn} \le 1$, and that $D_{nn} = 1$ in absence of any spin-dependent force.

$\bar{p}p \rightarrow \bar{\Lambda}\Lambda$ analysis by the Tübingen group: In Ref. [407], Furui and Faessler also considered $^3P_0$ versus $^3S_1$ model for creating or annihilating a quark-antiquark pair. They pointed out that the former can digest a change of angular momentum when going from $\bar{N}N$ to $\bar{\Lambda}\Lambda$, mimicking a kind of tensor force. They concluded that $^3P_0$ is favoured.

The paper considered an alternative mechanism with K, K*, K** ($J^P=2^+$) exchange. They found that the quark model with $^3P_0$ differs very little from a Yukawa model with K and K*.

The authors made an interesting comparison between $K^+K^-$ versus $\pi^+\pi^-$ final states, or $\bar{\Sigma}^-\Sigma^+$ versus $\bar{\Delta}^-\bar{\Delta}^{++}$ final states, concluding—within the assumed mechanism—that s¯s creation is significantly suppressed. This is a hot subject, as we shall see when reviewing annihilation. For instance, when one starts from a perfectly unbiased initial state and with enough phase-space, one observes almost perfect SU(3) symmetry, as seen from the decay of J/Ψ into various baryon-antibaryon pairs.

Note that the authors predict (in their Fig. 7) the values of the spin-correlation coefficient in their favourite model. We reproduce their curves in Fig. 6.30.

s-channel picture of $\bar{p}p \rightarrow \bar{\Lambda}\Lambda$: An improved quark model approach was proposed by Roberts [227]. Instead of naive diagrams with q¯q annihilated and s¯s created, he introduced fully interacting four-quark states as intermediate states. Tuning the parameters gives a very good agreement with the data, as seen in Fig. 6.31 below. This approach can easily be extended to other hyperon-antihyperon channels.

### 6.5.4. Impact of further observables

At the Venice Conference [48], the PS185 collaboration presented some preliminary results [408] on the spin transfer observables $D_{nn}$ and $K_{nn}$, and even some rank-3 observables. The analysing power $A_n$ (left-right production asymmetry) is shown to differ somehow from the final-state polarisation $P_n$. Correlations have been measured, and found consistent with previous data.
<figure><img src="image_129.png" /><figcaption>Fig. 6.30. Spin correlations coefficients for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1.5 Gev/c, as predicted by Furui and Faessler in Ref. [407].</figcaption></figure>

An updated version of the plot of $K_{nn}$ and $D_{nn}$ shown at Venice [408] is given in Fig. 6.32 below. The data are just in between the quark-model and the kaon-exchange predictions, leading the author to the somewhat pessimistic conclusion: “This demonstrates that the dynamics of strangeness production are not well understood.”

## 6.6. Protonium and low energy $\bar{p}N$ scattering

### 6.6.1. $\rho$ parameter

There have been several discussions about the $\rho$-parameter, the real-to-imaginary ratio of the forward scattering amplitude. If one takes seriously the nominal value published by different experimental groups, and the point at zero energy deduced from protonium measurement, one observes a puzzling structure. See Fig. 4.12.

The authors of the Nijmegen PWA insisted on the difficulty in extracting this $\rho$-parameter from the raw data, and the underestimation of systematic errors. A comparison of the LEAR data with the values of $\rho$ calculated with their PWA is given in Fig. 6.33, taken from Ref. [279].

### 6.6.2. $\bar{p}p$ annihilation at low energy

Cross-sections at very low energies are used to extract scattering lengths. The annihilation cross-section has been measured down to 44 MeV/c [96,409,97]. At these low momenta interference with the Coulomb forces become important [410], and the cross-section does no longer scale with the relative velocity but rather with the squared velocity [411].

A first fit to low-energy scattering data using (3.96) was performed in Ref. [295]. Later, more experimental points were added, leading to the fit of Ref. [412] which is presented in Fig. 6.34. The fit
<figure><img src="image_130.png" /><figcaption>Fig. 6.31. Spin parameters for $\bar{p}p \to \bar{\Lambda}\Lambda$, as predicted by Roberts in Ref. [227].</figcaption></figure>

returned the $\bar{p}p$ S-wave scattering length as

$$ \mathrm{Im} a_0^{\mathrm{sc}} = -[0.69 \pm 0.01(\mathrm{stat}) \pm 0.03(\mathrm{sys})] \mathrm{fm}, \qquad (6.10) $$

in excellent agreement with the protonium result given in (5.26). The fit also yielded the P-wave scattering volume

$$ \mathrm{Im} a_1^{\mathrm{sc}} = -[0.75 \pm 0.05(\mathrm{stat}) \pm 0.04(\mathrm{sys})] \mathrm{fm}^3, \qquad (6.11) $$

which is also in very good agreement with the results from the protonium atom (5.27). The systematic errors in (6.10) or (6.11) come from normalisation uncertainties.

The results obtained from the annihilation cross-section at low momenta obviously extrapolate very well to the strong interaction widths of protonium levels. This beautiful agreement gives credit to both types of measurement and to the theoretical frame within which the data were analysed.

### 6.6.3. $\bar{p}d$ annihilation at low energy

The strong interaction shift and width of the 1S level of the $\bar{p}d$ atom [177] are of the same order of magnitude as those of the $\bar{p}p$ atom. The errors in the scattering lengths one can deduce from these
<figure><img src="image_131.png" /><figcaption>Fig. 6.32. Depolarisation and spin transfer parameters for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1.637 GeV/c, as measured at LEAR, compared to a kaon-exchange and a quark-model prediction [215].</figcaption></figure>

<figure><img src="image_132.png" /><figcaption>Fig. 6.33. Comparison between the $\rho$ parameter data from PS172 and PS173 and the predictions from the Nijmegen PWA.</figcaption></figure>

numbers are rather large and a comparison with data on the low-energy $\bar{p}d$ annihilation cross-section is not very enlightening.

The authors of Ref. [412] adopted a different strategy. They fixed the real part of the S-wave scattering amplitude (to which the annihilation cross-section is not sensitive) to the 1S level shift and the P-wave parameters to the average 2P level width. Then they performed a fit to the annihilation cross-section data with the imaginary part of the S-wave scattering length as only parameter. Their fit and the corresponding scattering data are shown in Fig. 6.35. The fit provided the imaginary part
<figure><img src="image_133.png" /><figcaption>Fig. 6.34. The total $\bar{p}p$ annihilation cross-section multiplied by the square of the incoming beam velocity. Data are from [96] (▲), [409] ( ), and [97] (★). The error bars represent the sum of the statistical and systematic errors. Not included is an overall normalisation error (3.4% for [96,409] and 2.5% for [97]). The theoretical curves are the result of a fit. The full line is the total annihilation cross-section, the dashed line represents the S-wave contribution.</figcaption></figure>

<figure><img src="image_134.png" /><figcaption>Fig. 6.35. The total $\bar{p}d$ annihilation cross-section multiplied by the square of the incoming beam velocity. Data are from [274]. The theoretical curves are the result of a fit. The full line is the total annihilation cross-section, the dashed line represents the S-wave contribution, the dotted one, the P-wave contribution.</figcaption></figure>

of the scattering length:

$$ \mathrm{Im} a_0^{\mathrm{sc}}(\bar{p}d) = -[0.62 \pm 0.02(\mathrm{stat}) \pm 0.05(\mathrm{sys})] \mathrm{fm}. \qquad (6.12) $$

The imaginary part of the $\bar{n}p$ (or equivalently $\bar{p}n$) scattering length was determined from $\bar{n}p$ annihilation data [274]. The authors quote

$$ \mathrm{Im} a_0^{\mathrm{sc}}(\bar{n}p) = -[0.83 \pm 0.07(\mathrm{stat})] \mathrm{fm}. \qquad (6.13) $$
Within a naive geometrical approach to annihilation, one could expect the imaginary part of the $\bar{p}d$ scattering length (6.12) to be approximately equal to the sum of the $\bar{p}n$ (6.13) and $\bar{p}\bar{p}$ (6.10) contributions:

$$ \mathrm{Im} a_0^{\mathrm{sc}}(\bar{p}d) \approx \mathrm{Im} a_0^{\mathrm{sc}}(\bar{p}n) + \mathrm{Im} a_0^{\mathrm{sc}}(\bar{p}\bar{p}) . \qquad (6.14) $$

The results show that this view is too naive: the three-body problem has to be faced properly. As we shall see, the Faddeev approach resolves this apparent discrepancy.

## 6.7. Strong interaction effects in protonium

The effects of strong interactions on the energy levels of protonium atoms were predicted long before protonium spectroscopy was accessible to measurement. A detailed comparison of potential-model calculations with experimental results will be made shortly. We first discuss qualitatively the physical significance of the LEAR data using very simple models.

### 6.7.1. The 1S upwards shift

The 1S ground-state level is shifted upwards by about 0.73 keV, i.e., experiences a repulsive energy shift due to strong interactions. This is surprising: the nuclear forces between proton and antiproton are attractive and one may expect additional binding. There are two simple explanations of this phenomenon:

1. We have seen in Section 3 that with a superposition of a Coulomb and short-range potential, when the latter supports a bound state, the atomic “1S” state, in the keV range, is in fact a radial excitation of the nuclear bound state. The node generates extra kinetic energy, and thus the energy is shifted upwards with respect to a pure Coulomb state. In other words, the observation of a positive value of $\Delta E_{1S}$ may be the consequence of the strong binding force of nuclear interactions leading to nuclear bound states with zero orbital angular momentum with $^1S_0$ and $^3S_1$ quantum numbers!

2. A more conservative interpretation focusses on the annihilation part of the interaction. Annihilation is very strong at short distances, and the protonium wave function vanishes or is very small for radii of less than a fm. The Coulomb interaction thus loses a highly attractive part and, again, the 1S levels are pushed upwards. We may use our simple model to examine this possibility. For large positive values of $\lambda$, the wave function vanishes in the central part, and indeed the energy shift adopts negative values. $\Delta E = 0.7$ keV is reached for a nuclear range of $a = 1.8$ fm, see Fig. 3.23. This is not unreasonable as an estimate for the range of nuclear interactions, but this is obviously too large a value for the annihilation range.

The sign and strength of the 1S-level strong interaction shift results in fact from a combination effect of long- and short-range mechanisms, and is well reproduced in realistic potential models.

### 6.7.2. 2P levels

We can also understand intuitively the large strong interaction widths of the 2P levels [147]. For $\ell = 1$ between proton and antiproton, the potential at large distances comprises the Coulomb part ($\propto -1/r$) and the centrifugal part ($\propto r^{-2}$). The sum forms a Coulomb well which is schematically depicted in Fig. 3.26. Without strong interaction, the centrifugal barrier diverges for $r \to 0$. Strong
<table><caption>Table 6.3
Energy shifts and widths of low-lying levels of protonium; comparison of models with experimental results</caption><thead><tr><th></th><th>P1</th><th>P2</th><th>DR1</th><th>DR2</th><th>KW</th><th>Exp.</th></tr></thead><tbody><tr><td>ΔE<sub>1S</sub></td><td>0.84</td><td>0.75</td><td>0.71</td><td>0.76</td><td>0.71</td><td>0.73 ± 0.03 keV</td></tr><tr><td>Γ<sub>1S</sub></td><td>1.01</td><td>1.02</td><td>0.93</td><td>0.95</td><td>1.05</td><td>1.06 ± 0.08 keV</td></tr><tr><td>ΔE(<sup>3</sup>P<sub>0</sub>)</td><td>-84</td><td>-72</td><td>-74</td><td>-62</td><td>-69</td><td>-139 ± 20 meV</td></tr><tr><td>Γ(<sup>3</sup>P<sub>0</sub>)</td><td>130</td><td>111</td><td>114</td><td>80</td><td>96</td><td>120 ± 25 meV</td></tr><tr><td>ΔE(<sup>3</sup>P<sub>2</sub>, <sup>3</sup>P<sub>1</sub>, <sup>1</sup>P<sub>1</sub>)</td><td>1.9</td><td>-0.8</td><td>0.5</td><td>0.6</td><td>-3.9</td><td>+4 ± 5.8 meV</td></tr><tr><td>Γ(<sup>3</sup>P<sub>2</sub>, <sup>3</sup>P<sub>1</sub>, <sup>1</sup>P<sub>1</sub>)</td><td>28</td><td>26</td><td>26</td><td>27</td><td>29.5</td><td>30.5 ± 2.0 meV</td></tr><tr><td>Γ(<sup>1</sup>P<sub>1</sub>)</td><td>26</td><td>28</td><td>26</td><td>28</td><td>26</td><td>51 ± 18 meV</td></tr><tr><td>χ<sup>2</sup></td><td>25.2</td><td>19.4</td><td>21.0</td><td>25.3</td><td>17.7</td><td>All data</td></tr><tr><td>χ<sup>2</sup>/N<sub>F</sub></td><td>2.9</td><td>1.4</td><td>1.7</td><td>1.7</td><td>0.9</td><td>Without ΔE(<sup>3</sup>P<sub>0</sub>)</td></tr></tbody></table>

The largest contribution to $\chi^2$ comes from the width of the $^3P_0$ state. Excluding this entry from the $\chi^2$ evaluation, there is good agreement between data and predictions, in particular for the Kohno-Weise model. For DR1, DR2, and KW models, the protonium calculations are from Ref. [265]. For the Paris potential, P1 refers to Schweiger et al. [230], who use a separable approximation to the Paris potential, while P2 corresponds to a direct calculation by Moussalam [413].

interaction forces are, however, attractive. The centrifugal barrier and the strong interaction potential form a barrier through which tunnelling is possible. As soon as a protonium atom tunnels through this barrier, it is sucked into the annihilation region from which there is no return: the atom annihilates. The annihilation probability is obviously dominated by the well-known long-range interaction between proton and antiproton. Details of short-range interaction are not relevant.

### 6.7.3. Strong interaction effects: predictions versus experiment

We compare in Table 6.3 the experimental results with the corresponding calculated quantities. The latter values are weighted means assuming a statistical population of the fine-structure levels. The comparison is rather good, in particular for the KW model. Only the experimental shift of the $^3P_0$ level is not correctly predicted. This quantity is only measured in one experiment; and a confirmation would of course be desirable. If the present data are taken seriously, the attraction in the $^3P_0$ channel is even stronger than in current potentials. May be, in the these models, the coherent tensor force due to pseudoscalar and vector exchanges is too much regularised at short distances.

The 1S results are also summarised in Fig. 6.36.

### 6.7.4. Isospin mixing

We do not need the full machinery of coupled radial equations (3.101) to compute the energy shifts and widths safely. The main advantage of Eqs. (3.101) is to provide the full wave function in addition to energy shifts and widths. In particular, one can estimate the role of $\bar{p}p \rightarrow \bar{n}n$ transitions in protonium.

The results reveal some surprises. The overall amount of the $\bar{n}n$ wave function $w(r)$ since the exponential damping at large distances $w(r) \sim \exp[-(4m_r\delta m)^{1/2}r]$, for the $\bar{n}n$ component is much more effective than the damping $u(r) \sim \exp(-kr)$, with $k = [-2m_r \text{Re}(E)]^{1/2}$, for the main $\bar{p}p$ component. The $\bar{n}n$ contribution to the normalisation is of the order of $10^{-6}$ or $10^{-7}$ for 1S, and
<figure><img src="image_135.png" /><figcaption>Fig. 6.36. Average experimental result for the 1S energy shift and width, compared to the predictions of the potential models P1 (); P2 (▼); DR1 (●); DR2 (○); KW (▲). The contour plot corresponds to 70% C.L.</figcaption></figure>

$10^{-9}$ or smaller for 2P. Also, the change of the energy shift $\Delta E$ due to the charge-exchange potential is generally very small, except perhaps for $^3P_0$.

However, the amount of $\bar{n}n$ is predicted to be considerable at short distances. In fact, for $r \lesssim 1$ fm, the $\bar{p}p$ and $\bar{n}n$ wave functions are of comparable magnitude, and when one reconstructs the isospin eigenstates, as per Eq. (3.103), one is very far from the situation with 50% $I=0$ and 50% $I=1$. This is seen in Table 3.5. Here the hadronic width is split via Eq. (3.105) into $I=0$ and $I=1$ components. Even with an isospin-independent annihilation potential, as in the simple models used in Table 3.5, the ratio $I=1$ to $I=0$ is far from the value $\Gamma_1/\Gamma_0$ would assume with a pure $\bar{p}p$ state. This effect, which was often underlined, could play an important role in the dynamics of annihilation. Isospin mixing is dramatic for two of the P states: the $^3P_0$ state is predicted to be a pure isoscalar state, the $^3P_1$ a nearly pure isovector state.

It is interesting to examine to which extent the isospin distortion is seen in annihilation data. Though this is the subject of the forthcoming review article [64], we give below a skeletal summary of the state of the art.

The first attempt to deduce the isospin ratios from experiment was made in [414] (see Table 6.4). The model assumed that the transition matrix element for annihilation into two mesons is entirely determined by the isospins involved. There are transitions from the $I=0$ initial state to two isoscalars and to two isovectors, and from the $I=1$ initial state to one isoscalar and an isovector. The transition matrix elements are then supposed to be independent of the quantum numbers of the initial state, apart from a normalisation which could be different for different initial states. The results are certainly model-dependent, and the errors are large. The isospin-mixing coefficients are found compatible with the predictions of potential models, but, due to the large errors, also with the absence of mixing effects. In annihilation from S states, the effect of isospin mixing is not very large; in annihilation from P states, the accuracy was rather limited. The subject was further discussed by Dover et al. [270] and more recently by the Crystal Barrel collaboration [365].

The best possibility to test the prediction of isospin mixing is the use of the $^3P_0$ initial state which is predicted to have only a very small isovector component. We now show that the isovector
<table><caption>Table 6.4
Ratio of isovector to isoscalar fraction of the protonium wave function for various initial states $^{2s+1}L_J$</caption><thead><tr><th rowspan="2">Initial state</th><th colspan="3">Potentials</th><th colspan="2">Data analysis</th></tr><tr><th>KW</th><th>DR1</th><th>DR2</th><th>Ref. [414]</th><th>Ref. [365]</th></tr></thead><tbody><tr><td><sup>1</sup>S<sub>0</sub></td><td>0.68</td><td>0.68</td><td>0.8</td><td>0.72<sup>+0.24</sup><sub>-0.18</sub></td><td>0.50<sup>+0.48</sup><sub>-0.29</sub></td></tr><tr><td><sup>3</sup>S<sub>1</sub></td><td>1.22</td><td>0.95</td><td>1.26</td><td>1.17<sup>+0.39</sup><sub>-0.28</sub></td><td>1.17<sup>+0.30</sup><sub>-0.23</sub></td></tr><tr><td><sup>3</sup>P<sub>0</sub></td><td>0.03</td><td>0.03</td><td>0.05</td><td>1.16 ± 0.34</td><td>0.41<sup>+0.11</sup><sub>-0.09</sub></td></tr><tr><td><sup>3</sup>P<sub>1</sub></td><td>9.4</td><td>9.7</td><td>6.5</td><td>9 ± 5</td><td></td></tr><tr><td><sup>1</sup>P<sub>1</sub></td><td>0.96</td><td>0.82</td><td>0.61</td><td>0.81 ± 0.51</td><td></td></tr></tbody></table>

The three theoretical values correspond to different $\bar{N}N$ potentials, as compiled in Ref. [265].

component of the $^3P_0$ is not small, using the following series of arguments:

1. The annihilation process $\bar{p}p \to \pi^0\pi^0$ is found to proceed via the $^3P_0$ state and not via the $^3P_2$ state.

2. The annihilation process $\bar{p}p \to \pi^0\eta$ is found to proceed via the same initial state as $\bar{p}p \to \pi^0\pi^0$.

3. The size of the branching ratio for this reaction is not due to an extremely large coupling to $\pi^0\eta$ from a small isovector $^3P_0$ component.

4. The conclusion is confirmed looking at a further channel.

First we recall that the $2\pi^0$ final state is produced from the $^3P_0$ initial state (see Section 5.5.4). Otherwise, we should observe 30% P-state annihilation in most channels while partial-wave analyses give about 10%.

The $\pi^0\pi^0$ over $\pi^0\eta$ ratio does not change when going from liquid $H_2$ to $H_2$ gas at NTP; the ratios are [363,415]

$$ \left( \frac{\mathrm{BR}(\bar{p}p \to \pi^0\eta)}{\mathrm{BR}(\bar{p}p \to \pi^0\pi^0)} \right)_{\mathrm{LH}_2} = 0.303 \pm 0.010 \ [363], \quad 0.407 \pm 0.056 \ [415], \qquad (6.15) $$

$$ \left( \frac{\mathrm{BR}(\bar{p}p \to \pi^0\eta)}{\mathrm{BR}(\bar{p}p \to \pi^0\pi^0)} \right)_{\mathrm{GH}_2} = 0.366 \pm 0.035 \ [415]. \qquad (6.16) $$

Since the fractional contribution of the $^3P_0$ state to annihilation increases dramatically with increasing pressure while the $^3P_2$ contribution remains constant, as seen in Fig. 5.21, the two reactions $\bar{p}p \to \pi^0\pi^0$ and $\bar{p}p \to \pi^0\eta$ need to proceed via the same initial state. The cascade arguments at the end of Section 6.7 require this to be the $^3P_0$ state. Of course, the reaction $\bar{p}p \to \pi^0\eta$ goes via the isovector part of the $\bar{p}p$ wave function.

We now compare (Table 6.5) the branching ratios for $\mathrm{BR}(\bar{p}d \to \pi^0\eta n)$ and $\mathrm{BR}(\bar{p}d \to \pi^-\eta p)$: the former ratio should be reduced compared to the latter one by the fraction of the isovector component in the $\bar{p}p$ subsystem (the $\bar{p}n$ subsystem is pure $I=1$). According to this argument, the isovector component in the $\bar{p}p$ wave function annihilation to $\pi\eta$ is $0.40 \pm 0.10$.

Finally, the same arguments can be made, with reduced accuracy, using annihilation into $\pi\eta'$. The isovector component is also needed at a smaller momentum (which may correspond to a larger annihilation range).
<table><caption>Table 6.5
Branching ratios for $\bar{p}p$ annihilation at rest in liquid $H_2$</caption><thead><tr><th>Annihilation</th><th>from I = 0</th><th>Ref.</th><th>Annihilation</th><th>from I = 1</th><th>Ref.</th></tr></thead><tbody><tr><td>BR(\bar{p}p → π<sup>0</sup>π<sup>0</sup>)</td><td>(6.93 ± 0.43) × 10<sup>-4</sup></td><td>[363]</td><td>BR(\bar{p}p → π<sup>0</sup>η)</td><td>(2.12 ± 0.12) × 10<sup>-4</sup></td><td>[363]</td></tr><tr><td>BR(\bar{p}p → π<sup>0</sup>π<sup>0</sup>)</td><td>(6.14 ± 0.40) × 10<sup>-4</sup></td><td>[415]</td><td>BR(\bar{p}p → π<sup>0</sup>η)</td><td>(2.50 ± 0.30) × 10<sup>-4</sup></td><td>[415]</td></tr><tr><td>BR(\bar{p}d → π<sup>0</sup>ηn)</td><td>(1.64 ± 0.10) × 10<sup>-4</sup></td><td>[363]</td><td>BR(\bar{p}d → π<sup>-</sup>ηp)</td><td>(4.06 ± 1.00) × 10<sup>-4</sup></td><td>[365]</td></tr><tr><td>BR(\bar{p}p → π<sup>0</sup>η')</td><td>(2.16 ± 0.25) × 10<sup>-4</sup></td><td>[363]</td><td>BR(\bar{p}p → π<sup>0</sup>η')</td><td>(1.23 ± 0.13) × 10<sup>-4</sup></td><td>[363]</td></tr></tbody></table>

In seeking a possible explanation for the apparent failure of the potential models we notice that, possibly, the prediction of potential models for the ratio of $I = 1$ to 0 decay widths only applies to the total hadronic width, with an average of five pions in the final state. The decay into two light mesons involves the short-range part of the wave function. Perhaps some balance between the $I = 0$ and 1 components is restored there, as mentioned in Section 4.1, since both isospin components experience an overall suppression due to the strong absorption. Of course, optical models where annihilation is treated globally, reach their limits there. It would be interesting to repeat protonium calculations with coupled-channel models and examine whether the $I = 1$ to 0 ratio depends on the mass of the mesons which are introduced in the various channels.

To conclude, isospin mixing in protonium remains rather enigmatic. It is rather firmly predicted in potential models, independent of the particular optical model which is chosen, since mainly driven by pion exchange, which is well established. The effect has been confirmed by different groups. Other predictions of potential models have been verified to a good precision, see Table 6.3, in particular concerning the large width and shift of the $^3P_0$ as compared to the other states. On the other hand, the isospin mixing is not seen in current analyses of annihilation data. We are aware that these analyses are partly based on a long chain of arguments which include our present understanding of cascade effects.

### 6.7.5. Antiprotonic deuterium

There are many common features between protonium and antiprotonic deuterium. However, the microscopic calculation within a potential model is somewhat more complicated in the latter case. If one looks only at the complex energy shifts, then one can use simple approximations for the (n, p, \bar{p}) 3-body problem. This is discussed for instance in [416], with references to earlier works. We are aware of two calculations of the complex energy shift of \bar{p}d atoms. Weighted with the hyperfine structure multiplicities, shift and width were calculated to $\Delta E_{1S} = 2.17$ keV, $\Gamma_{1S} = 1.24$ keV [216] and $\Delta E_{1S} = 1.64$ keV, $\Gamma_{1S} = 0.78$ keV [417], respectively. The latter calculation uses Fadeev equations and the 3-body problem seems to be treated in a more adequate manner. It is not inconsistent with the experimental values $\Delta E_{1S} = 1.05 \pm 0.25$ keV and $\Gamma_{1S} = 1.10 \pm 0.75$ keV which were quoted in (5.16).

## 7. Conclusions

In this review article we have presented the results obtained at LEAR on $\bar{N}N$ elastic scattering, charge and strangeness exchange, and also on protonium and antiprotonic deuterium. We have
reviewed experiments investigating strong interaction physics by $\bar{N}N$ scattering and studies of the protonium atom. Emphasis was on the ideas which drove the experiments, their realisation, the results, and the impact of the data on the theoretical description of $\bar{N}N$ interaction.

The lowest momenta in $\bar{N}N$ scattering were accessed by studying strong interaction shifts and widths in protonium atoms. These agree very well with the predictions of models based on meson-exchange potentials. From the strong interaction shifts and widths one can deduce S-wave scattering length and P-wave scattering volume. Cross-section measurements at very low energies allow extrapolation and independent determination of scattering lengths. Both methods give results in good agreement.

In the region of 100 MeV/c a cross-over from $1/v^2$ to $1/v$ scaling of the cross-section is observed. This transition region is required to exist because of the interplay between Coulomb and strong forces.

The momentum range from 150 to 2000 MeV/c was explored by many experiments: total and differential cross-sections as well as analysing power and a few two-spin observables were measured for elastic scattering, charge-, and strangeness exchange.

The early onset of higher partial waves was a lively debated subject in $\bar{N}N$ physics. P-wave contributions play a prominent role in $\bar{N}N$ scattering at momenta where NN scattering is still dominated by S wave amplitudes. The large P-wave contribution even at lowest momenta was qualitatively assigned to a particular P-wave enhancement of the $\bar{N}N$ forces, with the predicted consequence of $\bar{N}N$ bound states close to the $\bar{N}N$ threshold, but also to the suppression of the S wave due to the strong annihilation in this partial wave. Now the phenomenon is understood as combined effect of the long-range forces attracting the $\bar{N}N$ system into a region where annihilation is strong. The beautiful agreement between the P-wave scattering volume derived from the protonium atoms and the prediction of the one-pion exchange model demonstrates that the large P-wave contribution to annihilation is a genuine effect of G-parity-transformed nuclear forces, and that no special mechanism is needed to explain this effect.

The charge-exchange reaction has allowed for an important contribution of antiproton physics: the $g_{\pi\bar{N}N}^2$ coupling constant could be determined at LEAR, and turned out to be smaller than thought before. It would be important to improve on the precision of the $g_{\pi\bar{N}N}^2$ value extracted from $\bar{N}N$ data and assess its equality with the value extracted from NN and $\pi N$ data, an interesting test of C invariance of the strong interactions. The charge exchange reaction also identifies the contribution of $2\pi$ exchange, likely dominated by the $\rho$ meson.

It was not possible at LEAR to determine a sufficient number of observables which would have allowed a full partial wave analysis. So our knowledge of $\bar{N}N$ interactions is not as complete as in the nucleon-nucleon case. Still, a first partial wave analysis of $\bar{N}N$ scattering data was performed imposing meson-exchange and Coulomb interaction at large distances.

Meson-exchange models make a few rather general predictions for $\bar{N}N$: the $\bar{N}N$ interactions should be strongly attractive, tensor forces should be very strong in some partial waves; a strong isospin dependence is expected. These features are well reproduced by the data. On the other hand, meson-exchange contributions, when expressed in the form of a real potential, predict quasinuclear states above and below the $\bar{N}N$ threshold. Some of them are deeply bound isoscalar states, several are expected close to the $\bar{N}N$ threshold and a large number of $\bar{N}N$ resonances was predicted. The search for these states was certainly one of the main motivations to build LEAR, but the premises turned out to be incorrect. In the high-mass range, a series of resonances is observed; none of them
has a particularly large coupling to $\bar{N}N$. The non-observation of baryonium states underlines the importance of annihilation: even a small overlap of the $\bar{N}N$ wave function seems to lead to its collapse.

Strangeness exchange is a short-range phenomenon which was extensively studied at LEAR in the hope that the process could reveal the dynamics of constituent quarks. Due to the self-analysing power of the hyperons, many different polarisation variables are accessible experimentally, and several were measured at LEAR. In spite of this effort, it was not possible to identify the relevant degrees of freedom: there is neither a convincing case that hadronic physics alone is responsible for the observed phenomena, nor have quark models demonstrated superiority in explaining the data. Strong interaction physics, in this energy regime, remains puzzling and resists simple explanations.

The largest uncertainty in the predictions of the potential models stems from annihilation and the contribution of annihilation to the real part of potential. To reproduce the main features of the cross-sections, global isospin-independent Wood-Saxon potentials can be assumed, and the annihilation potential is “black” carrying no characteristic features. The success of black-sphere models, where the incoming wave vanishes at a boundary, underlines both the importance of annihilation and our ignorance about the short-range part of $\bar{N}N$ interactions. Still, the use of these simple models allowed to extract important physical information from the data, like the interaction ranges: charge exchange takes place preferentially at 2.5 fm, the mean radius for strong interaction is 1.5 fm and 1 fm for annihilation.

When the full variety of the existing data set (which includes accurate differential cross-sections and analysing power at many energies) is considered, much more sophisticated models are needed, with typically 30 or more free parameters. In these models the annihilation is no longer black, a spin, isospin, and energy dependence develops, and precise agreement with the data can be obtained only at the expense of simplicity. Understanding the annihilation process therefore is the key for a deeper insight into this realm of physics. In $\bar{N}N$ scattering and protonium only the global properties of annihilation could be determined, in the language of potential models. In the annihilation process three quarks and three anti-quarks may annihilate on each other, rearrange and may be recreated to form outgoing $q\bar{q}$ mesons. This process must be understood at the quark level; it is hard to imagine that a baryon-exchange picture can be applied in a region in which quark wave functions overlap and direct colour-colour interactions must come into the play. A takeover of quark physics, of the colour degrees of freedom, may be the reason why quasinuclear bound states were not found, in spite of considerable experimental efforts.

A prerequisite to further progress in this field is new data. The advent of LEAR has stimulated the work we have summarised in this report. New antiproton facilities are needed to stimulate experimental and theoretical work and try to find the answers to the many questions still left open. In particular, more experiments should be done to extend the set of observables and the energy range covered by LEAR. As underlined several times in this review, one needs further tests of our current ideas on the $\bar{N}N$ interaction. In particular, spin transfer and spin correlation measurements are necessary; for this purpose, an important goal is to produce polarised antiproton and antineutron beams. Already now, accurate measurements of $\bar{N}N$ scattering at very low energy could be done at the CERN AD. Super-LEAR type facilities, with low and medium-energy antiprotons are presently being proposed in Japan (JHF) and in Europe (GSI project), and on a longer time scale could allow to resume the study of this fascinating chapter of hadronic physics.
Acknowledgements

We would like to thank all the colleagues who shared our enthusiasm about this physics and with whom we had the pleasure to collaborate and interact over the past 20 years, in particular C.B. Dover, T.E.O. Ericson, L. Montanet, I.S. Shapiro, and J.J. de Swart. We also benefited from fruitful discussions with Mary Alberg, C. Amsler, Chr. Batty, T. Bressani, P. Dalpiaz, D. Gotta, D. Herzog, I. Iazzi, T. Johannson, K. Kilian, H. Koch, M. Kretschmar, F. Myhrer, K. Paschke, L. Simons, T. Shibata, J. Speth, R.G.E. Timmermans, R. Vinh Mau, Th. Walcher, and R.E. Welsh. Very special thanks are also due to all our collaborators in Grenoble, Mainz, and Trieste.

Several of our colleagues passed away during recent years. It is impossible to review this physics without remembering their important role and outstanding contributions. We would like to mention in particular G. Alberi, B. Deutsch, C.B. Dover, N. Hamann, R. Hess, K. Holinde, H. Poth and I.S. Shapiro.

One of the authors (J.-M. R.) gratefully acknowledges the generous support of the Alexander von Humboldt Stiftung and the enjoyable hospitality at the University of Bonn.

Note added in proof

While this review was already finished, we received some new results from the collaboration PS185, whose last run at LEAR made use of a polarised proton target. The depolarisation $D_{nn}$ and the spin transfer $K_{nn}$ are shown in Fig. 6.32. Besides these parameters, several other observables have been measured, leading to a reconstruction of the basic products of amplitudes and thus to an estimate of all observables. Details can be found in a thesis by K. Paschke at the Carnegie Mellon University (2001) and in a forthcoming article of the PS185 collaboration. These data will allow for extremely stringent tests of the current models.

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