Embed Size (px)
Citation preview
ANTIPROTON PHYSICS∗
Jean-Marc Richard1
1Institut de Physique des 2 Infinis de Lyon & Université de Lyon, UCBL,CNRS-IN2P3, 4, rue Enrico Fermi, Villeurbanne, France
December 16, 2019
Abstract
We review the physics of low-energy antiprotons, and its link with the nuclearforces. This includes: antinucleon scattering on nucleons and nuclei, antiprotonicatoms and antinucleon-nucleon annihilation into mesons.
Contents
1 A brief history 2
2 From the nucleon-nucleon to the antinucleon-nucleon interaction 52.1 The G-parity rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Properties of the long-range interaction . . . . . . . . . . . . . . . . . . . . 62.A Appendix: Isospin conventions . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Baryonium 83.1 Experimental candidates for baryonium . . . . . . . . . . . . . . . . . . . . 83.2 The quasi-nuclear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Baryonium in the hadronic-string picture . . . . . . . . . . . . . . . . . . . 103.5 Color chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Other exotics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
∗To appear in the collection of review articles The long-lasting quest for nuclear interactions: the past, thepresent and the future of the journal Frontiers in Physics, eds. Laura Eliss Marcucci and Rupert Machleidt
1
arX
iv:1
912.
0738
5v1
[nu
cl-t
h] 1
6 D
ec 2
019
4 Antinucleon-nucleon scattering 114.1 Integrated cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Angular distribution for elastic and charge-exchange reactions . . . . . . . 124.3 Antineutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Spin effects in elastic and charge-exchange scattering . . . . . . . . . . . . 144.5 Amplitude analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.6 Potential models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Hyperon-pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 Spin effects in annihilation into two pseudoscalar mesons . . . . . . . . . . 194.A Appendix: Constraints on spin observables . . . . . . . . . . . . . . . . . . 20
5 Protonium 225.1 Quantum mechanics of exotic atoms . . . . . . . . . . . . . . . . . . . . . . 235.2 Level rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Isospin mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 Day-Snow-Sucher effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.5 Protonium ion and protonium molecule . . . . . . . . . . . . . . . . . . . . 25
6 The antinucleon-nucleus interaction 256.1 Antinucleon-nucleus elastic scattering . . . . . . . . . . . . . . . . . . . . . 256.2 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Antiprotonic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 Antiproton-nucleus dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 Neutron-antineutron oscillations . . . . . . . . . . . . . . . . . . . . . . . . 28
7 Antinucleon annihilation 307.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Global picture of annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Branching ratios: experimental results . . . . . . . . . . . . . . . . . . . . . 347.5 Branching ratios: phenomenology . . . . . . . . . . . . . . . . . . . . . . . 357.6 Annihilation on nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.7 Remarkable channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Modern perspectives 38
9 Outlook 39
1 A brief history
In 1932, the positron, the antiparticle of the electron, was discovered in cosmic rays andconfirmed in the β+ decay of some radioactive nuclei. See, e.g., the Nobel lecture byAnderson [1]. It was then reasonably anticipated that the proton also has an antiparticle,
2
the antiproton.1 It was also suspected that the antiproton would be more difficult toproduce and detect than positrons in cosmic rays. The Bevatron project (BeV, i.e., billionof electron-volts, was then a standard denomination for what is now GeV) was launchedat Berkeley to reach an energy high-enough to produce antiprotons through the reaction
p+ A→ p+ A+ p+ p , (1)
where A denotes the target nucleus. For A = p, this is a standard exercise in relativistickinematics to demonstrate that the kinetic energy of the incoming proton should behigher than 6m, where m is the proton mass, and c = 1. This threshold decreases if thetarget is more massive. The Bevatron was completed in 1954, and the antiproton wasdiscovered in 1955 by a team lead by Chamberlain and Segrè, who were awarded theNobel prize in 1959.2
Shortly after the antiproton, the antineutron, n, was also discovered at Berkeley, andup to now, for any new elementary particle, the corresponding antiparticle has alsobeen found. The discovery of the first anti-atom was well advertised [2], but this wasnot the case for the earlier observation of the first antinucleus, antideuterium, because ofa controversy between an European team [3] and its US competitors [4]. In experimentsat very high energy, in particular collisions of heavy ions at STAR (Brookhaven) andALICE (CERN), one routinely produces light antinuclei and even anti-hypernuclei (inwhich an antinucleon is replaced by an antihyperon Λ) [5–7].
The matter-antimatter symmetry is almost perfect, except for a slight violation in thesector of weak interactions, which is nearly exactly compensated by a simultaneous vio-lation of the left-right symmetry, i.e., the product PC of parity P and charge-conjugationC is only very marginally violated. Up to now, there is no indication of any violationof the product CPT , where T is the time-reversal operator: this implies that the protonand antiproton have the same mass, a property now checked to less than 10−9 [8].
Many experiments have been carried out with low energy antiprotons, in particu-lar at Brookhaven and CERN in the 60s and 70s, with interesting results, in particularfor the physics of the mesons produced by annihilation. However, in these early ex-periments, the antiprotons were part of secondary beams containing many negatively-charged pions and kaons, and with a wide momentum spread.
In the 70s, Simon van der Meer, and his colleagues at CERN and elsewhere imaginedand developed the method of stochastic cooling [9], which produces antiproton beamsof high purity, sharp momentum resolution, and much higher intensity than in theprevious devices. CERN transformed the fixed-target accelerator SpS into a proton-antiproton collider, SppS, with the striking achievement of the discovery of the W± andZ0, the intermediate bosons of the electro-weak interaction. A similar scheme was lateradopted at Fermilab with higher energy and intensity, leading to many results, amongwhich the discovery of the top quark.
1The only doubt came from the magnetic moment of the proton, which is not what is expected for aparticle obeying the Dirac equation.
2The other collaborators were acknowledged in the Nobel lectures, but nevertheless Piccioni suedChamberlain and Segrè in a court of California, which dismissed the suit on procedural grounds
3
As a side product of the experiments at the SppS program, CERN built a low-energyfacility, LEAR (Low-Energy Antiproton Ring) which operated from 1982 to 1996, andhosted several experiments on which we shall come back later. Today, the antiprotonsource of CERN is mainly devoted to experiments dealing with atomic physics and fun-damental symmetries. In spite of several interesting proposals, no low-energy extensionof the antiproton program was built at Fermilab.
As for the intermediate energies, at the beginning of the CERN cooled-antiprotonprogram, a p beam was sent in the ISR accelerator to hit a thin hydrogen target. Theexperiment R704 got sharp peaks corresponding to some charmonium states, and inparticular a first indication of the –then missing – P-wave singlet state hc [10]. But ISRwas to be closed, and in spite of a few more days of run, R704 was interrupted. Theteam moved to Fermilab, and charmonium physics with antiprotons was resumed withantiproton-proton collisions arranged in the accumulation device (experiments E760-E835) [11].
Today, the techniques of production of sharp antiproton beams is well undercon-trol. There are projects to perform strong-interaction physics with antiprotons at FAIR(Darmstadt) [12] and JPARC in Japan [13]. In the 80s, an ambitious extension of LEARat higher energies, SuperLEAR [14], was proposed by L. Montanet et al., but was notapproved by the CERN management. A major focus of SuperLEAR was charm physics.But more than 30 years later, this physics has been largely unveiled by beauty factoriesand high-energy hadron colliders.
Presently, the only running source of cooled antiprotons is the very low energy AD atCERN (Antiproton Decelerator) and its extension ELENA (Extra Low ENergy Antipro-ton) with the purpose of doing atomic-physics and high-precision tests of fundamentalsymmetries. Some further decelerating devices are envisaged for the gravitation exper-iments [15]. Of course, standard secondary antiproton beams are routinely produced,e.g., at KEK in Japan.
Note also that in devices making antiproton beams, a non-negligible fraction of an-tideuterium is produced, which could be cooled and stored. The intensity would besufficient to perform strong-interaction measurements, but there is not yet any proposalfor an experiment with an antideuterium beam.
We shall discuss along this review many results obtained at LEAR and elsewhere.Already the measurements made at Berkeley during the weeks following the discoveryof the antiproton were remarkable. After more than 60 years, we realize today thatthey gave keys to the modern understanding of hadrons, but the correct interpretationwas too far from the current wisdom of the 50s. Indeed, from the work by Fermi andYang, on which more later, it was realized that one-pion exchange constitutes the long-range part of the antinucleon-nucleon interaction. The simplest model, just before thediscovery of the antiproton, would be one-pion exchange supplemented by a very short-range annihilation. This would imply for the charge-exchange (pp → nn), elastic (pp →pp) and annihilation (pp→ mesons) cross-sections a hierarchy
σce > σel > σan , (2)
4
the first inequality resulting from straightforward isospin algebra. What was observedat Berkeley is just the opposite! And it took us years to admit and understand this pat-tern, which is a consequence of the composite character of the nucleon and antinucleon.
The era of LEAR and SppS at CERN, and then the large pp collider of Fermilab willcertainly be reminded as the culmination of antiproton physics. At very high energy,the trend is now more on pp rather than pp collision, due to the higher intensity ofproton beams. Certainly very-low energy experiments will remain on the floor to probethe fundamental symmetries with higher and higher precision. The question is openon whether antiproton beams will be used for hadron physics, a field where electronbeams and flavor factories already provide much information.
Of course, the role of antimatter in astrophysics is of the highest importance. An-tiprotons and even antinuclei are seen in high-energy cosmic rays. The question is toestimate how many antinuclei are expected to be produced by standard cosmic rays, toestimate the rate of primary antinuclei. See, e.g., [16, 17]. Some years ago, cosmologicalmodels were built [18] in which the same amount of matter and antimatter was created,with a separation of zones of matter and zones of antimatter. In modern cosmology, it isassumed that an asymmetry prevailed, so that, after annihilation, some matter survived.
This review is mainly devoted to the low-energy experiments with antinucleons.Needless to say that the literature is abundant, starting with dedicated workshops [19–24] and schools [25–28].
Because of the lack of space, some important subjects will not be discussed, in par-ticular the ones related to fundamental symmetry: the inertial mass of the antiproton,its charge and magnetic moment, in comparison with the values for the proton; the de-tailed comparison of hydrogen and antihydrogen atoms; the gravitational propertiesof neutral atoms such as antihydrogen, etc. We will mention only very briefly, in thesection on antiprotonic atoms, the dramatically precise atomic physics made with theantiprotonic Helium.
2 From the nucleon-nucleon to the antinucleon-nucleoninteraction
In this section we outline the general theoretical framework: how to extrapolate ourinformation on the nuclear forces to the antinucleon-nucleon system. We present thebasics of the well-knownG-parity rule, with a remark about the definition of antiparticlestates.
2.1 The G-parity rule
In QED, the e−e− → e−e− and e+e− → e+e− amplitudes are related by crossing, butthere is a long way from the region s > 4m2
e, t < 0 to the one withs < 0, t > 4m2e
to attempt a reliable analytic extrapolation. Here, me is the electron mass and s and t
5
the usual Mandelstam variables.3 A more useful approach consists of comparing bothreactions for the same values of s and t. The e−e− → e−e− amplitude can be decomposedinto a even and odd part according to the C-conjugation in the t-channel, say
M(e−e− → e−e−) =Meven +Modd , (3)
and the e+e− → e+e− amplitude for the same energy and transfer is given by
M(e+e− → e+e−) =Meven −Modd . (4)
The first term contains the exchange of an even number of photons, and the last one theexchange of an odd number. At lowest order, one retrieves the sign flip of the Coulombpotential. This rule remains valid to link pp → pp and pp → pp amplitudes: the ex-change of a π0 with charge conjugaison C = +1, is the same for both reactions, whilethe exchange of an ω meson (C = −1) flips sign.
In [29], Fermi and Yang astutely combined this C-conjugation rule with isospin sym-metry, allowing to include the exchange of charged mesons, as in the charge-exchangeprocesses. Instead of comparing pp → pp to pp → pp or np → np to np → np, the G-parity rule relates amplitudes of given isospin I . More precisely, if the nucleon-nucleonamplitude is decomposed as
MI(NN) =MG=+1 +MG=-1 , (5)
according to the G-odd (pion, omega, . . . ) or G-even (ρ, . . . ) in the t-channel, then itsNN counterpart reads
MI(NN) =MG=+1 −MG=-1 . (6)
Note that there is sometimes some confusion between the C-conjugation and the G-parity rules, especially because there are two ways of defining the isospin doublet n, p.See appendix 2.A.
In current models of NN , the pion-exchange tail, the attraction due to isoscalar two-pion exchange, and the spin-dependent part of the ρ exchange are rather well identified,and thus can be rather safely transcribed in the NN sector. Other terms, such as the cen-tral repulsion attributed to ω-exchange, might contain contributions carrying the oppo-site G-parity, hidden in the effective adjustment of the couplings. Thus the translationtowards NN might be biased.
2.2 Properties of the long-range interaction
Some important consequences of the G-parity rule have been identified. First, the mod-erate attraction observed in NN , due to a partial cancellation of σ (or, say, the scalar-isoscalar part of two-pion exchange) and ω-exchanges, becomes a coherent attraction
3For a reaction 1 + 2 → 3 + 4, the Mandestam variables are given in terms of the energy-momentumquadrivectors as s = (p1 + p2)2, t = (p3 − p1)2 and u = (p4 − p1)2.
6
once ω-exchange flips sign. This led Fermi and Yang to question whether the mesonscould be interpreted as bound states of a nucleon and an antinucleon. This idea hasbeen regularly revisited, in particular at the time of bootstrap [30]. As stressed, e.g.,in [31], this approach hardly accounts for the observed degeneracy of I = 0 and I = 1mesons (for instance ω and ρ having about the same mass).
In the 70s, Shapiro et al., and others, suggested that baryon-antibaryon bound stateswere associated with new types of hadrons, with the name baryonium, or quasi-deuteron[32–34]. Similar speculations were made later for other hadron-hadron systems, for in-stance DD∗, where D is a charmed meson (cq) of spin 0 and D∗ an anticharmed meson(cq) of spin 1 [35]. Some candidates for baryonium were found in the late 70s, inter-preted either as quasi-nuclear NN states à la Shapiro, or as exotic states in the quarkmodel, and motivated the construction of the LEAR facility at CERN. Unfortunately,the baryonium states were not confirmed.
Another consequence of the G-parity rule is a dramatic change of the spin depen-dence of the interaction. At very low energy, the nucleon-nucleon interaction is dom-inated by the spin-spin and tensor contributions of the one-pion exchange. However,when the energy increases, or, equivalently, when one explores shorter distances, themain pattern is a pronounced spin-orbit interaction. It results from a coherent sum ofthe contributions of vector mesons and scalar mesons.4 The tensor component of theNN interaction is known to play a crucial role: in most models, the 1S0 potential isstronger than the 3S1 one, but in this latter partial wave,5 the attraction is reinforced byS-D mixing. However, the effect of the tensor force remains moderate, with a percentageof D wave of about 5% for the deuteron.
In the case of the NN interaction, the most striking coherence occurs in the tensorpotential, especially in the case of isospin I = 0 [36]. A scenario with dominant ten-sor forces is somewhat unusual, and leads to unexpected consequences, in particulara relaxation of the familiar hierarchy based on the hight of the centrifugal barrier. Forinstance, if one calculates the spectrum of bound states from the real part of the NN in-teraction, the ground-state is 1,3P0, and next a coherent superposition of 1,3S1 and 1,3D1,and so on. In a scattering process, there is no polarization if the tensor component istreated to first order, but polarization shows up at higher order. Thus, one needs morethan polarization measurements6 to distinguish the dynamics with a moderate spin-orbit component from the dynamics with a very strong tensor component.
2.A Appendix: Isospin conventions
There are two possible conventions for writing the isospin states of antinucleons [37].
4The origin is different, for vector mesons, this is a genuine spin-orbit effect, for scalar mesons, this isa consequence of Thomas precession, but the effect is the same in practice
5The notation is 2S+1LJ , as there is a single choice of isospin, and it will become 2I+1,2S+1LJ for NN6Actually more than spin measurements along the normal n to the scattering plane, such as the ana-
lyzing power An or the transfer Dnn or normal polarization
7
The natural choice is based on the charge conjugation operatorC, namely |p〉c
= C |p〉and |n〉
c= C |n〉. However, it transforms the 2 representation of SU(2) into a 2 which
does not couple with the usual Clebsch-Gordan coefficients. For instance, the isospinI = 0 state of NN reads in this convention
|I = 0〉 =|pp〉
c+ |nn〉
c√2
, (7)
which anticipates the formula for an SU(3) singlet,
|0〉 =|uu〉+ |dd〉+ |ss〉√
3, (8)
However, the 2 representation of SU(2) is equivalent to the 2 one, and it turns outconvenient to perform the corresponding rotation, that is to say, define the states by theG-parity operator, namely (without subscript) |p〉 = G |n〉 and |n〉 = G |p〉. With thisconvention, the isospin singlet is written as
|I = 0〉 =|nn〉 − |pp〉√
2, (9)
3 Baryonium
The occurrence of baryonium candidates in antiproton-induced reactions was a majorsubject of discussion in the late 70s and in the 80s and the main motivation to buildnew antiprotons beams and new detectors. The name “baryonium” suggests a baryon-antibaryon structure, as in the quasi-nuclear models. More generally “baryonium” de-notes mesons that are preferentially coupled to the baryon-antibaryon channel, inde-pendently of any prejudice about their internal structure.
Nowadays, baryonium is almost dead, but interestingly, some of the innovative con-cepts and some unjustified approximations developed for baryonium are re-used inthe current discussions about the new hidden-charm mesons XY Z and other exotichadrons [38].
3.1 Experimental candidates for baryonium
For an early review on baryonium, see [39]. For an update, see the Particle Data Group[40]. In short: peaks have been seen in the integrated cross sections, or in the angulardistribution (differential cross section) at given angle, or in some specific annihilationrates as a function of the energy. The most famous candidate was the S(1932), seen inseveral experiments [39]. The most striking candidate was the peak of mass 2.95 GeV/c2
seen in ppπ− [41], with some weaker evidence for peaks at 2.0 and 2.2 GeV/c2 in the pp
8
subsystem, suggesting a sequential decay B− → B+ π−, where B denotes a baryonium.Peaks were also seen in the inclusive photon and pion spectra of the annihilations pp→γX and pp→ πX at rest.
None of the experiments carried out at LEAR confirmed the existence of such peaks.However, some enhancements have been seen more recently in the pp mass distributionof the decay of heavy particles, such as J/ψ → γpp, B → Kpp or B → Dpp, see [42] andthe notice on non qq mesons in [40]. There is a debate about whether they corresponda baryonium states or just reveal a strong pp interaction in the final state. See, e.g., thediscussion in [43–45]. Also, as stressed by Amsler [46], the f2(1565) [47] is seen only inannihilation experiments, and thus could be a type of baryonium, 1,3P2 − 1,3F2 in thequasi-nuclear models.
3.2 The quasi-nuclear model
Today, it is named “molecular” approach. The observation that the real part of theNN interaction is more attractive than its NN counterpart led Shapiro et al. [32], Doveret al. [33], and others, to predict the existence of deuteron-like NN bound states andresonances. Due to the pronounced spin-isospin dependence of the NN interaction,states with isospin I = 0 and natural parity were privileged in the predictions. Theleast one should say is that the role of annihilation was underestimated in most earlystudies. Attempts to include annihilation in the spectral problem have shown, indeed,that most structures created by the real potential are washed out when the absorptivepart is switched on [48].
3.3 Duality
Duality is a very interesting concept developed in the 60s. For our purpose, the mostimportant aspect is that in a hadronic reaction a + b → c + d, there is an equiva-lence between the t-channel dynamics, i.e., the exchanges schematically summarizedas∑
i a + c → Xi → b + d, and the low-energy resonances∑
j a + b → Yj → c + d. Inpractice, one approach is usually more efficient than the other, but a warning was set byduality against empirical superpositions of t-channel and s-channel contributions. Forinstance, KN scattering with strangeness S = −1 benefits the hyperons as s-channelresonances, and one also observes a coherent effect of the exchanged mesons. On theother hand, KN is exotic, and, indeed, has a much smaller cross-section. In KN , thereshould be destructive interferences among the t-channel exchanges.
Though invented before the quark model, duality is now better explained with thehelp of quark diagrams. Underneath is the Zweig rule, that suppresses the disconnecteddiagrams. See, e.g., [49, 50] for an introduction to the Zweig rule, and refs. there. Thecase of KN , or any other non-exotic meson-baryon scattering is shown in Fig. 1. For theexotic KN channel the incoming antiquark is s, and it cannot annihilate. So there is nopossibility of forming a quark-antiquark meson in the t channel, nor a three-quark state
9
m m
B B
Figure 1: Duality diagram for non-exotic meson-baryon scattering
in the s-channel. In a famous paper [51], Rosner pointed out that as meson-exchangesare permitted in nucleon-antinucleon scattering (or any baryon-antibaryon system withat least one quark matching an antiquark), there should be resonances in the s-channel:baryonium was born, and more generally a new family of hadrons. The correspondingquark diagram is shown in Fig. 2. As stressed by Roy [50], duality suggests higher
B B
B B
Figure 2: Duality diagram for non-exotic antibaryon-baryon scattering
exotics.
3.4 Baryonium in the hadronic-string picture
This concept of duality is illustrated in the hadronic-string picture, which, in turn, issupported by the strong-coupling limit of QCD. See, e.g., the contribution by Rossi andVeneziano in [39]. A meson is described as a string linking a quark to an antiquark. Abaryon contains three strings linking each of the three quarks to a junction, which acts asa sort of fourth component and tags the baryon number. The baryonium has a junctionlinked to the two quarks, and another junction linked to the two antiquarks. See Fig. 3.The decay happens by string breaking and qq, leading either to another baryonium anda meson, or to baryon-antibaryon pair. The decay into two mesons proceeds via theinternal annihilation of the two junctions, and is suppressed.
Figure 3: String picture of a meson (left), a baryons (center) and a baryonium(right)
The baryonium of Jaffe was somewhat similar, with the string realized by the cigar-shape limit of the bag model [52]. Note that the suppression of the decay into mesonsis due in this model to a centrifugal barrier, rather than to a topological selection rule.
10
The orbitally excited mesons consist of a quark and an antiquark linked by a string, theexcited baryons are the analogs with a quark and a diquark, and the baryonia involve adiquark and an antidiquark.
3.5 Color chemistry
Chan Hong-Mo et al. [53] pushed the speculations a little further in their “color chem-istry”. They have baryonia with color 3 diquarks, which decay preferentially into abaryon-antibaryon pair rather than into mesons, also more exotic baryonia in which thediquark has color sextet. Then even the baryon-antibaryon decay is suppressed, andthe state is expected to be rather narrow. This was a remarkable occurrence of the colordegree of freedom in spectroscopy. However, there was no indication on how and whysuch diquark-antidiquark structure arises from the four-body dynamics.
3.6 Other exotics?
The baryonium story is just an episode in the long saga of exotics, which includes thestrangeness S = +1 “Z” baryons in the 60s, their revival under the name “light pen-taquark” [40]. The so-called “molecular approach” hadrons was illustrated by the pic-ture of the ∆ resonance as πN by Chew and Low [54], and of the Λ(1405) as KN byDalitz and Yan [55], with many further discussions and refinements.
As reminded, e.g., in [56], there is some analogy between the baryonium of the 70sand 80s and the recent XY Z spectroscopy. The XY Z are mesons with hidden heavyflavor that do not fit in the ordinary quarkonium spectroscopy [38]. One can replace“quasi-nuclear” by “molecular”, “baryon number” by “heavy flavor”, etc., to trans-late the concepts introduced for baryonium for use in the discussions about XY Z. Thediquark clustering in the light sector is now replaced by an even more delicate assump-tion, namely cq or cq clustering. While the X(3872) is very well established, some otherstates either await confirmation or could be interpreted as mere threshold effects. Beforethe XY Z wave, it was suggested that baryon-antibaryon states could exist with strangeor charmed hyperons. This spectroscopy is regularly revisited. See, e.g., [57] and refs.there.
4 Antinucleon-nucleon scattering
In this section, we give a brief survey of measurements of antinucleon-nucleon scat-tering and their interpretation, for some final states: NN , ΛΛ, and two pseudoscalars.Some emphasis is put on spin observables. It is stressed in other chapters of this bookhow useful were the measurements done with polarized targets and/or beams for ourunderstanding of theNN interaction, leading to an almost unambiguous reconstructionof the NN amplitude. The interest in NN spin observables came at workshops held toprepare the LEAR experiments [19,20,22], and at the spin Conference held at Lausanne
11
in 1980 [58]. A particular attention was paid to pp→ ΛΛ, but all the theoreticians failedin providing valuable guidance for the last measurements using a polarized target, asdiscussed below in Sec. 4.7. However, Felix Culpa,7we learned how to better deal withthe relationships and constraints among spin observables.
4.1 Integrated cross sections
As already mentioned, the integrated cross sections have been measured first at Berke-ley, shortly after the discovery of the antiproton. More data have been taken in manyexperiments, mainly at the Brookhaven National Laboratory (BNL) and CERN, at vari-ous energies. The high-energy part, together with its proton-proton counter part, probesthe Pomerantchuk theorem, Froissart bound and the possible onset of the odderon. See,e.g., [59] and refs. there.
As for the low-energy part, some values of the total cross section are shown in Fig. 4,as measured by the PS172 collaboration [60]. It can be contrasted to the annihilationcross section of Fig. 5, due to the PS173 collaboration [61]. When one compares thevalues at the same energy, one sees that annihilation is more than half the total crosssection. Meanwhile, the integrated charge-exchange cross section is rather small (just afew mb).
250 300 350 400
200
250
300
Figure 4: Total pp cross section (in mb), as measured by the PS172 collaborationat LEAR.
Let us stress once more that the hierarchy σann > σel of the annihilation and elasticcross-sections is remarkable. One needs more than a full absorptive core. Somehow, thelong-range attraction pulls the wave function towards the inner regions where annihi-lation takes place [62, 63].
4.2 Angular distribution for elastic and charge-exchange reactions
The elastic scattering has been studied in several experiments, most recently at LEAR,in the experiments PS172, PS173, PS198, . . . An example of differential distribution isshown in Fig. 6.
7“For God judged it better to bring good out of evil than not to permit any evil to exist”, Augustinus
12
200 300 400 500 600
100
150
200
Figure 5: Antiproton-proton annihilation cross section (in mb), as measured byth PS173 collaboration at LEAR.
-1. -0.5 0. 0.5 1.
2
6
10
14
Figure 6: Angular distribution in elastic pp→ pp scattering at 0.697 GeV/c, asmeasured by the PS198 collaboration [64].
The charge exchange scattering has been studied by the PS199-206 collaboration atLEAR. As discussed in one of the workshops on low-energy antiproton physics [19],charge exchange gives the opportunity to study the interplay between the long-rangeand short-range physics. An example of differential cross-section is shown in Fig. 7,published in [65]. Clearly the distribution is far from flat. This illustrates the role ofhigh partial waves. The amplitude for charge exchange corresponds to the isospincombination
M(pp→ nn) ∝M0 −M1 , (10)
The smallness of the integrated charge-exchange cross-section is due to a large can-cellation in the low-partial waves. But in the high partial waves, there is a coherentsuperposition. In particular the one-pion exchange gets an isospin factor +1 for M1,and a factor −3 forM0.
4.3 Antineutron scattering
To access to pure isospin I = 1 scattering, data have been taken with antiproton beamsand deuterium targets, but the subtraction of the pp contribution and accounting for theinternal motion and shadowing effects is somewhat delicate. The OBELIX collaboration
13
-0.9 -0.5 -0.1 0.3 0.7
0.3
0.6
0.9
1.2
1.5
1.8
Figure 7: Angular distribution for the charge-exchange reaction pp→ nn atincident momentum 1.083 Gev/c in the target frame [66]. Only thestatistical error is shown here. Large systematic errors have to beadded.
0. 0.25 0.5 0.75 1.
1
2
3
4
Figure 8: Angular distribution for the charge-exchange reaction pp→ nn atincident momentum 0.601 Gev/c in the target frame, as measured bythe PS206 collaboration [66].
at CERN has done direct measurements with antineutrons [67]. For instance, the totalnp cross-section has been measured between plab = 50 and 480 MeV/c [68]. The data areshown in Fig. 9 together with a comparison with the pp analogs. There is obviously nopronounced isospin dependence. The same conclusion can be drawn for the pp and npannihilation cross sections [69].
4.4 Spin effects in elastic and charge-exchange scattering
A few measurements of spin effects in NN → NN were done before LEAR, mainlydealing with the analyzing power. Some further measurements were done at LEAR,with higher statistics and a wider angular range. An example of measurement by PS172is shown in Fig. 10: the analyzing power of pp → pp at 679 MeV/c [70]. One can seethat the value of An is sizable, but not very large. It is compatible with either a mod-erate spin-orbit component of the interaction, or a rather strong tensor force acting atsecond order. PS172 also measured the depolarization parameter Dnn in pp → pp. This
14
50 100 150 200 250 300 350
300
400
500
600
700
Figure 9: Total np cross section (red), as measured by the PS201 collaboration,and comparison with the pp total cross-section (blue)
-0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9
-0.2
0.2
0.4
-1. -0.75 -0.5 -0.25 0.25 0.5 0.75 1.
-0.2
0.2
0.4
Figure 10: Analyzing power of pp→ pp left: at 672 MeV/c, as measured atLEAR by the PS172 collaboration [70], right: at 697 MeV/c by thePS198 collaboration [64]
parameter Dnn expresses the fraction of recoiling-proton polarization along the normaldirection that is due to the polarization of the target. Thus, Dnn = 1 in absence of spinforces. PS172 obtained the interesting result Dnn = −0.169 ± 0.465 at cosϑ = −0.169for the momentum plab = 0.679 GeV/c [71]. The effect persists at higher momentum, asseen in Fig. 11.
The charge-exchange reaction has been studied by the PS199-206 collaborations atLEAR. See, e.g., [72, 73]. In Fig. 12 is shown the depolarization parameter Dnn. Theeffect is clearly large. It is predicted thatD`` is even more pronounced, and interestingly,also K``, the transfer of polarization from the target to the antineutron. This means thatone can produce polarized antineutrons by scattering antitprotons on a longitudinallypolarized proton target.
4.5 Amplitude analysis?
Decades of efforts have been necessary to achieve a reliable knowledge of the NN inter-action at low energy, with experiments involving both a polarized beam and a polarized
15
-0.9 -0.7 -0.5 -0.3 -0.1 0.1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
Figure 11: Transfer of polarization Dnn in elastic pp scattering atplab = 1.089 GeV/c [71]
-0.1 0.1 0.3 0.5 0.7 0.9
-0.6
-0.4
-0.2
0.2
0.4
Figure 12: Depolarization parameter in the charge exchange reaction at0.875 GeV/c [72]
target. In the case of NN , the task is more delicate, as the phase-shifts are complex evenat very low energy, and there is no Pauli principle to remove every second partial wave.So, as we have much less observables available for NN than for NN , it is impossible toreconstruct the phase-shifts or the amplitudes: there are unavoidably several solutionswith about the same χ2, and one flips from one solution to another one when one addsor removes a set of data. This is why the fits by Timmermans et al. [74, 75] have beenreceived with some skepticism [76, 77].
Clearly the measurements of analyzing power and depolarization at LEAR shouldhave been pursued, as was proposed by some collaborations, but unfortunately notapproved by the CERN management. Now, we badly miss the information that wouldbe needed to reconstruct the NN interaction unambiguously, and estimate the possibleways to polarize antiprotons (spin filter, spin transfer).
16
4.6 Potential models
For the use in studies of the protonium and antinucleon-nucleus systems, it is conve-nient to summarize the information about the “elementary” NN interaction in the formof an effective NN potential. Early attempts were made by Gourdin et al. [78], Bryanand Phillips [79] among others, and more recently by Kohno and Weise [80], and theBonn-Jülich group [81–83]. Dover, Richard and Sainio [63,84,85] used as long range po-tential VLR the G-parity transformed of the Paris NN potential, regularized in a square-well manner, i.e., VLR(r < r0) = VLR(r0) with r0 = 0.8 fm, supplemented by a complexcore to account for unknown short-range forces and for annihilation,
VSR(r) = − V0 + iW0
1 + exp(−(r −R)/a). (11)
The short-range interaction was taken as spin and isospin independent, for simplicity.A good fit of the data was achieved with two sets of parameters
model DR1 R = 0 , a = 0.2 fm , V0 = 21 GeV , W0 = 20 GeV ,
model DR2 R = 0.8 fm , a = 0.2 fm , V0 = 0.5 GeV , W0 = 0.5 GeV .(12)
In [86], the annihilation part is not described by an optical model, but by two effectivemeson-meson channels. This probably gives a more realistic energy dependence. Insome other models , the core contains some spin and isospin-dependent terms, but thereare not enough data to constrain the fit. Some examples are given by the Paris groupin [87], and earlier attempts cited there. In [88], a comparison is made of the successiveversions of such a NN potential: the parameters change dramatically when the fit isadjusted to include a new measurement. The same pattern is observed for the latestiteration [87].
More recent models will be mentioned in Sec. 8 devoted to the modern perspectives,namely an attempt to combine the quark model and meson-exchanges, or potentialsderived in the framework of chiral effective theories.
4.7 Hyperon-pair production
The PS185 collaboration has measured in detail the reactions of the type pp → Y Y ′,where Y or Y ′ is an hyperon. We shall concentrate here on the ΛΛ channel, which wascommented on by many theorists. See, e.g., [89]. In the last runs, a polarized hydrogentarget was used. Thus pp → ΛΛ interaction at low energy is known in great detail,and motivated new studies on the correlations among the spin observables, which arebriefly summarized in Appendix 4.A.
The weak decay of the Λ (and Λ) gives access to its polarization in the final state,and thus many results came from the first runs: the polarization P (Λ) and P (Λ) (whichwere checked to be equal), and various spin correlations of the final state Cij , where i or
17
j denotes transverse, longitudinal, etc.8 In particular the combination
F0 =1
4(1 + Cxx − Cnn + C``) , (13)
corresponds to the percentage of spin singlet, and was found to be compatible with zerowithin the error bars. Unfortunately, at least two explanations came:
• According to the quark model, the spin of Λ is carried by the s quark, with the lightpair ud being in a state spin and isospin zero. The vanishing of the spin singletfraction is due to the creation of the ss pair in a spin triplet to match the gluonin perturbative QCD or the prescription of the 3P0 model, in which the createdquark-antiquark pair has the quantum number 0++.
• In the nuclear-physics type of approach, the reaction is mediated by K and K∗
exchanges. This produces a coherence in some spin-triple amplitude, analogousto the strong tensor force in the isospin I = 0 of NN . Hence, the triplet is favored.
It was then proposed to repeat the measurements on a polarized hydrogen target.This suggestion got support and was approved. In spite of a warning that longitudinalpolarization might give larger effect, a transverse polarization was considered as anobvious choice, as it gives access to more observables. A detailed analysis of the latestPS185 are published in [91, 92].
What retained attention was the somewhat emblematic Dnn which measures thetransfer of normal polarization from p to Λ (in absence of spin effects, Dnn = 1). Itwas claimed that the transfer observable Dnn could distinguish among the different sce-narios for the dynamics [93], with quark models favoring Dnn positive (except modelsmaking use of a polarized ss sea [94]), and meson-exchange Dnn < 0. When the resultcame with Dnn ∼ 0, this was somewhat a disappointment. But in fact, it was real-ized [95, 96] that Dnn ∼ 0 was a consequence of the earlier data! As reminded briefly inappendix 4.A, there are indeed many constraints among the various spin observables ofa given reaction. For instance, one can show that
C2`` +D2
nn ≤ 1 . (14)
This inequality, and other similar constraints, implied that Dnn had be small, just fromdata taken with an unpolarized target, while D`` had a wider permitted range.
A sample of the PS185 results can be found in Fig. 13.
8The data have been analyzed with the value of the decay parameter α of that time. The parameterα is defined, e.g., in the note “Baryon decay parameters” of [40]. A recent measurement by the BESIIIcollaboration in Beijing gives a larger value of α [90]. This means that the Λ polarization would be about17 % smaller.
18
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Pn
•• •
•• •
••
•• ••••••
| | | | |
−0.0
−0.1
−0.2
FS
•
• •• •
• • • •• ••••••
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Cmm
•
• ••
•
• • •
•
••
••••
•
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Cml
•
•• •
• ••
•••
••••••
−−0.8
|−0.8
−−0.4
|−0.4
−0.0
|0.0
−0.4
|0.4
−0.8
|0.8
Cnn
cos(ϑcm)
•• •
• •
•• •
•• •
•••••
−−0.8
|−0.8
−−0.4
|−0.4
−0.0
|0.0
−0.4
|0.4
−0.8
|0.8
Cll
cos(ϑcm)
• • ••
••
• ••
•••
••••
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
An
•• •
••
•• • • •
••••••
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Dnn
•• • •
• • • ••••••
•
••
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Dmm
• •
•
• •
•• • • •
••••••
−−0.8
|
−−0.4
|
−0.0
|
−0.4
|
−0.8
|
Dml
• • • ••
••
•
• • ••••••
−−0.8
|−0.8
−−0.4
|−0.4
−0.0
|0.0
−0.4
|0.4
−0.8
|0.8
Knn
cos(ϑcm)
•
• •
• •
• • • •••••
•••
−−0.8
|−0.8
−−0.4
|−0.4
−0.0
|0.0
−0.4
|0.4
−0.8
|0.8
Kmm
cos(ϑcm)
••
•• • •
•
••
• •
•••••
Figure 13: Some spin observables of the reaction pp→ ΛΛ
4.8 Spin effects in annihilation into two pseudoscalar mesons
The reactions pp→ π+π− (and to a lesser extent π0π0) and K+K− were measured beforeLEAR. For instance, some results can be read in the proceedings of the Strasbourg con-ference in 1978 [97]. However, some adventurous analyses concluded to the existenceof unnatural-parity broad resonances, the large-width sector of baryonium. Needlessto say that such analyses with few or no spin observables, were flawed from the verybeginning. The same methods, and sometimes the same authors, were responsible forthe misleading indication in favor of the so-called Z baryons with strangeness S = +1,the ancestor of the late light pentaquark θ(1540).
The LEAR experiment PS172 remeasured these reactions with a polarized target.This gives access to the analyzing power An , the analog of the polarization in thecrossed reactions such as π−p→ π−p. Remarkably,An is very large, in some wide rangesof energy and angle. See Figs. 14 and 15. There is a choice of amplitudes, actually thetransversity amplitudes, such that
An =|f ]2 − |g|2
|f ]2 + |g|2, (15)
In this notation, |An| ∼ 1 requires one amplitude f or g to be dominant. This was un-
19
derstood from the coupled channel effects [98, 99]. Alternatively, one can argue that theinitial state is made of partial waves 3(J − 1)J and 3(J + 1)J coupled by tensor forces.The amplitudes f and g correspond to the eigenstates of the tensor operator S12 (seeSec. 2), and the amplitude in which the tensor operator is strongly attractive tends tobecome dominant [100]
Figure 14: Some results on pp→ ππ polarization at LEAR
4.A Appendix: Constraints on spin observables
A typical spin observable X is usually normalized such that −1 ≤ X ≤ +1. But if oneconsiders two normalized observables X and Y of the same reaction, several scenarioscan occur:
• The entire square −1 ≤ X, Y ≤ +1 is allowed. Then the knowledge of X does notconstrain Y .
• X, Y is restricted to a subdomain of the square. One often encounters the unitcircle X2 + Y 2 ≤ 1. In such case a large X implies a vanishing Y . This is what
20
Figure 15: Some results on pp→ KK polarization at LEAR
happens for Dnn vs. some of Cij in pp → ΛΛ. Another possibility is a triangle, seeFig. 16.
-1.0 -0.5 0.0 0.5 1.0-1.0
-0.5
0.0
0.5
1.0
X
Y
-1.0 -0.5 0.0 0.5 1.0-1.0
-0.5
0.0
0.5
1.0
X
Y
Figure 16: Examples of constraints among spin observables: only the coloredarea is permitted
For instance, in the simplest case of πN → πN (or its cross reaction as in Eq. 15),
21
there is a set of amplitudes such that the polarization (or the analyzing power), and thetwo independent transfer of polarization) are given by
X =|f ]2 − |g|2
|f ]2 + |g|2, Y =
2 Re(f ∗ g)
|f ]2 + |g|2, Z =
2 Im(f ∗ g)
|f ]2 + |g|2, (16)
such that X2 + Y 2 + Z2 = 1 and thus X2 + Y 2 ≤ 1. For reactions with two spin-1/2particles, the algebra is somewhat more intricate [96].
At about the same time as the analysis of the PS172 and PS185 data, similar in-equalities were derived for the spin-dependent parton distributions, in particular bythe late Jacques Soffer, starting from the requirement of positivity. An unified presen-tation of the inequalities in the hadron-hadron and quark distribution sectors can befound in [96]. The domain allowed for three normalized observables X , Y , Z can befound in this reference, with sometimes rather amazing shapes for the frontier.
Perhaps a new strategy could emerge. Instead of either disregarding all spin mea-surements, or to cumulate all possible spin measurements in view of an elusive fullreconstruction, one could advocate a stage by stage approach: measure first a few ob-servables and look for which of the remaining are less constrained, i.e., keep the largestpotential of non-redundant information.
5 Protonium
Exotic atoms provide a subtle investigation of the hadron-nucleon and hadron-nucleusinteraction at zero energy. For a comprehensive review, see [101]. Let us consider(h−, A), where h− is a negatively charged hadron such as π− or K−, and A a nucleusof charge +Z. One can calculate the energy levels E(0)
n,` by standard QED techniques, in-cluding finite volume, vacuum polarization, etc. The levels are shifted and broadenedby the strong interactions, and it can be shown (most simply in potential models, butalso in effective theories), that the complex shift is given by
δEn,` = En,` − E(0)n,` ' Cn,` a` , (17)
where a` is the scattering length for ` = 0, volume for ` = 1, . . . of the strong hA in-teraction. Cn,` is a know constant involving the reduced mass and the `th derivative ofthe radial wave function at the origin of the pure Coulomb problem. Experiments onprotonium have been carried out before and after LEAR. For a summary, see, e.g., [69].The latest results are:
• For the 1S level, the average shift and width are [102] δE(1S) = 712.5± 20.3 eV (tobe compared with the Bohr energy E(1S) ' −12.5 keV), and Γ(1S) = 1054± 65 eV,with a tentative separation of the hyperfine level as δE(3S1) = 785±35 eV, Γ(3S1) =940 ± 80 eV and δE(1S0) = 440 ± 75 eV, Γ(1S0) = 1200 ± 250 eV. The repulsivecharcater is a consequence of the strong annihilation.
22
• For the 2P level, one can not distinguish among 1P1, SLJ3P1 and 3P2, but thisset of levels is clearly separated from the 3P0 which receives a larger attractiveshift, as predicted in potential models (see, e.g., [85,103] and a larger width. Moreprecisely [104], δE[2(3P2,
31P1, 3P1)] ' 0, Γ[2(3P2,31P1, 3P1)] = 38 ± 9 meV, and
δE[23P0] ' −139 ± 28 mEV, Γ[23P0] = 489 ± 30 meV. For the latter, the admixtureof the nn component is crucial in the calculation, and the wave function at shortdistances is dominated by it isospin I = 0 component [105].
5.1 Quantum mechanics of exotic atoms
Perturbation theory is valid if the energy shift is small as compared to the level spac-ing. However, a small shift does not mean that perturbation theory is applicable. Forinstance, a hard core of radius a added to the Coulomb interaction gives a small up-ward shift to the levels, as long as the core radius a remains small as compared to theBohr radius R, but a naive application of ordinary perturbation theory will give an in-finite correction! For a long-range interaction modified by a strong short-range term,the expansion parameters is the ratio of the ranges, instead of the coupling constant. Atleading order, the energy shift is given by the formula of Deser et al. [106], and True-man [107], which reads
δE ' 4 π |φn`(0)|2 a0 , (18)
where a0 is the scattering length in the short-range potential alone, and φn`(0) the unper-turbed wave function at zero separation. For a simple proof, see, e.g., Klempt et al. [69].The formula (18) and its generalization (17) look perturbative, because of the occurrenceof the unperturbed wavefunction, but it is not, as the scattering length (volume, . . . ) a`implies iterations of the short-range potential.
There are several improvements and generalizations to any superposition of a short-range and a long-range potential, the latter not necessarily Coulombic, see, e.g., [108].For instance, in the physics of cold atoms, one often considers systems experiencingsome harmonic confinement and a short-range pairwise interaction.
5.2 Level rearrangement
The approximation (18) implies that the scattering length a remains small as comparedto the Bohr radius (or, say, the typical size of the unperturbed wave function). Zel’do-vich [109], Shapiro [32] and others have studied what happens, when the attractiveshort-range potential becomes large enough to support a bound state on its own. Letthe short-range attractive interaction be λVSR, with λ > 0. When λ approaches andpasses the critical value λ0 for the first occurrence of binding in this potential, the wholeCoulomb spectrum moves rapidly. The 1S state drops from the keV to the MeV range,the 2S level decreases rapidly and stabilizes in the region of the former 1S, etc. See, forinstance, Fig. 17. Other examples are given in [101, 108].
23
3S
2S
1S
-0.4
-0.3
-0.2
-0.1
0.1 2 3 4 5 6
Figure 17: Rerrangement of levels: first three levels of the radial equation−u′′(r) + V (r)u(r) = E u(r) with V (r) = −1/r + λ a2 exp(−a r), herewith a = 100 and λ variable.
It was then suggested that a weakly bound quasi-nuclear NN state will be revealedby large shifts in the atomic spectrum of protonium [32]. However, this rearrangementscenario holds for a single-channel real potential VSR. In practice, the potential is com-plex, and the Coulomb spectrum is in the pp channel, and the putative baryonium in astate of pure isospin I = 0 or I = 1. Hence, the rearrangement pattern is more intricate.
5.3 Isospin mixing
In many experiments dealing with “annihilation at rest”, protonium is the initial statebefore the transition NN →mesons. Hence the phenomenological analysis include pa-rameters describing the protonium: S-wave vs. P-wave probability and isospin mixing.Consider, e.g., protonium in the 1S0 state. In a potential model, its dynamics is given by
−u′′(r)/m+ V11 u(r) + V12 v(r)− e2
ru(r) = E1,0 u(r) ,
−v′′(r)/m+ V22 v(r) + V21 u(r) + 2 δmu(r) = E1,0 v(r) ,(19)
where δm is the mass difference between the proton and the neutron, and the strong(complex) potentials are the isospin combinations
V11 = V22 =1
2(VI=0 + VI=1) , V12 = V21 =
1
2(VI=0 − VI=1) . (20)
The energy shift is well approximated by neglecting the neutron-antineutron compo-nent, i.e., v(r) = 0. But at short distance, this component is crucial. In most currentmodels, one isospin component is dominant, so that the protonium wave function isdominantly either I = 0 or I = 1 at short distances, where annihilation takes place. Thisinfluences much the pattern of branching ratios. For instance, Dover et al. [105] found in
24
a typical potential model that the 3P0 level consists of 95 % of isospin I = 0 in the anni-hilation region. For 3P1, the I = 1 dominates, with 87 %. See [105,110,111] for a detailedstudy of the role of the nn channel on the protonium levels and their annihilation.
5.4 Day-Snow-Sucher effect
When a low-energy antiproton is sent on a gaseous or liquid hydrogen target, it is fur-ther slowed down by electromagnetic interaction, and is captured in a high orbit of theantiproton-proton system. The electrons are usually expelled during the capture andthe subsequent decay of the antiproton toward lower orbits. The sequence favors cir-cular orbits with ` = n − 1, in the usual notation. Annihilation is negligible for thehigh orbits, and becomes about 1% in 2P and, of course, 100% in 1S. This was alreadypredicted in the classic paper by Kaufmann and Pilkuhn [112].
In a dense target, however, the compact pp atom travels inside the orbits of the or-dinary atoms constituting the target, and experiences there an electric field which, byStark effect, mixes the (` = n−1, n) level with states of same principal quantum numbern and lower orbital momentum. Annihilation occurs from the states with the lowest `.This is known as the Day-Snow-Sucher effect [113]. In practice, to extract the branch-ing ratios and distinguish S-wave from P -wave annihilation, one studies the rates as afunction of the target density.
5.5 Protonium ion and protonium molecule
So far, the physics of hadronic atoms has been restricted to 2-body systems such as ppor K−A. In fact, if one forgets about the experimental feasibility, there are many otherpossibilities. If one takes only the long-range Coulomb interaction, without electromag-netic annihilation nor strong interaction, many stable configurations exist, such as ppp,the protonium ion, or pppp, the heavy analog of the positronium molecule. Identifyingthese states and measuring the shift and witdh of the lowest level would be most inter-esting. Today this looks as science fiction, as it was the case when Ps2 = e+e+e−e− wassuggested by Wheeler in 1945. But Ps2 was eventually detected, in 2007.
6 The antinucleon-nucleus interaction
6.1 Antinucleon-nucleus elastic scattering
At the very beginning of LEAR, Garreta et al. [114] measured the angular distributionof p-A scattering, where A was 12C, 40Ca or 208Pb. Some of their results are reproducedin Fig. 18.9 More energies and targets were later measured.
9I thank Matteo Vorabbi for making available his retrieving of the data in a convenient electronic form
25
10 20 30 40 50
0.1
1.
10.
102
103
10 20 30 40 50
0.1
1.
10.
102
103
104
10 20 30 40
0.1
1.
10.
102
103
104
105
Figure 18: Angular distribution for p scattering on the nuclei 12C, 40Ca and208Pb at kinetic energy Tp = 180 MeV [114].
10 20 30 40 50
0.1
1.
10.
102
103
Figure 19: Angular distribution for p scattering on 18O and 16O at178.7 MeV [115].
The results have been analyzed by Lemaire et al. in terms of phenomenological opti-cal models [116], which were in turn derived by folding the elementary NN amplitudeswith the nuclear density, see, e.g., [117–119].
In particular, a comparison of 16O and 18O isotopes, see Fig. 19, reveals that there isvery little isospin dependence of the pN interaction, when averaged on spins.
Other interesting measurements of the antinucleon-nucleus interaction have beencarried out and analyzed by the PS179 and OBELIX (PS201) collaborations, with morenuanced conclusions about the isospin dependence of the interaction at very low energy.See, for instance, [120, 121].
6.2 Inelastic scattering
It has been stressed that the inelastic scattering pA → pA∗, where A is a known exci-tation of the nucleus A, could provide very valuable information on the spin-isospindependence of the elementary NN amplitude, as the transfer of quantum numbers isidentified. One can also envisage the charge-exchange reaction pA → nB(∗). See, forinstance, [122].
Some measurements were done by PS184, on 12C and 18O [116]. The angular dis-tribution for p + 12C → p + 12C
∗ is given in Fig. 20 for the case where 12C is the 3−
level at 9.6 MeV. In their analysis, the authors were able to distinguish among models
26
that were equivalent for the NN data, but have some differences in the treatment of theshort-range part of the interaction. This is confirmed by the analysis in [122, 123]. Un-fortunately, this program of inelastic antiproton-nucleus scattering was not thoroughlycarried out.
20 30 40 50 60
0.001
0.100
1
Figure 20: Angular distribution of the 12C(p, p)12C∗ reaction for the 3− excited
state at 9.6 MeV. The incident p has an energy of 179.7 MeV
6.3 Antiprotonic atoms
The physics is nearly the same as for protonium. A low energy antiproton sent towardsa target consisting of atoms of nucleus A
ZX, is decelerated by the electromagnetic in-teraction and captured in a high atomic orbit, and cascades down towards lower orbits.During this process, the electrons are expelled. The difference is that annihilation occursbefore reaching the S or P levels, actually when the size of the orbit becomes compara-ble to the size of the nucleus. Again, the Day-Snow-Sucher mechanism can induce someStark effect. Thus precocious annihilation can happen, depending on the density of thetarget.
A review of the experimental data is provided in [124, 125], where a comparison ismade with pionic and kaonic atoms. The models developed to describe antiproton-nucleus scattering (see Sec. 6.1) have been applied, and account rather well for theobserved shifts. As for the purely phenomenological optical potentials Vopt, the mostcommon parametrization is of the form
2µVopt = −4π(
1 +µ
m
)(bR + i bI)%(r) , (21)
where µ is the reduced mass of p-A, m the mass of the nucleon, %(r) the nuclear densityand bR + i bi an effective scattering length.
More refined models, aiming at describing simultaneously the data on a variety ofnuclei, are written as [124]
2µVopt = −4 π(
1 +µ
m
)(b0[%n(r) + %p(r)] + b1[%n(r)− %p(r)]) , (22)
27
where the complex b0,1 are the isospin-independent and isospin-dependent effectivescattering lengths, respectively. Further refinements introduce in (22) a “P-wave” term∇.α(r)∇, or terms proportional to the square of the density. Typical values are [124]
b0 = (2.7± 0.3) + i (3.1±+0.4) fm , b1 = (1.2± 1.6) + i (1.6±+1.3) fm , (23)
so that there is no firm evidence for a strong isospin dependence.It is important to stress that the potential Vopt is probed mainly at the surface. Its
value inside the nucleus hardly matters. The same property is seen in the low-energyheavy-ion collisions: what is important is the interaction at the point where the two ionscome in contact.
6.4 Antiproton-nucleus dynamics
Modeling the antiproton-nucleus interaction has been done with various degrees of so-phistication. We have seen in the last section that phenomenological (complex) poten-tials proportional to the nuclear density account for a wide body of data on antiprotonicatoms. A relativistic mean-field approach was attempted years ago by Bouyssy andMarcos [126] and revisited more recently [127]. Meanwhile, a Glauber approach hasbeen formulated [128] and applied to the elastic and inelastic scattering of relativisticantiprotons.
There is a persisting interest in the domain of very low energies and possible boundstates. For instance, Friedman et al. [129] analyzed the subtle interplay between the NNS- and P-waves when constructing the antiproton-nucleus potential. There has beenalso speculations about possible p − A states, in line with the studies on the molecularNN baryonium. For a recent update, see, e.g., [130].
One could also envisage to use antiprotons to probe the tail of the nuclear densityfor neutron-rich nuclei with a halo structure. For early refs., see [27]. Recently, thePUMA proposal suggests an investigation by low-energy antiprotons of some unstableisotopes, for which the conventional probes have limitations [131].
6.5 Neutron-antineutron oscillations
In some theories of grand unification, the proton decay is suppressed, and one expectsneutron-to-antineutron oscillations. An experimental search using free neutrons hasbeen performed at Grenoble [132], with a limit of about τnn & 10−8 s for the oscilla-tion period. Any new neutron source motivates new proposals of the same vein, see,e.g., [133].
An alternative is to use the bound neutrons of nuclei. The stability of, say, 16O,reflects as well the absence of decay of its protons as the lack of n → n conversionwith subsequent annihilation of the antineutron. It has been sometimes argued [134]that the phenomenon could be obscured in nuclei by uncontrolled medium corrections.However, the analysis shows that the neutrons oscillate mainly outside the nucleus, and
28
the subsequent annihilation takes place at the surface, so that, fortunately, the mediumcorrections are small.
The peripheral character of the nn oscillations in nuclei explains why a simple pic-ture (sometimes called closure approximation) does not work too well, with the neutronand the antineutron in a box feeling an average potential 〈Vn〉 or 〈Vn〉, resulting in a sim-ple 2 × 2 diagonalization. The true dynamics of nn oscillations relies on the tail of theneutron distribution, where n and n are almost free.
There are several approaches, see for instance, [135]. The simplest is based on theSternheimer equation, which gives the first order correction to the wave function with-out summing over unperturbed states. In a shell model with realistic neutron (reduced)radial wave functions un`J(r) with shell energy En`J , the induced n component is givenby
− w′′n`J(r)
µ+`(`+ 1)
µ r2+ Vn(r)w′n`J(r)− En`J w
′n`J(r) = γ un`J(r) , (24)
with µ the reduced mass of the n-(A − 1) system, Vn the complex (optical) n-(A − 1)potential, and γ = 1/τnn the strength of the transition. Once wn`J is calculated, one canestimate the second-order correction to the energy, and in particular the width Γn`J ofthis shell
Γn`J = −2
∫ ∞0
ImVn |wn`J(r)|2 dr = −2 γ
∫ ∞0
un`J(r) Imwn`J(r) dr , (25)
which scales asΓn`J ∝ γ2 . (26)
An averaging over the shells give a width per neutron Γ associated with a lifeftime T
T = Tr τ2nn , (27)
where Tr is named either the “reduced lifetime” (in s−1) or the “nuclear suppressionfactor”. The spatial distribution of the wn`J and the integrands in (25), the relative con-tribution to Γ clearly indicate the peripheral character of the process. See, e.g., [136] foran application to a simulation in the forthcoming DUNE experiment, and refs. there toearlier estimates. Clearly, DUNE will provide the best limit for this phenomenon.
For the deuteron, an early calculation by Dover et al. [137] gave Tr ' 2.5 × 1022 s−1.Oosterhof et al. [138], in an approach based on effective chiral theory (see Sec. 8), founda value significantly smaller, Tr ' 1.1 × 1022 s−1. However, their calculation has beenrevisited by Haidenbauer and Meißner [139], who got almost perfect agreement withDover et al. For 40Ar relevant for the DUNE experiment, the result of [136] is Tr '5.6× 1022 s−1.
29
7 Antinucleon annihilation
7.1 General considerations
NN annihilation is a rather fascinating process, in which the baryon and antibaryonstructures disappear into mesons. The kinematics is favorable, with an initial center-of-mass energy of 2 GeV at rest and more in flight, allowing in principle up to more thana dozen of pions. Of course, the low mass of the pion is a special feature of light-quarkphysics. We notice, however, that the quark model predicts that (QQQ) + (QQQ) >3(QQ) [140], so that annihilation at rest remains possible in the limit where all quarksare heavy. The same quark models suggest that (QQQ)+(qqq) < 3 (Qq) if the mass ratioQ/q becomes large, so that, for instance, a triply-charmed antibaryon (ccc) would notannihilate on an ordinary baryon.
One should acknowledge at the start of this section that there is no theory, nor evenany model, that accounts for the many data accumulated on NN annihilation. Actu-ally the literature is scattered across various subtopics, such a the overall strength andrange of annihilation, the average multiplicity, the percentage of events with hidden-strangeness, the explanation of specific branching ratios, such as the one for pp → ρ π,the occurrence of new meson resonances, etc. We shall briefly survey each of theseresearch themes.
7.2 Quantum numbers
An initial NN state with isospin I , spin S, orbital momentum L and total angular mo-mentum J has parity P = −(−1)L and G-parity G = (−1)I+L+S . If the system is neutral,its charge conjugation is C = (−1)L+S . A summary of the quantum numbers for the Sand P states is given in Table 1.
Table 1: Quantum numbers of the S and P partial waves (PW) of the NN system. Thenotation is 2I+1,2S+1LJ .
PW 1,1S03,1S0
1,3S13,3S1
1,1P13,1P1
1,3P03,3P0
1,3P13,3P1
1,3P23,3P2
JPC 0−+ 0−+ 1−− 1−− 1+− 1+− 0++ 0++ 1++ 1++ 2++ 2++
IG 0+ 1− 0− 1+ 0− 1+ 0+ 1− 0+ 1− 0+ 1−
So, for a given initial state, some transitions are forbidden or allowed. The resultfor some important channels is shown in Table 2. In particular, producing two identicalscalars or pseudoscalars requires an initial P-state.
The algebra of quantum numbers is not always trivial, especially if identical mesonsare produced. For instance, the question was raised whether or not the 1S0 state of pro-tonium with JPC = 0−+ and IG = 0 can lead to a final state made of four π0. An poll
30
Table 2: Allowed decays from S and P-states into some two-meson final states.
FS 1,1S03,1S0
1,3S13,3S1
1,1P13,1P1
1,3P03,3P0
1,3P13,3P1
1,3P23,3P2
π0π0√ √
π−π+√ √ √
π0η(′) √ √
η(′)η(′) √ √
K−K+√ √ √ √ √ √
KsKl
√ √
KsKs
√ √ √ √
π0ω(φ)√ √
η(′)ω(φ)√ √
π0ρ0√ √
η(′)ρ0√ √
π±ρ∓√ √ √ √ √
among colleagues gave an overwhelming majority of negative answers. But a transi-tion such as 1S0 → 4 π0 is actually possible at the expense of several internal orbitalexcitations among the pions. For an elementary proof, see [88], for a more mathematicalanalysis [141].
The best known case, already mentioned in Sec. 1, deals with ππ. An S-wave π+π−
with a flat distribution, or a π0π0 system (necessarily with I = 0 and J even) requires aninitial state 1,3P0. It has been observed to occur even in annihilation at rest on a dilutehydrogen target [142]. This is confirmed by a study of the J = 0 vs. J = 1 content of theππ final state as a function of the density of the target, as already mentioned in Sec. 5.4.
7.3 Global picture of annihilation
As already stressed, the main feature of annihilation is its large cross-section, whichcomes together with a suppression of the charge-exchange process. This is reinforced bythe observation that even at rest, annihilation is not reduced to an S-wave phenomenon.This is hardly accommodated with a zero-range mechanism such as baryon exchange.The baryon exchange, for say, annihilation into two mesons is directly inspired by elec-tron exchange in e+ e− → γ γ. See Fig. 21. After iteration, the absorptive part of the NNinteraction, in this old-fashioned picture, would be driven by diagrams such as the onein Fig. 22. Other contributions involve more than two mesons and crossed diagrams.
31
e+
e−
γ
γ
N
N
m
m
Figure 21: Left: e+e− annihilation into two photons. Right: NN annihilation
into two mesons
N
N
m
m
N
N
Figure 22: Typical contribution to the absorptive part of the NN amplitude
As analyzed, e.g., in [143,144], this corresponds to a very small range, practically a con-tact interaction. Not surprisingly, it was impossible to follow this prescription whenbuilding optical models to fit the observed cross-sections. Among the contributions,one may cite [63, 78–80]. Claims such as [145], that it is possible to fit the cross sectionswith a short-range annihilation operator, are somewhat flawed by the use of very largestrengths, wide form factors, and a momentum dependence of the optical potential thatreinforce annihilation in L > 0 partial waves.
In the 80s, another point of view started to prevail: annihilation should be under-stood at the quark level.10 This picture was hardly accepted by a fraction of the com-munity. An anecdote illustrates how hot was the debate. After a talk at the 1988 Mainzconference on antiproton physics, where I presented the quark rearrangement, Shapirostrongly objected. At this time, the questions and answers were recorded and printedin the proceedings. Here is the verbatim [148]: I.S. Shapiro (Lebedev) : The value ofthe annihilation range . . . is not a question for discussion. It is a general statement fol-lowing from the analytical properties of the amplitudes in quantum field theory . . . . Itdoes not matter how the annihilating objects are constructed from their constituents. Itis only important that, in the scattering induced by annihilation, an energy of at leasttwo baryons masses is transferred. J.M. Richard: First of all, for me, this is an important"question for discussion". In fact, we agree completely in the case of "total annihilation",for instance NN → φφ. The important point is that [baryons and] mesons are compos-ite, so, what we call "annihilation" is, in most cases, nothing but a soft rearrangement ofthe constituents, which does not have to be short range.
10Of course , the rearrangement was introduced much earlier, in particular by Stern, Rubinstein, Caroll,
. . . [146, 147], but simply to calculate ratios of branching ratios, without any attempt to estimate the cross
sections.
32
In the simplest quark scenario, the spatial dependence of “annihilation” comes fromthat this is not an actual annihilation similar to e+e− → photons, in which the initial con-stituents disappear, but a mere rearrangement of the quarks, similar to the rearrangementof the atoms in some molecular collisions. This corresponds to the diagram of Fig. 23.The amplitude for this process is 〈Ψf |H|Ψi〉, where H is the 6-quark Hamiltonian, Ψi
N
N
m
m
m
Figure 23: Rearrangement of the quarks and antiquarks from a baryon and an
antibaryon to a set of three mesons
the nucleon-antinucleon initial state, and Ψf the final state made of three mesons. See,for instance, [149–151], for the details about the formalism. One gets already a good in-sight on the spatial distribution of annihilation within the quark-rearrangement modelby considering the mere overlap 〈Ψf |Ψi〉 using simple oscillator wave functions. Forthe initial state, a set of Jacobi coordinates (here normalized to correspond to an unitarytransformation)
x =r2 − r1√
2, y =
2 r3 − r1 − r2√6
, z =r1 + r2 + r3√
3,
x′ =r5 − r4√
2, y′ =
2 r6 − r4 − r5√6
, z′ =r4 + r5 + r6√
3,
(28)
with the further change
r =z′ − z√
2, R =
z + z′√2
, (29)
which, to a factor, are the NN separation and the overall center of mass. The initial-statewave function is thus of the form
Ψi =
(α
π
)3
exp[−(α/2)(x2 + y2 + x′2 + y′2)
]ϕ(r) , (30)
where ϕ denotes the NN wave function. Similarly for the final state, one can introducethe normalized Jacobi coordinates
u1 =r4 − r1√
2, u2 =
r5 − r2√2
, u3 =r6 − r3√
2,
v1 =r1 + r4√
2, v2 =
r2 + r5√2
, v3 =r3 + r6√
2,
X =v2 − v1√
2, Y =
2v3 − v1 − v2√6
, R =v1 + v2 + v3√
3,
(31)
33
and the wave function
Ψf =
(β
π
)9/4
exp[−(β/2)(u2
1 + u22 + u2
3)]
Φ(X,Y ) . (32)
Integrating for instance over x′ − x and y′ − y, one ends with
Φ∗f (X,Y ) exp(−βr2/2) exp(−α(X2 + Y 2)/2)ϕ(r) , (33)
and after iteration, one gets a separable operator v(r) v(r′), where v(r) is proportionalto exp(−βr2/2) and contains some energy-dependent factors [149, 151]. As expected,the operator is not local. There is an amazing exchange of roles: the size of the baryon,through the parameter α, governs the spatial spread of the three mesons, while the sizethe mesons becomes the range of the separable potential. Schematically speaking, therange of “annihilation” comes from the ability of the mesons to make a bridge, to pickup a quark in the baryon and an antiquark in the antibaryon.
Explicit calculations show that the rearrangement potential has about the requiredstrength to account for the observed annihilation cross-sections. Of course, the modelshould be improved to include the unavoidable distortion of the initial- and final-statehadrons. Also one needs a certain amount of intrinsic quark-antiquark annihilation andcreation to explain the production of strange mesons. This leads us to the discussionabout the branching ratios.
7.4 Branching ratios: experimental results
Dozens of final states are available for NN annihilation, even at rest. When the energyincreases, some new channels become open. For instance, the φφ channel was usedto search for glueballs in the PS202 experiment [152]. However, most measurementshave been performed at rest with essentially two complementary motivations. The firstone was to detect new multi-pion resonances, and, indeed, several mesons have beeneither discovered or confirmed thanks to the antiproton-induced reactions. The secondmotivation was to identify some leading mechanisms for annihilation, and one shouldconfess that the state of the art is not yet very convincing.
Several reviews contain a summary of the available branching ratios and a discus-sion on their interpretation. See, e.g., [88,153]. We shall not list all available results, but,instead, restrict ourselves to the main features or focus on some intriguing details. Forinstance:
• The average multiplicity is about 4 or 5. But in many cases, there is a formationof meson resonances, with their subsequent decay. In a rough survey, one canestimate that a very large fraction of the annihilation channels are compatible withthe primary formation of two mesons which subsequently decay.
• In the case of a narrow resonance, one can distinguish the formation of a resonancefrom a background made of uncorrelated pions, e.g., ωπ from ππππ. In the case ofbroad resonances, e.g., ρπ vs. πππ, this is much more ambiguous.
34
• The amount of strangeness, in channels such as K +K, K +K∗, K +K + pions, isabout 5%.
• Charged states such as pn or np are pure isospin I = 1 initial state. In the case ofpp annihilation at rest, the isospin is not known, except if deduced from the finalstate, like in the case of πη. Indeed, pp is the combination (|I = 0〉 + |I = 1〉)/
√2.
But, at short distances, one of the components often prevails, at least in modelcalculations. In the particle basis, there is an admixture of nn component, which,depending on its relative sign, tends to make either a dominant I = 0, or I = 1. Forinstance, Kudryavtsev [154] analyzed the channels involving two pseudoscalars,and concluded that if protonium annihilation is assumed to originate from anequal mixture of I = 0 and I = 1, then annihilation is suppressed in one of theisospin channels, while a better understanding is achieved, if pp− nn is accountedfor.
7.5 Branching ratios: phenomenology
The simplest model, and most admired, is due to Vandermeulen [155]. It assumes adominance of 2-body modes, say NN → a + b, where a and b are mesons or mesonresonances, produced preferentially when the energy is slightly above the thresholds
1/2ab = ma +mb. More precisely, the branching ratios are parametrized as
f = Cab p(√s,ma,mb) exp[−A(s− sab)] , (34)
where A is an universal parameter, p the center-of-mass momentum and the constantCab assume only two values: C0 for non-strange and C1 for strange.
In the late 80s, following the work by Green and Niskanen [149, 150], and oth-ers, there were attempts to provide a detailed picture of the branching ratios, usingquark-model wave functions supplemented by operators to create or annihilate quark-antiquark pairs. A precursor was the so-called 3P0 model [156] introduced to describedecays such as ∆→ N + π.
There has been attempts to understand the systematics of branching ratios at thequark level. We already mentioned some early papers [146, 147]. In the late 80s andin the 90s, several papers were published, based on a zoo of quark diagrams. Some ofthem are reproduced in Fig. 24. The terminology adopted An or Rn for annihilationor rearrangement into n mesons. Of course, they are not Feynman diagrams, but just aguidance for a quark model calculation with several assumptions to be specified. On theone had, the R3 diagram comes as the most “natural”, as it does not involve any changeof the constituents. On the other hand, it was often advocated that planar diagramsshould be dominant, see, e.g., [157]. This opinion was, however, challenged by Pirnerin his re-analysis the the 1/Nc expansion, whereNc is the number of colors in QCD [158].
A key point is of course strangeness. The R3 diagram hardly produces kaons, exceptif extended as to include the sea quarks and antiquarks. On the other hand, the An
35
R3 R2
A3 A2
Figure 24: Some quark diagrams describing annihilation
diagrams tend to produce too often kaons, unless a controversial strangeness suppressionfactor is applied: at the vertex where a quark-antiquark pair is created, a factor f = 1is applied for q = u, d and f 1 for q = s. This is an offending violation of theflavor SU(3)F symmetry. For instance the decays J/ψ → pp and J/ψ → ΛΛ are nearlyidentical, especially once phase-space corrections are applied. The truth is that at low-energy, strangeness is dynamically suppressed by phase-space and a kind of tunnelingeffect [159]. This could have been implemented more properly in the analyses of thebranching ratios. An energy-independent strangeness suppression factor is probablytoo crude.
Note that a simple phenomenology of quark diagrams is probably elusive. A di-agram involving two primary mesons can lead to 4 or 5 pions after rescattering or thedecay of a resonance. Also the An diagrams require a better overlap of the initial baryonand antibaryon, and thus are of shorter range than the Rn diagrams. So the relative im-portance can vary with the impact parameter and the incident energy.
7.6 Annihilation on nuclei
There has been several studies of N -A annihilation. In a typical scenario, a primaryannihilation produces mesons, and some of them penetrate the nucleus, giving rise toa variety of phenomenons: pion production, nucleon emission, internal excitation, etc.See, e.g., [160]. Some detailed properties have been studied, for instance whether an-nihilation on nuclei produces less or more strange particles than annihilation on nucle-ons [161].
At very low energy, due to the large NN cross section, the primary annihilation takesplace near the surface. It has been speculated that with medium-energy antiprotons,thanks to the larger momentum and the smaller cross section, the annihilation couldsometimes take place near the center of the nucleus. Such rare annihilations with ahigh energy release (at least 2 GeV) and little pressure, would explore a sector of the
36
properties of the nuclear medium somewhat complementary to the heavy-ion collisions.See, e.g., [14, 162–164].
Note the study of Pontecorvo reactions. In NN annihilation, at least two mesonshave to be produced, to conserve energy and momentum. On a nucleus, there is thepossibility of primary annihilation into n mesons, with n− 1 of them being absorbed bythe remaining nucleons. An example is pNN → πN or φn [165, 166]. This is somewhatrelated to the pionless decay of Λ in hypernuclei [167].
7.7 Remarkable channels
Some annihilation channels have retained the attention:
• pp→ e+e− led to a measurement of the proton form factor in the time-like region.The reversed reaction e+e− → pp was studied elsewhere, in particular at Frascati.For a general overview, see [168,169], and for the results of the PS170 collaborationat CERN, [170].
• We already mentioned the pp → charmonium → hadrons, leading to a bettermeasurement of the width of some charmonium states, and the first indicationfor the hc, the 1P1 level of cc [10, 171]. In principle, while e+e− → charmoniumis restricted to the JPC = 1−− states, pp can match any partial wave. However,perturbative QCD suggests that the production is suppressed for some quantumnumbers. It was thus a good surprise that ηc(1S) was seen in pp, but the couplingturns out less favorable for ηc(2S) [11, 172].
• The overall amount of hidden-strangeness is about 5 % [88]. This is remarkablysmall and is hardly accommodated in models where several incoming qq pairs areannihilated and several quark-antiquark pairs created. Note that the branchingratio for K+K− is significantly larger for an initial S-wave than for P-wave [46].This confirms the idea that annihilation diagrams are of shorter range than thanthe rearrangement ones.
• pp → K0K0 in the so-called CPLEAR experiment (PS195) [173] gave the oppor-tunity to measure new parameters of the CP violation in the neutral K systems,a phenomenon first discovered at BNL in 1964 by Christenson, Cronin, Fitch andTurlay.11 The CPLEAR experiment found evidence for a direct T -violation (timereversal).
• Precision measurements of the pp → γ + X and pp → π + X in search of boundbaryonium, of which some indications were found before LEAR. The results ofmore intensive searches at LEAR were unfortunately negative. See, e.g., [174].
11Happy BNL director, as his laboratory also hosted the experiment in which the Ω− was discovered,
the same year 1964.
37
When combined to the negative results of the scattering experiments, this wasseen as the death sentence of baryonium. But, as mentioned in Sec. 3, this opinionis now more mitigated, because of the pp enhancements observed in the decay ofheavy particles.
• pp → ρ π has intriguing properties. Amazingly, the same decay channel is alsopuzzling in charmonium decay, as the ratio of ψ(2S) → ρ π to J/ψ → ρ π differssignificantly from its value for the other channels. See, e.g., [175] and refs. there.In the case of pp annihilation, the problem, see, e.g., [46], is that the the productionfrom 1,3S1 is much larger than from 1,3S0. Dover et al., for instance, concluded tothe dominance of the A2 type of diagram [176], once the quark-antiquark creationoperator is assumed to be given by the 3P0 model [156]. But the A2 diagram tendsto produce too often kaons!
• pN → K + X , if occurring in a nucleus, monitors the production of heavy hy-pernuclei. It was a remarkable achievement of the LEAR experiment PS177 byPolikanov et al. to measure the lifetime of heavy hypernuclei. See, e.g., [177].
• pp→ π+π− and pp→ K+K− by PS172 revealed striking spin effects (see Sec. 4.8)
• pp → φφ was used to search for glueballs (Experiment PS 202 “JETSET”) [152],with innovative detection techniques.
8 Modern perspectives
So far in this review, the phenomenological interpretation was based either on the con-ventional meson-exchange picture or on the quark model for annihilation. The formerwas initiated in the 50s, and the latter in the 80s. Of course, it is not fully satisfac-tory to combine two different pictures, one for the short-range part, and another forthe long-range, as the results are very sensitive to the assumptions for the matching ofthe two schemes. This is one of the many reasons why the quark-model description ofthe short-range nucleon-nucleon interaction has been abandoned, though it providedan interesting key for a simultaneous calculation of all baryon-baryon potentials. Oneway out that was explored consists of exchanging the mesons between quarks. Thenthe quark wave function generates a form factor. For NN , a attempt was made by En-tem and Fernández [103], with some phenomenological applications. In this paper, theannihilation potential is due to transition qq → meson → qq or qq → gluon → qq. Butthis remains a rather hybrid picture and it was not further developed.
Somewhat earlier, in the 80s, interesting developments of the bag model have beenproposed, where the nucleon is given a pion cloud that restores its interaction withother nucleons. This led to a solitonic picture, e.g., Skyrme-type of models for low-energy hadron physics [178]. A first application to NN was proposed by Zahed andBrown [179].
38
As seen in other chapters of this book, a real breakthrough was provided by the ad-vent of effective chiral theories, with many successes, for instance in the description ofthe ππ interaction. For a general introduction, see, e.g., the textbook by Donnelly etal. [180]. This approach was adopted by a large fraction of the nuclear-physics commu-nity, and, in particular, it was applied to the study of nuclear forces and nuclear struc-tures. Chiral effective field theory led to very realistic potentials for the NN interaction,including the three-body forces and higher corrections in a consistent manner [181,182].Thus the meson-exchange have been gradually forsaken.
In such modern NN potentials, one can identify the long-range part due to one-,two- or three-pion exchange, and apply the G-parity rule, to derive the correspondinglong-range part of the NN potential. The short-range part of the NN interaction is de-termined empirically, by fixing the strength of a some constant terms which enter theinteraction in this approach. This part cannot be translated as such to the NN sector.There exists for sure, analogous constant terms that describe the real part of the interac-tion. As for the annihilation part, there are two options. The first one consists of makingthe contact terms complex. This is the choice made by Chen et al; [183]. Another op-tion that keeps unitarity more explicit is to introduce a few effective meson channelsXi and iterate, i.e., NN → X → NN , with the propagator of the mesonic channel Xi
properly inserted [184]. Then some empirical constant terms enter now the transitionpotential V (NN → Xi). A fit of the available data determines in principle the constantsof the model [75]. The question remains whether the fit of the constant terms is unique,given the sparsity of spin observables. For a recent review on chiral effective theoriesapplied to antiproton physics, see [77, 185]. The phenomenology will certainly extentbeyond scattering data. One can already notice that the amplitude of [184], when prop-erly folded with the nuclear density, provides with an optical potential that accountsfairly well for the scattering data, as seen in Fig. 25 borrowed from [186].
9 Outlook
The physics of low-energy antiprotons covers a variety of topics: fundamental symme-tries, atomic physics, inter-hadronic forces, annihilation mechanisms, nuclear physics,etc.
New experiments are welcome or even needed to refine our understanding of thisphysics. For instance, a better measurement of the shift and width of the antiprotoniclines, and some more experiments on the scattering of antineutrons off nucleons or nu-clei. We also insisted on the need for more measurements on pp scattering with a longi-tudinally or transversally polarized target.
Selected annihilation measurements could also be useful, from zero energy to wellabove the charm threshold, and again, the interest is twofold: access to new sectors ofhadron spectroscopy, and test the mechanisms of annihilation. For this latter purpose,a through comparison of NN - and Y N -induced channels would be most useful, where
39
0 10 20 30 40 50 60
θc.m.
[deg]
10-2
10-1
100
101
102
103
104
dσ
/dΩ
[m
b/s
r]
LONLO
N2LO
N3LO
12C (p,p)
12C
180 MeV
Figure 25: Differential cross-section for elastic p scattering on 12C at 180 MeV.
The optical potential is computed from successive refinements in
the effective theory [186]
Y denotes a hyperon.The hottest sectors remain these linked to astrophysics: how antiprotons and light
antinuclei are produced in high-energy cosmic ray? Is there a possibility in the earlyUniverse of separating matter from antimatter before complete annihilation? Studyingthese questions require beforehand a good understanding of the antinucleon-nucleonand antinucleon-nucleus interaction.
Acknowledgments
I would like to thank many colleagues for several stimulating and entertaining discus-sions along the years about the physics of antiprotons, and in particular, my collabo-rators. I would like also to thanks the editors of Frontiers in Physics for initiating thisreview, and their referees for constructive comments. Special acknowledgments are dueto M. Asghar for remarks and suggestions. Toto
References
[1] Lars Brink, Gösta Ekspong, and Stig Lundqvist. Nobel lectures physics: includingpresentation speeches and laureates’ biographies. World Scientific, Singapore, 1967.The first 3 volumes were published by North-Holland, Amsterdam.
[2] G. Baur, G. Boero, S. Brauksiepe, A. Buzzo, W. Eyrich, R. Geyer, D. Grzonka,J. Hauffe, K. Kilian, M. Lo Vetere, M. Macri, M. Moosburger, R. Nellen, W. Oelert,S. Passaggio, A. Pozzo, K. Röhrich, K. Sachs, G. Schepers, T. Sefzick, R. S. Simon,
40
R. Stratmann, F. Stinzing, and M. Wolke. Production of antihydrogen. PhysicsLetters B, 368(3):251–258, Feb 1996.
[3] T. Massam, T. Muller, B. Righini, M. Schneegans, and A. Zichichi. Experimentalobservation of antideuteron production. Nuovo Cimento Serie, 39:10–14, September1965.
[4] D. E. Dorfan, J. Eades, L. M. Lederman, W. Lee, and C. C. Ting. Observation ofantideuterons. Phys. Rev. Lett., 14:1003–1006, 1965.
[5] Yu-Gang Ma. Observation of antimatter nuclei at RHIC-STAR. J. Phys. Conf. Ser.,420:012036, 2013.
[6] Jaroslav Adam et al. Precise measurement of the mass difference and the bindingenergy of hypertriton and antihypertriton. 2019. arXiv:1904.10520 [hep-ex].
[7] Benjamin Dönigus. Highlights of the production of (anti-)(hyper-)nuclei and ex-otica with ALICE at the LHC. J. Phys. Conf. Ser., 1271(1):012001, 2019.
[8] Masaki Hori et al. Two-photon laser spectroscopy of antiprotonic helium and theantiproton-to-electron mass ratio. Nature, 475:484–488, 2011.
[9] Dieter Mohl. Stochastic cooling of particle beams. Lect. Notes Phys., 866:pp.1–139,2013.
[10] C. Baglin et al. Charmonium spectroscopy by anti-proton annihilation with pro-tons from an internal H2 gas jet target. In International Europhysics Conference onHigh-Energy Physics, Bari, Italy, July 18-24, 1985, Proc. ed. by L. Nitti and G. Preparata,Laterza Bari, page 609, 1985.
[11] Claudia Patrignani. E835 at FNAL: Charmonium spectroscopy in anti-p p annihi-lations. AIP Conf. Proc., 717(1):581–590, 2004.
[12] Ingo Augustin, Hans H. Gutbrod, Dieter Kramer, Karlheinz Langanke, and HorstStocker. New Physics at the International Facility for Antiproton and Ion Research(FAIR) Next to GSI. In 4th International Conference on Fission and Properties of Neu-tron Rich nuclei (ICFN4) Sanibel Island, Florida, USA, November 11-17, 2007, 2008.
[13] S. Kumano. Spin Physics at J-PARC. Int. J. Mod. Phys. Conf. Ser., 40:1660009, 2016.
[14] P. Dalpiaz, R. Klapisch, P. Lefevre, M. Macri, L. Montanet, Dieter Mohl, AndreMartin, J.-M. Richard, H. J. Pirner, and L. Tecchio. Physics at SUPERLEAR. In TheElementary structure of matter. Proceedings, Workshop, Les Houches, France, March 24- April 2, 1987, page 0414, 1987.
[15] Audric Husson and David Lunney. An Antiproton Deceleration Device for theGBAR Experiment at CERN. 2019.
41
[16] Pascal Chardonnet, Jean Orloff, and Pierre Salati. The Production of antimatter inour galaxy. Phys. Lett., B409:313–320, 1997.
[17] M. Aguilar et al. Antiproton Flux, Antiproton-to-Proton Flux Ratio, and Prop-erties of Elementary Particle Fluxes in Primary Cosmic Rays Measured with theAlpha Magnetic Spectrometer on the International Space Station. Phys. Rev. Lett.,117(9):091103, 2016.
[18] J. J. Aly, S. Caser, R. Omnes, J. L. Puget, and G. Valladas. The Amount of Matterin a Matter-Antimatter Model of the Universe. Astron. Astrophys., 35:271, 1974.
[19] Ugo Gastaldi and Robert Klapisch, editors. Physics at LEAR with low-energy cooledantiprotons, Proc. LEAR Workshop, 9 - 16 May 1982, Erice, Italy, New York, NY, 1984.Plenum.
[20] Ugo Gastaldi, Robert Klapisch, Jean Marc Richard, and Jean Trân Thanh Vân,editors. Physics with antiprotons at LEAR in the ACOL era, Proc. 3th LEAR Workshop,Tignes, 19 - 26 Jan. 1985, Gif-sur-Yvette, 1985. Ed. Frontières, Ed. Frontières.
[21] S. Charalambous, C. Papastefanou, and P. Pavlopoulos, editors. Anti-proton 86.Proceedings, 8th European Symposium on Nucleon Anti-nucleon Interactions, Thessa-loniki, Greece, September 1-5, 1986, Singapore, Singapore, 1987. World Scientific,World Scientific.
[22] C. Amsler, G. Backenstoss, R. Klapisch, C. Leluc, D. Simon, and L. Tauscher, ed-itors. Physics at LEAR with low energy antiprotons Proc. LEAR Workshop, 6 - 13 Sep1987, Villars-sur-Ollon, Switzerland , Chur, 1988. Harwood.
[23] C. Amsler and D Urner, editors. Physics at SuperLEAR, Proc. SuperLEAR Workshop,Zurich, Switzerland, 9 - 12 Oct 1991, Bristol, 1992. IOP.
[24] Lawrence S. Pinsky. A report on the first Fermilab workshop on antimatterphysics at low energy. AIP Conf. Proc., 150:515–519, 1986.
[25] Ph. Bloch, P. Pavlopoulos, and R. Klapisch, editors. 1st International School ofPhysics with Low-energy Antiprotons "Ettore Majorana" Fundamental symmetries, 26Sep - 3 Oct 1986, Erice, Italy , New York, NY, 1987. Plenum.
[26] U. Gastaldi, R. Klapisch, and F. Close, editors. Spectroscopy of light and heavy quarks.Proceedings, 2nd Course of the International School of Physics with Low-Energy Anti-protons, Erice, Italy, May 23-31, 1987, 1989.
[27] F. Bradamante, J.-M. Richard, and R. Klapisch, editors. Proceedings, 3rd Interna-tional School of Physics with Low-energy Antiprotons "Ettore Majorana" Anti-proton -nucleon and anti-proton - nucleus interactions, New York, 1990. Plenum, Plenum.
42
[28] R. Landua, R. Klapisch, and J.-M. Richard, editors. Proceedings, 4th InternationalSchool of Physics with Low-energy Antiprotons - Medium-energy anti-protons and thequark - gluon structure of hadrons, New York, 1991. Plenum, Plenum.
[29] E. Fermi and Chen-Ning Yang. Are mesons elementary particles? Phys. Rev.,76:1739–1743, 1949.
[30] Maurice René Michel Jacob and Geoffrey Foucar Chew. Strong-Interaction Physics:a Lecture Note Volume. Frontiers in Physics. Benjamin, New York, NY, 1964.
[31] James S. Ball, Antonio Scotti, and David Y. Wong. O boson exchange model ofNN and NN interaction. Phys. Rev., 142:1000–1012, 1966.
[32] I. S. Shapiro. The Physics of Nucleon - Antinucleon Systems. Phys. Rept., 35:129–185, 1978.
[33] W. W. Buck, C. B. Dover, and J.-M. Richard. The Interaction of Nucleons with anti-Nucleons. 1. General Features of the NN Spectrum in Potential Models. AnnalsPhys., 121:47, 1979.
[34] C. B. Dover and J.-M. Richard. The Interaction of Nucleons With Anti-nucleons.2. Narrow Mesons Near Threshold: Experiment and Theory. Annals Phys., 121:70,1979.
[35] Nils A. Törnqvist. From the deuteron to deusons, an analysis of deuteron - likemeson meson bound states. Z. Phys., C61:525–537, 1994.
[36] C. B. Dover and J.-M. Richard. Strong Tensor Forces and the Possibility of HighSpin States in the Nucleon-Antinucleon System. Phys. Rev., D17:1770–1777, 1978.
[37] Alan Douglas Martin and Thomas D Spearman. Elementary-particle theory. North-Holland, Amsterdam, 1970.
[38] Nora Brambilla, Simon Eidelman, Christoph Hanhart, Alexey Nefediev, Cheng-Ping Shen, Christopher E. Thomas, Antonio Vairo, and Chang-Zheng Yuan. TheXY Z states: experimental and theoretical status and perspectives. 2019.
[39] L. Montanet, G. C. Rossi, and G. Veneziano. Baryonium Physics. Phys. Rept.,63:149–222, 1980.
[40] M. Tanabashi et al. Review of Particle Physics. Phys. Rev., D98(3):030001, 2018.
[41] C. Evangelista et al. Evidence for a Narrow Width Boson of Mass 2.95-GeV. Phys.Lett., 72B:139–143, 1977.
[42] M. Ablikim et al. Spin-Parity Analysis of pp Mass Threshold Structure in J/ψ andψ′ Radiative Decays. Phys. Rev. Lett., 108:112003, 2012.
43
[43] B. Loiseau and S. Wycech. Antiproton-proton channels in J/psi decays. Phys. Rev.,C72:011001, 2005.
[44] S. Wycech, J. P. Dedonder, and B. Loiseau. In search of baryonium. J. Phys. Conf.Ser., 374:012002, 2012.
[45] Xian-Wei Kang, J. Haidenbauer, and Ulf-G. Meißner. Near-threshold pp invariantmass spectrum measured in J/ψ and ψ′ decays. Phys. Rev., D91(7):074003, 2015.
[46] Claude Amsler. Nucleon-antinucleon annihilation at LEAR. 2019. Talk at theECT∗ Workshop on Antiproton Physics, Trento, June 17-21, 2019.
[47] See the review by the Particle Data Group, http://pdg.lbl.gov/2019/listings/rpp2019-list-f2-1565.pdf.
[48] F. Myhrer and A.W. Thomas. The Influence of Absorption on Possible Nucleon-Antinucleon Resonances and Bound States. Phys. Lett., 64B:59–61, 1976.
[49] Nathan Isgur and H. B. Thacker. Origin of the okubo-zweig-iizuka rule in qcd.Phys. Rev. D, 64:094507, Oct 2001.
[50] D. P. Roy. History of exotic meson (four quark) and baryon (five quark) states. J.Phys., G30:R113, 2004.
[51] Jonathan L. Rosner. Possibility of baryon - anti-baryon enhancements with un-usual quantum numbers. Phys. Rev. Lett., 21:950–952, 1968.
[52] R. L. Jaffe. Q2Q2 Resonances in the baryon - antibaryon system. Phys. Rev.,D17:1444, 1978.
[53] Hong-Mo Chan, M. Fukugita, T. H. Hansson, H. J. Hoffman, K. Konishi,H. Hogaasen, and Sheung Tsun Tsou. Color chemistry: A study of metastableMultiquark molecules. Phys. Lett., 76B:634–640, 1978.
[54] G. F. Chew and F. E. Low. Effective range approach to the low-energy p wavepion - nucleon interaction. Phys. Rev., 101:1570–1579, 1956.
[55] R. H. Dalitz and S. F. Tuan. The phenomenological description of -K -nucleonreaction processes. Annals Phys., 10:307–351, 1960.
[56] Giancarlo Rossi and Gabriele Veneziano. The string-junction picture of multi-quark states: an update. JHEP, 06:041, 2016.
[57] Saori Maeda, Akira Yokota, Emiko Hiyama, Makoto Oka, and Yan-rui Liu. YcNand ΛcNN Bound States in the Potential Model. JPS Conf. Proc., 17:112006, 2017.
44
[58] C. Joseph and J. Soffer, editors. Proceedings, International Symposium on High-Energy Physics with Polarized Beams and Polarized Targets, volume 38, Basel, Switzer-land, 1981. Birkhaeuser, Birkhaeuser.
[59] E. Martynov and B. Nicolescu. Evidence for maximality of strong interactionsfrom LHC forward data. Phys. Lett., B786:207–211, 2018.
[60] D. V. Bugg et al. pp total cross sections below 420 MeV/c. Phys. Lett., B194:563–567,1987.
[61] W. Bruckner et al. Measurements of the Anti-proton - Proton Annihilation Cross-section in the Beam Momentum Range Between 180-MeV/c and 600-MeV/c. Z.Phys., A335:217–229, 1990.
[62] O. D. Dalkarov and F. Myhrer. A Simple Model for Proton-anti-Proton Scatteringat Low-Energies. Nuovo Cim., A40:152, 1977.
[63] Carl B. Dover and J.-M. Richard. Elastic, charge exchange, and inelastic pp crosssections in the optical model. Phys. Rev. C, 21:1466–1471, Apr 1980.
[64] R. Bertini et al. Full Angular Distribution of the Analyzing Power in pp ElasticScattering at 697-MeV/c. Phys. Lett., B228:531–535, 1989.
[65] A. Ahmidouch et al. Charge exchangepp→ nn differential cross-sections between546 MeV/c and 1287 MeV/c. Phys. Lett., B364:116–120, 1995.
[66] R. Birsa et al. High precision measurement of the pp → nn charge exchangedifferential cross-section. Phys. Lett., B339:325–331, 1994. [Erratum: Phys.Lett.B405,389(1997)].
[67] S. Marcello. Five years of antineutron physics at LEAR. Nucl. Phys., A655:107–116,1999.
[68] F. Iazzi et al. Antineutron proton total cross-section from 50 MeV/c to 400 MeV/c.Phys. Lett., B475:378–385, 2000.
[69] E. Klempt, F. Bradamante, A. Martin, and J.-M. Richard. Antinucleon nucleoninteraction at low energy: Scattering and protonium. Phys. Rept., 368:119–316,2002.
[70] R. A. Kunne et al. Asymmetry in pp Elastic Scattering. Phys. Lett., B206:557–560,1988.
[71] R. A. Kunne et al. First measurement of D0n0n in pp elastic scattering. Phys. Lett.,B261:191–196, 1991.
[72] R. Birsa et al. First measurement of the depolarization parameter D0n0n of thepp→ nn charge exchange reaction. Phys. Lett., B302:517–522, 1993.
45
[73] R. Birsa et al. PS199: pp → nn spin physics at LEAR. In Proceedings, 4th Inter-national School of Physics with Low-energy Antiprotons - Medium-energy anti-protonsand the quark - gluon structure of hadrons: Erice, Italy, January 25 - 31, 1990, pages195–201, 1990.
[74] R. Timmermans, T. A. Rijken, and J. J. de Swart. Anti-proton - proton partial waveanalysis below 925-MeV/c. Phys. Rev., C50:48–73, 1994.
[75] Daren Zhou and Rob G. E. Timmermans. Energy-dependent partial-wave anal-ysis of all antiproton-proton scattering data below 925 MeV/c. Phys. Rev.,C86:044003, 2012.
[76] J.-M. Richard. Comment on ‘Antiproton proton partial wave analysis below925 MeV/c’. Phys. Rev., C52:1143–1144, 1995.
[77] Johann Haidenbauer. Talk at the ECT∗ Workshop on Antiproton Physics, Trento,June 17-21, 2019,https://indico.ectstar.eu/event/41/contributions/848/attachments/814/1048/Haidenbauer.pdf.
[78] M. Gourdin, B. Jancovici, and L. Verlet. On the nucleon-antinucleon interactions.Il Nuovo Cimento, 8(3):485–496, May 1958.
[79] R. A. Bryan and R. J. N. Phillips. Nucleon anti-nucleon potential from singlemeson exchanges. Nucl. Phys., B5:201–219, 1968.
[80] M. Kohno and W. Weise. Proton - Anti-proton Scattering and Annihilation IntoTwo Mesons. Nucl. Phys., A454:429–452, 1986.
[81] J. Haidenbauer, T. Hippchen, and K. Holinde. The Bonn NN potential and therange of NN annihilation. Nucl. Phys., A508:329C–334C, 1990.
[82] T. Hippchen, J. Haidenbauer, K. Holinde, and V. Mull. Meson - baryon dynam-ics in the nucleon - antinucleon system. 1. The nucleon - antinucleon interaction.Phys. Rev., C44:1323–1336, 1991.
[83] V. Mull and K. Holinde. Combined description of NN scattering and annihilationwith a hadronic model. Phys. Rev., C51:2360–2371, 1995.
[84] C. B. Dover and J.-M. Richard. Spin Observables in Low-energy Nucleon Anti-nucleon Scattering. Phys. Rev., C25:1952–1958, 1982.
[85] J.-M. Richard and M. E. Sainio. Nuclear Effects in Protonium. Phys. Lett.,110B:349–352, 1982.
[86] P. H. Timmers, W. A. van der Sanden, and J. J. de Swart. An antinucleon-nucleonpotential. Phys. Rev., D29:1928, 1984. [Erratum: Phys. Rev.D30,1995(1984)].
46
[87] B. El-Bennich, M. Lacombe, B. Loiseau, and S. Wycech. Paris NN potential con-strained by recent antiprotonic-atom data and antineutron-proton total cross sec-tions. Phys. Rev., C79:054001, 2009.
[88] Eberhard Klempt, Chris Batty, and Jean-Marc Richard. The Antinucleon-nucleoninteraction at low energy : Annihilation dynamics. Phys. Rept., 413:197–317, 2005.
[89] M. Kohno and W. Weise. Anti-hyperon - Hyperon Production in Low-energyAnti-proton - Proton Reactions. Nucl. Phys., A479:433C–449C, 1988.
[90] M. Ablikim et al. Polarization and Entanglement in Baryon-Antibaryon Pair Pro-duction in Electron-Positron Annihilation. Nature Phys., 15:631–634, 2019.
[91] B. Bassalleck et al. Measurement of spin transfer observables in pp → ΛΛ at1.637 GeV/c. Phys. Rev. Lett., 89:212302, 2002.
[92] K. D. Paschke et al. Experimental determination of the complete spin structurefor pp→ ΛΛ at 1.637 GeV/c. Phys. Rev., C74:015206, 2006.
[93] J. Haidenbauer, K. Holinde, V. Mull, and J. Speth. Can one discriminate betweenmeson exchange and quark - gluon transition mechanisms in the pp → ΛΛ pro-cess? Phys. Lett., B291:223–227, 1992.
[94] Mary A. Alberg, John R. Ellis, and Dmitri Kharzeev. The proton spin puzzle anddepolarization in pp→ ΛΛ. Phys. Lett., B356:113–117, 1995.
[95] Jean-Marc Richard. Remarks about spin measurements in p → ΛΛ. Phys. Lett.,B369:358–361, 1996.
[96] Xavier Artru, Mokhtar Elchikh, Jean-Marc Richard, Jacques Soffer, and Oleg V.Teryaev. Spin observables and spin structure functions: Inequalities and dynam-ics. Physics Reports, 470(1-2):1–92, Jan 2009.
[97] A Fridman, editor. 4th Antiproton Symposium, Proc. Antiproton Conference, Barr &Strasbourg, 1978, Paris, 1979. CNRS.
[98] G. Q. Liu and F. Tabakin. Coupled channels study of anti-proton proton reactions.Phys. Rev., C41:665–679, 1990.
[99] S. Takeuchi, F. Myhrer, and K. Kubodera. Maximum asymmetry phenomena inpp→ π−π+ and pp→ K−K+ reactions. Nucl. Phys., A556:601–620, 1993.
[100] M. Elchikh and J.-M. Richard. Meson exchange and nonlocality effects in protonanti-proton annihilation into two pseudoscalar mesons. Phys. Rev., C48:R17–R20,1993.
[101] A. Deloff. Fundamentals in hadronic atom theory. River Edge, USA: World Scientific(2003) 352 p, 2003.
47
[102] M. Augsburger, D.F. Anagnostopoulos, G. Borchert, D. Chatellard, J.-P. Egger,P. El-Khoury, H. Gorke, D. Gotta, P. Hauser, P. Indelicato, K. Kirch, S. Lenz,K. Rashid, Th. Siems, and L.M. Simons. Measurement of the strong interactionparameters in antiprotonic hydrogen and probable evidence for an interferencewith inner bremsstrahlung. Nuclear Physics A, 658(2):149 – 162, 1999.
[103] D. R. Entem and F. Fernandez. The NN interaction in a constituent quark model:Baryonium states and protonium level shifts. Phys. Rev., C73:045214, 2006.
[104] D. Gotta, D.F. Anagnostopoulos, M. Augsburger, G. Borchert, C. Castelli,D. Chatellard, J.-P. Egger, P. El-Khoury, H. Gorke, P. Hauser, P. Indelicato, K. Kirch,S. Lenz, N. Nelms, K. Rashid, Th. Siems, and L.M. Simons. Balmer α transitions inantiprotonic hydrogen and deuterium. Nuclear Physics A, 660(3):283 – 321, 1999.
[105] C. B. Dover, J.-M. Richard, and J. Carbonell. Isospin mixing in protonium andannihilation dynamics. Phys. Rev., C44:1281–1288, 1991.
[106] Stanley Deser, M. L. Goldberger, K. Baumann, and Walter E. Thirring. Energylevel displacements in pi mesonic atoms. Phys. Rev., 96:774–776, 1954.
[107] T. L. Trueman. Energy level shifts in atomic states of strongly-interacting particles.Nucl. Phys., 26:57–67, 1961.
[108] Monique Combescure, Avinash Khare, Ashok Raina, Jean-Marc Richard, and Ca-role Weydert. Level rearrangement in exotic atoms and quantum dots. Int. J. Mod.Phys., B21:3765–3781, 2007.
[109] Ya.B Zel’dovich. Energy levels in a distorted coulomb field. Sov.Phys. Solid State,1:1497, 1960.
[110] Jaume Carbonell, Jean-Marc Richard, and Slawomir Wycech. On the relation be-tween protonium level shifts and nucleon - anti-nucleon scattering amplitudes. Z.Phys., A343:325–329, 1992.
[111] S. Furui, G. Strobel, A. Faessler, and R. Vinh Mau. The isospin mixing effect in ppannihilation at rest into π+π− and KK. Nucl. Phys., A516:643–661, 1990.
[112] W. B. Kaufmann and H. Pilkuhn. Black Sphere Model for the Line Widths of pState Protonium. Phys. Rev., C17:215–218, 1978.
[113] T. B. Day, G. A. Snow, and J. Sucher. High-Orbital S-State Capture of π-Mesons byProtons. Phys. Rev., 118:864–866, 1960.
[114] D. Garreta et al. Elastic Scattering of Anti-protons From Carbon, Calcium andLead at 180-MeV. Phys. Lett., 149B:64–68, 1984.
48
[115] G. Bruge et al. Comparative Study of the Elastic Scattering of 178.4-MeV Anti-protons by the 16O and 18O Isotopes. Phys. Lett., 169B:14–16, 1986.
[116] S. Janouin et al. Optical Model Analysis of Anti-proton Nucleus Elastic Scatteringat 50-MeV and 180-MeV. Nucl. Phys., A451:541–561, 1986.
[117] S. Adachi and H. V. Von Geramb. Microscopic Analysis of Anti-proton - NucleusElastic Scattering. Nucl. Phys., A470:461–476, 1987.
[118] T. Suzuki. Analysis of anti-proton nucleus scattering with the medium correctedoptical potential. Nucl. Phys., A444:659–677, 1985.
[119] H. Heiselberg, A. S. Jensen, A. Miranda, G. C. Oades, and J.-M. Richard. Mi-croscopic Calculation of Anti-proton Nucleus Elastic Scattering. Nucl. Phys.,A451:562–580, 1986.
[120] F. Balestra et al. Evidence of Isospin Effects in Anti-proton - Nucleus Annihilation.Nucl. Phys., A491:572–586, 1989.
[121] E. Botta. Antineutron nucleus annihilation. Nucl. Phys., A692:39–46, 2001.
[122] C. B. Dover, M. E. Sainio, and G. E. Walker. Antinucleon as a probe of nuclear spinand isospin excitations. Phys. Rev., C28:2368–2373, 1983.
[123] C. B. Dover, M. A. Franey, W. G. Love, M. E. Sainio, and G. E. Walker. Antinu-cleon nucleus inelastic scattering and the spin dependence of the NN annihilationpotential. Phys. Lett., 143B:45–49, 1984.
[124] C. J. Batty, E. Friedman, and A. Gal. Strong interaction physics from hadronicatoms. Phys. Rept., 287:385–445, 1997.
[125] D. Gotta. Precision spectroscopy of light exotic atoms. Prog. Part. Nucl. Phys.,52:133–195, 2004.
[126] A. Bouyssy and S. Marcos. Relativistic mean field approach to the scattering ofantinucleons and nucleons by nuclei. Phys. Lett., 114B:397–402, 1982.
[127] T. Gaitanos and M. Kaskulov. Toward relativistic mean-field description of N-nucleus reactions. Nucl. Phys., A940:181–193, 2015.
[128] A. B. Larionov and H. Lenske. Elastic scattering, polarization and absorption ofrelativistic antiprotons on nuclei. Nucl. Phys., A957:450–476, 2017.
[129] E. Friedman, A. Gal, B. Loiseau, and S. Wycech. Antinucleon–nucleus interactionnear threshold from the Paris NN potential. Nucl. Phys., A943:101–116, 2015.
[130] Jaroslava Hrtánková and Jirí Mareš. Calculations of antiproton–nucleus quasi-bound states using the paris nn potential. Nuclear Physics A, 969:45 – 59, 2018.
49
[131] T Aumann, W Bartmann, A Bouvard, O Boine-Frankenheim, A Broche, F Butin,D Calvet, J Carbonell, P Chiggiato, H De Gersem, R De Oliveira, T Dobers,F Ehm, J Ferreira Somoza, J Fischer, M Fraser, E Friedrich, J-L Grenard, G Hupin,K Johnston, Y Kubota, M Gomez-Ramos, P Indelicato, R Lazauskas, S Malbrunot-Ettenauer, N Marsic, W Müller, S Naimi, N Nakatsuka, R Necca, G Neyens,A Obertelli, Y Ono, S Pasinelli, N Paul, E C Pollacco, D Rossi, H Scheit, R Seki,A Schmidt, L Schweikhard, S Sels, E Siesling, T Uesaka, M Wada, F Wienholtz,S Wycech, and S Zacarias. PUMA: antiprotons and radioactive nuclei. TechnicalReport CERN-SPSC-2019-033. SPSC-P-361, CERN, Geneva, Sep 2019.
[132] M. Baldo-Ceolin et al. A New experimental limit on neutron - anti-neutron oscil-lations. Z. Phys., C63:409–416, 1994.
[133] Camille Theroine. A neutron-antineutron oscillation experiment at the EuropeanSpallation Source. Nucl. Part. Phys. Proc., 273-275:156–161, 2016.
[134] P. K. Kabir. Limits on nn oscillations. Phys. Rev. Lett., 51:231, 1983.
[135] W. M. Alberico. Neutron antineutron oscillations: Theory. In Lepton and baryonnumber violation in particle physics, astrophysics and cosmology. Proceedings, 1st Inter-national Symposium, Lepton-baryon’98, Trento, Italy, April 20-25, 1998, pages 94–108,1998.
[136] Joshua L. Barrow, Elena S. Golubeva, Eduard Paryev, and Jean-Marc Richard.Progress and Simulations for Intranuclear Neutron–Antineutron Transformationsin 40
18Ar, arXiv:1906.02833. 2019.
[137] C. B. Dover, A. Gal, and J.-M. Richard. Neutron antineutron oscillations in nuclei.Phys. Rev., D27:1090–1100, 1983.
[138] F. Oosterhof, B. Long, J. de Vries, R. G. E. Timmermans, and U. van Kolck.Baryon-number violation by two units and the deuteron lifetime. Phys. Rev. Lett.,122(17):172501, 2019.
[139] Johann Haidenbauer and Ulf-G Meißner. Neutron-antineutron oscillations in thedeuteron studied with NN and NN interactions based on chiral effective fieldtheory. 2019. arXiv:1910.14423 [hep-ph].
[140] Shmuel Nussinov and Melissa A. Lampert. QCD inequalities. Phys. Rept.,362:193–301, 2002.
[141] Crtomir Zupancic. Decay of a spinless particle into four identical spinless bosons.Annals Phys., 235:245–317, 1994.
[142] S. Devons, T. Kozlowski, P. Nemethy, S. Shapiro, N. Horwitz, T. Kalogeropou-los, J. Skelly, R. Smith, and H. Uto. Observation of pp → 2π0 at Rest: EvidenceConcerning s State Annihilation. Phys. Rev. Lett., 27:1614, 1971.
50
[143] Maurice Lévy. Range of proton-antiproton annihilation. Phys. Rev. Lett., 5:380–381,Oct 1960.
[144] A. Martin. Range of the nucleon-antinucleon annihilation potential. Phys. Rev.,124:614–615, Oct 1961.
[145] J. Côté, M. Lacombe, B. Loiseau, B. Moussallam, and R. Vinh Mau. Nucleon-antinucleon optical potential. Phys. Rev. Lett., 48:1319–1322, May 1982.
[146] H. R. Rubinstein and H. Stern. Nucleon - antinucleon annihilation in the quarkmodel. Phys. Lett., 21:447–449, 1966.
[147] J. P. Carroll. Extended quark-rearrangement model for proton-antiproton annihi-lation at rest. Phys. Rev., 184:1793–1802, 1969.
[148] K. Kleinknecht and E. Klempt, editors. Proton anti-proton interactions and fundamen-tal symmetries. Proceedings, 9th European Symposium, Mainz, F.R. Germany, September5-10, 1988, volume 8, 1989.
[149] Anthony M. Green and J. A. Niskanen. From phenomenological to microscopicdescriptions of NN annihilation. Int. Rev. Nucl. Phys., 1:569–700, 1984.
[150] Anthony M. Green and J. A. Niskanen. Low-energy Anti-proton Physics in theEarly LEAR Era. Prog. Part. Nucl. Phys., 18:93–182, 1987.
[151] G. Ihle, H. J. Pirner, and J.-M. Richard. s Wave Nucleon - Anti-nucleon Interactionin the Constituent Quark Model. Nucl. Phys., A485:481–508, 1988.
[152] R. A. Eisenstein et al. The JETSET experiment at LEAR. In Proceedings, BNL Work-shop on Glueballs, Hybrids, and Exotic Hadrons: Brookhaven, Upton, USA, August29-September 1, 1988, pages 636–642, 1989.
[153] Claude Amsler and Fred Myhrer. Low-energy anti-proton physics. Ann. Rev.Nucl. Part. Sci., 41:219–267, 1991.
[154] Alexander Evgenyevich Kudryavtsev. Some comments on anti-n p annihilationbranching ratios into pi pi, anti-K K and pi eta channels. Phys. Atom. Nucl.,63:1969–1972, 2000. [Yad. Fiz.63,2061(2000)].
[155] J. Vandermeulen. NN Annihilation creates two mesons. Z. Phys., C37:563–567,1988.
[156] A Le Yaouanc, Lluis Oliver, O Pène, and Jean-Claude Raynal. Hadron transitionsin the quark model. Gordon and Breach, New York, NY, 1988.
[157] Henning Genz. The quark line rule in nucleon antinucleon annihilation. AIP Conf.Proc., 176:405–410, 1988.
51
[158] H. J. Pirner. NN Annihilation in the Large Nc Limit. Phys. Lett., B209:154–158,1988.
[159] Hans Gunter Dosch and D. Gromes. Flavor Dependence of Spontaneous PairCreation. Z. Phys., C34:139, 1987.
[160] J. Cugnon, P. Jasselette, and J. Vandermeulen. Dynamics of the anti-p nucleusannihilation. In Proceedings, 4th LEAR Workshop: Physics at LEAR with low energyantiprotons: Villars, Switzerland, September 6-13, 1987, pages 775–778, 1996.
[161] J. Cugnon and J. Vandermeulen. Double strangeness production in anti-protonannihilation on nuclei. Z. Phys., A338:349–356, 1991.
[162] Johann Rafelski. Deep anti-proton annihilations on nuclei. In Proceedings, 3rdInternational School of Physics with Low-energy Antiprotons "Ettore Majorana" Anti-proton - nucleon and anti-proton - nucleus interactions: Erice, Italy, June 10 - 18, 1988,pages 231–239, 1996.
[163] H. Koch. Hadron physics with antiprotons. Nucl. Instrum. Meth., B214:50–55, 2004.
[164] Ulrich Wiedner. Hadron physics with PANDA. Hyperfine Interact., 194(1-3):219–223, 2009.
[165] M. Chiba et al. Pontecorvo reactions in antiproton annihilation at rest in deu-terium to π0n , π0∆0, ηn and η∆0. Phys. Rev., D55:2577–2583, 1997.
[166] O. E Gorchakov et al. Measurement of the pd → φn Pontecorvo reaction for anti-proton annihilation at rest. Phys. Lett., B528:34–42, 2002.
[167] W. M. Alberico and G. Garbarino. Weak decay of Λ hypernuclei. Phys. Rept.,369:1–109, 2002.
[168] Achim Denig and Giovanni Salme. Nucleon Electromagnetic Form Factors in theTimelike Region. Prog. Part. Nucl. Phys., 68:113–157, 2013.
[169] S. Pacetti, R. Baldini Ferroli, and E. Tomasi-Gustafsson. Proton electromagneticform factors: Basic notions, present achievements and future perspectives. Phys.Rept., 550-551:1–103, 2015.
[170] G. Bardin et al. Measurement of the proton electromagnetic form-factor nearthreshold in the timelike region. Phys. Lett., B255:149–154, 1991.
[171] C. Baglin et al. Search for the p Wave Singlet Charmonium State in pp Annihila-tions at the CERN Intersecting Storage Rings. Phys. Lett., B171:135–141, 1986.
[172] M. Ambrogiani et al. Search for the η′c(21S0) Charmonium Resonance. Phys. Rev.,D64:052003, 2001.
52
[173] E. Gabathuler and P. Pavlopoulos. The CPLEAR experiment. Phys. Rept., 403-404:303–321, 2004.
[174] D. Hatzifotiadou et al. Some results of experiment PS182 at LEAR. In Anti-proton86. Proceedings, 8th European Symposium on Nucleon Anti-nucleon Interactions, Thes-saloniki, Greece, September 1-5, 1986, pages 199–204, 1986.
[175] Qian Wang, Gang Li, and Qiang Zhao. Open charm effects in the explanation ofthe long-standing ’ρπ puzzle’. Phys. Rev., D85:074015, 2012.
[176] C. B. Dover, P. M. Fishbane, and S. Furui. Dynamical Selection Rules in NN An-nihilation. Phys. Rev. Lett., 57:1538–1541, 1986.
[177] H. Nifenecker. Fission of heavy hypernuclei. Prog. Part. Nucl. Phys., 30:407–414,1993.
[178] I. Zahed and G. E. Brown. The Skyrme Model. Phys. Rept., 142:1–102, 1986.
[179] I. Zahed and G. E. Brown. A Nonperturbative Description of NN Annihilation.Z. Phys., A335:349–354, 1990.
[180] T. William Donnelly, Joseph A. Formaggio, Barry R. Holstein, Richard G. Milner,and Bernd Surrow. Foundations of Nuclear and Particle Physics. Cambridge Univer-sity Press, 2017.
[181] R. Machleidt and D. R. Entem. Chiral effective field theory and nuclear forces.Phys. Rept., 503:1–75, 2011.
[182] E. Epelbaum, H. Krebs, and U. G. Meißner. Improved chiral nucleon-nucleonpotential up to next-to-next-to-next-to-leading order. Eur. Phys. J., A51(5):53, 2015.
[183] G. Y. Chen and J. P. Ma. NN Scattering at NLO Order in An Effective Theory.Phys. Rev., D83:094029, 2011.
[184] Ling-Yun Dai, Johann Haidenbauer, and Ulf-G Meißner. Antinucleon-nucleoninteraction at next-to-next-to-next-to-leading order in chiral effective field theory.JHEP, 07:078, 2017.
[185] Johann Haidenbauer. Antinucleon-nucleon interaction in chiral effective field the-ory. EPJ Web Conf., 181:01028, 2018.
[186] Matteo Vorabbi, Michael Gennari, Paolo Finelli, Carlotta Giusti, and Petr Navrátil.Elastic Antiproton-Nucleus Scattering from Chiral Forces. 2019. arXiv:1906.11984[nucl-th].
53
