aa r X i v : . [ nu c l - t h ] J u l Extraction of baryonia from the lightest anti-protonic atoms
B. Loiseau
Sorbonne Universit´es, Universit´e Pierre et Marie Curie,Sorbonne Paris Cit´e, Universit´e Paris Diderot, et IN2P3-CNRS, UMR 7585,Laboratoire de Physique Nucl´eaire et de Hautes ´Energies, 4 place Jussieu, 75252 Paris, France
S. Wycech
National Centre for Nuclear Research , Warsaw, Poland (Dated: July 6, 2020)Anti-protonic hydrogen and helium atoms are analyzed. Level shifts and width are expressed interms of ¯ p -nucleon sub-threshold scattering lengths and volumes. Experimental data are comparedto results obtained from the 2009 version of the Paris N ¯ N interaction potential. Comparison with1999 version is also made. Effects of N ¯ N quasi-bound states are discussed. Atomic 2P hyperfinestructure is calculated for antiprotonic deuterium and the significance of new measurements isindicated. PACS numbers: 13.75.-n, 36.10-k, 25.80.-e, 25.40.Ve
I. INTRODUCTION
The lightest hadronic atoms offer a chance to test hadron-nucleon scattering amplitudes just below the thresholds.This energy region is of special interest in cases of bound or quasi-bound states in the hadron nucleon systems. Twocases of current interest the ¯ p and K − atoms are similar in this respect. Here we discuss the case of antiproton.Nucleon-antinucleon bound states (baryonia) have been searched for since the beginnings of LEAR era at CERN.Nothing definite was found, apparently due to two factors. First, these states are broad due to fast annihilationprocesses and experiments are confronted with heavy backgrounds. Second, the exclusion principle is not operativeand a large number of partial waves may be formed in the N ¯ N systems.High selectivity is needed to detect a clear signal. One way to reach selected states is a formation reaction.In this way a resonant-like behavior was observed in the decay J/ψ → γp ¯ p . The experiment performed by theBES Collaboration [1] found an enhancement in the ¯ pp close to the threshold energy equal to 1876 MeV. With amodel it was attributed to a sub-threshold peak at invariant mass of 1859 MeV. This was confirmed by the BES IIICollaboration [2], and the extrapolation has led to a peak at 1832 MeV. This may be, but not necessarily is, aconfirmation of a quasi-bound state. The valid proof requires a look into sub-threshold region and this also has beenachieved by the BES Collaboration [3] in the mesonic decays of J/ψ into γπ + π − η ′ . A sub-threshold enhancement,the X(1835), was found in this way.Interpretation, in the J/ψ → γp ¯ p process, of the close to ¯ pp threshold enhancement in terms of N ¯ N potentials [4, 5]leads to the conclusion that there exist a broad spin singlet S -wave quasi-bound state. However, the state obtained [5]by Bonn potential is formed in the isospin I=1. That generated by Paris potential [6] happens in I=0 at the energy-4.8 MeV with a width of 50 MeV, state which we named N ¯ N S (1870) [7]. Paris potential explains the X (1835) MeVenhancement as a result of an interference of this quasi-bound S -wave state with a background amplitude [7].Paris 2009 potential generates also a deeply bound, isospin 1 S -wave state at about -80 MeV binding. This state isfar away from the experimentally tested energy regions and may be an artifact of unknown interactions at very shortranges. Nevertheless, it exists in the model and we find that after some modifications it may play a significant role inthe antiprotonic-atomic physics.There is a class of selective experiments which also allow to test the sub-threshold energy region. These are themeasurements of atomic levels in very light atoms. The condition for a direct partial wave analysis is the resolution ofthe fine structure of these levels. So far, such resolution has been achieved only in the antiprotonic hydrogen [8, 9] andit is instrumental in fixing the N ¯ N potential at low energy [6]. Other atoms useful to study baryonia are antiprotonicdeuterium and antiprotonic helium, as, in these atoms, the experimental data exist. The advantage of such systemslies in their (relatively) simple nuclear structure and the fact that ¯ pN interactions happen on bound nucleons of welldefined separation energy. In this way one can study the ¯ pN interaction below the threshold, moreover at differentsub-threshold energies.In this work we study the level shifts and widths due to ¯ pN interactions in the lightest atoms (levels): H(2 P ), He(2 P , 3 D ) and He(2 P , 3 D ). The number of states is limited to those characterized by very low atomic-nuclearoverlaps. In these states, multiple scattering expansions converge very quickly and double scattering or interactionson two nucleons contribute small fractions of the level shifts. Experiments discussed are fairly old, [9–11], but havebeen analyzed only in a phenomenological way and never discussed in terms of a well established ¯ pN potential.Understanding of the simplest atoms is important for the PUMA project at CERN aiming at studies of antiprotoncapture from atomic states formed on unstable nuclei [12]. Relative frequency of ¯ pp and ¯ pn captures will be measuredwith the intention to determine neutron haloes in unstable nuclei. These nuclei are likely to contain loosely boundnucleons, thus kinematic conditions may be similar to antiprotonic-deuterium and He atoms. Knowledge of the levelwidths structure in these atoms would be helpful for this project. Related experiments were performed on stablenuclei [13]. In general the ratio of captures on neutrons and protons is well controlled, but it has been found thatsome irregularities happen for loosely bound protons [14]. One hopes that the light atom studies will allow to pinpointthe partial wave responsible for these effects and will give constraints on the parameters describing heavy antiprotonicatoms.
TABLE I: The ratios of N (¯ pn ) and N (¯ pp ) capture rates from atomic states. The second column gives the experimental numbersobtained in radiochemical experiments [14]. ¸ atom N (¯ pn ) /N (¯ pp ) Zr 2.6(3)
Sn 5.0(6)
Cd 0.5(1)
Sn 0.79(14)
Few selected ratios of N (¯ pn ) and N (¯ pp ) capture rates from atomic states are presented in Table I. Two normalcases Zr and
Sn indicate neutron haloes. The other two results
Cd and
Sn are anomalous and point toproton haloes which cannot be understood by standard nuclear structure models. Only part of the effect is relatedto a sizable differences ( ∼ N - ¯ N -quasi-bound state that enhances ¯ p - p absorptions over ¯ p - n ones in these nuclei. Existence of a fairly narrow (Γ ≤
10) MeV state in a P wave has beenpredicted in Paris potential models [6] and [15]. Its position is, however, not well determined by the scattering dataand studies of light atoms might resolve this question.In conclusion, we argue that there could well be two quasi-bound isospin 1 N ¯ N states, one in a S wave and another ina P wave. Both are predicted by the recent Paris potential but at incorrect energies. An update of this potential modelis required for understanding of the planned CERN and GSI proposals concerning low energy-antiproton research.This paper is organized as follows. Section II presents a formalism used to calculate the related complex shifts∆ E − i Γ /
2. These are expressed in terms of averages of S - and P -wave N ¯ N scattering amplitudes derived from the2009 [6] and 1999 [15] Paris N ¯ N -interaction potentials and arranged into a sum of multiple scattering series. At thismoment, the fine structures in these levels, are, in general, not resolved and the N ¯ N amplitudes have to be averagedover spin states. Calculated atomic levels using the 2009 and 1999 Paris-potential models are compared to data insection III. This section is also devoted to the extraction of baryonium energies and widths as indicated by the data.Finally we calculate the fine structure of 2 P states in deuterium which may be useful to pinpoint properties of thebaryonia. Detailed expressions of the equations used in our calculation are presented in an appendix. II. FORMALISMA. Relation between level shifts and scattering amplitudes
Experiments which detect the X rays emitted from hadronic atoms provide energy levels shifted in comparison tothe electromagnetic levels by ∆ E due to nuclear interactions. The level widths Γ due to antiproton annihilation arealso provided in this way. For a given main atomic quantum number n and angular momentum L these complexlevel shifts are related to the corresponding hadron-nucleus L -wave scattering parameter A L . The relation is usuallyobtained by expansion in A L /B L where B is the Bohr radius. The scattering amplitudes are measured experimentallyand interpreted in terms of Coulomb plus short-ranged potential solutions. Inner Coulomb corrections are includedinto A L . For S waves such a relation,∆ E nS − i Γ nS / E nS − ǫ nS = 2 πm r | ψ n (0) | A (1 + λ A /B ) , (1)is known as Deser-Trueman formula [16, 17]. Here, ψ n (0) is the atomic wave function at the origin and m r is theantiproton-nucleus reduced mass. Formula (1) is accurate to a second order in A /B , higher terms in this expansion Relation between level shifts and scattering amplitudes Z nuclei. It has has been obtained in the non-relativistic limit. Changes are to be introducedinto the electromagnetic energy ǫ nS which is composed of the Bohr’s atom energy, ǫ n = − m r ( α ) / n , corrected forrelativity, radiative effects and nuclear polarization, α being the fine structure constant. In the case of antiprotons, theweak singularity of the Dirac wave function has to be smeared over the nuclear density but this brings no noticeablechanges.The Bohr radius is given by 1 / ( α m r ). In the 1 S states one has λ = 3 . A ≈ E nL − i Γ nL / ǫ on n Π Li =1 (cid:18) i − n (cid:19) A L /B L +1 (1 + λ L A L /B L +1 ) , (2)is due to Lambert [18]. The second order correction and inner Coulomb corrections for N - ¯ N systems may be foundin Refs. [19, 20]. For 2 P levels λ = 1 .
866 and this correction is negligible. Orders of magnitude of the shifts arerepresented by the coefficients Ω nL defined asΩ nL = ǫ on n Π Li =1 ( 1 i − n ) /B L +1 , (3)which, for the simplest atoms, are given in Table II A. It offers the relation of the level shifts to the scatteringparameters ∆ E nL − i Γ nL / nL A L . (4) TABLE II: Bohr radii and Ω nL coefficients relating the level shifts to the nuclear scattering parameters for some light antipro-tonic atoms. atom B [fm] Ω S [keV/ fm] Ω P [meV/fm ] Ω D [ µ eV/fm ]¯ p H 57.63 0.8668 24.46 0.3591¯ p H 43.247 1.539 77.19 2.012¯ p He 19.22 15.59 3958 522.6¯ p He 18.02 17.77 5125 770.1
The measurements of level shifts are equivalent to the measurements of parameters involved in the scatteringamplitudes f , which, in the low energy expansion, are given in terms of initial k and final k ′ c.m. momenta by f ( k , E, k ′ ) = a ( E ) + 3 k · k ′ a ( E ) . (5)In the scattering on nuclei discussed above we use capital A and A . In the case of ¯ pN scattering we use a and a which at the threshold energy become the scattering scattering length and volume. To specify additionalquantum numbers in the ¯ pN channels we use notation I +1 2 S +1 L J , where S, L and J are the spin, angular and totalmomentum of the pair, and I being its isospin. At this point we remind that A L which enter relations (1) and (2)are due to all short range interactions and contain also the inner Coulomb corrections. With the procedure usedhere, the Coulomb field is due to the target nucleus and Coulomb effects are included in the atomic-wave functions.Hence the basic scattering lengths and volumes a and a for the ¯ p - N systems are calculated without Coulombcorrections. The scattering lengths are defined in the baryon-baryon convention with negative absorptive part. Thus A L = Re A L − i | Im A L | and a bound state close to threshold in S wave is signaled by a large positive Re A . To understand elementary interactions on bound nucleons one needs to know a ( E ) and a ( E ) in the un-physicalregion of sub-threshold energies. A procedure to calculate the relevant extrapolation is given in appendix A. B. Relation between level shifts and sub-threshold amplitudes
Let us consider the antiproton, bound into an atomic orbital, scattering on a nucleon, bound in a nucleus with aseparation energy E s . Atomic levels are calculated, here, in a quasi-three body system as represented in Figure 1, and Relation between level shifts and sub-threshold amplitudes R . Such a model makes sense only for peripheralantiprotons and we limit ourselves to such cases. Three-body Jacobi coordinates are essential and our notation isspecified in Appendix C. We intend to study energy dependence of elementary ¯ pN scattering parameters and it isimportant to know the energy involved in the c.m. of the interacting ¯ pN pair. It is located in the sub-threshold regionpartly due to the bindings, nuclear E s , atomic E a and partly to recoil energy, E r , of the ¯ pN subsystem with respectto the residual nucleus. The values and ranges of the energies in question, E cm = − E s − E a − E r , (6)are indicated in Table III for the simplest first order collisions. Our discussion is limited to cases where the Born termgives a predominant result. The details of the formalism and of the calculation are given in appendices A and C.Below we present the main formulas.The inspection of figure 1 shows that we have to use two related pairs of Jacobi coordinate systems: ( k , p ) pairuseful to describe antiproton-nucleon interactions and another ( k , p ) pair, useful to describe nuclear and atomicwave functions. Correspondingly, in coordinate representations, one has the ( r , r ) pair and the ( r , r ) one. Therelation between these last two pairs is given by r = c r − r , r = γ r + β r , (7)where c = M p M ¯ p + M N , γ = M N ( M ¯ p + M N + M R )( M ¯ p + M N )( M N + M R ) , β = M R M R + M N . (8)In this section, to simplify the notation, we use the two coordinates r ≡ r and ρ ≡ r that correspond to the¯ p - N interaction range and to the N - R internuclear distance, respectively (see figure 1). ρ p r , k R(3) N (2)¯p(1)
FIG. 1: Quasi-three-body system, 1: antiproton , 2: nucleon and 3: residual system. Jacobi coordinates, momentum: p , k and space: ρ, r . Let us consider the S -wave interactions first. The basic ¯ pN interaction potential is V ¯ pN ( E cm , S ) = 2 πµ e T ( r, E cm ) (9)where e T ( r, E cm ) is an off-shell scattering matrix given by Eq. (A2). Here, isospin indices are not specified, µ is thereduced mass in the ¯ pN system. For a while, we go to local zero range limit and obtain V ¯ pN ( E cm , S ) = 2 πµ a ( E cm ) δ ( r ) (10)where a ( E cm ) is the scattering amplitude at a given c.m. energy, a ( E cm ) = Z d r e T ( r, E cm ) , (11) Relation between level shifts and sub-threshold amplitudes S hydrogen atom reproducesDeser-Trueman formula (1) in the leading order, up to inner electromagnetic corrections.If the antiproton is bound into an atomic orbital, the energy shifts of upper levels (or levels of small atomic-nucleusoverlap) are generated by perturbation and in the leading order∆ E nL − i Γ nL / j h ψ L ϕ | V ¯ pN j ( E, S ) | ϕ ψ L i , (12)where the sum over j extends over all nucleons of the nucleus. In general, the right-hand side of Eq. (12) consistsin a complicated 4 dimension integral which reduces to easier integrations due to the simplicity of V ¯ pN i . Relevantmanipulations in momentum space are explained in appendix C. The main purpose of our formulation is to separatethe factor of atomic-nuclear overlap and the calculation of the length a ( E cm ) averaged over the recoil energy. Thewave function ϕ of the struck nucleon is determined in the relative N - R coordinate ρ (see Figure 1). The last pointallows us to use proper asymptotic form of the wave function given by the separation energy. It is important as wediscuss interactions localized at nuclear surfaces. The Coulomb-atomic-wave functions of given angular momentum L , denoted by ψ L , are given in terms of antiproton-nucleus coordinates. The detailed development of formula (12)which, requires re-couplings of two basic Jacobi coordinate systems, is discussed in the appendix.The recoil energy is given by E r = p / M where p is the total momentum of ¯ p - N pair and M = µ R, ¯ pN is thereduced mass of this pair and of the residual nucleus R . As p is not a good quantum number, formula (12) involvesan integral over p . In order to obtain an intuitive picture one has to make calculations in momentum space. First letus define an average over recoil energy as¯ a = Z a ( − E s − E a − p M ) | e F L ( p | d p R | e F L ( q ) | d q . (13)The extent of the recoil energies is determined by e F L ( p ) which is calculated in appendix C and given by the formula(C4). It turns out to be a Fourier transform of the atomic and nuclear overlap F L ( ρ ) = ϕ ( ρ ) ψ L ( β ρ ) (14)with β given in Eq. (8). It reflects the fact that atomic wave function is given in the R + N center of mass systemand nuclear wave function depends on the relative R - N coordinate. The level shifts are then expressed in terms ofaveraged scattering length and an integral over the overlap∆ E nL − i Γ nL / πµ X j ( a ) j Z d ρ | F L ( ρ ) | (15)The energies involved in Eq.(13) cover some un-physical sub-threshold region. The separation, average recoilenergies and the spread of recoil energies are given in Table III. These values and the separation energies indicatethat three atoms H, He, He cover disjoint energy regions from about -5 down to -40 MeV. Therefore, a model isrequired for the sub-threshold extrapolation.
TABLE III: Separation energies and recoil energy ranges (in MeV) entering antiproton-nucleon amplitudes involved in theleading interaction terms. The upper line specifies the atomic states in the lightest antiprotonic atoms. The two numbersin each column give the minus separation energies, and the minus average recoil energies, − E r . Numbers in curly bracketsindicate the approximate widths of the recoil energy distributions. The averages of proton and neutron energies are roundedup to 0.1 MeV. Atom \ State 1 S P D ¯ p H 0 0 0¯ p H -2.2-8.9 { } -2.2-5.4 { } -2.2-3.0 { } ¯ p He -6.6-11.0 { } -6.6-8.9 { } -6.6-7.3 { } ¯ p He -20.7-13.8 { } -20.7-13.7 { } -20.7-13.8 { } Antiprotonic deterium atoms P -wave-antiproton interactions we use another pseudo-potential V ¯ pN ( E, P ) = 2 πµ a ( E cm ) ←−∇ δ ( r ) −→∇ (16)where a ( E cm ) is the scattering volume at a given c.m. energy. Here, spin indices are not specified. The calculationsof the recoil energy and overlap integrals are performed in similar manner as for S waves and details are given inthe appendix C. Averaging over recoil is done as in Eq. (13), the weighting function being the same. Formulas foroverlaps are more involved and may also be found in appendix C. III. RESULTS OBTAINED WITH PARIS POTENTIAL MODEL
The lowest twelve states in the two H and He lightest elements are listed in Table II A where the appropriate overlapΩ nL coefficients are presented. The standard X-ray techniques allow a direct determination of the “lower” line shapesand the extraction of the “upper” level width by the intensity loss. The experimental data from deuterium and heliumconsists of 3 level shifts and 5 level widths, see Tables IV, V and VII. The results from hydrogen have already beenused to fix the low energy parameters of the Paris 2009 potential [6]. Here we discuss only 2 P shifts and 2 P, D widths which can be described by a corrected first order perturbation.The Paris potential is a semi-phenomenological model used to describe nucleon-antinucleon scattering. It is based onfitting some 4300 data. Two versions are discussed, Paris 1999 [15] built to describe ¯ pp elastic and inelastic scatteringdata and an updated version, Paris 2009 [6] including also np scattering and antiprotonic-hydrogen data. Withrespect to sub-threshold properties these two versions differ in the position of P quasi-bound state. Correspondingenergies and widths are ( E B , Γ /
2) = ( − . ,
9) MeV for Paris 09 and ( E B , Γ /
2) = ( − , .
5) MeV for Paris 99. Also,Paris 09 predicts an S quasi-bound state at energy E = − . S wave bound statein Paris 99. Another sizable difference is a ( P ) = 7 . − i . for Paris 2009 and 8 . − i .
12 fm for Paris 1999.The preference of a ( P ) cannot be definitely established [6] by the hydrogen data [9], which offers a sum of three finestructure lines.These differences could possibly be detected in terms of 4 P / component in the deuteron [see below Eq. (25)].However, Table III indicates that fine structure in the 2 P states in deuterium tests a narrow energy region around-7.6 MeV below the threshold not too far from the position of P quasi-bound state predicted by Paris 09. If thisresonance position is correct it will dominate the 4 P / component thus reducing chances to check a ( P ). A. Antiprotonic deterium atoms
The calculated and experimental shifts and widths are compared in table IV. The results indicates P -wave dominanceand strong sensitivity to the P quasi-bound state energy, see figure 2 for a visualization. Paris 09 solution yields anunacceptable attractive level shift and an excessive level width. That is due to strong overlap of the quasi-bound statewith the E cm region characteristic for the deuterium. It is evident that bound state position of -4.8 MeV given by theParis 09 model should be pushed down to at least to the [ − , −
8] MeV region which is below − . pN c.m. energy in the 2 P atomic state, see Table III. This would reduce overlap of the c.m. energies with thequasi-bound state reducing the excessive width of the 2 P atomic level. In addition, locating the quasi-bound statebelow the ¯ pN energy will produce the missing repulsion to the 2 P level. On the other hand the P quasi-boundstate generated at -17 MeV by Paris 99 model is too far from the ¯ pN c.m. energies and the absorption width ofthe 2 P level becomes too small. Hence, the experimental atomic levels in deuterium suggests the proper positionfor the P quasi-bound state to be located in between the two Paris solutions. Similar conclusion follows fromTable I indicating anomalies of neutron haloes obtained in the radiochemical measurements. In the two controversialnuclei Cd and
Sn one finds proton separation energy S ( p ) = 7 .
35 and 7 .
55 MeV and neutron separation energy S ( n ) = 10 .
87 and 10 .
79 MeV, respectively. In comparison to other tested nuclei, these two, present a sizable differencein the neutron and proton separation energies. These differences are comparable to the P quasi-bound state width.Neutron haloes, tested by the radiochemical method, involve extreme nuclear surfaces and the antiproton capturesoccur predominantly on the valence nucleons. Characteristic ¯ pn energies in their c.m. systems are about -16 MeV andcorresponding ¯ pp c.m. energies are about -11 MeV. At these c.m. energies Paris 09 potential yields relative capturerates R n/p ≃ . S and P waves in the ¯ pN systems. Actually, that value is commonly used in the analysesof the radiochemical experiments [14] but it leads to the anomalous proton haloes in Cd and
Sn nuclei. Theseanomalous results may be understood if the the resonant position is close to the ¯ pp c.m. energies and far from the Antiprotonic deterium atoms pn c.m. energies. Such situation may favor ¯ pp captures imitating proton nuclear haloes. To meet this, the P stateshould to be located in [-11, -9] MeV segment. TABLE IV: Results for 2 P -deuterium level corrections in eV calculated with the spin averaged amplitudes of the Paris 2009potential. The second and third column numbers are the real and twice the imaginary part of the average complex shift ∆ (27).Numbers in curly brackets are results with the Paris 1999 potential.order Shift Width S wave 100 { } { } P wave -9 { } { } Sum 91 { } { } Data [8, 9] 243 ±
26 489 ± B. Helium atoms
Experiments measure lower 2 P and and upper 3 D atomic levels. Bulk of the interaction happens in a low densityregion, although Helium nuclei have no real surface region. Multiple scattering is negligible in the 3 D states but sizablein the 2 P states. What matters most is a proper single nucleon wave function. Here, we use Eckart functions [21, 22]which have asymptotic given by separation energies E s and a phenomenological form at short distances ϕ ( r ) = [1 − exp( − e βr )] exp( − e αr ) r , (17)where e α = p µ R,N E s . The short distance parameter e β is extracted from electron or pion scattering experimentsand fixed to reproduce the zeros of charge form-factors. Thus for He, e α = 0 . , e β = 1 .
753 [23, 24], and for He , e α = 0 . , e β = 1 .
20 [22], all in fm − units. TABLE V: Leading order calculations in eV for 2 P and in meV for 3 D (widths only) level corrections in He obtained with thespin averaged amplitudes of the Paris 2009 potential. Numbers in curly brackets are obtained with the Paris 1999 potential.2P shift 2P width 3D width S wave 6.68 { } { } { } P wave -6.36 { } { } { } Sum 0.31 { } { } { } Data [11] 17 ± ± ± . Reference [25] finds (with a different N ¯ N potential) a 2% higher order corrections to the 2 P levels in deuterium.Such corrections are of the same magnitude with the Paris potentials and would not change our conclusions. Onthe other hand in Helium we find sizable multiple scattering corrections. A method to sum the multiple scatteringexpansion series is presented in appendix D. Here we present results. TABLE VI: As in Table V but only for 2 P level including higher order corrections.2P shift 2P width S wave 6.64 { } { } P wave -5.21 { } { } Sum 1.43 { } { } Data [11] 17 ± ± Helium atoms a and second order a which dependson the signs of real parts. The average scattering lengths yield repulsive shifts ( Re a >
0) but scattering volumesgenerate attraction (
Re a <
0) below the quasi-bound state and repulsion (
Re a >
0) above this state. Thus theinterference pattern of the single scattering and double scattering terms depends on the ¯ pN c.m. energy.Paris 99 model is fairly consistent with the Helium data while Paris 09 misses the repulsion in the 2 P atomic state.The downward shift of P required by the deuteron levels does not remove the inconsistency. Apparently there isanother source of the difficulty and the He atoms indicate a new possibility.
TABLE VII: As in Table V but for the He atoms.2P shift 2P width 3D width S wave 9.72 { } { } { } P wave -9.01 { -10.4 } { } { } Sum 0.708 { } { } { } Data [11] 18 ± ± ± . He atoms.2P shift 2P width S wave 8.92 { } { } P wave -8.70 { -10.9 } { } Sum 0.22 { } { } Data [11] 18 ± ± Tables VII and VIII compare first order and higher order results, respectively. Both fail to reproduce the repulsivelevel shifts. Difficulties of the Paris potential rise with the increasing distance from the nucleon-antinucleon threshold.The He atom presents an extreme case which is not matched even in heavy anti-protonic atoms. Because of theperipheral nature of nuclear antiproton captures the nucleon separation energies as high as 21 MeV are met very rarely.In some sense the He atom is located on the boundary of unknown N ¯ N sub-threshold c.m energies. Many modelsof N ¯ N have predicted deeply quasi-bound states which historically have been viewed with scepticism as nothing hasbeen determined in direct experiments. Furthermore the meson exchange interactions are not very reliable at shortdistances. The Paris 09 potential also generates such a deep and broad state at about -80 MeV in the isospin 1state. Visualization of the related amplitudes may be found in reference [26] which shows that this state dominatesthe S -wave sub-threshold scattering in this deep region. Now, the difficulty in the understanding of He atomic datawould be solved immediately if the position of this quasi-bound state is shifted up by some 20 MeV. Both, the missingrepulsion and somewhat weak absorption strength will reach consistency with the data. C. The R n/p ratios Different but related experiments had been performed in the early LEAR era. These studied captures of stoppedantiprotons in deuterium and helium chambers and measured relative rates of antiproton captures by neutrons andby protons. Such ratios are essential for studies of the neutron haloes in heavier nuclei. Table IX gives R n/p , the ratioof basic ¯ pn capture rate to ¯ pp capture rate. It indicates consistency of Paris 09 potential with the R n/p results in He and He. The R n/p has been calculated in 2 P atomic state that is likely to be the state of capture for 70% of allantiprotons that reach low level atomic states [27]. These findings are also correct if a sizable fraction of antiprotonabsorption occurs from higher atomic nP states.On the other hand, the deuterium calculations given in table IX have no direct relation to the data as the captureis likely to happen from many nS atomic states. The nS results put into brackets are given by the Paris potentials,but calculated under the assumption that captures in nS deuterium levels happen predominantly in the ¯ pN S waves.This assumption is based on calculations of Ref. [25]. Here, we are unable to offer the atomic cascade details and The R n/p ratios nP states relative to captures from the nS atomic states. Nevertheless, Table IXallows the conclusion that Paris 09 sets proper limits on the R n/p in deuterium too. Results with Paris 99 are alsosatisfactory although too high for the ¯ p He and ¯ p He atoms.
TABLE IX: The R n/p , ratios. Second column gives experimental results from antiprotons stopped in bubble chambers. Thirdand fourth columns give the ratios calculated with the two versions of the Paris potential. It is assumed that capture occursfrom nP atomic levels. The experimental ratio in He Ref. [30] is given per target 0 . nS states are given into brackets.atom Experiment Paris 09 Paris 99¯ p H [28] 0.81(3) 1.09 h . i h . i ¯ p H [29] 0.749(18) 1.09 h . i h . i ¯ p He [30] 0.70(14) .65 1.00¯ p He [30] 0.48(3) .48 0.59
D. Hyperfine structure
All N ¯ N potentials are strongly spin dependent. That generates hyperfine structure of atomic levels built by stronginteractions and superposed on the electromagnetic structure. Below we present the isospin and spin structure of thebasic amplitudes involved in the iso-spin and spin states.The isospin structure is given by A (¯ pp ) = [ a ( I = 0) + a ( I = 1)] / A (¯ pn ) = a ( I = 1) . (18)For 1 S atomic level, deuteron spin 1 adds to antiproton spin 1/2 to total spin doublet and total spin quartet states.For these, the appropriate amplitudes for S-wave antiproton-deuteron interaction are denoted A ( λS e J ) where λ = 2 , e J denotes total angular momentum of the three particles. The notation for nucleon-antinucleon pair has been defined in Sec. II A, viz. I +1 2 S +1 L J . The antiproton interacts on the neutron and protonof the deuteron and we just add both interactions and in the notation used below, in Eqs. (20) to (26), we suppressthe upper isospin subscript and, following Eq. (18), we define, a ( S +1 L J ) = 12 a ( S +1 L J ) + 32 a ( S +1 L J ) (19)One has A (2 S / ) = 34 a ( S ) + 14 a ( S ); A (4 S / ) = a ( S ) . (20)For P -wave interactions in 1S state one has expressions similar to those of Eq. (20), A (2 P / ) = 34 a ( P ) + 14 a ( P ); A (4 P / ) = a ( P ) . (21)In 2 P atomic levels, the amplitudes given by Eq. (18) and (19) are mixed in the total e J = 1 / , / , / S wave interactions is minute and the main effectcomes from the P wave antiproton nucleon interactions. For the two doublet and three quartet states in the 2 P atomic levels, one requires additional terms given by P wave N ¯ N amplitudes. In doublet the combinations are A (2 P / ) = (cid:2) a ( P ) + a ( P ) + 2 a ( P )] (cid:14) , (22) A (2 P / ) = (cid:20) a ( P ) + 12 a ( P ) + 52 a ( P )] (cid:30) , (23) Hyperfine structure A (4 P / ) = 23 a ( P ) + 13 a ( P ) , (24) A (4 P / ) = 56 a ( P ) + 16 a ( P ) , (25) A (4 P / ) = a ( P ) . (26)The hyperfine structure level shifts and widths fullfil the relation∆ = P k =1 (2 e J k + 1)∆ k P k =1 (2 e J k + 1) , (27)where the complex shifts ∆ k are given in Table X. The summation extends over all five states specified in equations(22) till (26) and involves both S and P wave interactions. We find that the average ∆ differs by less than 8 % fromthe shift obtained by the spin averaged ¯ pN scattering parameters displayed in Table IV. The numbers presented inTable X do not contain electromagnetic splitings which are also of the order of a hundred eV [8]. TABLE X: Hyperfine structure splittings, ∆ k = ǫ k − i Γ k / P states in deuterium.Paris 09 Paris 99k state ǫ k Γ k ǫ k Γ k P / - 21 670 204 4592 2 P /
33 203 101 3233 4 P /
81 282 89 1624 4 P /
241 1250 337 6305 4 P /
54 620 79 344
The experiment gives essentially no hyperfine structure splitting but offers a good check on the widths controlled bydirect line structure and the X-ray intensity loss. Both Paris potential generate splitting which is narrow in relationto the level widths but built at an incorrect average level shift (see Table IV).One level, the 4 P / is singled out and located separately. Being very broad it can also be missed in the spectrum.It reflects strong overlap of ¯ pN c.m. energy with the quasi-bound state spectral density. As discussed already, adownward, few MeV shift of the quasi-bound state, would still enlarge the width and reduce attraction removingpractically the fine structure splitting. It must be added that this quasi-bound state shift will also massively repulsethe average shift given in Table IV as required by the data. The scenario discussed here is plotted in Fig. 2.On the other hand Paris 99 gives a group of five lines that seem close to data both in terms of cumulated widthsand average shift. In conclusion we see that the data are not consistent with the -4.5 MeV P quasi-bound statebut may tolerate this state if its binding is larger. IV. CONCLUSIONS
Anti-protonic atomic levels characterized with very small nuclear-atom overlap are a powerful method to studyantiproton-nucleon amplitudes below antiproton-nucleon threshold. In this way one is able to study structure of theamplitudes in the region reaching down to some -40 MeV below the threshold.Comparison of the level shifts and widths in deuterium and helium atoms with the predictions of Paris potentialmodel yields following conclusions. • The Paris 2009 potential offers S -wave antiproton-nucleon amplitudes dominated, just below the threshold, bya broad quasi-bound state in S wave. Interactions in this waves generate repulsive atomic which are stronglyrepulsive too. On the other hand, the P -wave interactions are attractive, on average, unless there exist very deeplybound P wave quasi-bound states which does not happen in both models. The balance between S -wave repulsion and1 FIG. 2: Subthreshold amplitudes generating the 4 P / hyperfine structure component in deuterium. With Paris 09 solution thisamplitude is strongly dominated by the resonant a ( P ) amplitude. Relevant ¯ pN c.m. energies fall in the region − . ± P -wave attraction gives indication on the uncertain position of the P -quasi-bound state predicted by two Parispotential models. Alas, the repulsion due to S wave is not strong enough to explain the experimental, stronglyrepulsive, levels. The strongest discrepancy happens in He atom where the energies in the ¯ pN subsystems reachthe lower limit of about -40 MeV. The data require a strong repulsion which is not given by the model and someenhancement of the absorption strengths predicted by the model. All this demands a new phenomenon in the regionbelow -40 MeV. It, in fact, exists in the Paris 09 model and is given by the isospin 1 quasi-bound state predictedat -80 MeV. Consistency with the data requires a shift of this state up to about -60 MeV. • The level shifts in light atoms require some enhancement of nuclear repulsion. This observation is contrary to theone found in medium and heavy atoms where an attractive antiproton optical potentials are required. That happenswith phenomenological potentials [31] and optical potentials based on Paris ¯ pN amplitudes [32]. The scenario inlight and heavy nuclei differ as lower levels in heavier atoms involve higher nuclear densities and the resulting levelshifts are far from the Born approximations. Nuclear many body effects matter. It is well understood that exclusionprinciple operating in nuclear matter may dissolve or push up the ¯ pN quasi-bound state. This may induce the missingattraction. Reference [32] includes this effect on a Fermi gas model with Fermi momentum K f ∼ ρ / but finds it notstrong enough to bring agreement with the data. A more extreme approach was studied in references [33, 34] wherea different form of K f was used and a kind of effective self-consistent antiproton nucleon scattering amplitude T eff was calculated. That may generate required attraction but the price to pay is a sizable uncertainty in the methodused to calculate T eff at the nuclear surfaces. It is clear that the problem is not straightforward. In this work we tryto check the basic ¯ pN subthreshold amplitudes in a simple situation. Studies of heavier atoms and the physics of theinvolved nuclear surface structure would come next. • The P -wave antiproton-nucleon interaction of Paris potential generates a quasi-bound state in P state. Itsposition is unstable and occurs at −
17 MeV in the potential version Paris 99 and at − . H , He atomic levels, and understanding of the R n/p anomalies seen in radiochemical experiments require the state to be located in the [-11, -9] MeV region. • Older data related to bubble chamber measurements of relative np and pp capture rates R n/p are consistentwith Paris 09 in all elements. This reflects a fair success of the model in description of the average sub-thresholdannihilation rates. • The P quasi-bound state dominates the fine structure 4 P / component of 2 P atomic state in the anti-protonicdeuterium. Unfortunately it is likely to be very broad. Measurement of the 4 P / fine structure would be valuable2and would fix the energy of P quasi-bound state.To summarize: there is a strong indication that Paris 2009 may offer successful description of atomic, bubblechamber, and radiochemical data provided the position of P -wave baryonium is shifted down by about a few MeVand the deeply bound S state is pushed up by some 20 MeV. That requires an update of this potential model andthe work is in progress. Acknowledgements
We wish to thank Jaume Carbonell for helpful discussions and Detlev Gotta for comments and advice on experi-mental side. This study was initiated by Alexandre Obertelli and PUMA project at CERN. The collaboration wassupported by French-Polish COPIN agreement No 05-115.
Appendix A: Half-off-shell N ¯ N interactions For atomic calculations one needs the off-shell extension of the scattering amplitude in the energy as well as in themomentum variables. The most general extension for S waves is given by T ( k, E, k ′ ) = µ N ¯ N π Z ψ o ( r, k ) V N ¯ N ( r, E )Ψ( r, E, k ′ ) r dr, (A1)where Ψ( r, E, k ′ ) is the full outgoing wave calculated with the regular free wave ψ o ( r, k ) = sin( rk ) / ( rk ). The normal-ization factor is chosen to produce T ( k, E, k ′ ) equal to the scattering lengths. In equation (A1) the momentum k ′ isnot related to the energy E . The Fourier-Bessel double transform of T ( k, E, k ′ ) would generate a nonlocal e T ( r, E, r ′ )matrix in the coordinate representation. Such an involved calculations do not seem necessary as the experimentaldata are not that precise. We resort to a simpler procedure, standard in nuclear physics, and for an application inthe antiproton physics see Ref. [33]. The sub-threshold scattering amplitudes are calculated in terms of an effective e T ( r, E ) matrix defined in the coordinate representation by e T ( r, E ) = µ N ¯ N π V N ¯ N ( r, E ) Ψ( r, E, k ′ ( E )) ψ o ( r, k ′ ( E )) , (A2)with k ′ ( E ) = p µ N ¯ N E . In this equation Ψ( r, E, k ′ ( E )) is the solution of the Lippman-Schwinger equationΨ = ψ o + G + V Ψ or of an equivalent Schr¨odinger equation. To avoid confusion in the notation we specifythe Ψ in the simplest S wave state. It is the radial scattering solution regular at the origin which behaves asymptot-ically as Ψ( r, E, k ′ ( E )) ∼ i k ′ r [ e − ik ′ r − Se ik ′ r ] (A3)where S = exp( i δ ) is the scattering matrix. The e T ( r, E ) is a local equivalent of the nonlocal T matrix in thesense that its matrix elements fulfill the relation (obtained from Eqs. (A1) and (A2 with k ′ replaced by k ′ ( E )) T ( k, E, k ′ ( E )) = R dr r ψ o ( r, k ) e T ( r, E ) ψ o ( r, k ′ ( E )) valid in a narrow sub-threshold region where the last integral isconvergent. For positive energies this equation is not practical due to zeros in the denominator of Eq. (A2) whichoccur at multiplicities of k ′ = π/r . One could nevertheless use it for k ′ < π/r max where r max is the distance at whichthe potential is cut-off. In this narrow region one finds T ( k, E, k ′ ) = Z ψ o ( r, k ) e T ( r, E ) ψ o ( r, E, k ′ ) r dr, (A4)thus, in this case, the e T ( r, E ) reproduces the half- and full-off-shell amplitudes, allowing to obtain a numericalcontinuity in calculations. One needs also the Fourier transform of e T ( r, E ), T ( κ, E ) = Z d r e T ( r, E ) sin ( κr ) κr , (A5)3and the relevant S -wave scattering amplitudes is given by a ( E ) = T (0 , E ) = Z d r e T ( r, E ) . (A6)The main step is to use equation (A2) for negative energies with V N ¯ N ( r, E ) given by the Paris model. It is used inthe Schr¨odinger equation to calculate Ψ( r, E, k ′ ( E )). Formula (A2) requires numerical care and cutting the potentialtail at very large distances is recommended. This point, related to the question of relatively long ranged pion-exchangeforces particularly strong in the nucleon-antinucleon systems, is discussed below. Appendix B: The N ¯ N force range effects With no knowledge of the full off-shell T ( r ′ , E, r ) matrix, one cannot utilize the power of Faddeev 3-body equations.On the level of pseudo-potential method, used here, the question is: what is a better description, either the localzero-range potential of Eq. (10) or a type of ‘folded” potential given by Eq. (9), with the range given by the half-offshell e T ( E, r ). We tried both with the result that the folded potential with the range involved within e T ( E, r ) is notacceptable. In particular the range in the absorptive part in e T ( E, r ) is fairly long due to joint impact of short rangeeffects and long range pion exchange. The mean square radius of this range defined by R ms = vuuut Im hR d r e T ( r, E ) r i Im hR d r e T ( r, E ) i (B1)amounts to about 1 . . P states, the range effects are moderate and both Eqs. (10) and (9) yield comparable results.However, the range involved in e T ( E, r ) makes tremendous difference in the upper 3 D level widths given predominantlyby R ms . In those levels, the widths depend on the well known asymptotic of nuclear wave functions and are givenessentially by the first order scattering terms. The experimental data data rules out folded potential based on the e T ( r, E ) by several standard deviations.Similar result was found in references [33, 34] where a simple two term separable potential for the N - ¯ N interactionsis studied with some form-factors v ( r ) representing effects of pion exchange. The full off-shell T matrix involves terms T ( r ′ , E, r ) ∼ v ( r ′ ) t ( E ) v ( r ) where the t ( E ) matrix is based on the Dover-Richard N ¯ N potential. Such terms in thefew-nucleon or nuclear systems yield effective interaction range, determined by v ( r ) , roughly one half of the rangegiven by the half off-shell e T ( r, E ).We conclude that localised potential (10), with range effect hidden in a ( E cm ), is more realistic in the overallapproach and it was used in the calculations. However, with future N - ¯ N potentials, of properly fixed quasi-boundstates positions, it is advisable to use fully off-shell input which will adapt itself better to states of different atomicangular momenta. Appendix C: Averaging over recoil energy
Following Sec. II B we consider the 3-body system depicted in Fig. 1 and denote the antiproton, the struck nucleonand the residual nucleus as particles 1, 2 and 3, respectively. We adopt the Jacobi coordinates k ij , p m ( i, j, m = 1 , , i < j ) and in coordinate representation we use r ij , r m , but to simplify the notation, we specify only the two coordinates r and ρ shown in the figure.For S -wave zero-range interactions the relevant antiproton-nucleon operator in momentum space is b a = a (cid:18) E B − p µ , (cid:19) δ ( p − p ′ ) , (C1)where µ , is the reduced mass of the 12 pair and particle 3. With the wave functions e ϕ ( k ) and e ψ ( p ) one has thefirst order expectation value4 h b a i = Z d k d p d k ′ d p ′ e ϕ ( k ) e ψ ( p ) a (cid:18) E B − p µ , (cid:19) δ ( p − p ′ ) e ϕ ∗ ( k ′ ) e ψ ∗ ( p ′ ) . (C2)In order to exploit the δ ( p − p ′ ) function, we use another pair of Jacobi coordinates p = − k − c p ; k = − β k + b p , (C3)where b = 1 − cβ .Now we define a function e F f F ( p ) = Z d k e ϕ ( − β k + b p ) e ψ ( − k − c p ) , (C4)which allows to present expectation value h b a i as h b a i = a Z d p | e F ( p ) | (C5)where a = Z d p a (cid:18) E B − p µ , (cid:19) | e F ( p ) | / Z d p | e F ( p ) | . (C6)This corresponds to sub-threshold value of the scattering length averaged over some region of recoil energies. The lastexpression becomes more intuitive in the coordinate representation. One obtains Fourier transform of e F ( p ) expressedby F ( ρ ) = ϕ ( ρ ) ψ ( a ρ ) , (C7)i.e. it is an atomic nucleus overlap. Its Fourier transform determines the distribution of the N ¯ N -pair momentumrelative to the residual nucleus.For P -wave antiproton-nucleon interactions the relevant operator in momentum space is b a = a (cid:18) E B − p µ , (cid:19) δ ( p − p ′ ) k · k ′ . (C8)The steps from Eq. (C2) to Eq. (C7) may be repeated. As before the operator b a involves only internal coordinatesin the N ¯ N subsystem and the average over recoil is given again by formulas similar to Eqs. (C4), (C5) and (C6). Theoverlap formula is more involved and in the coordinate representation one obtains (see the definition of γ in Sec. II B) h b a i = a Θ ≡ a Z d ρ ∂ r [ ϕ ( − r / − ρ ) ψ ( γ r − β ρ )] | r =0 · ∂ r [ ϕ ∗ ( − r / − ρ ) ψ ∗ ( γ r − β ρ )] | r =0 . (C9)The overlap may be calculated in spherical coordinates using radial and transversal derivatives ∂ r , ∂ T . For S - wavenucleons only radial derivatives apply, for atomic wave functions one needs to calculate both derivatives. Mixedatomic-nuclear derivative contributes only to the radial term.Atomic contribution obtained with ψ ( ρ ) = N ( n, L ) Y mL ρ L exp( − ρ/nB ) becomesΘ A = Z d ρ π ( γ ) (cid:20) L + 12 L + 1 ( 1 nB ) + L + 12 L + 1 ( 2 L + 1 ρ − nB ) (cid:21) (cid:20) dψ r ( βρ ) dβρ (cid:21) ϕ r ( ρ ) (C10)Θ N = Z d ρ π (cid:18) dϕ r ( ρ ) dρ (cid:19) (C11)where ψ r ( βρ ) and ϕ r ( ρ ) are radial parts of the atomic wave functions , Θ ≃ Θ A + Θ N while the mixed term is small.5 Appendix D: Summation of multiple-scattering series
The perturbation expansion for the level shift is, up to Coulomb corrections, equivalent to multiple scatteringexpansion. This follows from Trueman formula [see Eqs. (1) and (2)]. The method of summation used here is basedon simple formula for meson scattering on two nucleons fixed at a distance R obtained by by Brueckner [35]. Thescattering length on such a pair becomes A = 2 a a/R . (D1)The term 2 a/R in the denominator sums the multiple scattering series. It has very simple interpretation, 2 is thenumber of scatterers, a is the scattering amplitude on a single nucleon and 1 /R is the propagator for the mesonbouncing off the fixed nucleons of the scattered particle. Expansion of the expression (D1) in terms of 2 a/R gives themultiple scattering series which is a geometric series. Equation (D1) is correct also for large 2 a/R when the multiplescattering expansion is divergent. This simple result indicates a possibility to sum the multiple scattering series bycomparing it to a geometric series and correcting the sum at every step of the expansion. Let us present it as a seriesfor the solution of a pseudo potential built as a multiple-scattering expansion with a potential given by Eq. (10). Wedenote single scattering term by h V i = h ψ ϕ | Σ i πµ a i | ϕ ψ i . (D2)The double scattering term involves G a propagator of the whole system in-between the subsequent collisions h V GV i = h ψ ϕ | [Σ i πµ a i ] G [Σ j πµ a j ] | ϕ ψ i (D3)and similarly for higher order terms. The first order sum is h V i ′ = h V i − h V GV ih V i . (D4)Expanding the denominator into a geometric series yields first two terms of the multiple scattering. To have exacthigher orders, one corrects the denominator. In this way one obtains a series in the denominator. Up to third order h V i ′′ = h V i C + C + C + .. (D5)where the expansion coefficients are C = −h V i / h V GV i , C = h V GV i − h V GV GV i / h V i and C = h V GV i h V i + 2 h V GV GV ih V GV ih V i − h V GV GV GV ih V i . (D6)For illustration, we present the case of a S -wave projectile scattering on N nucleons bound in S waves in the nuclearc.m. system. The wave function is a product of Gaussian wave functions yielding a density radius mean square R .At zero projectile energy the propagator is taken to be G = − m π | r − r ′ | | ϕ ih ϕ | (D7)where m is the reduced mass of the projectile and of the nucleus, while | ϕ i is the nuclear wave function. The termsof expansion may be calculated in an analytic way, and the expansion parameter in this case becomes x = N a mRµ , (D8)6where µ is the reduced mass of the projectile and of a nucleon of the nucleus. One obtains C = − . x , C = − . x and C = 0 . x . For light nuclei, like helium, the series converges rapidly. For the 2 P states, thisconvergence is much faster, the second order formula (D4) is sufficient for the present experimental precision.Formula (D4) was checked against a full three-body calculation for low energy η -deuteron scattering [36]. It givesprecision of a few % in a very demanding case of virtual S state close to threshold. As the deuteron is a looselybound object one needs a better propagator allowing also for the deuteron excitation to continuum. Helium is amuch stronger bound system and for the 2 P states such corrections are not necessary. What is necessary is a betterdescription of the nuclear wave function ϕ . This is achieved here by the 3-body model of interaction and the properasymptotic form of the wave function (17). In this model the multiple scattering is a series of V GV R + ... where V R is the potential for the antiproton interaction with the residual system. For S waves we take it as V R = 2 πµ Σ N − i a i δ ( r ¯ p − r i ) . (D9)Thus, in this approach, the second order scattering term becomes h V GV i = − m π (cid:20) πµ (cid:21) Σ Nj Σ N − i a i a j Z Z d r d r ′ ψ ( r i ) ϕ ( r i ) ϕ ( r ′ j ) ∗ | r i − r ′ j | ψ ∗ ( r ′ j ) (D10)and similar formula is used for the P -wave interactions. We find the mixed sequence S - following P -wave scatteringterms small. Hence, the summation via formula (D4) is used separately for S S and
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