Meson-nucleus potentials and the search for meson-nucleus bound states
MMeson–nucleus potentials and the search for meson-nucleusbound states
V. Metag, M. Nanova, , and E. Ya. Paryev, , II. Physics Institute, University of Giessen, Giessen 35392, Germany Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia Institute for Theoretical and Experimental Physics, Moscow 117218, RussiaJuly 26, 2017
Abstract
Recent experiments studying the meson-nucleus interaction to extract meson-nucleus potentialsare reviewed. The real part of the potentials quantifies whether the interaction is attractive orrepulsive while the imaginary part describes the meson absorption in nuclei. The review is focusedon mesons which are sufficiently long-lived to potentially form meson-nucleus quasi-bound states.The presentation is confined to meson production off nuclei in photon-, pion-, proton-, and light-ion induced reactions and heavy-ion collisions at energies near the production threshold. Toolsto extract the potential parameters are presented. In most cases, the real part of the potential isdetermined by comparing measured meson momentum distributions or excitation functions withcollision model or transport model calculations. The imaginary part is extracted from transparencyratio measurements. Results on K + , K , K − , η, η (cid:48) , ω , and φ mesons are presented and comparedwith theoretical predictions. The interaction of K + and K mesons with nuclei is found to beweakly repulsive, while the K − , η, η (cid:48) , ω and φ meson-nucleus potentials are attractive, however, withwidely different strengths. Because of meson absorption in the nuclear medium the imaginary partsof the meson-nucleus potentials are all negative, again with a large spread. An outlook on plannedexperiments in the charm sector is given. In view of the determined potential parameters, thecriteria and chances for experimentally observing meson-nucleus quasi-bound states are discussed.The most promising candidates appear to be the η and η (cid:48) mesons. Contents
Scope of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Transport simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.1
Off-shell transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 a r X i v : . [ nu c l - e x ] J u l .1.2 The collision term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.3
Comparison to experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2
Collision model based on the nuclear spectral function . . . . . . . . . . . . . . . . . . . 103.2.1
Near-threshold η (cid:48) meson photoproduction off nuclei . . . . . . . . . . . . . . . . 103.2.2 One-step η (cid:48) production mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Nucleon spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.4
Two-step η (cid:48) production mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Determination of the imaginary part of the meson–nucleus potential from measurementsof the transparency ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4
Determination of the real part of the meson–nucleus potential . . . . . . . . . . . . . . . 183.4.1
Real part of the meson–nucleus potential deduced from excitation function mea-surements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2
Real part of the meson–nucleus potential deduced from measurements of the mesonmomentum distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 K + –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.1 K + –nucleon scattering length and K + –nucleus potential . . . . . . . . . . . . . . 214.1.2 K + –nucleus potential. Other theoretical approaches . . . . . . . . . . . . . . . . 224.1.3 Determination of the K + –nucleus real potential . . . . . . . . . . . . . . . . . . 244.2 K –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1 In-medium neutral kaon potential. Theoretical expectations . . . . . . . . . . . . 274.2.2
Determination of the K –nucleus real potential . . . . . . . . . . . . . . . . . . . 284.3 K − –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.1 In-medium antikaon potential. Theoretical predictions . . . . . . . . . . . . . . . 304.3.2
Determination of the K − –nucleus real potential . . . . . . . . . . . . . . . . . . 354.4 η –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.1 The η -nucleon scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.2 The imaginary part of the η - nucleus potential . . . . . . . . . . . . . . . . . . 424.4.3 The real part of the η -nucleus potential . . . . . . . . . . . . . . . . . . . . . . . 434.5 η (cid:48) –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5.1 The η (cid:48) -nucleon scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5.2 The imaginary part of the η (cid:48) -nucleus potential . . . . . . . . . . . . . . . . . . . 444.5.3 The real part of the η (cid:48) -nucleus potential . . . . . . . . . . . . . . . . . . . . . . 454.6 The ω –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.6.1 The ω –nucleon scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.6.2 The imaginary part of the ω –nucleus potential . . . . . . . . . . . . . . . . . . . 494.6.3 The real part of the ω –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . 504.7 φ –nucleus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7.1 The φ –nucleon scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7.2 The imaginary part of the φ - nucleus potential . . . . . . . . . . . . . . . . . . 544.7.3 The real part of the φ - nucleus potential . . . . . . . . . . . . . . . . . . . . . . 56 meson–nucleus potentials in the charm sector . . . . . . . . . . . . . . . . . . . . . . . . 57 The search for mesic states 60 pionic and kaonic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.2
Search for kaonic clusters and K − nuclear quasi-bound states . . . . . . . . . . . . . . . 657.3 Search for η -mesic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.4 Search for η (cid:48) -mesic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.5 Search for ω -mesic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.6 Search for φ -mesic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.7 Search for mesic states in the charm sector . . . . . . . . . . . . . . . . . . . . . . . . 75
Scope of the review
The interaction of mesons with nuclei and the possible existence of meson–nucleus bound states haverecently been in the focus of numerous theoretical and experimental studies. This review attempts tosummarise these activities and to describe the current status of the field.Mesonic atoms where an electron is replaced by a π − or K − meson are well known and have beenstudied in high precision experiments. Information on the real and imaginary part of the meson-nucleuspotential has been extracted from fitting density dependent potentials to comprehensive sets of data ofstrong-interaction level shifts, widths and yields across the periodic table, including more recent resultson the binding energy and width of deeply bound pionic states [1-4]. For reviews and recent resultssee [5-11]. The dominant interaction in these cases is the Coulomb interaction while the effect of thestrong interaction is only a correction. These experiments probe the meson-nucleus potentials near thenuclear surface and thus preferentially at relatively small nuclear densities.The main focus of this review is on nuclear exotic states and thus on the interaction of nucleiwith neutral mesons where only the strong interaction is relevant or where - in the case of chargedmesons - the strong interaction is dominant. Experimental studies of this type have applied differentprojectiles for the meson production process ranging from photons, pions, protons, light ions to heavy-ions. The experiments have been performed over a broad range of energies from reactions near theproduction threshold up to ultra-relativistic heavy-ion collisions (see [12-14]). Here, we will concentrateon energies close to or slightly above the production threshold, the regime relevant for the possibleformation of meson-nucleus bound states, since only slow mesons can be captured by the nucleusprovided there is sufficient attraction. Furthermore, we concentrate on mesons sufficiently longlivedto form a meson–nucleus quasi-bound state. These are the pseudoscalar mesons K + , K , K − , η, η (cid:48) andthe vector mesons ω and φ for which parameters of the meson-nucleus potential have been extractedin numerous experiments. The enormous theoretical and experimental work on the ρ meson has beenreviewed in [12-14] and will not be discussed here further since the ρ meson is too short-lived to formmesic states.Extensions of these studies to the charm sector have been considered in a number of theoreticalpapers but represent a real challenge experimentally. The heavier the meson, the larger is the momentumtransfer in the production process and the more difficult it is to produce mesons sufficiently slow tobe captured by the nucleus. The possibilities and challenges regarding nuclear bound states involvingheavy flavour hadrons have been outlined in a very recent review by Krein et al.[15].3 .2 Motivation
Why is it interesting to study the interaction of mesons with nuclei? This interaction is an impor-tant testing ground for our understanding of Quantum Chromodynamics (QCD) as the theory of thestrong interaction in the non-perturbative regime. Research in this field was motivated in particular bytheoretical predictions that meson properties might change within nuclei due to a partial restorationof chiral symmetry [16-19]. Mesons are considered to be excitations of the QCD vacuum which has acomplicated structure with non-vanishing chiral-, gluon- and higher order quark-condensates. Thesecondensates are predicted to change within a strongly interacting medium and, as a consequence, alsothe excitation energy spectrum, i.e. the mass spectrum of mesons is expected to be modified. Thisidea fostered widespread theoretical and experimental activities which have been summarised in recentreviews [12-14].
Mass [GeV]spontaneousU(3) L x U(3) R breakingm i = 0 π , K, η , η η π , K, η U(1) A breakingm i = 0 SU(3) L x SU(3) R Goldstonebosons SU(3) F breakingm u ≈ d ≈ s ≈
95 MeV π K ηη ’ Figure 1: Symmetry breaking pattern for pseudoscalar mesons as described in the text, adaptedfrom [20].Pseudoscalar mesons are particularly suited for studying in-medium modifications, as shown inFig. 1 [20]. Spontaneous chiral symmetry breaking generates a pseudoscalar nonet ( π, K, ¯ K, η, η (cid:48) ) ofmassless Nambu-Goldstone bosons. The explicit breaking of the U(1) A symmetry selectively shifts upthe singlet η mass, leaving the SU(3) flavour octet of pions, kaons and η massless. Explicit chiralsymmetry breaking by introducing non-zero quark masses then leads to the experimentally observedmeson masses [16, 17, 20]. Since symmetry breaking has such a big effect on meson masses one wouldexpect corresponding effects in case of a partial restoration of this symmetry in a strongly interactingmedium. These effects may have implications going beyond hadron- and nuclear physics. In particular,an effective reduction of the K − mass in a nuclear medium might indicate the onset of K − condensationin dense nuclear matter as e.g., in the interior of neutron stars, as proposed in [21, 22].If spontaneously broken chiral symmetry were restored at high nuclear densities one would expectthe full spectral function of vector mesons with spin parity J π = 1 − to become degenerate with thatof their chiral partners with the same spin but opposite parity, the axial vector mesons with J π =1 + . Traces of this tendency should already show up at normal nuclear matter density and should thusbecome observable in photonuclear experiments and reactions induced by pions and protons.For long-lived mesons such as pions or kaons, meson beams can be used to study the meson–nucleusinteraction experimentally. This is not possible for short-lived mesons like η, η (cid:48) and ω, φ mesons wheresuch beams are not available. Here, one has to produce the mesons in a nuclear reaction and to study4heir interaction with nucleons or nuclei in the final-state. The interaction of mesons with a nuclear medium of density ρ N is usually described by a dispersionrelation based on the Klein-Gordon equation (see e.g. [7]) E (cid:48) − p (cid:48) − m − Π( E (cid:48) , p (cid:48) , ρ N ( r )) = 0 , (1)where Π( E (cid:48) , p (cid:48) , ρ N ( r )) is the meson self-energy which summarizes all the meson interactions in themedium. Here m, p (cid:48) , E (cid:48) and r are the free meson rest mass, its 3-momentum, its energy in the mediumand its distance to the center of the nucleus. To leading order in density ρ N ( r ),Π( E (cid:48) , p (cid:48) , ρ N ( r )) = − πf ( E (cid:48) , p (cid:48) ) (2)where f ( E (cid:48) , p (cid:48) ) is the (in medium generally off-shell) meson-nucleon forward scattering amplitude whichmay involve scalar and/or vector t-channel terms.A meson-nucleus potential may be defined through the self-energy as U ( E (cid:48) , p (cid:48) , ρ N ( r )) = Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 √ p (cid:48) + m . (3)This potential is complex: U ( r ) = V ( r ) + iW ( r ) (4)The real part V ( r ) encodes whether the interaction of the meson with the nucleus is attractive orrepulsive, while the imaginary part W ( r ) of the potential accounts for the absorption of the meson inthe medium through inelastic reactions.Combining Eqs. (3),(4) one obtains U ( r ) = Re Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 √ p (cid:48) + m + i Im Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 √ p (cid:48) + m . (5)In this review the focus is on slow mesons with momenta small compared to their mass: | p (cid:48) | ≤ m ,i.e. mesons that are so slow that they may even be captured by the nucleus in case of an attractiveinteraction; thus in the region of main interest Eq. 5 simplifies to V ( r ) = Re Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 m , W ( r ) = Im Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 m . (6)In this approximation and assuming that the in-medium meson width is small compared to its in-medium mass [23] (which turns out to be justified for all mesons considered in this review), the realpart of Eq. 1 takes the form E (cid:48) − p (cid:48) − m − mV ( r ) = 0 (7)Introducing an effective in-medium mass m ∗ = ( m + ∆ m ( ρ N )) , (8)where ∆ m is the density dependent mass modification of the meson due to the interaction with thenuclear medium, the real potential V(r) can be related to ∆ m . From Eq. 1 one obtains E (cid:48) − p (cid:48) − m ∗ = E (cid:48) − p (cid:48) − ( m + ∆ m ( ρ N )) = E (cid:48) − p (cid:48) − ( m + 2 m ∆ m ( ρ N )) . (9)5ere it has been assumed that the mass modification ∆ m is small compared to the meson mass, so thatterms quadratic in ∆ m can be neglected [24].Comparing Eqs. (7),(9), the real part V ( r ) of the meson-nucleus potential is found to be equal tothe in-medium meson mass shift ∆ m ( ρ N ( r )). Assuming a linear density dependence one gets V ( r ) = ∆ m ( ρ N ( r )) = ∆ m ( ρ N = ρ ) · ρ N ( r ) /ρ = V · ρ N ( r ) /ρ . (10)Correspondingly, the imaginary part W(r) of the potential and the in-medium meson width in thenuclear rest frame can be derived from the imaginary part of the self-energy [23, 25, 26] W ( r ) = −
12 Γ( E (cid:48) , p (cid:48) , ρ N ( r )) = Im Π( E (cid:48) , p (cid:48) , ρ N ( r ))2 m . (11)Assuming again a linear density dependence Γ( ρ N ) = Γ( ρ N = ρ ) · ρ N ( r ) /ρ = Γ · ρ N ( r ) /ρ oneobtains W ( r ) = −
12 Γ · ρ N ( r ) ρ . (12)The quantities V and Γ are determined from the data as averages over a momentum range, typically0 ≤ | p (cid:48) | ≤ m .In this review we discuss the determination of the real and imaginary parts of the meson–nucleuspotential for pseudo-scalar mesons K + , K , K − , η, η (cid:48) and the vector mesons ω and φ . Most of the theoretical predictions have been calculated under the idealised assumption of completeequilibrium e.g. for a meson at rest embedded in infinitely extended nuclear matter with constantdensity and temperature, a scenario far from reality. In meson production experiments in the 1-3 GeVenergy range, the kinematics of the reaction leads to meson recoil momenta on average comparableto their mass; on the way out of the nucleus mesons see the nuclear density profile with a fall-offat the surface. Any density dependent mass shift or broadening is thereby smeared out due to thevariation in density. A link between the theoretical predictions and the experimental observables isprovided by either transport calculations like, e.g. Giessen-Boltzmann-Uehling-Uhlenbeck (GiBUU)[27], Coupled-Channel-Boltzmann-Uehling-Uhlenbeck (CBUU) [28], Relativistic Quantum MolecularDynamics (RQMD) [29, 30, 31], Relativistic-Boltzmann-Uehling-Uhlenbeck (RBUU) [32], Isospin Quan-tum Molecular Dynamics (IQMD) [33], Hadron String Dynamics (HSD) [34] or calculations within acollision model based on nuclear spectral functions [35], and Monte Carlo simulations [36, 37]. For ameaningful comparison of theoretical predictions with experimental data to extract in-medium prop-erties of hadrons, it is indispensable to investigate with such calculations how non-equilibrium effectsor reaction dynamics modify the theoretically predicted initial signals. These calculations are brieflydescribed in the following subsections.
Transport simulations
Transport calculations simulate photon-, electron-, neutrino-, hadron- and heavy-ion-induced reactionson nuclei. The relevant degrees of freedom are mesons and baryons which propagate in mean fieldsand scatter according to cross sections which are provided as external input. The simulations arebased on a set of semi-classical kinetic equations which explicitly describe the dynamics of a hadronicsystem in space and time, i.e. the hadronic degrees of freedom in nuclear reactions are taken intoaccount, including the propagation, elastic and inelastic collisions and decays of particles. In additionto nucleons, all established nucleon resonance as well as light pseudoscalar and vector mesons are6aken into account in the propagation through the nuclear medium. Here we restrict ourselves to abrief presentation of the basic ideas and confine ourselves for simplicity to non-relativistic transport,however, including relativistic kinematics. Details can be found in the original literature cited above.As an example the concept of GiBUU model calculations [38] will be discussed.The transport equations describe the classical time evolution of a system with N particle speciesunder the influence of a self-consistent mean-field potential and a collision term. For each particlespecies i involved one obtains a differential equation of the type:( ∂∂t + ∂H i ∂(cid:126)p ∂∂(cid:126)r − ∂H i ∂(cid:126)r ∂∂(cid:126)p ) F i ( (cid:126)r, (cid:126)p, W, t ) = G i S i (1 ± f i ) − L i F i ( i = 1 , ....., N ) , (13)where (cid:126)r and (cid:126)p are the spatial and momentum coordinates of the particle i . F i ( (cid:126)r, (cid:126)p, W, t ) denotes the so-called spectral phase-space density of particle species i which is the product of the ordinary quasiparticlephase-space density f i and the spectral function S i [27]: F i ( (cid:126)r, (cid:126)p, W, t ) = f i ( (cid:126)r, (cid:126)p, W, t ) S i ( W, (cid:126)r, (cid:126)p ) (14)The spectral function S i gives directly the probability to find a particle of species i and pole mass m i with invariant mass W and momentum (cid:126)p at position (cid:126)r : S i ( W, (cid:126)r, (cid:126)p ) = 2 π W Γ tot ( W, (cid:126)r, (cid:126)p )( W − m i ) + W Γ ( W, (cid:126)r, (cid:126)p ) . (15)Here, Γ tot is the total width of a particle of species i in its rest frame. The Hamilton functionH i ( (cid:126)r, (cid:126)p, W, f , f , ..f i , ..f N ) is given by the relativistic expression for the single particle energy: H i ( (cid:126)r, (cid:126)p, W, f , ...f i , ...f N ) = (cid:113) ( W + U i ( (cid:126)r, (cid:126)p, W, f , ...f i , ...f N )) + (cid:126)p , (16)where U i is the coordinate- and momentum-dependent effective scalar potential seen by particle i .Since all particle species mutually interact the integro-differential equations are coupled by the collisionintegrals which describe the gain G i and loss L i of each particle species in inelastic collisions. Theseequations are solved numerically by the so-called test-particle method. Off-shell transport
Of particular importance for the study of in-medium properties of hadrons is the implementation of anoff-shell transport, describing the propagation of quasi-particles with broad mass spectra, taking thedensity dependence of the spectral function into account. As the particle propagates through regions ofvarying density the spectral function changes accordingly, implying changes in the mass distribution.In particular, it has to be ensured that particles leaving the nucleus return to their vacuum spectralfunction, i.e. stable particles need to come back to their vacuum mass listed in [39] even though theyhad a broad (and shifted) mass distribution in the medium. A practical way to simulate off-shell effectsis to define a scalar potential as ˜ U i ( (cid:126)r i ) = ( W ∗ i − W i ) ρ N ( (cid:126)r i ) ρ N ( (cid:126)r i cr ) , (17)where ρ N ( (cid:126)r i ) is the local nucleon density and the asterisk refers to the mass determined at the instantof the test-particle production at the point with spatial coordinate (cid:126)r i cr . W i is a mass chosen randomlyaccording to the vacuum spectral function. This potential enters the single-particle Hamiltonian as aconventional scalar potential, thus guaranteeing energy conservation. The effective in-medium mass ofthe test-particle during its propagation through the nucleus is then given as W i = W i + ˜ U i ( (cid:126)r ) , (18)7hich yields the correct asymptotic value of the test-particle mass since the potential vanishes outsideof the nucleus. Thus, the mass spectrum of a physical particle, represented by a bunch of test-particles,takes the form of the vacuum spectral function when all test-particles have escaped from the nuclearenvironment. For a more rigoruos formulation of relativistic off-shell transport the reader is referred to[40, 41, 42]. The collision term
The so-called collision term in the right hand side of Eq. 13 influences the time evolution of the particlephase-space density via the creation and destruction of particles in scattering and decay processes. Forsimplicity we restrict us here to the quasiparticle limit, i.e. S i → F i → f i . The right hand sideof Eq. 13 then reads G i (1 ± f i ) − L i f i , (19)where (1 ± f i ) is a Pauli blocking or Bose enhancement factor for particle i being a fermion or a boson,respectively. In this simplified case - the complete formulae for the general case are given in [27] - theloss term L i can be written as L i ( (cid:126)r, (cid:126)p, t ) = 1(2 π ) (cid:88) j (cid:88) ab (cid:90) d p j (cid:90) d Ω cm dσ ij → ab d Ω cm v ij f j ( (cid:126)r, (cid:126)p j , t ) P a P b + (cid:88) cd (cid:90) d Ω cm d Γ i → cd d Ω cm P c P d , (20)where dσ ij → ab /d Ω cm and d Γ i → cd /d Ω cm are the angle differential cross sections and decay widths of theprocesses ij → ab and i → cd , respectively. The terms P x = 1 − f x ( (cid:126)r, (cid:126)p x , t ) with X = a, b, c, d introducePauli blocking of final-states in the case where a, b, c, d are fermions or P x = 1 otherwise; v ij is therelative velocity of the collision partners i and j . The final states ab or cd and the collision partners j are summed over in order to obtain the total loss rate L i . The expression for the particle gain G i hasa similar form.These ingredients are characteristic for any transport simulation although there are differences inthe implementation, depending on the optimisation for certain classes of reactions.Codes focusing on high-energy reaction include sub-nucleonic degrees of freedom to account for thephenomena like color transparency, hadron formation time or the transition to a quark-gluon plasmain ultra-relativistic heavy-ion reactions. For a detailed description of the different transport codes thereader is referred to the original literature [27-31,33,34,43-46]. Comparison to experiment
For a comparison to experimental data it is important that transport calculations take nuclear many-body effects into account which evolve dynamically in the course of the reaction. In particular theytreat- initial state effects: the absorption of incoming beam particles- non equilibrium effects: varying nucleon density and temperature- absorption and regeneration of mesons- fraction of decays outside of the nucleus- final-state interactions: distortion of momenta of decay products.As a result, a full space-time evolution of the many-particle system is obtained starting from the initialstate of the reaction to the final-state particles as they are observed in the experiment.As an example for the importance of transport calculations in the comparison of theoretical pre-dictions with experimental observables we discuss the line shape of the ω meson reconstructed fromthe ω → π γ decay in photoproduction off nuclei. As presented in Section 4.6, a reduction of the ω mass at normal nuclear matter density by 120 MeV has been predicted in QCD sum rule calculations8igure 2: π γ invariant mass spectra from ω decays for γ + Nb reactions at incident photon energies of0.9–1.3 GeV calculated with the GiBUU code [27] for four different in-medium modification scenarios:a.) no medium modification (vacuum spectral function), b.) collisional broadening by 150 MeV [51], c.)mass shift by -16%, d.) collisional broadening (150 MeV) and mass shift (-16%). The total spectrum isshown as well as contributions from in-medium ( ρ N > . ρ ) and vacuum ( ρ N < . ρ ) decays into the π γ channel, with and without pion final-state interaction (FSI). The calculated curves are folded withthe experimental resolution of the CBELSA/TAPS detector of σ = 19 MeV. The figure is taken from[50].[19]. Within the Nambu-Jona-Lasinio model a similar mass shift of -(100-150) MeV has been claimedtogether with a broadening by ≈
40 MeV [47]. The latter calculation has, however, been criticized in[48]. In contrast, no mass shift but a broadening by 60 MeV has been obtained in a resonance couplingmodel [49].Applying the GiBUU transport code, Weil et al. [50] have studied to what extent these predictedin-medium modifications can be observed under realistic experimental conditions in a photon-inducednuclear reaction. Fig. 2 a.) shows that in case of no in-medium modifications only about 20-30% of all ω → π γ decays occur in the nuclear medium for a photoproduction experiment on a Nb target nearthe production threshold. Due to the reaction kinematics, the ω mesons recoil from the nucleus withaverage momenta comparable to their mass and the decay length l dec = βγcτ = pm cτ ≈ cτ =22 fm thusbecomes larger than nuclear dimensions. Assuming a collisional broadening by 150 MeV [51] the π γ mass distribution for in-medium decays is smeared out, leading to the ω line shape given in Fig. 2 b.).Fig. 2 c.) shows the ω signal to be expected without collisional broadening but for a mass shift by-16% at normal nuclear matter density and Fig. 2 d.) displays the expected line shape for collisionalbroadening and mass shift.It should be noted that only a fraction of the in-medium decays occur in the center of the nucleusand that the invariant mass reconstructed from the 4-momenta of the decay products is sensitive to thenuclear density at the decay point. Every nucleus has a diffuse surface with a slowly dropping density9istribution. Assuming a density-dependent mass shift, ω meson decays at the nuclear surface willfurther smear out the in-medium peak. While these distortions of the originally predicted theoreticalsignal apply for all ω decay modes, also for e.g. the ω → e + e − decays, the ω → π γ channel has anadditional drawback. The 4-momentum vectors of the π and thus also the π γ invariant mass will bedistorted by elastic scattering or the π may even be absorbed within the nucleus due to the stronginteraction. Information on decays near the center of the nucleus will thus not reach the detector. Onlydecays near the nuclear surface will be registered where the mass shifts are smaller. The sensitivityof the meson line shape to in-medium modifications is thus reduced as evident from Fig. 2 d.). Thisexample shows how crucial it is to check whether theoretically predicted signals can really be observedunder realistic experimental conditions.In parallel to these rather involved and ambitious transport simulations simplified descriptions ofmeson production reactions have been developed which describe the main features of these reactionsbut do not take into account density dependent changes of the meson spectral function; i.e. the off-shelltransport, discussed above, is replaced by a simpler treatment, where instead of the local effective mesonmass their average in-medium mass is used. These are the so-called collision model calculations whichare described in the next session. Collision model based on the nuclear spectral function
Extensive calculations of the production of K + , K − , η , ω mesons and antiprotons in proton–, pion–,and photon–nucleus reactions at incident energies near or below the free nucleon–nucleon thresholdshave been performed in the framework of folding models [52-64]. These investigations are based on boththe direct and two-step particle production mechanisms, using different paramterisations for internalnucleon momentum distributions and for the free elementary cross sections. In these folding modelsonly the internal nucleon momentum distribution has been used and the off-shell propagation of thestruck target nucleon has been neglected or only crudely taken into account, but it could be significantin processes that are limited by phase-space such as near-threshold heavy meson production. As is wellknown [65-70] the off-shell behavior of a bound nucleon is described by the nucleon (nuclear) spectralfunction P A ( p t , E ), which represents the probability to find in the nucleus a nucleon with momentum p t and removal (binding) energy E and thus contains all the information on the structure of a targetnucleus. The knowledge of the spectral function P A ( p t , E ) is needed for calculations of cross sections ofvarious kinds of nuclear reactions. In particular, it has been widely adopted for the analysis of inclusiveand exclusive quasielastic electron scattering by nuclei [71-73]. It was found that even at very highmomentum and energy transfer in the scattering process, the target nucleus cannot be simply describedas a collection of A on-shell nucleons subject only to Fermi motion, but the full nucleon momentumand binding energy distribution has to be considered. The one- and two-step folding model, basedon the nucleon spectral function, has been developed in [35,74-87] to analyze the kaon, φ , η (cid:48) , J/ψ meson as well as Λ(1115) and Λ(1520) hyperon production in proton– and photon–nucleus collisionsin the near-threshold energy regime with the aim of obtaining information about in-medium hadronproperties. The main ingredients of this model are illustrated by discussing as an example the inclusiveproduction of η (cid:48) mesons off nuclei in photon-induced reactions. A detailed comparison of the calculated η (cid:48) meson production cross sections with the corresponding experimental data will be given in Sec.4.5 ofthis review. Near-threshold η (cid:48) meson photoproduction off nuclei One-step η (cid:48) production mechanism In the incident photon energy range up to 2.6 GeV, the following direct elementary processes, whichhave the lowest free production threshold ( ≈ η (cid:48) photoproduction on nuclei:10 + p → η (cid:48) + p, (21) γ + n → η (cid:48) + n. (22)The medium modification of the η (cid:48) mesons, participating in the production processes (21),(22), isaccounted for by using for simplicity their average in-medium mass < m ∗ η (cid:48) > defined as: < m ∗ η (cid:48) > = (cid:90) d rρ N ( r ) m ∗ η (cid:48) ( r ) /A, (23)where ρ N ( r ) and m ∗ η (cid:48) ( r ) are the local nucleon density and η (cid:48) effective mass inside the nucleus, respec-tively. Assuming in line with [35] that m ∗ η (cid:48) ( r ) = m η (cid:48) + V ρ N ( r ) ρ , (24)Eq. 23 can be readily rewritten in the form < m ∗ η (cid:48) > = m η (cid:48) + V < ρ N >ρ . (25)Here, m η (cid:48) is the η (cid:48) vacuum mass, < ρ N > and ρ are the average and saturation nucleon densities,respectively. V is the η (cid:48) scalar potential depth (or the η (cid:48) in-medium mass shift) at density ρ to beextracted from the data for η (cid:48) photoproduction off nuclei (see Section 4.5).The total energy E (cid:48) η (cid:48) of the η (cid:48) meson inside the nuclear medium is expressed via its average effectivemass < m ∗ η (cid:48) > and its in-medium momentum p (cid:48) η (cid:48) by: E (cid:48) η (cid:48) = (cid:113) ( p (cid:48) η (cid:48) ) + ( < m ∗ η (cid:48) > ) . (26)Since in the considered case of γA η (cid:48) photoproduction the created η (cid:48) meson is expected to propagateout of the nucleus in the field of conservative nuclear forces, this energy is assumed to be equal to thetotal vacuum energy E η (cid:48) of the η (cid:48) meson, having the vacuum momentum p η (cid:48) [35]: E (cid:48) η (cid:48) = E η (cid:48) = (cid:113) p η (cid:48) + m η (cid:48) . (27)In the collision model [35, 83] the medium modification of the final nucleons has also been taken intoaccount by using, by analogy with equation Eq. 25, their average in-medium momentum-dependentmass < m ∗ N ( p (cid:48) N ) > : < m ∗ N ( p (cid:48) N ) > = m N + V SC NA ( p (cid:48) N ) < ρ N >ρ . (28)Here, m N is the nucleon vacuum mass and V SC NA ( p (cid:48) N ) is the scalar momentum-dependent nuclear nucleon-potential at saturation density. It depends on the in-medium nucleon momentum p (cid:48) N and can bedetermined from the relation [83] V SC NA ( p (cid:48) N ) = (cid:113) m N + p (cid:48) N m N V SEP NA ( p (cid:48) N ) , (29)where V SEP NA ( p (cid:48) N ) is the Schroedinger-equivalent potential for nucleons at normal nuclear matter density.This potential is momentum-dependent and can be parametrized as a function of the momentum relativeto the nuclear matter at rest by [83]: V SEP NA ( p (cid:48) N ) = (cid:18) V − V e − . p (cid:48) N (cid:19) ; V = 50 MeV , V = 120 MeV , (30)11here the momentum | p (cid:48) N | is measured in GeV/c. In the reactions (21),(22), the total energy E (cid:48) N of theoutgoing nucleon in the nuclear interior can be expressed in terms of its effective mass < m ∗ N ( p (cid:48) N ) > defined above and its in-medium momentum p (cid:48) N as in the free particle case, i.e.: E (cid:48) N = (cid:113) p (cid:48) N + [ < m ∗ N ( p (cid:48) N ) > ] . (31)When the nucleon escapes from the nucleus with momentum p N its total energy E N becomes equalto that corresponding to its bare mass m N : E N = (cid:113) p N + m N , i.e. it is assumed that E (cid:48) N = E N . Inthe production of slow mesons the recoiling protons are highly energetic with momenta much largerthan the momenta of the surrounding nucleons. In this case it is natural to assume that the energyrequired to bring the nucleon back on shell is taken from the energy of this nucleon itself and not fromthe nucleus.Neglecting the distortion of the incident photon in nuclear matter and describing the η (cid:48) mesonfinal-state absorption by the in-medium cross section σ η (cid:48) N , the inclusive differential cross section for theproduction of η (cid:48) mesons with vacuum momentum p η (cid:48) off nuclei in the primary photon–induced reactions(21,22) can be represented as a product of the effective number of participating target nucleons I V [ A ],defined by Eq. (11) in [35], and the off-shell differential cross sections of these reactions, averaged overthe internal momentum and the binding energy of the struck nucleon [35]: dσ (prim) γA → η (cid:48) X ( E γ ) d p η (cid:48) = I V [ A ] × (cid:34) ZA (cid:42) dσ γp → η (cid:48) p ( p γ , p (cid:48) η (cid:48) ) d p (cid:48) η (cid:48) (cid:43) A + NA (cid:42) dσ γn → η (cid:48) n ( p γ , p (cid:48) η (cid:48) ) d p (cid:48) η (cid:48) (cid:43) A (cid:35) d p (cid:48) η (cid:48) d p η (cid:48) . (32)In the collision model [35] it is assumed that these cross sections are equivalent to the respective on-shell cross sections calculated for the off-shell kinematics of the elementary processes (21 ,22), using the η (cid:48) and nucleon in-medium masses < m ∗ η (cid:48) > and < m ∗ N ( p (cid:48) N ) > , respectively. For the on-shell differentialcross sections for η (cid:48) production off the ”free” proton and neutron the respective paramterisations ofthe measured cross sections [88, 89] are used in [35]. Before considering the two-step η (cid:48) productionmechanism, the nucleon spectral function will be discussed, which is a crucial point in the collisionmodel [35,74-87]. Nucleon spectral function
Considering the ground-state nucleon-nucleon (
N N ) correlations, which are generated by the short-range and tensor parts of the realistic
N N interaction, the spectral function P A ( p t , E ) can be representedin the following form [65, 66, 70]: P A ( p t , E ) = P ( p t , E ) + P ( p t , E ) , (33)where P includes the ground and one-hole states of the residual ( A −
1) nucleon system and P themore complex configurations (mainly 1 p –2 h states) that arise from the 2 p –2 h excited states generatedin the ground state of the target nucleus by N N correlations. For the single-particle (uncorrelated) part P ( p t , E ) of the nucleon spectral function the harmonic oscillator spectral function and the Fermi-gasmodel spectral function have been employed in case of C and for other medium or heavy target nuclei,respectively.For the correlated part P ( p t , E ) of the nucleon spectral function the simple analytical expression,given in [74, 76], is adopted. The internal nucleon momentum distributions used in calculations ofspecific particle production off carbon are shown in Fig. 3. Recent work on nucleon-nucleon short-rangecorrelations and the nucleon spectral function has been summarised in [92].12igure 3: Internal nucleon momentum distribution for C. The dashed and dotted lines are themany-body correlated and uncorrelated momentum distributions, presented in [65, 67] and definedas (cid:82) P ( p t , E ) dE and (cid:82) P ( p t , E ) dE , respectively. Light solid curve: total nucleon momentum distri-bution given by the sum of the shell-model momentum distribution [74] and the correlated momentumdistribution parametrized in [74] by two exponents and shown by the dot-dashed curve. Heavy solidline: total many-body momentum distribution, presented in [65, 90]. Squares represent the existingexperimental data [91]. For further details see [74]. The momentum distributions n ( p t ) are normalisedaccording to ∞ (cid:82) n ( p t ) p t dp t = 1. For heavier target nuclei the total nucleon momentum distribution inthe considered collision model is given by the sum of the Fermi-gas model momentum distribution andthat given by the dot-dashed line shown above. The figure is taken from [74]. With kind permission ofThe European Physical Journal (EPJ). 13 .2.4 Two-step η (cid:48) production mechanism In the incident photon energy range of interest, the following two-step production process may contributeto the η (cid:48) production in γA interactions [35]. In the first inelastic collision with an intranuclear nucleonan initial photon can produce pions through the elementary reaction γ + N → π + N. (34)Then, these pions, which are assumed to be on-shell, produce the η (cid:48) meson on another nucleon of thetarget nucleus via the elementary subprocess with the lowest free production threshold momentum (1.43GeV/c) π + N → η (cid:48) + N. (35)For kinematic reasons the elementary processes γN → πN , γN → πN , πN → η (cid:48) N π are expected toplay a minor role in η (cid:48) production in γA reactions at photon energies E γ ≈ γp → π + π − p , γp → π + π n , γn → π + π − n and γn → π − π p have been included inthe calculations [35] of the η (cid:48) production on nuclei.Accounting for the medium effects on the η (cid:48) mass on the same footing as employed in calculatingthe η (cid:48) production cross section off the target nucleus in the primary processes (21,22), and ignoring, forthe sake of numerical simplicity, the medium modifications of the outgoing nucleon in the subprocess(35), the η (cid:48) production total cross section for γA reactions from this subprocess can be expressed by: σ (sec) γA → η (cid:48) X ( E γ ) = I (sec) V [ A ] (cid:88) π (cid:48) = π + ,π ,π − (cid:90) d p π × (36) × (cid:34) ZA (cid:42) dσ γp → π (cid:48) X ( p γ , p π ) d p π (cid:43) A + NA (cid:42) dσ γn → π (cid:48) X ( p γ , p π ) d p π (cid:43) A (cid:35) × (cid:20) ZA (cid:104) σ π (cid:48) p → η (cid:48) N ( p π ) (cid:105) A + NA (cid:104) σ π (cid:48) n → η (cid:48) N ( p π ) (cid:105) A (cid:21) . The cross section is given by the product of the effective number of N N pairs I (sec) V [ A ] (defined byEq.(44) in [35]) involved in the two-step η (cid:48) production processes under consideration and the integral overthe intermediate pion three-momentum of the inclusive differential cross sections for pion productionfrom the primary photon-induced reaction channel (34) and the in-medium total cross sections forthe production of η (cid:48) mesons with the reduced mass < m ∗ η (cid:48) > in πN collisions [35]. The inclusivedifferential cross sections for pion production in process (34) are calculated following the approach [93].The elementary η (cid:48) production reactions π + n → η (cid:48) p , π p → η (cid:48) p , π n → η (cid:48) n and π − p → η (cid:48) n have beenincluded in the calculations of the η (cid:48) production on nuclei. Their total cross sections are given in [35].In Fig. 4, adapted from [35], the resulting A–dependencies of the total η (cid:48) production cross sections forthe one-step and two-step η (cid:48) production mechanisms in γA ( A = C, Al, Ca, Nb,
Pb, and
U)collisions are shown, calculated for E γ = 1 .
85 GeV. The role of the secondary pion–induced reactionchannel πN → η (cid:48) N turns out to be negligible compared to that of the primary γN → η (cid:48) N processesfor all considered target nuclei. This gives confidence that the channel πN → η (cid:48) N can be ignored inthe analysis of data on η (cid:48) photoproduction off nuclei obtained by the CBELSA/TAPS Collaborationdiscussed in Section 4.5. Determination of the imaginary part of the meson–nucleus potential from mea-surements of the transparency ratio
The imaginary part of the meson–nucleus potential is a measure for the absorption of the meson in themedium. It is related to the in-medium width of the meson m through Eq. 12 and can be extracted14 E g = 1 . 8 5 G e V sg A-> h ’X [ m b] A g N - > h ’ N p N - > h ’ N s h ’N = 1 3 m bN b P b UC aA lC Figure 4: A–dependences of the total cross sections of η (cid:48) production by 1.85 GeV photons from primary γN → η (cid:48) N channels and from secondary πN → η (cid:48) N processes in the full phase-space in the scenariowithout η (cid:48) and nucleon mass shifts. The absorption of η (cid:48) mesons in nuclear matter is calculated assumingan inelastic cross section σ η (cid:48) N = 13 . T A = σ γA → mX A · σ γN → mX . (37)It compares the production cross section per nucleon of meson m off a nucleus with mass number Awith the production cross section on a free nucleon N . The role of the nucleus is twofold: it serves asa production target as well as an absorber. T A quantifies the loss of the meson flux in a nuclear targetwhich, in turn, is governed by the imaginary part of the meson in-medium self-energy or width. Incase of no absorption, T A = 1, if secondary production processes can be neglected. In the low densityapproximation the in-medium width Γ in the nuclear rest frame at nuclear density ρ N is related to themeson–nucleon inelastic cross section byΓ( ρ N ) = Γ · ρ N ρ = ¯ hc · β · σ inel · ρ N . (38)Throughout this review the in-medium width of mesons will always refer to the nuclear rest frame. Thewidth in the meson eigen-system is then given byΓ eigen = γ · Γ , (39)where γ is the relativistic factor γ = E/m , the ratio of the particle energy E to its restmass m .To avoid systematic uncertainties in the comparison with experimental data, e.g. due to differencesin the initial state interaction, partially unknown meson production cross sections on the neutron orsecondary meson production, the transparency ratio for a heavy target with mass number A is frequently15igure 5: Nuclear transparency ratio as a function of the nuclear mass number A for ω photoproductionfor different multiplicative factors K inel to vary the in-medium width, starting from Γ = 37 MeV forK inel = 1.0. The calculation has been performed for an incident photon energy of 1.5 GeV. The figureis taken from [94].normalised to the transparency ratio for a light nucleus like Carbon: T C A = 12 · σ γA → mX A · σ γ C → mX . (40)Thereby nuclear effects not related to the absorption of mesons largely cancel. Transport calculations[94], Monte Carlo simulations [95] and collisional model calculations [35] have been performed to studythe sensitivity of the transparency ratio for ω and η (cid:48) mesons to the in-medium width and inelastic crosssections. As an example Fig. 5 shows the transparency ratio (37) obtained from GiBUU simulationsfor ω photoproduction as a function of the target mass number for different ω in-medium widths.After a rather steep fall off for A ≤
50, the transparency ratio almost levels off for higher massesA. In the latter mass region, a variation in the in-medium width by a factor 4 roughly leads to achange in the transparency ratio also by a factor 4. In Monte Carlo simulations [95] Kaskulov et al.studied the impact on the transparency ratio by in-medium decays involving strongly interacting decayproducts. Specifically they investigated the ω → π γ decay. In the experiment one tries to determinethe absorption of ω mesons; however, even if the ω is not absorbed, the detector may not see the ω mesonbecause the final-state interaction of the π prevents the reconstruction of the ω invariant mass fromthe 4-momentum vectors of the decay products: the π may either undergo elastic scattering, leading toan energy loss of the pion in the laboratory and thus to a strongly distorted π γ invariant mass, or the π meson itself may be absorbed. In both cases the ω meson is lost in the analysis although it was notabsorbed in the nucleus. It has been shown [95, 96] that the first class of events can be suppressed bya lower cut T π ≥
150 MeV on the kinetic energy of the outgoing pions. Fig. 6 demonstrates, however,that the impact of the π final-state interaction on the ω transparency ratio is only relevant for small ω in-medium widths of ≤
20 MeV. Note, however, that all experimentally deduced ω in-medium widthsare much larger (see Section 4.6). 16igure 6: Nuclear mass dependence of the transparency ratio T C A , normalised to Carbon, for ω photo-production confining the photon energy range to 1.45–1.55 GeV without (left panel) and with (rightpanel) taking the final-state interaction of π mesons into account. A cut of T π ≥
150 MeV on thekinetic energy of the outgoing pion has been applied to suppress contributions from elastic π scattering.The different curves correspond to different in-medium widths of the ω meson. The figure is taken from[95]. With kind permission of The European Physical Journal (EPJ). E g = 1 . 9 G e V TAC
A 681 01 1 . 51 31 51 7 s h ’N [ m b ] Figure 7: Transparency ratio T C A , normalised to Carbon, for η (cid:48) photoproduction as a function of thenuclear mass number A for different in-medium absorption cross sections at an incident photon energyof 1.9 GeV. The figure is adapted from [35]. 17ollision model calculations have been performed [35] to study the dependence of the η (cid:48) transparencyratio on the size of the inelastic cross section. Varying σ inel from 6 to 17 mb, i.e. by a factor ≈
3, thetransparency ratio changes by about a factor 1.4, as follows from Fig. 7.The examples discussed above show that transport calculations and collision model calculationsshow qualitatively very similar results which give confidence that a comparison with experimental datawill yield reliable information on the imaginary part of the meson–nucleus potential. In the followingsections we will present and discuss the potential parameters extracted from photon-, proton- andheavy-ion induced reactions for individual pseudoscalar and light vector mesons, using the approachesdiscussed above.
Determination of the real part of the meson–nucleus potential
As discussed below, the real part of the meson–nucleus potential has been determined for many mesonsby comparing experimental observables like the production cross sections, momentum or transversemomentum distributions of the produced mesons or their sideward flow with corresponding predictionsby transport calculations. These calculations are performed assuming different values for the real andimaginary part of the potential, thereby allowing the sensitivity of these observables to meson in-mediummodifications to be studied. The potential parameters are then extracted by optimising the agreementbetween the experimental distributions and the transport model results.
Real part of the meson–nucleus potential deduced from excitation function measurements
Using transport calculations, it has been shown in Section 3.1 and in [50] that it may be difficultto directly determine the in-medium mass of a meson - and thus the real part of the meson-nucleuspotential - by reconstructing the invariant mass from the 4-momentum vectors of the meson decayproducts: the small fraction of meson decays within the nuclear volume, the radial dependence of thenuclear density, the increase in width by inelastic reactions and final-state interactions may distortthe true in-medium mass distribution of the meson. In [50] alternative approaches have therefore beenworked out to access the in-medium meson mass such as the measurement of the excitation function, i.e.the cross section for meson production as a function of the incident beam energy or the measurementof the meson momentum distribution. In contrast to the line shape analysis, which is sensitive tothe nuclear density at the meson decay point, these methods are sensitive to the nuclear density atthe production point and hence applicable for all mesons, irrespective of their lifetime. As pointedout in [94], the excitation function is sensitive to the in-medium modification of the meson since adownward mass shift would lower the meson production threshold and increase the production crosssection at a given incident beam energy due to the enlarged phase-space. Fig. 8 shows as an exampleGiBUU calculations [50] of the cross section for photoproduction of ω mesons off C and Nb for differentin-medium modification scenarios. Even in case of no in-medium modification the calculations givenon-vanishing cross sections below the threshold for ω photoproduction off a free nucleon ( E thr = 1106MeV), due to the Fermi motion of nucleons in nuclei. If a photon with E γ ≤ √ s in the centre-of-mass system may exceed the energy neededfor ω production, giving rise to the subthreshold yield. The subthreshold cross section is only slightlyenhanced by allowing for an in-medium broadening of the ω meson. The scenarios including a massshift, however, exhibit a much stronger cross section enhancement which even persists near and abovethe free nucleon production threshold, but diminishes for higher incident photon energies.A corresponding result has been obtained in collision model calculations for photoproduction of η (cid:48) mesons [35]. Again a sizeable cross section enhancement is predicted in the subthreshold region andsomewhat above for an in-medium lowering of the η (cid:48) mass (see Fig. 9).18igure 8: Excitation function for photoproduction of ω mesons in the ω → π γ decay channel off Cand Nb calculated with the GiBUU model [27]. The curves show the results for four in-mediumscenarios: (i) red curve: no in-medium modification, (ii) green dashed curve: collisional broadening by150 MeV, (iii) dotted magenta curve: mass shift by -16%, (iv) blue dashed curve: collisonal broadeningand mass shift. The vertical dashed line indicates the ω photoproduction threshold on a free proton ofE γ = 1.108 GeV. The figure is taken from [50]. C o l l . B r o a d e n i n g ( s h ’N = 1 1 m b ) C o l l . B r o a d e n i n g + h ’ M a s s S h i f t = - 1 0 0 M e V a t d e n s i t y r sg C-> h ’X [ m b] sg Nb-> h ’X [ m b] g C - > h ’ X E g [ G e V ] C o l l . B r o a d e n i n g ( s h ’N = 1 1 m b ) C o l l . B r o a d e n i n g + h ’ M a s s S h i f t = - 1 0 0 M e V a t d e n s i t y r g N b - > h ’ X E g [ G e V ] Figure 9: Excitation function for photoproduction of η (cid:48) mesons off C (left panel) and Nb (rightpanel). The solid and dashed curves are collision model calculations assuming a collisional broadeningof the η (cid:48) meson corresponding to an inelastic cross section of 11 mb without and with an η (cid:48) mass shiftof -100 MeV at normal nuclear matter density, respectively. The arrows indicate the energy of thephotoproduction threshold on a free proton. The figure is adapted from [35].19igure 10: π γ momentum distributions from ω decays calculated with the GiBUU model [27] for γ +Nbat E γ = 0 . Real part of the meson–nucleus potential deduced from measurements of the meson momentumdistribution
Transport model calculations have demonstrated that also the momentum distribution of mesons pro-duced off nuclei is sensitive to the in-medium properties of the meson [50]. The simulations showthat, due to the kinematics of the in-medium meson production process, mesons with a lower mass are- on average - produced with a lower total energy. Furthermore, when leaving the nucleus the mesonhas to get back to its free vacuum mass. The missing mass has to be generated at the expense of thekinetic energy. Consequently, this energy → mass conversion lowers the meson momentum and shiftsthe momentum distribution to lower average values. Furthermore, the enhanced meson production crosssection at and below the free production threshold in case of a mass drop will also enhance the mesonyield at low momenta. It should be noted, however, that the meson momentum distribution is alsosensitive to the meson angular distribution. In any analysis determining the in-medium properties ofthe meson from the momentum distribution one has to make sure that the experimental meson angulardistribution is reproduced by the calculation. As shown in Fig. 10, the maximum of the momentumdistribution and also the average momentum are shifted by about 25-50 MeV for the scenarios involvinga mass shift by -16% while the collisional broadening scenario is hardly distinguishable from the vacuumcase.The corresponding effect is also found in collision model calculations as shown in Fig. 11, displayingthe momentum differential cross section for η (cid:48) photoproduction off Carbon and Nb for scenarios withcollisional broadening and with or without an additional in-medium η (cid:48) mass shift by -10% at normalnuclear matter density. 20 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 00 . 111 0 C o l l . B r o a d e n i n g ( s h ’N = 1 1 m b ) C o l l . B r o a d e n i n g + h ’ M a s s S h i f t = - 1 0 0 M e V a t d e n s i t y r g C - > h ’ XE g = 1 . 5 - 2 . 2 G e V p h ’ [ G e V / c ] C o l l . B r o a d e n i n g ( s h ’N = 1 1 m b ) C o l l . B r o a d e n i n g + h ’ M a s s S h i f t = - 1 0 0 M e V a t d e n s i t y r d sg C-> h ’X/dp h ’ [ m b/(GeV/c)] d sg Nb-> h ’X/dp h ’ [ m b/(GeV/c)] E g = 1 . 5 - 2 . 2 G e V g N b - > h ’ Xp h ’ [ G e V / c ] Figure 11: Momentum differential cross sections for η (cid:48) meson photoproduction from the primary γN → η (cid:48) N channel for photon energies of 1.5–2.2 GeV off C (left panel) and off Nb (right panel) calculatedwith the collision model [35] for the same in-medium scenarios as in Fig. 9. The figure is adapted from[35]. K + –nucleus potential K + –nucleon scattering length and K + –nucleus potential Kaon and antikaon properties in a strongly interacting environment are a subject of considerable currentinterest in the hadron and nuclear physics community (see, e.g., [7, 33] for recent reviews), especiallyin connection with the questions of the partial restoration of chiral symmetry in hot/dense nuclearmatter and of the possible existence of a kaon condensate in neutron stars. Since the K + N interactionis relatively weak and smooth at low energies ( σ tot K + N ≈
12 mb for p lab ≤
800 MeV/c and there are noresonances with strangeness S = 1), the real part of the kaon potential in a nuclear medium at theseenergies can be well estimated from the isospin-averaged kaon–nucleon scattering length in free space¯ a KN = − .
255 fm using the impulse ( tρ ) approximation [97, 98], i.e. V K + ( ρ N ) = − πm K (cid:18) m K m N (cid:19) ¯ a KN ρ N ≈ +31MeV ρ N ρ , (41)where m K and m N are the vacuum kaon and nucleon masses, respectively, and the saturation densityis taken to be ρ = 0 .
16 fm − . With this potential, the total kaon energy E (cid:48) K + in the nuclear interior ofordinary nuclei can be expressed in terms of its in-medium mass m ∗ K + , defined as: m ∗ K + ( ρ N ) = m K + V K + ( ρ N ) (42)and its in-medium three-momentum p (cid:48) K + as in the free particle case [98-101]: E (cid:48) K + = (cid:113) p (cid:48) K + + m ∗ K + . (43)21Figure 12: The energy of kaons and antikaons in nuclear matter (at p (cid:48) K + = 0) as a function of densityfor the soft equation of state (EOS) within different approaches. The figure is taken from [97]. K + –nucleus potential. Other theoretical approaches The kaon energy in a medium can also be obtained from the mean-field approximation to the chiralLagrangian [21,98-100,102-104], i.e. E (cid:48) K + ( p (cid:48) K + , ρ N ) = p (cid:48) K + + m K − Σ KN f ρ S + (cid:32) ρ N f (cid:33) / + 38 ρ N f , (44)where f = 93 MeV is the pion decay constant, Σ KN is the KN sigma term, which depends on thestrangeness content of a nucleon, and ρ S is the scalar nucleon density. Here, it should be noted that thisexpression includes terms arising from a scalar and a vector interaction of kaons with nucleons. Withoutthe vector interaction the effective in-medium K + mass would decrease due to scalar attraction. Thevector interaction ultimately leads to repulsion of the K + meson, while - due to a sign change becauseof G-parity - it provides the main contribution to the K − attraction in the nuclear medium [105], aswill be discussed in section 4.3. At low densities, the kaon in-medium mass, defined as its energy forzero three-momentum, is according to Eqs. (8),(10) then given by m ∗ K + ( ρ N ) = m K − Σ KN f m K ρ S + (cid:32) ρ N f m K (cid:33) / + 38 ρ N f ≈ m K + V K + ( ρ N ) , (45)where [77] V K + ( ρ N ) = − Σ KN f m K ρ S + 38 ρ N f . (46)Since the exact value of Σ KN and the size of the higher-order corrections - leading to different scalarattractions for kaon and antikaon - are not very well known (thus, Σ KN may vary from 270 to 45022igure 13: The kaon mass shift as a function of the nucleon density. The circles show the results basedon the vacuum kaon-nucleon scattering amplitude, while the squares correspond to the self-consistentresult. The curves smoothly connect the points. The figure is taken from [108].Figure 14: The kaon spectral function for different momenta: 0 (solid line), 200 MeV/c (dotted line),400 MeV/c (dashed line), and 600 MeV/c (dash-dot-dot line). Black lines show the results at saturationdensity ρ , gray lines at twice saturation density. The figure is taken from [108].MeV), the quantity Σ KN has been treated in [102, 103] as a free parameter which is adjusted separatelyfor K + and K − to achieve good fits to the experimental K + and K − spectra in heavy-ion collisions[106, 107]. Using the value of the ”empirical kaon sigma term”, obtained in [102, 103], and accountingfor that ρ S ≈ . ρ N at ρ N ≤ ρ [99], Eq. 46 can be rewritten in the form [77]: V K + ( ρ N ) = +22MeV ρ N ρ . (47)Other sophisticated studies within the relativistic mean-field (RMF) approach, the chiral perturbationtheory (ChPT) incorporating different values for Σ KN [97] and in the framework of the coupled-channel23odel of Waas et al. [23] showed that the K + mesons indeed feel a weak repulsive potential of about 30–40 MeV at saturation density ρ for low momenta (see table 2 and Fig. 12). This value is in accordancewith those given above.Figure 15: The ratio of the calculated differential K + spectra from p +Au to p +C reactions at 1.5, 1.75and 2.3 GeV in comparison to the ANKE data [123]. The lines, starting from the left, correspond tocalculations employing repulsive kaon potentials of 0, 10 and 20 MeV at ρ . The figure is taken from[111]. With kind permission of The European Physical Journal (EPJ).Korpa and Lutz [108] have also studied the kaon properties in cold isospin-symmetric nuclear matterby solving self-consistently the in-medium Bethe-Salpeter equation. Their results for the K + mass shiftas a function of nucleon density are shown in Fig. 13. The circles here show the mass shift obtainedwith vacuum kaon–nucleon scattering amplitude, while the squares correspond to the full self-consistentsolution, based on the in-medium scattering amplitude. One can see that the repulsive self-consistentmass shift exceeds that based on the vacuum kaon–nucleon scattering amplitude and reaches the valueof 36 MeV at density ρ . There is no structure in the kaon spectral function, which has a form of amodestly broadened quasiparticle peak (see Fig. 14). Its position is a few MeV below the value givenby the (cid:113) p (cid:48) K + + m ∗ K + for momenta | p (cid:48) K + | >
200 MeV/c, where m ∗ K + is the peak’s position for p (cid:48) K + = 0(cf. Eq.( 43)). Thus, K + mesons ( K + = ¯ su ) at low momenta feel a slightly repulsive nuclear potentialof about 20–40 MeV in nuclear matter at normal nuclear matter density ρ . The dispersion analysisof [109], which uses as input the vacuum K + N scattering amplitude, finds that this potential is alsorepulsive ( ≈
20 MeV at density ρ ) and shows only a moderate momentum dependence. Therefore,using in practical calculations [28, 77, 109, 110, 111] the quasiparticle dispersion relation (Eq. 43) withmomentum-independent kaon scalar potential V K + ( ρ N ) = V K + ρ N /ρ with V K + ≈ K + in-medium mass, appears well justified (cf. also the results of Ref. [108]). Determination of the K + –nucleus real potential The predictions of the theoretical approaches, outlined above, on kaon properties in dense matter can betested by analyzing proton–nucleus and heavy–ion collision data on kaon spectra, excitation functions(see Section 3.4) and collective flow [34,99-101,112-115]. The latter observable was also recognised as24igure 16: The K + spectra from CBUU calculations (solid lines) in comparison to the data from theANKE Collaboration [123, 127] for p +C reactions at 1.0, 1.2, 1.5, 2.0 and 2.3 GeV . The calculationsinclude both the momentum-dependent nucleon potential and a repulsive K + potential of +20 MeV atdensity ρ . The figure is taken from [111]. With kind permission of The European Physical Journal(EPJ).a promising probe of the kaon potential in dense matter, since it should push the K + mesons awayfrom the nucleons and attract K − mesons towards the nucleons. The analysis of KaoS [106,107,116-119], FOPI [120-122] and ANKE [123] on K + production in heavy–ion and proton–nucleus reactionsestablished in the framework of transport approaches [34, 103, 124, 125, 126] and [28, 44, 111] that the K + meson indeed feels a moderately repulsive nuclear potential of about 20–30 MeV at normal nuclearmatter density ρ , in agreement with the above theoretical estimates. As an example of such analysis,Fig. 15 shows the ratio of differential K + spectra from p +Au to p +C collisions at 1.5, 1.75 and 2.3 GeVcalculated within the CBUU transport model in comparison to the data from [123]. One can see thatat low momenta ( p lab <
230 MeV/c) the production of K + mesons on the gold target is suppressedwith respect to their creation on the carbon target. This observation can be explained by the combinedrepulsive effect of both the Coulomb and nuclear fields, which accelerate kaons before they escape fromthe nucleus. The CBUU calculation, which includes a repulsive kaon potential of 20 MeV, providesa good description of the experimental data over the full momentum range at all beam energies. A χ -fit of the data with the calculated ratio of the differential K + spectra for the adopted scenarios forkaon potential of 0, 10 and 20 MeV gives V K + ( ρ ) = 20 ± K + spectra fromC, Cu, Ag and Au targets taken at COSY-Juelich [123] also gives the same kaon potential at normalnuclear matter density (see Fig. 16).Another example of such an analysis is given in Fig. 17 which shows the differential experimentaldata on the transverse momentum and centrality dependence of the K + meson and proton sideward25igure 17: Sideward flow v versus transverse momentum p t for protons (triangles) and K + mesons(dots) in the rapidity range -1.2 ≤ y (0) ≤ -0.65 for semi-central (left) and central (right) Ru+Rucollisions at 1.69 A GeV [122]. The curves and shaded area show the predictions of the RBUU model[32] for K + and proton, respectively. K + potentials of 0, 15, and 20 MeV are applied in the modelcalculations. The figure is taken from [122].Figure 18: Sideward flow v versus transverse momentum p T for K + mesons (dots) in peripheral (a)and central (b) Ni+Ni collisions at 1.91 A GeV in the rapidity range -1.3 < y (0) < -0.5. The dataare compared to HSD [34] and IQMD [129] transport model predictions with (solid lines) and without(dashed lines) an in-medium kaon potential of 20 ± K + mean-field potential of about 20 MeV at density ρ . Inline with this result, recent findings on the azimuthal emission pattern of K + mesons in Ni+Ni collisionsat a beam kinetic energy of 1.91 A GeV, obtained by the FOPI Collaboration [128], also support theexistence of a repulsive kaon–nucleus potential of 20 ± v is overpredicted.In summary, the currently available theoretical predictions and experimental data on K + produc-tion off nuclear targets indicate consistently a slightly repulsive in-medium K + potential of about 20–30MeV at density ρ and low momenta. K –nucleus potential In-medium neutral kaon potential. Theoretical expectations
Figure 19: K + and K energies at zero momentum versus baryon density for symmetric nuclear matter(dashed curves), for ρ p /ρ B = 0.25 (dashed dotted curves) and for neutron matter (solid curves). Thefigure is taken from [131].Similar to the K + meson, the real part of the K in-medium nuclear potential V K is expected tobe repulsive as well due to its quark content ( K = ¯ sd ). Both these mesons have one light quark ( u in K + and d in K ) which repulsively interacts with the quarks ( u and d ) in the nuclear mediumdue to Pauli blocking. In this intuitive physical picture in-medium modifications of K mesons shouldbe close to those for K + mesons in symmetric nuclear matter with equal densities of protons andneutrons, ρ p = ρ n = ρ N /
2. This is also true from the hadronic point of view that the K self-energyΠ K in nuclear matter can be obtained from the K + self-energy Π K + by an isospin transformation as27igure 20: The ratio of K (K + ) yields produced by pions (protons) on heavy and light targets as afunction of the laboratory momentum. The full squares depict the ratio of K s produced on Pb and Ctargets [134]. The ratio of K + yields measured in proton-induced reactions on Au and C targets at 2.3GeV is represented by full circles [123]. The results of the HSD model [34, 135] for different strengthsof the K potential are depicted by solid (black), dashed (red), and dotted (blue) lines. The figure istaken from [134].Π K =Π K + [130]. Indeed, according to the predictions of the Quark-Meson Coupling (QMC) model byTsushima et al. [131], in which the scalar ( σ ) and the vector ( ρ , ω ) mesons are assumed to coupledirectly to the nonstrange quarks and antiquarks in the K and ¯ K mesons, one expects the neutral kaonin-medium mass m ∗ K to be equal to m ∗ K + of K + , i.e. m ∗ K = m ∗ K + , as illustrated in Fig. 19 (see also[132, 133]). On the other hand, as we observe here, the K + and K mesons are not mass degeneratein an asymmetric nuclear medium. In the QMC model, this is a consequence of the ρ meson, whichinduces different mean-field potentials for each member of the isodoublets, K and ¯ K , when they areembedded in asymmetric nuclear matter. Determination of the K –nucleus real potential Neutral kaons can help to get a deeper insight into the kaon potential in the nuclear medium. Ascompared to K + mesons, they have the essential advantage that possible medium effects are not shadedby the additional repulsive Coulomb interaction, which for heavy nuclei like Au amounts approximatelyto 17 MeV, i.e. it is of the same order of magnitude as the repulsive K + nuclear mean-field potential.These medium effects have been studied in the K meson production on several nuclear targets in π − -induced reactions at 1.15 GeV/c momentum by the FOPI Collaboration at SIS/GSI [134]. Compariingthe ratio of the measured K s momentum distributions from π − +Pb and π − +C reactions [134] withHSD transport model calculations [34, 135], a repulsive K –nucleus potential of about 20 MeV at anormal nuclear matter density is suggested, as is illustrated by Fig. 20.It is clearly seen that the calculation including a +20 MeV K nuclear potential at ρ (with a lineardependence of the potential on nuclear density), depicted by the dashed line in Fig. 20, reproducesqualitatively the observed dependence of the K s differential yield ratio both at low ( p lab <
170 MeV/c)and at high ( p lab ≈
650 MeV/c) kaon momenta. Whereas the HSD calculation without this potential,shown by the solid curve in Fig. 20, misses completely the data at low momenta. A χ -fit of the data28igure 21: Left: transverse momentum p t dis-tributions of K s mesons (full triangles) for Ar+ KCl collisions at 1.756 AGeV [136] in com-parison with IQMD model calculations [137] fora repulsive K –nucleus potential of 46.1 MeV(dashed curves) and without potential (dottedcurves) for different rapidity bins. Right: ratioof IQMD model calculations and experimentaldata for both scenarios as a function of p t forthe same rapidity bins. The figure is taken from[136].with the calculated ratio of the differential K s spectra for the employed scenarios for K potential of0, 10, 20 and 30 MeV (cf. also dotted lines in Fig. 20) gives V K ( ρ ) = 20 ± K + A interaction has been extracted as well from K + production in pA reactions as studied by the ANKE Collaboration (see Section 4.1).On the other hand, the data on K s transverse momentum spectra and rapidity distributions inAr+KCl reactions at a bombarding energy of 1.756 A GeV, collected recently by the HADES Collab-oration, point to the existence of a stronger repulsive in-medium K potential of about 40 MeV atsaturation density ρ for kaons at rest [136] as illustrated by Fig. 21. The IQMD simulations reproducerather well the measured K s p t distributions for all rapidity bins and over the full momentum rangeassuming a repulsive K potential of 46.1 MeV (dashed curves), whereas the calculations without thepotential overestimate the experimental data, especially at low transverse momenta ( <
400 MeV/c), asis nicely seen in the ratio plots given in Fig. 21 (Right). Note, however, that the IQMD calculationsinvolve less baryon resonances than HSD.The HADES Collaboration has also recently reported a measurement of inclusive K productionin p+Nb collisions at a beam kinetic energy of 3.5 GeV [138]. The obtained K data (phase-spacedistributions and the ratio of momentum spectra) were compared to theoretical calculations with theGiBUU transport model [27] leading to an estimate of the strength of the repulsive K potential of 40 ± ρ [138]. This value is consistent with that inferredalso by the HADES Collaboration from the analysis of K meson production in heavy–ion reactions [136](see above). As an example for such a comparison, Fig. 22 shows the ratios of measured and calculated29igure 22: Ratio of K momentum spectra measured in p+Nb collisions at 3.5 GeV [138] in twoneighbouring bins of the polar angle (black circles) and GiBUU transport model simulations [27] with(cyan) and without (blue) an in-medium ChPT repulsive momentum-dependent kaon potential definedas the difference of the in-medium (cf. Eq. 44) and vacuum kaon energies and described in detail in[138] (it amounts to ≈ +35 MeV for the kaon at rest and for density ρ ). Both the momentum and thepolar angle are in the laboratory reference frame. The figure is taken from [138]. K momentum spectra (obtained within the GiBUU transport model) for p+Nb collisions at 3.5 GeVbeam energy in two adjacent bins of the lab polar angle. The GiBUU calculation, which includes theChPT repulsive kaon potential, gives indeed a better description of the considered experimental data.Thus, the presently available experimental data on K meson production in pion–nucleus, proton–nucleus and heavy–ion collisions strongly support the existence of a repulsive in-medium K potentialof about 20–40 MeV at density ρ for neutral kaons at rest, consistent with that obtained for the K + nuclear potential (see Section 4.1). K − –nucleus potential In-medium antikaon potential. Theoretical predictions
The study of the ¯ KN two-body interaction at energies close to or below the kaon–nucleon threshold bothin free space and in nuclear matter as well as antikaon interaction with nuclei has received considerableinterest in recent years [7,8,10,23,108,109,131,139-149]. This interest has been motivated by the hopeof getting valuable information on a possible K − condensation in the interior of neutron stars [21, 22]as well as of exotic nuclear objects such as antikaon-nuclear bound states, i.e. K − –few–nucleon systems(antikaonic clusters) or bound states of a K − in heavier nuclei (see Section 7.2). But in spite of largeefforts, contrary to the K + and K mesons, the K − meson ( K − = s ¯ u ) properties in nuclear matter aremuch less known and still controversially debated.The free K − p interaction at threshold and above threshold energies is repulsive and well constrained[150] by the SIDDHARTA K − hydrogen data [151, 152] as well as by the low-energy scattering andreaction data [153] (and references therein). This is reflected in the negative real part of the elastic K − p forward scattering amplitude (which is an average of the isospin I = 0 and I = 1 components)at and above threshold, as is demonstrated by the left panel of Fig. 23. From the strong interactioninduced energy shift and width of the 1s state in kaonic hydrogen, Ikeda et al. [146, 147] extract a free30igure 23: Extrapolation of the real (left panel) and imaginary (right panel) parts of the amplitude K − p → K − p scattering to the subthreshold energy region generated in two NLO chiral-model fits [150].The K − p threshold values marked by solid dots with error bars were obtained from SIDDHARTA K − hydrogen data [151, 152]. The figure is taken from [150]. K − p scattering length of a K − p = ( − .
65 + i .
81) fm.However, at subthreshold energies the free K − p interaction is attractive (real part of the K − p forwardscattering amplitude is positive) and is extremely model dependent. As demonstrated in Fig. 23,two different NLO chiral-model fits to the available experimental data on low-energy K − p reactionsand SIDDHARTA K − hydrogen data generate essentially different K − p scattering amplitudes in thesubthreshold region. Here, the K − p interaction is dominated by the presence of the Λ(1405) resonancein the isospin I = 0 channel just 27 MeV below the K − p threshold. The interaction in the isospin I = 1 channel, corresponding to the two-body K − n free space interaction, is not affected by theΛ(1405) resonance and is moderately attractive both above and below threshold [148]. As long asthe 1s energy shift and width in kaonic deuterium is not measured - corresponding experiments arein preparation [154, 155] - the K − n scattering length has to be calculated. Using the Kyoto ¯ KN potential, a K − n = (0 .
58 + i .
78) fm has been obtained in [156]. The existence of the subthresholdΛ(1405) resonance leads to the pronounced peak in imaginary part of the free K − p forward scatteringamplitude, shown in the right panel of Fig. 23, and to the change of sign of its real part at energiesclosely below the K − p threshold. The Λ(1405) resonance is dynamically generated and corresponds totwo-poles in the complex energy plane of the I = 0 K − p and π Σ scattering S -wave amplitudes as isestablished in [150,157-159]. The lower (broader) mass pole (at about 1330 MeV) mainly couples to the π Σ channel, while the higher pole located at about 1430 MeV is more coupled to the ¯ KN channel. Forthis reason the line shape of the Λ(1405) resonance depends on the initial state of the reaction and onthe observed decay channel. The position of its lower pole in the complex energy plane is less well knownthan the position of the higher one [150], although the parameters of the two poles have recently beenfurther constrained [159, 160] by including recent photoproduction data of the γp → K + Σ π reaction[161] in the analysis. The remaining uncertainty in the position of the lower pole has an importantinfluence on the subthreshold extrapolation of the elastic K − p forward scattering amplitude (cf. Fig. 23)– a crucial input for our understanding of antikaon properties in nuclear matter.While antikaonic atoms probe K − N c.m. energies down to about 30–50 MeV below the K − N thresh-old, K − - nuclear 1s quasi-bound states would reach substantially lower energies [8, 10, 11, 148, 162],where the role of the Λ(1405) subthreshold resonance is expected to be dominant in the construction31igure 24: (Left) real part and (Right) imaginary part of best-fit K − potentials for kaonic Ni atomsas a function of nuclear density. Solid curves and long dashed curves are based on the single-nucleonamplitudes [146, 147] plus a phenomenological term proportional to ( ρ/ρ ) α , dashed curves are for apurely phenomenological density dependent best-fit potential [7]. All three potentials lead to equallygood fits to 65 kaonic atom data points. The figure is taken from [11].of medium-modified subthreshold K − N scattering amplitudes and relevant nuclear antikaon potentials.The Λ(1405) resonance experiences very strong modifications in the nuclear medium at densities around(0.1–0.2) ρ (it dissolves) mainly due to Pauli blocking of the intermediate proton states [23, 140]. Paulieffects reduce the attractive force which binds K − and p in the Λ(1405) resonance and shift it to higherenergies [23, 139, 140]. This strongly affects the in-medium K − p scattering amplitude close to threshold[23, 139, 140, 149], which together with the in-medium (smooth) K − n (I = 1) amplitude determines theantikaon self-energy in nuclear matter. Since the Λ(1405) has a K − –proton bound state component,the K − and nucleon self-energies should be accounted for in the in-medium dynamics of the Λ(1405)or in the subthreshold behavior of the in-medium ¯ KN scattering amplitudes. This means that the K − in-medium properties and the Λ(1405) states in matter should be treated self-consistently [8,108,142-144,148,149]. It was shown first in [142] that such self-consistent treatment leaves the position of theΛ(1405) resonance practically unchanged, due to a compensation of its repulsive mass shift resultingfrom Pauli effects with the attraction felt by the K − meson, however, with an increased decay width. Inthis case the real part of the in-medium K − p scattering amplitude remains positive (attractive) in theenergy region around the threshold in agreement with phenomenological analyses of antikaonic atomsand experiences a sharp increase when going to subthreshold energies [148]. This means that the K − p interaction becomes much stronger at energies ∼
30 MeV below the K − p threshold with respect to itsstrength at threshold.The real part of the K − –nuclear potential, relevant for antikaonic atoms, can be estimated from thereal part of the effective in-medium density-dependent isospin-averaged K − N scattering length ¯ a eff ( ρ N )using the single–nucleon approximation [8, 98, 140] V K − ( ρ N ) = Π K − ( ρ N )2 m K = − πm K (cid:18) m K m N (cid:19) ¯ a eff ( ρ N ) ρ N . (48)The effective scattering length ¯ a eff was extracted in [163] from the analysis of antikaonic atom data,viz. ¯ a eff ( ρ N ) = − .
15 + 1 . (cid:32) ρ N ρ (cid:33) . fm . (49)With this effective scattering length and for saturation density ρ = 0 .
16 fm − , the K − potential is32igure 25: Antikaon energies at zero momentum and the difference of the calculated chemical potentialsfor the neutron and proton, µ n − µ p versus baryon density for symmetric nuclear matter (dashed curves),for ρ p /ρ B = 0.25 (dashed dotted curves) and for neutron matter (solid curves). The figure is taken from[131].given by V K − ( ρ N ) = − − .
15 + 1 . (cid:32) ρ N ρ (cid:33) ..... ρ N ρ MeV . (50)This potential (50) is strongly attractive and amounts to -190 MeV in the nuclear interior [163, 164],in contrast to a weakly repulsive K − –nuclear potential of V K − ( ρ ) ≈ +5 MeV which would be obtainedfrom Eq. (48) for the free K − N scattering length ¯ a ¯ KN = ( a K − p + a K − n ) ≈ ( − .
04 + i .
80) fm. Asexplained above, this contradiction between a repulsive potential deduced from the vacuum isospinaveraged antikaon-nucleon scattering length and an empirical attractive potential from heavy atom X-ray data, can be explained by the medium modification of the K − p scattering amplitude in the isospinI = 0 channel through the in-medium modification of the Λ(1405) resonance.It should be noted, however, that the antikaonic atom data probe the surface of the nucleus andthus do not provide strong constraints on the K − –nucleus potential at normal nuclear matter density(see Fig. 24). In fact, more recently Friedman and Gal [11] have investigated the model-dependenceof extrapolating K − - nucleus potentials to nuclear saturation density ρ . These potentials, basedon different K − N amplitudes, fit kaonic atom data equally well and reproduce the experimentallydetermined fraction of K − absorption on single nucleons. As shown in Fig. 24 the K − -nucleus potentialcan reliably be determined only up to nucleon densities of about 30% of ρ for the real part and up to60% for the imaginary part, respectively. For higher densities the parameterisations diverge, indicatingthat antikaonic atom data are insensitive to the nuclear interior. The limited sensitivity of kaonic atomX-ray data to the K − - nucleus potential in the interior of the nucleus is also demonstrated by the factthat Baca et al. [165] - using a shallow chiral potential - reproduce the experimental energy shifts andwidths over the full periodic table - although with a worse χ than in [7]. For Z ≤
30, Hirenzaki etal. [6] find a reasonable description of kaonic x-ray data with a shallow potential of V K − ( ρ N = ρ ) ≈ -(30-40) MeV. 33ince the Λ(1405) does not contribute in the isospin I = 1 channel, the attractive K − n scatteringlength a K − n is practically density-independent [7, 98, 148] and the real part of the K − potential inpure neutron matter at low energies can be estimated from the free K − n scattering length a K − n =(0 .
58 + i .
78) fm [147, 156] using Eq. (48), V K − ( ρ n ) = −
70 MeV ρ n ρ . (51)With the K − potential (48), the in-medium total antikaon energy E (cid:48) K − at low in-medium K − momentum p (cid:48) K − can be determined as in [98-101] E (cid:48) K − = (cid:113) p (cid:48) K − + m K + 2 m K V K − ( ρ N ) . (52)In the case when 2 | V K − ( ρ N ) | /m K <<
1, Eq. (52) can be rewritten in the form E (cid:48) K − = (cid:113) p (cid:48) K − + m ∗ K − (53)corresponding to the free one but incorporating the effective antikaon mass m ∗ K − defined as (compareto Eq. (42) used for K + mesons) m ∗ K − ( ρ N ) = m K + V K − ( ρ N ) . (54)It should be noted that the use of equation (53) instead of formula (52) is well proven for antikaonpotentials satisfying the condition | V K − | ≤
100 MeV, as our estimates showed. In view of Eq. (54), theresult (51) for K − potential in neutron matter scales the K − mass as m ∗ K − ( ρ n ) = m K (cid:32) − α ¯ K ρ n ρ (cid:33) (55)with α ¯ K = 0 .
14, which is in good agreement with the respective value of 0.13 obtained in the coupled-channel approach [23].Similar to kaons, the antikaon energy in a medium can be also determined from the mean-fieldapproximation to the chiral Lagrangian [21,98-100,102-104]. It is expressed by the formula (44) inwhich one has to make the following substitutions: E (cid:48) K + → E (cid:48) K − , p (cid:48) K + → p (cid:48) K − and ρ N f → − ρ N f . Thenat low densities, the K − meson in-medium mass is defined by formulae (45) and (46) in which the term ρ N f and kaon potential V K + are replaced, correspondingly, by - ρ N f and the antikaon potential V K − .Proceeding in the same way as for the K + potential (Eq. (47) in Section 4.1.2) and using the empiricalvalue for Σ KN from [102, 103] one obtains the K − –nuclear potential [77]: V K − ( ρ N ) = −
126 MeV ρ N ρ . (56)It should be pointed out that at densities ρ N ≤ ρ this potential is in line with those obtained withinthe microscopic studies [97, 23, 140], and shown before in Fig. 12. It is also roughly in line withthe calculations by Tsushima et al. [131] within the Quark-Meson Coupling model that predicts adecrease of the effective K − mass in symmetric nuclear matter at zero momentum by ≈
144 MeV, as isillustrated by Fig. 25. It is seen from this figure, as in the case of K + and K mesons, considered above(cf. Fig. 19), that the energies of the K − and ¯K mesons are not degenerate in asymmetric nuclearmatter. The antikaon potential (56) scales the K − mass in nuclear matter in line with Eq. (55) with α ¯ K ≈ .
26 and ρ n → ρ N , which agrees reasonably well with that of α ¯ K ≈ K − productionin pA collisions, taken at the ITEP/Moscow and COSY/J¨ulich accelerators, in the framework of thenuclear spectral function approach [77, 78, 86]. We note in passing that the ”free particle” dispersionrelation (53) for the K − energy in matter with the effective momentum-independent antikaon mass(and momentum-independent scalar potential V K − ) m ∗ K − ( ρ N ) = m K (cid:32) − . ρ N ρ (cid:33) = m K −
110 MeV ρ N ρ , (57)has also been employed in [110] in the study of φ photoproduction from nuclei. However, the dispersionanalysis of [109] finds that the K − scalar potential shows a strong momentum dependence. It turnsout to be ≈ -140 MeV at density ρ and for zero momentum, while it decreases rapidly in magnitudefor higher momenta and saturates at ≈ -50 MeV for high momenta. Therefore, to achieve a betterunderstanding of the K − properties in the nuclear medium and to test the predictions of the microscopicmodels involved, one needs to determine the momentum dependence of the in-medium antikaon potentialexperimentally. In principle, this can be done by measuring the K − meson spectra from pA reactionsat bombarding energies close to the threshold in free pN collisions [77, 78, 86, 109] (see also below).A lot of self-consistent coupled-channel calculations of the K − self-energy in nuclear matter have beenperformed based on chiral Lagrangians [143,144,168-171] or on meson-exchange potentials [172, 173].They predict a relatively shallow low-energy K − –nucleus potentials with a central depths of the orderof -50 to -80 MeV, showing different momentum dependences at finite momenta ranging up to 500MeV/c [168-173]. Recently, a chirally motivated meson-baryon coupled-channel model [148, 174], whichaccounts for the subthreshold in-medium K − N s -wave scattering amplitudes, produced K − potentialdepths in the range of -(80–90) MeV in antikaonic atoms at nuclear matter density. However, taking K − absorption on two nucleons such as K − N N → Y N into account to improve the agreement withantikaonic X-ray data by adding a ρ N -dependent phenomenological term, the antikaon potential becomestwice as deep [8, 10, 148, 162].The situation with the imaginary part W K − of the K − –nuclear potential at low energies is alsocontroversial. Models based on a chiral Lagrangian [143,144,169-171] or on meson-exchange potentials[172, 173], predict central depths of the order of W K − ( ρ N = ρ ) ≈ -25 to -60 MeV at threshold.On the other hand, adding again a ρ N -dependent phenomenological term, simulating two–nucleon K − N N → Y N and multi–nucleon absorptive processes, to the single–nucleon potential from chirallymotivated models [8, 10, 148, 162] yields an imaginary K − potential depth in the range -(70–80) MeV.The momentum dependence of this potential has been also investigated in [169-173] in the momentumrange from 0 to 500 MeV/c. It shows both flat [169, 170, 173] and non-flat [171, 172] behavior at thesemomenta and at densities ρ N ≤ ρ .Thus, one may come to the conclusion that the situation with the K − –nucleus potential both atthreshold and at finite momenta is still very unclear presently and more theoretical and experimentalwork is needed to clarify it. Determination of the K − –nucleus real potential Experimental information about in-medium properties of antikaons can be deduced from the study oftheir production both in heavy–ion and proton–nucleus collisions at incident energies near or below thefree nucleon–nucleon threshold (2.5 GeV). The dropping K − mass scenario will lead to an enhancementof the K − yield in these collisions due to in-medium shifts of the elementary production thresholds tolower energies. Indeed, an antikaon enhancement in C+C, Ni+Ni interactions at beam energies pernucleon below the N N threshold has been observed by the KaoS and FRS Collaborations at SIS/GSIover two decades ago [106, 107, 175, 176]. This phenomenon has been attributed to the in-medium35igure 26: Inclusive invariant K − productioncross section as a function of the antikaon mo-mentum in the nucleus–nucleus cms for Ni+Nicollisions at 1.8 A GeV [106] and 1.85 A GeV[176, 177] in comparison with CBUU transportmodel calculations [34]. The dot-dashed linecorresponds to a calculation with bare kaonmasses, whereas the solid and dashed linesshow the results with an antikaon self-energyaccording to m ∗ K − ( ρ N ) = m K [1 − α ¯ K ( ρ N /ρ )]with α ¯ K = 0 . α ¯ K correspond to an at-tractive K − potentials of V K − ( ρ ) ≈ −
100 MeVand V K − ( ρ ) ≈ −
120 MeV for antikaons at rest,respectively. The figure is taken from [34].Figure 27: Sideward flow v versus rapidity for protons (a, triangles), K + (b, dots) and K − mesons(c, squares) in Ni+Ni collisions at 1.91 A GeV [128] in comparison with HSD [34] and IQMD [129]transport model calculations with (solid lines) and without (dashed lines) in-medium particle potentials.A repulsive K + potential of 20 ± ρ and a linear dependence onbaryon density has been employed in both HSD and IQMD calculations. For K − mesons an attractivepotential with V K − ( ρ , p (cid:48) K − = 0) = −
45 MeV has been used in the IQMD model, while in the HSDapproach K − mesons are treated as off-shell particles using the G-matrix formalism corresponding to V K − ( ρ , p (cid:48) K − = 0) = −
50 MeV. The star symbol for K − mesons at midrapidity in (c) is from the highstatistics data in the momentum range < >
5. The figure is takenfrom [128]. K − mass reduction [34, 98, 102, 103, 124, 135]. An analysis of inclusive K − momentum spectra takenat KaoS [106, 107, 175] and FRS [176] (see Fig. 26) in the framework of the CBUU transport model[34, 135] suggests a momentum-independent downward in-medium mass shift (or an attractive effectivescalar K − potential V K − ( ρ )) of about -(100–120) MeV at density ρ , assuming a linear dependence ofthe mass shift on the nuclear density. A similar K − potential of V K − ( ρ ) ≈ −
110 MeV for kaons atrest has been extracted by Li et al. [102, 103, 124] from a comparison of the KaoS data [106, 107] with36igure 28: The measured K − / K + ratio as a function of the kinetic energy in the center-of-mass systemfor Al+Al collsions at 1.9 A GeV [178]: (a) not-corrected and (b) corrected for the 17% contribution of K − mesons from φ decays. Error bars represent the statistical uncertainties. Shaded rectangles depictthe estimation of systematic errors. Curves are the results of HSD transport model [34] predictionswithout any in-medium effects (dotted curves), with only a K + potential with a linear dependence ondensity with V K + ( ρ ) = 40 MeV (dashed curves) as well as with the in-medium effects for both kaonsand antikaons ( V K + ( ρ ) = 40 MeV and V K − ( ρ , p (cid:48) K − = 0) ≈ −
50 MeV) (solid curves). The figure istaken from [178]. With kind permission of The European Physical Journal (EPJ).relativistic transport model calculations. The extracted antikaon potential V K − ( ρ ) is comparable topotentials predicted by adopting the relativistic mean-field or effective chiral Lagrangian approach [97]as well as the coupled-channel model [23] (see Fig. 12, lower curves, and table 1).On the other hand, recent findings on the azimuthal emission pattern of K − mesons in Ni+Nicollisions at a beam kinetic energy of 1.91 A GeV, obtained by the FOPI Collaboration [128], implyan only weakly attractive K − –nucleus potential (see Fig. 27) as predicted by both (HSD and IQMD)transport model calculations. Furthermore, the IQMD transport approach, which describes correctlyalso the data on the sideward flows of protons and K + mesons (cf. Fig.18 as well), suggests a K − –nucleuspotential of V K − ( ρ ) = − ±
10 MeV for particles at rest [128].However, there is a general problem with extracting information on the K − -nucleus potential fromheavy-ion data. The shape of the K − momentum spectra and the K − yield are strongly distorted byfeeding from φ → K + K − decays. Most of the transport model calculations underestimate the φ yieldand thereby do not properly account for these distortions (the contribution of the φ decay channel φ → K + K − to K + meson yield is negligible because of the much larger K + production cross section).Recently the FOPI Collaboration has measured the φ/K − ratio in Al + Al and Ni + Ni collisions at1.9 AGeV to be 0.34 ± +0 . − . (syst) [178] and 0.36 ± ≈ K − mesons stem from feeding through the φ → K + K − decay. This observation isconfirmed by a recent HADES measurement of Au-Au collisions at 1.23 AGeV who report a φ/K − ratio of 0.52 ± φ/K − ratios on the determination ofthe K − -nucleus potential is demonstrated in Fig. 28 which shows the K − /K + ratio as a function of37igure 29: Invariant production cross sec-tion for K + (open symbols) and K − mesons(full symbols) in inclusive p –C (left) and p –Au(right) collisions at 1.6, 2.5, and 3.5 GeV (fromtop to bottom) as a function of laboratory mo-mentum. The curves are fits to the data [46].The figure is taken from [46].the kaon kinetic energy. Ignoring the feeding from φ decays, a K − - nucleus potential of -50 MeV wouldbe obtained in comparison to HSD model transport calculations, assuming V K + ( ρ ) = 40 MeV [34],while a considerably weaker K − potential is needed to reproduce the data once the feeding is correctedfor. The production of K − mesons via φ meson decays is sizable and should be taken into account inattempts to extract the K − potential from comparisons between K − data in nucleus - nucleus collisionsand transport model simulations. This example shows how important it is to check whether transportcalculations reproduce also other observables before conclusions regarding meson-nucleus potentials canbe drawn.The in-medium modifications of K − meson properties have also been studied experimentally inproton–nucleus reactions at beam energies close to or below the production threshold in N N collisionsover the last years (see, for example, [46, 58, 64, 166, 167]). The advantage of such reactions comparedto heavy–ion collisions is that the processes of hadron production and propagation proceed in cold staticnuclear matter of well-defined density at zero temperature. The in-medium effects are somewhat smallerbut still comparable to those in heavy–ion collisions. Inclusive differential production cross sections of K ± mesons and K − /K + ratios have been measured in p +C and p +Au reactions at 1.6, 2.5, and 3.5GeV proton beam energy by the KaoS Collaboration [46], see Figs. 29, 30. An analysis within theBUU transport model [181] of the ratios of K − and K + inclusive differential yields presented in Fig. 30has shown (cf. Fig. 30) that these data are consistent with an in-medium momentum-independent K − nuclear potential of about -80 MeV at normal nuclear density [46]. Measurements of inclusive antikaonmomentum distributions from 0.6 to 1.3 GeV/c at a laboratory angle of 10.5 ◦ in p Be and p Cu interactionsat 2.25 and 2.4 GeV beam energy have been performed at the ITEP/Moscow accelerator [58]. Areasonable description of these data has been achieved in the framework of a folding model, assumingvacuum K + , K − masses [58]. These calculations, based on the target nucleon momentum distributionand on free elementary cross sections, consider incoherent primary proton–nucleon, secondary pion–38igure 30: Ratio of invariant produc-tion cross sections of K − over K + mesons for inclusive p-Au (left) andp-C (right) collisions as a function oftransverse mass. The data (solid cir-cles) were taken at 2.5 GeV and wereintegrated over laboratory angles be-tween θ lab = 36 ◦ and 60 ◦ . The solid anddashed curves depict results of BUUtransport model calculations [181] in-cluding strangeness exchange as well asa repulsive K + nucleus potential of +25MeV. The K − nucleus potentials usedin the calculations are indicated in thefigures. The figure is taken from [46].nucleon antikaon production processes and processes associated with the creation of antikaons via thedecay φ → K + K − of intermediate φ mesons. It has also been shown for the first time [58] that the K − production for momenta < pN → pN φ , φ → K + K − channel on alight nucleus like Be, while on a Cu nucleus the main contribution to the cross sections comes from thischannel and the πN → N KK − process. This indicates the importance of accounting for the effect ofthe φ feed-down in attempts to extract the antikaon potential from K − spectra (cf. the same conclusionas drawn above). The K − excitation functions in p Be and p Cu interactions have also been determinedfor a K − momentum of 1.28 GeV/c for bombarding energies < K + , K − masses [64]. On the other hand, these data do not contradict the results of calculations withinthe spectral function approach [78] for incoherent primary proton–nucleon and secondary pion–nucleonantikaon production processes (but without accounting for the φ feed-down effect on K − yield whichis insignificant at this high momentum [64, 58]), carried out in the scenario with zero kaon potentialand antikaon potential of about -126 MeV at density ρ . This can be explained [78] by the fact thatat a high K − momentum of 1.28 GeV/c the calculations are not so sensitive to the strength of theantikaon potential. So, one must admit that to make progress in understanding the strength of the K − interaction in the nuclear medium, it is necessary to carry out detailed measurements with taggedlow-momentum K − mesons. In line with the above mentioned, the K − mesons must not stem from φ → K + K − decays so that they bring ”genuine” information about the K − yield.Such measurements have recently been performed by the ANKE Collaboration at the COSY/J¨ulichaccelerator, where the production of K + K − pairs with invariant masses corresponding to both thenon- φ [167] and φ [182, 183] regions (see also Section 4.7 below) has been studied in proton collisionswith C, Cu, Ag, and Au targets at an initial beam energy of 2.83 GeV. The K − momentum dependenceof the coincident differential cross section has been measured for laboratory polar angles θ K ± ≤ ◦ over the momentum range of 0.2–0.9 GeV/c for these four target nuclei. The data presented in Fig. 31are compared with detailed model calculations, based on the nuclear spectral function for incoherentprimary proton–nucleon and secondary pion–nucleon K + K − creation processes for different scenarios ofthe K − nuclear potential [86]. In general, the cross sections calculated for K − potential depths of -60,-126, and -180 MeV at density ρ follow the data for all target nuclei for antikaon momenta above about0.4 GeV/c; the data exclude small in-medium K − mass shifts. On the other hand, the data at lowerantikaon momenta are described reasonably well with almost no K − potential and are overestimatedby all the calculations with a nonzero antikaon potential. This suggests [167] that the model missessome aspects of the absorption of low-momentum K − mesons and/or their production in nuclear matter.39igure 31: Double-differential cross sections for the production of non-resonant K + K − meson pairs inthe ANKE acceptance in the collisions of 2.83 GeV protons with (a) C, (b) Cu, (c) Ag, (d) Au targetsas a function of the K − laboratory momentum. The data are averaged over K ± angles θ K ± ≤ ◦ andover K + momenta in the range 200 ≤ p K + ≤ c . The curves represent model calculations [86]for K − potential depths U = V K − ( ρ ) = 0 MeV (long dashed), -60 MeV (dot long dashed), -126 MeV(short dashed), and -180 MeV (dot short dashed), respectively. The solid lines are spline functionsthrough the experimental data points. The figure is taken from [167].Figure 32: Ratio of the measured integrated cross section for non-resonant K + K − pair production onC, Cu, Ag, Au to the corresponding calculated cross section assuming four values of the K − potentialdepth at nuclear matter density: U = V K − ( ρ ) = 0, -60 ,-126, and -180 MeV. The curve represents athird-order polynomial fit of all ratios. The shaded band indicates the 1 σ confidence interval. The pairof vertical dotted lines corresponds to the regions where the ratio is unity within the errors given bythe fit. The color code is identical to that in Fig. 31. The figure is taken from [167].40herefore, to determine the antikaon nuclear potential, the ratio of the measured, momentum integrated K − cross section for the nonresonant K + K − pair production on a given nucleus A to the correspondingcross section calculated within the model for different potential strengths, has been considered in [167],rather than the differential cross sections themselves, which are shown in Fig. 31. This approachhas the advantage of decreasing both the statistical and systematic uncertainties. The cross sectionratios are shown in Fig. 32. The condition σ exp /σ cal = 1 is achieved for V K − ( ρ ) = − (cid:16) +15 − (cid:17) MeV.Accounting for the overall systematic uncertainties in the data leads to the extended error band of V K − ( ρ ) = − (cid:16) +50 − (cid:17) MeV [167]. The antikaon potential depth extracted in this way corresponds toan average K − momentum of about 0.5 GeV/c, where the main strength of the measured differentialdistributions is concentrated [167]. Within the uncertainties quoted, this value obtained for the antikaonpotential depth is consistent with those extracted in [34, 102, 103, 124, 135] from the KaoS [106, 107, 175]and FRS data [176], from the FOPI A–A collision data [128, 178] as well as from the p A KaoS data [46](see above). It also agrees with a potential of the order of -100 MeV at saturation density (but for zero K − momentum), predicted in [97, 23] and shown in Fig.12 (lower curves). Furthermore, it is consistentwith the moderate K − –nucleus potential of the order of -50 to -80 MeV predicted by calculations basedon a chiral Lagrangian [143,144,168-171] or on meson-exchange potentials [172, 173], as well as with asmaller potential of about -28 MeV at density ρ , extracted at an antikaon momentum of 0.8 GeV/c[184]. However, it is difficult to reconcile these values with the very deep potentials of order -200 MeVclaimed in experiments studying in-flight ( K − , N ) reactions on C and O at 1 GeV/c [185, 186]. Theanalysis of the latter data has, however, been questioned in [187].Summarising the above considerations, one may conclude that the K − –nucleus interaction is strongand attractive but the presently available results do not permit to draw unambiguous conclusions onthe exact strength of the K − –nucleus potential at low energies. η –nucleus potential While the interaction of kaons with nucleons and nuclei can be studied using kaon beams, η and η (cid:48) mesons with lifetimes ≤ · − s are too short-lived to generate a particle beam. Information onthe η –nucleon and η – nucleus interaction can thus only be extracted from final-state interactions in η production off nucleons and nuclei.Numerous investigations of η production in pion, proton and photon- induced reactions have beenperformed along with many theoretical studies. Detailed experimental information on these reactionsexists and has been summarised in several recent reviews, see e.g. [188, 189, 190]. In this review onlylittle can be added to this wealth of information. For completeness the main results will be discussed. The η -nucleon scattering length The interaction of an η meson with a nucleon near threshold is dominated by the S (1535)1 / − res-onance which is located just 49 MeV above the production threshold and has a width of 150 MeV.Since the ηN → πN, ππN and γN channels are always open, the ηN scattering length is complex; theimaginary part reflects the absorptive part of the cross section. All these channels have to be takeninto account and thus coupled channel calculations are required to extract the ηN scattering length.Analysing data on η production in photon-, pion-, and proton- induced reactions with different theoret-ical models, a broad spread of values has been deduced for the η – nucleon scattering length a ηN in therange of 0.18 fm ≤ Re ( a ηN ) ≤ ≤ Im ( a ηN ) ≤ ηN scattering amplitude using different interaction models are compiled in Fig. 33. Anexperiment is planned at the ELPH facility to determine the real part of the η − N scattering length toa precision of ± γd → ηpn and γp → pn cross sections under specific kinematicconditions [197]. The results of these measuremnets are eagerly awaited.41igure 33: Energy dependence of the real (Left) and imaginary (Right) parts of the free ηN scatteringamplitude in different interaction models GW [191] (dashed), CS [192] (solid), M2 [193] (dot-dashed)and GR [194, 195] (dotted). The vertical line denotes the ηN threshold. The figure is taken from [196].The ηN interaction is found to be strong and attractive in the s-wave. The rather large values ofthe scattering length led Haider and Liu [198] to predict possible exotic states of η mesons and nucleibound by the strong interaction. The current status of searches for η mesic states will be discussed inSection 7.3. The imaginary part of the η - nucleus potential Inclusive η photoproduction off nuclei ( C, Ca, Cu, N b, nat
P b ) has been studied at MAMI, ELSAand LNS (Tohoku) [199-202] for incident photon energies of 0.7 to 2.2 GeV. From the measured crosssections the transparency ratios in Fig. 34 (Left) have been deduced (normalised to carbon) showntogether with corresponding data for ω [51] and η (cid:48) [203] mesons which will be discussed in detail inthe next subsections. The data points for the η show only a weak drop with increasing nuclear massreaching a transparency ratio of 70% for a heavy nucleus like Pb. In interpreting this result one has toconsider the possibility that the transparency ratio may be distorted by secondary production processeswhere in a first step the incident photon produces a pion which then produces an η meson on anothernucleon of the same nucleus with the much larger strong interaction cross section. Because of theenergy balance, η mesons produced in secondary reactions will have a smaller kinetic energy than thoseproduced directly by the incident photon. As discussed in [200] secondary production processes canthus be suppressed by a cut T η ≥ ( E γ − m η ) / T η for each incident photon energy E γ . Applying this cut the transparency ratiofor η mesons drops to 40% for heavy nuclei while the transparency ratios for ω and η (cid:48) mesons are notaffected by this cut within errors. This shows that for the η meson the transparency ratio has to becorrected for contributions from secondary production processes while two-step processes do not seemto play an important role in the production of the heavier ω and η (cid:48) mesons. Analysing the corrected η transparency ratio as described in Section 3.3, the η –nucleon inelastic cross section is found to beconsistent with earlier Glauber - type analyses of near threshold data, giving σ ηinel ≈
30 mb [199],consistent with predictions in [204, 205].This cross section corresponds to a mean free path of ≈ η momentum range of 150 - 380 MeV/ c covered in [199]this inelastic cross section leads to an η in-medium width of 25 - 55 MeV within the linear densityapproximation (Eq. (38)). The imaginary part of the η –nucleus potential is thus in the range of -10 to-30 MeV.An equivalent approach to deduce information on the absorption of mesons in nuclei is to investigatethe scaling of the meson production cross section with nuclear mass number. In quasi-free productionthe incident photon interacts with one individual nucleon off which the η meson is produced while therest of the nucleons in the nucleus only acts as spectators. As long as initial state interactions, i.e.photon shadowing [206] can be neglected, which holds for E γ ≤ . η production cross section with the number of nucleons A, i.e. with the nuclear volume. Anydeviation from such a linear dependence expressed by a coefficient α different from 1.0 in σ = A α ( T ) (59)reflects the absorption of the produced meson in the nucleus and thus provides information on theimaginary part of the η –nucleus interaction. Since the meson absorption may be energy dependent,the coefficient α may vary as a function of the kinetic energy T of the meson. If only nucleons onthe surface of the nucleus would contribute to meson production because mesons produced in theinterior of the nucleus are absorbed due to a large absorption cross section, then α ≈ /
3. Thus, forphotoproduction α is expected to be in the range 2 / ≤ α ≤ .
0. Analysing photoproduction of η mesons off nuclei, the dependence of α on the kinetic energy of η mesons has been deduced [199, 200]after correction for secondary production processes (Eq. (58)) and is shown in Fig. 34 again togetherwith corresponding results for ω [51] and η (cid:48) mesons [203]. The data on π mesons [207] show that nucleiare transparent for π mesons with kinetic energies below 30 MeV but they get strongly absorbed whenthe π kinetic energy becomes sufficient to excite the ∆(1232) and higher baryon resonances in collisionswith a nucleon. For the η meson the α values are close to 2/3, consistent with surface production andthe mean free path of ≈ ω meson shows strong absorption as well while the η (cid:48) meson exhibits a much weaker attenuation in normal nuclear matter. Details for η (cid:48) and ω mesonswill be discussed in Sections 4.5 and 4.6, respectively. The real part of the η -nucleus potential Although numerous measurements of the excitation function for η photoproduction off nuclei have beenreported [199-202] no attempts have been made as yet to extract information on the real part of the η –nucleus potential in the spirit of Section 3.4.1. No quantitative values for the η –nucleus potential canthus be quoted. The interaction is known to be attractive and theoretical expectations will be discussedin the context of possible η -nucleus bound states in Section 7.3. η (cid:48) –nucleus potential The η (cid:48) -nucleon scattering length The η (cid:48) N scattering length has been estimated from the study of the pp → ppη (cid:48) cross section nearthreshold at COSY [208]. A refined analysis of this reaction, comparing the cross section with thatof the pp → ppπ reaction, concluded that the scattering length should be of the order of magnitudeof that of the πN interaction. The analysis of the new COSY-11 data updated the results to values: Re ( a pη (cid:48) ) = 0 ± .
43 fm and Im ( a pη (cid:48) ) = 0 . +0 . − . fm[209]. This indicates a rather weak η (cid:48) N interaction.Nevertheless, a lowering of the η (cid:48) mass by 100-150 MeV has been predicted in calculations within theNambu-Jona-Lasinio model (NJL) [24, 210, 211]. 43igure 34: (Left) Transparency ratio for η, ω and η (cid:48) mesons as a function of the nuclear mass numberA. The transparency ratio with a cut on the kinetic energy of the respective mesons (Eq. (58)) tosuppress secondary production processes is shown with full symbols. The incident photon energy is inthe range from 1500 to 2200 MeV. The lines are fits to the data. (Right) Dependence of the parameter α (Eq. (59)) on the kinetic energy T of the mesons for π , η, ω and η (cid:48) mesons. The figures are takenfrom [203]. The imaginary part of the η (cid:48) -nucleus potential As discussed in Sec. 3.3 the experimental approach to determine the in-medium width and the imaginarypart of the meson-nucleus potential (Eq. (12)) is the transparency ratio measurement. First experi-mental data on the η (cid:48) transparency ratio for several nuclei have been reported by the CBELSA/TAPSCollaboration [203]. The obtained in-medium width is Γ( ρ N = ρ ) ≈ −
30 MeV corresponding toan inelastic η (cid:48) N cross section of σ inel ≈ π, η, ω ) demonstrates the relatively weak interactionof the η (cid:48) -meson with nuclear matter. As discussed in Sec.3.3 the two-step meson production coulddistort the absorption measurement. The effect of these processes, via possible pion-induced reaction γN → πN → η (cid:48) N , has been studied by applying a cut on the kinetic energy of the η (cid:48) ’s [203]. Theresults show definitely no contribution to the η (cid:48) production via this channel in the observed incidentphoton energy because σ πN → η (cid:48) N is only ≈ π ≈ η (cid:48) absorption crosssection and the in-medium width of the η (cid:48) meson. The inelastic cross sections σ inel has been derived,using Eq. (38), and is shown as a function of the η (cid:48) momentum in Fig. 35 (Left). The observed meanvalue of (13 ±
3) mb is slightly larger but consistent with the earlier result of (10.3 ± πN and ηN coupled channels. The predictions seem to underestimate theexperimentally determined inelastic η (cid:48) cross section. This may not be surprising since multi-particleproduction, probably dominant because of the large η (cid:48) mass, has not been considered in [214].The resulting momentum dependence of the in-medium η (cid:48) width has been converted into the de-pendence of the imaginary part of the η (cid:48) -nucleus potential as a function of the available energy in themeson- Nb system, as presented in [213] . The obtained spectrum is shown in Fig. 35 (Right) comparedto the previous measurement with CBELSA/TAPS detector system [203]. The data have been fittedand extrapolated towards the production threshold. For the η (cid:48) meson the extrapolation towards the44 [MeV/c] ' h p0 500 1000 1500 2000 2500 [ m b ] ' h i n e l s PLB 710 (2012) 600EPJA 52 (2016) 297 [MeV] thr s- ' h s0 500 1000 1500 ) [ M e V ] ' h -( I m U PLB 710 (2012) 600EPJA 52 (2016) 297
Figure 35: (Left) Inelastic η (cid:48) -nucleon cross sections deduced from Eq. (38) as a function of the mesonmomentum (red stars) [213] in comparison to earlier measurements (open crosses) [203]. The solid blackcurve is a fit to the data and the shaded area indicates a confidence level of ± σ of the fit curve takingstatistical and systematic errors into account. The thick (red) error bars represent the statistical errors.The thin (black) error bars include the systematic errors added in quadrature. The error weightedmean value of the η (cid:48) absorption cross section is (13 ±
3) mb. The blue curve represents the inelastic η (cid:48) -nucleon cross section calculated in [214]. (Right) Imaginary part of the η (cid:48) -nucleus optical potentialas a function of the available energy in the meson- Nb system (red stars) [213] in comparison to earliermeasurements (open crosses) [203]. The solid curve is a Breit-Wigner fit to the present data. Theshaded areas indicate a confidence level of ± σ of the fit curve taking statistical and systematic errorsinto account. The figures are taken from [213]. With kind permission of The European Physical Journal(EPJ).production threshold yields an imaginary potential of -(13 ± ± η (cid:48) scattering length Im ( a η (cid:48) N ) = (0.16 ± η (cid:48) N scattering length from ananalysis of near threshold η (cid:48) production in the pp → ppη (cid:48) reaction [209], but almost overlaps within theerrors. The real part of the η (cid:48) -nucleus potential Recently many studies have focused on the η (cid:48) meson. Its especially large mass compared to the massof the other pseudoscalar mesons has to be attributed to chiral and flavor symmetry breaking effects(see Section 1.2). Due to a reduction of the chiral condensate in the nuclear medium a drop in theU A (1) breaking part of the η (cid:48) mass might be expected [211, 215], causing an η (cid:48) mass shift of ≈ -120 MeVat nuclear matter density ρ . This prediction is, however, in conflict with earlier calculations withinthe Nambu-Jona-Lasinio-model which expect almost no change in the η (cid:48) mass as a function of nucleardensity [216]. Further model calculations, linear σ model [217] and QMC model [218], claim mass shiftsof the η (cid:48) of -80, and -40 MeV at ρ , respectively. It is obvious that these contradictory theoreticalpredictions call for an experimental clarification.The experimental approaches to determine the real part of the potential, which have been appliedfor η (cid:48) mesons and discussed in Sec. 3.4, are the measurement of the (i) excitation function of the mesonand (ii) of the meson momentum distributions. Both these approaches are sensitive to the productionpoint of the meson. The η (cid:48) ’s as long-lived mesons ( τ =1000 fm/c) decay predominantly outside of thenucleus and therefore experimental approaches sensitive to the decay point of the meson, like a lineshape analysis, are not applicable. The excitation function and momentum distribution of η (cid:48) mesonshave been measured in photon induced reactions on C [219] and Nb [220] in the energy range of 1200-45 γ [MeV] σ η ’ [ µ b ] V( ρ = ρ ) = 0 MeVV( ρ = ρ ) = -75 MeVV( ρ = ρ ) = -100 MeVV( ρ = ρ ) = -150 MeVV( ρ = ρ ) = -50 MeVV( ρ = ρ ) = -25 MeV E γ [MeV] σ η ’ [ µ b ] σ tot σ diff E γ thr E γ [MeV] σ η ’ [ µ b ] E γ thr -1 [GeV] g E b ] m [ ' hs Nb tot s diff s = 13 mb inel' h s ) = 0 MeV r = r V( ) = - 25 MeV r = r V( ) = - 50 MeV r = r V( ) = - 75 MeV r = r V( ) = -100 MeV r = r V( ) = -150 MeV r = r V( ' thr hg E Figure 36: Measured excitation function for η (cid:48) meson off C (left) [219] and Nb (right) [220], incomparison to theoretical calculations for different scenarios. The experimental data are extracted byintegrating the differential cross sections (full circles) and by direct measurement of the η (cid:48) yield inincident photon energy bins (open circles). The calculations are for σ η (cid:48) N =11 mb (for C data) and for σ η (cid:48) N =13 mb (for Nb data), and for potential depths: V =0 MeV (black line), -25 MeV (green), -50MeV (blue), -75 MeV (black dashed), -100 MeV (red) and -150 MeV (magenta) at normal nucleardensity, respectively, and using the full nucleon spectral function. The dot-dashed blue curve (in theleft spectrum) is calculated for correlated intranuclear nucleons only.2600 MeV. The experiments have been performed with tagged photon beams from the ELSA electronaccelerator using the Crystal Barrel and TAPS detectors. Differential and total η (cid:48) production crosssections have been measured and compared to model calculations using the collision model based onthe nucleon spectral function, described in Sec. 3.2. Calculations are performed for different scenariosassuming depths of the η (cid:48) real potential at normal nuclear matter density of V =0, -25, -50, -75, -100and -150 MeV, respectively, and including σ inel =11 mb, consistent with the result of transparency ratiomeasurements [203]. The measured excitation functions for η (cid:48) mesons on C and Nb solid targets areshown in Fig. 36. An enhancement of the total cross section below the production threshold (1445MeV) can be clearly seen in both cases, due to increased phase-space caused by the in-medium masslowering of the meson (see Sec. 3.4). Within the model, the comparison indicates an attractive potentialof -(40 ±
6) MeV at normal nuclear matter density from the C data [219] and of -(40 ±
12) MeV for the η (cid:48) -Nb potential. The obtained results do not indicate any dependence of the potential parameters onthe nuclear mass number A.It has been investigated whether the observed cross section enhancement relative to the V =0 MeVcase could also be due to η (cid:48) production on dynamically formed compact nucleonic configurations - inparticular, on pairs of correlated nucleon clusters - which share energy and momentum. Applying theparametrisation of the spectral function given by [74] to the theoretical calculations for different sce-narios and comparing with the experimental data demonstrated that the correlated high momentumnucleons contribute only about 10-15% to the η (cid:48) yield in the incident energy regime above 1250 MeVFig. 36 (left) [219]. The observed cross section enhancement can therefore be attributed mainly to thelowering of the η (cid:48) mass in the nuclear medium.As a consistency check for the deduced η (cid:48) -potential depth, the momentum distribution of η (cid:48) mesons,46 η ’ [GeV/c ] d σ η ’ / dp η ’ [ µ b / G e V / c ] C data E γ =1500-2200 MeVV( ρ = ρ ) = 0 MeVV( ρ = ρ ) = -75 MeVV( ρ = ρ ) = -25 MeVV( ρ = ρ ) = -100 MeVV( ρ = ρ ) = -150 MeVV( ρ = ρ ) = -50 MeV -1 [GeV/c] ' h p0.5 1 1.5 2 2.5 b / G e V / c ] m [ ' h / dp s d Nb =1.3 - 2.6 GeV g E = 13 mb inel' h s ) = 0 MeV r = r V( ) = - 25 MeV r = r V( ) = - 50 MeV r = r V( ) = - 75 MeV r = r V( ) = -100 MeV r = r V( ) = -150 MeV r = r V( Figure 37: Left: Momentum distribution for η (cid:48) photoproduction off C for the incident photon energyrange 1500-2200 MeV. The calculations are for σ η (cid:48) N =11 mb and have been reduced by a factor 0.75 tomatch the data at high momenta [219]. Right: Momentum distribution for η (cid:48) photoproduction off Nbfor the incident photon energy range 1.3-2.6 GeV. The calculations are for σ η (cid:48) inel =13 mb and have beenmultiplied by a factor 0.83 [220]. In both spectra the theoretical curves are for potential depths V =0, -25, -50, -75, -100 and -150 MeV at normal nuclear density.which is also sensitive to the potential depth, has been investigated by CBELSA/TAPS Collabora-tion [219, 220] as well. A comparison of the measured and calculated momentum distributions in theincident photon energy range of 1500-2200 MeV for C and in the incident photon energy range of 1300-2600 MeV for Nb is shown in Fig. 37 [219, 220]. The momentum resolution of the CB/TAPS detectorsystem is around 25-50 MeV/c, smaller than the chosen bin size of 100 MeV/c, and this does not affectthe sensitivity of the potential depth determination. The comparison of data and calculations againseems to exclude strong η (cid:48) mass shifts. A χ -fit of the data with the calculated momentum distributionsfor the different scenarios over the full range of incident energies gives a potential depth of -(32 ± η (cid:48) -C and -(45 ±
20) MeV for η (cid:48) -Nb.The depth of the potential determined for the real part of the η (cid:48) -C and η (cid:48) -Nb interactions is com-pared in Fig. 38 [220]. The values deduced by analysis of the excitation functions and the momentumdistributions do agree for both nuclei within errors. Obviously there is no evidence for a strong vari-ation of the potential parameters with the nuclear mass number. The combined fit of the C- andNb-data gives a weighted average of V( ρ N = ρ ) = − (39 ± stat ) ± syst )) MeV reported in [220]and shown in Fig. 38. Furthermore, it has been concluded that the mass of the η (cid:48) meson is loweredby about 40 MeV in nuclei at saturation density, within the errors quoted for the potential depth.The results for V( ρ N = ρ ) are consistent with predictions of the η (cid:48) -nucleus potential depth within theQMC model [218] and with calculations in [221] but does not support larger mass shifts as discussed in[210, 211, 217]. The results confirm the (indirect) observation of a mass reduction of the η (cid:48) meson ina strongly interacting environment. The attractive η (cid:48) -nucleus potential may be sufficient to allow theformation of bound η (cid:48) -nucleus states. The search for such states is encouraged by the relatively smallimaginary potential of the η (cid:48) of ≈ -10 MeV [203]. However, because of the relatively shallow η (cid:48) -nucleuspotential, the search for η (cid:48) -mesic states has turned out to be more difficult than initially anticipated onthe basis of theoretical predictions [210, 211, 217].47 [MeV] 'A h V80 - - - - excitation functionmom. distributionweighted average CNb Figure 38: Depths of the real part of the η (cid:48) -nucleus potential determined by analyzing the excitationfunction and the momentum distributions for C [219] (full black circles) and for Nb [220] (red triangles).The weighted overall average is indicated by a blue square and the shaded area. The vertical hatchedlines mark the range of systematic uncertainties. The figure is taken from [220]. The ω –nucleus potential The ω –nucleon scattering length The strength of the ω –nucleon interaction can be extracted from photoproduction experiments off theproton near the meson production threshold which provides information on the ωN scattering length.Following Strakovsky et al. [222] , the ω photoproduction cross section near threshold (see Fig. 39) canbe parametrized as σ ( q ) = a q + a q + a q , (60)assuming a contribution of s-, p-, and d-waves. q is the ω momentum in the c.m. system. Very close tothreshold the higher order terms can be neglected and the linear term is determined by the s wave onlywith the total spin of 1/2 and/or 3/2. Within the vector meson dominance model (VDM) the crossFigure 39: Photoproduction cross section for ω mesons off the proton near threshold as a function ofthe ω momentum q in the c.m. system. The red solid curve shows the fit of the data with Eq. (60).The figure is taken from [222]. .48igure 40: (Left) Imaginary part of the ω meson propagator in nuclear matter compared to the onein the vacuum [224]. (Right) Spectral function of an ω meson at rest [49]. The results are shown fornucleon densities ρ = 0 , ρ = ρ = 0 .
16 fm − , and ρ = 2 ρ ( ρ = ρ N ).section near threshold is related to the modulus of the ω – p scattering length a ωp through [223] σ ( γp → ωp ) th = qk · απ γ · | a ωp | , (61)where k is the c.m. momentum of the incident photon at the production threshold, α is the fine structureconstant and γ = 8 . ± .
14 is the strength of the γ − ω coupling, derived from the ω → e + e − decaywidth. Combining Eqs. (60), (61), one obtains | a ωp | = γ π (cid:115) ka α = (0 . ± .
03) fm , (62)using the value of a obtained by fitting the experimental excitation function. This result shows anappreciable strength of the ω – N interaction. Since the sign could not be determined it is not clearwhether the interaction is attractive or repulsive. Furthermore, the real and imaginary parts of a ωp cannot be given separately. This information, however, becomes available when studying the ω –nucleusinteraction. The imaginary part of the ω –nucleus potential Theoretical predictions for the imaginary part of the ω –nucleus potential range from -20 to -100 MeV(see table 1), including calculations starting from an effective Lagrangian [47], calculations treatingexplicitly the coupling of the ω meson to nucleon resonances [49, 224], calculations within the chiralunitary approach [225] and standard hadronic many-body calculations [226]. As an example Fig. 40shows the results obtained in [49, 224]. In both calculations the coupling of the ω meson to the nucleonresonances leads to a splitting of the ω strength into an ω mode and a particles-hole mode at lower mass.In [49] the strength function is, however, dominated by the strong broadening. Cabrera and Rapp [226]and Ramos et al. [225] have studied the ω in-medium width as a function of the ω momentum. Bothgroups attribute the main contribution to the in-medium ω width to the ω → ρπ channel whereby thedressing of the ρ and π propagator in the medium is found to be essential. Cabrera and Rapp [226] findonly a moderate momentum dependence while Ramos et al. [225] predict an almost linear increase upto ω momenta of 600 MeV/ c .Experimentally - as outlined in Section 3.3 - the imaginary part of the meson–nucleus potentialcan be derived from the measurement of the transparency ratio Eqs. (37),(40). Results of the first49igure 41: Transparency ratio for ω mesons normalised to carbon as a function of the nuclear massnumber A in comparison with Monte-Carlo calculations [95] and GiBUU simulations [94]. The figureis taken from [51].measurement for the ω meson are shown in Fig. 41. A comparison to Monte-Carlo calculations [95] andGiBUU simulations [94] gives an in-medium width of 130-150 MeV in the nuclear rest frame for ω mesonswith an average 3-momentum of ≈ . c , indicating a strong in-medium broadening compared tothe free ω width of 8.4 MeV [39]. More recently the momentum dependence of the transparency ratiohas been studied in finer momentum bins [213]. The inelastic ω -nucleon cross sections deduced fromEq. (38) is shown in Fig. 42 (Left) as a function of the meson momentum. Extrapolating to vanishing ω momentum, the imaginary part of the ω –nucleus potential (corresponding to half of the in-mediumwidth) has been determined to -(48 ± ± ω meson has also been observed in experiments studying the ω → e + e − decay channel. Comparing dilepton invariant mass spectra in the p + Nb and p + p reactions at 3.5GeV, the HADES Collaboration [227] observes an appreciable decrease in the strength of the ω signal.An even more dramatic absorption of ω mesons corresponding to an in-medium width of more than 200MeV has been reported by the CLAS Collaboration [228]. The real part of the ω –nucleus potential Early predictions of large mass shifts of vector mesons of the order of -100 to -150 MeV [18, 19, 47,230] (see table 1) initiated widespread theoretical and experimental activities. As an example, resultsobtained by using QCD sum rules [19] and predictions within the QMC model [230] are shown in Fig. 43.Other calculations, considering the coupling of the ω meson to nucleon resonances [49], however, predictalmost no mass shift but a considerable broadening (see Fig. 40 (Right)). These conflicting theoreticalpredictions called for an experimental clarification.The real part of the ω –nucleus potential has been determined by measuring the near threshold ex-citation function and the momentum distribution for photoproduction off C and Nb. Correspondingresults are shown in Fig. 44. Fitting the excitation function data with GiBUU simulations for differ-50 [MeV/c] w p0 500 1000 1500 2000 2500 [ m b ] w i n e l s PRL 114 (2015) 199903EPJA 52 (2016) 297 [MeV] thr s- w s0 500 1000 1500 2000 ) [ M e V ] w -( I m U PRL 114 (2015) 199903EPJA 52 (2016) 297
Figure 42: (Left) Inelastic ω -nucleon cross sections deduced from Eq. 38 as a function of the mesonmomentum (red stars) [213] in comparison to earlier measurements (open crosses) [51]. (Right) Imag-inary part of the ω –nucleus optical potential as a function of the available energy in the meson– Nbsystem (red stars) [213] in comparison to earlier measurements (open crosses) [51]. The solid curvesare Breit-Wigner fits to the present data. The shaded areas indicate a confidence level of ± σ of thefit curve taking statistical and systematic errors into account. The thick (red) error bars represent thestatistical errors. The thin (black) error bars include the systematic errors added in quadrature. Thefigures are taken from [213]. With kind permission of The European Physical Journal (EPJ).Figure 43: (Left) Vector meson masses as a function of nuclear density obtained by QCD sum rules [19](figure adapted from [229]). (Right) Predictions of the ω mass as a function of the nuclear density fordifferent parameter sets in the QMC model [230]. 51nt potential depths gives a potential of − (42 ± ±
20 (syst)) MeV in contrast to theoreticalpredictions of large mass shifts of -(100–150) MeV [47]. This result is supported by the ω momentumdistribution (see Fig. 44, Right) which again seems to exclude the scenarios assuming large mass dropsin the medium. This data set corresponds to an average ω momentum of about 600 MeV/ c .An attempt has been made in [233] to measure the ω potential depth for even lower momenta whichcan be accessed by requiring the participant nucleon in coincidence. Detected at forward angles, it takes [GeV] γ E0.9 1 1.1 1.2 1.3 1.4 b ] µ / A [ γ π σ -2 -1 C data
GiBUU) = 0 MeV ρ = ρ V( ) = -20 MeV ρ = ρ V( ) = -40 MeV ρ = ρ V( ) = -55 MeV ρ = ρ V( ) = -94 MeV ρ = ρ V( ) = -125 MeV ρ = ρ V( thr ωγ E [MeV/c] γ π p0 100 200 300 400 500 600 700 800 900 1000 a . u . CNb
Figure 44: (Left) Measured excitation function for ω meson photoproduction off C in comparison toGiBUU transport calculations for several in-medium modification scenarios [229, 231]. (Right) Accep-tance corrected ω momentum distribution for incident photon energies from 900 to 1300 MeV and for C and Nb targets, compared to the theoretical predictions for different in-medium modificationsscenarios: no modification (solid red line), collisional broadening (dashed green line), collisional broad-ening plus mass shift (dashed blue line) and mass shift (magenta line). All distributions are normalisedto the same area [232]. With kind permission of The European Physical Journal (EPJ).over most of the momentum of the incoming photon beam, leaving the ω meson almost at rest relativeto the nucleus. In this situation its momentum (or kinetic energy) distribution is particularly sensitiveto the real part of the meson–nucleus potential: if the interaction is attractive mesons emerging fromthe nucleus will be further slowed down while for a repulsive interaction the mesons will be acceleratedcompared to a scenario with vanishing interaction. The peak in the kinetic energy distribution isthus sensitive to the strength and sign of the meson–nucleus potential [211], as quantitatively givenin Fig. 45 (Right). The left panel of Fig. 45 shows the measured kinetic energy distribution of ω mesons registered in coincidence with protons detected at 1 ◦ − ◦ in photoproduction off C. Thekinetic energy distribution peaks at 60 . ± V ωA ( ρ N = ρ ) = − (15 ± ± V ωA ( ρ N = ρ ) = − (29 ±
19 (stat) ± ω –nucleusattraction is rather weak.Attempts have been made to extract information on the real and imaginary part of the ω - nucleuspotential from a measurement of the ω line shape. Such an analysis is subject to the problems discussedin Section 3.1.3. Early claims of an ω in-medium mass drop [236] were not confirmed neither in a refinedanalysis of the data [237] nor in high statistics measurements near the production threshold [232, 238].Ozawa et al.[239], analyzing the ρ, ω, φ → e + e − decay, found evidence that spectral shapes of vectormesons are modified at normal nuclear matter density. Naruki et al. [240] reported a drop of the ρ ω mesons off C in coincidencewith protons in Θ p = 1 ◦ − ◦ as a function of the total energy of the π γ pairs minus 782 MeV. Thedata have been fitted with the Novosibirsk function [234]. (Right) Correlation between the potentialdepth and the peak position in the kinetic energy distribution. The (blue) points represent the peakposition in the kinetic energy distribution for the different scenarios [233]. The (blue) solid curve is afit to the points. The red dashed area corresponds to the peak position of (60.5 ±
7) MeV. The figuresare taken from [233]. [MeV] A w V80 - - - - excitation function coinc. w p-average C Figure 46: Depths of the real part of the ω –nucleus potential determined by analyzing the excitationfunction and the momentum distributions for C [235]. The weighted overall average is indicated by ablue square and the shaded area. The vertical hatched lines mark the range of systematic uncertainties.The figure is taken from [235]. 53nd ω mass by 9.2 ± .
2% at normal nuclear matter density without any in-medium broadening of themesons. This result is surprising since hadrons in the medium must experience inelastic collisions whichshorten their time of existence in the medium and thus increase their width, as found in this review forall other mesons. Furthermore, this result is in conflict with results of the CLAS Collaboration whoreported no mass shift ( ≤
5% at 95% confidence level) for the ρ and ω meson [241, 242]. The resultsfrom the line shape analyses thus appear to be inconclusive.Summarizing the above experimental information on the real and imaginary part of the ω –nucleusinteraction one obtains V ωA ( ρ N = ρ ) = − (29 ±
19 (stat) ± − i (48 ± ± ω –nucleus bound states since thepotential depth is not very large and the imaginary part of the potential is larger than or comparable tothe real one; if there were bound states at all, they would be very broad and thus difficult to separateexperimentally from the background (see Section 7.5). φ –nucleus potential The φ –nucleon scattering length As for the ω meson, the photoproduction of φ mesons near threshold has been studied to extractthe φ –nucleon scattering length [223]. At threshold Titov et al. derive a differential cross sectionof dσdt ≈ . µb/ GeV which is consistent with the estimate for the φ –nucleon scattering length of a φN = − (0 . ± .
02) fm, obtained in QCD sum rule calculations [243]. From measurements at higherincident photon energies ( E γ = 4.6-6.7 GeV) Behrend et al. deduce an inelastic φ –nucleon cross sectionof ≈ φ –nucleon interaction is found to be rather weak. This is consistent with therather small mass drop of about -2 to -3% at nuclear matter density (see Fig. 43) predicted by Hatsudaand Lee [19], Cabrera et al. [245, 246] and Cobos-Martinez et al. [247]. More recently an even smallerreduction of the φ meson mass in the nuclear medium of less than -2 % has been predicted [248, 249]. The imaginary part of the φ - nucleus potential The transparency ratio for the φ meson has been measured in photoproduction [228, 250] and p + Acollisions [182, 183] to determine the absorption of φ mesons in nuclei. The data - normalised to thetransparency ratio for the carbon target - consistently show a decrease of the transparency to about 40%for medium mass nuclei and to 30% for very heavy nuclei (see Fig. 47). The normalization to carbonreduces the sensitivity of the measured cross sections to differences in the initial state interaction forphoton- and proton- induced reactions, to secondary production processes and/or uncertainties in themeson production cross section off the neutron. To extract the inelastic in-medium φ - nucleon crosssection and φ width the data are compared to calculations by M¨uhlich and Mosel [251], Magas et al.[252],Paryev [80], and Cabrera et al., [26]. Ishikawa et al. [250] deduce an inelastic in-medium φ - nucleoncross section of 35 +17 − mb, roughly a factor 4 larger than σ φN in free space. The analysis of the same datawithin the GiBUU transport model yields a similar value of ≈
27 mb [251] for the inelastic φ - nucleoncross section which corresponds in the linear density approximation (Eq. (38)) to an in-medium widthin the nuclear rest frame of ≈
75 MeV at normal nuclear matter density for an average φ momentum of1.8 GeV/c. Wood et al. [228] deduce an in-medium φ - nucleon cross section in the range of 16 - 70 mb(see Fig. 47 Left bottom panel) within a Glauber model analysis. The transparency ratio data obtainedin p + A collisions (see Fig. 47 Right) by Polyanskiy et al. [182] have been compared to calculationsby Magas et al. [252] and Paryev [80]. Taking into account contributions from two-step productionprocesses which increase the transparency ratio, in-medium widths of 45 +17 − MeV and 50 +10 − MeV havebeen deduced, respectively. (The value of 73 +14 − MeV, quoted in the original literature [182] refers to the φ eigen-system). All these measurements consistently show that - with in-medium widths of the orderof 40-60 MeV in the nuclear rest frame - the φ meson is strongly broadened in the medium by about54igure 47: (Left top panel) Transparency ratio normalised to carbon reported by Ishikawa et al. [250]for photoproduction of φ mesons in comparison to GiBUU simulations by M¨uhlich and Mosel [251] fordifferent absorption cross sections. The figure is taken from [251]. (Left bottom panel) Transparencyratio for φ mesons in the γ + A → φ + X → e + e − + X reaction in comparison with Glauber calculationsfor different in-medium φ – nucleon cross sections [228]. (Right) Transparency ratio for φ mesonsobtained in p + A collisions [182] compared to calculations by Magas et al. [252] (a) and Paryev [80](b) for different in-medium widths. In Fig. (a) the widths are given as multiples of the φ width at restof 24 MeV. In Fig.(b) the widths are given in the φ meson eigen-system. The figure is taken from [182].55igure 48: (Left) Momentum dependence of the transparency ratio normalised to carbon for Cu, Ag,and Au targets. (Middle) In-medium width of the φ meson in the nuclear rest frame at saturationdensity ρ as a function of the φ momentum. The different symbols refer to different model calculationsused in the extraction of the φ width from the data. (Right) The φN absorption cross section as afunction of the φ momentum. The figures are taken from [183].an order of magnitude as compared to its free width of 4.3 MeV [39]. The imaginary potential of the φ - nucleus interaction is thus in the range of -(20 - 30) MeV.These experimental results are close to theoretical predictions. The φ width increases in the mediumbecause of the opening of inelastic reaction channels. In addition, including the in-medium interactionof the decay kaons in the nuclear many body system, the φ spectral function is further broadened.Using a coupled channel approach based on a chiral effective Lagrangian Klingl et al. predict an in-medium φ width of 45 MeV [132]. M¨uhlich and Mosel [251] estimate an in medium width of 40 MeV atsaturation density. In a more recent calculation Gubler and Weise obtain a φ width of 45 MeV [248, 249],emphasising that the interaction of kaons from φ → K ¯ K with the surrounding nuclear medium leadsto an asymmetric line shape of the φ meson peak structure. Within the QMC model a width of 33 -37 MeV is predicted [247]. In addition to the in-medium φ → ¯ KK decay, Cabrera et al. [253] discusscontributions to the φ width from the coupling to nucleon resonances yielding an additional broadeningof the order of 40-50 MeV.It should be noted, however, that all these values are much larger than the in-medium width of ≈
15 MeV reported in a line shape analysis of the φ → e + e − signal by Muto et al. [254] who performeda comparative study of φ meson production off C and Cu nuclei with proton beams of 12 GeV (seebelow).Hartmann et al. [183] investigated the momentum dependence of the φ transparency ratio. Fig. 48shows the transparency ratio measured in proton induced reactions on Cu, Ag, and Au targets, nor-malised to C. Comparing to model calculations [80, 252] the in-medium φ width in the nuclear restframe has been deduced and is found to increase almost linearly with the φ momentum from 20 to 60MeV over the momentum range of 0.6-1.6 GeV/ c , corresponding to an imaginary part of the φ - nucleuspotential in the range of - (10 - 30) MeV. The effective φN absorption cross section, obtained withinthe linear density approximation (Eq. (38)), increases from 14 to 25 mb. All data consistently indicatea strong broadening of the φ meson in the nuclear medium. The real part of the φ - nucleus potential The only experimental information on the real φ - nucleus potential and thus on a possible in-mediummass shift of the φ meson is based on the line shape analysis of the φ → e + e − signal measured in56igure 49: e + e − invariant mass distribution near the φ mass obtained in p+Cu collisions at 12 GeVfor slow ( βγ ≤ .
25) and fast ( βγ ≥ .
75) recoiling φ mesons. The solid histogram represents a fitwith the expected φ → e + e − line shape and a quadratic background. The dashed curve represents thebackground. No difference in line shape is observed for the corresponding measurement on a C target.The figure is taken from [254].the KEK E325 experiment in p+C, Cu collisions at 12 GeV [254]. Mass shifts are expected to beobservable only in heavy nuclei and for very slow mesons which have a higher probability to decaywithin the nucleus than fast ones. For the light C target no difference in the line shape of the invariantmass peak is observed for φ mesons recoiling with different velocities. For the heavier Cu nucleus Fig. 49(Left) shows an excess on the low mass side of the φ mass peak for slow φ mesons ( βγ ≤ . φ mass by 35 MeVand an increase of the φ width by a factor 3.6 at normal nuclear matter density. The latter value ismuch smaller than the indirect determinations of the in-medium φ width, discussed above. A loweringof the φ mass by 35 MeV is, however, in the range of some theoretical predictions [19, 245], whileKlingl et al. [132] and Gubler and Weise [248, 249] expect smaller mass modifications. From the onlyavailable measurement, the real part of the φ -nucleus potential is thus ≈ -35 MeV. Since the KEK E325experiment [254] is the only one where an in-medium broadening and a mass shift of a meson have beenreported, it is of utmost importance to verify this result with better statistics. An experiment (E16)is planned at J-PARC to improve the statistics of the φ → e + e − line shape measurement [255] by 2orders of magnitude, using a new detector system with large angular coverage and improved resolution.Φ mesons will be produced with 30 GeV protons. Slow φ mesons going backwards in the center-of-masssystem will be selected for this study. The results are highly awaited. meson–nucleus potentials in the charm sector Normal nuclear matter consists of nucleons and thus only of quarks and not of antinucleons (antiquarks).As discussed in Sections 4.1,4.2,4.3, this leads to different properties of kaons K ( qs ) and antikaons57igure 50: (Left) Total energies of D − and D + mesons at zero momentum calculated within the QMCmodel for nuclear matter, plotted as function of the baryon density in units of the saturation density ρ = 0 .
15 fm − . (Right) Total cross section for D + and D − meson production in p Au annihilation asfunction of the antiproton energy. The curves represent calculations for free (dashed) and in-medium(solid) masses of the D mesons. For comparison the free pN → D − D + cross section is also shown. Thefigure is taken from [260]. With kind permission of The European Physical Journal (EPJ). K ( qs ) (with q = u, d ) in nuclear matter. Mesons with a light quark ( K + , K ) experience repulsionin nuclear matter while mesons with a light nonstrange antiquark ( K − , K ) experience attraction.Analogously, one would expect in the charm sector attraction for D mesons ( D + ( cd ) and D ( cu )) andrepulsion for D mesons ( D − ( dc ) and D ( uc )). These intuitive arguments, however, turn out to beoversimplified. The properties of pseudoscalar D mesons have been investigated in several theoreticalstudies with partially conflicting results. Some investigations consider hadronic degrees of freedom likeself-consistent unitarized coupled channel calculations [256], explicitly incorporating Heavy Quark SpinSymmetry (HQSS) [257, 258]. These studies have recently been summarized in [259]. Other calculationsare based on quark and gluon degrees of freedom like the QMC model [260, 261, 262] or QCD sum ruleanalyses [263, 264, 265]. For an overview over heavy hadrons in nuclear matter see [266].In the unitarized coupled channel approach the open-charm in-medium spectral features like D, D masses and widths are calculated using the effective open-charm-nucleon interaction in matter in aself-consistent treatment. The spectral functions exhibit structures due to mixing with resonant-holestates. The potentials for D and D mesons are found to be attractive with depths ranging from -15 to-45 MeV but strongly energy dependent close to the open-charm meson mass [257].In contrast, using the QMC model, Sibirtsev et al. [260] find a dramatic lowering of the D + massby ≈ - 150 MeV at normal nuclear matter density, while the D − meson is shifted up in mass by about15 MeV, reducing the threshold for D + D − pair production in antiproton-proton annihilation by 135MeV in nuclei compared to the D + D − production in free pp collisions. As for the mesons discussedin the previous sections, such a change in the production threshold, leading to an enhanced D + D − production, could be verified by measuring the excitation function in the pp → D + D − reaction on58igure 51: Density dependence of D + and D − meson peak positions from a QCD sum rule analysis.The shaded areas indicate the theoretical uncertainties. The figure is taken from [265].Figure 52: (Left) Excitation function calculated for production of J/ψ mesons in the p + Nb reaction.The curves represent calculations for an absorption cross section σ J/ψN = 3.5 mb and a formationlength l J/ψ = 0, assuming
J/ψ in-medium mass shifts as depicted in the inset. The vertical dashed lineindicates the energy threshold for
J/ψ production off the free nucleon. (Right) Transparency ratio T A ,normalised to carbon, for J/ψ mesons as function of the nuclear mass number A in the scenario withoutmass shift for different absorption cross sections σ J/ψN and formation lengths l J/ψ as indicated in theinset. The curves are to guide the eye. The figures are taken from [84].59uclei, as suggested in Fig. 50. The difference in the D + and D − meson production rates is due to the D + absorption in nuclear matter.QCD sum rule studies have led to somewhat inconsistent results. While Hayashigaki [263] finds amass shift of the D meson of - 50 MeV at normal nuclear matter density, Hilger et al. [264] obtainan opposite mass shift of +45 MeV. Suzuki et al. [265] claim a positive mass shift for both D + and D − mesons (see Fig. 51). For the D + − D − mass difference throughout negative values are predictedat normal nuclear matter density: -65 MeV [264] and -15 MeV [265], which are however, much smallerthan the mass difference of -164 MeV claimed in the QMC model calculation [260].For the J/ψ ( cc ) meson the QMC model gives a mass drop of -16 to -24 MeV at normal nuclearmatter density [267]. The sensitivity of the excitation function for J/ψ production in proton- andphoton induced reactions off nuclei to in-medium mass shifts of the
J/ψ meson has been studied in[84, 268] (see Fig. 52). Furthermore, the extraction of the in-medium inelastic
J/ψN cross section fromtransparency ratio measurements has been discussed which provides information on the imaginary partof the
J/ψ -nucleus potential. Molina et al. [269] calculate the transparency ratio for photon-induced
J/ψ - production at 10 GeV as foreseen in the JLAB upgrade and expect that 30-35% of the
J/ψ -mesonsproduced in heavy nuclei are absorbed inside the nucleus.It is evident that this spread of theoretical predictions calls for an experimental clarification. Suchmeasurements will, however, have to await new experimental possibilities at J-PARC and in the futureat FAIR.
In this subsection the theoretical predictions and experimental results for the real and imaginary partsof meson-nucleus potentials, discussed in the above sections, have been compiled in two tables. This ismeant to provide a quick overview for the reader.Figs. 53, 54 provide a graphical representation of the experimental data given in table 2. One seesthat the real part of K + and K - nucleus potentials are weakly repulsive, while the K − , η , η (cid:48) , ω and φ -nucleus potentials are attractive, however, with widely different strengths. Because of meson absorptionin the nuclear medium the imaginary part of meson-nucleus potentials are all negative, again with alarge spread. The smallest imaginary part is found for the η (cid:48) mesons.In many experiments only the real part or only the imaginary part of the meson-nucleus potentialhave been determined. However, for the η, η (cid:48) , ω and φ mesons there are experiments which report areal and an imaginary part of the potential. The relative strength of the real and imaginary part is animportant information for the observation of meson-nucleus bound states (see Section 7) and can beread off from Fig. 55. For the η, η (cid:48) and φ mesons the modulus of real part | V | is found to be largerthan the modulus of the imaginary part | W | . Note, however, that the small imaginary part reportedfor the φ meson in [254] is in conflict with the much larger values reported in [183]. An attractive meson-nucleus interaction, if strong enough, opens the possibility of having a mesonbound in a nucleus to form a short-lived meson-nucleus quasi-bound state. Such states are of greatinterest in contemporary nuclear and hadronic physics. From the nuclear physics point of view thesestates are exotic configurations of nuclei as they correspond to states with excitation energies of severalhundred MeV up to 1 GeV. For hadron physics the study of these states provides a unique possibility60able 1: Compilation of theoretical predictions for the real ( V ) and imaginary ( W ) meson-nucleus potentials at normal nuclear matter density.meson V [MeV] W [MeV] σ inel [mb] model ref. K + ≈ +20 - - - [97] K + ≈ + 40 - - - [23] K + + 36 - - - [108] K − -(180-200) -(70-80) - NLO30 [7-11] K − ≈ -100 - - - [97] K − ≈ -120 - - - [23] K − -(40-50) ≈ -50 - Chiral [144] K − -(80-90) - - χ meson-baryon [148] K − -(80 -120) ≈
30 - χ meson-baryon [174] η - - ≈
30 - [204] η - - ≈
30 - [205] η (cid:48) ≈ -150 - - NJL [211] η (cid:48) ≈ η (cid:48) ≈ -80 - - lin σ [217] η (cid:48) ≈ -40 - - QMC [218] ω ≈ ω ≈ -120 - - QCD sum rule [19] ω ≈ -(100-150) ≈ -20 - L eff [47] ω ≈ ≈ -30 - res. coupl. [49] ω ≈ -30 ≈ -20 - res. coupl. [224] ω - -(50 - 100) - χ unitary [225] ω - -(75 - 100) - many body [226] ω ≈ -100 QMC [230] φ ≈ -30 - - - [19] φ -35 -(20-25) - - [245] φ ≈ -8 ≈ -15 - - [246] φ ≥ -20 -45 - χ EFT/QCD sum rule [248, 249] D -25 -14 - unitarized coupled channel [257] D, D ∗ -62 - - QMC [267] D + ≈ - 150 - - QMC [260] D + +23 - - QCD sum rules [265] D − +20 - - QMC [260] D − +38 - - QCD sum rules [265] J/ψ ≈ -(19-24) - - QMC [262] J/ψ ≈ -(16-24) - - L eff [267]61able 2: Compilation of experimentally deduced real ( V ) and imaginary ( W ) meson-nucleuspotentials at normal nuclear matter density. The potential values marked with an asteriskhave been extrapolated to meson momentum zero, otherwise the potential values have beendetermined as average over a momentum range, mainly 0 (cid:28) p ≤ m . When given separatelyin the original literature the first error refers to the statistical error and the second one tothe systematic error.meson V [MeV] W [MeV] σ inel [mb] reaction collaboration ref. K + ≈
25 - - p+A KaoS [46] K + K + ± K + ≈
30 - - Ni+Ni - [126] K + ± K ± π − + A FOPI [134] K ≈
40 - - Ar + KCl HADES [136] K ± K − ≈ −
80 - - p + A KaoS [46] K − − (45 −
50) - - Ni + Ni FOPI [128] K − − +50 − - - p + A ANKE [167] K − -160 ...-190 ≈ - 60 - C, O( K − , N ) KEK E548 [186] η -(10 - 30) 30 ± γ + A A2 [199] η − (54 ± − (20 ±
2) - p + d ANKE, COSY11 [270, 271] η (cid:48) - -(10 ± ± γ + A CBELSA/TAPS [203] η (cid:48) - − (13 ± ± ∗ ± γ + C, Nb CBELSA/TAPS [213] η (cid:48) -(37 ± ±
10) - - γ + C CBELSA/TAPS [219] η (cid:48) -(41 ± ±
15) - - γ + Nb CBELSA/TAPS [220] ω - -(35 - 50) ≈ γ + A CBELSA/TAPS [51] ω - -(48 ± ± ∗ - γ + C, Nb CBELSA/TAPS [213] ω -(29 ± ±
20) - - γ + C, Nb A2 [231] ω -(15 ± ±
20) - - γ + C CBELSA/TAPS [233] ω ≈ -75 0 - p + A KEK E325 [240] ω - ≤ -100 MeV - γ + A CLAS [242] φ - -(20 - 30) - p + A ANKE [182] φ - -(10 - 30) 14-25 p + A ANKE [183] φ - -(23-100) 16-70 γ + A CLAS [228] φ - - 35 +17 − γ + A LEPS [250] φ ≈ -35 -7.5 - p + A KEK E325 [254]62 [ M e V ] V − − − − + K K - K η ' η ω φ Figure 53: Compilation of the experimentally determined real part of the meson-nucleus potential for K + , K , K − , η, η (cid:48) , ω and φ mesons. The data are taken from table 2. The vertical bars represent theerrors or the range of values quoted. The horizontal dashed lines indicate average values. The CLASresults [228] are not included in the figure because of the large experimental uncertainties quoted bythe authors. [ M e V ] W − − − − − - K η ' η ω φ Figure 54: Compilation of the experimentally determined imaginary part of the meson-nucleus potentialfor K − , η, η (cid:48) , ω and φ mesons. The data are taken from table 2. The vertical bars represent the errors orthe range of values quoted. The horizontal dashed lines indicate average values. The CLAS results [228]are not included in the figure because of the large experimental uncertainties quoted by the authors.63 [MeV] potential depth |V0 10 20 30 40 50 60 70 80 | [ M e V ] i m a g i n a r y p a r t | W w ' h hf Figure 55: Compilation of the experimentally determined real and imaginary parts of the meson-nucleuspotential of η [270, 271], η (cid:48) [213, 219, 220], ω [213, 233, 229], and φ [254] mesons. Thick (thin) errorbars correspond to statistical (systematic) errors. The figure is updated from [235].to investigate the properties of mesons and their possible modification at finite nuclear densities. Thekey to the formation of such a quasi-bound state is to chose a reaction kinematics with low momentumtransfer to the meson (recoil-less production). The observation of the mesic state will be facilitated ifthe potential depth is sizable and large compared to the width of the state, i.e. : | V |(cid:29)| W | . (63)Furthermore the energy spacing of quasi-bound states should be larger than the widths of the states, acondition which becomes more difficult to fulfil in heavier nuclei.Most of the experiments search for resonance-like structures in the excitation energy range of theresidual nucleus where the meson-nucleus bound states are expected either by looking for decays of themesic states or by missing mass spectroscopy. These measurements, however, suffer from the multi-pion background from competing reactions. An alternative approach is to measure the production ofthe respective meson near the production threshold, which can normally be identified with a sufficientsignal-to-background ratio. The disadvantage is that one has to extrapolate in energy to the mesic poleand will not be able to distinguish between a quasi-bound or anti-bound (virtual) state. In the followingsections experimental searches for meson-nucleus bound states for various mesons will be summarisedand compared to corresponding theoretical predictions. pionic and kaonic atoms The existence of pionic and kaonic atoms is clearly established experimentally. Here, one of the atomicelectrons is replaced by a negatively charged pion or kaon. These exotic states have been formed bystopping π − or K − in a target. The mesons are captured in an outer atomic orbit and then cascade downto lower atomic levels by the emission of characteristic X-rays until at some lower principal quantumnumber the mesons are absorbed due to their interaction with the nucleus. The lifetimes of π − and K − mesons are long compared to the stopping times and lifetimes of atomic states so that well-definedexotic atomic states can be formed. The observables of interest are the binding energies and the widthsof these states which are affected by the strong interaction in the inner orbits. The level shifts and64idths of atomic levels have been analysed to extract information on the meson-nucleus interactionand in particular on its density dependence since the overlap of atomic wave functions with the nucleuscovers a wide range of nuclear densities [7-11].A breakthrough in these studies has been the observation of deeply bound π − s and 2 p states whichhave been directly populated in recoil-free reactions, avoiding the cascading down from higher orbits [1-4]. The existence of these states had been predicted in [272-274]. These deeply bound states correspondto atomic orbits very close to the nucleus and are thus more sensitive to the meson-nucleus interaction.The information on the strong interaction of low energy mesons extracted from these studies has beensummarised by Friedman and Gal [7]. Fitting phenomenological density dependent potentials to K − atomic data, they deduce deep potentials with a depth of about -200 MeV for K − mesons in nuclearmatter at saturation density, while chiral Lagrangian [142-144,168-171] or meson-exchange calculations[172, 173] predict much shallower K − - nucleus potentials with a central depth of the order of -50 to-80 MeV. Friedman and Gal [8, 9] emphasise that in order to achieve good fits to kaonic atom data thepotential constructed from in-medium chirally motivated K − N amplitudes have to be supplementedby a phenomenological term taking K − multi-nucleon interactions into account. Note, however, themodel dependence of extrapolating potentials to normal nuclear matter density, recently pointed outin [11] and discussed in Section 4.3. It is remarkable that Hirenzaki et al. [6] do find an equally goodreproduction of the kaonic atom data using a shallow chiral potential.The existence of deeply bound pionic states has been explained by the superposition of the attractiveCoulomb interaction with the repulsive s-wave pion-nucleus interaction such that the pions are boundin a potential pocket at the nuclear surface which gives rise to a halo-like pion distribution around thenucleus [275]. Kaonic atoms have a different origin. In contrast to the repulsive s-wave π − –nucleusinteraction, the K − -nucleus interaction is found to be attractive but the strength of the imaginary partof the K − -nucleus interaction suppresses the atomic wave function in the nuclear interior and expels itto the nuclear surface, again leading to a halo-like meson-nucleus configuration [7].This review, however, focuses on the possible existence of meson-nucleus states exclusively bound viathe strong interaction. Despite of experimental efforts for over about 30 years no convincing evidencefor meson-nucleus states exclusively bound by the strong interaction has as yet been found apart fromsome promising indications. Search for kaonic clusters and K − nuclear quasi-bound states As discussed in Section 4.3, low energy KN scattering [276] and X-ray spectroscopy of kaonic atoms[7, 151, 152, 277, 278] have shown that the KN interaction is strongly attractive. Similar to the caseof the η meson, where the strong S (1535)1 / − resonance governs the near threshold ηN interaction,the Λ(1405)1 / − resonance, located 27 MeV below the K production threshold, dominates the nearthreshold anitkaon-nucleon interaction. The nature of the Λ(1405) state has been widely discussed inthe literature. For a recent summary see [159]. As discussed in Section 4.3, it is considered to be amolecular state emerging from the interference of a Σ π and a K − N pole. The interpretation as a K − p bound state led Akaishi and Yamazaki [279, 280] to predict the existence of kaonic nuclear bound states.The simplest such state would be a bound K − N N cluster. Depending on the KN interaction model,widely scattered binding energies of 10 - 100 MeV and widths of the order of 30 -110 MeV have beenpredicted [280-288].Experimentally, the field is very controversial. Several experiments have reported peak structuresabout 100 MeV below the K − N N production threshold. In ¯ p He annihilations at rest the OBELIXcollaboration observed a structure in the Λ p invariant mass which - if interpreted as the decay of a K − pp cluster - would correspond to a binding energy of 160 ± ≤ ± . K − in Li, Li, and C targets, the FINUDA Collaboration at DAΦNE reported a structurein the back-to-back Λ p invariant mass spectrum corresponding to a binding energy of 115 +6 − MeV and65igure 56: (Left) Λ p invariant mass distribution for the in-flight He( K − , Λ p ) n reaction at a K − mo-mentum of 1.0 GeV/ c with simulated spectra for K − absorption on three or two nucleons. The figureis taken from [299]. (Right) The same data in comparison with a calculated Λ p invariant mass dis-tribution, assuming contributions from the decay of a bound KN N state (lower peak) and from theformation of an unbound Λ(1405) p system (higher peak). The figure is taken from [300].a width of 67 +14 − MeV [290]. This structure has later been interpreted as being due to K − two-nucleonabsorption followed by final state interactions of the produced particles with the daughter nucleus [291].However, later studies by the FINUDA collaboration [292] of correlated Λ t pairs in the absorption of K − at rest in light nuclei revealed that even more nucleons may be involved in the absorption process.Studying pp collisions at 2.85 GeV, the DISTO group claimed a structure in the Λ p invariant masswith a binding energy of 103 ± ± ± ± K − pp ” like structure in the Σ p channel with binding energy of 95 +18 − (stat) +30 − (syst) MeVand width of 162 +87 − (stat) +66 − (syst) MeV was reported by the E27 experiment at J-PARC, investigatingthe d( π + , K + ) reaction at 1.69 GeV/ c [294]. In contrast, no significant structures were found by theHADES Collaboration in p + p reactions [295-297] nor in a photon induced inclusive reaction studiedat LEPS [298].On the other hand, more recently the E15 experiment at J-PARC has reported a structure in theΛ p invariant mass near the K − pp threshold in the in-flight He( K − , Λ p ) n reaction at a K − momentumof 1.0 GeV/ c [299], as shown in Fig. 56. The Λ pn final-state is reconstructed from the momenta of thecharged particles ( π − pp ) measured in a cylindrical drift chamber while the neutron was kinematicallyidentified as missing mass particle. The structure in the Λ p invariant mass resides on a background ofevents attributed to kaon absorption on three- and/or two- nucleons. A Dalitz-plot analysis shows thatthis structure is associated with forward going neutrons and with a small momentum transfer to theΛ p system. The structure has a mass of 2355 +6 − (stat) ±
12 (syst) MeV and a width of 110 +19 − (stat) ±
27 (syst) MeV and could thus be a signal of a
KN N bound state, located ≈
15 MeV below the K − pp threshold. This interpretation is supported by a recent theoretical study [300], assuming two possiblereaction scenarios: (i) the Λ(1405) resonance is generated after the emission of an energetic neutronfrom the absorption of the initial K − without forming a bound state with the remaining proton; (ii)after the emission of the energetic neutron a KN N bound state is formed, decaying subsequently intoa Λ p pair. The contributions from both scenarios give rise to a two-peak structure in the Λ p invariantmass distribution as shown in Fig. 56 (Right) where the calculations are compared to the experimentaldata of Fig. 56 (Left). With the statistics of the present data this two-peak structure cannot be resolvedbut overall the shape of the experimental invariant mass spectrum is well described by the calculation.66igure 57: The real (Left) and imaginary (Right) parts of the K − − Pb potential, calculated self-consistently for the single-nucleon K − potential (KN) and two different versions (FD, HD) of the densitydependence of the K − multi-nucleon potential. The shaded areas indicate the uncertainties. The KM1model applies the Ikeda-Hyodo-Weise chiral K − N potential [146, 147] plus a phenomenological termproportional to ( ρ/ρ ) α with α = 1, as shown in Fig. 24 of this review. The figure is taken from [301].A high statistics follow-up run will hopefully bring further insight.The multi-nucleon kaon absorption which accounts for the background in Fig. 56 plays a decisiverole in the search for K − - nuclear quasi bound-states, as recently pointed out in [301]. Hrt´ankov´a andMareˇs find that K − - nuclear states in many-body nuclear systems, if they exist at all, will have hugewidths, considerably exceeding the K − binding energies (see Fig. 57) because of the large multi-nucleonkaon absorption contributions. This will make the observation of K − -mesic states very difficult. Infact, the search for K − − nucleus bound states pursued by the AMADEUS Collaboration [302] has asyet not provided evidence for such exotic states but for the importance of K − multi-nucleon absorption[303-305]. Search for η -mesic states As discussed in Section 4.4, the large scattering length and the strong s-wave attraction due to thenear threshold S (1535)1 / − resonance led Haider and Liu to predict the existence of exotic stateswith an η meson bound to a nucleus in 1986. Motivated by this work many experimental searches forthese exotic states and numerous theoretical investigations have been performed since then which haverecently been comprehensively summarised in dedicated reviews [188, 189]. Here, only the main resultsand most recent developments will be discussed.The model dependence of the ηN scattering length, discussed in Section 4.4, manifests itself in abroad spread of recent predictions for binding energies and widths of η –nuclear states shown in Fig. 58.The binding energies increase with the nuclear mass number A and saturate at larger values of A,depending on the model for the ηN amplitude. For some models the widths are only 2-4 MeV, makingan observation feasible, provided a suitable production reaction is found. Other models give too largewidths or are too weak to generate η –nucleus bound states. It should be noted that all predicted widthsare considerably smaller than the widths extracted from η production reactions at above thresholdenergies (see 4.4.2) since they are calculated from the subthreshold imaginary amplitudes shown in67igure 58: (Left) Binding energies and (Right) widths of 1 s η -nuclear states in selected nuclei calculatedself-consistently using different ηN scattering amplitudes shown in Fig. 33 of this review. The figure istaken from [196].Fig. 33 of the present review.Experimental searches of structures in the expected bound state region were undertaken with pion-[306], photon- [307-309], proton- [310, 311] and deuteron beams [312-315], however, with inconclusiveresults. The strongest claim for the discovery of an η -mesic state has been made by the COSY-GEMCollaboration [310]. They have studied the p + Al → He+ η ⊗ Mg → He p π − X reaction at a protonenergy of 1745 MeV. In this two-nucleon transfer reaction the forward going He was detected withhigh resolution in the Big-Karl magnetic spectrometer and η mesic states with binding energies 0 MeV ≤ B η ≤
30 MeV could be produced under almost recoil-free conditions with a momentum transfer q ≤
30 MeV/ c . The decay of η mesic states was expected to occur via the ηn → N ∗ (1535) → π − p channel, leading to a π − and a proton emitted almost back-to-back in the laboratory system. Underthese conditions the corresponding excitation energy spectrum shows an enhancement about 13 MeVbelow the η production threshold with a width of ≈
10 MeV(see Fig. 59), while no structure is observedwithout the back-to-back requirement. This observation is remarkably close to the prediction of a 1s η -mesic state in Mg with a binding energy of 12.6 MeV [195]. In a refined analysis of the COSY-GEMdata, Haider and Liu [316] take into account that the η produced in the intermediate state can alsobe inelastically scattered by the residual nucleus and emerge as a pion, without being captured by thenucleus. Allowing for the interference with this non-resonant reaction amplitude a binding energy of B η ≈ η He system has also been studied in η production experiments close to the threshold in thereactions d + p → η He [317, 318, 319] and γ + He → η + He [308, 309]. Corresponding excitationfunctions are shown in Fig. 60. In both reactions a very sharp rise of the cross section above thethreshold was observed which was taken as an indication for the possible existence of an η He quasi-bound state extremely close to threshold [318]. The d + p data which are much superior in statisticshave been analysed by Wilkin et al. [320] and more recently by Xie et al. [270]. While Wilkin et al.68 c oun t s / M e V -50 -40 -30 -20 -10 0 B η (MeV) [MeV] Figure 59: Counts for the p + Al → He p π − X reaction as a function of the excess energy in the η ⊗ Mg system [310, 311]. The solid curve represents a fit to the data with a Gaussian and a constantbackground. The dashed curve is a fit with a Breit-Wigner function taking the interference with anon-resonant reaction amplitude into account [316]. The figure is adapted from [188].Figure 60: (Left) Total cross section for the dp → η He reaction below an excess energy of 2 MeV. Thedata are from [317, 318]. The curve represents a fit to the data performed by [270]. The figure is takenfrom [270]. (Right) Total cross section for the γ He → η He reaction (red points) from [309] comparedto data (green triangles) from [308]. Solid (dashed) curves represent plane wave impulse approximation(PWIA) calculations with a realistic (isotropic) angular distribution for the γn → nη reaction. Insert:ratio of measured and PWIA cross sections. The figure is taken from [309].69eport a pole in the production amplitude at Q = -0.3 MeV very close to the threshold with a widthof 0.4 MeV, Xie et al. find a Breit-Wigner type structure at a mass of -0.3 MeV with a width of 3MeV, but a pole in the continuum. This is at the verge of binding. From their analysis of the ANKEand COSY 11 data Xie et al. deduce an attractive η He potential of U = [-(54 ± ± η He and η He systems Barnea et al. [321] find in pionless effectivefield theory calculations that deeper potentials would be required corresponding to a real part of the ηN scattering length close to 1 fm and exceeding 0.7 fm, respectively.Selecting correlated π p pairs, a peak in the π p invariant mass spectrum was observed in the γ He → π pX reaction and interpreted as evidence for the decay of a mesic η ⊗ He state [308]. In alater experiment [309] this signal was confirmed with much improved statistical significance, but couldbe attributed to a structure arising from quasi-free pion production. In an attempt to search for η Hemesic states, back-to-back pion-nucleon pairs were also measured in the d + d → η ⊗ He → π n He, π − p He reaction [313, 315]. No unambiguous signal was found and upper limits on the cross sectionsof ≈ π n He and ≈ π − p He channel were deduced, respectively.30 years of experiments without an absolutely convincing and unambiguous signal have shown howcumbersome the search for η -mesic states really is. New approaches and higher statistics measurementsare needed to make progress. Recently high statistics data on the p + d reaction have been takenby the WASA-at-COSY Collaboration dedicated to the search for η He bound states in three decaychannels: pppπ − , He 2 γ , and ppn . The data analysis is still in progress [311]. In addition, the π η photoproduction off He has recently been measured at MAMI in an attempt to identify η − He boundstates [322]; also here the data analysis is ongoing. Utilising the high intensity π beam at J-PARC ahigh statistics search for η -mesic nuclei is going to be performed [323]. Search for η (cid:48) -mesic states As discussed in Sec. 4.5 the mass reduction of the η (cid:48) meson corresponds to an attractive interactionbetween an η (cid:48) meson and a nucleus. Furthermore, due to the larger real part of the η (cid:48) nucleus potentialcompared to its imaginary part the η (cid:48) meson has become a promising candidate for the search of η (cid:48) -bound states. Binding energies of η (cid:48) -mesic states have been calculated based on the NJL model[24, 211, 324, 325] and on a chiral unitary model [221]. As an example, bound-state spectra of the η (cid:48) -mesic states assuming an attractive potential with a real part of V = -100, -150, -200 MeV and animaginary part W = -20 MeV are shown in Fig. 61 (Left) [324]. There are several bound states ina small nucleus ( C) due to the strong attraction, well separated because of the assumed relativelysmall imaginary part of the η (cid:48) -nucleus potential. In contrast, as it can be seen in this figure, potentialshaving an imaginary part comparable with the real part, V = -100 MeV and W = -50 MeV, providebound states with larger widths than the binding energy and are overlapping. In this case it wouldbe hard to observe bound states as clear peaks in the formation spectra. Reactions like ( γ, p ), ( π + , p ),( p, d ) have been proposed for the formation of η (cid:48) -mesic states for nuclei such as C and Ca . Thecalculated spectrum of the C(p, d) C ⊗ η (cid:48) reaction for the formation of the η (cid:48) -nucleus system at aproton kinetic energy energy of 2.5 GeV and at deuteron angle θ d = 0 is shown in Fig. 61 (Right) [325]as a function of the excitation energy and for the η (cid:48) - potential parameters (V ,W )=-(150, 5) MeV.One can see clearly separated peaks corresponding to η (cid:48) bound states in the s , p and d states. Spectrafor other combinations of (V ,W ) and different η (cid:48) scattering lengths have been calculated showing thatthe width of each peak becomes wider as W and/or | a pη (cid:48) | increase. It has been pointed out that thespectra for nuclei heavier than Ca will exhibit an overlap of different nucleon-hole- η (cid:48) configurationsthat smear out the individual peaks and the structures get less prominent [221, 324].These theoretical predictions and the experimental data [219] encouraged the search for η (cid:48) boundstates. The first pioneering experiment searching for η (cid:48) bound states was performed in 2014 at the70 B i nd i ng E n e r gy [ M e V ] (-100,-20) (-150,-20) (-200,-20) (-100,-50)V [MeV] s p s p d s p d s p Figure 61: (Left) Bound state spectra of the η (cid:48) meson in C in units of MeV. Boxes denote thebinding energies and indicate the widths of the bound states. The letters s, p and d label the angularmomentum states. The potential of the η (cid:48) meson in the nucleus is assumed in the form of Eq. 4 with thepotential depths at normal nuclear matter density V =-100, -150, and -200 MeV with a fixed imaginarypotential W = -20 MeV. Also shown is the result with W = -50 MeV and V =-100 MeV [324]. (Right)Calculated spectra of the C(p,d) C ⊗ η (cid:48) reaction for the formation of η (cid:48) -nucleus systems with protonkinetic energy T p =2.5 GeV and deuteron angle θ d =0 as a function of the excitation energy E ex . The η (cid:48) production threshold is E . The η (cid:48) -nucleus optical potential is taken to be (V , W )=-(150, 5) MeV. Thethick solid line shows the total spectrum and dashed lines indicate subcomponents. The neutron-holestates are indicated as ( nl j ) − n and the η (cid:48) states as l η (cid:48) [325].Fragment Separator (FRS) at GSI using the C(p,d) reaction at an incident proton energy of 2.5 GeV[326]. Similar to the experiment for pionic atoms [4], ideas for the formation of η (cid:48) -mesic states inalmost recoil-less kinematics have been developed and studied in simulations [327]. The proton beamhits the C target and ejects a deuteron with high momentum, while the η (cid:48) meson produced with lowmomentum could be bound to the C. In the experiment only the deuteron momentum distributionhas been measured, applying missing mass spectrometry. The FRS has been used as spectrometer witha specially developed ion optical setting. In Fig. 62 (Left) the measured excitation spectrum of the C(p,d) reaction near the η (cid:48) emission threshold is shown [326]. Unfortunately, no narrow structure hasbeen observed in spite of the extremely good statistical sensitivity. For positive excitation energies theincreasing contributions from quasi-free η (cid:48) production is observed. An upper limit for the formationcross section of η (cid:48) -mesic nuclei of ≈
20 nb/(sr MeV) near the threshold has been deduced. The spectrumhas been compared with the theoretical spectra from [325] for different potential parameters ( V , W )and is shown in Fig. 62 (Right). For each potential parameter combination the spectrum has beenfitted by a polynomial describing the background and a theoretical spectrum scaled by a factor µ ,folded with the spectral resolution σ exp ( σ exp = 2 . ± µ have been evaluated at the 95% confidential level (CL). As can be seen in Fig. 62 (Right), a stronglyattractive potential of V ≈ -150 MeV predicted by the NJL model [211] is rejected for a relativelyshallow imaginary potential [213].The experimental results of Fig. 62 (Right) are shown again in Fig. 63 in comparison to the theoreticalpredictions for η (cid:48) -nucleus potentials based on NJL [211], linear σ [217] and QMC [218] models, and theexperimental potential values determined by CBELSA/TAPS [213, 220, 235]. In conclusion the FRSat GSI experiment had only a limited sensitivity for a relatively weak attraction as predicted by theQMC model [218] and deduced by the η (cid:48) photoproduction experiments [213, 220, 235]. The shallow71 − − − − − M e V ) ] b / ( s r μ ) [ d E Ω d / ( σ d c Deuteron momentum [MeV/ 27802800282028402860288029002920 C oun t s d , p C( [MeV] E − ex E − − − − − R e s i due − p ) d , p D( | [MeV] V |50 100 150 200 | [ M e V ] W | μ Figure 62: Left: (top panel) Excitation spectrum of C measured in the C ( p, d ) reaction at a protonenergy of 2.5 GeV. The abscissa is the excitation energy relative to the η (cid:48) production threshold E =957 .
78 MeV. The gray curve represents a fit of the data with a third-order polynomial. The upperhorizontal axis is the deuteron momentum scale. (Inset) Deuteron momentum spectrum measured inthe elastic D ( p, d ) p reaction using a 1.6 GeV proton beam. (Bottom panel) Fit residues with envelopesof 2 standard deviations. Right: Contour plot of µ (solid curves), upper limit of the scale parameter µ at the 95% confidence level, on a plane of real and imaginary potential parameters ( V , W ). Thelimits have been evaluated for the potential parameter combinations ( V , W ) in (-50,-100,-150,-200) x(-5,-10,-15,-20) and (-60,-80) x (-5,-10,-15) MeV and linearly interpolated in between. Dashed curvesshow a band of µ =1 contour indicating systematic errors. Regions for µ ≤ η (cid:48) -mesic nuclei. A semi-exclusivemeasurement has been considered for the Super-FRS at FAIR [325, 326]. An alternative approachis the photoproduction of η (cid:48) mesons in the C( γ ,p) reaction, again in almost recoil-free kinematics.The predictions based on NJL model by [24, 211] and the estimated formation cross section motivatedexperiments at LEPS2 facility (Spring8) [328, 329] and BGO-OD at ELSA [235, 330], where the missingmass spectroscopy is combined with detecting the decay particles of the η (cid:48) -mesic states. The BGO [331]and BGO-egg [329] detectors are well suited for detecting photons from decay particles of the η (cid:48) -mesicstate, e.g. η (cid:48) N → ηN → γγN [214, 325]. As an alternative, the ( π ± , p ) reaction has been proposed forthe formation of η (cid:48) -mesic states for a measurement at J-PARC [221]. Sofar no experimental results areavailable from LEPS2, BGO-OD and J-PARC. Search for ω -mesic states The experimental results discussed in Section 4.6 have shown that the ω –nucleus attraction is quiteweak. Nevertheless, motivated by theoretical predictions for the existence of ω –nucleus bound states in[332-334], an experimental search for ω -mesic states has been performed [233]. As proposed in [332-334],the C ( γ, p ) reaction has been studied in recoil-less kinematics. The optimum incident photon energyneeded to produce an ω at rest in the nucleus is around 2.75 GeV. At this incident photon energy theproton detected at forward angles takes over the momentum of the incident photon, leaving the ω at72 [MeV] potential depth |V0 20 40 60 80 100 120 140 160 180 | [ M e V ] i m a g i n a r y p a r t | W ' η QMC σ lin NJL ' η p a c h i r a l u n i t a r y excluded Figure 63: The imaginary part versus the real part of the η (cid:48) -nucleus potential. The vertical lines denotethe model calculations for the real part of the η (cid:48) optical potential: (from left to right) QMC model[218], linear σ model [217] and NJL model [211]. The dashed curve is the result of a study in the chiralunitary approach, accounting for η (cid:48) absorption by pairs of nucleons [221]. The box labeled a pη (cid:48) indicatesthe potential parameters corresponding to the pη (cid:48) scattering length reported by Czerwinski et al. [209].The red point with thick (statistic) and thin (systematic) error bars represents the experimental resultfor V and W for the η (cid:48) -nucleus potential measured by CBELSA/TAPS [213, 220, 235]. The blueline is the µ =1 line in Fig. 62 (Right) and the dashed area represents the region of V ,W parametercombinations excluded by the experiment of Tanaka et al. [326].rest in the laboratory system (see Fig. 64) so that it can be captured by the nucleus in case of anattractive ω -nucleus interaction. If the ω mass in the nucleus were reduced by 100 MeV, the optimumincident energy would be lowered to around E γ = 1 . ω meson is lower than 300 MeV/c (comparable to the average nucleonFermi momentum) for all envisaged mass drops as long as the outgoing proton is confined to laboratoryangles of 1 ◦ - 11 ◦ . This is the kinematics chosen for the CBELSA/TAPS experiment [233]. Protonswere identified in the TAPS forward wall and π γ pairs from ω mesons or the decay of a quasi-bound ω mesic state were detected with the Crystal Barrel detector. This approach allowed the simultaneousstudy of quasi-free ω production above the threshold as well as the search for decays of ω -nucleus statesin the bound state regime below the production threshold.Fig. 65 (Left) shows the theoretically predicted excitation energy spectrum of the residual nucleus B near the ω production threshold, decomposed into contributions from the ω in s or p states boundto different proton hole states. The corresponding experimental spectrum is shown in Fig. 65 (Right).There is some yield of correlated π γ pairs in the bound state region but no statistically significantstructure can be observed. This may not be surprising in view of the strong in-medium broadeningof the ω meson discussed in Section 4.6.2. Correcting for the effective branching ratio for in-medium ω → π γ decay, modified by the increased in-medium width to about 1.5%, the π γ yield in the range-90 ≤ E π γ −
782 MeV ≤ - 20 MeV corresponds to a population cross section of (22 ±
7) nb MeV − sr − which is comparable to the theoretically expected formation cross section of ω -mesic states. The tailingin the total energy distribution into the bound state region may, however, simply be a consequenceof the large in-medium broadening of the ω meson. Conclusive results can only be obtained in anexperiment with much higher statistics. Such an experiment, using the A ( π − , n ) ω reaction and lookingfor π γ pairs from ω decays and decays of ω -nucleus bound states is in preparation, taking advantageof the high intensity π − beams at J-PARC [335]. 73igure 64: Momentum transfers as a function of the incident photon energy E γ in the ( γ, p ) reaction.The solid, dashed, and dotted lines show the momentum transfers at ω energies E ω = m ω , E ω = m ω − E ω = m ω −
100 MeV, respectively. The thick and thin lines refer to θ lab = 0 ◦ and 10.5 ◦ ,respectively. The figure is taken from [334].Figure 65: (Left) Missing energy spectra for the C( γ, p ) ω ⊗ B reaction at E γ = 2.75 GeV. Dottedlines represent the contributions from two particular combinations of bound ω and proton-hole states.The quasi-free ω production cross section is also shown. The figure is taken from [332]. (Right)Kinetic energy distribution of the ω meson off Carbon (black stars) compared with the kinetic energydistribution of the ω meson off the free proton (full blue circles). The LH data are normalised to theC data in the peak of the total energy distribution. The experimental distributions are compared toMonte Carlo simulations (LH : blue histogram; C: red histogram), taking the Fermi motion of nucleonsinto account for the C target. All distributions request the detection of a proton in the polar angularrange 1 ◦ − ◦ and are normalised to the fitted peak height for C. The Monte Carlo simulations arefolded with the experimental resolution of σ E ≈
16 MeV. The figure is taken from [233].74igure 66: Theoretical predictions for the real (Left) and the imaginary part (Middle) of the φ mesonoptical potential as a function of the radial distance r for the φ − B system, obtained from the φ self-energy reported in [246]. The solid, dashed, dotted, and dotted-dashed curves indicate the potentialstrength for the φ meson energies Re (E -m φ ) = 0 MeV, -10 MeV, -20 MeV, and - 30 MeV, respectively.(Right) Differential cross section for the formation of φ –nucleus bound states plotted as a function ofthe φ meson energy in the C ( p, φ ) reaction at p p = 1 . c for the φ - nucleus potential given inthe left and middle part of the figure. The dashed and dotted curves represent the contributions fromdifferent proton hole states. The vertical line indicates the φ meson production threshold. The figuresare taken from [336]. Search for φ -mesic states The discussion in Section 4.7 has shown that the φ –nucleus attraction is expected to be rather weak.Some theoretical calculations give even smaller potential depths than the experimentally claimed masslowering of about 30 MeV at normal nuclear matter density [254]. Fig. 66 shows the calculated realand imaginary part of the φ B - potential which is too shallow to support a φ –nucleus bound state.Bound states are only expected for heavier nuclei. Only when the strength of the real potential isscaled up by a factor 4 to match the reported experimental mass drop [254], bound φ –nucleus statesmay also be expected in lighter nuclei. Theoretical predictions for the imaginary part of the potentialare comparable to the real part (see Fig. 66, Middle), implying rather broad states with a width in therange of ≈
20 MeV [336]. Similar results have recently been obtained within the QMC model [337].This leads to structure-less excitation energy spectra exhibiting some tailing into the bound state regionas shown in Fig. 66 (Right).In spite of these not really encouraging predictions, an interesting experiment (E29) [338] with anantiproton beam has been proposed at J-PARC to study φ -mesic states. Using double φ productionoff nuclei, the forward going φ meson in the primary reaction pp → φφ is used to tag the productionof the backward going φ meson which may be captured by the nucleus. The φ - nucleus bound statemay decay via the process φ + p → K + + Λ where the reaction products are expected to emerge almostback-to-back. By registering the φ → K + K − decay of the forward going φ meson in coincidence withthe back-to-back emitted Λ and K + , the formation as well as the decay of the φ -nucleus state canbe measured. C and Cu are to be used as targets. It will be interesting to see the results of thisexperiment.
Search for mesic states in the charm sector
Already in 1990 Brodsky et al. [339] predicted the existence of ( cc ) - nucleus bound states due to aQCD van der Waals type attractive interaction arising from multiple-gluon exchange. More recently75igure 67: Binding energies and widths for different D - nucleus states, predicted in unitarized coupled-channel calculations. The figure is taken from [257].the possible existence of J/ψ - nucleus bound states has been addressed in quark-meson couplingmodel calculations [267], predicting binding energies in the range of 5-15 MeV. Quenched lattice QCDcalculations find potentials for the η c − N and J/ψ − N systems which are weakly attractive at shortdistances and exponentially screened at large distances [340]. More recent lattice QCD calculationsreport binding energies of ≤
40 MeV [341].Using unitarized coupled channel calculations, Garcia-Recio et al. [257] investigated also the exis-tence of mesic nuclei with open charm mesons. As shown in Fig. 67 they predict binding energies of D - nuclear states of the order of 10 MeV with comparable widths throughout the nuclear chart. TheCoulomb interaction prevents bound states for D + mesons. For D mesons not only D − but also D mesons are predicted to bind to nuclei [258].Experimentally, the main problem is the large momentum transfer associated with charm production.It is highly unlikely that momenta higher than 1 GeV/ c can be taken up by the whole nucleus throughFermi-motion or re-scattering effects. The probability for a nucleus to change momentum and stayintact is given by the square of its form factor F A ( q A ). The question of minimising the momentumtransfer has been studied by Faessler [342]. Antiproton induced reactions provide maximum energyrelease at low momentum transfer. As most favourable reaction he investigates reactions of the type: pp → XY (64)If one of the particles X, Y goes forward, the other one will go backwards in the center-of-mass systemand will thus have a small momentum in the laboratory so that it can be captured by the nucleus. Inthe high energy limit , i.e. for energies √ s (cid:29) m X the minimum momentum of particle X is p min ( X ) ≈ m X − m p m p (65)In case of D meson pair production the laboratory momentum of the backward (in c.m.) produced D meson is still as large as 1.4 GeV/c according to Eq. (65). This kinematics has also been considered76y Yamagata-Sekihara et al. [343] who calculate formation spectra for D − ⊗ B and D ⊗ B withdifferential cross sections in the pb/(sr MeV) range for an antiproton beam impinging on a C target.Larger cross sections might be achieved in reactions producing D ∗ D meson pairs such as p + p → D ∗− + D + ,D ∗− + ( Z, A ) → π + D − ⊗ ( Z, A ) (66)After pion emission the charmed meson may be slow and can get trapped by the nucleus. The PANDAdetector will be highly suited for such investigations making use of the high quality antiproton beamsat FAIR [344].
In this review we have compiled experimental results on the interaction of K + , K , K − , η, η (cid:48) , ω and φ mesons with nuclei, reported in photon-, proton-, and pion-induced reactions and heavy-ion collisions,and compared them to theoretical predictions. We have focused the discussion on the energy regimenear the production threshold where under optimised kinematic conditions the mesons may be so slowthat they could be captured by the nucleus to form a meson-nucleus bound state if there were sufficientattraction. We have confronted the experimental results with theoretical predictions and have foundthat in many cases the experimentally observed in-medium modifications are smaller than theoreticallypredicted.All mesons exhibit a broadening in the nuclear medium and thus a non-zero imaginary potential,ranging from about -13 MeV for the η (cid:48) meson to about -50 MeV for the ω meson at normal nuclearmatter density and low meson momenta. In most of the cases the broadening has been indirectly inferredfrom transparency ratio measurements and the observation of meson absorption via inelastic channelswhich reduce the meson in-medium ”lifetime”. A direct observation of an in-medium broadening hasonly been reported for the φ meson by the KEK E325 experiment [254], described in Section 4.7, andfor the ρ → e + e − [241] , µ + µ − [345] decays of the short-lived ρ meson, not discussed in this review.Regarding the real part of the meson-nucleus potential, there seems to be consensus that the K + andK meson experience a repulsive potential of about 20-40 MeV at normal nuclear matter density, whilethe K − and η (cid:48) mesons feel an attractive potential, leading to a mass drop by about 60 and 40 MeV,respectively. For the η and ω meson no evidence for strong mass modifications has been established.The φ meson is the only case where a mass shift and a broadening has been claimed, however, only inone experiment. It is of utmost importance to verify this experimental finding.The determination of meson-nucleus potentials has paved the way for the search for meson-nucleusbound states, exclusively bound by the strong interaction. These states are of interest from the nuclearphysics point of view as they correspond to highly excited nuclear states with excitation energies ofthe order of several hundred MeV up to 1 GeV. For hadron physics these states provide a uniquelaboratory for studying the properties of mesons and their possible modification in a strongly interactingenvironment at finite nuclear density in a quasi-static configuration.The formation and observation ofsuch states is favoured by a strong real part of the meson-nucleus potential corresponding to a deeppotential well. Furthermore, the modulus of the imaginary part of the potential should be muchsmaller than the modulus of the real part, leading to relatively narrow bound states which do notoverlap and can more easily be distinguished from background experimentally. Mesic states have beensearched for the K − , η, ω and η (cid:48) meson. Experimental searches have so far not provided definitive andunambiguous experimental evidence for such states, although several indications have been reported.Most of the experiments suffer from low statistics or limited sensitivity because of insufficient backgroundsuppression. A new generation of experiments is required where the formation of the mesic state can be77agged and identified and its decay simultaneously measured via a characteristic decay mode. Currentefforts focus on the K − , η , η (cid:48) and φ meson. Results from such experiments planned at J-PARC andGSI/FAIR are eagerly awaited. We would like to thank all collaborators and colleagues in the field of meson-nucleus interactions whohave given helpful advice during the preparation of this review, in particular Wolfgang Cassing, LauraFabbietti, Bengt Friman, Avraham Gal, Satoru Hirenzaki, Kenta Itahashi, Bernd Krusche, PawelMoskal, Eulogio Oset, Laura Tolos and Wolfram Weise. Special thanks go to Ulrich Mosel for hisclose and long-time collaboration and many stimulating discussions on in-medium physics. Some ofthe data on η, η (cid:48) and ω mesons, summarised in this review, were taken together with the colleaguesfrom the CBELSA/TAPS collaboration in Bonn and the A2 collaboration in Mainz. These experimentswould not have been possible without the support from the Deutsche Forschungsgemeinschaft withinSFB/TR16. References [1] H. Gilg et al. , Phys. Rev.
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