# Antinucleon nucleon interaction at low energy scattering and protonium

E. Klempt, J.-M. Richard, F. Bradamante, A. Martin

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E. Klempt, J.-M. Richard, F. Bradamante, A. Martin. Antinucleon nucleon interaction at low energy scattering and protonium. Physics Reports, 2002, 368, pp.119-316. 10.1016/S0370-1573(02)00144-8. in2p3-00011702

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Distributed under a Creative Commons Attribution 4.0 International License
Antinucleon–nucleon interaction at low energy:
scattering and protonium

Eberhard Klempt$^a$, Franco Bradamante$^b$, Anna Martin$^b$, Jean-Marc Richard$^{c,d,*}$

$^a$Institut für Strahlen- und Kernphysik der Rheinischen Friedrich-Wilhelms Universität, Nußallee 14-16,
D-53115 Bonn, Germany

$^b$INFN and University of Trieste, Via A. Valerio 2, 34127 Trieste, Italy

$^c$Institut des Sciences Nucléaires, Université Joseph Fourier-CNRS-IN2P3, 53 avenue des Martyrs,
F-38026 Grenoble Cedex, France

$^d$Institut de Physique Nucléaire, Université Claude Bernard-CNRS-IN2P3, 4 rue Enrico Fermi,
F-696222 Villeurbanne Cedex, France

We present nu cleon–antinu cleon scattering experiments performed at the Low Energy Antiproton Ring (LEAR) of CERN. The data are reviewed and the underlying physics is discussed, in particular by comparison with the predictions of cu rrent models based on meson exchange and short-range absorption. A detailed description is given of protonium, which gives information on the interaction at zero energy and is the initial state when annihilation occu rs at rest.

# 1. Introduction

1.1. From the Dirac equation to the discovery of the antiproton

The history of antimatter is well known and hardly needs to be presented in detail. Several
interesting books have been written on the birth and the development of particle physics, with many
entertaining anecdotes. One may quote here Pais [1], Segrè [2], Cahn and Goldhaber [3], Lederman
[4], etc.

We wish to remind only a few points. First, the antiparticle associated with the electron was
thought to be possibly the proton. The large mass of the proton, as compared to that of the electron,
could have been due to its carrying the inertia of the Dirac sea. In modern particle physics, we are
accustomed to symmetry breaking phenomena of such large magnitude. A serious problem with this
hypothesis, however, was to understand how the hydrogen atom could survive internal annihilation.
Of course, the discovery of the positron by Anderson in 1932 [5] clarified the situation.

A second problem arose after the measurement of the magnetic moment of the proton by O. Stern
in 1933, with the result μ ≃ 2.8μ$_{N}$, where μ$_{N}$=eħ/(2m$_{p}$) is the value expected for a Dirac particle [3].
It was then not completely clear whether the proton possesses an associated antiparticle, since it does
not obey the Dirac equation strictly. The answer was given by the famous experiment by Chamberlain

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jean-marc.richard@isn.in2p3.fr (J.-M. Richard).</footnote>
et al. at Berkeley in 1955 [6]. We have eventually understood that the abnormal magnetic moment of the proton arises from its composite nature, and that particle-antiparticle symmetry is more general than a peculiar property of the Dirac equation.

Another surprise came with the first measurement of antiproton cross-sections [7]. The annihilation part is much larger than the elastic one for low antiproton momenta. We shall often return to this observation along this review. Let us briefly say here that this large inelastic cross-section is another consequence of the composite structure of the proton, i.e., its finite size.

## 1.2. Antiproton beams and facilities

In the early experiments performed at Berkeley, BNL, CERN or KEK, antiprotons were used just after being produced, in the form of secondary beams with low intensity, ill-defined energy, and a large contamination by negatively charged mesons. It was only in the last 20 years that new devices were elaborated to provide antiproton beams with high purity, intensity and momentum resolution, at CERN and at Fermilab.

At CERN, in 1968, Simon van der Meer proposed the “stochastic cooling technique” to improve beam quality and intensity in storage rings, and in the following years experiments (ICE, initial cooling experiment) demonstrated its feasibility. In 1978, the decision to go ahead with the Antiproton Project was taken at CERN, and an antiproton accumulator (AA) was built to apply stochastic cooling to antiprotons. The AA started up in 1980, and in 1981 antiprotons were sent to the ISR (Intersecting Storage Ring, dismounted in 1983) and soon afterwards to the Super Proton Synchrotron (SPS), converted into a proton-antiproton collider (the “SppS” collider). The p̅ beam energy was first 270 GeV, and then increased up to 315 GeV. The collider experiments at CERN were stopped in 1990.

In 1982, the Low Energy Antiproton Ring (LEAR) was launched for completion. This new machine, which will be described in some detail in Section 2, provided very high-quality antiprotons beams with momenta between 100 MeV/c and 2 GeV/c in the period 1983–1996, when it was stopped.

An proton-antiproton collider with centre-of-mass energy $s^{1/2}$ in the range from about 2 to 8 GeV, SuperLEAR, was proposed at CERN, mainly to study CP violation with $\bar{\Lambda}\Lambda$ systems and heavy quark spectroscopy. The proposal was discussed by the CERN-SPSLC committee, but not recommended for approval. Presently, at CERN the only p̅ facility is antiproton decelerator (AD); it provides antiprotons with momentum from 300 to 100 MeV/c, without slow extraction.

Building on the CERN innovation and experiences, Fermilab constructed an antiproton source. The first pp collider run began late in 1986; with its centre-of-mass energy of 1.8 TeV (900 GeV on 900 GeV) it is today the world's highest-energy accelerator. A crucial parameter is the luminosity, and an upgrade of the antiproton source and of the Main Injector has been done recently. Fermilab has today the world's highest-intensity source (the production rate is $10^{11}$ p̅/h) and work is going on for further improvement. Up to November 2000, precision experiments using p̅ could be performed by putting a hydrogen gas target in the Accumulator, where the antiprotons have about 8 GeV kinetic energy: this medium-energy programme is now terminated, and for the next 5 years the antiprotons will only be used for the Tevatron Run II.

In the future (very likely only after Run IIB), it is possible that a new, small, low-energy p̅ facility will be built at Fermilab. The motivations for such a facility come both from nuclear and particle physics, and from long-term future technical projects, like pp-annihilation-fuelled interstellar travel
[8,9]. Both a low-energy ring (from 2 GeV/c down to a few tens of MeV/c) and a larger storage ring (1–10 GeV/c) are being considered.

Two other projects are presently being pursued. The first in Japan, where the Japan Hadron Facility of KEK and the Neutron Science Project of JAERI (Japan Atomic Energy Research Institute) have been merged to one project: the JAERI-KEK Joint Project for high-intensity proton-accelerator facility. The Project was approved for construction in December 2000 by the Japanese Government. The accelerator for the joint project comprises a 15 μA, 50 GeV proton synchrotron, to explore a broad range of topics in nuclear and particle physics [10,11]. Secondary beams of antiprotons will have the highest intensity in the world in this energy domain, and ideas to store them in LEAR-like facilities will be pursued in the long term.

The second project has been proposed at GSI [12], as part of the hadron facility. It is based on a new accelerator ring, using the present SIS18 accelerator as injector, but with a rigidity more than an order of magnitude larger (protons will be accelerated to 50 GeV/c). The facility will comprise ring(s) to accumulate, store and cool intense, highest-quality primary and secondary beams, from short-lived exotic nuclei to antiprotons, for research in hadronic, nuclear, atomic and plasma physics.

### 1.3. Physics with antiprotons, an overview

The physics directly or indirectly associated with antiproton beams covers an enormous field. Already mentioned are particle-antiparticle symmetry, and the $\bar{p}p$ cross-sections, whose low-energy behaviour is discussed in Section 4.

Antiproton-proton colliders opened a new era in high-energy physics, reaching the TeV range. At these high energies, there is a considerable interest in studying the diffraction peak and the energy dependence of the cross-sections. Colliders are more famous, however, for their contribution to the physics of intermediate bosons, jets or heavy flavours. The $W^\pm$ and $Z^0$ bosons were discovered at the CERN collider and the top quark at the Fermilab Tevatron, where the Higgs boson is presently searched for actively.

The quality of the new antiproton beams motivated experiments which were not conceivable earlier. In the CERN ISR, antiprotons interacted with a thin hydrogen jet target to form charmonium mesons. This experiment [13,14], later resumed at the Fermilab accumulator [15], gave very accurate information on heavy quark-antiquark bound states.

Many experiments have been performed at LEAR. They dealt with many facets of physics. Relevant to this review are experiments which measured $\bar{p}p$ and $\bar{n}p$ total, elastic, and annihilation cross-sections, and also $\bar{p}p$ charge-exchange and strangeness-exchange reactions. In many instances, precise differential cross-sections were first measured, as well as some spin parameters.

The first results at LEAR were obtained in experiments where antiprotons were scattered on nuclei. The results on elastic and inelastic cross-sections stimulated many theoretical papers [16,17]. Annihilation on nuclei was compared with $\bar{N}N$ annihilation [18], and heavy hypernuclei were produced by annihilation of antiprotons on heavy nuclei [19]. More recent experiments deal with fission and multifragmentation induced by antiprotons, and annihilation on peripheral neutrons [20].

Strong interactions at zero energy were studied with antiprotonic atoms. The topics involve some atomic physics to understand how antiprotons are captured and cascade down to low-lying orbits where they experience strong interactions. New metastable states of the $(He, \bar{p}, e^-)$ system have been
found [21] and will be further studied [22]. Let us also mention that antiprotons can be used for some solid-state experiments, where they are sometimes more appropriate than muons.

A large fraction of the LEAR experimental programme was devoted to spectroscopy, in particular to the search of exotic meson states (qqq̄q baryonium states, qqg hybrids, gg glueballs) produced in the $\bar{N}N$ annihilation process, both at rest and in flight. The structure of the proton in the time-like region was probed by measuring the rare annihilation channel $p\bar{p} \to e^+e^-.$

Fundamental symmetries were tested, with the comparison of the inertial mass of the antiproton with that of the proton, as a test of CPT, and the investigation of CP asymmetries in the decay of neutral kaons by the so-called CPLEAR collaboration [23]. New measurements of inertial masses and gravitational tests are planned at the new AD facility of CERN [24]. The comparison of hydrogen and antihydrogen atomic spectroscopy will also probe symmetries with high accuracy.

The role of antimatter in astrophysics remains controversial. Some early cosmological models suggested that antimatter was as abundant as matter in the Universe. Modern cosmology tends to favour scenarios where antimatter has disappeared, thanks to baryon number violation, CP violation and absence of thermodynamical equilibrium in the early Universe. Antimatter had already disappeared when nucleosynthesis began: otherwise the present abundance of $^3$He should be much larger, due to antiproton annihilation on $^4$He, which yields $^3$He with high probability, as shown by the experiment PS179 at LEAR [25]. For a recent discussion, see, e.g., [26]. Still, antiprotons and antinuclei are searched for in cosmic rays, perhaps as a result of dark matter interaction with galaxies. A new generation of balloon, satellite or space-station experiments has started [27,28].

## 1.4. Nucleon-antinucleon interaction at low energy

In this review, we shall concentrate on the aspects dealing with strong interaction physics at low energy. This includes long- and short-range $\bar{N}N$ forces, the possibility of $\bar{N}N$ bound states or resonances and the physics of protonium atoms. Both experimental and theoretical aspects will be discussed.

The challenge of antiproton scattering and annihilation at low energy consists of combining long-range physics, efficiently described by conventional meson-exchanges, and short-range physics, where direct interaction between quarks and antiquarks presumably takes place.

The situation is similar to the one encountered in NN physics, where long-range meson exchanges have to be supplemented by phenomenological hard cores, which still await a satisfactory description in terms of quark dynamics. Sophisticated potential models have been constructed (Paris, Bonn, Nijmegen, etc.), which summarise our theoretical knowledge of meson exchanges. The short-range parts of the potentials have been parametrised on the data, and the overall result is that predictions are in excellent agreement with the impressive NN data base.

Investigation of the $\bar{N}N$ sector allows to test on a new system the basic approach to NN physics (necessity of all exchanges, values of the coupling constants, symmetry requirements), in the presence of a short-range dynamics (annihilation) which plays a much more important role than in NN physics, and which is considerably more difficult to describe than the NN core repulsion.

The present approach to $\bar{N}N$ interaction is somewhat hybrid. The long-range part is described in terms of meson exchanges, the same as those used in the meson theory of nucleon-nucleon (NN) forces. The short-range part, and in particular the strong annihilation component is either mocked up by an ad hoc imaginary potential or tentatively described in terms of quarks. Thus, the
phenomenological analyses aim at measuring the role of each part of the interaction. The theory
should provide a unified picture of long- and short-range forces. In spite of the inherent difficulties
of the problem (too many partial waves, complex phase-shifts), a phase-shift analysis of $\bar{N}N$ data
has been carried out. It will be reviewed in Sections 3 and 6.

Optical-potential models incorporate our best knowledge of Yukawa forces, but describe annihi-
lation empirically, and thus cannot provide much insight on the microscopic mechanisms operat-
ing at short distances. On the other hand, considerations based on flavour symmetry or on simple
quark diagrams suggest relations between branching ratios for annihilation into various channels.
To test these relations, one should take into account the probability of finding N and $\bar{N}$ overlap-
ping. This probability, which is likely to vary with spin and isospin, depends on the long-range
dynamics.

$\bar{N}N$ physics is rich of many facets which are complementary and intimately connected. To extract the physics, one better applies some filters. For instance, some specific spin observables enhance the role of long-range forces, while in cross-sections short-range dynamics plays a more important role.

When comparing the long-range NN and $\bar{N}N$ potentials, one notices a much stronger attraction
in the latter case, at least in some channels (the $\bar{N}N$ potential turns out to depend on spin and
isospin significantly). This led Shapiro and his collaborators [29] to speculate about the possibility
of deuterium-like bound states and resonances of the $\bar{N}N$ system. A comprehensive review was given
some years ago by Buck et al. [30].

The existence of such states, sometimes called quasi-nuclear bound states or resonances, or bary-
onia, heavily relies on the hope that annihilation does not wash out the wave function built by the at-
tractive elastic potential. The range of annihilation is thus a crucial quantity: simple baryon-exchange
diagrams suggest a very short range, but huge form-factor corrections have to be applied; in the
quark approach, the range is governed by the size of the quark distributions of the incoming nucleons
and outgoing mesons, and turns out to be about 1 fm.

1.5. *A guide to the related literature*

The most direct and precise information on low-energy antiproton physics can be found in the
Proceedings of the specialised Conferences and Workshops.

The series of European Antiproton Conferences was begun by L. Montanet at Chexbres [31] and
continued regularly up to the Mainz conference [32–39].

The first LEAR Workshop was organised in 1979 at Karlsruhe [40] by Helmut Poth. Further
Workshops were held [41–43] to discuss the machine developments, the planned experiments, and
the interpretation of the results.

The European Symposia and the LEAR Workshops were merged in a new series of Low Energy
Antiproton Physics (LEAP) conferences. The first one has been organised at Stockholm [44], the
second in 1992 at Courmayeur, Italy [45], the third one in 1994, at Bled, Slovenia [21], the fourth
at Dinkelsbühl, Germany in 1996 [46], the fifth at Villasimius, Italy, in 1998 [47], and the sixth one
at Venice in 2000 [48]. Intermediate workshops were organised in Russia [49–51].

More pedagogical introductions can be found in the Proceedings of the Low Energy Antiproton
Schools organised at Erice irregularly [52–55]: the first one was devoted to fundamental symmetries,
the second to hadron spectroscopy, the third on $\bar{N}N$ and $\bar{p}$-nucleus interaction, and the fourth was
more oriented towards the physics issues of SuperLEAR.
The physics of SuperLEAR was discussed in particular at Tignes [42], and then presented in a document which was published in the Proceedings of a Workshop held at Les Houches [56]. A more updated and complete survey can be found in the Proceedings of the Zurich Workshop [57]. See also [58].

Finally, we shall also quote the reviews on $\bar{p}$ physics by Walcher [59], Amsler and Myhrer [60], Dover et al. [61], Amsler [62], and Eades and Hartmann [63].

## 1.6. *Outline*

This review is organized as follows. In Section 2, we present the antiproton beam available at CERN in the LEAR facility and discuss the set-up of the experiments mainly or partially devoted to study $\bar{N}N$ interaction. The formalism of elastic, charge-exchange and strangeness exchange $\bar{N}N$ scattering as well as protonium is presented in Section 3. The scattering data are presented and discussed in Section 4. Protonium formation and spectroscopy is the subject of Section 5. Section 6 contains a comparison between data and the most recent theoretical calculations. Conclusions are proposed in Section 7.

A second review article, presently in preparation, will be devoted to annihilation dynamics [64]. This overview of strong interaction physics at LEAR is split into two parts only for the sake of convenience, and for complying with the constraint of keeping a reasonable size. The physics of scattering and annihilation are, however, intimately correlated. Annihilation occurs from a protonium or $\bar{N}N$ scattering state which is strongly distorted by initial-state interaction. Also exotic mesons can be viewed either as composite of quarks, antiquarks and gluons or as hadron-hadron molecules. So a good knowledge of both quark-gluon dynamics and hadron-hadron interaction is required to analyse new meson states.

# 2. Beams and major experiments

In this section we sketch the technological achievements of LEAR with respect to conventional antiproton beams. The experiments at LEAR which investigated $\bar{N}N$ interactions are introduced and their design and performance are reviewed.

## 2.1. *The LEAR facility at CERN*

The possibility to store intense antiproton beams and to use them both in a high-energy proton-antiproton collider and at low energy, was put forward in 1976 by the Novosibirsk group [65,66], soon after the feasibility of both stochastic and electron cooling had been demonstrated. As apparent from Fig. 2.1, which shows the $\bar{p}$ yield (i.e., the number of $\bar{p}$'s which can be captured into a beam channel of a given acceptance) as a function of the momentum of the $\bar{p}$'s, the yield has a broad maximum at $3.5 \text{ GeV/c}$ (the incident proton momentum is $23 \text{ GeV/c}$, a typical CERN PS operation energy) and falls off by many orders of magnitude for lower momenta (dashed curve). On the other hand, if $3.5 \text{ GeV/c}$ antiprotons were stored in a ring, cooled to reduce phase-space, and then decelerated to the desired energy, one could dispose of the same high flux at all energies. Cooling
<figure><img src="image_2.png" /><figcaption>Fig. 2.1. Momentum spectrum of antiprotons produced at 0° with 23 GeV/c protons on a lead target. The number of $\bar{p}$ per interacting proton is normalised to 1 msr solid angle and ±1% momentum bite.</figcaption></figure>

is an essential part in the process: if the stored beam is decelerated without any cooling, the density decreases due to the adiabatic increase of the emittance.

The LEAR project was approved at CERN in 1980, and in July 1983 first beams were delivered to the users. Fig. 2.2 shows a schematic layout of the PS accelerator. The particles created by the extracted PS beam hitting the production target (typically a 3 cm diameter, 10 cm long metal rod through which a current of up to 200 kA is pulsed to focus the produced $\bar{p}$'s) are stored in the antiproton accumulator (AA), at a rate of $6 \times 10^6$ per pulse. In the AA, the $\bar{p}$'s are stochastically cooled and stacked into an intense stored beam, typically $3 \times 10^{11}$ $\bar{p}$'s, with vertical and horizontal emittance of about $2\pi$, from which spills of $1 \times 3 \times 10^9$ $\bar{p}$'s are extracted and sent to the CERN PS, where they are decelerated at 609 MeV/c and transferred to LEAR.

Under the strong pressure to increase the $\bar{p}$ flux to the SPS $\bar{p}p$ Collider program, the AA complex was stopped in 1987 to construct a new Antiproton Collector (ACOL) which allowed to separate the $\bar{p}$ collection function from the $\bar{p}$ stacking, still performed in AA. The use of two rings and important improvements of the antiproton production target system resulted in a net flux gain of a factor 10. The new AA-ACOL complex was operational at the end of 1987: about $6 \times 10^{10}$ $\bar{p}$'s could be accumulated per hour, in stacks of about $10^{12}$ $\bar{p}$'s. All together, the $\bar{p}$ CERN complex has performed remarkably well, as testified by the LEAR running statistics shown in Fig. 2.3. Up to 1991 LEAR was operated in parallel with the SPS $\bar{p}p$ Collider, but since 1992 LEAR was the only user. The program was terminated at the end of 1996.

A very important feature of LEAR was the ultra-slow extraction system, which allowed the users to dispose of essentially DC beams. The extraction system could provide continuous spills of up
<figure><img src="image_3.png" /><figcaption>Fig. 2.2. Layout of the PS accelerator complex.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2.3. LEAR running statistics. The major gain occurring in 1988 is due to the Antiproton Collector entering into operation.</figcaption></figure>

to 15 h, i.e., $10^3$ times longer than the spills obtained in previous machines, corresponding to less than one particle per turn (on average) leaving the machine. This system was invented for LEAR [67], and is still regarded as the most innovative contribution of LEAR to accelerator physics and technology.
Other features worth mentioning are:

* beams were provided between 105 and 2000 MeV/c momentum;

* both stochastic and electron cooling were used;

* provisions were made for internal targets;

* three (later four) beam splitters on the extracted p̄ beam line allowed to deliver antiprotons simultaneously to four (five) experiments.

Fig. 2.4 shows the arrangement of the LEAR ring in the South Hall of the CERN PS, the extracted beam lines, and the locations of the experiments in the year 1983.

## 2.2. Physics motivation of the LEAR scattering experiments

In the 1970s, several $(qq\bar{q}\bar{q})$ baryonium candidates had been observed, with masses either below the $\bar{N}N$ threshold or above. At the time when LEAR was proposed and constructed the scientific community had a strong interest in studying p̄p scattering in the region of incident p̄ momenta around 500 MeV/c, where many experiments had reported the presence of a narrow state, the S(1936). This state was observed as a bump over a smooth background in the p̄p total and annihilation cross-section. The observation came both from counter experiments and from bubble chamber measurements. For a review on the S(1936) and other baryonium candidates in the pre-LEAR era see, e.g., Ref. [68].

At LEAR several experiments were proposed to study baryonium states of mass smaller than two-nucleon mass, by spectroscopy studies of p̄p and p̄d annihilation at rest. In a complementary way, a number of experiments were proposed to study the formation of baryonium in p̄p scattering in the entire momentum range.

The first scattering experiments to be proposed and carried out (PS172 and PS173) did energy scans of the integrated cross-sections for momenta smaller than 600 MeV/c to confirm the existence of the S-meson, measure its width and formation cross-section, and possibly identify new states. As it is well known, far from confirming the original observations, these two experiments provided conclusive evidence against the existence of the S-meson. It should be remembered, however, that shortly before LEAR entered into operations the evidence for this state was questioned, in experiments at BNL and KEK.<fnref n="1" />

Confirming the S-meson was only one point in the experimental programme of the scattering experiments. Independent of the existence of narrow baryonium states, a rich spectrum of baryonium resonances, with typical strong-interaction width of about 100 MeV, was expected to exist and several possible candidates had been observed in the momentum region from 1 to 2 GeV/c. These observations were done by studying the two-meson annihilation channels p̄p → π⁻π⁺ and p̄p → K⁻K⁺. The scattering matrix of these reactions depends on only two complex amplitudes, which can be reconstructed in a phase-shift analysis if the differential cross-section and the analysing power are measured at a sufficient number of energies. This programme was proposed by PS172 at LEAR in the momentum interval from about 500 to 1500 MeV/c (to overlap with the previous measurements).

<footnote n="1">¹ See, for instance, Ref. [69].</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 2.4. The LEAR extracted beam lines and the experimental areas in 1983, at the start-up of the physics programme.</figcaption></figure>
It was carried through successfully and subsequent amplitude analyses of the data have suggested a number of high-spin resonances.

In parallel to $\bar{p}p \to \pi^-\pi^+$ and $\bar{p}p \to K^-K^+$, PS172 has measured the elastic channel $\bar{p}p \to \bar{p}p$: a comparison of the coupling of the resonances to $\bar{N}N$ and to the mesons was regarded as necessary to access the nature of the state, since $(qq\bar{q}\bar{q})$ baryonium states were expected to couple more strongly to $\bar{N}N$ than to multi-meson channels. For a correct isospin-value assignment, a similar investigation of the $\bar{p}p \to \bar{n}n$ charge-exchange channel was proposed (experiment PS199, measurement of differential cross-section and analysing power in the momentum range 500 to 1300 MeV/c), approved, and performed.

The search of baryonium states was undoubtedly the main motivation of the scattering experiments at LEAR. Later, annihilation channels as a function of the $\bar{p}$ momentum were studied in a comprehensive way by the Crystal Barrel Collaboration. A large number of meson resonances were identified. For none of these resonances a particularly strong coupling to $\bar{N}N$ was reported.

A second strong motivation for the scattering experiments was the understanding of the $\bar{N}N$ reaction dynamics and its comparison with the known NN interaction. As discussed in detail in this review, potential models, based on well-known meson exchanges and supplemented by phenomenological description of annihilation, resulted in a number of definite predictions for cross-sections, spin observables, and initial-state interaction in annihilation processes. Many experiments, PS172, PS173, PS198, PS199, either had the study of $\bar{N}N$ reaction dynamics as important part of their programme, or were fully dedicated to it. A special mention has to be made of the PS185 experiment, devoted to the study of the dynamics of strangeness production. The threshold for $\bar{p}p \to \bar{A}A$ is at 1.435 GeV/c. The Lorentz boost provided by the large $\bar{p}$ incident momentum makes the study of these reactions near to threshold an interesting experimental problem.

The use of polarised targets and of the intense $\bar{p}$ beam of LEAR allowed to detect large spin effects in PS172, PS198 and PS199, and new proposals were put forward. At the Cogne meeting in 1990 [70], these proposals were not approved by the CERN Committee, and the investigation of the $\bar{N}N$ reaction dynamics was stopped at LEAR.

With two exceptions. Experiment PS206 was approved to measure the $\pi NN$ coupling constant from a precision measurement of the differential cross-section of $\bar{p}p \to \bar{n}n$. Experiment PS201 (OBELIX), whose main objectives were spectroscopy studies, could use the general-purpose apparatus to obtain a variety of cross-section data, in different channels, particularly at very low energy.

## 2.3. Description of the antiproton scattering experiments

Given the momentum range of interest, all the proposed experiments consisted of detectors typical of particle physics (scintillation counters, multi-wire proportional chambers, drift chambers, limited streamer tubes, etc.), and in spite of the capability of LEAR of delivering antiprotons with very low momentum, no dedicated experiment was proposed to explore $\bar{N}N$ scattering at momenta smaller than 200 MeV/c. The FILTEX Collaboration considered [71] the possibility of exploring this low-energy region, by using a polarised atomic hydrogen gas target in the LEAR ring, but the project was not encouraged [70]. In the ACOL era, only the PS201 experiment (a Bologna-Brescia-Cagliari-Dubna-Frascati-Legnaro-Padua-Pavia-Turin-Trieste-Udine collaboration, about 100 physicists) has measured cross-sections down to about 1 MeV $\bar{p}$ energy, using special techniques, as will be described in Section 2.3.3.
<table><caption>Table 2.1
Survey of low-energy antiproton and antineutron cross-section and scattering measurements at LEAR</caption><thead><tr><th>Measurement</th><th>Incoming p̄ momenta (MeV/c)</th><th>Experiment</th></tr></thead><tbody><tr><td colspan="3"><em>Integrated cross-sections</em></td></tr><tr><td>σ<sub>tot</sub> (p̄p)</td><td>222–599 (74 momenta)<br>181, 219, 239, 261, 287, 505, 590</td><td>PS172<br>PS173</td></tr><tr><td>σ<sub>ann</sub> (p̄p)</td><td>177–588 (53 momenta)<br>38 to 174 (14 momenta)</td><td>PS173<br>PS201</td></tr><tr><td>σ<sub>tot</sub> (n̄p)</td><td>80–293 (18 momenta)</td><td>PS201</td></tr><tr><td>σ<sub>ann</sub> (n̄p)</td><td>50–400 (18 momenta)</td><td>PS201</td></tr><tr><td colspan="3"><em>p̄p elastic scattering</em></td></tr><tr><td>ρ</td><td>233, 272, 550, 757, 1077<br>181, 219, 239, 261, 287, 505, 590</td><td>PS172<br>PS173</td></tr><tr><td>dσ/dΩ</td><td>679–1550 (13 momenta)<br>181, 287, 505, 590<br>439, 544, 697</td><td>PS172<br>PS173<br>PS198</td></tr><tr><td>A<sub>0n</sub></td><td>497–1550 (15 momenta)<br>439, 544, 697</td><td>PS172<br>PS198</td></tr><tr><td>D<sub>0n0n</sub></td><td>679–1501 (10 momenta, 23 points)</td><td>PS172</td></tr><tr><td colspan="3"><em>p̄p charge-exchange</em></td></tr><tr><td>dσ/dΩ</td><td>181–595 (several momenta)<br>546, 656, 693, 767, 875, 1083, 1186, 1287<br>601.5, 1202</td><td>PS173<br>PS199<br>PS206</td></tr><tr><td>A<sub>0n</sub></td><td>546, 656, 767, 875, 979, 1083, 1186, 1287</td><td>PS199</td></tr><tr><td>D<sub>0n0n</sub></td><td>546, 875</td><td>PS199</td></tr><tr><td colspan="3"><em>p̄p → ȲY</em></td></tr><tr><td>Several observables</td><td>1424–1922 (several momenta)</td><td>PS185</td></tr></tbody></table>

Six experiments were specifically devoted to $\bar{N}N$ scattering:

* PS172 (SING), an Amsterdam–Geneva–Queen Mary College–Surrey–Trieste collaboration (about 20 physicists);

* PS173, a Heidelberg–Laval–Mainz–Rutgers collaboration (about 15 physicists);

* PS185, a Carnegie Mellon–CERN–Erlangen–Freiburg–Urbana–Jülich–Uppsala–Vienna collaboration (about 25 physicists);

* PS198, a Karlsruhe–Lyon–Saclay–PSI Villigen collaboration (about 25 physicists);

* PS199 (POLCEX), a Cagliari–CERN–Geneva–Saclay–Trieste–Turin collaboration (about 35 physicists);

* PS206 (CEX), a Cagliari–CERN–Geneva–Saclay–Trieste–Turin collaboration (about 30 physicists).

The PS172 and PS173 experiments were designed, constructed, and operated in the pre-ACOL era of LEAR, the other ones ran with ACOL. PS185 took data over the full LEAR lifetime. Table 2.1 shows a survey of the data taken by the various experiments.
<figure><img src="image_6.png" /><figcaption>Fig. 2.5. Layout of the C2 beam line: Q denotes quadrupoles, D bending magnets and $C_H$ and $C_V$ collimators in the horizontal and vertical planes. The first and second carbon scatterers are located at the two focal points $F_1$ and $F_2$.</figcaption></figure>

From Table 2.1 it should be apparent that the experiments at LEAR could perform only an exploratory work and no systematic study of the $\bar{N}N$ dynamics. In particular, proposals to polarise the LEAR $\bar{p}$ beam, either by filtering one spin component by recirculation through a polarised jet target (FILTEX), or by a coherent Stern–Gerlach mechanism using a “spin-splitter” [72–74], a combination of two quadrupoles separated by a solenoidal magnet, were not accepted.

Scattering data have been obtained also by the experiment E760 (collaboration Irvine–Fermilab–Ferrara–Genoa–Northwestern University–Penn State University–Turin) at the Fermilab Antiproton Accumulator Ring.

### 2.3.1. *Experiment PS172*

Experiment PS172 was designed to perform several different measurements, both with a liquid hydrogen target and a polarised proton target. For most of the measurements an energy scan was foreseen, which was done by setting LEAR to the desired energies. This procedure was inconvenient for the total cross-section measurement, because of the number of steps required for a fine scan. For these reasons, only a few energies were selected within the range of the scan and the beam was slowed down with a suitably designed carbon degrader.

*The beam line and the measurement of $A_{\bar{p}C}$:* The C2 beam line was designed according to the needs of the measurements of the total cross-section and of the analysing power in $\bar{p}C$ elastic scattering ($A_{\bar{p}C}$).

The layout of the beam line is shown in Fig. 2.5. There, $F_1$ and $F_2$ give the positions of the first and second focal points. The layout was symmetric between $F_1$ and $F_2$ (where the experimental apparatus was located), so the magnification was 1. Momentum analysis was performed by the horizontal collimator $C_H$ put at the intermediate focus. The main characteristics of the beam are:

* large angular acceptance, $\pm 18$ mrad horizontally and $\pm 36$ mrad vertically, giving a good intensity of the degraded beams;

* good momentum resolution, $(\Delta p/p \simeq \pm 1\%)$, necessary to reject antiprotons inelastically scattered off carbon;

* achromatism, to minimise the beam spot, obtained with the use of the quadrupole $Q_4$ in between the two momentum-analysing bending magnets $D_3$ and $D_4$.

In order to perform the measurement of $A_{\bar{p}C}$, the first two bending magnets $D_1$ and $D_2$ gave the possibility of sweeping the extracted LEAR beam in the horizontal plane. This option was used to measure $A_{\bar{p}C}$, by hitting the carbon scatterer in $F_1$ at an angle varying from $-9^\circ$ to $+9^\circ$. The large
<figure><img src="image_7.png" /><figcaption>Fig. 2.6. PS172: Schematic side view of the set-up used for the measurement of the $\bar{p}p$ total cross-section.</figcaption></figure>

angular acceptance of the beam line guaranteed a reasonable intensity for the scattered beams. The beam at $F_1$ was monitored continuously using a MWPC with 1 mm wire spacing and a scintillation counter B0.

The analysing power $A_{\bar{p}C}$ was measured in order to assess the possibility of polarising antiprotons by scattering off light nuclei. A large $A_{\bar{p}C}$ would have allowed to set up a polarised $\bar{p}$ beam in analogy with what is routinely done with protons. Unfortunately, the value of $A_{\bar{p}C}$ [75,76] turned out to be too small.

**$\sigma_{\text{tot}}(\bar{p}p)$ measurements:** PS172 measured the $\bar{p}p$ total cross-section using the traditional transmission technique at 29 $\bar{p}$ momenta between 388 and 596.8 MeV/c [77] and at 45 $\bar{p}$ momenta between 221.9 and 413.2 MeV/c [78], in few days of data taking. The different momenta were obtained using extracted $\bar{p}$ beams of 388, 599, 352, and 432 MeV/c, and carbon degraders of different thicknesses in F1.

The experimental set-up for the higher momenta measurements is shown schematically in Fig. 2.6; for the lower-momentum measurements some small changes were introduced.

The incoming beam was defined by the coincidence B of the signals from three thin (0.5 mm) scintillation counters, B0, B1 and B2 (only B1 is shown in the figure). B2 had a diameter of 1 cm and was placed in front of the target. B0 was placed in front of the degrader at the first focus $F_1$, 20 m upstream of the target. The time-of-flight between B0 and B1 eliminated the small (less than 0.1%) contamination of pions (and decay muons) produced in the degrader. The beam focusing was continuously monitored by two multi-wire chambers PC1 and PC2.

A liquid hydrogen target (LHT) was used, which was emptied for background measurements. The target consisted of two cells (the first one had $8.33 \pm 0.04$ cm length and 3.5 cm diameter, the second one $1.17 \pm 0.03$ cm length and 5 cm diameter), which could be filled and emptied independently. The temperature was continuously monitored.

A box of scintillation counters S1-S5 surrounded the target except for holes at the top and for the entrance and exit beam windows. The box covered 90% of the solid angle around the target.

The transmitted beam was measured in thin scintillation counters in air light guides: a circular one, T, and three overlapping concentric annular ones, A1-A3. The latter extended up to an external radius of 10.8 cm; rings were chosen to minimise $\bar{p}$ annihilation in the array. The signals from these detectors were added electronically to form a set of four transmission rates $B \cdot T_i$ ($T_1=T$, $T_2=T+A1$, etc.) corresponding to different maximum values of the momentum-transfer square $|t|$. The efficiency of counter T was monitored using the E1 and E2 counters.
The angular range between the backward hole of the target veto counters and the transmission array was covered by two annular counters (S6 and S7). The $S_i$ counters recorded predominantly events with annihilation into charged pions and the OR of the signals from all of them (S) was used to derive a second set of transmission rates $B \cdot \bar{S} \cdot T_i$, with a slope in $|t|$ dominated by elastic scattering.

At each incoming antiproton momentum, the two sets of transmission rates were corrected for accidental coincidences and vetoing, for energy losses in the targets, and for residual hydrogen gas present in the empty target. The partial cross-sections obtained from both sets were corrected for Coulomb-nuclear interference and single Coulomb scattering. The linear extrapolation to $|t|=0$ gave two measurements of the total cross-section which turned out to be in very good agreement.

The quoted normalisation error is ±0.7% at higher energies (long target), and ±0.9% at low energies (short target). At momenta below 285 MeV/c, the correction for straggling and nuclear attenuation in the apparatus became relevant and, at the lowest momenta, the error on the correction was larger than the statistical error.

*ρ measurements:* The ρ parameter defined as ratio of real to imaginary part of the elastic scattering amplitude was determined in dedicated measurements of the $\bar{p}p$ elastic scattering differential cross-section at small angles, in the Coulomb-nuclear interference region. Data were taken at five momenta (233, 272, 550, 757, and 1077 MeV/c) and at the target centre) under two different running conditions.

The lower energy measurements at 233 and at 272 MeV/c [79] were performed with a dedicated set-up, similar to that used for the $\sigma_{tot}$ measurements and shown in Fig. 2.6. For these data the set of annular transmission counters behind the target was replaced by a set of four multi-wire proportional chambers (PC3, PC4, PC5 and PC6). The “short” liquid hydrogen target was used. Scattered $\bar{p}$'s were detected by a counter (R), placed behind the last wire chamber. This counter was made up by four partly overlapping rectangular scintillation counters, and had a square hole of 6 × 6 cm² for the outgoing beam.

The trigger for elastic events consisted in a first level trigger (given by the coincidence B0·B1·R) and in a second level trigger, which consisted of a software cut on the calculated distance between the coordinates measured in the last multi-wire chamber and the nominal beam axis. After the cut, the acceptance for events with a scattering angle of 6° was larger than 80%, and the trigger rate was reduced by a factor of 10 at 233 MeV/c, giving a typical rate of 250 events per second. Events with trigger B0·B1 were also collected to determine the angular resolution for full and empty target runs and the angular acceptance of the elastic trigger.

In the analysis, the scattering angle was determined from the measured directions of the incoming and outgoing $\bar{p}$ as obtained from the two sets of multi-wire proportional chambers. Events due to $\bar{p}p$ annihilation in the liquid hydrogen target were rejected by identifying the pions using the time-of-flight between B1 and R. For the data at 233 MeV/c the time-of-flight between B1 and S1-S5 was also used to reject annihilation events.

The momentum spread of the incoming beam, determined from the B0-B1 time-of-flight spectra, was smaller than ±1.2 MeV/c. The average momentum loss in the liquid hydrogen target was 10 at 272 MeV/c and 14 at 233 MeV/c. Energy straggling calculations indicated that less than 0.5% of the antiprotons stopped before reaching the R-counters at both momenta, making negligible the effect on the measured angular distributions.
<figure><img src="image_8.png" /><figcaption>Fig. 2.7. PS172: Schematic top view of the set-up used for the measurement of small angle $\bar{p}p$ elastic scattering differential cross-section at 757 and 1077 MeV/c.</figcaption></figure>

At these two low $\bar{p}$ momenta, the experimental angular resolution was dominated by multiple scattering (25.7 mrad with target full at 272 MeV/c); to fit the measured differential cross-section a “folding” procedure had to be applied. The elastic $\bar{p}p$ differential cross-section was published in the range where the acceptance was higher than 70% (54 points with $1.37 \times 10^{-3} < -t < 6.60 \times 10^{-3}$ GeV$^2$ and 69 points with $1.07 \times 10^{-3} < -t < 8.98 \times 10^{-3}$ GeV$^2$, at 233 and 272 MeV/c, respectively); the statistical errors are less than 10%.

The data at 550 MeV/c, and those at 757 and 1077 MeV/c [80], were collected during two separate runs, under somewhat different experimental conditions, in parallel with the measurement of the analysing power in $\bar{p}C$ elastic scattering $A_{\bar{p}C}$. The geometrical arrangement of the detectors was optimised for the $A_{\bar{p}C}$ measurements; in particular the “scattered” $\bar{p}$ beam (either at 5° and 8°) was used. The carbon scatterer was 5.2 g/cm$^2$ thick and the scattered beam intensity was between 300 and 1000 $\bar{p}$/s, depending on angle and extracted beam momentum (600.8, 800 and 1100 MeV/c). The apparatus, shown in Fig. 2.7, was very similar to the one used for the low-energy measurements. It consisted of the liquid hydrogen target (LHT), telescopes of multi-wire proportional chambers (PCs in the figure), and scintillation counters. It allowed to measure scattering events on either the liquid hydrogen target, or on a carbon target (C), and to extract the (eventual) $\bar{p}$ beam polarisation from the azimuthal asymmetry of the events. The scintillator boxes S1–S5 and P1–P5 ensured the scattering process on both targets to be elastic. The events on which the scattering took place on the liquid hydrogen target were used to extract $\rho$ for the $\bar{p}p$ elastic scattering.

The differential cross-section was measured in the range $0.39 \times 10^{-3} < -t < 65.5 \times 10^{-3}$ GeV$^2$, $0.38 \times 10^{-3} < -t < 43.0 \times 10^{-3}$ GeV$^2$, and $0.60 \times 10^{-3} < -t < 80.7 \times 10^{-3}$ GeV$^2$, at 550, 757, and 1077 MeV/c, respectively. The corresponding total detection and reconstruction efficiencies were estimated to a few per cent to be 75%, 89%, and 76%, respectively. Also in this case, a “folding” procedure had to be applied to measure the $\rho$ parameter.

*Differential cross-section and analysing power in $\bar{p}p$ elastic scattering:* The most important physics objective of PS172 was an energy scan of the differential cross-section and the analysing power of the two-body annihilation channels $\bar{p}p \to \pi^-\pi^+$ and $\bar{p}p \to K^-K^+$, with the aim of revealing the existence of $\bar{p}p$ s-channel resonances. A pentanol polarised target was used. Data have been published for 20 momenta between 360 and 1550 MeV/c [81], and analysed in several papers (see e.g., Refs. [82–84]).

The same apparatus was also used to measure (in parallel) the same observables for the elastic $\bar{p}p$ reaction. The $A_{0n}$ data are given for 15 momenta, between 497 and 1550 MeV/c [85]. Differential
<figure><img src="image_9.png" /><figcaption>Fig. 2.8. PS172: Schematic side view of the set-up used for the measurement of the $\bar{p}p$ elastic scattering differential cross-section and analysing power.</figcaption></figure>

cross-section data for the $\bar{p}p \to \bar{p}p$ reaction are given at 13 momenta, down to 679 MeV/c [86]: at lower momenta absorption of the outgoing particles in the target material led to large systematic uncertainties and the results were not published.

The experimental apparatus is sketched in Fig. 2.8. The pentanol target was a cylinder, 3 cm long and 1 cm diameter, with a hydrogen content corresponding to 3.7 cm of liquid hydrogen. The transverse proton polarisation, typically 75±4%, was reversed every few hours. The target was placed in the nose of a cryostat, keeping the target temperature below 1 K, and in a locally homogeneous field of 2.5 T, provided by a C-shaped dipole magnet (M in the figure). A liquid hydrogen and “dummy” target, made of Teflon, were also used for absolute normalisation and to determine the background.

The beam signal was given by the coincidence of three scintillation counter (B0, S1, and S2 or S3), and the incoming particle trajectories were measured by two multi-wire proportional chambers (the J and C chambers).

The scattered and recoil particles were detected by the multi-wire proportional chambers J, C, and either R or L. The J and C chambers were operated in a high magnetic field region; their positions were chosen to optimise the measurement of the outgoing particle momenta from the deflection in the field of the dipole magnet. An array of trigger scintillation counters placed around the R and L chambers completed the set-up. The R and L chambers and the scintillation counters could be rotated around the magnet axis according to the beam deflection in the magnetic field.

The background due to scattering on quasi-free target nucleons was substantially reduced using only events with both recoil and scattered particle detected. In the PS172 case, this request reduced the geometrical acceptance both in the scattering angle $\vartheta_{cm}$ and in the azimuthal angle $\varphi$. The acceptance, given by the minimal energy needed by the final particles to traverse the target and the detectors, was $-0.28 < \cos\vartheta_{cm} < 0.28$ at 497 MeV/c and $-0.84 < \cos\vartheta_{cm} < 0.80$ at 1550 MeV/c; only events with $|\varphi| < 6^\circ$ and $15^\circ$ were used for the $d\sigma/d\Omega$ and $A_{0n}$ measurements, respectively.
Corrections above 10% had to be applied to take into account the background subtraction, the chamber inefficiencies, and the absorption of the secondary particles; all these points are relevant for the differential cross-section measurement. The quoted systematic errors are about 10% in the differential cross-section and 4.5% in the analysing power.

In parallel to the $A_{0n}$ measurements, the depolarisation parameter $D_{0n0n}$ was measured [87] using a standard polarimeter with a Carbon slab (shown in Fig. 2.8) as analyser for the proton polarisation. The polarimeter was built by a variable number (typically 6) of 1 cm thick C plates and seven MWPC (three upstream and four downstream of the carbon plates), each with horizontal and vertical wires.

The polarimeter was positioned to analyse the polarisation of recoil protons coming from $\bar{p}p$ elastic scattering in the polarised target, in the angular range in which the proton is detected by the J, C and R chambers.<fnref n="2" /> From the left-right asymmetries in pC elastic scattering, measured for two different orientations of the transverse spin of the polarised target protons, it is possible to extract the spin correlation parameter $D_{0n0n}$. This measurement was severely limited in statistics: the $\bar{p}$ was scattered in the backward hemisphere where the cross-section is small, and the measurement had to be performed in the 1 or 2 days allocated at each momentum for the measurement of $A_{0n}$ in $\bar{p}p \rightarrow \pi^{-}\pi^{+}, K^{-}K^{+}, \bar{p}p$.

### 2.3.2. Experiment PS173

The experiment PS173 was set up to measure with good precision cross-sections for antiproton-proton scattering in the low-energy domain. Total and the annihilation cross-sections, as well as differential elastic and charge-exchange cross-sections were determined in the range from 180 to 600 MeV/c.

The experiment used the same C2 beam line designed for PS172, and, during the runs, the apparatus was located in the focus F2 (see Fig. 2.5). The degrader placed in the first focus $F_1$ provided a fine momentum scan (in 5 and 10 MeV/c steps). The beam momentum was measured by time-of-flight between the $F_1$ focus and the beam detectors positioned at $F_2$.

Fig. 2.9 shows the projection of the apparatus in the horizontal plane. Details can be found in Ref. [88].

The beam impinged on a liquid-hydrogen target placed in the centre of the 1 m diameter vacuum tank with 1 cm thick aluminium walls. The incoming $\bar{p}$ was identified by a coincidence between a scintillation counter $F_1$ placed in the first focus, close to the degrader, and two thin (50 µm) scintillators placed in front of the target (SD and TD); the dimensions of SD and TD were chosen to define a beam with $\pm 1^\circ$ divergence and a 5 mm diameter spot size at the target position.

Two target cells of different thickness (a vertical cylinder, 2 cm in diameter, and a 7 mm thick planar disk, respectively) filled with liquid hydrogen were used, depending on the beam momentum. Data collected with empty targets were used for background measurements.

Two 3 mm thick scintillator detectors (BA1 and BA2), located on the beam axis downstream from the target, were used in some measurements to veto antiprotons that did not interact strongly in the target. These detectors were also used for the measurement of the total cross-section.

<footnote n="2">² The original idea of analysing the scattered $\bar{p}$ (which would have lead to the measurement of the spin transfer parameter $K_{n00n}$) was abandoned since the $\bar{p}$C elastic scattering analysing power turned out to be small.</footnote>
<figure><img src="image_10.png" /><figcaption>Fig. 2.9. Schematic view of the horizontal plane of the PS173 experiment apparatus.</figcaption></figure>

In the horizontal plane of the forward hemisphere, a cylindrical multi-wire proportional chamber (MWPC) with two wire planes was located in front of a thin window. It measured charged scattered particles with a resolution between 0.2° and 0.3° in the scattering angle. The MWPC covered scattering angles up to 75° in the horizontal plane and ±15° (full acceptance ±10°) in the vertical direction.

In the same plane, the apparatus was completed by a scintillator hodoscope (FHD, forward hodoscope) and a calorimeter-like detector array (ANC, antineutron calorimeter). The FHD consisted of 32 pieces of plastic scintillators (200 × 50 × 3 mm³), positioned 66 cm from the target, and was used for particle identification by energy loss and time-of-flight measurements. The ANC consisted of 32 modules; each module contained 50 slabs of 6 mm thick plastic scintillator and 50 plates of 4 mm thick iron, with a total thickness of 2.5 absorption lengths for 1.5 GeV/c antineutrons. The ANC was used to detect and identify $\bar{n}$ annihilating in it; the $\bar{n}$-$\gamma$ separation was done using time-of-flight and energy-loss measurements.

Charged mesons from antiproton annihilation were detected by the FHD and an upper, a lower, and a backward plastic scintillator hodoscope (UHD, LHD, and BHD) surrounding the vacuum tank. The solid angle covered by the hodoscopes was 73% of $4\pi$. One hundred and twenty-six lead glass blocks above and below the scattering chamber (not shown in the figure) detected the $\gamma$-rays from $\pi^0$ decays.

$\sigma_{\text{ann}}(\bar{p}p)$ measurement: PS173 measured the $\bar{p}$ annihilation cross-section $\sigma_{\text{ann}}(\bar{p}p)$ at 53 incident $\bar{p}$ momenta between 180 and 590 MeV/c [89,90]. The extracted beam momenta were 605, 547, 527, 467, 397, 305, 243, and 190 MeV/c; the intermediate momenta were obtained using the carbon
degrader or a varying number of 200 µm thick polyethylene foils (for momenta below 300 MeV/c) located in $F_1$.

The incoming beam was defined by the coincidence $F_1 \cdot SD \cdot TD$ and the beam profile was monitored by the MWPC. Both liquid hydrogen targets were used, in different momentum ranges.

The charged mesons produced by $\bar{p}$ annihilation in the target were detected in the hodoscopes. The $\gamma$-rays from $\pi^0$ decays were converted to electron-positron pairs in the wall of the vacuum tank with the average probability of 6%, and were then detected by the same hodoscopes. Annihilation channels with neutral particles only in the final state were detected if at least one $\gamma$-ray was converted in the region covered by the hodoscopes. Events with $\bar{p}$ elastically scattered into the FHD were eliminated by TDC-ADC correlation. The “reaction” trigger required a “beam” signal, no signal in the beam veto-counter BA1, and at least one hit in the hodoscopes.

The effect of beam instabilities on the measured annihilation cross-section was evaluated to be 0.6% (point-to-point systematic error); after all corrections, the precision of the beam-flux measurement was estimated to be 1%. A correction to the cross-section was applied to account for $\bar{p}$ elastic scattering at large angles followed by $\bar{p}$ annihilation in the target; this correction is larger than the statistical error only below 300 MeV/c (where the corresponding estimated systematic error becomes not negligible). The corrections to the annihilation cross-section due to the overall charged-meson acceptance and to the all-neutral channels detection efficiency (evaluated at 297 MeV/c) were estimated to be 10.4 ± 1.0% and 3.0 ± 1.1%, respectively. Further systematic errors were due to the uncertainty about the target thickness (±1% and ±4% for the thick and the thin target, respectively) and the density of the liquid hydrogen (±0.7%). Summarising: the point-to-point systematic errors due to beam instabilities and $\bar{p}$ elastic scattering followed by $\bar{p}$ annihilation in the target range from 0.5 to 6.1 mb. The overall normalisation errors are 2.2% and 4.4% for the thick and the thin target, respectively.

**$\bar{p}p$ elastic differential cross-section measurement:** Results have been published at 181, 287, 505, and 590 MeV/c [91,92]. The p and $\bar{p}$ were detected in the FHD and in the MWPC. Forward $\bar{p}$ detected in the FHD were identified by time-of-flight and energy loss. At large scattering angles, the $\bar{p}$ annihilated in the target and pions coming from annihilation were detected in the hodoscopes; the p was identified by the coincidence of a signal in the FHD and a pion signal in one of the four hodoscopes. At intermediate angles, both p and $\bar{p}$ came out of the target and could be detected in the FHD and the MWPC, giving a clear angular-correlation signal. In this case, the $\bar{p}$ was identified by the detection of its annihilation products in the slabs of either the ANC or the FHD next to the FHD-slab in which the $\bar{p}$ was detected.

The scattering angle was measured from the coordinates of the hit in the MWPC and the geometrical centre of the target. The beam axis was determined and monitored with an accuracy of ±0.3°. The angular resolution due to beam divergence, uncertainty in the reaction vertex in the target, and spatial resolution of MWPC, was estimated to be better than 1° over the full angular range. Multiple scattering was evaluated to be negligible for the published data.

Corrections for MWPC efficiency (98–99%) and geometrical acceptance were applied.

**$\rho$ measurement:** The $\rho$ parameter was determined at 181, 219, 239, 261, 287, 505, and 590 MeV/c [93,92], by measuring the forward differential elastic cross-section and using the Coulomb-nuclear interference method. The beam momenta were obtained by using directly the extracted $\bar{p}$ beam (202, 309, and 609 MeV/c) or by degrading its momenta to the desired value.
In the data analysis, corrections were applied for the FHD acceptance; the probability of $\bar{p}$ ab-
sorption in the window of the vacuum chamber and in MWPC was evaluated to be 0.15–2.5%,
depending on beam energy and scattering angle.

To measure $\rho$, only data at angles outside the multiple Coulomb scattering region were used at all momenta (40–50 points), so that this effect did not need to be considered. In the differential cross-section fit, three free parameters were used ($\rho, \sigma_{\text{tot}}(\bar{p}p)$, and the slope $b$ of the nuclear amplitude); results were given for all of them.

**$\bar{p}p$ charge-exchange differential cross-section:** The $\bar{p}p$ differential charge-exchange cross-section was measured [94] at four $\bar{p}$ momenta, 590, 505 (degraded beam), 287, and 183 MeV/c, by measuring the angular distribution of the antineutrons with the ANC calorimeter.

In the analysis, the $\bar{n}$ were defined asking for a hit with the correct TOF in the ANC and no early
$\pi^{\pm}/K^{\pm}$ or $\bar{p}$ signal in the FHD. If any other hodoscope was fired, the signal had to be compatible
with pions from $\bar{n}$ annihilation in the ANC. This cut had to eliminate events with a $\bar{p}$ scattered at
large angle and annihilated in the vacuum-tank wall, since a $\pi$ could also not be detected in the
FHD or MWPC sector corresponding to the fired ANC module. To reject events with the $\bar{p}$ going
through the gap of two FHD modules, it was required to have no MWPC hit within $\pm 8$ mm from
the calculated $\bar{n}$ trajectory.

The $\bar{n}$ scattering angle was measured with a precision ranging from $\pm 4.4^\circ$ (forward direction, first module) to $\pm 2.7^\circ$ if only one ANC module fired; for the less than 35% of the events in which two or more ANC modules fired, the angular resolution was not affected appreciably. To avoid edge effects, the vertical acceptance was restricted to 75% of the geometrical height of the calorimeters. The TOF separation between $\gamma$ and $\bar{n}$ was considered sufficient to neglect $\gamma$ contamination in the $\bar{n}$ sample.

The n detection efficiency of the ANC was sufficiently low to assume pure $\bar{n}$ detection, since in
the kinematic region in which both n and $\bar{n}$ could be detected less than 5% of coincidences with
the correct correlation angle were found. This assumption could have been a source of systematic
error in the backward hemisphere where neutrons largely outnumber antineutrons. The detection
efficiency of the $\bar{n}$ in the ANC was estimated to be $98 \pm 2\%$. Module-to-module variations of
about 10% were found and corrected for. The quoted overall normalisation error is 5%, due to the
uncertainty in the $\bar{n}$ detection efficiency and to uncertainties in the cuts for background reduction.
At the smallest angle, a systematic error of 5% due to $\bar{p}p$ annihilation in the beam veto counters
was estimated.

### 2.3.3. Experiment PS201

The aim of the second generation LEAR experiment PS201 (OBELIX) was the study of meson spectroscopy, as well as low-energy $\bar{p}$ and $\bar{n}$ annihilation on nucleons and nuclei to investigate nuclear dynamics effects.

A detailed description of the experimental apparatus can be found in [95]. It consisted of a
magnetic spectrometer (shown in Fig. 2.10) covering a solid angle of about $3\pi$, a cylindrical target
located at the centre of the spectrometer, and a thin detector (about 80 mm of scintillator), placed
closed to the beam pipe window, to measure the incoming $\bar{p}$ beam.

The spectrometer consisted of the Open Axial Field magnet, whose magnetic field was about
0.6 T in an open volume of about 3 m³, and of four sub-detectors arranged inside and around
<figure><img src="image_11.png" /><figcaption>Fig. 2.10. Schematic view of the PS201 experiment set-up. The numbers indicate the main components of the apparatus: the Open Axial Field magnet (1), the SPC (2,4), the TOF (3), the AFSJet (5), the HARGD (6).</figcaption></figure>

the magnet:

* spiral projection chambers (SPC): an imaging vertex detector with three-dimensional readout for charged tracks and X-ray detection. This detector allows to take data with a large fraction of P-wave annihilation.

* a time-of-flight system (TOF): two coaxial barrels of plastic scintillators consisting of 30 (84) slabs positioned at a distance of 18 cm (136 cm) from the beam axis; a time resolution of 1 ns FWHM is achieved. This device provides information about multiplicity and topology of annihilation events; it was used in the first level trigger.

* a jet drift chamber (AFSJet, axial field spectrometer jet chamber): it was used for tracking and particle identification by $dE/dx$ measurement. The chamber was split into two half-cylinders (160 cm in diameter, 140 cm long) with a total of 3280 wires.

* a high-angular-resolution gamma detector (HARGD), consisting of four modules made by layers of 3 × 4 m² lead converter foils enclosed by planes of limited streamer tubes.

The target could be filled with different gases at different pressures or liquids, according to the measurements to be performed.

Several measurements of the $\bar{p}p$ and $\bar{n}p$ integrated cross-sections were performed. These measurements extend the existing data-base down to very low energies (about 1 MeV). The experimental techniques are described in the following.

**$\sigma_{\text{ann}}(\bar{p}p)$ measurement:** The $\bar{p}p$ annihilation cross-section was measured by PS201 at 14 incoming $\bar{p}$ momenta in the range 37.6–174.4 MeV/c [96,97].

The data were taken using an extracted $\bar{p}$ beam with 105 and 201 MeV/c, in two different data taking periods using the same procedure. The beam was degraded in mylar sheets before entering
the gaseous hydrogen target (75 cm length and 30 cm diameter) whose pressure could be varied
to allow the incident $\bar{p}$ beam to stop near or upon the end window of the target tank. Using no
degrader or mylar sheets, nine different $\bar{p}$ momenta at the entrance of the target were selected. For
the data of Ref. [96], five more bins in the incoming $\bar{p}$ momentum were obtained by dividing the
target along the beam axis in several fiducial regions.

The $\bar{p}p$ annihilation cross-section into charged particles was measured by counting the number $N_{ev}$ of annihilation events in flight within a fiducial volume inside the target, and the number $N_{\bar{p}}$ of $\bar{p}$ not interacting in the target and annihilating, at rest, near or in the end wall of the target tank. To measure $N_{\bar{p}}$, a scintillator disc positioned close to the end wall of the target tank was used, together with the TOF system. The number $\bar{p}$'s crossing the fiducial volume, corrected for the efficiency of the counting system, was used as incident beam rate to extract the annihilation cross-section.

For $\bar{p}p$ annihilating in flight, the coordinates of the annihilation point were measured with an
uncertainty of 1 cm using the tracking system of the spectrometers; the annihilation time relative
to the beam detectors was measured with a total uncertainty of 1 ns by the TOF system detecting
the charged annihilation products. The correlation between the vertex coordinate of the reconstructed
annihilation point along the beam axis and the annihilation time was used to determine the incident
beam momentum (in good agreement with Monte-Carlo calculation) and to reject in-flight annihila-
tion events of $\bar{p}$ with momentum in the low-energy tail of the distribution (3$\sigma$ cut). Only annihilation
events inside cylindric fiducial volumes at a suitable distance from the entrance mylar window of
the target were considered.

Corrections were applied to take into account several effects, like annihilation detection efficiency,
all-neutral annihilation channels, background due to annihilation on the target walls whose vertex
was (wrongly) reconstructed inside the fiducial volume, the cut on the correlation between the recon-
structed annihilation point and the annihilation time, and the efficiency of the beam counting system.

The overall normalisation error (to be added to the quoted systematic error) was estimated to
be 3.4% [96] and 2% [97]; it is mainly due to the corrections for the apparatus efficiency and for
all-neutral annihilation, and to the uncertainty in the target density. The quoted systematic errors on
$\beta\sigma_{ann}$ range from about 1.2% at 174.4 MeV/c to 14% at 37.6 MeV/c, to be compared with statistical
errors between 3.8 and 1.9%.

*The $\bar{n}$ beam:* To perform measurements with low-energy $\bar{n}$'s, the OBELIX collaboration put into operation a facility for the production of a collimated $\bar{n}$ beam [98–100]. The $\bar{n}$ beam was produced via the charge-exchange reaction $\bar{p}p \rightarrow \bar{n}n$ on a liquid $H_2$ target, a technique already used at AGS by Armstrong et al. [101] to produce $\bar{n}$ beams of momenta between 100 and 500 MeV/c, and at LEAR by experiment PS178 [102].

In PS201, the “$\bar{n}$ production target” was a 40 cm long liquid-hydrogen target, positioned 2 m upstream of the centre of the main detector, on the nominal $\bar{p}$-axis. The target thickness was chosen as to stop the $\bar{p}$ beam (incident momentum 406 MeV/c) in the target. Charge-exchange events in the production target were selected asking no signal in the veto box of scintillators surrounding the target. The veto box could detect charged particles produced in $\bar{p}$ interaction in the target, and $\gamma$'s produced in annihilation into neutral particles and converted in a 5 mm thick lead shields wrapping the target. The $\bar{n}$ produced by charge-exchange in the forward direction were collimated using a suitable shaped lead shield. The resulting $\bar{n}$ beam had an intensity of $3-5 \times 10^{-5} \bar{n}/\bar{p}$, and momenta between 50 and 400 MeV/c.
Because of the structure of the apparatus, the $\bar{n}$ beam could not be tagged by detecting the associated n, thus the momentum of each $\bar{n}$ was unknown. For $\bar{n}$'s annihilating in the reaction target, the annihilation point and time were measured; with an iterative procedure, the $\bar{n}$ momentum could be estimated from the $\bar{p}$ and $\bar{n}$ time-of-flight, with an error ranging from about 2 at 50 MeV/c to about 18 at 400 MeV/c.

The $\bar{n}$ flux was monitored by $\bar{n}$ annihilating in a nuclear target, 30 cm downstream the reaction target, or a $\bar{n}$ detector put at the end of the apparatus. The $\bar{n}$ beam intensity was evaluated by the measured $\bar{p}$ flux through Monte-Carlo simulation.

**$\sigma_{\text{tot}}(\bar{n}\bar{p})$ measurement:** The $\bar{n}\bar{p}$ total cross-section has been measured at 18 $\bar{n}$ momenta between 54 and 380 MeV/c using the transmission technique in a thick target [100]. The reaction target was a 25 cm long liquid-hydrogen target, sitting in the centre of the spectrometer. For $\bar{n}$ annihilating into charged mesons, the annihilation vertex was reconstructed. The $\bar{n}$ momentum was evaluated from the time-of-flight of the charged particles produced in the annihilation and from the annihilation position. The annihilation data were then grouped into 18 sets, corresponding to different $\bar{n}$ momentum bins, 10–20 MeV/c wide, according to the momentum resolution. In each bin the total cross-section was evaluated from the measured $z$ distribution (where $z$ is the depth in the target along the beam axis) of the annihilating $\bar{n}$, inside a fiducial region with a frustum of cone shape of small aperture (about $1.3^\circ$). The position of the cone was defined taking into account beam misalignments. Corrections due to annihilation events occurring after scattering were estimated with a dedicated Monte-Carlo to be 2–5% (at the lowest momentum).

The quoted systematic errors are between 10 and 26 mb (less than 10% at the higher momenta and about 5% at 60 MeV/c); the statistical errors are of the same order at higher momenta and about twice at the lower momenta.

**$\sigma_{\text{ann}}(\bar{n}\bar{p})$ measurement:** The cross-section for $\bar{n}\bar{p}$ annihilation was measured at 18 $\bar{n}$ momenta between 50 and 400 MeV/c. The data are not yet published; information can be found in a Thesis [103] or in Conference proceedings [104,99,105].

The cross-section was extracted from the number of annihilation events in the liquid-hydrogen reaction target. The statistical errors are small, and the systematic error is about 10%, mainly due to the uncertainties in determination of the $\bar{n}$ flux.

### 2.3.4. Experiment PS198

Experiment PS198 has measured differential cross-section and analysing power in $\bar{p}\bar{p}$ elastic scattering at 439, 544, and 697 MeV/c [106,107] in the full angular range. The measurements were performed using a solid polarised proton target, and a one-arm magnetic spectrometer to select elastically scattered $\bar{p}$.

The experimental set-up is sketched in Fig. 2.11. The incoming beam with an intensity of a few times $10^5$ $\bar{p}/s$ was monitored by the scintillation counter F, 0.3 mm thick, and by the “antihalo” scintillation counter HF, 0.5 mm thick and with a circular hole of 12 mm diameter. Additional monitoring was performed with the scintillation counter M, placed downstream of the target, outside the acceptance of the spectrometer.

The polarised target consisted of a 5 mm thick slab of pentanol and was operated in the frozen-spin mode. The 0.7 T vertical magnetic field needed to hold the proton polarisation was produced by
<figure><img src="image_12.png" /><figcaption>Fig. 2.11. Schematic top view of the PS198 experiment set-up.</figcaption></figure>

a superconducting split-coil magnet. The polarisation ranged between 68% and 85% and was measured with an error of ±4%.

The forward final particle (the $\bar{p}$ for $\vartheta_{cm} < 90^\circ$, the p for $\vartheta_{cm} > 90^\circ$) was detected and its momentum analysed with the magnetic spectrometer SPESII [108]. To cover the full angular range, the spectrometer was rotated and set at the chosen scattering angles. The detection system of the spectrometer consisted of four MWPC (CH0-CH3), all of them with horizontal and vertical wires, and of a scintillation counter S. Protons and antiprotons were discriminated from the other particles (mainly $\pi^-$) produced by $\bar{p}$ interactions in the target by means of time-of-flight, measured by F and S. Using this technique, the complete reaction kinematics was reconstructed, and antiprotons scattering off free protons of the target could be distinguished from those scattered off quasi-free target nucleons, on the basis of the reconstructed missing mass. Around $\vartheta_{cm} = 90^\circ$, where the energy of the detected particle was minimal, the angular and energy straggling deteriorated the missing mass resolution. To improve the signal to background ratio in this angular region and to check the background evaluation, a detector R, consisting of scintillator slabs and a MWPC, was added to detect the recoil particle, thus improving considerably the kinematic reconstruction.

The systematic error in the differential cross-section (5%, included in the quoted error) is due to background, acceptance, efficiency, and absorption evaluation. The quoted error does not include the overall normalisation error of 10%. The total systematic error in $A_{0n}$ is of the order of 8%, the major contribution being the uncertainty about the target polarisation.

### 2.3.5. Experiment PS199

Experiment PS199 was proposed to measure spin effects in the charge-exchange reaction $\bar{p}p \rightarrow \bar{n}n$ at low energy, in particular the analysing power $A_{0n}$ and the polarisation transfer parameter $D_{0n0n}$, using a solid-pentanol polarised target. The differential cross-section was also extracted from the
<figure><img src="image_13.png" /><figcaption>Fig. 2.12. Top view of the experiment PS199 set-up.</figcaption></figure>

same data and from the calibration data collected using a liquid hydrogen target at one $\bar{p}$ momentum (693 MeV/c).

The experimental apparatus is shown in Fig. 2.12. The incoming $\bar{p}$'s were detected by two small scintillation counters (not shown in the figure). The beam direction in the horizontal plane (indicated by the continuous line in the figure) was monitored by the scintillation counter hodoscope HB.

The pentanol polarised target (PT) had 12 cm length and 1.8 cm diameter, and was operated in the frozen-spin mode. During data taking, its polarisation was about 75%. To reduce systematic effects, the spin orientation was reversed typically after four 1-h spills. For calibration purposes, the polarised target could be replaced by a liquid hydrogen target, 12 cm length and 3.2 cm diameter. The background from charge-exchange events on the bound nucleons of the polarised target was evaluated using data collected with a dummy target (DT), having the same mass and density as the PT but all H atoms replaced by F atoms. The considerable amount of material in the target caused a large energy loss of the $\bar{p}$ beam in the target: at 905 MeV/c extracted beam momentum, the mean reaction momentum was 875 MeV/c and the total range was about 40 MeV/c. A scintillator veto box surrounded the target and rejected at the trigger level annihilation and elastic scattering events, as well as non-interacting $\bar{p}$'s. The azimuthal acceptance was limited to $\pm 15^\circ$ by the polarised target magnet (PTM) coils.

The neutron detectors $NC_1$, $NC_2$ and $NC_3$ were made of vertical plastic scintillator bars [109]. Each bar (8 cm wide and 20 cm thick) was viewed at its end by two photomultipliers (PM); the coincidence between the two PMs defined a bar hit. In the off-line analysis, the n candidates were
identified by requiring one bar (or at most two adjacent bars) be hit in one NC hodoscope. The neutron coordinates were given by the bar number in the horizontal plane. The vertical coordinate was given by the difference between the time measured by the top and the bottom PM of each bar; it was determined with a precision of 3.7 cm. The neutron time-of-flight was given by the mean of the TOFs of all the PMs of the hit bars. Only neutron bars showing a stable behaviour (monitored with a laser system during all data taking) were considered.

The antineutron detectors (ANC₁ and ANC₂) were designed to have good efficiency and an excellent $n/\bar{n}$ separation. The $\bar{n}$'s were identified by reconstructing the trajectories of the charged products of their annihilation. The geometry of the $\bar{n}$ detectors was chosen to have a maximum antineutron annihilation rate on relatively thick absorbers, which were sandwiched by tracking detectors to reconstruct the trajectories of charged particles. An antineutron-annihilation event was identified in the off-line analysis looking for a “star” pattern in the detectors. With Monte-Carlo events, the resolution in the transverse coordinates of the annihilation point was estimated to be somewhat smaller than 1 cm, and about 2 cm in the z-coordinate.

Each ANC [110] was built up using five identical basic units, the “modules”, separated by four iron slabs, 30 mm thick, which made up most of the mass of the detector. The distance between the iron slabs was 190 mm. Each “module” was made of four planes of vertical Limited Streamer Tubes (LST) with one plane of scintillation counters in between, and was closed by two 6.35 mm thick Al walls. The LST planes were used to reconstruct the trajectories of the charged particles produced in the $\bar{n}$ annihilation. Each plane had an active surface of 166 × 200 cm² and was equipped with 192 strips to measure the horizontal coordinates and 160 strips for the vertical coordinates. The planes were made up with PVC eight-tube chambers filled with a 30:70 Ar-isoC₄H₁₀ gas mixture at atmospheric pressure.

The scintillation counter planes were hodoscopes of six vertical scintillator slabs 10 mm thick, 33 cm wide, and 166 cm long. Each slab was viewed by two PMs and a scintillation counter hit was given by the mean-time coincidence of the two PMs.

In the off-line analysis, the time-of-flight of the antineutron was computed from the mean of the TOFs of the scintillation counters of the two modules sandwiching the absorber where the vertex of the annihilation “star” was reconstructed. A TOF cut allowed the rejection of $\gamma$ and $\pi$ produced in the target.

At the trigger level, a $\bar{n}$ signal was defined as at least two fired counters in one ANC.

*Measurement of $A_{0n}$ and $d\sigma/d\Omega$:*$ $A_{0n}$ was measured at 8 $\bar{p}$ momenta between 546 and 1287 MeV/c, during two short runs in 1989 and 1990 [111–114].

The useful charge-exchange events were characterised by a neutron detected in one neutron counter and an antineutron seen by the corresponding $\bar{n}$ detector (NC₁-ANC₁ and NC₂-ANC₂ for the forward and backward angular range, respectively). The requirement to detect both particles in the final state implied that very forward and very backward scattering events could not be measured. Only at 875 MeV/c incident $\bar{p}$ momentum, a measurement was done over a broad angular range detecting only the $\bar{n}$ [114].

All events with a valid beam signal, with no signal from the veto box surrounding the target, with at least one bar hit in a NC and with a $\bar{n}$ signal in the corresponding ANC were recorded on tape.

The data collected for the $A_{0n}$ measurement were also used to extract the $\bar{p}p \to \bar{n}n$ differential cross-section at all $\bar{p}$ momenta but 1000 MeV/c [115]. This was possible thanks to a calibration
<figure><img src="image_14.png" /><figcaption>Fig. 2.13. Top view of the experiment PS199 set-up used for the measurement of $D_{0n0n}$.</figcaption></figure>

method for the ANC detectors [116] which gave the $\bar{n}$ detection efficiency with an error of few per cent. The quoted overall normalisation error of about 10% was mainly due to the uncertainty in the polarised target length.

Using the calibration data collected with a liquid hydrogen target, the differential cross-section was measured at 693 MeV/c over a wider angular range [111].

*Measurement of $D_{0n0n}$:* The $D_{0n0n}$ parameter was measured at 546 and 875 MeV/c [117,118], with two similar experimental set-ups. As in the case of $A_{0n}$, the measurement of $D_{0n0n}$ was performed detecting both the $n$ and the $\bar{n}$. $D_{0n0n}$ was then measured by analysing the polarisation of the final-state neutrons. The apparatus used for the measurement at 875 MeV/c is shown in Fig. 2.13 and is very similar to the one used to measure $A_{0n}$: the polarised target, and the $n$ and $\bar{n}$ detectors are the same, they are only differently arranged.

The data were collected in an about 10 days run at each energy, almost equally divided among PT spin up, PT spin down, and DT data.

The relevant detectors for the $D_{0n0n}$ measurement were the $\bar{n}$ counter $ANC_1$ and neutron hodoscopes ND and NG which made up the neutron polarimeter; $ANC_3$ was a LST module, used in the off-line analysis to reject events with charged particles produced in ND. $ANC_2$ was used to measure in parallel $A_{0n}$ and $d\sigma/d\Omega$ in the backward hemisphere [114]. The useful events for the $D_{0n0n}$ measurement were those with an $\bar{n}$ produced by charge-exchange on a free hydrogen of the
polarised target and detected in ANC₁, and the associated n interacting elastically on a hydrogen nucleus of the counter ND and then detected by NG. The polarisation of the n produced in the charge-exchange reaction could thus be measured from the left-right asymmetry of the scattered neutrons, since the analysing power of the np elastic scattering reaction is known. Only a small percentage of the sample (10%) were events in which the neutron was elastically scattered off a proton in ND. They were selected using all three measured quantities, namely the n scattering angle, the time-of-flight of the scattered particle and the energy *T*<sub>p</sub> released in ND. A precise energy calibration of each neutron-counter bar was performed before, during and after the run measuring the ADC spectra of the 4.4 MeV γ-rays from an Am/Be neutron source and the ADC spectra of cosmic ray muons crossing the hodoscopes. The stability of the counters during data-taking was monitored using a system based on a N₂ laser. The effective analysing power of the n polarimeter was estimated with a Monte Carlo programme which took into account all the interactions of neutrons in the scintillator, and using previous measurements for a similar polarimeter.

To eliminate the systematic errors due to the non-uniform efficiency of the NG counter, and to
possible geometrical effects, D<sub>0n0n</sub> was extracted using an estimator which did not require a uniform
efficiency of the polarimeter, it only required the detection efficiency to be stable during the data
taking.

Using the same data, a measurement of the product of the $\bar{n}p$ analysing power times the spin parameter $K_{n00n}$ was performed at 875 MeV/c [119].

2.3.6. *Experiment PS206*

The objective of PS206 [120,121] was to accurately measure at a few energies the differential cross-section of the charge-exchange reaction $\bar{p}p \to \bar{n}n$. It was proposed and performed by a large fraction of the PS199 Collaboration, using much of the detectors and the experimental method of that experiment, with a few notable differences. A dedicated liquid hydrogen target was built for this experiment, the $\bar{p}p \to \bar{n}n$ differential cross-section was measured from the $\bar{n}$'s angular distribution alone, and the associated neutrons were detected only over a smaller angular range, to determine the efficiency of the $\bar{n}$ detectors.

The measurements were performed at two extracted $\bar{p}$ momenta, 612 and 1206 MeV/c. The layout of the experiment for the measurements at 612 MeV/c is shown in Fig. 2.14. For the measurement at 1206 MeV/c a slightly different arrangement of the detectors was adopted.

The antineutron detectors ANC₁ and ANC₂ were used to measure the angular distribution in the forward region; ANC₃ detected both n and $\bar{n}$ in the backward hemisphere. Thus, almost all the angular range was covered, extending down to the forward direction $\theta = 0$. The continuous line in Fig. 2.14 indicates the $\bar{p}$ beam direction, the dashed line the direction of $\bar{n}$'s produced at zero degree in the target. The incoming $\bar{p}$'s were defined by two scintillation counters B₀ (50 × 70 cm²) and B₁ (a 1 cm diameter, 5 mm thick). The total beam flux was measured by B₀, while the coincidence B₀ × B₁ defined the beam entering the target. Three multi-wire proportional chambers (PBC₁, PBC₂, and PBC₃) monitored the beam direction and position. A C-shaped magnet (MN) was used to sweep the $\bar{p}$ leaving the target away from the 0° direction, allowing to measure the differential cross-section in the very forward region. The scintillation counter hodoscope HB monitored the beam direction in the horizontal plane. The beam momenta at the target centre were 601.5 and 1202 MeV/c, with a total spread of about 14 and 5 MeV/c in the two cases. The 1σ beam divergence,
<figure><img src="image_15.png" /><figcaption>Fig. 2.14. Top view of the experiment PS206 set-up.</figcaption></figure>

6.8 mrad in both planes at 601.5 MeV/c and 3.3 mrad at 1202 MeV/c, was dominated by multiple scattering.

The liquid hydrogen target (LHT), 10.61 cm long and 3 cm diameter, could be filled and emptied in a few minutes; inside the target, a diode measured its temperature in order to monitor the filling operations and to evaluate the residual hydrogen density in the “empty” target.

To reject annihilation and elastic events at the trigger level, the target was surrounded by a scintillation counter box on all sides, except that of the incoming beam. The forward and the two lateral counters ensured that no trigger caused by charged particles in the ANCs was accepted. The upper and lower vetoes were made up of four layers of scintillator, interleaved with a total of 1 radiation lengths of lead to reject annihilation events into neutrals.

The $\bar{n}$ detectors $ANC_1$ and $ANC_2$ were essentially the same which had already been used in experiment PS199. Each detector was made up by three identical units, a unit consisting of a 3 cm thick iron slab, sandwiched between two modules of four planes of limited streamer tubes (LST) and one plane of scintillation counters, already described in Section 2.3.5.

$ANC_3$ was a new counter, designed to detect with reasonable efficiency both neutrons and antineutrons, and which could thus be used both for calibration purposes and to measure the $\bar{n}$ angular distribution in the backward hemisphere. $ANC_3$ consisted of a neutron counter (NG) sandwiched by two modules, identical to those making up $ANC_1$ and $ANC_2$. The neutron counter was a hodoscope of 27 vertical scintillator bars, 8 cm wide and 20 cm thick; the bar heights range from
110 to 130 cm. The bars were viewed from each end by a photomultiplier; time and amplitude information was recorded. A system based on a $N_2$ laser had been used during all runs to monitor the NG bars as well as the ANC scintillation counters.

For the measurement at 1202 MeV/c, the antineutron detectors $ANC_1$ and $ANC_2$ were put at about 6 m away from the target in order to separate, on the basis of the measured time-of-flight, $\bar{n}$'s produced in the $\bar{p}p \to \bar{n}n$ reaction from $\pi^0$ and $\gamma$ coming from $\bar{p}p$ annihilation in the target, much in the same way as at the low-energy measurement.

For the first time, large samples of data were collected for the charge-exchange reaction. For background evaluation, data were taken alternating full target (FT) and empty target (ET) runs. With the full target about $75 \times 10^6$ events at 601.5 MeV/c and $17 \times 10^6$ events at 1202 MeV/c were collected, corresponding to a useful beam of $38 \times 10^9$ and $21.5 \times 10^9$ $\bar{p}$ respectively; the statistics with empty target were $5 \times 10^6$ and $3 \times 10^6$ events on tape and $8.7 \times 10^9$ and $15.6 \times 10^9$ incoming $\bar{p}$ at 601.5 and 1202 MeV/c, respectively.

The measurement of the $\bar{p}p \to \bar{n}n$ charge-exchange differential cross-section required the measurement of the $\bar{n}$ angular distribution, and the measurement of the $\bar{n}$ detection efficiency. To identify the $\bar{n}$'s, one looked at “star” topologies in the ANC detectors, much in the same way as in PS199. As compared to PS199, the $\bar{n}$ definition had been further improved in two ways:

* a “sphericity” cut was applied to the “star” topology, to distinguish an $\bar{n}$ annihilation star (the distribution of secondaries in ANC is almost isotropic) from $\gamma$-stars (the corresponding $e^+$ and $e^-$ tracks are almost collinear) or $\pi$-stars. The $\gamma$- or $\pi$-stars could have been produced by pions from an undetected $\bar{n}$ annihilation in the ANC and might result in a false $\bar{n}$-star, far from the $\bar{n}$-annihilation point.

* To improve the measurement of the $\bar{n}$ time-of-flight, a more restricted definition was introduced, which used the tracks belonging to the star topology to identify the scintillators that should have fired. Only scintillators crossed by these tracks were taken into account, and the measured $\bar{n}$ TOF was defined as the average of the “good” scintillator TOFs. Further TOF cuts were applied to reject cross-talk events between $ANC_1$ and $ANC_2$.

To evaluate the ANC efficiency for $\bar{n}$'s, PS206 applied the same procedure already adopted by PS199, relying on the “associate particle” method. A detailed investigation of many systematic effects, the improved layout of the experiment, and the much larger data sample available for the analysis allowed to reduce the error of the method by almost a factor of three, as compared to PS199. A precision in the absolute normalisation of 2% and 4% at 601.5 and 1202 MeV/c, respectively is quoted, while over most of the angular range the point-to-point error is lower than 1%.

### 2.3.7. Experiment PS185

The aim of this experiment was to study how strangeness is produced, by studying hyperon-antihyperon final states. The cross-section, angular distribution and final-state polarisation were measured. The last runs, PS185/3, benefited from a polarised proton target, as described in the proposal [122]; its data are still being analysed at the time we finish this review. They include the spin transfer from proton to $\Lambda$ and to $\bar{\Lambda}$.

The data on strangeness-exchange scattering have been published in Refs. [123–129]. Results on CP tests can be found in Ref. [130]. The collaboration also took data on $K\bar{K}$ channels [131].
<figure><img src="image_16.png" /><figcaption>Fig. 2.15. Overview of the PS185 detector at LEAR. 1: target, 2: multi-wire proportional chambers, 3: drift chambers, 4: hodoscope, 5: solenoid with drift chambers, 6: limited streamer tubes, 7: silicon microstrips.</figcaption></figure>

A schematic view of the set-up is shown in Fig. 2.15. The antiproton beam was sent on a C-CH₂ target which was thick, to obtain high luminosity, and segmented to provide a good determination of the reaction vertex.

The target was divided into five modules, each with a thickness of 2.5 mm. Four target cells were built of polyethylene (CH₂) (high proton density). This structure allowed fine momentum scales in 800 keV/c bins. In order to control the carbon background of the other modules, one cell consisted of pure carbon.

The detection system included tracking chambers, multi-wire proportional chambers and drift chambers and allowed the reconstruction of the charged-particle tracks. Within the LEAR range ($p_{\bar{p}} \le 2$ GeV/c) the hyperons are always emitted into a forward cone. In fact, also the decay proton (or antiprotons) are confined within a limited forward cone ($\le 42°$), fully within the acceptance of the tracking system and of the triggering hodoscope. The charge of each particle was determined from a set of three additional drift chambers inside a magnetic field of typically $B = 0.09$ T. The hodoscope was used to reduce the background, in particular from neutral kaon decays. The limited streamer tubes were not necessary to study the $p_{\bar{p}} \to \Lambda\Lambda$ reaction, but were used for $K\bar{K}$ final states or to measure $\bar{\Lambda}\Lambda$ production on carbon.

The $\Lambda$ hyperon was identified by its dominant $p\pi^-$ decay mode (about 64%), which produced a characteristic $V^0$ signature in the detector. In principle, the reaction $p\bar{p} \to \bar{\Lambda}\Lambda$ is fully identified from the complete reconstruction of the $p\pi^-\bar{p}\pi^+$ final state.
<figure><img src="image_17.png" /><figcaption>Fig. 2.16. Experiment E760: schematic view of the apparatus for detecting recoil protons (a), and top view of the solid-state detectors set-up (b).</figcaption></figure>

Thanks to parity violation, the decay $\Lambda \rightarrow p\pi^{-}$ is not isotropic in the $\Lambda$ rest frame, but correlated to its spin. The proton is emitted preferentially parallel to the $\Lambda$ spin, and the antiproton opposite to the $\bar{\Lambda}$ spin. The distribution is of the type $I(\vartheta) \propto 1 + \alpha P \cos\vartheta$, where P is the (anti)-hyperon polarisation, and $\alpha = \pm 0.642 \pm 0.013$. So measuring the proton and pion momentum with high statistics gives the polarisation of the outgoing $\Lambda$.

As for other hyperons, $\Sigma^0$ was identified through its $\Lambda\gamma$ decay, which has almost 100% branching ratio. The photon needs not be measured as long as the precision on charged-particle tracking is sufficient. The charged hyperons were reconstructed through the decays $\Sigma^{+} \rightarrow p\pi^{0}$ ($\sim 52\%$), $\Sigma^{+} \rightarrow n\pi^{+}$ ($\sim 48\%$) and $\Sigma^{-} \rightarrow n\pi^{-}$ ($\sim 99.85\%$) and their antibaryon analogues.

### 2.3.8. Experiment E760 at Fermilab

Experiment E760, at the Fermilab Antiproton Accumulator ring, was devoted to high-resolution studies of charmonium states formed in $p\bar{p}$ annihilation. The $p\bar{p}$ forward elastic scattering parameters between 3.7 and 6.2 GeV/c have also been measured [132], using an apparatus incorporated in the luminosity monitor of the experiment. The monitor was designed to perform precision measurements of the $p\bar{p}$ differential cross-section in the very small $t$ region by measuring the recoil protons, to extract the luminosity from the shape of the differential cross-section, with a systematic error of a few percent. The analysis of the shape of the small angle differential cross-section provided also precise values for the $\rho$ parameter.

Details on the apparatus, shown in Fig. 2.16, can be found in Refs. [132,133]. The circulating $p$ beam (typically $4 \times 10^{11} \bar{p}$, about 8 mm diameter) intersected an internal $H_2$ gas jet target (typical density $3.5 \times 10^{13}$ protons/cm$^2$, 7 mm diameter). The solid-state detectors measuring the recoil protons of the $p\bar{p}$ elastic scattering events were placed in a pan (shown in Fig. 2.16(b)) located at the bottom of a 150 cm long tapered vacuum chamber suspended vertically from the beam pipe. The detector system consisted of one fixed detector for luminosity monitoring positioned at an angle $\alpha = 90^\circ - \theta = 3.547^\circ$ (where $\theta$ is the recoil angle), and five detectors with an active area of about
1 × 5 cm². The latter were located on a carriage which could be moved in a direction parallel to the beam. The “movable” detectors were positioned about 1° apart along a line parallel to the beam, and typically covered the recoil angle range α = 1.1–5°. At 5 GeV/c this angular range corresponds to −t ranging from 0.0009 to 0.019 GeV² and to recoil proton energies between 0.4 and 11 MeV. The energy calibration of the detectors was performed using a ²⁴⁴Cm alpha source; the energy resolution of the different detectors ranged between 60 and 110 keV and was found to be reproducible to within 0.2%.

The basic points of the measurements were to measure in each solid-state detector the spectrum
of the kinetic energy distribution of the recoil protons, subtract the background, and evaluate the
mean value of the recoil energy, and thus α and t. Only at −t < 0.003 GeV², the recoil angle had to
be determined from the known distance between the detector and another detector placed at larger
−t values. Then, the relative differential cross-section was extracted from the number of detected
protons corrected for the slightly different solid angles seen by each detector, and normalised to the
fixed detector counts.

The error on the recoil angle was estimated to be less than ±0.006°. The systematic error due to the uncertainties in the values of the relative areas of the different detectors (less than 0.1%) was also taken into account in extracting the p̄p forward elastic scattering parameters.

The angular distribution was measured at six incident p̄ momenta between 3.70 and 6.23 GeV/c, corresponding to the masses of charmonium resonances. To perform the scanning over the width of the charmonium resonances, the beam momentum variations were always less than 75 MeV/c, and were considered negligible for the p̄p forward elastic scattering parameters measurements. Also, the perturbation caused by the resonances on the elastic scattering parameters are expected to be completely negligible because the cross-section for the reaction p̄p → (c̄c)R → p̄p is known to be by 5–7 orders of magnitude smaller than the p̄p elastic scattering cross-section.

The data were fitted considering as free parameters σtot, the slope b, and ρ; because of the large correlation between the parameters, a fit was also performed which used for σtot the best estimate from the world data in the region of interest. In the two cases, the systematic errors on ρ were estimated to be 0.004 and 0.005, respectively. Correspondingly, the total error on ρ ranged between 0.007 and 0.024 and between 0.007 and 0.012.

2.4. Experiments on p̄p and p̄d atoms

2.4.1. Pre-LEAR experiments

The motivation to search for X-rays from p̄p atoms came from three different sources or “tradi-
tions”. These traditions and the results of early experiments [134–136] had a significant impact on
the experimental techniques chosen at LEAR. It seems, therefore, adequate to spend a few sentences
on the “pre-history” of protonium atoms.

One of the roots of the experimental searches for the X-ray spectrum of antiprotonic hydrogen goes back to the discovery of heavy antiprotonic atoms by the group of Backenstoss, Bamberger, Koch and Lynen [137]. Pionic and kaonic atoms were studied extensively at that time, and the same experimental techniques proved to be useful for antiprotonic atoms, too. The good resolution of solid-state detectors allowed to determine line shifts and broadenings due to strong interactions. But the resulting strong interaction parameters were influenced by the nuclear environment. The demand to determine the “free” p̄p scattering length required the use of H₂ as target.
Studies of antiproton–proton annihilation at rest in bubble chamber experiments [138] motivated also the study of p̄p atoms. From data on p̄p annihilation into K$\bar{K}$, it had been deduced that most antiprotons annihilate from S-wave orbitals of p̄p atoms and that the contributions of P states should be negligible. The dominance of S-wave capture was explained as cascade effect [139,140]. S-wave dominance was later challenged by Kalogeropoulos and his coworkers, who argued in favour of very large P-wave contributions to p̄p [141] and p̄d [142] annihilation. A deeper understanding of the atomic cascade seemed necessary.

Last not least, the Lamb-shift experiments on muonic helium by Zavattini and collaborators [143] had a significant impact. These experiments focused the attention on the importance of cascade processes and on the need to minimise the interaction of light exotic atoms with neighbouring molecules. Indeed, no muonic lines had been observed after stopping muons in liquid hydrogen [144] but Bailey and collaborators did see lines from pionic hydrogen when stopping pions in hydrogen gas [145].

The early experiments on antiprotonic hydrogen confirmed the idea that low-density targets are mandatory for a successful search for radiative traditions of p̄p atoms. Two experiments which used a liquid [135] or a high-density H₂ gas [136] target and solid-state detectors failed to observe p̄p atomic X-rays. It required the use of a low-density gas target to observe at least the *Balmer* series of p̄p [134] and p̄d [146] atoms and thus to establish the formation and observability of protonium atoms. The low stop rate of the low-density target was compensated by use of a cylindrical multi-wire proportional chamber with large solid angle for X-ray detection. The Balmer series was observed with a yield of (6 ± 3)% per annihilation but no K X-rays were found. The result thus confirmed the large annihilation probability of the 2P protonium states, predicted by Kaufmann and Pilkuhn [147].

When LEAR came into operation three experiments were proposed, all three using H₂ gas at low density. In this section, we discuss experimental techniques; physics results will be presented in Section 5.

### 2.4.2. PS171: The Asterix experiment

The Antiproton STop Experiment with tRigger on Initial X-rays (ASTERIX) experiment was designed to study p̄p annihilation from P-wave orbitals of protonium atoms formed by stopping antiprotons in H₂ gas. The main emphasis of the experiment was (q̄q) spectroscopy and the search for glueballs, hybrids, baryonia and other boson resonances [148]. However, the detector was also designed to contribute to the physics of the protonium atom [149]. The detector is fully described in [150]. Physics results on protonium are published in [151–154]. Results on antiproton annihilation from P states of the protonium atom can be found in [155–166].

Protonium spectroscopy relied on the central detector of a general-purpose particle spectrometer with cylindrical proportional chambers and a homogeneous 0.8 T magnetic field. Fig. 2.17 shows the central components of the detector.

Antiprotons with an incident momentum of 105 MeV/c were moderated by passing through the LEAR exit window, a variable air gap, and a 50 µm entrance scintillator which identified incoming antiprotons, and then entered a H₂ target at standard temperature and pressure (STP). At the end of the target, a thick scintillator was mounted to facilitate beam tuning. The distance between the thin entrance (T2) and the exit (T4) counter was 72 cm. Two further beam defining counters (T1, T3)
<figure><img src="image_18.png" /><figcaption>Fig. 2.17. The central drift chamber of the Asterix experiment and the antiproton defining scintillators.</figcaption></figure>

were used only in the initial phase of the experiment. For 105 MeV/c antiproton beam momentum, the residual range of antiprotons entering the target led to a well-localised stop distribution in the centre of the H₂ target.

The H₂ target was surrounded by an X-ray drift chamber (XDC) and separated from it by a thin (6 µm) aluminised mylar cylinder of 16 cm diameter. The mylar cylinder was held at a voltage of -10 kV and acted as cathode. The XDC was continuously flushed with an (50:50) argon-ethane gas mixture; likewise the H₂ gas was exchanged continuously to minimise a possible argon contamination from the XDC gas diffusing through the mylar foil. A slight overpressure of 1 mbar on the H₂ side (controlled to better than 0.1 mbar) stretched the mylar foil to a perfect cylindrical shape.

The 90 anode wires (Ni-Cr, 32 µm diameter, 1.5 kΩ/m) were kept at ground potential, 270 field wires (Cu-Be, 100 µm diameter) at -2.1 kV shaped the electric field in 90 “cells” into which the XDC was segmented. An outer Al container provided the mechanical stability of the construction. Each XDC cell was defined by five field wires and one sense wire collecting the charge deposited along a road from the mylar cylinder to the cell. Due to the presence of the 0.8 T magnetic field, the roads were curved. Hence charges, deposited along a straight track originating from the target centre, were collected at different sense wires.

The readout of the XDC was based on the UA1 central detector electronics which recorded the pulse-height history on each wire as a function of time (in 32 ns time bins and for a time period of 4 µs). The sense wires were read out on both ends; from the drift time and the signal ratio on the two wire ends, the conversion point of X-rays could be reconstructed in space. The XDC had, for the measured stop distribution, a large solid angle (90% of $4\pi$), a high detection efficiency even for low X-ray energies (30% for the $L_\alpha$ line at 1.74 keV) but only a rather modest energy resolution of 25% at 5.5 keV.

The XDC allowed an efficient discrimination of X-rays against the dominating background from charged particles, see Fig. 2.18. Charged particles ionise the counter gas all along their path. In presence of the magnetic field, several cells show ionisation due to the effect of the Lorentz angle. X-rays lose their energy locally very close to the conversion point. A short pulse is detected in one cell while the neighbouring cells are free of charge. The localisation of the energy deposit can be used to define X-rays.
<figure><img src="image_19.png" /><figcaption>Fig. 2.18. Antiproton–proton annihilation into four charged particles preceded by an X-ray transition. Tracks of charged particles deposit energy along their path, X-rays only locally. The short penetration depth and the small energy indicate that a Balmer or Paschen X-ray is observed.</figcaption></figure>

A special hard-wired trigger was designed to select events with isolated charge deposits in the chamber and thus to enhance the fraction of events on tape with X-rays. A trigger on the absence of charged particles removed the background associated with charged particles.

### 2.4.3. PS174: The cold-gas experiment

The cold-gas experiment used a gas target which provided the possibility to reduce the temperature or the pressure of the gas and to vary its density over a wide range, from 10 times to 1/8 times STP density. The target was filled with H₂, D₂ and He gas; in this report only the results using H₂ or D₂ are discussed.

Fig. 2.19 shows the apparatus. Antiprotons entered (from left) through a Be window and a final thin scintillator providing sufficient light so that very slow antiprotons at the end of their range can be detected. The moderator thickness can be tuned by rotating a 50 µm thick mylar foil to optimise the stop distribution for detection of X-rays. The antiproton momenta were reduced in the course of time: 300, 200 and 105 MeV/c antiprotons were delivered to the experiment.

Stopped antiprotons form antiprotonic hydrogen (or deuterium) atoms which emit X-rays. The volume in which antiprotons stopped was viewed at by Si(Li) detectors with excellent energy resolution (320 eV FWHM at 6.4 keV) but small solid angle, typically $10^{-3}$ of $4\pi$. Si(Li)'s consist of solid material; high-energy γ-rays have a large probability to scatter off the detector material via Compton scattering. The kicked electron leaves ionisation in the detector thus producing a continuous background. This background can be reduced (but not eliminated) by surrounding the Si(Li) detector with a NaI(Tl) annulus as Compton shield vetoing scattered high-energy γ-rays.

With Si(Li) detectors, a high resolution can be achieved; their disadvantage is the small solid angle and the low background-rejection power. In a later stage, the PS174 collaboration used also two gas
<figure><img src="image_20.png" /><figcaption>Fig. 2.19. Apparatus used in experiment PS174. For part of the data, a GSPC (not shown) viewed the target from the side.</figcaption></figure>

scintillation proportional detectors (GSPD). The main advantage of gas counters lies in their low mass, resulting in a low probability for Compton scattering. Energy deposits from charged particles can easily be reduced by pulse shape analysis. On the other hand, gas counters have a limited energy resolution due to the smaller number of electron-ion pairs created; about 32 eV are needed for one pair leading to an expected resolution of 15% at 8 keV.<fnref n="3" /> These electrons normally drift towards a wire where gas amplification occurs. The statistics of the gas amplification deteriorates the theoretical resolution in normal proportional counters, but this can be avoided: in the GSPD the primary electron cloud is drifted through a gas at a velocity that optical excitation of the gas takes place but no ionisation. The amount of light produced in this process is only limited by the experimental set up; the resolution is essentially given by the primary electron-ion pair statistics. Practically, a resolution of 850 eV at 11 keV was reached. A detailed description of the GSPC can be found in [167]. The results on H₂ and D₂ are published in [168–170].

<footnote n="3">³ The creation of a number of ion pairs is not a completely random process since energy must be conserved. Therefore, the theoretical resolution is better by a factor *F*, the Fano factor, than the above naive argument suggests.</footnote>
<figure><img src="image_21.png" /><figcaption>Fig. 2.20. The inverse cyclotron in the setup for PS207. Si(Li) or CCD detectors counters view at the target center along the axis of the magnetic field provided by two superconducting coils. Scintillation counters determine the time at which annihilation took place. On the opposite side, a two-arm crystal spectrometer was set up, equipped with three Bragg crystals each reflecting to a separate CCD detector.</figcaption></figure>

### 2.4.4. PS175 and PS207: the inverse cyclotron

In the inverse cyclotron experiment, a dense antiproton stop distribution was reached even at very low densities. The apparatus is shown in Fig. 2.20. Antiprotons were decelerated in a dedicated low-energy beam line with optimised emittance. The beam entered the H₂ chamber through a 12.5 µm thick Kapton window separating the beam line from the experiment, and passed through a thin scintillator and a set of mylar foils. The range of incoming particles was wound up in a focusing magnetic field. Two superconducting coils provided a field gradient exerting a force towards the symmetry plane and the actual equilibrium orbit. The antiprotons continued to loose energy in collisions with the H₂ gas, with decreasing cyclotron orbits and betatron oscillation amplitudes, until they came to rest. A large fraction (~90%) of the incoming antiproton flux was stopped in the target gas.

If a target density of, e.g., 30 mbar would have been chosen in the experiments above, beam straggling would have led to a stop distribution along the beam axis of 45 cm; the beam would have grown to a transverse width of 65 cm. The use of a cyclotron to decelerate antiprotons allowed
a concentration of the stop distribution in a volume of about 100 cm³. Later, in experiment PS207, a special low-energy beam line was built with very good emittance and the stop distribution was contained in a volume of about 1 cm² × 1.5 cm. The development of the inverse cyclotron and the performance which was reached was a major technical breakthrough which also had a considerable impact on the scientific programme at PSI [171].

Three detectors were used in PS175 to detect X-rays, two Si(Li) detectors and one X-ray drift chamber. The first Si(Li) was mounted in a guard-ring configuration to reduce background from Compton scattering and from annihilation products. Only the centre area of the detector was used to collect X-rays while charge deposited in the outer guard ring vetoed the event. The low background allowed clear identification of X-rays from the Balmer series with high resolution (280 eV at 6.4 keV). The second Si(Li) had a larger sensitive area (300 mm²) and less resolution (560 eV at 6.4 keV). Its main purpose was to search for the Kα transition.

The X-ray drift chamber had 16 anode wires and 16 cathodes strips and covered an active area of 16 × 16 cm². Off-line pulse shape analysis and the request for isolated ionisation clusters reduced the charged-particle background. A resolution of 11% (FWHM) at 9 keV was achieved.

The inverse cyclotron trap was used for a second series of experiments (PS207). Charged-coupled X-ray detectors with CCD pixel sizes of 22 µm² were used to detect the Balmer and Lyman series'. CCD's also allow a powerful background rejection, with no compromise in energy resolution (which was 320 eV at 9 keV). They need long readout times; many events had to be accumulated before readout took place. Hence they cannot be used for trigger purposes. In parallel, precision measurements on the energy profile of the Balmer series were carried out by use of two crystal spectrometers with a resolution of about 30 MeV. In the spherically bent quartz or silicon crystal, photons were reflected under the Bragg condition, and detected in CCD pixel detectors. An efficiency for X-ray detection and reconstruction of up to 10⁻⁶ was reached. The spectrometer is fully described in [172]. Results from experiment PS175 on antiprotonic hydrogen and deuterium are published in [173,174], from experiment PS207 in [175–178].

### 3. Theoretical background

In this section, we present the kinematics of nucleon–antinucleon elastic, charge- and strangeness-exchange scattering. We define the various spin observables and list the relations among them. We briefly summarise the state of understanding of nucleon–antinucleon interaction when LEAR came into operation. The G-parity rule is derived, and potential models are briefly introduced. The role of strong interaction in antiprotonic hydrogen and deuterium is described and linked to very-low-energy scattering.

#### 3.1. Kinematics

##### 3.1.1. Elastic scattering

The notations for the incoming and outgoing 4-momenta are summarised in Fig. 3.1. From the 4-momenta $\tilde{p}_i$ and $\tilde{n}_i$, one computes the Mandelstam variables

$$s = (\tilde{p}_1 + \tilde{p}_2)^2, \quad t = (\tilde{p}_1' - \tilde{p}_1)^2, \quad u = (\tilde{p}_2' - \tilde{p}_1)^2, \qquad (3.1)$$

which fulfil the relation $s + t + u = 4m^2$, where m is the nucleon mass.
<figure><img src="image_22.png" /><figcaption>Fig. 3.1. Kinematics of $\bar{N}N$ scattering, showing momenta and helicities of the incoming and outgoing particles.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 3.2. Relation between the invariant mass of the initial state and the momentum of the incoming antiproton.</figcaption></figure>

In the centre-of-mass frame (c.m.), the values are

$$ \tilde{p}_i = [E, \vec{p}_i], \quad \tilde{p}'_i = [E, \vec{p}'_i], \qquad (3.2) $$

with $\vec{p}_2 = -\vec{p}_1$, $\vec{p}'_2 = -\vec{p}'_1$, $\vec{p}_1^2 = \vec{p}_2^2 = p^2$, and

$$ s = 4E^2 = 4(p^2 + m^2), \quad t = -2p^2(1 - \cos\vartheta_{\text{cm}}), \qquad (3.3) $$

where $\vartheta_{\text{cm}} = (\hat{p}_1, \hat{p}'_1)$ is the scattering angle. In the laboratory frame where the initial proton is at rest,

$$ s = 2m^2 + 2mE_{\text{lab}} = 2m^2 + 2m\sqrt{m^2 + p_{\text{lab}}^2}. \qquad (3.4) $$

The relation between the c.m. energy $\sqrt{s}$ and the momentum $p_{\text{lab}}$ is illustrated in Fig. 3.2. Eq. (3.4) can be inverted into the useful relations

$$ E_{\text{lab}} = \frac{s - 2m^2}{2m}, \quad p_{\text{lab}} = \frac{\sqrt{s(s - 4m^2)}}{2m}. \qquad (3.5) $$
<figure><img src="image_24.png" /><figcaption>Fig. 3.3. Ellipse drawn by the momentum $\vec{p}'_1$ of the scattered antiproton for given $s$ in a given half-plane. The figure corresponds to $s = 5$ in units where $m = 1$.</figcaption></figure>

The relation between the angle and momentum of a particle in the final state can be obtained by writing energy-momentum conservation as $\tilde{p}'_2 = \tilde{p}_1 + \tilde{p}_2 - \tilde{p}'_1$ and squaring. One obtains

$$m - \sqrt{m^2 + p_1'^2} + \sqrt{\frac{s - 4m^2}{s}} p_1' \cos \vartheta_1 = 0. \qquad (3.6)$$

Thus, for given $s$ in a given plane, the momentum $\vec{p}'_1 = \{p'_1, \cos \vartheta_1\}$ of the scattered antiproton draws an ellipse passing through the origin, as shown in Fig. 3.3. In particular, (i) any acute angle $0 < \vartheta_1 < \pi/2$ is possible; (ii) there is only one value of the momentum $p'_1$ associated with any given $\vartheta_1$ in this interval. The momentum $p'_1$ experiences values between 0 and $p_{\text{lab}}$. The eccentricity of this ellipse is $2m/\sqrt{s}$.

### 3.1.2. Charge and strangeness exchange

At very low energy, one should account for the neutron-to-proton mass difference when describing the charge-exchange reactions. The strangeness-exchange reactions involves hyperon masses in the final state. We shall restrict ourselves to equal masses in the final state. The generalisation to, e.g., $\bar{p}p \rightarrow \bar{\Sigma}^0 \Lambda$ is straightforward.

In units where the proton mass is set to $m = 1$, the masses of interest are: $m(n) = 1.00138$, $m(\Lambda) = 1.189$, $m(\Sigma^+) = 1.268$ and $m(\Sigma^-) = 1.276$ [179].

The threshold for the reaction $m + m \rightarrow M + M$ is $s = 4M^2$ if $M > m$. This corresponds to $p_{\text{lab}} = 98.7, 1435.3, 1853.1$, and $1898.9$ MeV/c, for $\bar{p}p \rightarrow \bar{n}\bar{n}$, $\bar{\Lambda}\Lambda$, $\bar{\Sigma}^-\Sigma^+$ and $\bar{\Sigma}^+\Sigma^-$, respectively.

In such a reaction $m + m \rightarrow M + M$, the momentum $p_2$ of one of the final state particle runs between the extreme values

$$p = \frac{\sqrt{s}}{4m} \left[ \sqrt{s - 4m^2} \pm \sqrt{s - 4M^2} \right]. \qquad (3.7)$$

In a given plane, $\vec{p}_2$ draws an ellipse, still with eccentricity $2m/\sqrt{s}$. The origin is outside if $M > m$, as shown in Fig. 3.4. The polar equation of the ellipse is

$$\cos \vartheta_1 = \frac{\sqrt{M^2 + p_1'^2} - m}{p_1' \sqrt{(s - 4m^2)/s}}. \qquad (3.8)$$
<figure><img src="image_25.png" /><figcaption>Fig. 3.4. Ellipse drawn by the momentum $\vec{p}'_1$ of the outgoing $\bar{\Lambda}$, for given $s$ in a given half-plane. The figure corresponds to $s = 6$ in units where $m = 1$ and $M = 1.1$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 3.5. Relation between the maximal scattering angle (in degrees) and the incident momentum (in MeV/c) for the reaction $\bar{p}p \to \bar{\Lambda}\Lambda$.</figcaption></figure>

Thus, as compared to the elastic case, there are two major differences:

(i) the range of scattering angle is restricted to $(0, \vartheta_M)$, where

$$ \cos \vartheta_M = \sqrt{\frac{s(M^2 - m^2)}{M^2(s - 4m^2)}}, \quad \sin \vartheta_M = \sqrt{\frac{m^2(s - 4M^2)}{M^2(s - 4m^2)}}. \qquad (3.9) $$

The relation between the maximal angle $\vartheta_M$ and the incident momentum $p_{\text{lab}}$ is illustrated in Fig. 3.5.

(ii) for a given angle $0 < \vartheta < \vartheta_M$, there are two possible values of the final-state momentum $p_2$.

## 3.2. Amplitudes and observables

### 3.2.1. Isospin formalism

In the limit where the neutron-to-proton mass difference can be neglected, as well as Coulomb corrections, the $\bar{N}N$ system obeys isospin symmetry: antiproton–neutron (or c.c.) is pure isospin
$I = 1$, while $\bar{p}\bar{p}$ and $\bar{n}\bar{n}$, with $I_3 = 0$, are combinations of $I = 1$ and $I = 0$, namely

$$ |\bar{p}\bar{p}\rangle = \frac{|I=1\rangle + |I=0\rangle}{\sqrt{2}}, \quad |\bar{n}\bar{n}\rangle = \frac{|I=1\rangle - |I=0\rangle}{\sqrt{2}}, \qquad (3.10) $$

so that the elastic and charge-exchange amplitudes are given by

$$ \mathcal{T}(\bar{p}\bar{p} \rightarrow \bar{p}\bar{p}) = \frac{1}{2}(\mathcal{T}_{\bar{N}\bar{N}}^{1} + \mathcal{T}_{\bar{N}\bar{N}}^{0}), \quad \mathcal{T}(\bar{p}\bar{p} \rightarrow \bar{n}\bar{n}) = \frac{1}{4}(\mathcal{T}_{\bar{N}\bar{N}}^{1} - \mathcal{T}_{\bar{N}\bar{N}}^{0}). \qquad (3.11) $$

Otherwise, the $\bar{p}\bar{n} \rightarrow \bar{p}\bar{n}$, $\bar{p}\bar{p} \rightarrow \bar{p}\bar{p}$ and $\bar{p}\bar{p} \rightarrow \bar{n}\bar{n}$ scattering processes should be treated in a formalism where proton and neutron are different particles.

Note that the relative sign in the above equations is a matter of convention. The choice adopted here differs from the current way of writing a SU(n) singlet as $|0\rangle \propto u\bar{u} + d\bar{d} + s\bar{s} + \dots$. For a comprehensive discussion of isospin wave functions for antiparticles, see, e.g., Ref. [180].

### 3.2.2. Spin amplitudes, elastic case

The description of the reaction $\bar{N}\bar{N} \rightarrow \bar{Y}Y'$, where Y and Y' are spin-1/2 baryons, involves 16 helicity amplitudes. Those are $\mathcal{T}_{\pm\pm\pm} = \mathcal{T}(\lambda_1, \lambda_2; \lambda'_1, \lambda'_2)$, if one uses the notations of Fig. 3.1.

In the elastic case $\bar{p}\bar{p} \rightarrow \bar{p}\bar{p}$, symmetry considerations reduce this number to 5 amplitudes, as for the well-studied cross-channel reaction $pp \rightarrow pp$. They can be chosen as

$$ \begin{aligned} \mathcal{T}_1 &= \mathcal{T}_{++++}, \\ \mathcal{T}_2 &= \mathcal{T}_{+--+}, \\ \mathcal{T}_3 &= \mathcal{T}_{+--}, \\ \mathcal{T}_4 &= \mathcal{T}_{+-+}, \\ \mathcal{T}_5 &= \mathcal{T}_{+---}. \end{aligned} \qquad (3.12) $$

There are many other sets of amplitudes, which are linear combinations of these $\mathcal{T}_i$. One of them is proposed by Lehar et al. [181]

$$ \begin{aligned} a &= (\mathcal{T}_1 + \mathcal{T}_2 + \mathcal{T}_3 - \mathcal{T}_4)(\cos \vartheta_{cm})/2 - 2\mathcal{T}_5 \sin \vartheta_{cm}, \\ b &= (\mathcal{T}_1 - \mathcal{T}_2 + \mathcal{T}_3 + \mathcal{T}_4)/2, \\ c &= (-\mathcal{T}_1 + \mathcal{T}_2 + \mathcal{T}_3 + \mathcal{T}_4)/2, \\ d &= (\mathcal{T}_1 + \mathcal{T}_2 - \mathcal{T}_3 + \mathcal{T}_4)/2, \\ e &= (-\mathcal{T}_1 - \mathcal{T}_2 - \mathcal{T}_3 + \mathcal{T}_4)(i \sin \vartheta_{cm})/2 + 2\mathcal{T}_5 \cos \vartheta_{cm}. \end{aligned} \qquad (3.13) $$

In the forward direction, one should satisfy $\mathcal{T}_4 = \mathcal{T}_5 = 0$, i.e.,

$$ e(0) = 0, \quad a(0) - b(0) = c(0) + d(0). \qquad (3.14) $$

The amplitudes $a, b, \dots$ can be defined directly as [181]

$$ \begin{aligned} \mathcal{T} = (a+b)I + (a-b)\vec{\sigma}_1 \cdot \hat{n} \vec{\sigma}_2 \cdot \hat{n} + (c+d)\vec{\sigma}_1 \cdot \hat{k} \vec{\sigma}_2 \cdot \hat{k} \\ + (c-d)\vec{\sigma}_1 \cdot \hat{p} \vec{\sigma}_2 \cdot \hat{p} + e(\vec{\sigma}_1 + \vec{\sigma}_2) \cdot \hat{n}, \end{aligned} \qquad (3.15) $$
where the unit vectors are defined as

$$ \hat{p} = \frac{\vec{p}_1 + \vec{p}'_1}{|\vec{p}_1 + \vec{p}'_1|}, \quad \hat{k} = \frac{\vec{p}'_1 - \vec{p}_1}{|\vec{p}'_1 - \vec{p}_1|}, \quad \hat{n} = \frac{\vec{p}_1 \times \vec{p}'_1}{|\vec{p}_1 \times \vec{p}'_1|}. \qquad (3.16) $$

In the limit of exact isospin symmetry, parity, time-reversal and G-parity relations hold for each isospin channel, and thus a set of five amplitudes $\mathcal{T}_i$ or $\{a, b, \dots\}$ can be introduced for each isospin state, $I = 0$ and $1$.

The symmetry-violating amplitudes are discussed, e.g., in Ref. [182].

### 3.2.3. Observables, elastic case

The general formalism is described, e.g., in Ref. [181], where all possible beam and target polarisations are considered, as well as all possible spin measurements in the final state.

For $\bar{p}\bar{p}$ scattering, the total integrated cross-section includes an elastic and a charge-exchange parts. At low energy, the remainder is the annihilation cross-section, i.e.,

$$ \sigma_{\text{tot}} = \sigma_{\text{el}} + \sigma_{\text{ce}} + \sigma_{\text{ann}}. \qquad (3.17) $$

For $\bar{p}n$ or $\bar{n}p$ scattering, we simply have

$$ \sigma_{\text{tot}} = \sigma_{\text{el}} + \sigma_{\text{ann}}. \qquad (3.18) $$

The total cross-section $\sigma_{\text{tot}}$ is related to the forward amplitude ($\vartheta_{\text{cm}} = 0$) by the optical theorem. It reads [181], in terms of the polarisation $\vec{P}_t$ of the target and $\vec{P}_b$ of the beam

$$ \begin{aligned} \sigma_{\text{tot}} &= \sigma_{0,\text{tot}} + \sigma_{1,\text{tot}}\vec{P}_t \cdot \vec{P}_b + \sigma_{2,\text{tot}}\vec{P}_t \cdot \hat{p}\vec{P}_b \cdot \hat{p}, \\ \sigma_{0,\text{tot}} &= \frac{2\pi}{p} \text{Im}[a(0) + b(0)], \\ \sigma_{1,\text{tot}} &= \frac{2\pi}{p} \text{Im}[c(0) + d(0)], \\ \sigma_{2,\text{tot}} &= -\frac{4}{p} \text{Im}[d(0)]. \end{aligned} \qquad (3.19) $$

The observables measured in elastic and charge-exchange antiproton-proton scattering are restricted to the differential cross-section $I_0 = d\sigma/d\Omega$, the analysing power $A_{000n}$, sometimes abbreviated as $A_{0n}$ or $A_n$, the depolarisation $D_{0n0n} = D_{nn}$, and some indirect information on the transfer of polarisation $K_{n00n} = K_{nn}$ for charge-exchange. These observables are given by

$$ \begin{aligned} I_0 &= |a|^2 + |b|^2 + |c|^2 + |d|^2 + |e|^2, \\ I_0 A_n &= 2 \text{Re}(a^* e), \\ I_0 D_{nn} &= |a|^2 + |b|^2 - |c|^2 - |d|^2 + |e|^2, \\ I_0 K_{nn} &= |a|^2 - |b|^2 + |c|^2 - |d|^2 + |e|^2. \end{aligned} \qquad (3.20) $$
More observables can be deduced from the formulas displayed in (3.28) for $\bar{p}p \to \bar{\Lambda}\Lambda$, using the prescription given below in Eq. (3.26).

The analysing power $A_n$ measures the azimuthal dependence of the angular distribution

$$d(\vartheta_{cm}, \varphi_{cm}) \propto I_0(1 + A_n P_{2,n} \cos \varphi_{cm}). \qquad (3.21)$$

$D_{nn}$ reflects how much of the initial polarisation of the proton target normal to the scattering plane remains in the recoil nucleon, and $K_{nn}$, how much is transferred to the scattered antinucleon, namely

$$P'_{2,n} = A_n + D_{nn}P_{2,n}, \quad P'_{1,n} = A_n + K_{nn}P_{2,n}, \qquad (3.22)$$

if $\vec{P}_i$ denotes the polarisation the $i$th particle (with the usual convention 1 = beam, 2 = target, 1' = scattered, 2' = recoil).

### 3.2.4. Spin amplitudes for charge exchange

At high energy, the neutron-to-proton mass difference can be neglected, and the spin observables are given by the same expressions as for the elastic case. The only difference lies in the isospin combinations (3.11) used for the amplitudes.

If one analyses experiments close to the threshold, or if one suspects for any other reason that isospin symmetry might be violated, then one should introduce a sixth amplitude and use the formalism given in the next subsection.

### 3.2.5. Spin amplitudes for strangeness exchange

The case of unequal masses in the final state, as for $\bar{p}p \to \bar{\Sigma}^0\Lambda + \text{c.c.}$ is rather straightforward. The data for this channel are, however, too meager to deserve a detailed formalism. We shall thus restrict ourselves here to reactions of the type $m + m \to M + M$. As compared to the elastic case, time-reversal invariance is lost, and six amplitudes are needed, instead of five.

A recent discussion of the $\bar{p}p \to \bar{\Lambda}\Lambda$ spin formalism has been done in Refs. [183,184], where references to earlier works can be found.

At each energy and angle, the transition matrix can be decomposed into [181,185]

$$\begin{aligned} \mathcal{T} = (a' + b')I + (a' - b')\vec{\sigma}_1 \cdot \hat{n}\vec{\sigma}_2 \cdot \hat{n} + (c' + d')\vec{\sigma}_1 \cdot \hat{k}'\vec{\sigma}_2 \cdot \hat{k}' \\ + (c' - d')\vec{\sigma}_1 \cdot \hat{p}'\vec{\sigma}_2 \cdot \hat{p}' + e'(\vec{\sigma}_1 + \vec{\sigma}_2) \cdot \hat{n} + g'(\vec{\sigma}_1 \cdot \hat{k}'\vec{\sigma}_2 \cdot \hat{p}' + \vec{\sigma}_1 \cdot \hat{p}'\vec{\sigma}_2 \cdot \hat{k}') \end{aligned} \qquad (3.23)$$

where the kinematical unit vectors are defined from the momentum $\vec{p}_1$ of $\bar{p}$ and $\vec{p}'_1$ of $\bar{\Lambda}$:

$$\hat{p}' = \frac{\vec{p}_1}{|\vec{p}_1|}, \quad \hat{n} = \frac{\vec{p}_1 \times \vec{p}'_1}{|\vec{p}_1 \times \vec{p}'_1|}, \quad \hat{k}' = \hat{n} \times \hat{p}' \qquad (3.24)$$

and are adapted to describe the final-state spins. Since

$$\hat{p} = \hat{p}' \cos(\vartheta_{cm}/2) - \hat{k}' \sin(\vartheta_{cm}/2), \quad \hat{k} = \hat{p}' \sin(\vartheta_{cm}/2) + \hat{k}' \cos(\vartheta_{cm}/2), \qquad (3.25)$$

the elastic amplitudes (3.15) correspond to the special case $a' = a, b' = b, c' = c, e' = e$, and

$$d' = d \cos \vartheta_{cm}, \quad g' = d \sin \vartheta_{cm}. \qquad (3.26)$$
### 3.2.6. Spin observables for strangeness production

The rank-1 and rank-2 observables are defined (to an overall factor) as

$$
\begin{align*}
I_0 &= \mathrm{Tr}[\mathcal{T}\mathcal{T}^\dagger], \\
P_n I_0 &= \mathrm{Tr}[\vec{\sigma}_1 \cdot \hat{n}\mathcal{T}\mathcal{T}^\dagger], \\
A_n I_0 &= \mathrm{Tr}[\mathcal{T}\vec{\sigma}_2 \cdot \hat{n}\mathcal{T}^\dagger], \\
C_{ij} I_0 &= \mathrm{Tr}[\vec{\sigma}_1 \cdot \hat{i}\vec{\sigma}_2 \cdot \hat{j}\mathcal{T}\mathcal{T}^\dagger], \\
D_{ij} I_0 &= \mathrm{Tr}[\vec{\sigma}_2 \cdot \hat{i}\mathcal{T}\vec{\sigma}_2 \cdot \hat{j}\mathcal{T}^\dagger], \\
K_{ij} I_0 &= \mathrm{Tr}[\vec{\sigma}_1 \cdot \hat{i}\mathcal{T}\vec{\sigma}_2 \cdot \hat{j}\mathcal{T}^\dagger], \tag{3.27}
\end{align*}
$$

corresponding to differential cross-section, polarisation, analysing power, spin correlation in the final state, baryon depolarisation and baryon-to-antibaryon polarisation transfer, respectively.

More explicitly, in terms of the amplitudes

$$
\begin{align*}
I_0 &= |a'|^2 + |b'|^2 + |c'|^2 + |d'|^2 + |e'|^2 + |g'|^2, \\
P_n I_0 &= 2 \operatorname{Re}(a'e^{*}) + 2 \operatorname{Im}(d'g^{*}), \\
A_n I_0 &= 2 \operatorname{Re}(a'e^{*}) - 2 \operatorname{Im}(d'g^{*}), \\
C_{nn} I_0 &= |a'|^2 - |b'|^2 - |c'|^2 + |d'|^2 + |e'|^2 + |g'|^2, \\
C_{xx} I_0 &= -2 \operatorname{Re}(a'd^{*} + b'c^{*}) - 2 \operatorname{Im}(g'e^{*}), \\
C_{zz} I_0 &= 2 \operatorname{Re}(a'd^{*} - b'c^{*}) + 2 \operatorname{Im}(g'e^{*}), \\
C_{xz} I_0 &= -2 \operatorname{Re}(a'g^{*}) - 2 \operatorname{Im}(e'd^{*}), \\
D_{nn} I_0 &= |a'|^2 + |b'|^2 - |c'|^2 - |d'|^2 + |e'|^2 - |g'|^2, \\
D_{xx} I_0 &= 2 \operatorname{Re}(a'b^{*} + c'd^{*}), \\
D_{zz} I_0 &= 2 \operatorname{Re}(a'b^{*} - c'd^{*}), \\
D_{xz} I_0 &= 2 \operatorname{Re}(c'g^{*}) + 2 \operatorname{Im}(b'e^{*}), \\
K_{nn} I_0 &= |a'|^2 - |b'|^2 + |c'|^2 - |d'|^2 + |e'|^2 - |g'|^2, \\
K_{xx} I_0 &= -2 \operatorname{Re}(a'c^{*} + b'd^{*}), \\
K_{zz} I_0 &= -2 \operatorname{Re}(a'c^{*} - b'd^{*}), \\
K_{xz} I_0 &= -2 \operatorname{Re}(b'g^{*}) + 2 \operatorname{Im}(e'c^{*}). \tag{3.28}
\end{align*}
$$

To project out the spins of the particles, we follow here the usual convention that for $\bar{\Lambda}$, the axes $\{\hat{x}, \hat{n}, \hat{z}\}$ coincide with $\{\hat{k}, \hat{n}, \hat{p}\}$, while for p or $\Lambda$, the axes $\{\hat{x}, \hat{n}, \hat{z}\}$ coincide with $\{-\hat{k}, \hat{n}, -\hat{p}\}$.

In principle (i.e., with enough statistics), a polarised target gives access to some rank-3 observables, of the type

$$ C_{0xij} I_0 = \mathrm{Tr}[\vec{\sigma}_1 \cdot \hat{i}\vec{\sigma}_2 \cdot \hat{j}\mathcal{T}\vec{\sigma}_2 \cdot \hat{\alpha}\mathcal{T}^\dagger]. \tag{3.29} $$
For instance,

$$
\begin{align}
C_{0nzz}I_0 &= 2 \operatorname{Re}(d'e'^*) - 2 \operatorname{Im}(a'g'^*) , \nonumber \\
C_{0nxx}I_0 &= -2 \operatorname{Re}(d'e'^*) + 2 \operatorname{Im}(a'g'^*) , \nonumber \\
C_{0nzx}I_0 &= -2 \operatorname{Re}(g'e'^*) - 2 \operatorname{Im}(a'c'^* + b'd'^*) , \tag{3.30}
\end{align}
$$

$C_{0nnn}$ being equal to $A_n$.

3.2.7. Constraints on observables

Each spin observable X or Y is typically normalised to −1 ≤ X ≤ + 1. However, the allowed domain for a pair (X, Y) of observables is usually smaller than the unit square (|X| ≤ 1, |Y| ≤ 1). Inequalities can be derived, which restrict the domain. They are, of course, automatically fulfilled in any theoretical model, where the amplitudes a ′ , b ′ ,... are first calculated and then used to compute the observables. However, these inequalities represent non-trivial constraints when the observables are extracted from data. Similar relations have been written for other reactions, for instance photo- production of vector mesons off nucleons [186].

We just list a few examples below. For a more comprehensive list, and a possible derivation,
see Ref. [183].

(i) Linear relations:

$$
\begin{align}
& 2|A_n| - C_{nn} \le 1, \nonumber \\
& I_0(1 - C_{nn} + C_{xx} + C_{zz}) \ge 0. \tag{3.31}
\end{align}
$$

Note that the second relation is nothing but the spin-singlet fraction,

$$
F_0 = \frac{1}{4} (1 + C_{xx} - C_{yy} + C_{zz}) = \frac{1}{2I_0} |b' - c'|^2, \quad (3.32)
$$

being positive. The normalisation is such that $F_0 = 1/4$ in absence of any spin-dependent interaction.

(ii) Quadratic relations:

$$
\begin{align}
C_{zz}^2 + D_{nn}^2 &\le 1, \nonumber \\
\left( \frac{D_{nn} - K_{nn}}{2} \right)^2 &+ (2F_0 - 1)^2 \le 1, \tag{3.33}
\end{align}
$$

the latter relating $D_{nn}$, $K_{nn}$, $C_{nn}$, $C_{xx}$ and $C_{zz}$. As a consequence, $D_{nn} = K_{nn}$ in both limits of a pure spin-singlet ($F_0 = 1$) or pure spin-triplet ($F_0 = 0$) reaction.

3.3. Possibility of reconstructing the amplitudes from the data

3.3.1. General considerations

Extracting the amplitudes from data is a rather delicate subject. Consider first the case of spinless
particles. Measuring the differential cross-section at a given energy provides |f|², where f is the
scattering amplitude. There are difficulties to access the phase from the angular distribution [187], even if one uses elastic unitarity globally or under the form of a partial wave expansion,

$$f = \sum_{\ell} (2\ell + 1) \frac{\exp(2i\delta_{\ell}) - 1}{2ik} P_{\ell}(\cos\vartheta), \qquad (3.34)$$

where $\delta_{\ell}$ is real and the expansion finite. The absolute phase of $f$ can sometimes be derived, for instance from its interference in the extreme forward region with the Coulomb amplitude.

More severe difficulties occur in case of inelasticity ($\delta_{\ell}$ parameters, now complex) or particles with spin (several amplitudes involved). Low-energy $\bar{N}N$ scattering cumulates both handicaps.

The case of two amplitudes is very much documented, as it corresponds to $\pi N$ scattering or its crossed reaction $\bar{N}N \to \pi\pi$. The angular distribution gives $I_0 = |f|^2 + |g|^2$ and the polarisation (or analysing power) $P_n = 2 \text{Re}(fg^*)/I_0$. If the latter is extreme ($P_n = \pm 1$), $f = \pm g$ is implied. In general, the value of this parameter is not maximal, and several choices might be considered. For instance, $P_n = 0$ can be achieved with $f = 0$, or $g = 0$, or more generally with $f \propto ig$. In short, reconstructing the amplitudes from the data becomes easier in situations where some spin parameters are extreme.

Still a few common-sense considerations hold. One cannot determine unambiguously 5 or 10 complex amplitudes from the measurement of just two or three observables. It took a long time to achieve the “tour de force” of reconstructing the five amplitudes in the NN case. See, e.g., Ref. [188]. Several subtle measurements were required, with beam and target both polarised.

Now, the $\bar{N}N$ case looks by far more difficult than the NN one. First, there is no Pauli principle to remove every second partial wave. Second, the phase-shifts are complex already at threshold (or, equivalently, the partial-wave $S$-matrix contains both a real phase and an inelasticity parameter). Moreover, there are much fewer data than in the NN case. So there is no hope to achieve an unambiguous amplitude or even phase-shift analysis.

Still, integrated cross-sections are now available in a wide range of energy. For the elastic $\bar{p}p \to \bar{p}p$ and the charge-exchange reaction $\bar{p}p \to \bar{n}n$, we also have detailed angular distributions and polarisations, as well as some indication on the depolarisation parameter. For $\bar{n}p$ or $\bar{p}n$ scattering, we are restricted to the integrated cross-sections. Several observables have been measured for hyperon pair production.

It remains that one can always fit the data within some models which summarise the best of our knowledge on the physics of the $\bar{N}N$ interaction and contain several free parameters. In these circumstances, the resulting amplitudes and the corresponding complex phase-shifts are compatible with the data. Such analyses are thus often quoted for comparison with the data, and they provide a possible extrapolation for observables which have not yet been measured. Of course, it is more delicate to appreciate to which extent a set of amplitudes obtained by tuning a specific model can be considered as unique.

### 3.3.2. Elastic or charge-exchange case

Let us examine more precisely here what can be learned from the data on elastic $\bar{p}p$ or charge exchange. As it will be seen in the next chapter, the only observables which have been measured consist of the differential cross-section $I_0 = d\sigma/d\Omega$, the analysing power $A_n$ (sometimes called polarisation) and the depolarisation $D_{nn}$. Though a lot of physics can be extracted (pion coupling, early onset of P-waves, etc.), we are far from envisaging a full reconstruction of the five amplitudes.
<figure><img src="image_27.png" /><figcaption>Fig. 3.6. One-photon exchange, the driving mechanism for $e^−e^−$ interaction.</figcaption></figure>

For the ease of discussion, let us replace Lehar’s amplitudes (3.15) $a$ and $e$ by their normalised sum and difference, namely

$$ \{a, e\} \rightarrow \{\tilde{a} = (a + e)/\sqrt{2}, \tilde{e} = (a - e)/\sqrt{2}\}, \qquad (3.35) $$

and omit the tilde mark in this subsection. These new $a$ and $e$ correspond to transversity amplitudes [181]. The available observables read

$$ \begin{aligned} I_0 &= |a|^2 + |b|^2 + |c|^2 + |d|^2 + |e|^2, \\ I_0 A_n &= |a|^2 - |e|^2, \\ I_0 D_{nn} &= |a|^2 + |b|^2 - |c|^2 - |d|^2 + |e|^2, \end{aligned} \qquad (3.36) $$

this defining a possible normalisation. In the optimal cases, i.e., at energies and angles where these three observables are measured, one can at best extract from the data the quantities

$$ |a|^2 + |b|^2/2, \quad |e|^2 + |b|^2/2, \quad |c|^2 + |d|^2, \qquad (3.37) $$

or linear combinations. Improving the accuracy on $A_n$ or $D_{nn}$ will not enable one to disentangle $|c|^2$ from $|d|^2$, for instance, unless it is found that $|c|^2 + |d|^2 = 0$, which would imply $c = d = 0$.

### 3.3.3. Hyperon-pair production

We restrict ourselves here to the $\bar{\Lambda}\Lambda$ case, as channels involving a $\Sigma$ or a $\bar{\Sigma}$ do not benefit of the same amount of data on spin observables. Would the question of statistics be disregarded, the situation would be different from the $\bar{\mathbb{N}}\mathbb{N}$ case, with the risk of having too many observables, and thus the problem of compatibility among those observables.

As pointed out in Ref. [184], experiments carried out with a transversally polarised target lead to many rank-2 observables (correlations in the final state, transfer from p to $\Lambda$ or to $\bar{\Lambda}$), and even some rank-3 observables. There are enough relations to fix $a'$, $b'$, $c'$, $d'$, $e'$ and $g'$, to an overall phase.

## 3.4. General properties of the $\bar{\mathbb{N}}\mathbb{N}$ interaction

The best known particle-antiparticle system consists of an electron $e^−$ and a positron $e^+$, which is described by comparison with the two-electron system. In this latter case, the interaction is known to be mediated by the exchange of a photon, as pictured in Fig. 3.6. This diagram is then iterated in the Schrödinger equation or in some more elaborate relativistic equation, and supplemented by small higher-order corrections.
<figure><img src="image_28.png" /><figcaption>Fig. 3.7. Basic mechanisms for $e^{+}e^{-}$ interaction: elastic photon exchange (left), virtual annihilation into one photon (centre), annihilation into two photons (right). (What is pictured is not annihilation itself, but its feedback on the $e^{+}e^{-}$ amplitude.)</figcaption></figure>

The C-conjugation rule states that, since the exchanged photon has $C = -1$, the corresponding diagram for $e^{+}e^{-}$ changes sign. In other words, the repulsive $1/r$ potential becomes an attractive $V \propto -1/r$ for the electron-positron case. This is, however, not the only change. At the same lowest order, there is the annihilation diagram of Fig. 3.7, centre. The two particles merge into a virtual photon which then decays back to $e^{+}e^{-}$. This is a contact interaction. Another annihilation diagram is shown in Fig. 3.7, right. It corresponds to the feed-back into the $e^{+}e^{-}$ amplitude of the annihilation processes $e^{+}e^{-} \to n\gamma$, pictured here for $n=2$. This annihilation part is, however, a small correction, due to its short range $(2m_e)^{-1}$, where $m_e$ is the mass of the electron. For instance, the positronium atom has well-defined levels. The S-wave levels eventually decay into photons, but with a lifetime which is much larger than the classical period of rotation. States with angular momentum $l \ge 1$ are practically stable with respect to annihilation. They decay by ordinary radiative transitions to lower states.

At first sight, this dynamics is easily translated for the $\bar{N}N$ system, with electrons, positrons and photons being replaced by nucleons, antinucleons and pions, respectively. More attraction is expected in $\bar{N}N$ than in NN, leading to speculations on "Baryonium" states, the analogues of positronium atoms.

This is, however, far from being that simple. A minor complication comes from isospin or, say, the existence of two nucleons. As will be seen in Section 3.5, the C-conjugation rule has to be rewritten with the G-parity operator. More serious is the strength of the coupling constant, and the variety of mesons. The NN interaction cannot be reduced to one-pion exchange: multiple exchanges, as well as the exchange of heavier meson resonances should be introduced. Moreover, we do not know precisely the nature of the short-range NN forces. This difficulty is of course, translated into a similar uncertainty for the elastic part of the $\bar{N}N$ interaction. But the worse is still to come: annihilation cannot be handled as a short-range correction: it is a violent process, which takes place up to about 1 fm and presumably washes out the bound-state spectrum generated by the attractive real potential.

A possible way out is that the correspondence between electromagnetic- and strong-interaction physics should be stated at the quark level rather than at the nucleon level. Then electrons and photons of QED are replaced by quarks and gluons, respectively. This is certainly the right approach at high energy and high momentum transfer. For our low-energy physics, one cannot elude the problem of confinement, and the many-body aspects: $\bar{N}N$ annihilation is not an elementary process, it is the analogue of a molecular-collision process in QED, with several possible final states.

We shall discuss below these aspects of the $\bar{N}N$ interaction and look what understanding was reached when LEAR came into operation.
### 3.5. The G-parity rule for amplitudes

It is well known in quantum field theory that the same S-matrix describes the reaction $a+b \to c+d$ and its crossed reactions such as $a + \bar{c} \to \bar{b} + d$. However, the empirical knowledge of the direct reaction, in a kinematical domain {$s > 4m^2, t < 0$} (for equal masses), does not give easy access to the domain {$s < 0, t > 4m^2$}, as no analytic continuation can be used reliably for such a huge jump.

Another point of view is adopted in the so-called “G-parity rule”: one compare the $a+b \to c+d$ and $\bar{a}+b \to \bar{c}+d$ reactions in the same kinematical domain.

The G-parity $G = C \exp(-i\pi I_2)$ associates the C-conjugation with a rotation in isospin space, such that $[180] G | p \rangle = | \bar{n} \rangle$ and $G | n \rangle = | \bar{p} \rangle$, linking states with same isospin quantum numbers. Non-strange meson systems are eigenstates with $G = +1$ or $-1$. In particular, for a system of n pions, $G = (-1)^n$.

The mesons which contribute to the NN forces can also be exchanged between a nucleon and an antinucleon to build the long-range part of the $\bar{N}N$ interaction. The precise relation between the NN and $\bar{N}N$ amplitudes is the G-parity rule: in a given isospin state, $I=0$ or $1$, the contribution of the exchange of a meson or set of mesons $\mu$ to the $\bar{N}N$ amplitude differs from its NN analogue by a factor $G_\mu = \pm 1$, which is the G-parity of the meson system $\mu$.

$$ \mathcal{T}_{\text{NN}}^{I} = \sum_{\mu} \mathcal{T}_{\mu} \Rightarrow \mathcal{T}_{\bar{N}N}^{I} = \sum_{\mu} G_{\mu} \mathcal{T}_{\mu}. \qquad (3.38) $$

Remarks are in order:

(i) This rule links isospin eigenstates. It generalises the familiar “C-conjugation rule”, which states that $\mathcal{T}(e^+e^-) = -\mathcal{T}(e^+e^+)$ for the leading diagram, since the photon which is exchanged has $C = -1$. The C-conjugation rule links pp to $\bar{p}p$, np to $\bar{p}n$. The G-parity rule is nothing but an astute combination of the C-conjugation rule with isospin invariance.

(ii) The G-parity rule is by no means restricted to the one-boson-exchange approximation, where only single, narrow mesons are involved. It works for instance for two-pion-exchange ladder or crossed diagrams, with or without excitation of nucleon resonances in the intermediate states.

The derivation of the G-parity rule can be elaborated as follows. Both amplitudes, $\mathcal{T}_1$ for NN and $\mathcal{T}_2$ for $\bar{N}N$, are formally written in terms of their t-channel content

$$ \begin{aligned} \mathcal{T}_1 &= \langle \bar{N}(-p_1)N(p_2) | \mathcal{T} | N(n_1)\bar{N}(-n_2) \rangle, \\ \mathcal{T}_2 &= \langle N(-p_1)\bar{N}(p_2) | \mathcal{T} | N(n_1)\bar{N}(-n_2) \rangle. \end{aligned} \qquad (3.39) $$

The kinematics is defined in Fig. 3.1. Since strong interactions are invariant under G-parity, $\mathcal{T} = G\mathcal{T}G$, each amplitude can be split into two parts, with $G = +1$ or $-1$ in the t-channel, without interference between them:

$$ \mathcal{T}_i = \mathcal{T}_i^+ + \mathcal{T}_i^- . \qquad (3.40) $$

In particular,

$$ \mathcal{T}_1^{\pm} = \langle \bar{N}(-p_1)N(p_2) | G\mathcal{T}G | N(n_1)\bar{N}(-n_2) \rangle. \qquad (3.41) $$
The first $G$, acting on the left, transforms the bra into the desired $\langle N(-p_1)\bar{N}(p_2) \rangle$, while the second $G$ can be replaced by its eigenvalue$\pm$, with the desired result $\mathcal{T}_1^{\pm} = \pm \mathcal{T}_2^{\pm}$.

The $G$-parity rule is rather general, and reflects basic symmetries of strong interactions. This is why the rule is first formulated in terms of scattering amplitudes. In the particular case, where the amplitude is generated from a potential, the rule implies that the potential mediated by a meson (or set of mesons) of $G$-parity $G_\mu = \pm 1$ is multiplied by $G_\mu$ when translated from NN to $\bar{N}N$, in a given isospin state $I$. For instance, the one-pion-exchange potential changes sign when going from NN and $\bar{N}N$, and so does its Born approximation when it generates an amplitude. Its first iteration (second term in the Born expansion) is identical for NN and $\bar{N}N$, the next iteration flips its sign, etc.

As we shall see along this review, the $G$-parity rule (3.38) induces dramatic changes in the long-range potential. First, the $\bar{N}N$ potential is on the average more attractive than the NN one. This is important for understanding the observed cross-sections and the speculations about possible $\bar{N}N$ bound states and resonances. Secondly, the spin and isospin dependence is different: the more pronounced effects are expected in the $I=0$ channel, and affect mainly the tensor component of the potential.

### 3.6. Potential models

The $G$-parity rule is the starting point of phenomenological studies of low-energy $\bar{N}N$ interaction based on potential models. Schematically, one starts from a realistic NN potential, apply the $G$-parity rule to its meson-exchange tail, and replace the short-range NN core by a complex short-range part to account both for the short-range elastic forces and for the strong annihilation component.

Potentials models are very useful tools

(i) to get a hint on the strength and range of annihilation,

(ii) to test the role of various components of the interaction ($\pi$-exchange, $\rho$-exchange, absorption, etc.) in various observables,

(iii) to make quantitative comparisons with NN interaction,

(iv) to use information on the “elementary” $\bar{N}N$ interaction for studying antinucleon–nucleus interaction.

It should remain clear, however, that there is no fundamental reason for believing that the $\bar{N}N$ interaction should reduce to a potential, especially if this potential is restricted to be local and energy independent.

Any microscopic derivation of the absorption potential from the internal quark structure gives, indeed, a kernel that is (i) separable rather than local and (ii) highly sensitive to the difference between the incoming energy and the location of the main thresholds (e.g., the threshold for two vector mesons and a pseudoscalar). See, for instance, [189].

#### 3.6.1. Current NN potentials

From the 1950s to the 1970s, a rather successful description of the NN interaction was achieved with “semi-phenomenological” potentials. They consist of a long-range part which include meson exchanges and a short-range part which describes empirically the observed repulsion with a few
parameters. These potentials have been tuned to reproduce the NN scattering data with good accuracy and used in nuclear-structure calculations. For a review, see, e.g., [190].

### 3.6.2. Pion-exchange and more elaborate nucleon-nucleon potentials

There are several reviews on the meson-exchange approach to the NN interaction. We thus refer to the literature [190–192] for a detailed survey and restrict ourselves here to what is necessary for extrapolating to the $\bar{N}N$ case.

Following Yukawa, the exchange of a particle of mass $\mu$ leads to a potential of range $\hbar c/\mu c^2$ (hereafter simplified into $\mu^{-1}$), namely

$$V(r) \propto \frac{\exp(-\mu r)}{r}, \qquad (3.42)$$

when spin and isospin complications are omitted. In momentum space, the one-pion-exchange (OPE) potential reads as

$$V_{\text{OPE}} = \frac{g^2}{\vec{q}^2 + \mu^2}. \qquad (3.43)$$

In the actual situation, the pion is pseudoscalar and isovectorial, and gives a characteristic spin and isospin dependence. The OPE potential is an operator acting on isospin state and Dirac spinor of each nucleon

$$V_{\text{OPE}} \propto \gamma_5^{(1)} \gamma_5^{(2)} \vec{\tau}_1 \cdot \vec{\tau}_2 \frac{1}{\vec{q} + \mu^2}, \qquad (3.44)$$

where $\vec{\tau}_i$ is the isospin operator for nucleon (i) and $\vec{q} = \vec{p}_2 - \vec{p}_1$ is the momentum transfer. After non-relativistic reduction, one obtains in position space a combination of spin-spin and tensor potentials, namely,

$$V_{\text{OPE}} = (V_{\text{SS}} \vec{\sigma}_1 \cdot \vec{\sigma}_2 + V_{\text{T}} S_{12}) \vec{\tau}_1 \cdot \vec{\tau}_2,$$

$$V_{\text{SS}} = \frac{g^2}{4\pi} \frac{\mu^2}{4m^2} \frac{\exp(-\mu r)}{r},$$

$$V_{\text{T}} = \frac{g^2}{4\pi} \frac{\mu^2}{4m^2} \frac{\exp(-\mu r)}{r} \left( 1 + \frac{3}{\mu r} + \frac{3}{\mu^2 r^2} \right), \qquad (3.45)$$

where the tensor operator is

$$S_{12} = 3\vec{\sigma}_1 \cdot \hat{r} \vec{\sigma}_2 \cdot \hat{r} - \vec{\sigma}_1 \cdot \vec{\sigma}_2, \qquad (3.46)$$

and the isospin coefficient

$$\vec{\tau}_1 \cdot \vec{\tau}_2 = 1 \quad \text{for } I = 1,$$

$$\vec{\tau}_1 \cdot \vec{\tau}_2 = -3 \quad \text{for } I = 0. \qquad (3.47)$$

This potential $V_{\text{OPE}}$ successfully explains the peripheral NN phase-shifts at low energy, as well as the quadrupole deformation of the deuteron. Now, some departures from the simple OPE model are
<figure><img src="image_29.png" /><figcaption>Fig. 3.8. Box diagram, part of the two-pion-exchange contribution to NN interaction.</figcaption></figure>

seen in NN phase-shifts of lower angular momentum $\ell$, or in peripheral partial waves when energy increases. This indicates the need for additional terms that have a range shorter than OPE.

A first improvement consists of allowing the nucleons to exchange mesons with larger mass and new quantum numbers, leading to additional potentials that have a shorter range and exhibit new kinds of spin-isospin dependence. Strictly speaking, one should distinguish the one-boson-exchange models (OBEM) with a direct parametrisation of the scattering *amplitude* in terms of *t*-channel poles

$$M_{\text{OBEM}} = \sum_{i} \frac{g_i}{\vec{q}^2 + \mu_i^2}, \qquad (3.48)$$

from the one-boson-exchange potentials (OBEP) where these exchanges are written for the *potential*

$$V = \sum_{i} g_{i} \frac{\exp(-\mu_{i} r)}{r}, \qquad (3.49)$$

and thus are iterated in the Schrödinger equation. Eqs. (3.48) and (3.49) should be understood with spin and isospin factors which are specific for each type of mesons, and play an important role. One needs for instance vector mesons to generate spin-orbit forces for $I = 1$ and to cancel the tensor part of OPEP at short distances in the isospin $I = 0$ channel. The existence of vector mesons was in fact anticipated by scrutinising NN scattering data and their spin dependence [1,193].

These OBE models were elaborated in the 1960s and 1970s, typically. In the 1950s and 1960s, there were attempts to construct a field theory of nuclear forces, in analogy with QED, but based on the Yukawa coupling $g\gamma_5$ of the pseudoscalar pion field to the nucleons. This turned out eventually unsuccessful, but the studies revealed the role of two-pion exchange (TPE) in the intermediate range attraction. This is why all modern NN potentials include TPE contributions explicitly.

In fact, $\rho$ or $\sigma^4$ exchanges are already part of TPE, $\rho$ and $\sigma$ being resonances of the $\pi\pi$ system. Also, since TPE includes the box (ladder) diagram of Fig. 3.8 with nucleons in the intermediate states, it is natural to account for these ladder or crossed diagrams where one or two $\Delta$ or $N^*$ are excited. All these contributions to TPE are incorporated, without double counting, in the dispersion-theoretical calculation whose formalism was written down by Amati et al. [194,195] and later on used to compute the medium-range NN potential [190,191].

In the approach based on dispersion relations, one has given up the idea of an expansion in powers of the pion-nucleon coupling constant $g$. In particular, TPE contains diagrams where the

<footnote n="4">⁴ In the early literature on nuclear forces, $\epsilon$ or $\sigma$ denotes a scalar and isoscalar meson exchanged between the nucleons. Today the amplitude for two pions in a scalar and isoscalar state below ~ 1.2 GeV is often designated as $\sigma$. If it has to be identified with an actual meson, then it should be renamed $f_0$.</footnote>
exchanged pions rescatter several times with each others or with the nucleons. Instead, one achieves an expansion according to the range of the various contributions: OPE provides the tail of the potential and TPE the medium part. This gives a good description of the NN phase-shifts with $\ell > 1$ at low energy.

In potentials designed for practical purpose, the short-range part of the NN interaction is described empirically by cutting off meson exchanges at distances $r \lesssim 1$ fm and adding a phenomenological core to account for the observed repulsion. This is the basis of the Paris, Bonn and others semi-phenomenological potentials [191,192] that are widely used for nuclear-physics calculations. Some potentials have been updated regularly to reach better accuracy [196].

In early studies of the NN interaction, the short-range repulsion was attributed to the exchange of heavy vectors mesons. It is now explained by the direct interaction of the quarks which build up the nucleons. There is an abundant literature on this subject [197,198] and one can say that a semi-quantitative understanding of the short-range NN potential has been achieved. There are two major mechanisms: the exclusion principle between quarks and the chromomagnetic interaction.

To summarise, we have in hand potentials describing very accurately NN scattering and the deuteron properties which can be applied to nuclear systems. However, the theoretical ground is uncomfortably split into two parts. The Yukawa picture accounts for the long-range part in terms of nucleons and pions, while the short-range part is described directly at the quark level. A synthesis is badly needed. One should in principle derive microscopically the Yukawa model out of QCD, to end with a unified picture in terms of quarks. Alternatively, one could reformulate QCD at low energy in terms of pions and solitons. One attempt is the Skyrmion model. However, the bosonisation of QCD remains to be demonstrated rigorously.

More promising, perhaps, is the approach with effective Lagrangians which incorporate the basic symmetries of QCD. Applications to the NN interaction have already been done, and there is already a rich literature on the subject, which can be traced back from Refs. [199–205].

### 3.6.3. Long-range nucleon-antinucleon potential

Once the long-range NN potential is described in terms of exchange of mesons or set of mesons, schematically

$$V^I(\text{NN}) = \sum_m V_m, \qquad (3.50)$$

there is no difficulty in writing down the $\bar{\text{NN}}$ analogue

$$V^I(\bar{\text{NN}}) = \sum_m G_m V_m, \qquad (3.51)$$

except that $G_m$ should be clearly identified. For instance, the $\omega$-exchange contribution is sometimes thought as a resummation of many contributions of mass 0.8 GeV or higher. If this is $\pi\rho$, the conclusions are not changed as this system has the same G-parity $-1$ as $\omega$. If this is part of four-pion exchange, then the translation from NN to $\bar{\text{NN}}$ is misleading.

Sometimes the boson-exchange NN potential is regularised à la Pauli-Villars [206], i.e.,

$$\frac{1}{q^2 + \mu^2} \rightarrow \frac{1}{q^2 + \mu^2} - \frac{1}{q^2 + \Lambda^2}, \qquad (3.52)$$
Table 3.1

Sign of meson-exchange contributions to central (C), spin-spin (SS), spin-orbit (LS) and tensor (T) components of the NN and $\bar{N}N$ potentials.

<table><thead><tr><th rowspan="2">Exch. meson</th><th colspan="4">NN I = 0</th><th colspan="4">NN I = 1</th><th colspan="4">N̄N I = 0</th><th colspan="4">N̄N I = 1</th></tr><tr><th>C</th><th>SS</th><th>LS</th><th>T</th><th>C</th><th>SS</th><th>LS</th><th>T</th><th>C</th><th>SS</th><th>LS</th><th>T</th><th>C</th><th>SS</th><th>LS</th><th>T</th></tr></thead><tbody><tr><td>π</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>-</td><td>0</td><td>-</td></tr><tr><td>η</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td><td>+</td></tr><tr><td>ρ</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td><td>+</td><td>-</td><td>-</td></tr><tr><td>ω</td><td>+</td><td>+</td><td>-</td><td>-</td><td>+</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>a<sub>0</sub></td><td>+</td><td>0</td><td>+</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>+</td><td>0</td><td>+</td><td>0</td></tr><tr><td>σ</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td><td>-</td><td>0</td></tr></tbody></table>

the short-range term elegantly conspiring to parametrise a realistic core in terms of elementary Yukawa functions. Nothing guarantees that for the short-range interaction, the same regularising term should be applied to $\bar{N}N$, using the G-parity factor. In absence of a microscopic derivation, the G-parity content of the core is simply unknown.

The translation of NN forces toward the $\bar{N}N$ system has been performed by several authors, including Fermi and Yang [207], and Ball and Chew [208], the latter been motivated by the first measurements of antiproton cross-sections. The potentials by Gourdin et al. [209], Bryan and Philips [210], Dover and Richard [211], Kohno and Weise [212], just to mention a few examples, will be discussed shortly.

### 3.6.4. Spin-isospin dependence of the long-range potential

We now discuss the spin and isospin dependence of the long-range $\bar{N}N$ potential. For simplicity, we shall often adopt the language of the one-boson-exchange approximation, but most of the analysis remains valid in more elaborated models.

Consider any $J^P$ exchange. Thanks to the exchange degeneracy of the meson spectrum, there are always two mesons, one with $I=0$ and another with $I=1$, with the same C-conjugation (for the $I_3=0$ states) and different G-parity. For instance, the vector-meson exchange contribution to NN being

$$V_V = V_\omega + \vec{\tau}_1 \cdot \vec{\tau}_2 V_\rho, \qquad (3.53)$$

it reads

$$ \begin{aligned} V_V^{I=0} &= V_\omega - 3V_\rho, \\ V_V^{I=1} &= V_\omega + V_\rho. \end{aligned} \qquad (3.54) $$

Coherences appear in the isospin $I=1$ channel. Now, if one looks at the cumulated effect of scalar, vector and pseudoscalar exchanges, the most pronounced coherence occurs in the spin-orbit potential, as seen in Table 3.1. It is well known, indeed, that protons experience strong spin-orbit forces when scattered on protons or on nuclei. The spin-spin component of the NN interaction is less visible, since the Pauli principle forbids the existence of the triplet state if the singlet is allowed, or vice versa.
<figure><img src="image_30.png" /><figcaption>Fig. 3.9. Comparison of the NN and $\bar{N}N$ spin-orbit potentials in the Paris model.</figcaption></figure>

<caption>Fig. 3.10. Comparison of the NN and $\bar{N}N$ tensor potentials in the Paris model.</caption>

In the $\bar{N}N$ case, Eq. (3.54) becomes

$$ 
\begin{aligned}
V_V^{I=0} &= -V_\omega - 3V_\rho, \\
V_V^{I=1} &= -V_\omega + V_\rho.
\end{aligned} 
\qquad (3.55) 
$$

Coherences now take place in the $I=0$ channel, with a larger strength. From Table 3.1, the most pronounced effect occur in the tensor component [213]. This has many consequences on the phenomenology of $\bar{N}N$ interaction in potential models: spin effects in elastic or charge-exchange scattering, spin effects in specific reactions such as $\bar{p}p \to \Lambda\Lambda$ or $\bar{p}p \to K^-K^+$, distortion of the protonium wave function, enhancement of some branching ratios, etc. We shall come back very often on possible signatures of this strong tensor interaction.

A quantitative illustration of these coherences is provided in Figs. 3.9 and 3.10. We use the static version of the Paris potential, and display the NN and $\bar{N}N$ spin-orbit and tensor potentials for both $I=0$ and $I=1$ isospin. In Fig. 3.9, one sees the large spin-orbit component of the proton-proton potential, with $I=1$. In Fig. 3.10, one notices the large and positive tensor potential for $\bar{N}N$ in isospin $I=0$ states.

Testing tensor forces in scattering experiments require delicate measurements. If one would treat tensor forces at first order in DWBA<fnref n="5" /> starting from central forces only, there would be no polarisation. In actual calculations, tensor forces produce polarisation from their second- and higher-order contributions, but polarisation alone cannot distinguish current models with strong tensor force and weak spin-orbit from possible alternative models with weak tensor forces and moderate spin-orbit

<footnote n="5">⁵ Distorted-wave Born approximation.</footnote>
<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /><figcaption>Fig. 3.11. Meson-exchange diagram for $\bar{p}p \to \bar{\Lambda}\Lambda$.</figcaption></figure>

<caption>Fig. 3.12. Quark pair and creation for $\bar{p}p \to \bar{\Lambda}\Lambda$. The strange quarks or antiquarks are shown with thick lines.</caption>

potential. The best tests rely on observables to which tensor forces contribute already at first order. As the tensor operator $S_{12}$ involves two spin operators (see Eq. (3.46) of Section 3), these observables are of rank 2. An example is the transfer of longitudinal polarisation [214].

In this latter paper [214], it was shown that the spin dependence of the long-range potential, when blindly associated with a spin-independent optical potential, produces substantial spin effects in $\bar{N}N$ scattering. The question is to which extent scattering experiments can test this spin dependence unambiguously and give information on the spin dependence of short-range forces.

### 3.7. Strangeness-exchange reactions

When $\bar{\Lambda}\Lambda$ or other hyperon-antihyperon pairs are produced near threshold, a nuclear-physics approach seems justified, where the reaction is described by exchange of strange mesons, in the same way as the charge-exchange reaction $\bar{p}p \to \bar{n}n$ is mediated by charged mesons. This corresponds to the diagram of Fig. 3.11.

As mentioned earlier, the pseudoscalar character of the pion gives a specific spin dependence in the charge-exchange reaction $\bar{p}p \to \bar{n}n$. A similar pattern can thus be expected for $\bar{p}p \to \bar{\Lambda}\Lambda$. However, the $K^*$ is closer in range to K than $\rho$ to $\pi$, and thus the corrections due to vector and higher exchanges could be more important in $\bar{p}p \to \bar{\Lambda}\Lambda$ than in $\bar{p}p \to \bar{n}n$.

Since K or $K^*$ mesons are relatively heavy, the production of hyperons is a rather short-range process and, instead of summing over all possible kaon excitations in the t-channel, one might think of a simple quark process, as pictured in Fig. 3.12: a pair of ordinary quarks annihilate and a pair of strange quarks is created. Gluons are not shown, but are crucial to actually generate the process.

While PS185 was taking data and seeking further spin observables, new ideas were developed, motivated by experiments on the structure functions of the nucleon. A s¯s pair might be extracted from the nucleon or antinucleon sea instead of being created during the reaction, as schematically pictured in Fig. 3.13. This point of view was proposed in particular by Ellis et al. [215], who pointed out that a similar mechanism would produce an abundant violation of the OZI<fnref n="6" /> rule in annihilation.

The channel $\bar{\Sigma}^0\Lambda$ forces an isospin $I=1$ initial state, unlike $\bar{\Lambda}\Lambda$ which filters $I=0$. In a naive quark model, the spin of $\Lambda$ is carried by the s quark, which is associated with a spin 0(ud) diquark, while the spin of a $\Sigma$ is opposite to that of its constituent s quark.

<footnote n="6">⁶ Okubo–Zweig–Iizuka, see, e.g., Ref. [216] for references.</footnote>
<figure><img src="image_33.png" /><figcaption>Fig. 3.13. Possible sea-quark contribution to $\bar{p}p \to \bar{\Lambda}\Lambda$. The strange quarks or antiquarks are shown with thick lines.</figcaption></figure>

The final states $\bar{\Sigma}^+\Sigma^-$ and $\bar{\Sigma}^-\Sigma^+$ bear essential differences. In a Yukawa- or Regge-exchange language, the production of the latter can be understood in terms of a neutral kaon in the *t*-channel, and thus its cross-section should be of the same order of magnitude as for $\bar{\Lambda}\Lambda$ or $\bar{\Sigma}^0\Lambda$. The production of $\bar{\Sigma}^+\Sigma^-$ requires exotic exchanges, namely mesonic systems with one unit of strangeness and two units of charge. It is thus expected to be suppressed. This was clearly seen in pre-LEAR data (for References see, e.g., the bibliography in Ref. [129]). The same difference can be seen in the quark-diagram approach: $\bar{\Lambda}\Lambda, \bar{\Sigma}^0\Lambda$ or $\bar{\Sigma}^-\Sigma^+$ can be reached by annihilating a single $u\bar{u}$ or $d\bar{d}$ pair replaced by an $s\bar{s}$ one. For the final state $\bar{\Sigma}^+\Sigma^-$, the simple quark diagram of Fig. 3.12 does not operate: one needs more pairs created or annihilated.

## 3.8. Phenomenological description of annihilation

Only the long-range part of the $\bar{N}N$ interaction can be translated from its NN analogue. It remains to include short-range elastic forces and annihilation. We review here the different approaches and the salient results obtained in early models. More refined studies incorporating the LEAR results will be presented in Section 6.

### 3.8.1. The range of annihilation and early optical models

Before antiprotons were produced and their cross-sections first measured, in the years 1955–1956, one would probably have guessed that the $\bar{p}p$ interaction is mostly elastic, with a small correction due to annihilation, as in the $e^+e^-$ case. Why such a (wrong) prediction? The elastic forces have a range given by the inverse pion mass and thus act up to a few fermis, while annihilation was expected to have a very short range, the inverse nucleon mass, i.e., about 0.1 fm, as the range of $e^+e^-$ annihilation is the inverse electron mass.

A typical annihilation diagram, representing $\bar{N}N \to \pi\pi$ is shown in Fig. 3.14 (left), as well as its contribution to the $\bar{N}N$ amplitude (right). This amplitude has an imaginary part, because the sum of the meson masses is smaller than the centre-of-mass energy $\sqrt{s}$. There are two baryons in the *t*-channel, of invariant mass larger than $2m$, thus the range is $r \le (2m)^{-1} \approx 0.1$ fm. The same result holds for intermediate states with a larger number of mesons. For a more rigorous discussion of the range of annihilation, see, e.g., Ref. [217].

The striking surprise was [7] that the total antiproton cross-section on matter is very large (as compared to the proton one), and that more than half of this cross-section is due to annihilation. This annihilation cross-section $\sigma_{\text{ann}}$ cannot be understood from a simple S-wave correction. It requires angular-momentum $\ell > 0$ contributions.
<figure><img src="image_34.png" /><figcaption>Fig. 3.14. Simplest annihilation diagram in an old-fashioned hadronic approach: the nucleon and the antinucleon and transformed into a two-pion state by exchange of two nucleons. Shown are the basic process (left) and its contribution to the $\bar{N}N$ amplitude (right).</figcaption></figure>

<table><caption>Table 3.2
Comparison of the imaginary potentials of several optical-model fits of $\bar{N}N$ scattering</caption><thead><tr><th>Model</th><th>Ref.</th><th>a (fm<sup>-1</sup>)</th><th>R (fm)</th><th>V<sub>1</sub> (GeV)</th></tr></thead><tbody><tr><td>BP1</td><td>[210]</td><td>6</td><td>0</td><td>62</td></tr><tr><td>BP2</td><td>[210]</td><td>6</td><td>0</td><td>8.3</td></tr><tr><td>DR1</td><td>[211]</td><td>5</td><td>0</td><td>20</td></tr><tr><td>DR2</td><td>[220]</td><td>5</td><td>0.8</td><td>0.5</td></tr><tr><td>KW</td><td>[212]</td><td>5</td><td>0.55</td><td>1.6</td></tr></tbody></table>

The simplest phenomenological method to describe these data consists of supplementing the meson-exchange potential (regularised at short distance) by an empirical complex potential, which represent the sum of $\bar{N}N \rightarrow \bar{N}N$ diagrams with meson intermediate states, and other short-range processes.

The method of optical models is very successful in nuclear physics [218]. The idea, when studying a specific reaction, say $a+A \rightarrow a+A$, consists of not wasting time and computer resources to try to parametrise each open channel such as $a+A \rightarrow a' + A^*$ or $a+A \rightarrow b + B$, where $A^*$ stands for an excited level and $B$ another nucleus with different quantum numbers. What matters, for the reaction of interest [218], is that the presence of these channels strongly distorts the incoming $a+A$ wave. An empirical potential reproducing this distortion provides a realistic picture of the scattering process.

Early attempts to describe $\bar{N}N$ annihilation by an optical model were proposed by Ball et al. [208], Gourdin et al. [209], etc. A rather celebrated model, remaining a reference for later studies, is the potential developed by Bryan and Phillips [210]. The starting point is the NN potential of Bryan and Scott with includes several one-boson-exchange contributions. The short-range $\bar{N}N$ interaction is parametrised as

$$W(r) = \frac{V_R - iV_I}{1 + \exp(a(r - R))}. \qquad (3.56)$$

An impressively large strength is required to fit the data, in both variants BP1 and BP2 of the model, as seen in Table 3.2. Independent of the precise value of the parameter $a$, the Bryan-Philips (and earlier) fits imply that the annihilation potential should be important up to about 1 fm. An illustration is provided in Fig. 3.15, whose drawing is inspired by Ref. [219]. The annihilation potential, chosen
<figure><img src="image_35.png" /><figcaption>Fig. 3.15. Comparison of the imaginary optical potential $W(r)$, corresponding to the fit of Ref. [210], with the spin-spin and tensor components of the pion-exchange potential.</figcaption></figure>

as the renowned Bryan-Philips model (static version), is compared to the spin-spin and tensor components of the one-pion-exchange potential. One clearly needs something like $r \gtrsim 1.5$ fm to become sure that the annihilation potential is negligible.

If one reduces the range of the absorptive potential $W(r)$ in the Bryan-Philips model, then the fit to the cross-section quickly deteriorates. In particular, one cannot account for the ratio of annihilation to elastic cross-sections, nor for the smallness of the charge-exchange cross-section. Remember that in a naive model with one-pion exchange only, treated in the Born approximation, the charge-exchange process will be four times larger than the elastic cross-section, due to the favourable isospin factors! One needs large corrections due to higher-order iterations, exchange of heavier mesons, and absorption, to restore the proper hierarchy of cross-sections.

To summarise, the relatively long-range character of the imaginary potential, needed to reproduce the observed integrated cross-sections, is the most striking feature of these optical-model fits.

Later on, a strong support to a large value of the annihilation range came from the detailed study of protonium annihilation. With a zero-range absorption, annihilation at rest would be restricted to S-waves. The observation of the $\pi^0\pi^0$ decay of protonium [141] gave the first indication for P-wave annihilation at rest. It was followed by the observation of several other channels with a clear signature of P-wave annihilation. Globally, P-states of protonium have an annihilation width much larger than their radiative decay to S-states.

Many years after the pioneering experiments on antiproton annihilation and the first phenomenological analyses, we understand better what is underneath: the finite size of the nucleon (and antinucleon), eventually due to its composite structure.

### 3.8.2. Phenomenology of optical models

Shortly before LEAR came into operation, Dover and Richard [211] revisited the Bryan-Philips model. The long range is a slightly simplified version of the Paris potential, which includes two-pion exchange and excitation of $\pi$N resonances in the intermediate states. The potential is cut off at short distances and supplemented by an empirical Wood-Saxon core of the same type, see Eq. (3.56). This corresponds to models DR1 and DR2 [220] in Table 3.2. A similar procedure was used by Kohno and Weise [212] (model KW) and others [221].
<figure><img src="image_36.png" /><figcaption>Fig. 3.16. Comparison of various imaginary potentials used for describing $\bar{N}N$ annihilation. We restrict ourselves here to local models, Bryan-Philips (BP1), Dover-Richard (DR1 and DR2), and Kohno-Weise (KW), Bydzovsky et al. (BMN).</figcaption></figure>

Though every microscopic derivation of the optical potential leads to an interaction that is strongly non-local and energy dependent, most phenomenological models use a local ansatz, for simplicity. The real and the imaginary parts of the isoscalar and isovector components in the core (3.56) are chosen to be identical. The possibility of adding some spin or isospin dependence is discussed in Section 6.

The values of $a$, $R$ and $V_1$ adopted by Bryan and Philips [210] (BP), Dover, Richard and Sainio [211,220] (DR) and Kohno and Weise [212] (KW) are summarised in Table 3.2. The parameters look rather different, but the imaginary potentials almost coincide in the region of interest, near 1 fm. The comparison of the above imaginary potentials is made in Fig. 3.16. Also shown here is the average imaginary potential of Ref. [222]:

$$ \mathrm{Im} V = -\frac{1}{2} [w_0 \exp(-(\alpha_0 r)^2) + w_1 \exp(-(\alpha_1 r)^2)] , \quad (3.57) $$

with $w_0 = 3$ GeV, $w_1 = 0.6$ GeV, $\alpha_0 = 1.80 \text{ fm}^{-1}$ and $\alpha_1 = 1.47 \text{ fm}^{-1}$. Note that in Ref. [222], the absorptive potential is allowed to be isospin dependent. It is found that data suggest a somewhat stronger absorption for isospin $I=0$ than for $I=1$. This presumably explains why in Fig. 3.16, this $\bar{p}p$ absorptive potential is slightly weaker than the others, if the full strength is needed in one isospin channel only.

In Fig. 3.16, we observe at a different scale the same phenomenon of surface interaction as in low-energy heavy-ion collisions, where the scattering is entirely determined by the nuclear potential in a very small interval of the distance between the two ions. Similarly, the low-energy $\bar{N}N$ data are not sensitive to the value of the absorptive potential at very small distances. What really matters is the strength of annihilation near 1 fm. This correlation between the range and the strength of the imaginary potential was noticed in Ref. [222].

Note that the values of the parameters $V_R$ in different models cannot be compared independently of the cut-off procedure which is adopted for the short range of the meson-exchange potential.
It was noticed in Ref. [211] that increasing the attraction allows one to use smaller values of $V_1$ and tends to slightly improve the overall fit of integrated cross-sections. This illustrates the mechanism proposed by Shapiro [29] for annihilation: the attractive potential focuses the wave function toward the region where absorption acts.

### 3.8.3. Boundary condition models

In practice, the method consists of a phenomenological prescription for the logarithmic derivative of the radial wave function in each partial wave, at a distance $r = R$. Then the long-range behaviour, given by a homogeneous second-order equation, is entirely determined by the knowledge of $R du(r)/dr|_{r=R}$.

A boundary-condition model was used, for instance, by Lomon et al. [223], to account for the short-range repulsion of the NN potential. Years later, a microscopic interpretation was proposed by Jaffe and Low [224]. The boundary condition achieves a transition between the inner part of the wave function, described in terms of quarks, and the outer part, which accounts for the relative motion of two well-separated hadrons.

In the $\bar{N}N$ case, the inner part of the wave function contains the rich dynamics of multiquark states ($q^n\bar{q}^n$). Thus, the boundary-condition models are not too far from those by Dorokhov [225] and Roberts [226], who later extended this approach to strangeness exchange [227]. These authors describe $\bar{N}N$ scattering as a superposition of broad s-channel resonances.

A simple boundary-condition model was used by Ball and Chew [208], but in practice, when combined with WKB approximation, it led to discontinuous inelasticity parameters, as a function of energy. It was sufficient, however, to account for the large magnitude of the observed cross-sections. This method has been regularly revisited. The boundary condition model was used in particular by Kaufmann and Pilkuhn [147], and Dalkarov and Myhrer [228].

### 3.8.4. Coupled-channel models

Here the various annihilation channels are mimicked by a few coupled channels. Usually, no diagonal term is introduced in these new channels, i.e., the interaction among mesons is neglected. Annihilation manifests itself as a coupling between the main channel (nucleon-antinucleon) and the mesonic channels [229–232].

A reasonable description of scattering data has been obtained by various groups. It is interesting that the transition operators $V_i$ from $\bar{N}N$ to the mesonic channels can be chosen as relatively short ranged. As explained, e.g., by Liu and Tabakin [229], the contribution of a mesonic channel to the optical potential can be derived from the coupled equations

$$ (2m + T_{\bar{N}N} + V_{11} - \sqrt{s})\Psi_1 = -V_{12}\Psi_2, \\ (2\mu + T_{\mu} + V_{22} - \sqrt{s})\Psi_2 = -V_{21}\Psi_1. \qquad (3.58) $$

Solving formally the second equation as

$$ \Psi_2 = -(2\mu + T_{\mu} + V_{22} - \sqrt{s})^{-1} V_{21} \Psi_1 \qquad (3.59) $$

and inserting the result into the first equation gives an effective potential

$$ V_{\text{eff}} = V_{11} - V_{21}(2\mu + T_{\mu} + V_{22} - \sqrt{s})^{-1} V_{21}, \qquad (3.60) $$
which is highly non-local, as it contains the kinetic operator $T_μ$ of the meson sector and explicit dependence upon the energy $\sqrt{s}$. In other words, the meson propagator can be appreciably extended, and tend to spread the annihilation region.

Usually, the channels that are introduced represent an average over many channels with comparable phase-space. There are more specific models, with a few explicit physical channels, such as $\bar{p}p \to \pi^-\pi^+, K^-K^+$, and either fictitious channels or an optical model to account for the rest of annihilation [232]. That way, not only $\bar{N}N$ observables can be studied, but also those of the specific reactions such as $\bar{p}p \to \pi^-\pi^+$ or $\bar{p}p \to K^-K^+$.

### 3.8.5. Microscopic derivation of the absorptive potential

In principle, the optical potential can be entirely determined once one knows the transition amplitude from $\bar{N}N$ to every possible mesonic channel. A prototype calculation was given by Ihle et al. [189], who simply assumed that annihilation is dominated by rearrangement diagrams, and neglected most of the interaction between the three mesons in the final states. They ended with an annihilation potential which is not too far from being realistic and at least exhibits the appropriate range.

Note that optical potentials obtained in explicit models are far from being local. With simple Gaussian wave functions for the quark distribution inside hadrons, one usually ends with a separable operator [233,234]. For simplicity, empirical optical potentials are parametrised in terms of local operators. The energy dependence which is sometimes introduced reflects the need for non-local corrections.

In the above references, and others, for instance, Ref. [235], annihilation is described at the quark level. The baryon-exchange process is used in Ref. [236]. The short-range character of the potential is somewhat hidden by large form-factor corrections. Baryon-exchange diagrams were also advocated by the Paris group to motivate the parametrisation of their short-range potential [237].

Note also a series of papers by the Bonn group [236,232,238] where a fraction of the annihilation is taken from the assumed dynamical mechanisms, and the rest of annihilation is treated phenomenologically.

### 3.8.6. Annihilation range revisited

Many debates arose about the range of annihilation. Let us quote for instance our late friend I.S. Shapiro in the discussion following the talk by one of us [239] at the Mainz Conference [39]:

> The value of the annihilation range (that is equal in order of magnitude to the Compton length of the annihilating baryons) is not a question for discussion. It is a general statement following from the analytical properties of the amplitudes in quantum field theory... It does not matter how the annihilating objects are constructed from their constituents...

The arguments based on the analytical properties of Feynman graphs (see, e.g., Refs. [217,240]) cannot be challenged. However, as addressed at Erice [54] by Brodsky [241] and Shapiro [240], and also at Mainz [239,242], there is a semantic question about what is meant by “annihilation” and on how to get proper guidance from the comparison with QED.

If the rule links nucleons to electrons, and mesons to photons, then what we call “$\bar{N}N$ annihilation” is comparable to $e^+e^- \to$ photons, and should, indeed, be of very short range, about $m^{-1} \sim 0.1$ fm, as already said.
<figure><img src="image_37.png" /><figcaption>Fig. 3.17. Possible diagram for “annihilation” into ordinary mesons.</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>Fig. 3.18. Possible diagram for the transition to K⁺K⁻.</figcaption></figure>

However, nucleons are composite, as well as mesons. At the quark level, the analogy with positronium annihilation into photons is not relevant anymore for the so-called “annihilation” into ordinary mesons. In the language of atomic collisions, most of the transitions from baryon-antibaryon to mesons are not “annihilation” reactions, they are rearrangements, with sometimes partial annihilation.

This distinction can be made more clearly by considering muonium ($\mu^+e^−$) and antimuonium ($\mu^−e^+$) scattering. The process

$$ (\mu^+e^-) + (\mu^-e^+) \rightarrow (\mu^+\mu^-) + (e^+e^-), \qquad (3.61) $$

which is energetically possible even at rest, is simply governed by the spatial overlap of the initial and final-state wave functions. It has nothing to do with annihilation.

Now, the process

$$ (\mu^+e^-) + (\mu^-e^+) \rightarrow \gamma's + (\mu^+\mu^-), \qquad (3.62) $$

requires the initial electrons on the top of each other, and thus is of shorter range. Note that the muons can be a little separated, of the order of the ($\mu^+\mu^−$) Bohr radius.

Finally, the (rare) reaction

$$ (\mu^+e^-) + (\mu^-e^+) \rightarrow \gamma's, \qquad (3.63) $$

corresponds to a complete annihilation. It implies $\mu^+$ very close to $\mu^−$ and $e^+$ very close to $e^−$, and thus a perfect overlap of the initial atoms. Is is extremely short ranged at the atomic scale. As noted by Shapiro [240] on a basis of a calculation by Brodsky [241], one also ends with a very short range if one starts with the rearrangement process (3.61) and imposes eventual annihilation of both ($\mu^+\mu^−$) and ($e^+e^−$) atoms in the final state. What matters is the final state one considers.

Similarly, $\bar{N}N \rightarrow pions$ “annihilation” might proceed by rearrangement of the initial quarks and antiquarks into meson resonances, as pictured in Fig. 3.17. In the harmonic oscillator model, one can write down simple expressions for the transition amplitude [233], and realise that the crucial parameter is the size of the mesons, i.e., their ability to make a “bridge” to pick up a quark in the nucleon and an antiquark in the antinucleon. (The size of the baryons governs the spatial spread of the mesons in the final state.)

Now, processes like $\bar{N}N \rightarrow K^+K^−$ (see Fig. 3.18) should be of shorter range, since some of the initial constituents have to disappear. It is often proposed that a fraction of annihilation events into pions proceed with at least one quark-antiquark pair merging into gluons.
<figure><img src="image_39.png" /><figcaption>Fig. 3.19. Possible diagram for annihilation into $\phi\phi$.</figcaption></figure>

The extreme case corresponds to $\bar{N}N \to \phi\phi$, studied by the JETSET collaboration [243]. This is genuine “complete” annihilation, and should be of very short range. It corresponds to the diagram of Fig. 3.19.

It would be interesting to test somehow that final states with or without strangeness have different ranges, and, perhaps, behave differently when extrapolated from $\bar{N}N$ annihilation to annihilation in nuclei [244].

A detailed study of annihilation is beyond the scope of the present paper. What matters to $\bar{N}N \to \bar{N}N$ scattering process are the global properties of annihilation, which most likely occur via quark rearrangements, and are governed by the size of the mesons. It is not surprising therefore that most parametrised potential models suggest an annihilation radius of the order of 1 fm.

### 3.9. Possibility of quasi-nuclear bound states

The NN interaction is rather weak: phase-shifts are small, and there is only one bound state with a very small binding energy $E \sim 2$ MeV. The attraction provided by $\sigma$-exchange is partly compensated by the repulsion due to $\omega$-exchange. In the $\bar{N}N$ case, the potential induced by $\omega$-exchange flips sign since $G(\omega) = -1$, leading to a coherent attraction. Thus, the $\bar{N}N$ potential is likely to support a rich spectrum of bound states and resonances.

The first explicit calculations were performed in the 1960s, in the “bootstrap” approach to strong interactions [245]. In this very appealing, and unfortunately unsuccessful theory, hadrons are built out of hadrons interacting through hadron exchanges. In an approximation to the full bootstrap picture, one has tried to describe mesons as $\bar{N}N$ states, and this was not too successful, as analysed for instance by Ball et al. [246]. In fact, the $\bar{N}N$ potential is isospin dependent, due to its strong isovector exchange ($\pi, \rho$ exchange in particular), and the $\bar{N}N$ model fails in accounting for the observed degeneracy of the meson spectrum (exchange degeneracy in the Regge-pole language): except in the pseudoscalar sector and perhaps in the scalar sector, isospin $I=0$ and 1 mesons have nearly the same mass.

Shapiro and collaborators [29] came back to the $\bar{N}N$ model of mesons, but with a different point of view. In their approach, $\bar{N}N$ states do not refer to ordinary mesons such as $\rho$ or $\omega$, but to new mesons lying near the $\bar{N}N$ threshold and strongly coupled to the $\bar{N}N$ channel. These states are called “quasi-nuclear $\bar{N}N$ bound or resonant states”, or “nuclear baryonia”. Such states can be seen in radiative or pionic transition from atomic protonium, or, for these above the threshold, as resonances in $\bar{N}N$ cross-sections. A review of the work of the Russian school can be found in [29].

Further calculations of the $\bar{N}N$ spectrum were performed by other groups, in particular by Dover et al. A comprehensive review can be found in [30], and a comparison with other approaches to baryonium (string models of quarks, multiquarks), in [247]. See also [248].
While early calculations tend to give $\bar{N}N$ states in almost every low partial wave, the spectra displayed in [30] show clear selection rules: the most favourable channels are those with isospin $I=0$ and natural parity $P=(-1)^J$, i.e., the coupled $\ell=J-1$ and $J+1$ partial waves, where tensor forces act at best.

The role of tensor forces was emphasised in [213]. In the case of deuteron, tensor forces are crucial to achieve binding, but the quadrupole distortion remains moderate, with typically a D-wave percentage of 5% or 6%. In $I=0$ baryonium, we are far from the dominance of the lowest angular momentum, $\ell=J-1$, based on centrifugal-barrier arguments. We have a coherent mixture of the two partial waves, in a combination $|\alpha_J\rangle$ corresponding to an eigenstate of the tensor operator with eigenvalue $S_{12} = -4$, while the orthogonal combination $|\beta_J\rangle$ experiences a repulsion, with $S_{12} = 2$. They are

$$ |\alpha_J\rangle = \sqrt{\frac{J+1}{2J+1}}|J-1\rangle + \sqrt{\frac{J}{2J+1}}|J+1\rangle, \\ |\beta_J\rangle = -\sqrt{\frac{J}{2J+1}}|J-1\rangle + \sqrt{\frac{J+1}{2J+1}}|J+1\rangle. \qquad (3.64) $$

(This basis is also relevant for discussing spin effects for annihilation into two pseudoscalar mesons.)

So far, annihilation is neglected, or treated as a perturbation. This is clearly not justified, even so one can argue that the $I=0$, natural parity states have a wave function with a large radius. A calculation of the spectrum with the full $\bar{N}N$ potential, including the imaginary part, was performed by Myhrer and Thomas [249]. They use the Bryan-Phillips potential [210], and found that the nice spectrum of bound states produced by the real part is completely washed out when the annihilation part is switched on. Widths exceed several hundreds of MeV, typically, and thus $\bar{N}N$ states cannot account for the narrow structures that were claimed to be found at that time.

The conclusions of Myhrer and Thomas might be attenuated in several ways. Firstly, the brute-force annihilation they used can be replaced by imaginary potentials that are slightly weaker, and have a slightly shorter range, but still fit total cross-sections. Secondly, non-local effects, or energy-dependent effects can be significant. For instance, in any microscopic derivation of optical potential, channels with vector mesons are found to contribute significantly. When one looks at possible baryonia below the $\bar{N}N$ threshold, these channels are suppressed by simple phase-space considerations, and one is left with an absorption that is weaker than the one governing the scattering experiments. See, e.g., [250,251]. In these references, it is remarked that if there are baryonia below the threshold, a fraction (with the largest range) of $\bar{N}N$ annihilation at rest goes through baryonium plus pion, and due to the lack of phase-space, this component of annihilation does not act on baryonium itself.

Explicit calculations of the $\bar{N}N$ spectrum with these refinements lead, indeed, to conclusions more moderate than those of Myhrer and Thomas. Most states are washed out by annihilation, but some moderately broad (50 MeV typically) structures survive, near the threshold and in some specific channels. See, for instance, Ref. [252].

Similar calculations have been carried out for $\bar{N}N$ resonances, i.e., baryonia above the threshold. Here, absorption corresponds to the actual annihilation. Moreover, elastic decay (into $\bar{N}N$) is sufficient to produce a large width as soon as one goes above threshold. Some plots are given in [247], where it is clear that the elastic width becomes very large as soon as the resonance move above the