Pionization experiments in light nuclei

Introduction

Antiprotonic atoms and very low-energy scattering of antiprotons off nuclei offer a number of advantages for nuclear and low-energy particle physics:

First, the studies of atomic level shifts and widths allow us to test hadron-nucleon scattering amplitudes below the threshold, as one or both particles are bound. In this way one could trace exotic baryonia (quasi-bound states in the two-body systems). Second, atomic level shifts and detection of atomic decay modes of these systems are well established methods to study the structure of nuclear surfaces. Third, the nuclei and in particular light nuclei allow to test antiproton-neutron interactions which complement the knowledge of isospin dependence of $N\bar{N}$ forces.

In this work we concentrate on a specific form of antiprotonic studies consisting in measurements of charged pion spectra created in nuclear annihilation of antiprotons (“pionization”). The purpose of old and ongoing experiments has been the detection of neutron haloes in nuclei. A pioneering experiment on Carbon target measured the total charge $Q$ of all $\pi$ mesons which managed to leave the nucleus [1]. The charge distribution $P[Q]$ was determined and the usefulness of the method for nuclear studies was proven. Subsequent annihilation experiments in a freon chamber [2] found evidence of neutron halos in heavier nuclei. A ratio of antiproton annihilation on neutrons versus annihilation on protons in Carbon nucleus was found in a hydrogen chamber [3].

Nowadays, a modern version of this type of experiments incorporating an electronic detection is prepared at CERN in PUMA project [4]. It will study pionization in antiprotonic atoms of radioactive nuclei.

The early experiments were not analyzed in terms of nuclear structure. The necessary knowledge of $\bar{p}N$ and $\pi N$ interactions and the properties of antiprotonic atoms did not exist at that time. There were, and to some extent there are still, two specific questions :

1. the ratios of $\bar{p}n$ and $\bar{p}p$ annihilation rates $R_{n/p}$

2. the atomic states of antiproton from which nuclear captures happens

Only with this knowledge, the nuclear region where the antiproton capture happens may be determined, and the neutron halo be quantified. Below we present briefly the status of current knowledge both experimental as well as theoretical. While the experiments in middle and heavy nuclei are well understood, the captures in light nuclei present problems. We try to understand these, check several theoretical models of $\bar{N}$-$N$ interactions and suggest some improvements. Reliable $\bar{N}N$ potentials are needed and the forthcoming experiments should be helpful.

Apart from the pionization experiments, the majority of low-energy antiproton experiments involve X-ray measurements which yield atomic level width and shifts due to nuclear interactions of $\bar{p}$. These are related to subthreshold $\bar{p}-N$ amplitudes. A global analysis of the data is successful on a phenomenological level [6] but an analysis in terms of a theoretical model [7] shows that nuclear attraction of antiprotons is not understood. On the other hand, the strengths of antiproton absorption is better controlled and an average best fit value of $R_{n/p}$ turns out to be about 1. Concerning question $i$) the $R_{n/p}$ ratio the low-energy anti-protonic experiments give $R_{n/p}=0.47(.04)$ in ${}^{3}\mathrm{He}$ [10] $R_{n/p}=0.48(10)$ in ${}^{4}\mathrm{He}$ [9] and $R_{n/p}=0.63$ in C [3], $R_{n/p}=0.81(3)$ in Deuterium [11] and $R_{n/p}\simeq1$ in heavy nuclei [5]. Sizable differences happen in light nuclei and reflect different nucleon-antiproton angular momenta and center-of-mass energies at the annihilation moment. Determination of the latter becomes our main task.

Each experiment offers a specific method to answer the relevant questions $i$) and $ii$). The capture orbitals may be inferred from rates of antiproton reactions leading to specific final nuclear states compared to total capture rates, this method was used in radiochemical experiments [5]. However, in these of experiments the $R_{n/p}$ was considered as a parameter to be taken from other sources. For pionization studies, reference [8] offers determination the capture states from the charge spectra of emitted mesons. The lightest nuclei allow for a kind of mixed procedure which we present and develop in this work. It is based on the analysis of final state mesonic interactions with the residual nuclei. It allows also to extract $R_{n/p}$ directly from experimental data.

In the pionization measurements performed in Carbon, Deuterium and Helium the $R_{n/p}$ were extracted directly from final mesonic charges, but the capture states remained unknown. Unfortunately, these ratios are subject to final state interactions, apparently small in light but non negligible in heavier nuclei. To find these one has to know the capture states and the two uncertainties $i$), $ii$) have to be discussed jointly.

The present paper is organized as follows:

Section II introduces the pionization results, i.e. the $\pi$-meson charge distributions obtained in ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ nuclei. These results are compared to those of the elementary reactions $\bar{p}p\to$ mesons and $\bar{p}n\to$ mesons. Differences of the elementary distributions and those obtained in Helium are used to establish the rates of two final state reactions of the mesons: absorptions and charge-exchanges. Next, these rates are calculated for three expected antiproton-nucleus angular momentum states: S, P and D. Subsequent comparison of the phenomenological and the calculated results is used to determine the relative weights of these states in the nuclear capture process.

Section III discusses the predictions of three different $\bar{N}N$ interaction models and tests these against the $R_{n/p}$ and level shift measurements. Some technicalities related to the final state description are presented in Appendix.

Pionic annihilation of antiprotons in helium

We will first consider the measurement in ${}^{3}\mathrm{He}$ [10], which offers the most precise and extensive information. Pionic annihilation experiments in Helium were performed in a steam chamber [9, 10]. The results are compared here with elementary Hydrogen and Deuterium data from references [12] and [13].

The primary reaction of antiproton-nucleon annihilation into $\pi$ mesons happens at the nuclear surface. The reactions on protons are known with a considerable precision and are the basis of our analysis. The normalized distribution $\Pi(n)$ of the number of pions emitted in $\bar{p}p$ annihilation

(”pion spectrum”) covers charge and neutral mesons and extends from multiplicity $n=2$ till $n=8$ [14]. Their value depends on the $\bar{p}p$ relative energy. The results corresponding to antiproton annihilation at rest presented in Table I. They follow from chamber experiments in Ref. [15] and are normalized to 100% inside the error bars.

$n$	2	3	4	5	6	7	8
$\Pi(n)$	0.38(0.03)	7.8(0.4)	17.5(3.0)	45.8(3.0)	22.1(0.15)	6.1(1.0)	0.3(0.1)

The measurement performed in ${}^{3}\mathrm{He}$ detects charged mesons and final charged baryons. The results of the reaction

consists of the frequencies of charged pions and the number $N_b$ of charged baryons (predominantly protons, but these may also be deuterons). To describe the frequencies we use symbol $P_{N_b}[N_{+},N_{-}]$ where $N_{\pm}$ denotes respectively the number of positively and negatively charged pions. Numbers of neutral mesons are not determined. Each baryonic sector with $N_b=0,1,2$ is presented below in a separate table.

In first approximation, ${}^{3}\mathrm{He}$ results given in Table II are due to initial ($\bar{p}p$) annihilations. Inspection of the three last columns indicates that the differences of the two basic ($\bar{p}p$) experiments are sizable, but still this sector offers the largest statistics and the best known elementary reactions. In further analysis we use the weighted average of the two ($\bar{p}p$) measurements.

$N_{+}$	$N_{-}$	$P_{1}[N_{+},N_{-}]$	${}^{3}\mathrm{He}$	$\bar{p}p$	ref. [13]	ref. [12]
0	0	4.31(.47)	0.0570 (.006)	0.0407(4)	0.029	0.064(.004)
1	1	34.58(1.80)	0.4469(.0232)	0.4147(15)	0.398(.018)	0.423(010)
2	2	35.31(2.17)	0.4563(.0280)	0.4883(15)	0.527(.017)	0.469(.011)
3	3	3.16(.70)	0.00409(.009)	0.0447(5)	0.046(.01)	.0441 (.003)

The next $N_{b}=0$ sector is due to loss of the final state proton generated by $\pi^{-}$ meson produced in the dominant $N_{b}=1$ sector

Another process that links these two sectors is an annihilation reaction,

but we show in Appendix that it is negligible in the ${}^{3}\mathrm{He}$ case.

$N_{+}$	$N_{-}$	$P_{0}[N_{+},N_{-}]$
1	0	0.59(.22)
2	1	1.95(.46)
3	2	1.11(.32)
4	3	0.09(.09)

Extraction of capture states in Helium

The purpose of this section is to establish the states from which antiprotons are captured by the nucleus. To achieve this, we compare the charge spectrum obtained in Helium with the basic charge spectra obtained in $\bar{p}p$ reactions. The differences are attributed to final state interactions of mesons and residual nuclei. In a next step, a comparison of experimental and calculated differences is used to extract the angular momentum of capture states.

The X-ray cascade in antiprotonic atoms indicates nuclear captures to happen at distant nuclear surfaces. Mesons are created in an initial $\bar{p}N$ annihilations at small 0.8 $fm$ region around some point $\mathbf{r}$. It is located inside a shell with a thickness of 2-3 $fm$ and an average radius of about $R_{1/2}+0.6$ fm, where $R_{1/2}$ is a half nuclear density radius. Next, these mesons propagate in all directions. Some cross the residual nucleus and interact. The processes of interest are mesonic absorptions and charge exchanges. These change the initial spectrum of pions that would be emitted in a free annihilation. Comparison of two spectral distributions the free one and one obtained in nuclei permits to establish the nuclear region where $\bar{p}$ was absorbed. Detailed analysis is somewhat complicated by the initial mesonic spectra, which contain also mesonic resonances: $\rho$ and $\omega$. However, the latter decay, within one $fm$ distance, into final $\pi$ mesons which are detected. An analysis of reference [16] shows that angular averaging over mesonic trajectories allows us, to a good approximation, to discuss the $\pi$ mesons as born at an initial point $\mathbf{r}$. In the next section, the experimental spectra are discussed. Changes due to interactions with a final system are calculated in Appendix A.

Single charge-exchange

The probability of charge-exchange reaction (3) per single meson is denoted by $\lambda$. The relation of multiplicity probabilities in sectors $N_b=0$ and $N_b=1$ is driven by $\lambda$ and one has for each number of $\pi^{+}\pi^{-}$ pairs. :

where $N_-$ is the number of negative mesons in a given initial channel in the $N_b=1$ sector. Data from Tables II and III allow to solve for $\lambda$. Corresponding values are given in the third column of Table IV.

The calculated $\lambda^{(i)}_L$ values depend strongly on the number of mesons involved in a given transition. As explained and calculated in Appendix, the charge-exchange reactions are dominated by the $N\pi$ coupling to $\Delta(1233)$ resonance. It is the energy overlap of the $\Delta(1233)$ spectrum and mesonic energy spectra that determines $\lambda$. The mesonic energy spectra depend on the total number of charged and neutral mesons emitted in a given channel. Experiment [10] detects charged mesons and the spectrum is given in Table V. These serve to calculate an energy spectrum of a single meson. The numbers of neutral mesons have to be imported from another experiment [17]. The latter are shown in Table VI. The spectrum in the $MP[0,m]$ sector involving small fraction of events where no charge pions are emitted is not known. In calculations we use $MP[1,m]$ instead. The related uncertainty is minute in comparison to uncertainties of the dominant sectors.

$K$	0	1	2	3
$MP[K]$	0.029(0.005)	0.398(0.018)	0.527(0.017)	0.046(0.010)
$MN[K]$	0.055	0.734	0.189	0.022

	$MP[0,m]$	$MP[1,m]$	$MP[2,m]$	$MP[3,m]$
$m=0$		0.061(3)	0.144(7)	0.285
$m=1$		0.150(17)	0.522(53)	0.628
$m=2$		0.349 (16)	0.1807	0.083
$m=3$		0.333	0.1325	0
$m=4$		0.073	0	0
$m=5$		0.034		0

The last three columns in Table IV give the calculated values of $\lambda^{(i)}_L$, following the method discussed in Appendix A. The first step of this calculation finds likely densities of antiproton - nucleon annihilation region which is determined by overlap of the atomic wave function and a wave function of the struck nucleon. These densities depend strongly on the angular momentum of the atomic state $L$. The next step is a calculation of the chance that a meson formed in a state of given $L$ changes sign in the final state interactions. We follow a prescription given in ref. [16] which shows that the large energy released in the annihilation allows a closure approximation. It permits to describe residual nucleus in terms of optical potential with strength related to charge exchange cross sections. Average meson energy is about 450 MeV high enough to present the mesonic wave functions in terms of an eikonal. As the calculated values $\lambda_{L}^{(i)}$ depend strongly on $L$ this gives a chance to extract the angular momenta of antiproton capture states.

transition	$N_{\pi}$	$\lambda^{(i)}_{Exp}$	$\lambda^{(i)}_0$	$\lambda^{(i)}_1$	$\lambda^{(i)}_2$
$i=1$ : $K=1\to K'=0$	4	1.7(0.6)	4.47	2.65	1.40
$i=2$ : $K=2\to K'=1$	5	2.7(0.6)	8.62	5,19	2.76
$i=3$ : $K=3\to K'=2$	7	11.6(3.2)	13.6	8.37	4.52

Experimental errors of $P_0[N_+,N_-]$ are large and to increase the confidence level we consider also the second order transitions involving two charge-exchange reactions.

Double charge-exchange

Consider two parallel charge-exchange reactions confined to the $N_b=1$ sector,

These are of second order in $\lambda$ and may show up only in the most precise parts of the data. The initial antiproton-proton annihilation sector allows for that. For an initial state with $K$ $(\pi^+,\pi^-)$ pairs, the final state consist of $K-1$ pairs. Comparison of the primary $(p,\bar{p})$ distribution with that obtained in Helium shows a loss of mesons in some $K$ states and a gain of mesons in other $K$ states. The loss of rates is given by

The inspection of Table II indicates that the largest loss happens in $K=2$ states. The difference of Helium and Hydrogen results is $0.4563-0.4883=-0.0320$, it is recovered as a gain in the $K=1$ state: $0.4469-0.4147=0.0322$. This, and other two $K$ states allow us to extract the charge-exchange parameters presented in Table VII. In this Table, we compare the rates $\lambda_{exp}$ extracted from the experimental data by using the two methods: single charge charge-exchange and double charge-exchange. The latter are fairly close to results obtained from the first order transition between $N_b=1$ and $N_b=0$ sectors. However, large errors arise as the procedure involves subtractions of two large uncertain experimental quantities. In ${}^{4}\mathrm{He}$ atoms discussed in next sections the double charge-exchange is the only detectable interaction. That destroys the precision.

	$K=3$	$K=2$	$K=1$
one exchange	$1.7(0.6)$	$2.7(0.6)$	$11.(3.2)$
two exchanges	$1.17(1.5)$	$4.05(4.00)$	$9.72\ (18.0)$

Extraction of capture states

To calculate the probabilities of capture from a given $L$ state - $w_L$ - we use the calculated $\lambda_L^{(i)}$ values from Table IV and assume a distribution saturated with a mixture of L=0, 1 and 2 states

Next we determine their respective weights $w_L$ with a best fit to the three experimental transitions IV. The detailed procedure of this fit is given in Appendix. For ${}^{3}\mathrm{He}$ this yields $w_{L=2}=0.85+0.15-0.32$, $w_{L=1}=0.15+0.17-0.15$ and $w_{L=0}=0.00+0.10$. Comparison of calculated and phenomenological charge exchange rates indicate predominant captures from $D$ states. This result is consistent with the measurements by Schneider et al. [20] who observe X-ray transitions to $2P$ and $3D$ atomic states. However, this measurement finds an absolute yield of the lowest unperturbed $4\to3$ transition to be only $35\pm9\%$. That shows importance of molecular effects which induce orbital mixing leading to captures from high $n$ states. For nuclear halo studies this question is not relevant as in all cases the antiproton wave functions at nuclear distances take the form $\Psi(r)\sim r^L\cdot f(r)$ where $f(r)$ describes nuclear effects that are roughly the same in all reactions. The wave function normalization is irrelevant, unless resonant mixtures of atomic states happen.

Another problem is: how many events go by the in-flight captures? It is not known, but again it is the $\Psi(r) \sim r^{L}$ behavior that matters, The capture state distribution obtained here is close to calculations of Reifenroether et al [18] who find 8% from S-waves 49% from P-waves and 53% from D-waves in gas at 1 atm pressure. The X-ray measurement was done in a gas at 600 mb, while the pionization experiment was done at normal pressure. The results by PUMA will be different anyway as the atomic capture will happen in vacuum. What matters is the detection of charge baryons allows to fix the capture states.

The Helium experiments offer also some additional information. Table VIII presents the multiplicities following initial antiproton-neutron annihilations. However, there are two problems that hinder our analysis: larger experimental errors and lower capture rates. No significant gain of precision may be obtained.

$N_{+}$	$N_{-}$	$P_{2}[N_{+},N_{-}]$	norm	$\overline{p}n$	[13]	[12]
0	1	2.72(.50)	0.1440 (.026)	0.116	0.065	0.168(.008)
1	2	12.13(1.22)	0.6431(.06468)	0.667	0.733	0.595(.014)
2	3	3.86(.770)	0.2047 (.0408)	0.209	0.189	0.230(.009)
3	4	0.15(0.11)	0.0079 (.0059)	0.041	0.0121	0.07(.020)

Captures in ${}^{4}\mathrm{He}$

Pionization in ${}^{4}\mathrm{He}$

The ${}^{4}\mathrm{He}$ pionization experiment does not detect charged baryons, apparently due to the strong binding of the latter. The former sectors $N_{b}=1$ and $N_{b}=2$ that is initial ($p\overline{p}$) captures and ($n\overline{p}$) captures are lumped together and no equivalence to the $N_{b}=0$ sector exists [10]. The charged pions distributions for ${}^{4}\mathrm{He}$ are simply denoted by $P$. Again, as the ($p\overline{p}$) data are more precise that the ($n\overline{p}$) data and we consider captures of final zero charge that predominantly are due to the initial ($p\overline{p}$) annihilations. As the analysis in terms of single charge-exchange reaction is not easy, one has to use the double exchange transitions (6) and primary $p\overline{p}$ capture data shown in Table IX. The related charge-exchange rates are collected in Table X.

$K$	$N_{+}$	$N_{-}$	$P$	$P_{\overline{p}p}$
0	0	0	0.0496(.0052)	0.0407(4)
1	1	1	0.4664(0.0165)	0.4147(15)
2	2	2	0.4383(0.0252)	0.4883(15)
3	3	3	0.0420(0.0142)	0.0447(5)

transition	$N_{\pi}$	$\lambda_{\mathrm{Exp}}^{(i)}$	$\lambda_{0}^{(i)}$	$\lambda_{1}^{(i)}$	$\lambda_{2}^{(i)}$
$i=1$ $K=1\to K^{\prime}=0$	4	$10.3(9.1)$	11.4	8.45	5.7
$i=2$ $K=2\to K^{\prime}=1$	5	$11.3\ (8.7)$	18.4	13.9	9.5
$i=3$ $K=3\to K^{\prime}=2$	7	$5.8(14.0)$	25.1	19.3	13.4

Table X indicates a dominant $L=0$ captures happening in ${}^{4}\mathrm{He}$ atoms which, according to Appendix B, gives the populations $w_{L=0}=0.77+0.13-0.14$, $w_{L=1}=0.00+0.18$ and $w_{L=2}=0.23+0.26-0.23$. Possibly, it is due to stronger mixing of atomic states generated by Stark effect in the early stage of cascade and absorptions from $S$ states and high main quantum number $n$. The findings of Schneider [20] indicate 65% of absorptions to happen in high $n$ possibly $S$ states and such result is not excluded. However, the difference of ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ makes it hard to accept. More likely it is related to uncertainties of our method which requires final state effects larger than experimental errors. An estimate better than that leading to Table X should consider inverse reactions $n\pi^{0}\to p\pi^{-}$, $n\pi^{+}\to p\pi^{0}$. This approach would however involve much uncertain data: the rates of $\pi^{0}$ formation and $\overline{p}n$ pionization spectra, and hence a reliable calculation cannot be fulfilled. The extraction of capture orbits will have to be achieved by other methods, possibly via comparison of ${}^{4}\mathrm{He}$ and ${}^{3}\mathrm{He}$ atomic cascade calculations.

TABLE X: Probabilities of charge-exchange transitions $\bar{\pi}^{-}p \to\pi^{0}n$ and $\pi^{+}n \to\pi^{0}p$ in $^{4}\mathrm{He}$. $K(K^{\prime})$ are numbers of $(\pi^{+}\pi^{-})$ pairs in the initial (final) state. $N_{\pi}$ are the average numbers of mesons involved in the transition. The numbers given in column three are experimental $\lambda$ numbers. The last column gives $\lambda$ obtained with eikonal calculations performed for the three expected angular momenta of the capture states. All $\lambda$ values are multiplied by $10^{2}$.

Testing $\bar{N}N$ Interaction Models

Extraction and understanding the $R_{n/p}$ value

Table VIII presents pionization in $^{3}\mathrm{He}$ initiated by antiproton-neutron absorption. Let us notice that differences between Helium and $\bar{p}n$ spectra are much smaller then experimental errors. Hence, the type of analysis performed in the last section would produce very uncertain results. Still, these results are essential, showing the isospin dependence of the annihilation. It yields the ratio of $\bar{p}n$ and $\bar{p}p$ capture rates $R_{n/p}=0.465(.0415)$, which is very important in the nuclear structure studies. This value is obtained in ref. [10] on the basis of experimental results given in Tables II and VIII, as a ratio of $N_{b=2}$ and $(N_{b=1}+N_{b=0})$ events. Below, we check it against corrections due to $N_{b=1}\rightleftarrows N_{b=2}$ balance. The transition $N_{b=2}\to N_{b=1}$ and the return one are given by reactions

The balance of the two sectors depends on $P[N_{+},N_{-}]$ given in Tables II, VIII and the content of neutral mesons given in table VI. The values of experimental $\lambda$ obtained in section II A 1 and formulas analogous to eq. (5) yield a minute and insignificant enhancement $0.001(.05)$ which leads to $R_{n/p}=0.467(0.035)$.

An open problem is understanding of the $R_{n/p}$ in terms of nucleon - antinucleon potentials. Below we analyze several results. The first question is: what is the relevant energy in the $\bar{N}N$ system that leads to the experimental value of $R_{n/p}$? To answer we follow reference [19] which discusses light anti-protonic atoms as quasi three body systems: antiproton (1), struck nucleon (2) and a residual nucleus (3). This allows to determine energies in the $\bar{N}N$ center of mass system as

where: $E_{S}$ is a separation energy of the nucleon, $E_{A}$ is a (small) antiproton binding energy and $E_{\mathrm{REC}}$ is a recoil energy of $\bar{N}N$ pair with respect to the residual nucleus. Calculations performed in ref. [19] yield the values shown in table XI. It turns out that $R_{n/p}$ are generated at subthreshold energies of about $-34$ MeV in $^{4}\mathrm{He}$ and about $-14$ MeV in $^{3}\mathrm{He}$.

The second question is: what are spin states of the $\bar{N}N$ system at the annihilation? In atomic D waves the result is given by expressing the atomic spin state in terms of internal spin of the $\bar{N}N$. The absorption in this state is essentially given by Born term and we operate on spin averaged amplitudes. This allows S,P,D internal angular momenta, but D-waves are negligible and the weights of S and P waves is calculable by 3-body kinematics with known wave functions. Ref. [19] gives the relative weights of the internal $\bar{N}N$ states as shown in table XI.

TABLE XI: Separation and recoil energies in $L=2$ atomic states. The second column gives averages of proton and neutron separation energies MeV. Third column gives average recoil energies [MeV], numbers in parentheses are approximate widths of recoil energy distributions. The last two columns presents weights $w(L)$ of $\bar{N}$, $\bar{P}$ captures in relative S and P waves obtained with Paris potential.

	$E_{S}$	$\langle E_{\mathrm{REC}}\rangle$	$w(\mathrm{S})$	$w(\mathrm{P})$
$^{3}\mathrm{He}$	6.6	7.3(1)	0.38	0.62
$^{4}\mathrm{He}$	20.2	13.7(10)	0.41	0.59

The last question is to find annihilation rates in S and P waves at subthreshold energies. According to ref. [19] we define the off-shell below-threshold $\bar{N}N$ T-matrix in a given partial wave $\alpha\equiv\{TLSJ\}$ as

where

is the the $\bar{N}N$ reduced potential in the partial wave $\alpha$, $\mu_{\bar{N}N}=m_N/2$ is the $\bar{N}N$ reduced mass, $\hat{j}_{L}(z)$ is the reduced regular spherical Bessel function [34], $\chi_{\alpha}^{+}$ is the outgoing part of the corresponding reduced regular radial scattering solution $u_L\equiv\hat{j}_{L}(qr)+\chi_{\alpha}^{+}(r)$ [35]. This definition is such that the integrated quantity

coincides at $E=0$ with the usual scattering length: $t_L(E=0)\equiv-a_L$.⁴ For $E<0$, $t_L(E)$ constitutes a natural extrapolation of the usual scattering amplitude to subthreshold energies.

It is also interesting to introduce the ”moments of the off-shell T-matrix” by

where for K=0 one has $t_L^0(E)=t_0(E)$. We have computed the S- and P-waves amplitudes $t_L^0(E)$ for the $\bar{N}N$ optical potentials in configuration space that we have examined in [28], i.e. Paris 2009 [29] Dover-Richard model 2 [30] and Kohno-Weise model [31]. The results for the L=0 ($t_0^0$) and L=1 ($t_1^0$) amplitudes as functions of the c.o.m. energy are displayed in Figure 1 for Paris potential, in Figure 2 for Kohno-Weise (KW) and in Figure 3 for Dover-Richard model 2 (DR23). Different colors correspond to different partial waves in the isospin basis. The real parts are in solid lines and the imaginary pars in dashed lines. Notice that for all partial waves, and in all the energy range, one has $\operatorname{Im}[t_L(E)]>0$, which corresponds at $E=0$ to $\operatorname{Im}(a_L)<0$.

For the S-wave (left panel) the singular character of the Paris potential is manifested, due to the presence of a near-threshold quasi-nuclear state in the ${}^{11}S_0$ channel. This state is absent in KW and DR model and hence the amplitudes have a smooth behavior as a function of the energy, in sharp contrast with Paris.

For P-waves (right panel) a strong energy dependence in Paris potential is displayed by the ${}^{33}\mathrm{P}_{1}$ partial wave, also due to the narrow bound state. A common feature of the three models is the large values of the ${}^{13}\mathrm{P}_{0}$ amplitudes.

Our aim is to compute – for the lowest angular momentum states – the ratio $R_{L}(E)$ between the imaginary part of the amplitudes (11) for T=1 [$t_{L}(T = 1)$] and the isospin-averaged values $t_{L}(\overline{N N})$ defined as

Still, for an easier comparison with the available experimental data it is interesting to take the ”spin-averaged ” values of the amplitudes, i.e.

for S waves and P-waves respectively. Corresponding results are displayed in Figure 4.

One is then left to compute

The results of (15) for the Paris and KW potential are displayed in Figure 5. Results presented in Figure 4, are summarized in table XII. The elementary ratios for S- and P-waves follow the spin-averaged subthreshold absorptive amplitudes calculated for three models. Next these were weighted by probabilities of S- and P-wave captures, W(S) and W(P), given in table XI. D wave contributions are negligible.

${}^{3}\mathrm{He}$	$R(S)$	$R(P)$	$R(D)$	$R_{n/p}$
Paris	0.51	1.40	1.00	0.72
K-W	0.99	0.80	0.77	0.89
D-R	1.04	0.90	1.17	0.95
${}^{4}\mathrm{He}$
Paris	0.62	1.18	0.72	0.94
K-W	1.00	0.88	0.78	0.89
D-R	1.03	0.91	1.18	0.99

With the captures from atomic $D$-states, it is the P-wave $\bar{N}N$ absorption which dominates the $R_{n/p}$ ratio. Inspection of Figure 4 and Table XIV shows inconsistency of all models with the experimental ${}^{3}\mathrm{He}$ result of 0.47(.04). Also, the experimental ${}^{4}\mathrm{He}$ result of 0.48(.10) is not reproduced by any potential. On the other hand, all models are consistent with the pionization and X-ray experiments in heavier nuclei [5], where $R_{n/p}$ are determined predominantly by P-wave interactions.

	Paris	DR	KW	Data
$\epsilon(2P)$	33	84	- 48	243(33)
$\Gamma(2P)$	517	345	380	489(30)

At present, Paris model is the only potential that offers small $R_{n/p}$ values, and only in the S-wave. It comes as a result of an $I=0$ baryonium state formed in the $N\bar{N}$ system. As shown in Figure 5, the presence of this quasi-bound state pulls the ratio down to the values obtained in Helium. The S-wave baryonium in question has an independent confirmation in $J/\psi$ decays [25]. It also offers an explanation of X(1835) state [25]. On the other hand, the $R_P$ ratio in P-wave spoils consistency with the Helium results. It is due to a P-wave baryonium located close to the threshold. There is no experimental support for this result. Apparently it is excluded by the $R_{n/p}$ result in Helium. Also, as shown in ref[19], it is inconsistent with level shifts in light antiprotonic atoms. The old pionization experiments used Helium gas, while the planned PUMA experiment will be performed in vacuum. The atomic capture states are likely to be different and possibly the coming experiment will solve the puzzle.

Atomic level widths and shifts due to nuclear interactions

Finally we give results for atomic level shifts and level widths calculated with the three discussed potentials. The method of multiple scattering series summation discussed in reference [19] was used. It involves a quasi three-body separation of the system into : (1) = the antiproton (2) = an interactive nucleon and (3) = a residual nucleus. For the initial function we use normalized relative (3)-(2) wave functions

where $\widetilde{\alpha}=\sqrt{2\mu_{R,N}E_s}$ and $E_s$ is the separation energy. The short distance parameter $\widetilde{\beta}$ is extracted from electron or pion scattering experiments and fixed to reproduce the zeros of charge form-factors. For ${}^{3}\mathrm{He}$, $\widetilde{\alpha}=0.45$, $\widetilde{\beta}=1.753$ [23], and for ${}^{4}\mathrm{He}$, $\widetilde{\alpha}=0.846$, $\widetilde{\beta}=1.20$ [24], all in $\mathrm{fm}^{-1}$ units.

The level shifts are dominated by the first collision. Summation of multiple collisions presented in Appendix § 12 leads to results compared to experimental data. Results are displayed in Table XIII (for He) and XIV (for ${}^{2}\mathrm{H}$). As one can see, all models produce fairly proper level widths, but they fail badly to reproduce the shifts, in some cases (Paris or KW) by one order of magnitude. Summation of higher numbers of collisions produces corrections: small in ${}^{3}\mathrm{He}$ (Table XIII) and negligible in ${}^{2}\mathrm{H}$ (Table XIV). On the other hand, the multiple scattering collisions in ${}^{4}\mathrm{He}$ strongly damp the $2P$ level widths. In this atom the antiproton reaches much denser parts of the nucleus and this damping is due to repulsion exerted by the nucleus. An apparent paradox is created in ${}^{4}\mathrm{He}$ and also in heavier atoms, [7]. The level shifts are repulsive but fitting all data requires an attractive optical potential. Some possibilities to understand it exist: one is related to strong medium effects changing the basic $N\bar{N}$ interactions, another possibility is a fairly narrow $N\bar{N}$ state at about 30-40 MeV below the threshold. At the moment there is no explanation. The upper $3D$ level widths are systematically above the calculated ones. The difference is not related to $D$ wave $\bar{p}N$ contribution which generates a 0.02 meV extra width. It may indicate slightly longer-ranged absorptive $\bar{p}N$ potential. However, these widths are not measured directly and depend on difficult to control cascade calculations. New experiments with high resolution detectors would be helpful to clarify the problem.

	Paris	DR	KW	Data
${}^{3}\mathrm{He}$
$\epsilon(2P)$	1.0	7.7	-2.1	17(4)
$\Gamma(2P)$	26.2	17.5	13.0	25(9)
$\Gamma(3D)$	1.59	1.48	1.57	2.14(18)
${}^{4}\mathrm{He}$
$\epsilon(2P)$	-1.5	10.8	1.9	18(2)
$\Gamma(2P)$	26.1	18.8	26.0	45(5)
$\Gamma(3D)$	1.80	1.88	1.97	2.36(10)

Concluding Remarks

The ratio of antiproton-neutron to antiproton-proton captures $R_{n/p}$ was calculated with several $N\bar{N}$ interaction models. In medium and heavy antiprotonic atoms, this ratio is dominated by $P$-waves nucleon-antinucleon interactions and is close to unity in all studied models. It is consistent with all types of nuclear absorption reactions: pionizations, radioactive final state measurements and level widths of atomic states. Thus, one may expect the $R_{n/p}$ to be independent of atomic cascade details and states of nuclear captures. As a consequence one expects it to describe nuclear captures also in antiprotonic atoms formed in vacuum.

The results obtained in He are different. The pionization experiments yield $R_{n/p}$ to be about one half and a comparable result is obtained only by the Paris potential and in an $S$-wave nucleon - antinucleon state. As shown in this work pionization data in Helium are consistent with dominant captures in $D$ wave atomic states. That indicates an alternative: either a change of Paris potential (suppressing $P$-wave interactions) or in experiments with gas targets the majority of nuclear absorptions happening in high $nS$, $nP$ atomic states. However, the second option is much less likely. It is the PUMA experiment, which will be performed in vacuum, that may give a more definite answer.

The three presented nucleon-antinucleon potentials fare well in comparison to experimental level widths. The Dover-Richard potential is compatible also with the level shifts. The other two: Paris and Kohno-Weise fail badly in this respect.

Acknowledgements

This work was supported by the IN2P3-COPIN agreement no 22-154 and realized at National Center for Nuclear Research in Warsaw and at IJClab in Orsay. We thank the warm hospitality of both centers and the support of the CNRS/IN2P3 French-Polish agreement.

Final state interactions

Antiprotons are absorbed by the nucleus from high angular momentum states. At nuclear distances the repulsion due to centrifugal barriers makes atomic wave functions falling down as $\Psi(r) \sim r^{L} \cdot f(r)$ where $r$ is the radial coordinate and $L$ is the angular momentum. This makes nuclear captures to happen at distant peripheries, typically at less then 5% of the central nuclear density. Functions $f(r)$ describe deformation of atomic wave functions due to nuclear interactions and for most of peripheral cases $f(r) \simeq1$.

Antiproton annihilation and related emission of mesons is a complicated quantum process, The basic transition matrix element for the pionization is

where $\Psi$ are wave functions of a final $A-1$ and the initial $A$ nuclei, $\psi_{L}$ is the atomic wave function of initial antiproton and $\Psi_{\pi}$ is a wave function of the final state mesons.

Multi-body calculations are involved and approximations are necessary. We refer to refs. [16] and [8] for details and present main formulas below. The first approximation is the assumption of completeness of the final states. It is allowed by 2 GeV energy release which goes into kinetic energies of final particles. It allows to turn phase space integral of mesonic functions into a classical integral over the nuclear density and the atomic density overlap. The reaction rate becomes

where the sum over single particle nucleon states generate nuclear density $\rho(r)$, and pionization rate $W(r)$ is an aggregate describing rate of pionization and effects of final state mesonic interactions

where $\prod_i$ denotes product of all mesonic wave functions $\phi^{-}$.

The analysis of data in light nuclei requires calculations of charge-exchange probability $\lambda$ and absorption probability $\omega$ per single emitted meson. All functions $\phi^{-}$ involve final state interactions. The mesons are formed on the nuclear surfaces and propagate in all directions. Average mesonic momenta are large $k \sim400$ MeV/c and allow for eikonal description. Wave functions for each outgoing pion are thus given by

and the transmission probability is calculated as

where the exponential damping of mesonic wave function is determined by a semi-classical function $S$ which sums up the damping of mesonic wave as it propagates across the nucleus

where $U$ is the potential describing absorption, elastic or inelastic scattering. Coordinate $\mathbf{r}$ refers to the nuclear center. Boundary conditions are set in such a way that for distant captures $T(\mathbf{r},\mathbf{k})\to1$ as $r\to\infty$. For Klein–Gordon equation the nuclear optical potentials are determined by cross sections of elementary reactions with the normalization

In practice we expand the root in eq. (22) and consider only inelastic, or absorptive cross sections corresponding to

In order to obtain an average chance for a single meson to be emitted one has to calculate its wave function and average the transmission rate over capture region and over mesonic momenta $\mathbf{k}$ and directions of flight. This involves integration over mesonic phase space $dL^{cm}$ in each multiplicity and sum over specific transition matrices frequencies $|t_{pN\to M}|^2=M_{pN\to M}$ where frequencies $M$ are given in Table VI and Table V of the main text.

The extent of summation depends on the nature of experiment analyzed. Thus, experiments detecting cold residual nuclei [5] require summation of all final mesonic channels and all modes of pionic reactions [16]. The pionization experiments [2, 4] require all channels but separate: charge-exchange and meson absorption modes [8]. On the other hand, the pionization experiments in light nuclei [9, 10] detect specific mesonic states of given number of charged mesons. Here we follow the last scenario presented in more detail in ref. [8].

The analysis of data in light nuclei requires calculations of charge-exchange probability $\lambda$ and absorption probability $\omega$ per single produced meson.

The average transmission probability is calculated as

where $\alpha$ denotes $\pi^+$, $\pi^-$, $\pi^0$ and some specific final state, $\int dL^{cm}$ denotes Monte Carlo integration over mesonic phase space for a given multiplicity discussed in reference [16] and normalization overlap is

Probabilities for captures (or charge-exchanges) become $1-T_{\alpha}$.

In the next subsections we discuss the two main absorption modes which determine spectra of final mesonic charges.

Charge-exchange processes

The pion charge-exchange reaction

was studied up to 250 MeV pion kinetic energies in ref. [22]. The cross section follows the shape of $\Delta(1233)$ which dominates the reaction

where $T_{\pi}$ is a kinetic energy, $T_r=167$ MeV and the width at the resonance is 110 MeV. It was extrapolated to 300 MeV by SAID [22]. As we need to extrapolate it further to about 400 MeV the experimental $\Delta(1233)$ profile was followed. On the distant right shoulder of $\Delta$ our extrapolation follows $\Gamma=70+T_{\pi}/18$ MeV.

The energies of the pions emitted in the $NN$ annihilation depend on the number of emitted pions. Average values are shown in Table XV.

$N(\pi)$	2	3	4	5	6	7	8	9
$E_{\mathrm{kin}}$	793	482	327	233	171	127	93	67

Maximal cross sections happen close to the resonance peak $T_r\simeq160-170$ MeV that is in the emission of six mesons. In the lowest charge channel of Helium experiment (one $\pi^+\pi^-$ pair supplemented by two $\pi^0$) $N(\bar{\pi})=4$. The kinetic energies are much above the resonance and the charge-exchange probability $\lambda$ is very small. For higher charge states ($N(\bar{\pi})=5,7$) measured in the Helium capture the overlap of mesonic energy spectrum and $\Delta(1233)$ is much larger, $\lambda$ also becomes larger.

Reaction (A11) and its inverse are the same by isospin symmetry and differ only slightly, by the final phase space. Hence, equation (A12) describes also other mesonic charge-exchange reactions involved in the final states.

True two-nucleon absorptions.

We use theoretically motivated, energy dependent two-nucleon capture approach. The general understanding is that the $U_{NN}$ term describes two nucleon capture initiated by the excitations of $\Delta(1233)$. A direct model of such potential was offered by Muttaz Nuseirat et al [33]. This potential is local

with parameters: $\lambda_o=46.35\,\mathrm{fm}^4$, $E_o=215$ MeV and $\Gamma=77$ MeV. The parameters were tested in light nuclei and for pion kinetic energies $T_\pi<180$ MeV/c, in $Z=N$ situations. We find effects of absorptions to be very small in Helium 3 and slightly larger in He 4 with small effect on the final spectra.

Fit procedure

Given the data set corresponding to the charge-exchange transitions $K=1\to K'=0$ (i=1), $K=2\to K'=1$ (i=2) and $K=3\to K'=2$ (i=3), displayed in Table IV (for $^3$He) and IX (for $^4$He), we aim to determine three real positive numbers $0<w_L<1$ (L=0,1,2) minimizing the $\chi^2$ with respect to the experimental data defined by.

with the constraint

and so the number of (independent) degrees of freedom $N_d=2$.

Their values correspond to the weight of the L=0,1,2 contribution to the considered transitions, assuming that these angular momenta exhaust all possibilities. They have been obtained by numerically searching the minima of (B1) on a two-dimensional plane (positive quadrant) $w_0+w_1=1$ with $w_2$ given by (B2). Results are:

• For $^3$He:

with $\chi^2(\mathrm{minima})=2.33$

• For $^4$He:

with $\chi^2(\mathrm{minima})=0.90$

We have plotted in Figure 6 the isoline map of the $\chi^2(w_0,w_1)$ surface as a function of $\alpha_L\equiv\sqrt{w_L}$ (to obtain quasicircular isolines). As one can see the solution is unique (single minima) but the surface is quite flat resulting into large uncertainties.

One can estimate the error $\Delta_i$ in each independent parameter $w_i$ with the $\Delta\chi^2=1$ rule, that is by looking for independent variations of one unit around the minima $(\bar{w}_0,\bar{w}_1,\bar{w}_2)$ in the $\chi^2$ given by (B1)

The final result of this analysis gives:

• For ${}^{3}\mathrm{He}$

$\bar{w}_0 = 0.00 + 0.08 \qquad\bar{w}_1 = 0.15 \pm0.14 \qquad\bar{w}_2 = 0.85 + 0.15 - 0.26$

• For ${}^{4}\mathrm{He}$

$\bar{w}_0 = 0.77 \pm0.11 \qquad\bar{w}_1 = 0.00 + 0.14 \qquad\bar{w}_2 = 0.23 + 0.21 - 0.23$

Summation of multiple-scattering series

The perturbation expansion for the level shift is equivalent to multiple scattering expansion for meson-nucleus scattering amplitudes. For high angular momenta discussed here Coulomb corrections to the shifts are small and the method of summation used here follows references [19], [27]. In comparison to optical potentials it prohibits double scattering on the same nucleon, permits excitations of the nucleus and allows for correlations. The solution is built as a multiple-scattering expansion with nucleon-antinucleon scattering amplitude

where $a$ is a spin average of $S$-wave antiproton nucleon amplitudes at proper center of mass energy $E_{cm}$ and $b$ is a similar average for $P$ wave amplitudes.

The scattering series is summed into a quasi geometric series that converges rapidly. We denote the single scattering term by $\langle V \rangle$

where $\Phi$ is an atomic wave function of antiproton, $\Phi$ is a nuclear (single particle) wave function and $\varphi$ are wave functions of struck nucleons.

The first sum for atomic level shifts becomes

Expanding the denominator of equation C4 into geometric series yields the first two terms of the multiple scattering. To obtain higher orders, one calculates triple antiproton-nucleon collisions and terms related to struck nucleon-residual nucleus collisions. The latter may be included into an effective operator $G'$ (see reference [27]) and corrects the denominator

where the expansion coefficients are $D_1=-\langle VG'V\rangle/\langle V\rangle$ and $D_2=-(\langle VG'V\rangle/\langle V\rangle)^2\langle VG'VGV\rangle/\langle V\rangle$. In the upper $2P$ levels of interest one finds $|D_1|\sim0.15$; $|D_2|\sim0.01$ and only the first term $D_1$ is needed. For the $S$ levels triple collisions have to be calculated to reach a similar precision. For detailed formulas and kinematical factors we refer to calculations performed in the lightest atoms [19]. The double scattering term involves a propagator of the whole system $G$. It includes excitation of the system to continuum. In principle, refined effects like correlations and nuclear excitations are hidden there. However, since $\langle VGV\rangle/\langle V\rangle$ is small in atomic P,D states we stop at expression $\Delta E'$.