threshold: narrow structures of $\bar{N}N$ type cannot exist for a centre-of-mass energy greater than 10 or 20 MeV above the threshold.

### 3.10. Colour chemistry

At the time when several candidates for baryonium were proposed, the quasi-nuclear approach, inspired by the deuteron described as an NN bound state, was seriously challenged by a direct quark picture. We define here baryonium as a meson with preferential coupling to baryon-antibaryon channels, and let theory propose either a quasi-nuclear model of baryonium or a multiquark picture.

Among the first contributions to multiquark spectroscopy, there is an interesting remark by Jaffe [253] that $q^2\bar{q}^2$ S-wave are not that high in the spectrum, and might even challenge P-wave ($q\bar{q}$) to describe scalar or tensor mesons. The debate is still open, and even more confused by the possible contributions of gluonia, hybrids or ($s\bar{s}$) configurations in these sectors, with many mixing scenarios.

It was then pointed out [254] that orbital excitations of these states, of the type ($q^2$)—($\bar{q}^2$), have preferential coupling to $\bar{N}N$. Indeed, simple rearrangement into two ($q\bar{q}$) is suppressed by the orbital barrier, while the string can break into an additional $q\bar{q}$ pair, leading to ($q^3$) and ($\bar{q}^3$).

Chan and collaborators [255,256] went a little further and speculated about possible internal excitations of the colour degree of freedom. When the diquark is in a colour $\bar{3}$ state, they obtained a so-called “true” baryonium, basically similar to the orbital resonances of Jaffe. However, if the diquark carries a colour 6 state (and the antidiquark a colour $\bar{6}$), then the “mock-baryonium”, which still hardly decays into mesons, is also reluctant to decay into N and $\bar{N}$, and thus is likely to be very narrow (a few MeV, perhaps).

This “colour chemistry” was rather fascinating. A problem, however, is that the clustering into diquarks is postulated instead of being established by a dynamical calculation. An analogous situation existed for orbital excitations of baryons: the equality of Regge slopes for meson and baryon trajectories is natural once one accepts that excited baryons consist of a quark and a diquark, the latter behaving as a colour $\bar{3}$ antiquark. The dynamical clustering of two of the three quarks in excited baryons was not shown before 1985 [257]. Diquark clustering is even more questionable for sextet than for triplet states. If confinement has anything to do with colour, one naively expects forces to depend on the colour configuration. See, e.g., Ref. [258] for a discussion on mock-baryonium.

### 3.11. Introduction to antiprotonic hydrogen and deuterium

The lowest momenta for $\bar{p}p$ scattering are obtained by stopping antiprotons in hydrogen, by forming antiprotonic hydrogen atoms (called protonium) and by deducing from X-ray transitions the proton-antiproton scattering lengths. Protonium atoms possess several features in common with the well-known positronium atoms and the name “protonium” had been chosen to emphasise this similarity. Both systems consist of a particle and its antiparticle. In both cases, the constituents are intrinsically stable, but annihilate each on the other. Essential features of both positronium and protonium atoms can be understood from the *Bohr* formula. The energy levels $E_n$ and classical radii $r_n$ are functions of the principal quantum number $n$. The energy levels are proportional to the
<table><caption>Table 3.3
QED transition energies in $\bar{p}p$ and $\bar{p}d$ atoms</caption><thead><tr><th>\bar{p}p</th><th></th><th>\bar{p}d</th><th></th></tr></thead><tbody><tr><td>2P → 1<sup>3</sup>S<sub>1</sub></td><td>9405.7 eV</td><td>2P → 1<sup>4</sup>S<sub>3/2</sub></td><td>12505 eV</td></tr><tr><td>2P → 1<sup>1</sup>S<sub>0</sub></td><td>9405.7 eV</td><td>2P → 1<sup>2</sup>S<sub>1/2</sub></td><td>12505 eV</td></tr><tr><td>3D → 2<sup>3</sup>P<sub>2</sub></td><td>1736.79 eV</td><td>3D → 2<sup>4</sup>P<sub>5/2</sub></td><td>2316.48 eV</td></tr><tr><td>3D → 2<sup>3</sup>P<sub>1</sub></td><td>1736.85 eV</td><td>3D → 2<sup>4</sup>P<sub>3/2</sub></td><td>2316.38 eV</td></tr><tr><td>3D → 2<sup>3</sup>P<sub>0</sub></td><td>1737.00 eV</td><td>3D → 2<sup>4</sup>P<sub>1/2</sub></td><td>2316.46 eV</td></tr><tr><td>3D → 2<sup>1</sup>P<sub>1</sub></td><td>1736.70 eV</td><td>3D → 2<sup>2</sup>P<sub>3/2</sub></td><td>2316.52 eV</td></tr><tr><td></td><td></td><td>3D → 2<sup>2</sup>P<sub>1/2</sub></td><td>2316.52 eV</td></tr></tbody></table>

reduced mass $m_r$, which is the combination $m_1m_2/(m_1 + m_2)$ of the constituent masses $m_i$ and is equal to $m_p/2$ in case of protonium. The radii are inversely proportional to $m_r$:

$$E_n = -\frac{1}{2} m_r c^2 \left(\frac{\alpha}{n}\right)^2, \quad r_n = \frac{\hbar c}{m_r c^2} \frac{n^2}{\alpha} \qquad (3.65)$$

Here, $\alpha = 1/137.036$ is the fine structure constant, and $\hbar c = 197.33$ MeV fm. The first Bohr radius $r_1$ is often denoted as $a_0$ in the literature; we use $a_0$ for the S-wave scattering length. More specifically, for $\bar{p}p$ and $\bar{p}d$ atoms,

$$E_n(\bar{p}p) = -12.491 \frac{1}{n^2} \text{keV}, \quad r_n(\bar{p}p) = 57.6 n^2 \text{fm}, \qquad (3.66)$$

$$E_n(\bar{p}d) = -16.653 \frac{1}{n^2} \text{keV}, \quad r_n(\bar{p}d) = 43.2 n^2 \text{fm}. \qquad (3.67)$$

Corrections due to QED and relativistic effects have to be applied to find the true electromagnetic binding energies. They have been calculated to be

$$E(n, \ell) = E_n \left[ 1 + \left(\frac{\alpha}{n}\right)^2 \left( \frac{m_r}{2(m_1 + m_2)} - \frac{3}{4} + \frac{n}{\ell + 1/2} (1 + a_v + a_{lj}) \right) \right], \qquad (3.68)$$

by Barmo et al. [259]. The values of $a_v$ and $a_{lj}$ are tabulated in this reference. The most important correction is due to vacuum polarisation which increases the binding energy of the $\bar{p}p$ atom by 42 eV. The finite size of the proton makes a shift into the opposite direction, of -3.2 eV. In $\bar{p}d$ atoms, the vacuum polarisation of 67 eV is partly compensated by the finite size effect of -48 eV. The corrections were reevaluated recently [260]; in Table 3.3 we give final QED transition energies relevant for this review.

The QED fine-structure splittings of the 2P states are very small, of the order 100 MeV. Strong interaction effects are—as we shall see—of the same order of magnitude. The level splittings of the higher-$n$ levels are even smaller and not resolvable with present-day techniques. The transition energies listed in Table 3.3 are calculated using mean 3D-level energies, weighted with their statistical frequencies.

Fig. 3.20 shows the energy levels of protonium and some radiative transitions. Of particular importance are transitions to the 1S ground state, often called *Lyman* series or *K-line* series.
<figure><img src="image_40.png" /><figcaption>Fig. 3.20. Energy levels of antiprotonic hydrogen atoms showing the definition of K, L, M series. Effects due to strong interactions are also indicated.</figcaption></figure>

The 2P → 1S transition is named Lyman-α or K$_{α}$ line; its expected energy without corrections due to strong interactions is 9.406 keV. The 3P → 1S transition is called Lyman-β or K$_{β}$ line, and its energy is 1.737 keV higher. The 4P → 1S transition—the Lyman-γ or K$_{γ}$ line—carries 2.345 keV more energy than the K$_{α}$ line. The nD → 2P transitions are called Balmer series or L-series, with Greek suffixes α, β, γ, δ, ..., ∞ to characterise individual lines. The Paschen series or M-series feeding the 3D levels has energies from 0.608 keV (M$_{α}$) to 1.389 keV (M$_{∞}$). Transitions of type (n, ℓ = n − 1) → (n − 1, ℓ = n − 2) are called circular transitions. They take place between two atomic states with circular classical orbits. Transitions between states with smaller angular momenta are called parallel transitions. They play no significant role in the experiments which are described later in this review.

Of course, experiments on p̅p and p̅d atoms are carried out not to study quantum electrodynamics but strong interaction effects. Before discussing results it is useful to estimate the magnitude which we might expect for these effects. Strong interactions—given, e.g., by the widths of the Δ(1232) or the ρ(770)—have a typical strength of 100–150 MeV. The pion mass µ$_{π}$ appears as a natural scale parameter. In p̅p atoms, strong interactions are smaller due to the large volume of the p̅p atom. We may expect hadronic shifts and half-widths to be of the order

$$
\Delta E \sim \frac{1}{2} \Gamma \sim \frac{\mu_{\pi}}{n^3} \left( \frac{a_{\text{hadr}}}{r_1} \right)^{2/3}, \qquad (3.69)
$$

where a<sub>hadr</sub> ~ 1 fm and r<sub>1</sub> is the first Bohr radius. The wave functions of atomic states have a tail

$$
\Psi_n \propto \exp(-\sqrt{-2m_r E_n} r / \hbar), \quad (3.70)
$$

i.e., extends in a volume that grows as n³. This is why the density at short distances has a n⁻³ dependence, which will be made precise in Section 3.12.
<table><caption>Table 3.4
Expected hadronic and radiative widths in $\bar{p}p$ atoms</caption><thead><tr><th>Hadronic width</th><th>Radiative width</th></tr></thead><tbody><tr><td>Γ<sub>1S</sub>/2 ~ 700 eV</td><td>Stable</td></tr><tr><td>Γ<sub>2P</sub>/2 ~ 30 meV</td><td>Γ<sub>X</sub> = 0.379 meV</td></tr><tr><td>Γ<sub>3D</sub>/2 ~ 2 μeV</td><td>Γ<sub>X</sub> = 38.9 μeV</td></tr></tbody></table>

In Table 3.4 we list the expected size of strong interaction parameters for the S-wave ground states, for the 2P states and for the 3D states, and a comparison with the radiative widths $\Gamma_X$, whose expression will be given below. We show the imaginary part of the energy shift, $\mathrm{Im}(\Delta E) = \Gamma/2$; the real part is expected to be of similar magnitude. The energy shift and width of the 1S states are directly measurable. A direct measurement of the 2P level width is not possible with conventional techniques like solid-state detectors; the expected shifts and widths are, however, within the range of modern X-ray diffraction spectrometers. Strong interaction effects in the 3D and other $\ell \ge 2$ states are unmeasurably small, even when $^3D_1-^3S_1$ mixing is taken into account [261]. This is why, once all relativistic and electromagnetic corrections are taken into account, the measured change of energy $\epsilon$ of photon transitions is identified with the shift of the lowest state in the transition, namely

$$ \epsilon = -\Delta E. \qquad (3.71) $$

The values given in Table 3.4 show that for P levels of $\bar{p}p$ atoms, annihilation is expected to be more likely than emission of X-rays by two orders of magnitude! In addition, collisions of the $\bar{p}p$ atom with neighbouring molecules will lead to premature annihilation reducing the yield of X-rays even further. These two effects made experimental searches for $K_\alpha$ radiation extremely difficult. The difficulties were only overcome when LEAR came into operation.

The width of the 2P levels can also be measured indirectly. The total intensity feeding one of the 2P levels is larger than the intensity of Lyman-$\alpha$ radiation. Protonium atoms in the 2P levels mostly annihilate instead of emitting $K_\alpha$ radiation to the 1S ground state. The intensity balance of radiative transition and annihilation width can be determined from the yield $Y(K_\alpha)$ of $K_\alpha$ radiation compared to the yield $Y(L_{tot})$ with which the 2P levels are populated. This ratio is related to the radiative width $\Gamma_X$ of the 2P fine structure levels and their annihilation widths $\Gamma(^{2s+1}P_J)$:

$$ \begin{aligned} r &= \frac{Y(K_{\alpha})}{Y(L_{\text{tot}})} \\ &= \frac{\Gamma_X}{12} \left[ \frac{3}{\Gamma_X + \Gamma(^1P_1)} + \frac{1}{\Gamma_X + \Gamma(^3P_0)} + \frac{3}{\Gamma_X + \Gamma(^3P_1)} + \frac{5}{\Gamma_X + \Gamma(^3P_2)} \right]. \end{aligned} \qquad (3.72) $$

The equation assumes that the fine-structure levels are populated statistically. If the P levels all have identical widths, the summation yields

$$ Y(K_{\alpha})/Y(L_{\text{tot}}) = r'\Gamma_X/(\bar{\Gamma}_{2P} + \Gamma_X), \qquad (3.73) $$

from which $\bar{\Gamma}_{2P}$ can be deduced. In both relations (3.72) and (3.73), it is tacitly assumed that parallel transitions $nP \to 2S$ do not contribute to $Y(L_{tot})$. This assumption is justified by the preference of the protonium cascade to populate circular orbits.
### 3.12. Quantum mechanics of protonium

In this section, we review how to calculate the properties of protonium in a given potential model. Some results hold beyond this particular framework. The energy shifts and widths can be computed with simple tools, like the Trueman formula. However, the microscopic calculation of branching ratios requires the knowledge of the wave function at short distances. One should thus solve accurately the wave equation with a superposition of long-range Coulombic and short-range nuclear potentials. The corrections due to relativity and QED and these due to strong interactions are essentially additive and thus can be computed independently. This is why we may use the Schrödinger framework to estimate the effect of strong interactions.

#### 3.12.1. Results on pure Coulomb systems

Before introducing strong-interaction corrections at short distances, we recall here some useful results on pure Coulomb bound states.

**Scaling:** Any Coulomb problem with a reduced mass $m_r$ and coupling $e^2$ in the potential $-e^2/r$ can be reduced to the case where $\hbar = 2m_r = e^2 = 1$, for which the radial equation reads as

$$u''_{n,\ell}(r) + \left[ \frac{\ell(\ell+1)}{r^2} + E_n - \frac{1}{r} \right] u_{n,\ell}(r) = 0, \quad (3.74)$$

with $E_n = -1/(4n^2)$. In the actual problem, the energy scale is $2m_r e^4/\hbar^2$ and the unit of distance is $\hbar^2/(2m_r e^2)$.

The radial wave function is defined as usual from the wave function by

$$\Psi_{n,\ell,m}(\vec{r}) = \frac{u_{n,\ell}(r)}{r} Y_{\ell}^{m}(\vartheta, \varphi). \quad (3.75)$$

The normalised S-state wave function of principal quantum number $n$ are given by

$$u_{n,0} = \frac{r \mathcal{L}(n-1, 1, r/n)}{\sqrt{2n^5}} \exp\left(-\frac{r}{2n}\right), \quad (3.76)$$

where $\mathcal{L}$ denotes a Laguerre polynomial.

The normalised P states are

$$u_{n,1} = \frac{r^2 \mathcal{L}(n-2,3,r/n)}{\sqrt{2n^7(n^2-1)}} \exp\left(-\frac{r}{2n}\right). \quad (3.77)$$

**Radiative widths:** De-excitation of highly excited protonium states formed in vacuum occurs via radiative transitions. They populate preferentially “circular” states with maximum angular momentum $\ell = n - 1$, since the angular momentum changes by one unit only, $\ell_f = \ell_i \pm 1$, while the transition probability increases with the energy difference $\Delta E_{if}$ between initial and final state:

$$\Gamma_{\text{rad}} = \frac{4\alpha}{3\hbar} \frac{1}{m_p^2} (\Delta E_{if})^3 (R_{n_i}^{n_f})^2, \quad (3.78)$$

with

$$R_{n_i}^{n_f} = \int_0^\infty u_{n_i, \ell_i}(r) u_{n_f, \ell_f}(r) r \, dr. \quad (3.79)$$
Radiative transitions from high $n, \ell$ states are very slow; from the $n=30$ level, the cascade time is of the order of $\mu$s.

*Value at the origin:* For S states, Eqs. (3.75) and (3.76) imply

$$|\Psi_{n,0}(0)|^2 = \frac{|u'_{n,0}(0)|^2}{4\pi} = \frac{1}{8\pi n^3}. \qquad (3.80)$$

For P states, the analogue is the second derivative of $u_{n,1}$, with the result

$$|u''_{n,1}(0)|^2 = \frac{n^2 - 1}{18n^5}. \qquad (3.81)$$

Thus, the annihilation widths are expected to scale as with principal quantum number according to

$$\begin{aligned} \Gamma_{nS} &= \frac{1}{n^3} \Gamma_{1S}, \\ \Gamma_{nP} &= \frac{32}{3} \left( \frac{1}{n^3} - \frac{1}{n^5} \right) \Gamma_{2P}. \end{aligned} \qquad (3.82)$$

Eq. (3.80) can be derived from the Schwinger rule, which states that $u'_{n,0}(0)^2$ is proportional to the expectation value of the derivative of the potential [262]. Here, this derivative is $1/r^2$, and its expectation value is linked by the Feynmann-Hellmann theorem to the $\ell$-dependence of the binding energy, $\langle r^{-2} \rangle \propto (\partial E_{n,\ell}/\partial \ell)_{\ell=0}$. For P states, the derivation of Eq. (3.81) is a little more laborious.

*Nodal structure:* The structure of radial excitations is important, as annihilation often occurs from $nP$ states with $n > 2$ rather than from $2P$ or from $nS$ states with $n > 1$ rather than from the $1S$ ground state.

In a confining potential such as the harmonic oscillator $Kr^2/2$, the binding energy $E_n$ increases with the number of nodes $n$ and dominates the behaviour of the radial wave function $u_n$ at short distances. In units where $K/2 = 2m_r/\hbar^2 = 1$, the radial equation of S states,

$$u''_{n} + (E_{n} - r^{2})u_{n} = 0, \qquad (3.83)$$

gives energies $E_n = 3 + 2n$ and wave functions $u_n \propto \sin((3 + 2n)^{1/2}r)$ at large $n$ and small $r$, i.e., more and more oscillations closer and closer to the origin as $n$ increases.

The situation is different in the Coulomb case.<fnref n="7" /> In Eq. (3.74), the binding energy $E_n = -1/(4n^2)$ becomes negligible compared to the Coulomb potential at short distances as $n$ increases. For $\ell=0$, the ground state $u_{1S} \propto r \exp(-r/2)$ is nodeless and peaks at the Bohr radius which in our temporary units is $r_1 = 2$. The first excitation $u_{2S} \propto r(4-r) \exp(-r/4)$ has a node at $b_2 = 4$, this ensuring the orthogonality with $u_{1S}$. The first node $b_n$ of $u_{nS}$ should move to smaller $r$ as $n$ increases, from general principles. It quickly converges towards its $n \to \infty$ limit, which is $b_\infty \simeq 3.67$, corresponding to the first zero of a Bessel function that is solution of

$$u''_{\infty}(r) + u_{\infty}(r)/r = 0, \quad u_{\infty}(0) = 0. \qquad (3.84)$$

This is illustrated in Fig. 3.21.

<footnote n="7">⁷ We thank André Martin for an enlightening discussion on this point.</footnote>
<figure><img src="image_41.png" /><figcaption>Fig. 3.21. Radial wave functions $u_{n,0}(r)$ for $n = 1-4$ (left), and $u_{n,1}(r)$ for $n = 2-5$ (right). Natural units are used, so that the Bohr radius $r_1 \sim 50$ fm corresponds to $r_1 = 2$.</figcaption></figure>

For P states, $u_{2P} \propto r^2 \exp(-r/4)$ is nodeless and peaks at $r = 8$. The first radial excitation, $u_{3P} \propto r^2(r-12) \exp(-r/4)$ has a node at $b_3 = 12$. The first node moves to $b_4 \simeq 11.06$ for $n = 4$, then to $b_5 \simeq 10.71$, and quickly reaches its asymptotic value $b_\infty \simeq 10.18$, this again corresponding to a zero of a Bessel function. The first P states are shown in Fig. 3.21.

In short, at small distances the radial functions exhibit a universal shape as $n$ increases, besides the overall normalisation factor corresponding to $u'_{n,0}(0)$ and $u''_{n,1}(0)$. In particular, the nodes never reach the region of small $r$ where one-pion-exchange and other strong interaction effects take place. We thus expect annihilation rates to exhibit a universal $n^{-3}$ behaviour, and thus the branching ratios to be independent of $n$.

### 3.12.2. Hadronic widths and shifts

*Simple Trueman formula:* We have to study the changes that undergo the solution of the radial equation

$$u''(r) - \frac{l(l+1)}{r^2} u(r) + 2m_r(E-V)u(r) = 0, \qquad (3.85)$$

where the pure Coulomb potential $V^c = -\alpha/r$ is replaced by the total potential $V = V^c + V^n$ with a nuclear piece $V^n$. For protonium the reduced mass is $m_r = m_p/2$. One typically gets shifts of the order of 1 keV for the 1S level, small compared to the Bohr energy $E^c = -12.49$ keV. This does not mean, however, that $V^n$ can be treated perturbatively. A first-order estimate,

$$\Delta E \equiv E - E^c = \int_0^\infty u_c(r)^2 V^n dr, \qquad (3.86)$$

would overestimate $|\Delta E|$ by orders of magnitude! In the limit of a hard core of radius $a$, the integration (3.86) gives an infinite shift when one uses the undistorted wave function $u_c(r)$ but vanishes exactly when $u_c(r)$ is replaced by the “true” wave function $u(r)$, which vanishes for $r < a$.

The ordinary expansion in terms of powers of the additional potential is not applicable here. What is appropriate is the “radius perturbation theory” [263,264], where the expansion parameter is the ratio $a/r_1$ of the scattering length $a$ in the nuclear potential $V^n$ to the Bohr radius $r_1$ of the atom.
<figure><img src="image_42.png" /><figcaption>Fig. 3.22. Comparison of the 1S radial wave functions with a pure Coulomb potential ($u_c$) and a Coulomb potential added to a hard core of radius $a = 1$ fm.</figcaption></figure>

At first order, one gets the famous Trueman formula [263,265], which reads as

$$ \Delta E = \pi m_r |\Psi_c(0)|^2 \frac{a}{r_1}, \qquad (3.87) $$

for S-wave states. There is an analogue for P-waves, where the first derivative of the radial wave function $du_c(r)/dr|_{r=0} = \sqrt{4\pi}\Psi_c(0)$ is replaced by the second derivative and $a$ by the scattering volume. A poor-man derivation is as follows. At zero energy, the nuclear potential is equivalent to a hard core of radius $a$, where $a$ is the scattering length, first supposed to be real and positive and later on allowed to be negative or complex. The two radial wave functions $u_c(r)$ (pure Coulomb) and $u(r)$ (total) are submitted to the same equation with different boundary conditions

$$ \begin{gathered} u_c'' - \frac{l(l+1)}{r^2} u_c + 2m_r(E^c - V^c)u_c = 0, \quad u_c(0) = u_c(\infty) = 0, \\ u'' - \frac{l(l+1)}{r^2} u + 2m_r(E - V^c)u = 0, \quad u(a) = u(\infty) = 0. \end{gathered} \qquad (3.88) $$

For $r > a$, $uu_c'' - u''u_c = 2m_r(\Delta E)uu_c$, hence

$$ \Delta E = \frac{1}{2m_r} \frac{[uu_c' - u'u_c]_a^\infty}{\int_a^\infty uu_c dr}. \qquad (3.89) $$

In first approximation, the denominator is equal to 1; $u'(a) = 2r_1^{-3/2}$ if one accepts that $u(r)$, at large distances, is nothing but $u_c(r)$ translated by $a$ (see Fig. 3.22); $u_c'(a) = 2ar_1^{-3/2} + O(a^2/r_1^2)$. Then one gets the desired formula.

*Improved Trueman formula:* A more rigorous derivation makes use of the low-energy expansion of the scattering amplitude for $V = V^c + V^n$ and introduces the binding energy as a pole in this
amplitude. This results into the more accurate formula

$$ \Delta E = E_{n,l} - E_{n,l}^{c} = -E_{n,l}^{c} \left(\frac{4}{n}\right) \frac{a_{l}^{sc}}{r_{1}^{2l+1}} \alpha_{n,l} \left(1 - \frac{a_{l}^{sc}}{r_{1}^{2l+1}} \beta_{n,l} + \dots \right), \quad (3.90) $$

where

$$ \alpha_{n,0} = 1, \quad \alpha_{n,l} = \prod_{s=1}^{l} \left( \frac{1}{s^2} - \frac{1}{n^2} \right) \quad (l > 0), \quad (3.91) $$

and the $\beta_{n,l}$ are numerical coefficients. For S and P waves, they take the values

$$ \beta_{n,0} = 2 \left( \log n + \frac{1}{n} - \Psi(n) \right), \quad \beta_{n,1} = \alpha_{n,1} \beta_{n,0} - \frac{4}{n^3}, \quad (3.92) $$

$\Psi$ being the digamma function [266].

In Eq. (3.90), $a_l^{sc}$ is the *Coulomb corrected* scattering length (volume, etc.) for angular momentum $l$. It is defined as usual from the low-energy expansion of the reaction matrix $K$ as a function of the c.m. momentum $p$ as

$$ K^{-1}(p) = -\frac{1}{a_l} + \frac{\rho_l p^2}{2} + \dots \quad (3.93) $$

Unlike the case of a short-range interaction, where $K^{-1} = p^{2l+1} \cot \delta_l(p)$, the Coulomb-corrected reaction matrix is defined as

$$ K^{-1}(p) = p^{2l+1} g_l(\eta) [C_0^2(\eta) \cot \delta_l^{sc}(p) + 2\eta h(\eta)], \quad (3.94) $$

where $\eta = -1/(pr_1)$ is the Coulomb parameter, $\delta_l^{sc}$ the Coulomb-corrected phase-shift (measuring the matching of the radial wave function to the regular and irregular asymptotic Coulomb wave functions), and

$$ \begin{aligned} g_0(\eta) &= 1, \\ g_l(\eta) &= \left[ \frac{1}{(2l+1)!!} \right]^2 \prod_{s=1}^{l} \left( 1 + \frac{\eta^2}{s^2} \right) \quad (l = 1, 2, \dots), \\ C_0^2(\eta) &= \frac{2\pi\eta}{\exp(2\pi\eta) - 1}, \\ h(\eta) &= \frac{\Psi(i\eta) + \Psi(-i\eta) - \log(\eta^2)}{2}. \end{aligned} \quad (3.95) $$

Eq. (3.90) gives the protonium energy level shifts as a function of the $\bar{p}p$ scattering length (or volume). See, e.g., [265,267].

For illustration, we display in Fig. 3.23 the results obtained for a hard-core potential. Natural units are used, so the shift $\Delta E$ has to be compared to the unperturbed $E_1 = -1/4$, and a core of radius 1 fm corresponds to $a \approx 0.04$ if the Bohr radius is $r_1 = 2$. The naive estimate $a/2$ is improved to $a^c/2$ in terms of the Coulomb-corrected scattering length. The exact result is easily obtained: the
<figure><img src="image_43.png" /><figcaption>Fig. 3.23. Comparison of the naive Trueman formula (3.87), the Coulomb-corrected formula (3.90) and the exact calculation for a hard core of radius a. Natural units are used.</figcaption></figure>

energy is such that the Coulomb wave function with proper vanishing at large distance has a node at the border of the core.

In the presence of Coulomb interaction, the S-wave scattering length is also related to the annihilation cross-section at low energy by

$$p^2 \sigma_{\text{ann}}^{\text{sc}}(\text{S wave}) = \frac{8\pi^2}{1 - \exp(2\pi\eta)} \frac{\text{Im}(-a_0^{\text{sc}}/r_1)}{|1 + i pw(\eta)a_0^{\text{sc}}|^2}, \quad (3.96)$$

where $w(\eta) = C_0^2(\eta) - 2i\eta h(\eta)$.

Antiproton-proton interactions are known to have contributions from P waves even at very low energies. This contribution can be obtained by replacing the scattering length $a_0^{\text{sc}}$ by $a_1^{\text{sc}}(1 + \eta^2)p^2$ where $a_1^{\text{sc}}$ is the corresponding P-wave spin averaged scattering volume [268]:

$$p^2 \sigma_{\text{ann}}^{\text{sc}}(\text{P wave}) = 24\pi^2 \frac{1 + 1/\eta^2}{1 - \exp(2\pi\eta)} \frac{\text{Im}(-a_1^{\text{sc}}/r_1^3)}{|1 - pw(\eta)(1 + 1/\eta^2)\text{Im} a_1^{\text{sc}}/r_1^3|^2}. \quad (3.97)$$

There is some confusion in the literature about the validity of the Trueman formula, with a tentative clarification in a survey by Carbonell et al. [265]. We first note a lack of unified conventions for defining $a$, $E$, and $\Delta E$. Secondly, there are claims for the Trueman formula being inaccurate. The problem comes in fact from the Coulomb corrections to the scattering length, which are often omitted or badly computed. In particular, some popular prescriptions for calculating $a_\ell^{\text{sc}} - a_\ell$ turn out to be inadequate [265]. In fact, once the Coulomb corrected scattering length $a_\ell^{\text{sc}}$ is properly computed, the Trueman formula (3.90) turns out to be very precise.

**Pathological cases:** The most famous exception would be a situation where a nuclear bound state lies very close to the threshold. The scattering length becomes large and radius perturbation theory diverges. A first analysis of this situation is based on a potential [269,242]

$$V(\lambda) = V^c + \lambda V^n, \quad (3.98)$$

where $V^c = -\alpha/r$ is the usual Coulomb term and $V^n$ the strong-interaction potential assumed to be real and attractive. Note that Zel'dovich's work [269] was inspired by a problem encountered in solid-state physics.
<figure><img src="image_44.png" /><figcaption>Fig. 3.24. Energy levels of an atom for a Coulomb potential modified by a square well of radius $a = 0.04$ and increasing depth $V$. Natural units are used, with the unperturbed 1S energy is $-1/4$ and the Bohr radius $r_1 = 2$. Also shown are the unperturbed levels 1S, 2S and 3S, and the critical value of the strength $V$ for binding one or two S states in the square well.</figcaption></figure>

For small values of the strength $\lambda$, the Coulomb spectrum is slightly shifted downwards. When $\lambda$ approaches the critical value $\lambda_0$ for which $\lambda V^n$ supports a bound state, there is a sudden change: the former atomic 1S state, in the keV range, becomes a nuclear state with binding energy in MeV or even tens or hundreds of MeV. The former 2S state quickly becomes what looks like a modified 1S state, with a node at short distance. Atomic spectroscopy shows a repulsive energy shift $\Delta E$ associated with an attractive potential! The observation of a positive value of $\Delta E_{1S}$ may be the consequence of the strong binding force leading to nuclear bound states in the $^1S_0$ and $^3S_1$ partial waves!

In the transition region near $\lambda_0$, the spectrum is completely disorganised. If $\lambda$ is further increased, the scattering length $a(\lambda)$ becomes negative again. Another sharp transition occurs at $\lambda_1$ for which $\lambda V^n$ starts supporting a second bound state. This is illustrated in Fig. 3.24. It corresponds to a Wood-Saxon potential of depth $-V$ and radius $a = 0.04$ in natural units, i.e., about 1 fm for protonium.

In the actual situation, the short-range part includes a strong annihilation potential and mixes $\bar{n}$ to $\bar{p}$ in the wave function, so that the pattern is more involved than in the above models. Still, oscillations are seen in the protonium wave function at short distances [270], reminiscent of the nodes one gets with one-channel, real potentials.

The role of absorption on the Zel'dovich-Shapiro rearrangement of atomic levels has been carefully studied by Gal et al. [271], who used an optical potential

$$V^n \propto -(b_r + ib_i)\rho(r), \qquad (3.99)$$

where $\rho(r)$ is a typical nuclear density. For very small $b_i$, the rearrangement persists, though the real part of the energy changes less rapidly than in Fig. 3.24, which corresponds to $b_i = 0$. When $b_i$ increases, the rearrangement disappears: the $nS$ level is “shaken” when $g_r$ passes near its critical
<figure><img src="image_45.png" /><figcaption>Fig. 3.25. First levels of the atomic spectrum in a model with Coulomb potential and point interaction of strength $g_r + ig_i$, as a function of $g_r$, for $g_i = 5$.</figcaption></figure>

value, but the nS energy remains in the region of the unperturbed nS. In short, the “atomic” and the “nuclear” spectra become independent.

The numerical analysis of Ref. [271] has been checked analytically by Raina and Richard [272], who used a model where the Coulomb potential is supplemented by a point interaction. When the strength $g_r - ig_i$ of the point interaction is real, or when $g_i$ is very weak, then the spectrum reproduces the Zel'dovich effect of Fig. 3.24, with however, only one nuclear bound state. For larger $g_i$, the atomic spectrum, as a function of $g_r$ is given in Fig. 3.25. The nuclear state, instead of being generated from the atomic spectrum as in Fig. 3.24, is disconnected [272].

*Direct numerical estimate of the complex shift:* Another approach to $\Delta E$ consists of solving directly Eq. (3.85) with appropriate numerical algorithms. One should only care that $V^c$ and $V^n$ require different meshes in the discretisation. A direct numerical method looks a little heavy if $\Delta E$ only is needed. However, it also provides the values of $u(r)$, and it is easily generalisable to the case of coupled equations. The first coupling one has to consider comes from charge exchange. Strong interactions contains a $\bar{p}p \rightarrow \bar{n}n$ component, and this results into the following equations [147], written here for $^1S_0$.

$$ \Psi = \frac{u(r)}{r} |\bar{p}p\rangle + \frac{w(r)}{r} |\bar{n}n\rangle, \qquad (3.100) $$

$$ \begin{gathered} u'' + 2m_r \left[ E + \frac{\alpha}{r} + V_{\bar{p}p} \right] u = 2m_r V_{ce} w, \\ w'' + 2m_r \left[ E - 2\delta m + V_{\bar{n}n} \right] w = 2m_r V_{ce} u, \end{gathered} \qquad (3.101) $$

where $\delta m = m_n - m_p$ is the neutron-to-proton mass difference, and the nuclear potentials are $V_{\bar{p}p} = V_{\bar{n}n} = (V_0 + V_1)/2$ and $V_{ce} = (V_1 - V_0)/2$ in terms of the potentials in isospin eigenstates. For $^1P_1$ and $^3P_0$, one gets the same equations with an orbital barrier $2/r^2$. For $^3S_1$ and $^3P_2$, one should also account for orbital mixing due to tensor forces and there are altogether four coupled equations.

Several authors have calculated the complex energy shifts of protonium, using a variety of realistic optical potentials. We reproduce in Table 3.5 part of the compilation by Carbonell et al. [265], for the Dover-Richard and Kohno-Weise potentials introduced earlier in this chapter.
<table><caption>Table 3.5
Energy shift and half-width, and ratio of *I* = 1 to 0 widths for the lowest states of protonium, as calculated from the Dover-Richard and Kohno-Weise potentials</caption><thead><tr><th>State</th><th>Units</th><th>Model</th><th>Re(ΔE)</th><th>Γ/2</th><th>Γ<sub>1</sub>/Γ<sub>0</sub></th></tr></thead><tbody><tr><td rowspan="3">1<sup>1</sup>S<sub>0</sub></td><td rowspan="3">kev</td><td>DR1</td><td>0.54</td><td>0.51</td><td>0.68</td></tr><tr><td>DR2</td><td>0.58</td><td>0.52</td><td>0.80</td></tr><tr><td>KW</td><td>0.50</td><td>0.63</td><td>0.68</td></tr><tr><td rowspan="3">2<sup>1</sup>S<sub>0</sub></td><td rowspan="3">ev</td><td>DR1</td><td>68</td><td>66</td><td>0.68</td></tr><tr><td>DR2</td><td>73</td><td>67</td><td>0.80</td></tr><tr><td>KW</td><td>65</td><td>78</td><td>0.68</td></tr><tr><td rowspan="3">1<sup>3</sup>SD<sub>1</sub></td><td rowspan="3">kev</td><td>DR1</td><td>0.77</td><td>0.45</td><td>0.79</td></tr><tr><td>DR2</td><td>0.82</td><td>0.46</td><td>0.79</td></tr><tr><td>KW</td><td>0.78</td><td>0.49</td><td>0.90</td></tr><tr><td rowspan="3">2<sup>1</sup>P<sub>1</sub></td><td rowspan="3">meV</td><td>DR1</td><td>-26</td><td>13</td><td>0.96</td></tr><tr><td>DR2</td><td>-24</td><td>14</td><td>0.61</td></tr><tr><td>KW</td><td>-29</td><td>13</td><td>0.82</td></tr><tr><td rowspan="3">2<sup>3</sup>P<sub>0</sub></td><td rowspan="3">meV</td><td>DR1</td><td>-74</td><td>57</td><td>0.03</td></tr><tr><td>DR2</td><td>-62</td><td>40</td><td>0.05</td></tr><tr><td>KW</td><td>-69</td><td>48</td><td>0.03</td></tr><tr><td rowspan="3">2<sup>3</sup>P<sub>1</sub></td><td rowspan="3">meV</td><td>DR1</td><td>36</td><td>10</td><td>9.4</td></tr><tr><td>DR2</td><td>36</td><td>8.8</td><td>6.5</td></tr><tr><td>KW</td><td>29</td><td>11</td><td>9.7</td></tr><tr><td rowspan="3">2<sup>3</sup>PF<sub>2</sub></td><td rowspan="3">meV</td><td>DR1</td><td>-4.8</td><td>15</td><td>0.63</td></tr><tr><td>DR2</td><td>-5.9</td><td>16</td><td>0.66</td></tr><tr><td>KW</td><td>-8.5</td><td>18</td><td>0.43</td></tr></tbody></table>

A few comments are in order:

1. The calculated widths are of the order of magnitude expected from the very simple ansatz in Eq. (3.69).

2. There is reasonable agreement among the models and also, as we shall see in Section 6, with data.

3. The effect of orbital mixing (S-D or P-F mixing) is not dramatic.

The complex energy shifts is essentially equivalent to the (Coulomb corrected) scattering length or volume. Table 3.6 displays a partial compilation of scattering lengths.

Simple (i.e., not adjusted to the recent LEAR data) potential models tend to give similar predictions, which are in reasonable agreement with the protonium data, via the Trueman formula. Early potential models, such as BP, DR1, DR2 or KW in Table 3.6 had an isospin-independent annihilation. This seems sufficient to reproduce the main features of $\bar{p}p$ and $\bar{p}p \to \bar{n}n$ data. The work of Bydzovsky et al. [222] and Pirner et al. [273] also include the constraint from $\bar{n}p$ absorption at low energy [274].
Table 3.6

<table><caption>Comparison of $\bar{p}p$ scattering lengths, computed from early potentials models: static Bryan-Phillips (BP), Dover, Richard and Sainio (DR1, DR2), Kohno and Weise (KW), Bydzovsky, Mach and Nichitiu (BMN), and from simple or more refined analyses of scattering data</caption><thead><tr><th>State</th><th>Source</th><th>Ref.</th><th>Re(a<sub>sc</sub>) (fm)</th><th>Im(a<sub>sc</sub>) (fm)</th></tr></thead><tbody><tr><td><sup>1</sup>S<sub>0</sub></td><td>Pot. BP</td><td>[210]</td><td>1.10</td><td>-0.72</td></tr><tr><td><sup>1</sup>S<sub>0</sub></td><td>Pot. DR1</td><td>[265]</td><td>0.62</td><td>-0.63</td></tr><tr><td><sup>1</sup>S<sub>0</sub></td><td>Pot. DR2</td><td>[265]</td><td>0.68</td><td>-0.68</td></tr><tr><td><sup>1</sup>S<sub>0</sub></td><td>Pot. KW</td><td>[265]</td><td>0.57</td><td>-0.77</td></tr><tr><td><sup>3</sup>SD<sub>1</sub></td><td>Pot. BP</td><td>[210]</td><td>0.68</td><td>-0.85</td></tr><tr><td><sup>3</sup>SD<sub>1</sub></td><td>Pot. DR1</td><td>[265]</td><td>0.91</td><td>-0.57</td></tr><tr><td><sup>3</sup>SD<sub>1</sub></td><td>Pot. DR2</td><td>[265]</td><td>0.98</td><td>-0.59</td></tr><tr><td><sup>3</sup>SD<sub>1</sub></td><td>Pot. KW</td><td>[265]</td><td>0.92</td><td>-0.63</td></tr><tr><td>Average</td><td>Pot. BMN</td><td>[222]</td><td>0.82</td><td>-0.67</td></tr><tr><td>Average</td><td>Eff. range exp.</td><td>[275]</td><td>0.1</td><td>-0.7</td></tr><tr><td>Average</td><td>Eff. range exp.</td><td>[273]</td><td>0.4</td><td>-0.6</td></tr><tr><td>Average</td><td>Eff. range exp.</td><td>[276]</td><td>0.6</td><td>-0.7</td></tr><tr><td>Average</td><td>Fit scatt. data</td><td>[277]</td><td>0.52</td><td>-0.56</td></tr></tbody></table>

*Spin dependence of 2P shifts:* The strong spin dependence of the $\bar{N}N$ interaction is responsible for the differences in Table 3.5 between the $^1P_1$, $^3P_0$, $^3P_1$ and $^3P_2$ energy shifts. There are also noticeable differences for the widths which can be understood in a simple semi-classical picture.

In Fig. 3.26 are shown the effective potentials (including the centrifugal barrier) for $^1P_1$ and $^3P_0$. It is clearly seen that tunnelling towards the annihilation region through the centrifugal barrier is much easier in the latter case.

### 3.12.3. Isospin mixing

Optical potentials include an isovector part, in particular pion exchange, leading to isospin mixing in the protonium wave functions at small distances.

The isospin content, as defined in Eq. (3.100), can be recombined into

$$ \Psi = u_1 \mathcal{I} = 1 \rangle + u_0 |\mathcal{I} = 0 \rangle \qquad (3.102) $$

with

$$ u_1 = \frac{u(r) + w(r)}{\sqrt{2}}, \qquad u_0 = \frac{u(r) - w(r)}{\sqrt{2}}. \qquad (3.103) $$

Assuming an overall normalisation

$$ \int_{0}^{\infty} (|u_0(r)|^2 + |u_1(r)|^2) dr = 1, \qquad (3.104) $$

the annihilation width in isospin $\mathcal{I}$ can be computed as

$$ \Gamma_{\mathcal{I}} = -2 \int_{0}^{\infty} \mathrm{Im} V_{\mathcal{I}}(r) |u_{\mathcal{I}}(r)|^2 dr. \qquad (3.105) $$

This was done by Kaufmann and Pilkhun [147], Richard and Sainio [220], Gutsche et al. [278], and others. Carbonell et al. [265] have estimated the ratio of the integrated isovector to isoscalar annihilation widths $\Gamma_1/\Gamma_0$ with various potentials. Their results are reproduced in Table 3.5.
<figure><img src="image_46.png" /><figcaption>Fig. 3.26. The P-wave effective radial potential is the sum of the Coulomb contribution and the centrifugal term (dotted line). It is supplemented by the $\bar{p}p$ strong potential (average of $I = 0$ and 1) shown here for the $^3P_0$ (solid line) and $^1P_1$ (dashed line) partial waves. We use the Paris model, but the effect is largely model independent, since dominated by pion exchange. Annihilation proceeds by tunnelling between the Coulomb attractive tail (magnified in the insert) and the short-range region. The height and width of the barrier depends dramatically on the quantum numbers.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Fig. 3.27. Effective radial potential, sum of the Coulomb, centrifugal and strong-interaction contributions, for the $I = 0$ and 1 components of the $^3P_0$ partial wave. The dotted line is obtained when strong interaction is switched off. The Paris model is used, but the isospin dependence is largely model independent, since dominated by one-pion exchange. Note that the Coulomb potential has also off-diagonal components in this isospin basis.</figcaption></figure>

The differences between the $I = 0$ and 1 components of the $^3P_0$ width can be understood from Fig. 3.27. The meson exchange potential is attractive for $I = 0$ and repulsive for 1. So the tunnelling from the Coulomb attraction to the annihilation region is much easier for $I = 0$ than for $I = 1$.
Remarks are in order:

1. Isospin mixing (ñn admixture) has no significant impact on the complex energy shift. But very important effects are predicted for the wave function, and hence for the annihilation properties.

2. The isospin-mixing effects are particularly large for two levels, the $2^3P_0$ state which is predicted to be an almost pure isoscalar state, and the $2^3P_1$ state which is dominantly of isovector character.

# 4. Antinucleon-nucleon scattering data

In this section, we highlight the salient features of scattering data. We do not attempt to collect all pieces of the available experimental information, nor even to list all relevant references in the bibliography. One can refer for instance to the papers by Timmermans et al. [279] and Pignone et al. [277], which contain a comprehensive bibliography (we shall comment the corresponding analyses in Section 6). Most pre-LEAR data are reviewed in the 1984 compilation by the High-Energy Reactions Analysis Group [280]. There have been $\bar{N}N$ scattering experiments performed at KEK, BNL, FNAL and CERN. Some results of these experiments will be presented in the next sections, together with the results of the LEAR experiments. Their technical aspects were described in Section 2.

## 4.1. Integrated cross-sections

### 4.1.1. $\bar{p}p$ cross-sections

Some representative measurements of the $\bar{p}p$ total, annihilation, elastic, and charge-exchange integrated cross-sections are shown in Fig. 4.1 as a function of the $\bar{p}$ incident momentum.

The data are from:

* Total cross-section:

* Experiment PS172 [77,78]. The systematic and statistical errors are added quadratically; in addition, there is an overall normalisation error of 0.7–0.9% at higher $\bar{p}$ momenta.

* Experiment PS173 [93,92]. Systematic errors were not quoted separately; the data are from the $\rho$-parameter measurement.

* Hamilton et al. [281] (BNL AGS). The quoted absolute normalisation uncertainty is 1.5%; the relative difference with the PS172 data [77] is about 3.4%.

* Ganguli et al. [282] (CERN PS). The systematic errors, estimated to 2 mb, have not been added. Among the data sets which are not shown one may mention:

* Nakamura et al. [283] (KEK), in agreement with Hamilton et al. [281] data. The relative difference with the PS172 data [77] is about 4.3%.

* Chaloupka et al. [284] (CERN PS, a bubble chamber experiment), in agreement with Hamilton et al. [281], plus evidence for the S meson.

* Annihilation cross-section:

* Experiment PS173 [90]. The systematic and statistical errors are added quadratically; the quoted overall normalisation error varied from 2.2% to 4.4%. A correction for annihilation in all-neutral channels (+3.0 ± 1.1% at 297 meV/c) was applied.
<figure><img src="image_48.png" /><figcaption>Fig. 4.1. Representative $\bar{p}p$ total, annihilation, elastic, and charge-exchange cross-section data versus the incoming antiproton momentum.</figcaption></figure>

o Experiment PS201 [96,97]. The systematic and statistical errors are added quadratically; for the two sets of data the quoted normalisation errors are 3.2% and 2%, respectively.

o Bizzarri et al. [285] (CERN PS, a bubble chamber experiment). The values were extracted from $\bar{p}d$ annihilation data, looking for events with even or 0 prongs (i.e., charged mesons) in the final state, without corrections for “shadowing” nor charge-exchange events. The contribution of the charge-exchange cross-section was estimated to be about 9%.

o Ganguli et al. [282]. The data were obtained from their total cross-section measurements, subtracting the interpolated measurements of the elastic and charge-exchange cross-section from Coupland et al. [286] and Alston-Garnjost et al. [287].

o Kalogeropoulos-Tzanakos [288] (BNL). The values were extracted from $\bar{p}d$ annihilation data, looking for events with even or 0 prongs in the final state, without corrections for “shadowing” nor charge-exchange events.
Other data exist, but are not shown in the figure:

o Chaloupka et al. [284]:

o Brando et al. [289] (BNL AGS). They measured several points from 308 to 591 MeV/c; the normalisation is from previous published data:
* Hamilton et al. [281]. They measured several points at closely spaced $\bar{p}$ momenta in the range 355–1066 MeV/c; the measured annihilation cross-sections into charged particles are not corrected for background from large angle $\bar{p}p$ elastic scattering and neutral interactions in the veto box.

* Elastic cross-section:

* Experiment PS172 [86]. The integrated elastic $\bar{p}p$ cross-sections were estimated by fitting the differential cross-section data with Legendre Polynomials; the quoted normalisation error is about 10%.

* Chaloupka et al. [284].

* Coupland et al. [286]. The measurement was obtained from the Eisenhandler et al. [290] differential cross-section data; the quoted normalisation error is 4%.

* Charge-exchange cross-section:

* Alston-Garnjost et al. [287] (BNL AGS). The quoted systematic error varies from 5% (higher momenta) to 10% and it is mainly due to the absorption correction.

* Hamilton et al. [291] (BNL AGS). The quoted systematic error varies from 3% (higher momenta) to 5%. The point to point uncertainty is less than 1%. For the two lowest momentum points large uncertainties exist in the mean interaction momentum.

Not shown in the figure are the data from Cutts et al. [292], in agreement with the other measurements, but probably with large systematic errors.

Comments on those integrated cross-sections are in order:

(i) The total cross-section shows the $1/k^2$ behaviour, typical of diffraction.

(ii) The cross-sections are large. This is due to the long-range interaction. A simple black disk would be restricted by its own size $(2\pi(\hbar/m_\pi c)^2 \approx 120 \text{ mb})$.

(iii) The annihilation cross-section $\sigma_{\text{ann}}$ is larger than the elastic one, $\sigma_{\text{el}}$. A black disk would give comparable $\sigma_{\text{ann}}$ and $\sigma_{\text{el}}$, a result reminiscent from the Babinet theorem in optics. Again, we have a combined effect of the strong absorption, and of long-range forces [29]: the latter focuses the wave function towards the short-distance region where the former act. This is tentatively realized in optical models with real and imaginary components, and in some boundary-condition models, to be discussed in the next chapter.

(iv) As mentioned in Section 3.8, the integrated cross-sections can be reproduced in potential models provided the annihilation potential is strong enough up to 0.8–1 fm. If the range is smaller one can obtain the correct values at one energy by adjusting the strength of Im $V$, but the energy dependence of the cross-section will not be reproduced.

(v) The charge-exchange cross-section provides the strongest constraint on the strength of the potential. Since $\sigma_{\text{ce}} = |\mathcal{T}_{I=0} - \mathcal{T}_{I=1}|^2$ a dramatic cancellation occurs, and the cross-section is only due to the tail of $\pi$- and $\rho$-exchange. With isovector exchange only, the two isospin amplitudes would have opposite signs, and thus add coherently, leading to a charge-exchange cross-section four times larger than the elastic one! Such a coherence presumably holds for the high partial waves, which are dominated by one-pion exchange. To get a small cross-section, one needs an effective cancellation between the $I=1$ and 0 amplitudes in the low partial waves. This reflects that annihilation is almost isospin blind. This is probably not accidental. As there are many contributions to short-range annihilation, the equality of the two isospin amplitudes should reflect a kind of symmetry.
<figure><img src="image_49.png" /><figcaption>Fig. 4.2. $\bar{p}p$ total and annihilation cross-section data (a) and the corresponding $\beta\sigma$ values (b) versus the incoming antiproton momentum.</figcaption></figure>

(vi) Structures have been sometimes seen when scanning the cross-sections as a function of the incoming energy $\sqrt{s}$, and tentatively interpreted as NN or $qq\bar{q}\bar{q}$ type of baryonium. They are not confirmed by the most recent measurements. The most famous baryonium candidate was the S(1936), searched for with detailed cross-section scans around an incident $\bar{p}$ momentum of 500 MeV/c. The latest experiments did not confirm the original observations (for a review see [293]).

(vii) As experiments have been done with unpolarised beams and targets, or with at most a polarised target, no information on $\Delta\sigma_L = \sigma_{\leftrightarrow} - \sigma_{\rightarrow\rightarrow}$ nor on $\Delta\sigma_T = \sigma_{\uparrow\downarrow} - \sigma_{\uparrow\uparrow}$ exists. If one of these $\Delta\sigma$ is large, one could use a polarised proton target to filter one spin component, and possibly get a polarised $\bar{p}$ beam [71].

The low-energy data of the $\bar{p}p$ total and annihilation cross-sections are shown in Fig. 4.2(a). Fig. 4.2(b) shows the behaviour of $\beta\sigma$. To be noticed, the very fast rise of the annihilation cross-section for momenta smaller than 100 MeV/c. From general principles [294] it was known that the total reaction cross-section near threshold in a hadronic system with Coulomb attraction should have exhibited a $1/v^2$ behaviour, $v$ being the velocity of the incident particle. Fig. 4.2(b) shows the behaviour of $\beta\sigma$: the very low momentum PS201 data indeed show very clearly the breakdown of the $1/v$ law and the onset of this new regime, and agree with the theoretical expectation [295].

### 4.1.2. $\bar{n}p$ and $\bar{p}n$ cross-sections

For obvious experimental reasons, the data are much poorer in the $I=1$ sector, corresponding to $\bar{p}n$ or $\bar{n}p$ scattering.

Very few data exist in the $\bar{p}n$ channel, from the pre-LEAR period. They are extracted from $\bar{p}d$ measurements with bubble chambers. In Fig. 4.3 the $\bar{p}n$ data are compared from:

* Bizzarri et al. [285]: the values were extracted from $\bar{p}d$ annihilation data, looking for events with an odd number of prongs in the final state;

* Kalogeropoulos-Tzanakos [288]: the same technique was used; are compared with the $\bar{n}p$ annihilation cross-section from PS201 [103].
<figure><img src="image_50.png" /><figcaption>Fig. 4.3. The $\bar{n}p$ (open points) and the $\bar{p}n$ (closed points) annihilation cross-sections versus the incoming antinucleon momentum.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>Fig. 4.4. The $\bar{n}p$ total (a) and annihilation (b) cross-sections $\sigma_{tot}$ and $\sigma_{ann}$ versus the incoming antineutron momentum.</figcaption></figure>

The $\bar{n}p$ total and annihilation (no elastic data exist) cross-sections are shown in Fig. 4.4(a) and (b) as a function of the $\bar{n}$ incident momenta. The data are from:

* Total cross-section:

* Experiment PS201 [100]. The systematic and statistical errors are added quadratically; no normalisation error is quoted.

* Armstrong et al. [101]. The total $\bar{n}p$ cross-section was measured using the transmission method. The $\bar{n}$ were produced by $\bar{p}p$ charge-exchange reaction on scintillation counters. The systematic and statistical errors added quadratically in the figure.
<figure><img src="image_52.png" /><figcaption>Fig. 4.5. The $\bar{n}p \beta \sigma_{\text{tot}}$ (a) and $\beta \sigma_{\text{ann}}$ (b) measured values versus the incoming antineutron momentum.</figcaption></figure>

* Annihilation cross-section:

o Experiment PS201 [103]. The systematic and statistical errors are added quadratically; the quoted normalisation errors are about 10%.

o Armstrong et al. [101]. The annihilation cross-section was obtained from the total cross-section by subtracting the elastic cross-section, evaluated using a parametrisation from the potential model of Ref. [211].

While the $\bar{n}p$ total cross-section data from PS201 and Armstrong et al. are in fair agreement, the more precise annihilation cross-section data do not agree, and in the low momentum end ($\sim 100$ MeV/c) the PS201 data suggest rather low values. The different behaviour of the data is even more apparent in Fig. 4.5, where the quantities $\beta \sigma_{\text{tot}}$ and $\beta \sigma_{\text{ann}}$ are plotted: while the $\beta \sigma_{\text{ann}}$ data of Armstrong et al. are consistent with a constant value of about 40 mb, similar to the $\bar{p}p$ case, the more precise PS201 data show a strong decrease at small momentum, and deviate from the $1/v$ law.

The authors of Ref. [100] were not able to describe both their $\sigma_{\text{tot}}$ and $\sigma_{\text{el}}$ data with an effective-range-expansion technique. While the fits give good results when applied to either set of data, no parameters could be found to fit simultaneously the two sets. They interpret this situation as being due to an anomalous energy behaviour of $\sigma_{\text{el}}(\bar{n}p) = \sigma_{\text{tot}}(\bar{n}p) - \sigma_{\text{ann}}(\bar{n}p)$, which shows a profound dip in the momentum range 60–90 MeV/c. They tentatively relate this anomaly to the presence of a near-threshold resonance, and have proposed a new set of measurements of the elastic $\bar{p}p$ cross-section at the CERN Antiproton Decelerator [296].

### 4.1.3. Comparison of $\bar{p}p$ and $\bar{n}p$ cross-sections

All available $\bar{n}p$ cross-section data are compared with the corresponding $\bar{p}p$ data in Fig. 4.6: Fig. 4.6(a) and (b), the $\bar{p}p$ total cross-section results are from experiment PS172 [77,78], and the corresponding $\bar{n}p$ data from experiment PS201 [100] and from Armstrong et al. [101], respectively; in Fig. 4.6(c) and (d), the $\bar{p}p$ annihilation cross-section from experiment PS173 [90] and PS201 [96,97] are compared with the $\bar{n}p$ annihilation cross-section data from experiment PS201 [103].
<figure><img src="image_53.png" /><figcaption>Fig. 4.6. Comparison between the $\bar{p}p$ (black points) and the $\bar{n}p$ (open points) total ((a) and (b)) and annihilation ((c) and (d)) cross-sections.</figcaption></figure>

The rough equality of the $\bar{p}p$ and $\bar{n}p$ cross-sections, like the smallness of the $\bar{p}p \rightarrow \bar{n}n$ cross-section, hints at the fact that annihilation is almost isospin independent. A closer look at the total cross-sections suggests somewhat smaller values for the $\bar{n}p$ cross-section, although, in the overlap region with the $\bar{p}p$ measurements, the effect seems to be smaller for the PS201 data.

In the case of annihilation data, while there is essentially equality between the Armstrong et al. $\bar{n}p$ data and the $\bar{p}p$ data, a strong effect is visible in the low momentum region of the PS201 data, around 100 MeV/c. In this region the $\bar{n}p$ annihilation cross-section is measured to be only 2/3 of the corresponding $\bar{p}p$ cross-section. It has to be mentioned that data from $\bar{p}$-nucleus experiments also suggest that the $\bar{p}n$ annihilation cross-section is appreciably weaker than the $\bar{p}p$ one [297,298].
## 4.2. $\bar{p}p$ elastic scattering differential cross-sections

Many precise measurements existed for the $\bar{p}p$ elastic scattering differential cross-section even before LEAR entered into operation, but mostly at momenta of about 600 MeV/c or larger. For this reason only one experiment (PS173) was proposed at LEAR to measure the $\bar{p}p$ differential cross-section, and it was especially designed to measure at incident momenta lower than 600 MeV/c. Also the experiments PS172 and PS198 produced $\bar{p}p$ elastic scattering differential cross-section data, but they were obtained while measuring, with a polarised target, the analysing power $A_{0n}$. As a consequence, some of the most precise data are still non-LEAR data.

### 4.2.1. Shape

Fig. 4.7 shows a collection of data from many experiments, extending from the lowest momentum (181 MeV/c) to 1400 MeV/c. Almost the entire angular range is covered by each measurement. The two low-energy data sets are from experiment PS173 [92,91]; at 287 MeV/c they are compared with the pp data of Batty et al. [299] (open points). The PS173 data at 505 MeV/c are compared with the Sakamoto et al. data [300] at 504.7 MeV/c (open points). The data at 439 MeV/c are from experiment PS198 [107], as well as the data at 697 MeV/c [106], which are compared with the data at 679 MeV/c [86] from experiment PS172 (closed points). The data at 790 and 990 MeV/c are from Eisenhandler et al. [290]. Data from Sakamoto et al. and from Eisenhandler et al. exist also at many other values of the incident $\bar{p}$ momentum.

The sharp rise of the cross-section in the forward direction, visible in the data at the two lowest energies, is due to Coulomb scattering, and will be discussed in the next section.

Even the low-energy $\bar{p}p$ data show a strong angular dependence, not present in the pp case. For comparison, the data at 287 MeV/c (about 50 MeV kinetic energy) are plotted together with the corresponding pp data at the same energy. The pp data are essentially isotropic, corresponding to S-wave scattering, while the $\bar{p}p$ data show a strong P-wave component.

Some insight into the behaviour of the elastic scattering differential cross-section can be obtained by neglecting the spin and performing a simplified partial wave analysis, as in Refs. [91,301]. This analysis is possible only at low energy, where few partial waves are present; in Refs. [91,301] only data at incident $\bar{p}$ momenta less than 300 MeV/c were used. Important results of this analysis are that

* the S-wave elastic cross-section is suppressed ($\eta_0$, the absorption coefficient, is about 0.5): on the contrary, the S-wave inelastic cross-section is close to its geometrical limit;

* the P wave is large, and is present even at the lowest measured momentum (181 MeV/c).

The question whether the relative strength of the P wave is only due to the suppression of the S wave has been discussed in the literature. The general understanding is that one observes a strong P-wave enhancement, as a direct consequence of the nuclear interaction, and not necessarily related to the strong S-wave annihilation. For the authors of Refs. [240,302], the main reason for this enhancement is the existence of near-threshold bound or resonance states, due to the strong attractive nuclear forces between N and $\bar{N}$. Indeed, switching off the annihilation, the P-wave contribution to the $\bar{p}p$ elastic scattering is even larger than in the presence of annihilation. This result was obtained both using a unitary coupled-channel model, and an optical model. It seems therefore reasonable to conclude
<figure><img src="image_54.png" /><figcaption>Fig. 4.7. $\bar{p}p$ elastic differential cross-section data in mb/sr as a function of $\cos\vartheta_{cm}$ at several values of the incident $\bar{p}$ momentum (from 181 to 1400 MeV/c).</figcaption></figure>
<figure><img src="image_55.png" /><figcaption>Fig. 4.8. The $\bar{p}p$ elastic differential cross-section $d\sigma/dt$ as a function of $-t$ at four incoming $\bar{p}$ momenta between 287 and 990 MeV/c.</figcaption></figure>

that the P-wave enhancement is a consequence of the strong $\bar{N}N$ meson-exchange potential. This fact was first emphasised qualitatively by Dalkarov [228], a long time before the appearance of the LEAR data.

The strong P-wave enhancement at low energy is considered as one of the main results obtained by the scattering experiments at LEAR. It has also been observed in charge-exchange scattering, as it will be seen later. Moreover, the annihilation of protonium from P states is very large, as well as the P-wave contribution to $\bar{p}p$ annihilation in flight.

In the higher momentum range (above 800 MeV/c), the shape of the elastic cross-section is essentially due to diffraction. It is the shadow of the strong absorption of the incoming wave which is caused by annihilation. Due to isospin factors ($\vec{\tau}_1 \cdot \vec{\tau}_2 = +1$ for $I = 1$ and $\vec{\tau}_1 \cdot \vec{\tau}_2 = -3$ for $I = 0$) the $\pi$-exchange contribution is partially suppressed in the $\bar{p}p$ elastic channel ($\mathcal{T}_{I=0} + \mathcal{T}_{I=1}$). The situation is different at low momenta, where a strong anti-shrinkage of the forward peak is observed. This effect is clearly visible in Fig. 4.8, where differential cross-section data at a few energies are plotted as a function of $-t$. Parametrising the forward cross-section as $\exp(bt)$ the $b$-values shown in Fig. 4.9 are obtained, which clearly exhibit a strong rise at low energy. This fact has been interpreted [228] as a consequence of the strong attraction of the long-range $\bar{N}N$ meson-exchange potential, which pulls the $\bar{N}N$ wave function into the annihilation region. The net effect is that the effective absorption radius $R_{\text{eff}} = 2\hbar cb^{1/2}$ is larger than the annihilation radius, as defined in optical potential models (Section 3.8). As a consequence, at low energy the forward slope of the elastic differential cross-section, as well as its energy dependence, are given by the meson-exchange potential, and the $\bar{N}N$ scattering is not diffractive [60].
<figure><img src="image_56.png" /><figcaption>Fig. 4.9. Best-fitted slope values $b$ for the differential cross-section $\bar{p}p \rightarrow \bar{p}p$ in the forward region, as a function of the incident $\bar{p}$ momentum. The data are from Refs. [93, 79, 80, 303–305]. The curve is the calculation of the optical model of Ref. [228].</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>Fig. 4.10. $\bar{p}p$ elastic differential cross-section at about 990 MeV/c; the data are from PS172 (closed circles) and from Eisenhandler et al. (stars).</figcaption></figure>

<caption>Fig. 4.11. $\bar{p}p$ elastic differential cross-section in the range 679–690 MeV/c; the data are from PS172 (closed circles), Eisenhandler et al. (stars), and Sakamoto et al. (open circles).</caption>

As a final comment, we note that some of the data are not compatible with each other. This is a known problem, discussed repeatedly in the literature, and sometimes a matter of debate when it comes to use a data base to extract information on the parameters of the various $\bar{N}N$ models. Figs. 4.10 and 4.11 exhibit two typical situations faced by the community: the first one is “easily” solved by introducing a renormalisation factor, but the second one (see also [277]) is clearly a problem.
### 4.2.2. Extraction of $\rho$

In the extreme forward region ($\vartheta \simeq 0$), the shape of the $\bar{p}p$ differential cross-section reflects the interference of the strong amplitude $\mathcal{T}_S$ with the Coulomb one $\mathcal{T}_C$. Since the latter is known, we get access to the so-called “$\rho$” parameter, which is defined as

$$ \rho(s) = \mathrm{Re} \, \mathcal{T}_S / \mathrm{Im} \, \mathcal{T}_S, \qquad (4.1) $$

In Eq. (4.1) it is assumed that only the non-spin flip part of the strong amplitude contributes to the interference with the Coulomb amplitude. The strong amplitude is usually parametrised as an exponential

$$ f(t) = \frac{\sigma_{\text{tot}}}{4\pi}(i + \rho) \exp(bt/2), \qquad (4.2) $$

where $b$, the slope of the diffraction peak, is of the order of $20 \text{ GeV}^{-2}$ around $600 \text{ MeV/c}$, as seen in Fig. 4.9. The Coulomb amplitude is well known from the literature. A more refined treatment would require a spin dependence of the strong amplitudes, which can only be provided by a model or a phase-shift analysis, as stressed, e.g., in Refs. [306,279].

Usually $b$ and $\rho$ are extracted from a fit to the data. In some cases the fit is also used to estimate values for the total cross-section.

The results of scattering experiments at CERN, Brookhaven, and FNAL are plotted in Fig. 4.12. The data are from

* PS172: points at 233 and 272 MeV/c [79], and 550, 757, and 1077 MeV/c [80].

* PS173: seven points between 181 and 590 MeV/c [93]; $\sigma_{\text{tot}}$ is also extracted from the data.

* Ashford et al. (BNL): 11 points between 359 and 652 MeV/c [305].

* Cresti et al. (CERN PS): eight points between 353.3 and 578.3 MeV/c [304].

* Iwasaki et al. (KEK): six points between 430 and 687 MeV/c [307].

* Jenni et al. (CERN PS): six points between 1.174 and 2.607 GeV/c [303].

* E760 (FNAL): six points between 3.70 and 6.23 GeV/c [132]; $\sigma_{\text{tot}}$ fixed.

The point at $p_{\bar{p}}=0$ comes from the measurement of the $\bar{p}p$ atomic 1S state: as explained in Section 5, it is given by the ratio of the shift to the width of the protonium 1S level, which numerically turns out to be smaller than $-1$.

The rapid change of $\rho$ from about 0 for momenta larger than 200 MeV/c to less than $-1$ at $p_{\bar{p}}=0$ is usually interpreted in terms of the strong P-wave enhancement already mentioned in Section 4.2.1, and S- and P-wave interference effects [308].

As one can see, there is an intriguing structure near 200 MeV/c. Several authors have computed the effect of the nearby $\bar{p}p \to \bar{n}n$ threshold, and concluded it is negligible and could not be responsible for the structure (see, e.g., Ref. [309]). Others have tried to explain the bump in terms of an $\bar{N}N$ state below threshold (see, e.g., Ref. [310]).

Three remarks, however, should be made:

(i) there is some disagreement between the data;

(ii) as already pointed out, if there is any spin dependence in the amplitude (and this is likely to occur), it should be accounted for when extracting $\rho$;

(iii) in Eq. (4.1) it is assumed that $\mathrm{Re} \, \mathcal{T}_S$ and $\mathrm{Im} \, \mathcal{T}_S$ have the same $t$-dependence;

(iv) the unfolding of the experimental measuring errors or the effects of multiple scattering is a delicate procedure and the assessment of the systematic errors is not straightforward.
<figure><img src="image_58.png" /><figcaption>Fig. 4.12. The $\rho$ parameter as a function of the $\bar{p}$ momentum in MeV/c.</figcaption></figure>

### 4.3. $\bar{p}p$ charge-exchange differential cross-section

The charge-exchange reaction is an instructive observatory of the subtle interplay between long-range Yukawa-type of forces, and the strong short-range absorption [311,312]. Data were taken at LEAR, by the PS173, PS199, and PS206 collaborations, and also at BNL and KEK. Very much like in the $\bar{p}p$ elastic reaction, the data from the LEAR experiment (PS173) are the data at the lowest incident momentum.

#### 4.3.1. Shape

Typical data from various experiments are shown in Fig. 4.13, as a function of $\cos\vartheta_{cm}$. The data are from:

* PS173: closed squares at 181, 228, 240, 262, 300, 470, 499, 550 and 595 MeV/c. The data are from the HEPDATA Reaction Data Database [313]: the incident momenta bins differ slightly from those given in [94];

* Nakamura et al. (KEK): open circles at 392.4, 490.1, 591.2, 689.0, 780.5 MeV/c [314];
<figure><img src="image_59.png" /><figcaption>Fig. 4.13. $\bar{p}p$ charge-exchange differential cross-section data in mb/sr as a function of $\cos\theta_{cm}$ at several values of the incident $\bar{p}$ momentum (from 181 to 1130 MeV/c). The data are from experiment PS173 (closed squares), experiment PS199 (stars), and pre-LEAR experiments.</figcaption></figure>
<figure><img src="image_60.png" /><figcaption>Fig. 4.14. The $\bar{p}p$ charge-exchange differential cross-section at 601 MeV/c from experiment PS206. Also shown are the measurements from PS173 and Nakamura et al. The curve is a fit to the PS206 data.</figcaption></figure>

* Kohno et al. (CERN PS): open stars at 696 MeV/c [315];

* Colebourne et al. (CERN PS): closed triangles at 1130 MeV/c [316];

* Banerjee et al. (CERN PS): open squares at 760 MeV/c [317];

* Bogdanski et al. (CERN PS): open triangles at 730 MeV/c [318];

* PS199: closed stars at 693 MeV/c [111] and at 546 and 767 MeV/c [115].

Even at momenta as low as $p_{\bar{p}}=181$ MeV/c, the differential cross-section in Fig. 4.13 exhibits several structures. This is not surprising. The smallness of the integrated charge exchange cross-section is due to a strong cancellation of the low partial waves in the combination $\mathcal{T}_{\text{ce}} \propto \mathcal{T}_{I=1} - \mathcal{T}_{I=0}$. Then the high partial waves have a more important weight in the charge exchange than in the other isospin channels.

The most interesting structure shown by the $\bar{p}p \to \bar{n}n$ differential cross-section data is the sharp peak in the forward direction, followed by a dip-bump structure. These features can be seen best in Fig. 4.14, which shows the recent measurement at LEAR by experiment PS206 at 601 MeV/c [121]. This experiment has provided at two momenta (601 and 1202 MeV/c) the most precise measurements, extending down to $t=0$, and has allowed for the first time to measure accurately the shape of the forward peak. Very much like the forward peak of the np $\to$ pn reaction, this structure is interpreted as a manifestation of the nearby pion pole, which must contribute to the scattering amplitude with the Born term

$$ \frac{1}{s} g_{\pi NN}^2 \frac{t}{t - \mu_{\pi}^2}, \qquad (4.3) $$
where $s$ and $t$ are the Mandelstam variables, and $g_{\pi NN}^2 \simeq 14$ is the charged pion nucleon coupling constant.

Applying the Chew extrapolation method, the PS206 data have been used to extract $g_{\pi NN}^2$. The value of this fundamental constant has been a matter of debate over the past 15 years. A reanalysis of the NN data and a whole set of new experiments suggest a value of $g_{\pi NN}^2$ of 13.6 [319], about 5% lower than the “standard” value. The value of $g_{\pi NN}^2$ is important for our world: if it would be too small, the deuteron would be unbound and no nuclei would have been formed. If it would be too large, there would be no hydrogen-to-helium burning, and our sun would not shine [320]. The possibility to extract $g_{\pi NN}$ from the $\bar{p}p \to \bar{n}n$ data was already suggested in 1967 by Phillips [321], and it was first realised by the Njimegen group, in a global fit of the $\bar{N}N$ data base [322]. In Refs. [323,324] is was shown that $g_{\pi NN}$ could be extracted with good precision from the data of a single experiment, in a model independent way by extrapolating the data to the pion pole. Like in the NN case, a “low” value for $g_{\pi NN}^2$ was obtained, which has been a matter of concern and discussion [325].

Also interesting is the dip-bump structure in Fig. 4.14, which had already been seen in several previous experiments, and was known to depend on the energy: sometimes a simple shoulder, and sometimes a pronounced minimum. In the literature, several explanations of the dip-bump structure can be found:

(i) according to Phillips [321,210], the second maximum is a typical OPE effect, coming from a double-spin-flip amplitude;

(ii) Shibata [219] ascribes the shape of the charge-exchange differential cross-section to $\pi$-exchange ($V_{SS} + V_{T}$) only; basically it is the same explanation;

(iii) it turns out that the shape of the cross-section is also sensitive to the absorptive potential which accounts for the annihilation, and which interferes with the OPE amplitude.

Fits to the PS206 data with $\pi$-exchange only plus a smooth background [323,324,121] are excellent, as seen in Fig. 4.14. What the background is, it is still an open question. $2\pi$-exchange and $\rho$-exchange surely contribute to it.

### 4.3.2. Exchange structure from the charge-exchange scattering data

The success of the simple fit described in the previous section to the small angle cross-section data rises the obvious question “is this all?”, i.e., is annihilation plus OPEP enough to describe the $\bar{N}N$ scattering data?

The answer is no. In particular, as will be shown in Section 6, with such simple methods there is no way to reproduce $A_{0n}$ data. But even for what concerns the differential cross-section data, overall fits of the existing data at different energies seem to indicate the necessity of more structures in the amplitudes.

The effect of including $2\pi$-exchange in the extrapolating function has been checked and the results presented at the last LEAP conferences [326,327]. To take into account $2\pi$-exchange a $\rho$-pole term has been introduced.

Removing the $\pi$ and $\rho$ singularities by multiplying the data by $(t - \mu_{\pi}^2)^2 (t - \mu_{\rho}^2)^2$, excellent fits have been obtained to all the differential cross-section data from PS199 and PS206, at all energies and in all the $t$ range (up to $1.15 \text{ GeV}^2$), with smooth polynomials in $t$ as can be seen in Fig. 4.15. On the contrary, the fits which had the pion as the only singularity did not give good results for the
<figure><img src="image_61.png" /><figcaption>Fig. 4.15. The $\bar{p}p$ charge-exchange differential cross-section data from PS199 and PS206. The fits, described in the text, assume the $\pi$-pole and the $\rho$-pole as the only singularities.</figcaption></figure>

higher momenta PS199 data. This result has been interpreted by the authors as direct evidence for $2\pi$-exchange in the $\bar{p}p \to \bar{n}n$ reaction. The comparison between the fits and the data from PS199 [111,115,114] and PS206 [120,121] is shown in Fig. 4.15.

### 4.4. $\bar{N}N$ interaction radii from scattering data

An interesting analysis of the $\bar{p}p$ cross-sections at low energy was performed by the Heidelberg group [219,328], to describe in a model independent way the various interaction ranges. As shown, e.g., in Ref. [329], it is possible to derive from the potential in a straightforward way the interaction ranges. As an example, from the imaginary optical potential which simulates annihilation one can derive the “annihilation radius”. If $W$ is the strength of the absorption potential (negative), and $\psi$ the outgoing wave,

$$ \psi(r, \theta) = \sum_l (2l + 1) i^l R_l(r) P_l(\cos \theta), \qquad (4.4) $$
<figure><img src="image_62.png" /><figcaption>Fig. 4.16. The squared wave function $|R_1|^2(kr)^2$ and the annihilation probability $-W|R_1|^2(kr)^2$ for $^{13}P_1$. $R_1$ is the radial wave function for the $P_1$ state.</figcaption></figure>

in the notation of Ref. [329], then

$$ \sigma_{\text{ann}} = -\frac{2}{\hbar v} \int W |\psi|^2 d\Omega, \qquad (4.5) $$

and the annihilation radius $r_a$ can be defined as

$$ \int_{0}^{r_a} W |\psi|^2 r^2 dr = \int_{r_a}^{\infty} W |\psi|^2 r^2 dr. \qquad (4.6) $$

Similar expressions can be written for the other interaction ranges, starting from the corresponding potentials.

The Heidelberg group has fitted the parameters of various potentials to the low-energy cross-section data of PS173 (elastic scattering and charge exchange). In the context of an optical model, they have used different potentials, the real part, either a Wood-Saxon or a G-parity transformed OPE potential, to describe the meson-exchange potential, and an imaginary potential, again a Wood-Saxon or a Gaussian, to describe the annihilation. Good agreement with the data could be obtained with all these potentials, although the potentials, as always, could not be uniquely determined. Still, the interaction ranges evaluated from these various potentials agree remarkably with each other, so that the determination of these parameters seems really model independent.

From the fitted parameters an “annihilation radius” of about 1 fm was thus derived, as apparent from Fig. 4.16, which, as an example, shows the density probability $|R_1|^2(kr)^2$ and the annihilation probability $-W|R_1|^2(kr)^2$ computed for $^{13}P_1$ at 290 MeV/c [219]. The value obtained for $r_a$, about 1 fm, is much larger than the Compton wavelength associated with baryon exchange ($\hbar/2m_Nc \simeq 0.1$ fm) which originally was proposed to account for annihilation. The size of the annihilation range has been a matter of debate for many years, as seen in Section 3.8.6.

A value of about 1 fm is also consistently derived for the annihilation radius using more elaborate potentials. Even from the Bonn potential, which uses small Compton wavelength exchanged baryons to derive the annihilation potential, one obtains consistently an annihilation radius around 1 fm [330,331].
One can conclude that essentially in a model independent way the annihilation radius is about 1 fm. This result agrees with a simple picture where

$$ \langle r_a^2 \rangle = 2 \langle r_p^2 \rangle, \qquad (4.7) $$

with $r_p$ the proton and $r_a$ the antiproton electromagnetic radius.

In the analysis of the Heidelberg group, the range of the real potential consistently turns out to be roughly twice as large as the annihilation radius, and practically equal to the corresponding range of the NN potential (essentially OPEP). This result is not surprising, since the G-parity transformation affects OPEP only slightly, and the cross-section data are not sensitive enough to underline the difference between the NN and the $\bar{N}N$ real potential.

As a final remark, it is interesting to remind that the range of the real potential for $\bar{p}p$ elastic is markedly smaller than the corresponding range for $\bar{p}p$ charge exchange, hinting at the fact that OPE dominates this last reaction, while its contribution is partially cancelled in the elastic channel as already pointed out in Section 4.2.1: the charge-exchange potential is therefore the long-range OPEP, while the elastic $\bar{p}p$ potential is characterised by a medium range.

### 4.5. Analysing power of $\bar{p}p$ elastic scattering

The analysing power $A_{0n}$ has been measured at LEAR over most of the angular range by experiment PS172 [85] at 15 momenta, ranging from 497 to 1550 MeV/c, the full angular range by experiment PS198 at 439, 544, and 697 MeV/c [106,107]. Most of the data are shown in Fig. 4.17. A few nearby energies have been combined in a single plot. The agreement between the two sets of experimental data is good.

Some analysing power data were taken at the CERN PS [332], before LEAR entered into operation, but the error bars are almost an order of magnitude larger, as can be seen in the figure for the data at 910 MeV/c (“combined 0.88 and 0.95 GeV/c data”).

Although the trend of the differential cross-section data is reminiscent of a diffractive phenomenon, and can be reproduced by simple optical models, like the Frahn-Venter model [333], this is not the case for the analysing-power data. The $A_{0n}$ pattern is reminiscent of diffraction, showing pronounced minima (sometimes going almost to zero) at the diffraction minima seen in the differential cross-section, but, as noticed by the authors [85] the analysing power data are not compatible with the simple optical models, especially in the backward hemisphere. As discussed in Section 6, much more sophisticated models are needed to reproduce these data.

### 4.6. $\bar{p}p$ charge-exchange analysing power

No pre-LEAR data existed for the analysing power of the $\bar{p}p \to \bar{n}n$ charge-exchange channel.

At LEAR, the PS199 experiment measured $A_{0n}$ over most of the angular range at eight incident $\bar{p}$ momenta, ranging from 546 to 1287 MeV/c. The complete data set is shown in Fig. 4.18. The closed points are from Refs. [112,113]; the open points at 875 and 546 MeV/c are independent measurements from Refs. [114,118], respectively.

The analysing power values do not exceed $\pm 20\%$, thus setting limits on the role of spin-orbit type of forces in this reaction.
<figure><img src="image_63.png" /><figcaption>Fig. 4.17. The analysing power $A_{0n}$ as a function of $\cos\vartheta_{cm}$ in the $\bar{p}p$ elastic channel, at several incident $\bar{p}$ momenta, from 439 to 1550 MeV/c. The data are from experiment PS172 (open and closed circles), experiment PS198 (open squares) and from Ref. [332] (stars).</figcaption></figure>

At low energy the data exhibit a simple pattern, with a peak in the forward region and another in the backward region, but as the energy increases, a bump builds up in the central region.

Although some interesting analogies could be drawn between the $np \to pn$ and the $\bar{p}p \to \bar{n}n$ data [327], it is fair to say that a simple physical explanation of the observed polarisation signal is still
<figure><img src="image_64.png" /><figcaption>Fig. 4.18. The analysing power $A_{0n}$ as a function of $\cos\vartheta_{cm}$ in the $\bar{p}p$ charge-exchange channel, from PS199, at several incoming $\bar{p}$ momenta, from 546 to 1287 MeV/c.</figcaption></figure>

lacking, even at small $t$, where the $\pi$-exchange should be the dominating dynamical mechanism. A possible explanation is that, to first-order Born approximation, $\pi$-exchange does not contribute to $A_{0n}$. Polarisation is contributed by a spin-orbit $\vec{L} \cdot \vec{S}$ term generated either by a vector exchange (for instance $\rho$-exchange), or by iterated $\pi$-exchange.

## 4.7. Two-spin correlation data

As byproduct of the $A_{0n}$ measurement, the PS172 experiment at LEAR obtained some $D_{0n0n}$ data for the $\bar{p}p$ elastic reaction, mostly in the central angular region [87]. The data were extracted from
<figure><img src="image_65.png" /><figcaption>Fig. 4.19. $D_{0n0n}$ data from PS172 as a function of $\cos\vartheta_{cm}$ in the $\bar{p}p$ elastic channel, at incident $\bar{p}$ momenta from 679 to 1501 MeV/c.</figcaption></figure>

the measured polarisation of the scattered proton, by using a carbon polarimeter, as described in Section 2.

Data were taken at 10 incoming $\bar{p}$ momenta between 679 and 1501 MeV/c. In Fig. 4.19 they are plotted as a function of $\cos\vartheta_{cm}$. In the figure, the point at 1416 MeV/c has been shifted (from $\cos\vartheta_{cm} = -0.295$ to $\cos\vartheta_{cm} = -0.3$). The error bars are large, but we note appreciable departures from $D_{0n0n} = 1$, indicating that the spin-spin forces are important.

The depolarisation parameter $D_{0n0n}$ for the $\bar{p}p \to \bar{n}n$ reaction has been measured at 875 MeV/c [117] and 545 MeV/c [118]. This measurement was an important part of the PS199 programme, and two relatively long runs were dedicated to it. $D_{0n0n}$ was obtained from the measured polarisation of the scattered neutron. The measurement is difficult, and could be performed only in a limited angular
<figure><img src="image_66.png" /><figcaption>Fig. 4.20. $D_{0n0n}$ data for the $\bar{p}p \to \bar{n}n$ charge-exchange channel from PS199 versus $\cos\vartheta_{cm}$ at 545 MeV/c (a) and at 875 MeV/c (b).</figcaption></figure>

range. The results, shown in Fig. 4.20, clearly indicate that $D_{0n0n}$ is very different from 1, thus confirming the relevance of the tensor forces in the $\bar{N}N$ interaction. The same data are plotted as a function of $-t$ in Fig. 4.21. One notices a similar trend for $D_{0n0n}$, a small negative value (about -0.2) for $-t \le 0.2$ GeV$^2$. The $K_{n00n}$ parameter has also been measured for the charge-exchange reaction by PS199 at 875 MeV/c [119]. The measurement was done parasitically to the $D_{0n0n}$ run, detecting the $\bar{n}$ in the neutron polarimeter. Since the analysing power of the $\bar{n}p$ reaction is not known,<fnref n="8" /> the data shown in Fig. 4.22 are presented as $\alpha K_{n00n}$ where $\alpha$ is an unmeasured proportionality factor, related to the $\bar{n}p$ elastic scattering analysing power $A_{\bar{n}p}$ by the relation $A_{\bar{n}p} = \alpha q$ where $q$ is the momentum transfer. The quantity $\alpha$ was estimated by PS172 from the measurement of the analysing power of the $\bar{p}C$ elastic scattering to be $0.61_{-0.20}^{+0.18}$, a value consistent with $\bar{N}N$ potential-model calculations. The data, plotted as a function of $\cos\vartheta_{cm}$, are given in $(\text{GeV/c})^{-1}$.

## 4.8. Strangeness exchange reactions

### 4.8.1. Total cross-sections for $\bar{\Lambda}\Lambda$ production

In Refs. [130,124], the $\bar{\Lambda}\Lambda$ production is studied very close to the threshold at $s_0 = 4m_\Lambda^2$. The original motivation of the PS185 experiment was indeed to investigate the behaviour of the cross-section near threshold, to identify either a $(s - s_0)^{1/2}$ energy dependence, typical of S wave, or $(s - s_0)^{3/2}$ given by a P wave. See, e.g., Kilian's contribution at Tignes [336]. A gluon intermediate state associated with scalar diquarks suggests a $^3S_1$ dominance, while current pair-creation models favour $^3P_0$. This is discussed in several papers, for instance in Refs. [337,338].

<footnote n="8">⁸ A proposal from the PS199 Collaboration [334,335] to measure the analysing power of $\bar{n}p$ elastic scattering by scattering $\bar{p}$ on a polarised deuteron target was not accepted.</footnote>
<figure><img src="image_67.png" /><figcaption>Fig. 4.21. $D_{0n0n}$ data for the $\bar{p}p \to \bar{n}n$ channel as a function of $-t$ from experiment PS199.</figcaption></figure>

<figure><img src="image_68.png" /><figcaption>Fig. 4.23. LEAR data on the $\bar{p}p \to \bar{\Lambda}\Lambda$ cross-section at very low energy.</figcaption></figure>

The PS185 data are summarised in Fig. 4.23, as a function of the c.m. excess energy $\epsilon = \sqrt{s}-2M$, where $M$ is the $\Lambda$ mass. There is a sharp rise of the cross-section starting at 1.434 GeV/c. Above 1.5 GeV/c, the measured cross-section matches values obtained in earlier experiments, and listed in Ref. [124].

A puzzling structure occurred near 1 MeV excess energy in the 1989 data. It motivated further measurements in this threshold region. The data taken in 1994, also shown in Fig. 4.23, seemingly confirmed the structure near this momentum of 1438 MeV/c. However, more recent studies, reported for instance at the LEAP98 Conference [339], indicated a smooth behaviour of the cross-section in
<figure><img src="image_69.png" /><figcaption>Fig. 4.24. LEAR data on the $\bar{p}p \to \bar{\Lambda}\Lambda$ cross-section at low energy.</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>Fig. 4.25. Differential cross-section for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1476.5 and 1507.5 MeV/c.</figcaption></figure>

this energy range. The final analysis of the low-energy region [340] confirmed this result, as can be
clearly seen in Fig. 4.23.

A view at the cross-section of the $\bar{p}p \to \bar{\Lambda}\Lambda$ reaction in a wider energy range is provided in
Fig. 4.24.

### 4.8.2. Angular distribution for $\bar{\Lambda}\Lambda$ production

The angular distribution for $\bar{\Lambda}\Lambda$ production has been measured for several values of the antiproton
momentum. Ref. [130] displays the differential cross-section at $p_{\text{lab}} = 1476.5$ and 1507.5 MeV/c. The
results are reproduced in Fig. 4.25. Even at these low values of the excess energy, the distribution
is far from being flat: there is an abundant contribution of P-wave scattering. This is confirmed in
Ref. [124], where, for an excess energy of the order of 1 MeV or even lower, there is still a net
asymmetry between forward and backward hemisphere in the c.m. frame.
<figure><img src="image_71.png" /><figcaption>Fig. 4.26. Differential cross-section for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1726 and 1771 MeV/c.</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>Fig. 4.27. Differential cross-section for $\bar{p}p \to \bar{\Lambda}\Lambda$ at various energies.</figcaption></figure>

The angular distribution has later been measured at higher energy, in particular at 1.726 and 1.771 GeV/c [127]. The results are shown in Fig. 4.26. The PS185 often displayed the cross-section as a function of $s$ and $t$ in plots similar to that of Fig. 4.27. The plot emphasises that the forward peak does not start at a given angle, but at a particular value of $t$ ($-t \simeq 0.4 \text{ GeV}^2$, or $R \simeq 1/3 \text{ fm}$).
<figure><img src="image_73.png" /><figcaption>Fig. 4.28. Polarisation for $\bar{\Lambda}\Lambda$ production at $p_{\text{lab}} = 1476.5$ MeV/c.</figcaption></figure>

<caption>Fig. 4.29. Polarisation for $\bar{\Lambda}\Lambda$ production at $p_{\text{lab}} = 1507.5$ MeV/c.</caption>

### 4.8.3. Polarisation for $\bar{\Lambda}\Lambda$ production

The observed decay of $\Lambda$ or $\bar{\Lambda}$ gives an indication on its polarisation and hence on the polarisation and spin correlation parameters of the $\bar{p}p \to \bar{\Lambda}\Lambda$ reaction.

In Ref. [130], the polarisation is measured at 1476.5 MeV/c. The data are shown in Fig. 4.28. They include the polarisation of $\Lambda$, that of $\bar{\Lambda}$ and an average. There is a fair agreement between $\Lambda$ and $\bar{\Lambda}$ polarisations.

A similar comparison has been made at $p_{\text{lab}} = 1507.5$ MeV/c [130] (see Fig. 4.29). There is now a better agreement on $\Lambda$ and $\bar{\Lambda}$ polarisations. From now on, the data on polarisation will correspond to an average between $\Lambda$ and $\bar{\Lambda}$.

There are also data on polarisation at very low energy [124].

Data on polarisation at higher energy, namely $p_{\text{lab}} = 1.546$ and $1.691$ GeV/c are given in Ref. [126] and reproduced below in Fig. 4.31. These corresponding to $p_{\text{lab}} = 1.726$ and $1.771$ GeV/c [127] are shown in Fig. 4.32. A summary of polarisation data at various energies as a function of $t$ is given in Fig. 4.30.

The polarisation turns from negative to slightly positive values about at the same $t$-value ($-t \simeq 0.4$ GeV$^2$) where the forward peak of the differential cross-section starts to rise.

### 4.8.4. $\bar{\Lambda}\Lambda$ spin correlations

The spin correlation coefficients have been extracted for several values of the momenta. Early data [126] corresponding to $p_{\text{lab}} = 1546$ and $1695$ MeV/c are shown in Fig. 4.31. The measurements done at $p_{\text{lab}} = 1.726$ and $1.771$ GeV/c [127] are displayed in Fig. 4.32. Here $F_0$ is the spin-singlet fraction, introduced in Section 3.2. In principle, $F_0 > 0$, so data showing negative values illustrate how difficult these measurements are. Similarly a value $|C_{ij}| > 1$ is in principle not allowed.

At selected energies and angles, the correlation coefficients are remarkably large, saturating the simple unitarity limits, or even exceeding them, due to experimental uncertainties. As seen in Section 3, this implies strong constraints on the amplitudes and restricts drastically the range of variation of the remaining observables.
<figure><img src="image_74.png" /><figcaption>Fig. 4.30. Polarisation for $\bar{p}p \to \bar{\Lambda}\Lambda$ at various energies.</figcaption></figure>

The polarisation and the spin-correlation parameters have also been measured with high statistics at 1.642 and 1.918 GeV/c [127]. Large values of $|C_{yy}|$ or $|C_{zz}|$ are still observed at some angles. At the highest momentum, there are definitively spin-singlet contributions, i.e., $F_0 > 0$.

### 4.8.5. Further spin observables for $\bar{\Lambda}\Lambda$ production

The last runs of PS185 were devoted to measurements using a polarised hydrogen target, transverse to the beam.

The motivation was inspired by Holinde et al. [341,342], Ellis et al. [215] and others. Both the K-exchange mechanism of Fig. 3.11 and the s-channel gluon-exchange process of Fig. 3.12 can accommodate the trend of the data taken on an unpolarised target, in particular the small spin-singlet fraction. The correlation between initial- and final-state spin, as measured by $D_{nn}$ or $K_{nn}$, could be more selective.

A warning, however, is that existing data on the correlations coefficients $C_{ij}$, if taken seriously, already restrict the domain of variation of $D_{nn}$ and $K_{nn}$ [343, 183].

Data with a transversally polarised target remain anyhow extremely interesting, as they allow several consistency checks. In principle, without limitation on statistics, this would permit reconstruction of all six amplitudes (of course to an overall phase).<fnref n="9" />

The last data of PS185 are currently being analysed.

<footnote n="9">⁹ We thank B. Quinn and K. Paschke of the PS185 collaboration for useful correspondence on this subject.</footnote>
<figure><img src="image_75.png" /><figcaption>Fig. 4.31. Spin observables for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1546 and 1695 MeV/c.</figcaption></figure>

### 4.8.6. $\bar{\Lambda}\Sigma_0 + \text{c.c. production}$

The first measurement of $\bar{\Lambda}\Sigma_0 + \text{c.c.}$ by the PS185 collaboration was performed at 1.695 GeV/c [125]. The cross-section is $7.53 \pm 0.35$ µb. A Legendre analysis in Ref. [125] shows that the magnitude of the partial waves, as the angular momentum increases, decreases less rapidly than for $\bar{\Lambda}\Lambda$ production. The angular distribution is shown in Fig. 4.33.

Other measurements of the $\bar{\Sigma}^0\Lambda$ channel have been done at 1726 and 1771 MeV/c [128]. The differential cross-section is shown in Fig. 4.34. The $\bar{\Sigma}^0\Lambda$ and $\bar{\Lambda}\Lambda$ angular distributions exhibit a similar shape at these energies. A more precise comparison of the two reactions at the same excess energy is done in Ref. [128]. The $\bar{\Sigma}^0\Lambda$ differential cross-section, as a function of the invariant transfer $t$, exhibit a steeper forward peak, with a slope $b$ (defined as $I \propto \exp(-b|t|)$) of about 11 to $14 \text{ GeV}^{-2}$, as compared to $b \sim 8-10 \text{ GeV}^{-2}$ for $\bar{\Lambda}\Lambda$.

Some spin parameters have been measured for $\bar{\Sigma}^0\Lambda$ production [128]. However, the $\Sigma^0$ polarisation spin correlations in the final state and spin-singlet fraction have error bars of the order of 1 and thus are not very meaningful. At most, one can see that $F_0$ tends to be larger than in the $\bar{\Lambda}\Lambda$ case, namely $F_0 \sim 0.64 \pm 0.2$ when averaged over energies and angles. This suggests that both spin-triplet and spin-singlet states contribute to the reaction. The $\Lambda$ polarisation is measured more precisely. The results of [128] are reproduced in Fig. 4.35. As in the $\bar{\Lambda}\Lambda$ case, the magnitude is moderate, $\pm 0.3$ or smaller.
<figure><img src="image_76.png" /><figcaption>Fig. 4.32. Spin observables for $\bar{p}p \to \bar{\Lambda}\Lambda$ at 1726 and 1771 MeV/c.</figcaption></figure>

<figure><img src="image_77.png" /><figcaption>Fig. 4.33. Angular distribution for $\bar{\Lambda}\Sigma_0 + \text{c.c.}$ production at 1.695 GeV/c.</figcaption></figure>
<figure><img src="image_78.png" /><figcaption>Fig. 4.34. Angular distribution for $\bar{\Lambda}\Sigma_0$ + c.c. production at 1.726 and 1771 GeV/c.</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>Fig. 4.35. $\Lambda$ polarisation for $\bar{\Lambda}\Sigma_0$ + c.c. production at 1.726 and 1771 GeV/c.</figcaption></figure>

The very-low-energy behaviour of $\bar{p}p \to \bar{\Sigma}^0\Lambda$ + c.c. is analysed in Ref. [340] and compared to $\bar{p}p \to \bar{\Lambda}\Lambda$ at the same excess energy. Fig. 3 of this reference is reproduced in our Fig. 4.36. The trend is different, showing a larger P to S ratio in $\bar{p}p \to \bar{\Sigma}^0\Lambda$ + c.c. than in $\bar{p}p \to \bar{\Lambda}\Lambda$.

### 4.8.7. $\bar{\Sigma}\Sigma$ production

A measurement of the $\bar{p}p \to \bar{\Sigma}^-\Sigma^+$ cross-section has been performed at 1.922 GeV/c and is presented in Ref. [129]. As mentioned by the authors, this corresponds to an excess energy $\epsilon = 23$ MeV close to the $\epsilon = 25$ MeV of the $\bar{\Lambda}\Lambda$ measurement performed at 1.5075 GeV/c, and of the $\bar{\Sigma}^0\Lambda$ data taken at 1.726 GeV/c and already presented.

The integrated cross-sections is

$$ \sigma = 3.68^{+0.43+0.34}_{-0.43-0.22} \mu b, \qquad (4.8) $$

where the errors are due to statistics and systematics, respectively.
<figure><img src="image_80.png" /><figcaption>Fig. 4.36. Comparison of $\bar{p}\bar{p} \to \bar{\Lambda}\Lambda$ and $\bar{p}\bar{p} \to \bar{\Sigma}^0\Lambda + \text{c.c.}$ cross-sections at very small excess energy above their respective threshold.</figcaption></figure>

The ratio of cross-sections is

$$ \sigma(\bar{\Sigma}^{-}\Sigma^{+})/\sigma(\bar{\Lambda}\Lambda) = 0.14 \pm 0.02, \qquad (4.9) $$

at $\epsilon(\bar{\Lambda}\Lambda) = 25$ and $\epsilon(\bar{\Sigma}^{-}\Sigma^{+}) = 23$ MeV, while [128]

$$ \sigma(\bar{\Sigma}^{0}\Lambda)/\sigma(\bar{\Lambda}\Lambda) = 0.29 \pm 0.02, \qquad (4.10) $$

at the same $\epsilon = 25$ MeV. In a Yukawa picture, this implies a smaller coupling of kaons between a proton and a $\Sigma$ than between a proton and a $\Lambda$. A similar observation is made when studying the parameters required to fit the potential between hyperons and nucleons [344,345].

The production of $\bar{\Sigma}^{-}\Sigma^{+}$ is also compared to that of $\bar{\Sigma}^{+}\Sigma^{-}$ at this antiproton momentum $p_{\text{lab}} = 1.992 \text{ GeV/c}$. The ratio

$$ \sigma(\bar{\Sigma}^{-}\Sigma^{+})/\sigma(\bar{\Sigma}^{+}\Sigma^{-}) = 2.4^{+3.0}_{-1.3} (90\% \text{ CL}) \qquad (4.11) $$

is not very accurately determined and should be considered with care, since the excess energy are $\epsilon = 23$ and 7 MeV, respectively. It is consistent with measurements done at higher energy (see [129] for references).

The differential cross-section for $\bar{\Sigma}^{-}\Sigma^{+}$ production at $\epsilon = 23$ MeV excess energy is given in [129]. It is similar to the distribution of $\bar{\Lambda}\Lambda$ and $\bar{\Sigma}^{0}\Lambda$ at $\epsilon = 25$ MeV, but the forward peak is narrower and sharper. This angular distribution is reproduced in Fig. 4.37.

Some details about $\bar{\Sigma}^{+}\Sigma^{-}$ production have been given in recent conferences. There is no forward peak. But there seems to be an enhancement in the backward region. The differential cross-section
<figure><img src="image_81.png" /><figcaption>Fig. 4.37. Angular distribution for $\bar{\Sigma}^{-}\Sigma^{+}$ (left) and $\bar{\Sigma}^{+}\Sigma^{-}$ (right) production. In the latter case, the units on the vertical axis are arbitrary.</figcaption></figure>

given in Fig. 4.37, whose absolute normalisation is not determined, should be considered as preliminary. The results of the final analysis are expected shortly.

# 5. Protonium and antiprotonic deuterium

In this section, we describe the measurement of X-rays from antiprotonic hydrogen and deuterium atoms and discuss how the strong-interaction parameters are deduced from the data. Many interesting phenomena are associated with this search, a large number having been observed in the Asterix experiment. So, instead of introducing these phenomena as abstract concepts, we discuss them as they show up when presenting results from this experiment.

## 5.1. PS171: The Asterix experiment

The Asterix experiment was carried out by stopping antiprotons in gaseous $H_2$ at room temperature and pressure. A short description of the X-ray detector can be found in Section 2.4, a report on the performance of the full spectrometer in [150].

### 5.1.1. X-ray spectra with $\bar{p}p$ annihilation into charged particles

Fig. 5.1 shows the X-ray spectrum observed in coincidence with two charged particles after stopping 200 MeVc antiprotons. A double-peak structure is seen at low energies while the high-energy spectrum contains much fewer entries. The low-energy peak at 1.74 keV is due to the Balmer-$\alpha$ line, the smaller peak at 3.1 keV to a convolution of Balmer lines close to the series limit, denoted as $L_{\infty}$. Part of the 3.1 keV peak is due to argon fluorescence: charged particles (or high-energy X-rays) kick an electron out of the K shell of an Ar atom. The K shell is then re-populated by emission of a 3 keV photon. In a segmented detector the photon may escape the local detector segment and convert elsewhere. This contribution can be estimated from the X-ray spectrum observed when there are only neutral particles in the final state (see Fig. 5.3(a) below) for which the argon fluorescence contribution is much smaller.
<figure><img src="image_82.png" /><figcaption>Fig. 5.1. Energy spectrum of X-rays emitted by protonium atoms annihilation into final states with two charged particles. The line represents a fit taking into account the Balmer series of p̄p atoms, a (small) constant background, inner bremsstrahlung (solid line), and contributions from the Lyman series (dashed curve). The data on the right are multiplied by 20.</figcaption></figure>

The large peak in Fig. 5.1 due to the Balmer-α line is obviously not accompanied by a Lyman-α line of similar strength. The spectrum evidences at the first glance the importance of annihilation from the 2P levels of protonium atoms: p̄p atoms in 2P states annihilate instead of radiating Lyman-α X-rays. A fit to the data [152] shows some (weak) evidence for Lyman radiation with a fractional intensity of the Kα line of (1.0±0.4)×10⁻³, compared to a total yield of the Balmer series of 12±2%. These data are used—together with Eq. (3.73)—to determine an average 2P strong interaction width of 45 ± 18 meV.

The continuous background at higher energies originates from inner bremsstrahlung which is emitted in the spontaneous acceleration of charged particles in the annihilation process. The emission of bremsstrahlung quanta can be calculated in the soft-photon approximation [346]. For a final state characterised by a partial width $dΓ_0$, the charges $Q_i$ of the outgoing mesons and their four-momenta $\tilde{q}_i$, the differential width for emission of a bremsstrahlung photon (after summation over all polarizations) can be written as

$$d\Gamma_B = \left(\frac{\alpha}{4\pi^2}\right) dW_B \omega^{-1} d\omega d\Gamma_0, \qquad (5.1)$$

with

$$dW_B = -\sum_i \sum_j Q_i Q_j \frac{\tilde{q}_i \cdot \tilde{q}_j}{(\tilde{q}_i \cdot \tilde{k})(\tilde{q}_j \cdot \tilde{k})} \omega^2 d\Omega_k, \qquad (5.2)$$
<figure><img src="image_83.png" /><figcaption>Fig. 5.2. Angular distributions of X-rays originating from the atomic cascade of antiprotonic hydrogen atoms ($E < 5$ keV) and from inner bremsstrahlung ($E > 5$ keV). Histogram: data; solid line: Monte Carlo simulation.</figcaption></figure>

where $\tilde{k}$ is the four-momentum vector ($\omega, \vec{k}$) of the photon, and $d\Omega_k$ the solid angle element into which the photon is emitted. The photon energy distribution (5.1) exhibits the expected $\omega^{-1}$ behaviour, and Eq. (5.2) gives the angular correlation between bremsstrahlung and atomic X-rays.

Photons emitted radiatively as part of the atomic cascade lead to a small alignment of the $\bar{p}p$ angular momentum states; thus the direction of a primarily produced meson resonance is also correlated with the photon direction [347]. But in the subsequent decay of the resonance, this weak correlation is washed out; angular correlations between, e.g., Balmer X-rays and charged particles can be neglected.

Eqs. (5.1) and (5.2) can be evaluated in the rest system of the two charged particles assuming that they are pions; contamination by kaons is known to be small. The measured angular distributions depends on experimental cuts taken into account in the Monte Carlo simulations. In particular, to identify the X-ray, a minimum angular spacing is required between the direction of a X-ray and that of the nearest charged particle.

Fig. 5.2 (upper panel) shows the angular distributions for X-rays with energies below 5 keV (mostly originating from the atomic cascade). The angle $\theta$ is defined with respect to the direction of the higher-momentum particle; the angle $\phi$ is the polar angle, $\phi = \pi$ for X-rays which are found in the direction of the total laboratory momentum of the two charged particles. The solid lines represent an isotropic distribution in the laboratory system. Fig. 5.2 (lower panel) shows the angular distributions for X-rays above 5 keV (originating mostly from bremsstrahlung). It was calculated
from the measured particle momenta using formula (5.2) to calculate the direction of the photon emission. The differential bremsstrahlung-emission width was calculated for each individual event. After summation over all events, a theoretical bremsstrahlung spectrum is obtained.

The two processes—radiative transitions to lower energy levels with subsequent annihilation and annihilation with emission of inner bremsstrahlung—lead to the same final state, and interference effects may occur. They could be particularly disturbing for the Lyman series since in this case the amplitudes for radiative transitions and for bremsstrahlung are of comparable magnitude. The phase of the amplitude changes by 180° while crossing the resonance position while the phase of the hadronic transition amplitude remains, at least approximately, constant. Wrong results for the line centre and width could be obtained when constructive and destructive interferences on the two sides of the K<sub>α</sub> line are neglected. This effect could be particularly dangerous in low-statistics experiments in which the line shape cannot be unambiguously determined from data. In inclusive protonium experiments, many hadronic final states contribute likely having different, statistically distributed, hadronic phases. Hence there is a good chance that the superposition of all lines should result in an undistorted line shape. Only when exclusive final states are selected, interferences are likely to play a role.

### 5.1.2. X-ray spectra with $\bar{p}p$ annihilation into neutral particles only

The contributions from bremsstrahlung and the residual background can be suppressed quantitatively by two further techniques:

* selecting all-neutral events in which the $\bar{p}p$ atom annihilates into neutral particles with no associated bremsstrahlung;

* requiring two X-rays in coincidence.

The fraction of $\bar{p}$ annihilation in liquid H₂ into all-neutral final states is given by the ratio of frequencies $f$ [348]

$$ \frac{f(\bar{p}p \rightarrow \text{neutral particles})}{f(\text{all annihilations})} = 3.6 \pm 0.4\% . \qquad (5.3) $$

In all-neutral final states (defined by the absence of detected charged particles), a small bremsstrahlung contribution survives due to events in which two charged particles were produced but escaped detection. The coincidence requirement (together with an energy cut) displays only those events in which the atomic cascade reached the 2P level. Only the K<sub>α</sub> line is observed and thus the interpretation of the resulting X-ray spectrum is facilitated.

Results using 300 MeV/c antiprotons were reported in [151]. Fig. 5.3(a) shows the X-ray spectrum obtained by stopping 105 MeV/c antiprotons and requiring absence of any charged particle in the final state [153]. The spectrum is dominated by the Balmer series and does not yet allow unambiguous conclusions on the presence of the Lyman series. The situation improves when coincidences between two X-rays are required. In Fig. 5.3(b) only events are considered with two X-rays observed in coincidence; the energy of the more energetic X-ray is plotted. The low-energy peak at 1.74 keV originates from coincidences of a line belonging to the M series with an L<sub>α</sub> line; note the absence of L<sub>∞</sub>. The broad peak at about 8 keV stems from coincidences of an L X-ray with a K<sub>α</sub> line. The apparent width of the peak (~4 keV) is much broader than the experimental resolution of 2.4 keV
<figure><img src="image_84.png" /><figcaption>Fig. 5.3. X-ray spectrum of protonium for events with neutral particles only in the final state. (a) The spectrum shows the Balmer series but the residual background is still too large for an unambiguous identification of the Lyman series. (b) X-ray spectrum of protonium for all-neutral events for two coincident X-rays. The energy of the more energetic X-ray is shown. The peak at 8.67 keV is due to the Lyman-α line. The line shape theory of Ericson and Hambro is used to fit the data.</figcaption></figure>

indicating a sizable broadening due to strong interactions. When a cut is made to select events with one X-ray in the 6.9–10.9 keV energy interval, the energy distribution of the second X-ray shows contributions from the full Balmer series.

The proper line-shape theory for broad lines (for which $Γ \ll ϵ$ is not satisfied) is by no means trivial. To first order, the $K_α$ line shape should correspond to a Lorentzian distribution. The detector resolution has to be taken into account. The convolution of a Lorentzian line with a Gaussian resolution function is called Voigt function:

$$F(E) = \int_{-\infty}^{+\infty} \frac{A}{(E - E_0)^2 + (\Gamma/2)^2} \exp \left\{ -\frac{(E' - E)^2}{2\sigma^2} \right\} dE'. \quad (5.4)$$

When the detection efficiency varies across the line, the Voigtian line shape has to be folded with it. The numerator $A$ in (5.4) is constant only when the transition matrix element for the radiative transition does not depend on energy. This is, however, not the case. Electric dipole transitions prefer more energetic transitions suggesting that $A$ in (5.4) is proportional to $E^3$. However, this integral would diverge. The situation was analysed by Ericson and Hambro [349] who developed a line-shape theory for broad lines. From the requirement of probability conservation, they deduced that the term $A$ contains an energy dependence

$$I_{\text{EH}}(E) = \frac{(2-\beta)(1+\beta)^2}{\beta^2(2+\beta)^7(4-\beta)^2(3-\beta)^2} \left[ {}_2F_1\left(5, 1-\beta, 5-\beta; \frac{\beta-2}{\beta+2}\right) \right]^2, \quad (5.5)$$

where $β = (1/4 + E/12.49)^{-1/2}$ and ${}_2F_1$ is the hypergeometric function. In the proximity of the resonance energy $E_0$, the relation $I_{\text{EH}}(E)=E$ holds to a good approximation. A description of results
from fitting the data of Fig. 5.3 with different line-shape theories can be found in [350]. Eq. (5.5) gives the best description of the data, with a central energy of the line at $E_{1S} = 8.67 \pm 0.15$ keV. The photon energy is shifted to a lower value than expected from QED ($\epsilon_{1S}$ is negative), the 1S level is shifted to a higher energy ($\Delta E_{1S}$ is positive). Thus, a final result of

$$ \epsilon_{1S} = -\Delta E_{1S} = -0.74 \pm 0.15 \text{ keV}, \quad \Gamma_{1S} = 1.60 \pm 0.40 \text{ keV}, \qquad (5.6) $$

is obtained.

The yield of Balmer X-rays in all-neutral events is $13 \pm 2\%$. This is surprisingly close to the value $12 \pm 2\%$ obtained with charged particles in the final state. Annihilation from S states into all-neutral events proceeds dominantly only via the $^1S_0$ state (annihilation into any number of $\pi^0$ and $\eta$ requires positive C-conjugation) which has a statistical weight of 1/4. In P wave, annihilation into all-neutral final states is allowed from spin-triplet states, with a statistical weight of 3/4. In spite of this statistical argument the probability of the protonium atom to annihilate into all-neutral events from P states is only $(13 \pm 2)/(12 \pm 2)$ times larger from S states. We deduce

$$ \frac{f_{\text{P}(\bar{\text{p}}\text{p} \rightarrow \text{neutral particles})}}{f_{\text{P}(\text{all annihilations})}} = 3.9 \pm 1.0\%. \qquad (5.7) $$

The fraction of all-neutral events in which a $K_\alpha$ X-ray is emitted after a Balmer X-ray is detected is $r_K = 0.79 \pm 0.20\%$. We may now assume that all-neutral events come only from positive-parity states. (This is true for the majority of all-neutral events but not for those which contain strange particles or $\omega$-mesons decaying into $\pi^0\gamma$.) And we assume that the protonium spin is conserved in radiative transitions, that there are no intercombination lines. Then we can deduce the strong interaction width of the $2^1P_1$ level.

The total number of L X-ray followed by annihilation into neutral particles is $(13 \pm 2)\% \times (3.9 \pm 1.0)\%$. The number of $K_\alpha$ X-ray followed by all-neutral annihilation is $r_K$ times smaller. This gives

$$ \Gamma(2^1P_1) = 51 \pm 18 \text{ meV}. \qquad (5.8) $$

### 5.1.3. The cascade time

The Asterix collaboration determined the cascade time of $\bar{p}p$ atoms [154]. For 90 000 events with four tracks coming from a common vertex, the annihilation point was determined (with a resolution of $\pm 2$ mm). The data were split into events with vertices in the entrance counter T2 and in the exit counter T4 and, along the $H_2$ target, into 12 slices of 5 cm length (see Fig. 2.17). For each of these 14 positions, the mean time of the two inner proportional counters was determined.

Fig. 5.4 shows the mean time spent by antiprotons in the scintillation in the counter at the entrance of the target and their annihilation. The time increases with the distance travelled by antiprotons before capture, but those stopping immediately after the scintillation counter T2 need

$$ \tau = 5.1 \pm 0.7 \text{ ns} \qquad (5.9) $$

more time than those stopping in T2. The cascade time in solids is some ps only, hence the time difference $\tau$ corresponds to the time elapsed between capture in $H_2$ and annihilation, at normal pressure and temperature.
<figure><img src="image_85.png" /><figcaption>Fig. 5.4. The mean time at which annihilation occurs as a function of the annihilation vertex. Annihilation immediately after the entrance window is delayed by 5.1 ns compared to annihilation in the entrance window due to the time which elapsed from capture of an antiproton in gaseous H₂ and annihilation.</figcaption></figure>

The solid line in Fig. 5.4 is obtained by numerical integration of the Bethe-Bloch formula. The integration also yields the kinetic energy of antiprotons stopping in front of T4: when they entered the target they had $E_{\text{kin}} = 3$ MeV.

The Obelix experiment [351] performed similar measurements using H₂ gas at four different pressures. They determined the time antiprotons need to reach the downstream end of the target. Antiprotons stopping in gas need a long time and produce the Gaussian time distribution in Fig. 5.5. Antiprotons reaching the exit wall may have some residual kinetic energy. These antiprotons traverse the target at a larger speed and the measured time difference is smaller. These antiprotons produce the steep rise at short times in Fig. 5.5. The time-of-flight of antiprotons stopping on the wall surface is obtained by linear extrapolation to the very latest antiproton annihilation on the wall. The difference between the Gaussian peak and extrapolated value gives the cascade time. A finite time resolution gives entries at late times, neglecting it favours cascade times which are too short. Table 5.1 lists the final results.
<figure><img src="image_86.png" /><figcaption>Fig. 5.5. Annihilation time distribution of events for which the vertex is reconstructed close to the exit wall. In-gas annihilation show a (late) Gaussian distribution. Antiprotons with higher velocities reach the end wall and annihilate early; the latest antiprotons (defined by the linear fit) annihilate on the wall surface.</figcaption></figure>

<table><caption>Table 5.1
Cascade time for antiprotons stopping in H₂ as measured in the Asterix and Obelix experiments</caption><thead><tr><th>Pressure (mbar)</th><th>Atom</th><th>Cascade time (ns)</th><th>Reference</th></tr></thead><tbody><tr><td>LH<sub>2</sub></td><td>Σ<sup>-</sup>p</td><td>≤ 5 × 10<sup>-3</sup></td><td>[352]</td></tr><tr><td>STP</td><td>p̄p</td><td>5.1 ± 0.7</td><td>[154]</td></tr><tr><td>150 ± 1</td><td>p̄p</td><td>6.7 ± 1.1</td><td>[351]</td></tr><tr><td>9.8 ± 0.05</td><td>p̄p</td><td>34.3 ± 2.4</td><td>[351]</td></tr><tr><td>5.8 ± 0.05</td><td>p̄p</td><td>59.9 ± 6.0</td><td>[351]</td></tr><tr><td>3.4 ± 0.05</td><td>p̄p</td><td>84.1 ± 10.3</td><td>[351]</td></tr></tbody></table>

In liquid H₂, the cascade time is too short to be measured. However, for the Σ⁻p system, the time can be deduced which elapses from the moment where the Σ⁻ has a residual velocity of 0.004c (where the sum of the measured decay momenta does not yet vanish) to nuclear absorption [352]. This time is an upper limit for the cascade time and, likely, a good estimate for it. It is included in Table 5.1. For convenience, we summarise the results in Fig. 5.6.
<figure><img src="image_87.png" /><figcaption>Fig. 5.6. Cascade time of antiprotons in H₂. Dots: Obelix; square: Asterix; upper limit (↓): from Σ⁻ stopping in liquid H₂.</figcaption></figure>

## 5.2. PS174: The cold-gas experiment

The cold-gas experiment used a variable H₂ gas density over the range from 10 to 1/8 times STP. The first results were obtained using antiprotons with momenta of 300 MeV/c and a Si(Li) detector [168]. We discuss here the final results using Si(Li) detectors [169] and two gas-scintillation proportional detectors (GSPD) [167].

Fig. 5.7 shows the low-energy part of the X-ray spectrum of antiprotonic hydrogen and deuterium atoms. Contributions from individual lines of the Balmer series are clearly identified. Similar data were obtained for a wide range of H₂ densities; they provide valuable information about the cascade processes which precede the emission of X-ray lines. The intensities are listed in Tables 5.4 and 5.5.

The high-energy part of the X-ray spectrum [169] is shown in Fig. 5.8 for various H₂ densities. There are striking differences between the data sets: at the largest density, only one line is observed which can be identified with the limit of the Lyman series. At moderate densities a further line is seen, the Kα line, which becomes the strongest component at the lowest density. Below atmospheric pressure, background lines from antiprotonic oxygen and carbon show up due to inadequate gas tightness. In comparison to the Kα line, they have narrow widths. The data were fitted with a polynomial background and a complex of eight lines corresponding to Kα, Kβ, ..., K∞ transitions. Their relative strengths were taken from the results of cascade calculations (see Section 5.5). The published strong-interaction parameters were obtained with the Voigt function, the use of the Hambro-Ericson line shape not leading to any significant change [353]. The strong interaction parameters determined from the five data sets are mutually consistent and give average values of

$$ \epsilon_{1S} = -0.75 \pm 0.06 \text{ keV}, \quad \Gamma_{1S} = 0.90 \pm 0.18 \text{ keV}, \quad \Gamma_{2P} = 45 \pm 10 \text{ meV}. \qquad (5.10) $$

In parallel to the data with the Si(Li) detectors, the GSPD was also used to detect the X-rays. The fit to the results, shown in Fig. 5.9, gives [170]

$$ \epsilon_{1S} = -0.73 \pm 0.05 \text{ keV}, \quad \Gamma_{1S} = 1.13 \pm 0.09 \text{ keV}. \qquad (5.11) $$
<figure><img src="image_88.png" /><figcaption>Fig. 5.7. X-ray spectrum of protonium and antiprotonic deuterium formed in gas at 0.25ρ<sub>STP</sub> as observed in the cold-gas experiment by Baker et al. Individual peaks due to the Balmer series are clearly identified. The detection efficiency falls off rapidly at low energies: the L<sub>α</sub> line at 1.74 keV is the strongest line.</figcaption></figure>

The Balmer series was not observed because of the need of rather thick windows. Thus, no value was determined for Γ<sub>2P</sub>. A search for the Lyman series of p̄d atoms was not successful and only upper limits (95% confidence level) for the yield of K<sub>≥γ</sub> X-rays per stopped antiproton were given: 8 × 10<sup>-4</sup> and 5 × 10<sup>-4</sup> at 0.25 and 0.92ρ<sub>STP</sub>, respectively [170].

5.3. PS175 and PS207: The inverse cyclotron experiment

5.3.1. PS175

The high stopping power of the inverse cyclotron experiment allowed the use of very thin targets or of very low H₂ gas densities, where Stark mixing plays only a minor role. Early experiments had to use a beam with 300 MeV/c momentum [173]; we show only the final results [174].

Fig. 5.10 shows expanded views of a high-statistics run using a target pressure of 30 mbar H₂ (top) and D₂ (bottom). The comparison of the two spectra evidences a peak at above 8 keV, the K<sub>α</sub> line and a small contribution from K<sub>∞</sub>. No sign is seen from any p̄d K line.

The authors of Ref. [174] tried to extract information on the energy splitting between the ³S₁ and ¹S₀ ground states. The data are compatible with the assumption that the observed energy distribution (after background subtraction) is composed of two lines, but one line is sufficient to fit the data.
<figure><img src="image_89.png" /><figcaption>Fig. 5.8. X-ray spectrum of protonium atoms for different gas H₂ gas densities. The line at 11.78 keV is assigned to the Lyman series limit, the line at 8.66 keV observed at low H₂ densities to the Kα line.</figcaption></figure>

More data were recorded than shown here. The K lines were searched for with two Si(Li) detectors and an X-ray drift chamber. The three sets of data were combined into the final results

$$ \epsilon_{1S} = -0.727 \pm 0.023 \text{ keV}, \quad \Gamma_{1S} = 1.160 \pm 0.078 \text{ keV}. \qquad (5.12) $$

Data with lower statistical significance were recorded at pressures from 16 to 120 mbar. The intensities are listed in Tables 5.4 and 5.5.

### 5.3.2. PS207

In 1990, it was proposed to combine the cyclotron trap with a high-resolution crystal spectrometer to study the line splitting and broadening of the 2P hyperfine levels of p̄p and p̄d atoms [354]. We first discuss the results obtained using as X-ray detectors three charged coupled devices (CCDs) and not yet the crystal spectrometer. With these detectors, the Balmer series and the Lyman-α line of protonium [175,176] and of antiprotonic deuterium [177] atoms were studied.

Fig. 5.11 shows the energy spectrum observed when antiprotons were stopped in the cyclotron trap operated at 20 mbar of H₂ (a) and D₂ (b) gas. The low-energy range demonstrates remarkable achievements in technology: individual lines are clearly identified, the Lα being the strongest one. Even the M series limit is observed, in H₂ as a shoulder, in D₂ as a peak.
<figure><img src="image_90.png" /><figcaption>Fig. 5.9. X-ray spectrum of protonium atoms. The line at 11.78 keV is assigned to the Lyman series limit, the line at 8.66 keV observed at low $H_2$ densities to the $K_\alpha$ line.</figcaption></figure>

The high-energy part of the spectrum is contaminated by lines from electronic fluorescence and from heavier antiprotonic atoms. In addition, there are at least two further sources of background. One source is present in $\bar{p}p$ and $\bar{p}d$ data and called “standard CCD background”. It contains inner bremsstrahlung (as discussed in Section 5.1.1) and contributions from other sources. The “standard CCD background” was assumed in [176] to be the same for $H_2$ and $D_2$. The difference of the spectra for $H_2$ and $D_2$ can, however, not be assigned to the Lyman series. There is an additional source of background of unknown origin, present only in the $H_2$ data. This additional background was assigned to coherent interference between K lines and bremsstrahlung, and then subtracted incoherently by the requirement that all three CCDs gave compatible results on X-ray energies, widths and yields for the K-line series. The correlation of background fit and final result was not taken into account when the final errors were evaluated.

In Fig. 5.12 we present the energy spectra after background subtraction. The observation of $K_\alpha$ lines from $\bar{p}p$ is obvious. The data are fitted with a single Voigtian function. The results from this fit and two analogous fits to data from two further CCDs are displayed in Table 5.2.

The authors attempted to split the line into contributions from ortho- and para-protonium even though the data show no visible shoulder. A free fit with two energies, two widths (folded with experimental resolution) and two intensities does, indeed, not converge. Hence the authors decided to guide the fit by subsequentially freezing and releasing parameters. Clearly, the parameter space
<figure><img src="image_91.png" /><figcaption>Fig. 5.10. X-ray spectrum of protonium (top) and of p̄d atoms (bottom). The data are collected using a high-resolution guard-ring protected Si(Li) detector. The low-energy range is shown on the left, the high-energy range on the right.</figcaption></figure>

is not fully explored. With this warning we quote their result on the hyperfine splitting:

$$ ^{3}\text{S}_{1}: \epsilon_{1\text{S}} = -0.785 \pm 0.035 \text{ keV} \qquad \Gamma_{1\text{S}} = 0.940 \pm 0.080 \text{ keV}. \qquad (5.13) $$

$$ ^{1}\text{S}_{0}: \epsilon_{1\text{S}} = -0.440 \pm 0.075 \text{ keV}, \qquad \Gamma_{1\text{S}} = 1.200 \pm 0.250 \text{ keV}. \qquad (5.14) $$

The intensity ratio of the two hyperfine lines was determined to be

$$ Y(2\text{P} \rightarrow {}^{3}\text{S}_{1})/Y(2\text{P} \rightarrow {}^{1}\text{S}_{0}) = 2.75 \pm 0.06. \qquad (5.15) $$

The evidence for Kα lines from p̄d atoms is much weaker (Fig. 5.13(b)). There are several contaminant lines from gas impurities and from the target vessel; in particular the presence of a p̄O(7–6) line at the proposed Kα energy is very unfortunate. Nevertheless, it is possible to subtract the background contribution in such a way that an excess of events is seen in the region where the Kα line is expected. The results of fits to the three difference spectra listed in Table 5.2 are not
<figure><img src="image_92.png" /><figcaption>Fig. 5.11. X-ray spectrum of protonium (left) and $\bar{p}d$ atoms (right), from experiment PS207.</figcaption></figure>

<figure><img src="image_93.png" /><figcaption>Fig. 5.12. X-ray spectrum of protonium atoms after background subtraction and a fit using a Voigtian function. The dashed and dotted lines describe possible contributions from the spin triplet and spin singlet component.</figcaption></figure>

fully compatible. A systematic error is introduced to account for the correlation between the results of the fits and the background subtraction. The final result reads

$$ \epsilon_{1S} = -1.05 \pm 0.25 \text{ keV}, \quad \Gamma_{1S} = 1.10 \pm 0.75 \text{ keV}, \quad 80 \le \Gamma_{2P} \le 350 \text{ meV}. \qquad (5.16) $$
<table><caption>Table 5.2
Energy shift and width (in eV) of the 1S width of $\bar{p}p$ and $\bar{p}d$ in the three CCDs and the final result</caption><thead><tr><th></th><th>ε<sub>p̄p</sub></th><th>Γ<sub>p̄p</sub></th><th>ε<sub>p̄d</sub></th><th>Γ<sub>p̄d</sub></th></tr></thead><tbody><tr><td>CCD1</td><td>-(642.6 ± 61.3)</td><td>1109.4 ± 211.0</td><td>-(1077 ± 380)</td><td>1496 ± 762</td></tr><tr><td>CCD2</td><td>-(714.4 ± 23.8)</td><td>1023.3 ± 74.6</td><td>-(838 ± 243)</td><td>1130 ± 452</td></tr><tr><td>CCD3</td><td>-(751.7 ± 51.0)</td><td>1182.6 ± 176.0</td><td>-(1358 ± 98)</td><td>541 ± 205</td></tr><tr><td>Final</td><td>-(712.3 ± 20.3)</td><td>1053.5 ± 65.3</td><td>-(1050 ± 250)</td><td>1100 ± 750</td></tr></tbody></table>

<figure><img src="image_94.png" /><figcaption>Fig. 5.13. X-ray spectrum of $\bar{p}d$ atoms before (left) and after (right) background subtraction. The sum of contamination lines and of a polynomial function is used to subtract the background.</figcaption></figure>

We caution the reader that in our view the identification of the observed structure with the $K_α$ line from $\bar{p}d$ atoms is not unambiguously established. Also the yield of $(2.3 \pm 1.4) \times 10^{-3}$ at a target pressure of $0.02 \rho_{STP}$ seems rather high. Batty, using cascade calculations, estimates the yield to be lower by one order of magnitude [355].

In parallel to the search for K X-rays, the line profile of the Balmer-$\alpha$ radiation from antiprotonic hydrogen and deuterium atoms was measured with a crystal spectrometer. To combine highest energy resolution with a sufficient count rate, a Bragg spectrometer was set up, in Johann geometry, equipped with spherically bent crystals. Three two-dimensional position-sensitive pixel detectors (CCDs) were used for X-ray recording.

Fig. 5.14 shows the line profile of 3D to 2P transitions for one of the three detectors. The expected splitting of the D levels is negligibly small so that only four lines are expected, corresponding to transitions to the $2^3P_2$, $2^3P_1$, $2^3P_0$, and $2^1P_1$ levels. The data on $\bar{p}p$ exhibit a shoulder at the high-energy side. Its relative intensity was determined by a two-component fit to be $9.5 \pm 0.9\%$, in good agreement with the statistical population of 8.3% for the $2^3P_0$ level. This level is expected
<figure><img src="image_95.png" /><figcaption>Fig. 5.14. Balmer series of protonium atoms. The fine structure components $2^3P_2$, $2^1P_1$ have common strong interaction shifts and widths, the fine structure component $2^3P_0$ can be identified as individual contribution.</figcaption></figure>

to have a large strong-interaction shift. Hence, the high-energy tail is likely due to $3^3D_1 \rightarrow 2^3P_0$ transitions. The main part of the line is attributed to the (unresolved) transitions to the three hyperfine levels, labelled ($2^3P_2, 2^1P_1, 2^3P_1$).

The mean energy of the $D \rightarrow (2^3P_2, 2^1P_1, 2^3P_1)$ transitions and the energy of the individual $D \rightarrow 2^3P_0$ line were determined from a two-component fit to the measured line shape. The energy profile of the group was constructed from the individual contributions with positions and widths calculated from QED (see Table 3.3) and from NN potential models or using a single (broadened) Gaussian or Voigtian distribution. Both treatments lead to the same results on strong interaction parameters for the $2^3P_0$ line. The relative intensities were always fixed to the statistical values. Under these assumptions the authors derived

$$ \epsilon(2^3P_2, 2^1P_1, 2^3P_1) = +4.0 \pm 5.8 \text{ MeV}, \quad \Gamma(2^3P_2, 2^1P_1, 2^3P_1) = 38 \pm 9 \text{ MeV}, \qquad (5.17) $$

$$ \epsilon(2^3P_0) = +139 \pm 20 \text{ MeV}, \quad \Gamma(2^3P_0) = 120 \pm 25 \text{ MeV}. \qquad (5.18) $$

The energy shifts quoted in (5.17) and (5.18) are of hadronic nature. The determination of the average hadronic width $\Gamma(2^3P_2, 2^1P_1, 2^3P_1)$ relies on the (very reasonable) assumption that the small splittings within the multiplet are known with sufficient precision.

From the intensity ratio $r' = K_\alpha/L_{tot}$ (see Eq. (3.73)) the authors deduce a spin-averaged value

$$ \Gamma_{2P} = 44 \pm 8 \text{ meV}. \qquad (5.19) $$

The results on $r'$ and on the direct crystal spectrometer measurement were combined assuming that $\Gamma_X \ll \Gamma(2^3P_2) \approx \Gamma(2^1P_1) \approx \Gamma(2^3P_1)$ yielding:

$$ \Gamma(2^3P_0) = 120 \pm 25 \text{ meV}, \qquad (5.20) $$
$$ \bar{\Gamma}(2^3P_2, 2^1P_1, 2^3P_1) = 30.5 \pm 2.0 \text{ meV}, \qquad (5.21) $$

$$ \bar{\Gamma}_{2P} = 38.0 \pm 2.8 \text{ meV}. \qquad (5.22) $$

The last value uses our knowledge on the different fine-structure levels and is thus more reliable than (5.19), and more precise. Using the same set-up, data were taken also with D₂ as target gas. The splittings within the D levels were again neglected; hence a line quintuplet is expected. In a first attempt, QED splittings as given in [259] were used with a common resolution given by the p̅²⁰Ne calibration line. The fit did not reproduce the data. Much better agreement was obtained when the electromagnetic hyperfine splittings from [260] (Table 3.3) were used. These splittings are small enough to treat the whole multiplet as a single line which is fitted with one Voigt profile. Imposing the splittings from [260] and the hadronic shifts as given by [261] did not effect the final result. A common broadening of all substates was a free parameter in the fits. The relative intensities of the hyperfine transitions were frozen to represent a statistical population of the 2P sublevels. The three detectors gave consistent results; we quote the weighted average as final result for the spin-averaged hadronic shift (negative, i.e., repulsive) and broadening of the 2P levels of antiprotonic deuterium:

$$ \bar{\epsilon}_{2P} = -243 \pm 26 \text{ meV}, \qquad (5.23) $$

$$ \bar{\Gamma}_{2P} = 489 \pm 30 \text{ meV}. \qquad (5.24) $$

## 5.4. Summary of results on p̅p and p̅d atoms

The results on the strong interaction shift and width given in (5.10)–(5.12) and the mean value from Table 5.2 are fully compatible even though we believe the errors to be sometimes underestimated. Therefore, we give the linear average of the four measurements with a conservative estimate of the error:

$$ \epsilon_{1S} = -0.730 \pm 0.030 \text{ keV}, \quad \Gamma_{1S} = 1.060 \pm 0.080 \text{ keV}. \qquad (5.25) $$

Using the Trueman formula (3.90), we can relate these values to the complex S-wave scattering length

$$ a_o^{sc} = (0.88 \pm 0.04) - i(0.64 \pm 0.05) \text{ fm}. \qquad (5.26) $$

Similarly, we obtain the imaginary part of the P-wave scattering volume:

$$ \mathrm{Im} a_1^{sc} = -(0.77 \pm 0.06) \text{ fm}^3. \qquad (5.27) $$

The ratio of the real to imaginary part of strong-interaction amplitude is read as

$$ \rho(E=0) = \frac{2\epsilon_{1S}}{\Gamma_{1S}} = -1.38 \pm 0.12. \qquad (5.28) $$

In Table 5.3 we summarise the results on strong interaction parameters. They will be compared to theoretical predictions in Section 6.

The experiments gave X-ray yields over a wide range of pressures. These are reproduced in Tables 5.4 and 5.5.
<table><caption>Table 5.3
Strong interaction shifts and widths of antiprotonic hydrogen and deuterium atoms</caption><thead><tr><th colspan="2">Antiprotonic hydrogen atoms</th><th colspan="2">Energy width:</th></tr></thead><tbody><tr><td>Energy shift:</td><td>ε<sub>1S</sub> = -730 ± 30 eV<br>ε(2<sup>3</sup>P<sub>2</sub>, 2<sup>1</sup>P<sub>1</sub>, 2<sup>3</sup>P<sub>1</sub>) = +4.0 ± 5.8 meV<br>ε(2<sup>3</sup>P<sub>0</sub>) = +139 ± 20 meV</td><td>Γ<sub>1S</sub> = 1060 ± 80 eV<br>Γ(2<sup>3</sup>P<sub>2</sub>, 2<sup>1</sup>P<sub>1</sub>, 2<sup>3</sup>P<sub>1</sub>) = 30.5 ± 2.0 meV<br>Γ(2<sup>3</sup>P<sub>0</sub>) = 120 ± 25 meV<br>Γ(2<sup>1</sup>P<sub>1</sub>) = 51 ± 18 meV</td><td></td></tr><tr><td>Mean 2P level widths</td><td></td><td>Using (3.73):</td><td></td></tr><tr><td>Using (3.72):</td><td>Γ<sub>2P</sub> = 38.0 ± 2.8 meV</td><td>Γ̃<sub>2P</sub> = 44 ± 8 meV</td><td></td></tr><tr><td>S-wave scattering length</td><td>a<sub>o</sub><sup>sc</sup> = (0.88 ± 0.04) - i(0.64 ± 0.05) fm</td><td>P-wave scattering volume, imag. part</td><td></td></tr><tr><td>ρ-Parameter at threshold</td><td>ρ(E = 0) = -1.38 ± 0.12</td><td>Im a<sub>1</sub><sup>sc</sup> = -(0.77 ± 0.06) fm<sup>3</sup></td><td></td></tr><tr><td>Antiprotonic deuterium atoms</td><td></td><td></td><td></td></tr><tr><td>Energy shift:</td><td>ε<sub>1S</sub> = -1.05 ± 0.25 keV<br>ε̃<sub>2P</sub> = 243 ± 26 meV</td><td>Energy width:</td><td></td></tr><tr><td></td><td></td><td>Γ<sub>1S</sub> = 1.10 ± 0.75 keV<br>Γ̃<sub>2P</sub> = 489 ± 30 meV</td><td></td></tr></tbody></table>

<table><caption>Table 5.4
Intensities (in %) of L and K X-rays radiation from p̄p atoms for different target densities</caption><thead><tr><th rowspan="2">Density<br>ρ<sub>STP</sub></th><th colspan="3">L X-ray intensity</th><th colspan="3">K X-ray intensity</th><th rowspan="2">Ref.</th></tr><tr><th>L<sub>α</sub></th><th>L<sub>β</sub></th><th>L<sub>tot</sub></th><th>K<sub>α</sub></th><th>K<sub>β</sub></th><th>K<sub>tot</sub></th></tr></thead><tbody><tr><td>0.016</td><td>51.9 ± 11.0</td><td>9.2 ± 2.5</td><td>70.6 ± 11.6</td><td></td><td></td><td></td><td>[173]</td></tr><tr><td>0.016</td><td>53.2 ± 9.3</td><td>7.7 ± 2.4</td><td>71.9 ± 10.0</td><td></td><td></td><td></td><td>[174]</td></tr><tr><td>0.03</td><td>40.7 ± 8.5</td><td>3.7 ± 1.3</td><td>52.3 ± 8.9</td><td>0.62 ± 0.17</td><td></td><td>0.91 ± 0.19</td><td>[173]</td></tr><tr><td>0.03</td><td>40.2 ± 5.0</td><td>6.2 ± 1.3</td><td>55.1 ± 5.6</td><td>0.81 ± 0.15</td><td></td><td></td><td>[174]</td></tr><tr><td>0.06</td><td>34.7 ± 7.5</td><td>3.5 ± 1.5</td><td>47.7 ± 8.0</td><td></td><td></td><td></td><td>[173]</td></tr><tr><td>0.06</td><td>31.9 ± 3.8</td><td>4.8 ± 0.9</td><td>44.3 ± 4.1</td><td></td><td></td><td></td><td>[174]</td></tr><tr><td>0.120</td><td>26.5 ± 3.3</td><td>5.2 ± 1.0</td><td>38.9 ± 3.7</td><td></td><td></td><td></td><td>[174]</td></tr><tr><td>0.125</td><td>17.2 ± 6.5</td><td>8.4 ± 1.9</td><td>35.7 ± 7.0</td><td></td><td></td><td></td><td>[170]</td></tr><tr><td>0.25</td><td>10.3 ± 2.2</td><td>5.2 ± 0.6</td><td>24.0 ± 2.4</td><td>0.28 ± 0.08</td><td>0.03<sup>+0.05</sup><sub>-0.03</sub></td><td>0.52 ± 0.12</td><td>[169]</td></tr><tr><td>0.25</td><td></td><td></td><td></td><td>0.37 ± 0.05</td><td>0.09 ± 0.04</td><td>0.78 ± 0.08</td><td>[170]</td></tr><tr><td>0.30</td><td>9.5 ± 2.6</td><td>1.6 ± 1.1</td><td>17.8 ± 3.6</td><td></td><td></td><td></td><td>[173]</td></tr><tr><td>0.92</td><td>3.4 ± 0.8</td><td>2.4 ± 0.3</td><td>11.2 ± 1.0</td><td>0.10 ± 0.04</td><td>0.01<sup>+0.04</sup><sub>-0.01</sub></td><td>0.36 ± 0.07</td><td>[169]</td></tr><tr><td>0.92</td><td>3.4 ± 0.8</td><td>2.4 ± 0.3</td><td>11.2 ± 1.0</td><td>0.18 ± 0.04</td><td>&lt; 0.02</td><td>0.53 ± 0.06</td><td>[170]</td></tr><tr><td>1.0</td><td></td><td></td><td>13.0 ± 2.0</td><td>0.26 ± 0.14</td><td></td><td>0.65 ± 0.32</td><td>[151]</td></tr><tr><td>1.0</td><td>5.5 ± 1.5</td><td></td><td>12.0 ± 2.0</td><td>0.10 ± 0.05</td><td></td><td>0.14 ± 0.06</td><td>[152]</td></tr><tr><td>2.0</td><td>3.4 ± 0.9</td><td>1.5 ± 0.2</td><td>8.1 ± 1.0</td><td>0.04<sup>+0.09</sup><sub>-0.04</sub></td><td>&lt; 0.07</td><td>0.25 ± 0.17</td><td>[169]</td></tr><tr><td>4.0</td><td></td><td></td><td>6.0 ± 3.0</td><td>&lt; 0.6</td><td></td><td></td><td>[134]</td></tr><tr><td>10</td><td>&lt; 0.4</td><td>0.4 ± 0.2</td><td>1.6 ± 0.3</td><td>&lt; 0.06</td><td>&lt; 0.08</td><td>0.25 ± 0.17</td><td>[169]</td></tr></tbody></table>
<table><caption>Table 5.5
Intensities (in %) of L X-ray radiation from p̄d atoms for different target densities</caption><thead><tr><th rowspan="2">Density<br>ρ<sub>STP</sub></th><th colspan="3">L X-ray intensity</th><th rowspan="2">Ref.</th></tr><tr><th>L<sub>α</sub></th><th>L<sub>β</sub></th><th>L<sub>tot</sub></th></tr></thead><tbody><tr><td>0.016</td><td>53.2 ± 9.3</td><td>7.7 ± 2.4</td><td>71.9 ± 10.0</td><td>[174]</td></tr><tr><td>0.03</td><td>40.2 ± 5.0</td><td>6.2 ± 1.3</td><td>55.1 ± 5.6</td><td>[174]</td></tr><tr><td>0.06</td><td>31.9 ± 3.8</td><td>4.8 ± 0.9</td><td>44.3 ± 4.1</td><td>[174]</td></tr><tr><td>0.12</td><td>26.5 ± 3.3</td><td>5.2 ± 1.0</td><td>38.9 ± 3.7</td><td>[174]</td></tr><tr><td>0.25</td><td>19.0 ± 2.1</td><td>2.9 ± 0.2</td><td>29.1 ± 2.2</td><td>[169]</td></tr><tr><td>0.92</td><td>7.9 ± 0.9</td><td>1.6 ± 0.1</td><td>14.1 ± 1.0</td><td>[169]</td></tr><tr><td>2.0</td><td>5.0 ± 0.6</td><td>1.1 ± 0.1</td><td>9.0 ± 0.7</td><td>[169]</td></tr><tr><td>4.0</td><td></td><td></td><td>6.0 ± 3.0</td><td>[146]</td></tr><tr><td>10</td><td>1.0 ± 0.2</td><td>0.5 ± 0.1</td><td>2.4 ± 0.3</td><td>[169]</td></tr></tbody></table>

## 5.5. Cascade processes in p̄p and p̄d atoms

Most experiments on proton-antiproton annihilation at rest into exclusive final states were carried out by stopping antiprotons in a liquid hydrogen target. Annihilation at rest takes place from atomic orbits, when antiprotons with a kinetic energy of a few eV were captured by the Coulomb field of a proton or deuteron. The p̄p system annihilates only from a small number of states with given quantum numbers which can be determined or at least restricted by using selection rules or by observing the X-rays emitted in the course of the atomic cascade. The distribution of initial states can be changed by varying the target density. Hence we have a unique situation where annihilation processes can be studied with ab initio knowledge of the quantum numbers. In scattering experiments or in annihilation in flight, several partial-wave amplitudes contribute to the observables.

### 5.5.1. The capture process

Antiprotons stopping in H₂ or D₂ loose energy in collisions. Their energy loss per unit length is given by the Bethe-Bloch formula as long as their velocity is larger than αc, corresponding to p̄ energies of ~25 keV. In H₂ gas at STP, the range of, e.g., 3 MeV antiprotons is about 75 cm; the antiprotons need 40 ns before they come to rest. Range and energy loss calculated with the Bethe-Bloch equation are in good agreement with data [154] even though precision experiments reveal a small difference between energy-loss curves of protons and antiprotons [356].

Below 25 keV the p̄ continues to lose its energy by ionisation until its energy is in the few eV range. Then it is captured by the Coulomb field of a proton by Auger emission of an electron.

$$ \bar{p} + H_2 \rightarrow \bar{p}p(nl) + e^- + H. \qquad (5.29) $$

The capture process can be followed numerically using the Classical Trajectory Monte Carlo (CTMC) method. It describes a three-body problem (antiproton, proton and electron) using a classical Hamiltonian to derive equations of motion, which are solved for a statistical choice of the so-called micro-canonical variables. Fig. 5.15 shows the simulation of a capture process. The H atom is described by a classical p + e⁻ system with a radius corresponding to the first Bohr orbit. Phases and
<figure><img src="image_96.png" /><figcaption>Fig. 5.15. Simulation of the capture of a 5 eV antiproton by a H atom using the Classical Trajectory Monte Carlo method. The plot shows the respective distances between proton and antiproton and electron (from [357]).</figcaption></figure>

eccentricity are chosen randomly. After ejection of the electron, antiproton and proton are bound in a flat ellipse, corresponding to a classical radius of 0.5 Å and to a principal quantum number $n \sim 32$.

The cross-section for protonium formation rises steeply for very low energies and is $2-3 \text{ Å}^2$ for $\bar{p}$ energies below 13.6 eV [358]. Most antiprotons are captured when their energy is below the H ionisation energy; the principal quantum number $n$ is most often between 30 and 50 (sometimes even larger than 100) and the average orbital angular momentum $\ell$ about 20 [358]. Qualitatively, the preference for protonium capture into high Rydberg states can be understood when the overlap of electronic and antiprotonic wave functions is considered. Capture will occur with high probability, when the classical radius of protonium atoms is matched to the size of ground-state hydrogen atoms. The expectation value of the atomic radius is related to $n$ and $\ell$ via

$$ \langle r_{n,\ell} \rangle = \frac{a_0}{2}(3n^2 - \ell(\ell + 1)). \qquad (5.30) $$

The “best” choice of the principal quantum number is then in the range

$$ \sqrt{\frac{2m}{m_e}} < n_c < -3/2 + \sqrt{\frac{9}{4} + \frac{3}{2}\frac{2m}{m_e}}, \qquad (5.31) $$

or $32 < n_c < 36$. For the angular momentum states after capture, a statistical population seems plausible. Calculations show that the preferred distribution in $\ell$ has its maximum at about $n/2$ [359].

### 5.5.2. Collisions between protonium atoms and H₂ molecules

*Collisions.* Once formed, protonium atoms collide with H₂ molecules where they experience large electric fields inducing transitions from initial $(n, \ell)$ protonium states to other levels via dissociation of neighbouring molecules, Auger effect or Stark mixing. They are schematically represented in Fig. 5.16.
<figure><img src="image_97.png" /><figcaption>Fig. 5.16. Level scheme and atomic cascade of antiprotonic hydrogen.</figcaption></figure>

*Chemical effects.* In very high levels (for $n \ge 20$), $\bar{p}p$ atoms de-excite by dissociation of the colliding $H_2$ molecules:

$$ (\bar{p}p)^{n_i, \ell_i} + H_2 \rightarrow (\bar{p}p)^{n_f, \ell_f} + H + H, \\ \Gamma_{\text{chem}} = N v \pi (r_{n_i})^2 \quad \text{for } \delta E_{n_i \rightarrow n_f} \ge \Delta, \quad \ell_i = \ell_f. \qquad (5.32) $$

The rate for this effect is assumed to be given by the classical “size” of the $\bar{p}p$ atom and the collision frequency [140]. $\Delta = 4.7$ eV is the dissociation energy of $H_2$ molecules, $N$ is the hydrogen atom density, and $v$ is the protonium velocity.

*Auger effect.* For $n \sim 20$ the classical radius becomes too small to allow chemical effects to play a significant role. Yet one of the H atoms of a $H_2$ molecule can be ionised and an Auger process can take place.

$$ (\bar{p}p)^{n_i, \ell_i} + H_2 \rightarrow (\bar{p}p)^{n_f, \ell_f} + H + p + e^-, \\ \Gamma_{\text{Auger}} = \frac{16\pi}{3} \frac{N}{m_{\text{p}}^2} (R_{n_i}^{n_f})^2 (2\delta E + 1.39)^{-1/2} \quad \text{for } \delta E + 1.39 \ge 15.2 \text{ eV}. \qquad (5.33) $$

The Auger effect is induced by the electric field seen by the $\bar{p}p$ atom in the collision and is governed by the same matrix element $(R_{n_i}^{n_f})^2$ as radiative de-excitation. But while radiative transitions prefer large transition energies, Auger transitions occur most frequently with a minimal change in the principal quantum number. The energy gain is then just sufficient to knock out an electron. As in radiative transitions, angular momentum changes according to $l_f = l_i \pm 1$.

In collisions, peak electric field strengths of typically $\sim V/\AA$ are experienced (for an impact parameter of $1.5 \AA$) for about 20 fs. The electron density integrated over the collision time is $\sim 0.1 \rho_0$ fs where $\rho_0$ is the electron density of H atoms at $r = 0$. Protonium atoms are neutral; hence they move along straight lines if the attraction between the two collision partners is neglected. The path can be calculated using, e.g., the CTMC method [360]. For an impact parameter $b$ of $1.5 \AA$,
the minimal distance shrinks with increasing principal quantum number of the protonium atom. For $n=20$, the minimal distance is by a factor 2 smaller than the impact parameter, for $n=5$ this effect is negligible. The straight-line approximation therefore underestimates the effect of the electric field and in particular the Stark mixing probability.

*Stark mixing.* Stark mixing of states with different angular momenta is extremely important for the cascade of $\bar{p}p$ and $\bar{p}d$ atoms, as first demonstrated by Day et al. [139]. Many transitions between different nearly mass-degenerate angular momentum states occur in a single collision between a $\bar{p}p$ or $\bar{p}d$ atom and a $H_2$ molecule. The electric field induces Stark mixing transitions between different orbital angular momentum states having the same principal quantum number $n$. Since the direction of the electric field changes during the collision, not only transitions with $\Delta m = 0$ occur but also transitions in which $\Delta m = \pm 1$. In principle, the theory involves $n^2$ coupled Schrödinger equations with a time-dependent electric field. Leon and Bethe avoided this difficulty and used instead a shuffling model which takes into account the net effect of back and forth transitions between different $\ell$.

$$ \Gamma_{n,\ell \to n,\ell+1} = \frac{2\ell+1}{2\ell-1} \Gamma_{n,\ell \to n,\ell-1} = \pi N v \rho_0^2. \qquad (5.34) $$

In a microscopic model, the $n^2$ coupled differential equation are integrated numerically and transition rates from any initial state $(n, \ell)$ to the other states $(n, \ell')$ are determined.

### 5.5.3. The cascade

The microscopic cascade model of Reifenröther and Klempt [361] begins with an initial population $p_{n,\ell}$ of the protonium levels. All levels can radiate to lower levels or annihilate at any time. Collisions with different impact parameters may take place with their respective probabilities. Five different impact parameters are chosen in a way that the electric field strength in a collision reaches a maximum value of $10^{-2}, 10^{-1}, \dots, 10^2$ V/Å. The impact parameters corresponding to these field values and hence the collision frequencies depend on $n$. These collisions induce external Auger effect and Stark mixing. For the figures presented here, the Auger effect was enhanced by a factor 2. This adjustment leads to a better agreement with data.

The cascade model starts at $n=30$ and assumes an initial distribution in $\ell$ and calculates the depopulation of these states until the residual population of 0.1% is reached. Each X-ray emission or annihilation from an S or P state is recorded. Thus, from the X-ray yields, the fraction of S- and P-state capture and the cascade time are determined. The X-ray intensities listed in Table 5.4 are shown in Figs. 5.17 and 5.18 and compared with the calculation.

The cascade of $\bar{p}d$ atoms is very similar to that of protonium. Cascade calculations concentrated on the role of S-wave and P-wave capture [361]. The X-ray yields are reproduced in Fig. 5.19.

### 5.5.4. S versus P capture

Cascade models predict the density-dependent probability for a protonium atom to annihilate from an atomic S state or from a P state. This is an important issue since the dynamics of the annihilation process depends on the angular momentum state from which annihilation occurs.

The fraction of S- and P-state capture can be determined using selection rules. For instance, annihilation at rest into $K_S^0 K_1^0$ is allowed from the $^3S_1$ state, into $K_S^0 K_1^0$ from the states $^3P_0$ and $^3P_2$.
<figure><img src="image_98.png" /><figcaption>Fig. 5.17. L X-ray intensity of $\bar{p}p$ atoms as a function of target density. Left: $L_\alpha$ and $L_\beta$ intensity; right: sum of intensities of L-line series. The solid line is from the Mainz cascade model, the dotted line from Batty [364].</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>Fig. 5.18. K X-ray intensity of $\bar{p}p$ atoms as a function of target density. Left: $K_\alpha$ and $K_\beta$ intensity; right: sum of intensities of K-line series. The solid line is from the Mainz cascade model, the dotted line from Batty [364].</figcaption></figure>

The number of $K_s^0K_1^0$ and $K_s^0K_s^0$ events found in bubble chambers at BNL and CERN [348]:

$$787 \text{ events } \bar{p}p \rightarrow K_s^0 K_1^0, \\ 4 \text{ events } \bar{p}p \rightarrow K_s^0 K_s^0, \qquad (5.35)$$

show a strong preference for the annihilation into $K_s^0K_1^0$ and evidence the dominance of S-wave capture. It was therefore a great surprise when Devons et al. [141] found an unexpectedly large branching ratio for the reaction $\bar{p}p$ annihilation $\rightarrow \pi^0\pi^0$, forbidden from S-state orbitals.
<figure><img src="image_100.png" /><figcaption>Fig. 5.19. L X-ray intensity of p̅d atoms as a function of target density. Left: L_α and L_β intensity; right: sum of intensities of L-line series. The solid line is from the Mainz cascade model, the dotted line from Batty.</figcaption></figure>

After a long history of conflicting results, the Crystal Barrel Collaboration found a rate for $2\pi^0$ production of $(6.93 \pm 0.43) \times 10^{-4}$ [362] for antiprotons stopping in liquid H₂, fully compatible with the old findings of Devons et al. [141]. When compared to the $\pi^+\pi^-$ rate, a P-wave fraction of 45% to p̅p annihilation at rest in liquid H₂ can be derived [363].

This large discrepancy is derived from two rare channels, with frequencies of about 0.3% for p̅p → $\pi^+\pi^-$ and about 0.1% for p̅p → $K_s^0K_l^0$. It reflects a large coupling to $\pi\pi$ and a small coupling to $K\bar{K}$ from P states. With the measured rates for p̅p → $\pi\pi$ [156] and to $K\bar{K}$ [157] from P states, the P-state contribution reduces to ~30 ± 15%, and there is no more conflict between the results derived from $K\bar{K}$ and from $\pi\pi$.

In the derivation of the new P-state fraction, the assumption is made that, at the moment of annihilation, the statistical distribution of the fine-structure levels $2s+1$P_J states for large n is the same as for n = 2. This assumption is likely not true: in high-n levels, Stark mixing of atomic states is very strong. The $^3P_0$ level has a strong interaction width four times larger than the mean 2P width (compare (5.20) and (5.21)). High-n $^3P_0$ levels can be repopulated after annihilation via Stark mixing collisions and the $^3P_0$ levels have a larger chance to contribute to annihilation. There is practically no Stark mixing for n=2; p̅p atoms in the 2P fine-structure levels annihilate and the $^3P_0$ level is not refilled after annihilation. When this effect is taken into account, the fraction of P-state capture for antiprotons stopping in liquid hydrogen reduces from 28.8 ± 3.5% to 12 ± 2% [362,364]. This is a value compatible with most partial-wave analyses. Fig. 5.20 shows the fraction of P-state annihilation as a function of H₂ density.

Batty [364] also determined the fractional contributions of individual hyperfine structure states to annihilation as a function of the hydrogen density. He found that for any selected channel, the contributions of individual hyperfine states change by an enhancement factor $B_{HFS}$. Fig. 5.21 shows these factors as functions of the H₂ density.

We notice a substantial increase of the contribution of the $^3P_0$ state with increasing density. This increase is responsible for the large $\pi^0\pi^0$ branching ratio. In turn, this large branching ratio is only
<figure><img src="image_101.png" /><figcaption>Fig. 5.20. Fraction of P-state capture as a function of H₂ density. The lower line reproduces results from the Mainz cascade model; the upper line uses the Borie-Leon model. The “experimental” points are derived in [364].</figcaption></figure>

<caption>Fig. 5.21. Change in the population of protonium levels as functions of H₂ density.</caption>

compatible with other determinations of the P-state capture rate, when we assign the majority of $\bar{p}p$ annihilations into $2\pi^0$ to the $^3P_0$ and not to the $^3P_2$ initial state.

The fraction of P-state annihilation in $\bar{p}n$ annihilations is even more uncertain. From a comparison of $\bar{p}p$ annihilation into $\pi^0\pi^0$ in liquid H₂ and D₂, the P-state capture fraction in D₂ was estimated to be 22 ± 4% (after a cut on the proton momentum to ensure annihilation on a quasi-free nucleon) [365]. Batty [355] estimated the P-state annihilation frequency from $\bar{p}d$ annihilations into $\pi\pi$ and into $K\bar{K}$ and derived a fraction of 34 ± 4% P-state capture. From cascade calculations he estimated this fraction to be 40%. In summary, a P-state fraction of 30% for antiprotons stopping in liquid D₂ seems realistic.

# 6. Phenomenology of the nucleon-antinucleon interaction

In this section, we resume the discussion on the theoretical approaches to $\bar{N}N$ interaction entering elastic, charge- and strangeness-exchange scattering, and protonium. We focus on the progress on phenomenological analyses made possible by LEAR data.

## 6.1. *Comparison of scattering data with the predictions of the early optical models*

The basic features of the data, i.e., the shape of integrated cross-sections and the trend of the elastic and charge-exchange differential distributions, can be reproduced by the simple optical models introduced in Section 3, where the meson-exchange part is taken from NN scattering, via G-parity rule,
<figure><img src="image_102.png" /><figcaption>Fig. 6.1. Total, annihilation and charge-exchange cross-sections as a function of the $\bar{p}$ momentum. The curves are the calculations with the Kohno–Weise model [212].</figcaption></figure>

and supplemented by an energy- and state-independent imaginary absorptive potential, to simulate the effect of annihilation.

Some representative data are compared with various model calculations in Figs. 6.1–6.4.

### 6.1.1. Integrated cross-sections

The data shown in Fig. 6.1 exhibit a smooth behaviour as a function of the incident momentum. In particular, no narrow resonance emerges! This stems naturally from optical potentials, as illustrated by the Kohno–Weise model [212]. To maintain the proper annihilation-to-elastic ratio and the smallness of the charge-exchange cross-section as energy increases, one needs a wide enough annihilation core. Otherwise, one has to play with the energy-dependence and non-locality parameters, in more sophisticated models.

### 6.1.2. Differential cross-sections

A representative set of data is displayed in Fig. 6.2, for the elastic and charge-exchange cases.

The main trend of the elastic differential cross-section is rather well reproduced. Some adjustments are necessary to match exactly the shape of the charge-exchange differential cross-section. Minor changes in the models change the shoulder shape into a dip-bump structure, or vice versa. This effect was pointed out in particular by H. Poth (private communication) and Myhrer [60,238,366]. The changes could consist either in an adjustment of the $\rho$-meson coupling or an alteration of the shape of the annihilation potential. We also remark from Fig. 4.15 in Section 4, that the angular
<figure><img src="image_103.png" /><figcaption>Fig. 6.2. Differential cross-section in the elastic (left) and charge-exchange (right) channels. The data are from experiment PS173. The figures are from Ref. [219]: the solid line represents a simple optical model fitted to the data, the dotted line is the earlier Dover-Richard model [211], the dashed one, the Paris model [237].</figcaption></figure>

distribution of charge-exchange varies from one energy to another. The charge-exchange angular distribution at 287 MeV/c poses a more serious problem: no model can account for the ample decrease of the cross-section suggested by the data.

### 6.1.3. Elastic-scattering analysing power

The $\bar{p}p \rightarrow \bar{p}p$ analysing-power measurements from experiment PS172 are shown in Fig. 6.3. The continuous lines are the predictions of the first version of the Paris model [237], to be discussed in more detail later in this chapter. The dashed line corresponds to the predictions of the Dover-Richard model [211], also based on the G-parity transformed NN Paris potential, but with a simpler treatment of absorption.

Comments are in order. It is impressive how the overall trend of the data is predicted by these models. To obtain a better agreement with the data, it was necessary to tune more extensively the
<figure><img src="image_104.png" /><figcaption>Fig. 6.3. The $\bar{p}p \rightarrow \bar{p}p$ analysing power as measured by PS172 at 15 momenta, compared with the predictions of the Paris (continuous line) and Dover-Richard (dashed line) models.</figcaption></figure>
<figure><img src="image_105.png" /><figcaption>Fig. 6.4. Analysing power at 656 MeV/c, compared to early models: Dover-Richard (DR1,DR2), Kohno-Weise (KW), Myhrer-Dalkarov (M), and an early analysis by the Nijmegen group (N).</figcaption></figure>

parameters of the potential, including its non-local terms. The early Paris model [237] was adjusted
to fit pre-LEAR analysing-power data.

6.1.4. *Charge-exchange analysing power*

The analysing-power parameter $A_n$ of the charge-exchange reaction was measured for the first time at LEAR by experiment PS199. The data are compared in Fig. 6.4 with the model calculations which were published before the data appeared: the models DR1, DR2 and KW were already introduced in Section 3, as well as the boundary-condition model of Myhrer and Dalkarov, referred to as M; the label N corresponds to an early analysis of the Nijmegen group, on which more later. The broad range of predictions comes from this observable being very sensitive to small contributions to the scattering amplitudes, as already stressed in Section 4.6 ($\pi$-exchange, which is the dominating dynamical mechanism, does not contribute to the analysing power at first order).

Agreement with the data could not be obtained by simply tuning the absorptive part of these
simple optical models; much more refined models are needed to reproduce the data. Even in the
case of the latest Paris potential [367], which describes the short-range potential with 30 parameters
(see Section 6.3), the agreement with the data is not perfect. Note that the authors of this model
insisted on having an annihilation potential looking short ranged; as a consequence, they slightly
overestimate the integrated charge-exchange cross-section; so it is not too much a surprise that they
also face difficulties for the spin observables of this reaction.

Note that in their latest analysis, the Nijmegen group was able to produce good fits to these data.
We will come back to this point later.
<figure><img src="image_106.png" /><figcaption>Fig. 6.5. Analysing power below 900 MeV/c, compared to the predictions of the Myhrer–Dalkarov model.</figcaption></figure>

In Fig. 6.4, the best agreement between the measured data at 656 MeV/c and the model calculations is observed with the boundary-condition model of Myhrer and Dalkarov [228]. As seen in Fig. 6.5 showing the set of data measured by PS199 below 900 MeV/c, the agreement remains at the lowest energies, but only in the forward hemisphere.
## 6.2. Comparison with coupled-channel models

### 6.2.1. The Lebedev school

The Lebedev Institute group, led by I.S. Shapiro, carried out a pioneering work on the coupled-channel-model (CCM) description of $\bar{N}N$ scattering. A rather comprehensive summary has been given in Refs. [242,243], of which one can highlight the following points.

The motivation for a coupled-channel approach stems from a criticism of optical models. Unitarity is not explicitly fulfilled in optical models (though room is left for the annihilation channels which are integrated out). It is delicate to interpret and handle the non-orthogonal wave functions generated by optical models. It is even argued in [240] that “the optical model potentials take annihilation into account but do not reproduce the effect of the reappearance of the initial particles, so that the baryon-antibaryon wave functions would usually be underestimated”. To our knowledge, there is no such a criticism for the optical model approach to nuclear reactions, as initiated by Feshbach [218]. Anyhow, Shapiro in [240] has underlined that the CCM is more consistent.

In practice, the Lebedev group considered a simple CCM with a minimal number of mesons in the annihilation channels. To have non-relativistic kinematics, it was also assumed that the masses of the two mesons are equal to the $\rho$ mass.

Besides the coupling which are obtained by G-conjugation of the NN OBEP, two sets of parameters must be settled by fitting the data, namely

* the annihilation constant $\lambda_\ell$ for each angular momentum $\ell = 0, 1$, or 2.

* the cut-off distances $r_c(\ell)$ for the singular OBEP terms, for the different $\ell$, at which the NN potentials were put to 0. Depending on the quantum numbers, the values of $r_c$ varied between 0.50 and 0.72 fm.

The fitting procedure does not give a unique set of values for the annihilation constants $\lambda_\ell$. But if one retains only the parameters corresponding to a potential well that is not too deep, then the solution becomes unique. In the fit, only the annihilation cross-section, and the integrated $\bar{p}p \to \bar{p}\bar{p}$ and $\bar{p}\bar{p} \to \bar{n}\bar{n}$ cross-section data were used.

The fit reproduces quite well the angular dependence of the $\bar{p}p \to \bar{p}\bar{p}$ differential cross-section at 287 MeV/c, whose strong angular anisotropy results from an interference between the S and P waves, and the trend of the $\rho$ parameter as a function of the $\bar{p}$ momentum.

In Ref. [240], the Lebedev group gave a comparison between their results and the experimental $\bar{p}p$ scattering and atomic data. Though the agreement is rather encouraging, we should note that:

1. Only the very low-energy region was investigated.

2. The work was limited by presumably poor computing facilities. In particular, the influence of the mass of effective mesons, of the number of channels which are introduced, etc., was not studied. Also, the influence of the parameters on the angular distributions has not been examined in detail.

3. The spin observables were ignored.

Finally, they calculated the spectrum of the near-threshold $\bar{N}N$ quasi-nuclear levels, finding at least five near-threshold P levels, and confirming the theoretical expectation given in earlier papers by this group.
<figure><img src="image_107.png" /><figcaption>Fig. 6.6. Differential $\bar{p}p$ cross-sections at 780 MeV/c: (a) elastic, (b) charge-exchange, (c) $\pi^+\pi^-$, (d) $K^+K^-$.$\bar{N}$N scattering refers to one of the coupled-channel solutions of Ref. [229], where references for the data can also be found.</figcaption></figure>

<caption>Fig. 6.7. Analysing power for $\bar{p}p \rightarrow \bar{p}p$ (a) and $\bar{p}p \rightarrow n\bar{n}$ (b) at 780 MeV/c compared to one of the coupled-channel solutions of Ref. [229].</caption>

### 6.2.2. Liu and Tabakin

Liu and Tabakin have worked out an elaborated coupled-channel description of $\bar{N}N$ scattering [229]. They considered explicitly the $\pi^+\pi^-$ and $K^+K^-$ channels, calculated the angular distribution and analysing power for these channels, and compared them with the interesting data taken at LEAR [81].

Acceptable fits to the differential cross-section of both $\bar{N}N$ scattering and two-meson production data were obtained, as shown in Fig. 6.6, taken from Ref. [229].

The analysing power was estimated for $\bar{p}p \rightarrow \pi^+\pi^-$ and $K^+K^-$, and for the elastic and charge-exchange $\bar{N}N$ channels. See Fig. 6.7, again from Ref. [229]. It is a pity that such a promising model has never been applied to calculate more spin observables and compare them with the predictions of optical models.

### 6.2.3. The Bonn group

The Bonn group (more precisely, the Bonn–Jülich group) has achieved an impressive amount of work on $\bar{N}N$ physics, over many years.
Their goal was rather ambitious: develop for the short-range region a coherent model which is consistent with the long-range dynamics. This means describing annihilation in terms of baryon exchange, with the same baryon–meson coupling as in the Yukawa potential. For the sake of comparison, this group also considered some quark-model scenarios.

A pure hadron picture of $\bar{N}N$ scattering and annihilation might look at first rather obsolete or nostalgic of the physics of the 1950s and 1960s. On the other hand, it can be viewed as anticipating “effective theories” designed as an approximation to QCD.

As compared to Liu and Tabakin, the Bonn group has considerably enlarged the set of the two-meson channels explicitly included: besides the pseudoscalar mesons $\pi, \eta$ and K, they considered all possible contributions of the lowest-mass mesons with $J^{PC} = 0^{++}, 1^{--}, 1^{++}$ and $2^{++}$ quantum numbers for both isospin $I=0$ and $I=1$.

In practice, the Bonn group proceeded by steps, namely [232]:

* model A (BOX) is a mere phenomenological optical model based on the G-parity transformed of the celebrated Bonn NN potential,

* model C contains some explicitly calculated channels, whose strength is artificially enhanced to account for the missing channels. This leads to a very pronounced spin and isospin dependence.

* model D is presumably more realistic, since it contains contributions of two-meson intermediate states made of $\pi, \eta, \rho, \omega, a_0, f_0, a_1, f_1, a_2, f_2$ (via N or $\Delta$ exchange) and K, $K^*$ (via $\Lambda$ or $\Sigma, \Sigma^*$ exchange), as shown in Fig. 6.8. These two-body (in case of narrow mesons) or quasi-two-body (in case of resonances) channels represent only about 30% of the annihilation which is required to fit the data. The missing part is described by means of phenomenological, state-independent potential $V_{\text{opt}}$.

The formalism developed in Eqs. (3.58)–(3.60) should thus be slightly extended. The coupled equation for the $\bar{N}N$ scattering amplitude $\mathcal{F}^{\bar{N}N \to \bar{N}N}$ and the transition amplitudes $\mathcal{F}^{\bar{N}N \to M_1 M_2}$ for the annihilation in two mesons, which proceeds via baryon exchange, can be written as

$$ \begin{aligned} \mathcal{F}^{\bar{N}N \to \bar{N}N} &= V^{\bar{N}N \to \bar{N}N} + V^{\bar{N}N \to \bar{N}N} G^{\bar{N}N \to \bar{N}N} \mathcal{F}^{\bar{N}N \to \bar{N}N}, \\ \mathcal{F}^{\bar{N}N \to M_1 M_2} &= V^{\bar{N}N \to M_1 M_2} + V^{\bar{N}N \to M_1 M_2} G^{\bar{N}N \to \bar{N}N} \mathcal{F}^{\bar{N}N \to \bar{N}N}. \end{aligned} \quad (6.1) $$

The $\bar{N}N$ interaction $V^{\bar{N}N \to \bar{N}N}$ consists of an elastic and an annihilation part, say,

$$ V^{\bar{N}N \to \bar{N}N} = V_{\text{el}} + V_{\text{ann}}, \quad (6.2) $$

where $V_{\text{el}}$ is the G-parity transformed of the full NN potential, and

$$ V_{\text{ann}} = \sum_{ij} V^{\bar{N}N \to M_i M_j} G^{M_i M_j} V^{\bar{N}N \to M_i M_j} + V_{\text{opt}}. \quad (6.3) $$

The diagrams corresponding to the three pieces of the $\bar{N}N$ interaction are shown in Fig. 6.9.

Results obtained with the three Bonn models can be seen in Figs. 6.10–6.13, for the integrated cross-sections, angular distributions, and analysing power parameters.

Model C fails almost everywhere in reproducing the data. There is not enough absorption in this model. In particular, the integrated charge-exchange cross-section is coarsely overestimated, as apparent in Fig. 6.10. Models A and D reproduce fairly well the differential cross-sections, with the exception of the charge-exchange data below 300 MeV (Fig. 6.12). For what concerns analysing
<figure><img src="image_108.png" /><figcaption>Fig. 6.8. Bonn model: transition potentials included explicitly in the microscopic annihilation model.</figcaption></figure>

<caption>Fig. 6.9. Bonn model: elastic (a), microscopic annihilation (b), and phenomenological annihilation (c) part of the $\bar{N}N$ interaction model.</caption>

power, model A has some success with the elastic channel in the forward hemisphere (Fig. 6.11), while no model reproduces the charge-exchange data (Fig. 6.13).

### 6.2.4. The Nijmegen group

In the last decades, the Nijmegen group concentrated a large effort on the study of both the NN and the $\bar{N}N$ interaction. They started with the construction of semi-phenomenological potentials, with a meson-exchange tail and an empirical core. Later on, they used their expertise to perform a partial-wave analysis (PWA) of the scattering data, to be discussed below. The results of the PWAs were used to improve the potentials, so that there is an interplay between the two approaches.

In analysing the data, they followed a purely statistical approach: the model parameters are adjusted to the data by comparing the theoretical predictions with the experimental data using a least-squared-fitting procedure. In this procedure, the experimental data are scrutinised and filtered: if a contribution to the $\chi^2$ is too large according to rules which can be found, e.g., in Ref. [368], the corresponding data point or set of data is rejected.

They end in 1993 with the “Nijmegen $\bar{N}N$ database” [279], which they regard as important as the coupled-channel model or the PWA they produce simultaneously. More comments on the data selection will be given in Section 6.4, while presenting the PWA. The so-called CC84 Nijmegen coupled-channels model was constructed in 1984 by Timmers et al. [368]. Fitting to the then available pre-LEAR data resulted in a quite satisfactory fit with $\chi^2/N_d = 1.39$. An update of the model CC84 was made in 1991 [369] (“Nijmegen model CC93”). The results were reported at LEAP94 [370].

The $\bar{N}N$ coupled channels are not treated in the isospin basis, but in the $(\bar{p}\bar{p}, \bar{n}\bar{n})$ basis. This allows to introduce symmetry-breaking effects due to Coulomb interaction in the $\bar{p}\bar{p}$ channel and to the mass difference between the neutron and the proton and between the exchanged $\pi^0$ and $\pi^\pm$.
<figure><img src="image_109.png" /><figcaption>Fig. 6.10. Total, elastic, charge-exchange, and annihilation cross-sections for $\bar{p}p$ scattering compared with the Bonn models D (solid curve), A (dashed curve), and C (dotted-dashed curve) calculations.</figcaption></figure>

The $\bar{N}N$ channels are also coupled to annihilation channels. Those are mimicked by the Nijmegen group by two pairs of fictitious mesons with equal masses, one pair with total mass 1700 MeV/c², and another one with total mass 700 MeV/c², in both isospin $I=0$ and $I=1$. See Ref. [371].

In the neutral case, one ends up with six coupled channels. There are also six coupled channels for most initial $\bar{p}p$ partial waves, such as $^1S_0$, $^1P_1$, $^1D_2$, $^1F_3$, etc. Due to the tensor force, there are 12 coupled waves for each of the $\bar{p}p$ states with natural parity: $^3S_1 + ^3D_1$, $^3P_2 + ^3F_2$, $^3D_3 + ^3G_3$, etc.

The authors use a Schrödinger equation with relativistic kinematics in coordinate space. The interaction is then described by either a 6 × 6 or a 12 × 12 potential matrix, schematically

$$V = \begin{pmatrix} V_{\bar{N}N} & \tilde{V}_{ann} \\ V_{ann} & 0 \end{pmatrix}. \qquad (6.4)$$

The 2 × 2 (or 4 × 4) sub-matrix $V_{\bar{N}N}$ can be written as

$$V_{\bar{N}N} = V_c + V_{MM} + V_{OBE}, \qquad (6.5)$$

where for $V_c$ denotes the Coulomb potential with relativistic corrections, $V_{MM}$ the magnetic moment interaction, and $V_{OBE}$ the charge-conjugated Nijmegen NN potential Nijm78 [372]. The diagonal
<figure><img src="image_110.png" /><figcaption>Fig. 6.11. Elastic pp polarisation data compared with the Bonn models D (solid curve), A (dashed curve), and C (dotted-dashed curve) calculations.</figcaption></figure>

interaction in the annihilation channels is neglected. The annihilation potential $V_{ann}$ connects the $\bar{N}N$ channels to the two-meson annihilation channels. It is either a 2 × 4 matrix or a 4 × 8 matrix. This potential is decomposed as

$$V_{ann}(r) = \left( V_C + V_{SS}\vec{\sigma}_1 \cdot \vec{\sigma}_2 + V_T S_{12} m_a r + V_{SO} L \cdot S \frac{1}{m_a^2 r} \frac{d}{dr} \right) \frac{1}{1 + \exp(m_a r)} \quad (6.6)$$

The factor $m_a r$ is introduced in the tensor component to force its vanishing at the origin. The cut-off mass $m_a$ is taken to be the mass of the meson in the channel, either 850 or 350 MeV/c². This annihilation potential depends on the spin structure of the initial state.

For each isospin and for each meson channel five parameters are introduced: $V_C$, $V_{SS}$, $V_T$, $V_{SO}$, and $m_a$. This gives a model with altogether 4 × 5 = 20 parameters, which can be adjusted to fit the $\bar{N}N$ data. The best agreement is found with $\chi^2/N_d = 3.5$. The conclusion of the authors was that “although the old Nijmegen soft-core potential Nijm78 is a pretty good NN potential, it is definitely not the ultimate potential”.

Extra parameters were thus introduced: the coupling constants of the $\rho$, $\omega$, $\varepsilon(760)$, and $a_0(980)$ mesons, as well as that of the Pomeron. Varying these parameters within a reasonable range resulted
<figure><img src="image_111.png" /><figcaption>Fig. 6.12. Charge exchange $\bar{p}p \rightarrow \bar{n}n$ differential cross-section compared with the Bonn models D (solid curve), A (dashed curve), and C (dotted-dashed curve) calculations.</figcaption></figure>
<figure><img src="image_112.png" /><figcaption>Fig. 6.13. Charge exchange $\bar{p}p \rightarrow \bar{n}n$ polarisation compared with the Bonn models D (solid curve), A (dashed curve), and C (dotted-dashed curve) calculations.</figcaption></figure>

in an appreciable improvement of the fit. The results shown at the LEAP94 conference at Bled [21] indicated, indeed, a dramatically lower $\chi^2/N_d = 1.58$, on the 1993 data set [370]. In Fig. 6.14, the results of this coupled channel model are compared to the PS199 data on charge-exchange differential cross-section at 693 MeV/c and the analysing power at 656 MeV/c. The agreement with the data is very good.

### 6.3. Refined optical model: Paris potential

Early fits to $\bar{N}N$ data, such as those by Bryan and Phillips [210], Dover and Richard [214], or Kohno and Weise [212] were done by using a short-range potential which is spin and isospin independent, for simplicity. The spin-isospin dependence of the meson-exchange tail was, however, taken into account (except sometimes an unjustified neglect of the orbital-mixing component $\ell=J-1 \leftrightarrow \ell=J+1$ of the tensor force).

It was then natural to improve the phenomenological studies by introducing some channel dependence in the annihilation potential. We already mention in Section 3 the work of Bydzovsky et al. [222], where the absorptive potential is allowed to be isospin dependent. A step forward was made by the Paris group in Refs. [237, 373, 272, 367]: in this series of fits, besides some energy dependence,
<figure><img src="image_113.png" /><figcaption>Fig. 6.14. Charge-exchange differential cross-section and analysing power at 693 and 656 MeV/c. The data are from PS199; the curves are from the Nijmegen coupled-channel model [370].</figcaption></figure>

a complete spin dependence (central, spin-spin, spin-orbit and tensor terms) is introduced in the short-range potential for each isospin state. In particular, the imaginary potential is written in each isospin state as

$$ \mathrm{Im}V = \left[ g_C(1 + f_c T_L) + g_{SS}(1 + f_{SS} T_L)\vec{\sigma}_1 \cdot \vec{\sigma}_2 + g_T S_{12} + \frac{g_{LS}}{2m^2} \vec{L} \cdot \vec{S} \frac{1}{r} \frac{d}{dr} \right] \frac{K_0(2mr)}{r}, \quad (6.7) $$

where $T_L$ is the kinetic energy of the antiproton in the target frame, and $K_0$ the modified Bessel function, whose occurrence is inspired by considerations on the box diagram 3.8 of Section 3.

The meson-exchange potential is kept for distances larger than $r_c = 0.84$ fm. For smaller distances, it is replaced by a polynomial which matches continuously the external potential at $r_c$, and whose values at $r_2 = 0.6$ fm is a free parameter in the fit.

The authors display their best fit in each paper. The parameters of the successive versions of this Paris potential are compared in Tables 6.1 and 6.2.<fnref n="10" /> Our reading of Tables 6.1 and 6.2 is that several solutions lead to comparable low $\chi^2$, and that new data tend to promote a solution that was only ranked second or third on the basis of the previous data set.

In other words, the fit, however precise and interesting, is far from being unique. This is not too much a surprise. One needs several spin observables, involving complicated polarisation devices and final-spin measurements, to determine the full S-matrix of spin 1/2-spin 1/2 scattering. In the $\bar{N}N$ case, we are unfortunately restricted to polarisation (or analysing power) measurements and to a meager set of depolarisation data. In short, fitting the data does not determine the detailed spin and isospin dependence of the core uniquely.

<footnote n="10">¹⁰ We are indebted to B. Loiseau for his help in collecting these values.</footnote>
<table><caption>Table 6.1
Parameters of the imaginary part in the successive versions of the Paris potential [237,373,277,367]</caption><thead><tr><th>Parameter</th><th>1982</th><th>1991</th><th>1994</th><th>1999</th></tr></thead><tbody><tr><td>g<sup>0</sup><sub>C</sub></td><td>850.0</td><td>1109.0</td><td>180.0</td><td>125.0</td></tr><tr><td>g<sup>0</sup><sub>SS</sub></td><td>-570.0</td><td>-774.0</td><td>30.0</td><td>-4.0</td></tr><tr><td>g<sup>0</sup><sub>LS</sub></td><td>74.0</td><td>55</td><td>8.0</td><td>35.0</td></tr><tr><td>g<sup>0</sup><sub>T</sub></td><td>53.0</td><td>99.0</td><td>-6.0</td><td>2.0</td></tr><tr><td>f<sup>0</sup><sub>C</sub></td><td>0.019</td><td>0.011</td><td>0.022</td><td>0.020</td></tr><tr><td>f<sup>0</sup><sub>SS</sub></td><td>0.015</td><td>0.003</td><td>0.005</td><td>-0.037</td></tr><tr><td>g<sup>1</sup><sub>C</sub></td><td>660.0</td><td>548</td><td>64.0</td><td>78.0</td></tr><tr><td>g<sup>1</sup><sub>SS</sub></td><td>-474.0</td><td>-299.0</td><td>11.0</td><td>20.0</td></tr><tr><td>g<sup>1</sup><sub>LS</sub></td><td>74.0</td><td>7.0</td><td>7.0</td><td>12.0</td></tr><tr><td>g<sup>1</sup><sub>T</sub></td><td>23.0</td><td>126.0</td><td>9.0</td><td>5.0</td></tr><tr><td>f<sup>1</sup><sub>C</sub></td><td>0.019</td><td>0.033</td><td>0.050</td><td>0.033</td></tr><tr><td>f<sup>1</sup><sub>SS</sub></td><td>0.026</td><td>0.060</td><td>0.058</td><td>0.041</td></tr></tbody></table>

These parameters are the dimensionless strength factors $g_i^f$ or the slope $f_i^f$ (in MeV$^{-1}$) of the energy dependence, for isospin $I$.

<table><caption>Table 6.2
Values of the real potential $^1V_\alpha$ at $r_2 = 0.6$ fm, for each isospin $I$, in the successive versions of the Paris potential</caption><thead><tr><th>Potentials</th><th>1982</th><th>1991</th><th>1994</th><th>1999</th></tr></thead><tbody><tr><td><sup>0</sup>V<sup>a</sup><sub>0</sub>(r<sub>2</sub>)</td><td>-550.0</td><td>-677.0</td><td>-1287.0</td><td>-1014.0</td></tr><tr><td><sup>0</sup>V<sup>b</sup><sub>0</sub>(r<sub>2</sub>)</td><td>1.23</td><td>0.84</td><td>0.10</td><td>0.15</td></tr><tr><td><sup>0</sup>V<sup>a</sup><sub>1</sub>(r<sub>2</sub>)</td><td>-753.0</td><td>-796.0</td><td>-1936.0</td><td>-1613.0</td></tr><tr><td><sup>0</sup>V<sup>b</sup><sub>1</sub>(r<sub>2</sub>)</td><td>-1.66</td><td>-1.92</td><td>0.47</td><td>-0.23</td></tr><tr><td><sup>0</sup>V<sub>LS</sub>(r<sub>2</sub>)</td><td>749.0</td><td>701.0</td><td>328.0</td><td>151.0</td></tr><tr><td><sup>0</sup>V<sub>T</sub>(r<sub>2</sub>)</td><td>377.0</td><td>231.0</td><td>353.0</td><td>195.0</td></tr><tr><td><sup>1</sup>V<sup>a</sup><sub>0</sub>(r<sub>2</sub>)</td><td>-1670.0</td><td>-1037.0</td><td>-2972.0</td><td>358.0</td></tr><tr><td><sup>1</sup>V<sup>b</sup><sub>0</sub>(r<sub>2</sub>)</td><td>-0.15</td><td>0.075</td><td>-0.44</td><td>0.07</td></tr><tr><td><sup>1</sup>V<sup>a</sup><sub>1</sub>(r<sub>2</sub>)</td><td>-1107.0</td><td>-1238.0</td><td>-658.0</td><td>-291.0</td></tr><tr><td><sup>1</sup>V<sup>b</sup><sub>1</sub>(r<sub>2</sub>)</td><td>-1.80</td><td>-2.42</td><td>-0.44</td><td>-0.27</td></tr><tr><td><sup>1</sup>V<sub>LS</sub>(r<sub>2</sub>)</td><td>-391.0</td><td>-108.0</td><td>-341.0</td><td>-178.0</td></tr><tr><td><sup>1</sup>V<sub>T</sub>(r<sub>2</sub>)</td><td>104.0</td><td>183.0</td><td>-357.0</td><td>-221.0</td></tr></tbody></table>

Those values were free parameters adjusted to fit the data. The central and spin-spin components are recombined here into a singlet $V_0 = V_C - 3V_{SS}$ and triplet $V_1 = V_C + V_{SS}$ components, both with a static and an energy-dependent parts, $V_s = V_s^a + T_L V_s^b$. Units are MeV, except $V^b$, dimensionless.

It remains that the very good quality of the fit demonstrates that the LEAR data are compatible with the long- and medium-range interaction of the Paris model, i.e., with the current understanding of the meson-exchange dynamics. For illustration, we reproduce below some figures (Figs. 6.15–6.20) of the latest paper by the Paris group [367].
<figure><img src="image_114.png" /><figcaption>Fig. 6.15. $\bar{p}p \rightarrow \bar{n}n$ differential cross-section. The curves are from Ref. [367].</figcaption></figure>

<figure><img src="image_115.png" /><figcaption>Fig. 6.17. $\bar{p}p \rightarrow \bar{n}n$ differential cross-section. The data are from PS199, the curves are from Ref. [367].</figcaption></figure>

<caption>Fig. 6.18. $\bar{p}p \rightarrow \bar{n}n$ differential cross-section. The data are from PS206, the curves are from Ref. [367].</caption>

The agreement is extremely precise. Even the sharp structure in the elastic differential distribution is well reproduced. In the charge-exchange case, the shape of the angular distribution can evolve from a shoulder to a more pronounced dip-bump structure by tuning the parameters, as already stressed. The rise of the analysing power near 150°, at 546 MeV/c, raises some difficulties, as in other models. Fig. 6.18, for instance, illustrates how an earlier version of the Paris potential failed in predicting the forward $\bar{p}p \rightarrow \bar{n}n$ angular distribution. On the other hand, this early Paris potential was rather successful in predicting $D_{0n0n}$, as illustrated in Fig. 6.20.
<figure><img src="image_116.png" /><figcaption>Fig. 6.19. $\bar{p}p \to \bar{n}n$ analysing power. The data are from PS199, the curves are from Ref. [367].</figcaption></figure>

<caption>Fig. 6.20. $D_{0n0n}$ data from PS199. The curves are from Ref. [367].</caption>

## 6.4. Partial wave analysis

### 6.4.1. Method

The Nijmegen group is the only one to have performed a partial-wave analysis (PWA) of $\bar{N}N$ scattering including the LEAR data. An earlier attempt was done by Laloum [374], in a restricted energy range.

For almost 15 years, the Nijmegen group has worked on partial-wave analyses of NN data, developing rather sophisticated and accurate methods [375–377]. In the early 1990s, they applied their techniques to $\bar{p}p$ in exactly the same way as to their NN PWA. An account was presented at the NAN93 Conference [371]). The final result is published in Ref. [279].

The paper [279] is entitled “phase-shift analysis”, but it is close in spirit to potential models. The authors, indeed, use a Schrödinger equation with relativistic kinematics, with a tail potential including a refined Coulomb interaction and meson exchanges, and a combination of boundary condition and optical model to account for annihilation.

The boundary radius is fixed rather precisely at $R=1.3$ fm from the width of the diffraction peak. In each partial wave $L$, the long-range potential $V_L$ for $r > R$ is

$$V_L = V_{\bar{N}N} + V_c + V_{MM}, \qquad (6.8)$$

where the electromagnetic terms $V_c$ and $V_{MM}$ are defined in Section 6.2.4, and $V_{\bar{N}N}$ is the charge-conjugated Nijmegen NN potential, Nijm78 [372].

The boundary condition at $r = R$ may be energy dependent. For the uncoupled waves (like $^1S_0$, $^1P_1$, ..., and $^3P_0$, $^3D_2$, ...), an optical-potential picture is adopted, with a square-well optical potential for $r \le R$, the short-range potential $V_S$ being written as

$$V_S = U_S - iW_S. \qquad (6.9)$$
A slightly more complicated procedure is adopted for the natural-parity partial waves, which are
coupled by tensor forces.

A thorough minimisation, by tuning the parameters in the partial waves, led to a remarkably good fit to the subset of data which has been selected. As the phase-shifts are unavoidably computed on the way from the potential to the observables, the authors chose to present their work as a phase-shift analysis.

6.4.2. Data selection

The Nijmegen group had to face several problems in fitting the data. A large amount of work was devoted by the authors to collect the existing data, check their internal consistency and select a subset of data which in their opinion is suitable for a refined analysis. The Nijmegen group also used normalisation errors sometimes larger than those given by the experiments. The same has been done by the Paris group, in their last fits.

The most delicate issue was the construction of the Nijmegen $\bar{N}N$ data set, in which some measurements were not included. In the $\bar{p}p$ case, the Nijmegen rejected 744 data points, corresponding to 17% of their final dataset. The detailed description of the criteria applied for data selection can be found in Ref. [279]. They essentially say that the rejected data are not necessarily “bad” data, but had to be rejected to allow them to apply statistical methods satisfactorily. The authors applied a similar practice for analysing proton–proton scattering.

A striking feature of the Nijmegen analysis is that of the final data set only 22% of the data points comes from LEAR. The Nijmegen group has, however, acknowledged the important contribution of LEAR for charge-exchange and polarisation. Still, it is a little disappointing, given the unique potential of LEAR, and the claim that this new facility would have provided much better data than the previous ones. In particular, no experiment dedicated to precision measurements of the $\bar{p}p \to \bar{p}\bar{p}$ differential cross-section (these data constitute a large fraction of the $\bar{N}N$ data bases) was performed at LEAR. The reasons are diverse. The beam allocation to scattering experiments was certainly not sufficient. Other measurements were given more priority. The decision of CERN to give low priority to reaction-dynamics studies at LEAR, taken at Cogne at the beginning of the ACOL era [70], undoubtedly discouraged the experimental groups to propose scattering measurements.

6.4.3. Results of the PWA

The results of the Nijmegen PWA on the “accepted” data are really very good. They reached $\chi^2/N_{df} = 1.085$ for 3646 data points, corresponding to $N_{df} = 3503$. The free parameters include the model parameters (30) and normalisation parameters (113) introduced for the different measurements.

The experimental data are compared with the PWA results in Figs. 6.21–6.26 reproduced from Ref. [279].

*Integrated cross-sections:* In Fig. 6.21 the total cross-sections from PS172 and the annihilation cross-sections from PS173 (see Section 4 for references) are compared to the cross-sections calculated with the PWA.

*$\bar{p}p$ angular distribution:* An example of fits to the $\bar{p}p$ differential cross-section can be seen in Fig. 6.22, where the elastic differential cross-section data at $p_L=790$ MeV/c as measured by Eisenhandler et al. are plotted. The fit is excellent and reproduces the smooth trend of the data in the full angular range. As already pointed out in Section 4, and underlined also by the Paris group, there are several
<figure><img src="image_117.png" /><figcaption>Fig. 6.21. Total and annihilation cross-section as a function of the incoming $\bar{p}$ momentum. The curves are the result of the Nijmegen PWA, the data are from PS172 and PS173.</figcaption></figure>

<figure><img src="image_118.png" /><figcaption>Fig. 6.22. Differential cross-section for elastic scattering at 790 MeV/c. The data are from Eisenhandler et al.; the curve from the PWA.</figcaption></figure>

$\bar{p}p$ differential cross-section measurements which are incompatible. Moreover, the quoted errors are often so small that any smooth fit to the entire dataset unavoidably ends up with a very large $\chi^2$. The authors studied the different sets of data coming to the conclusion that the pre-LEAR data by Eisenhandler et al. are incompatible with the LEAR and KEK experiments; also, they could obtain reasonable fits to the LEAR data only at the expense of rejecting the data of Eisenhandler et al. and those of Sakamoto et al. The PS173 data could be reasonably fitted only by rejecting the very forward points, adding point-to-point systematic errors, and assuming a 5% normalisation error not quoted in the publications. Also for the PS198 data they had to add point-to-point errors. At the end, they used the Eisenhandler et al. data, even if with a word of caution. We can only agree with their conclusion that new dedicated experiments (which were essentially not performed at LEAR) “might shed some light on this issue”.

$\bar{p}p \rightarrow \bar{n}n$ angular distribution: The $\bar{p}p \rightarrow \bar{n}n$ differential cross-section data at 693 MeV/c from PS199 is compared with the PWA fit in Fig. 6.23. These experiments are considered by the authors as “one of the most constraining experiments in the database”. To fit these data, they had to take into account orbital momenta up to $\ell = 10$. When this PAW analysis was performed, the PS206 experiment was just approved, and no good-quality data in the dip-bump region were available. Before the PS206 data were published, the prediction at 601 MeV was asked,<fnref n="11" /> and the comparison can be seen in Fig. 6.24. The shape is almost perfect, and the difference in the absolute normalisation between the Nijmegen prediction and the PS206 final result is within the quoted errors.

<footnote n="11">¹¹ R.G.E. Timmermans, private communication.</footnote>
<figure><img src="image_119.png" /><figcaption>Fig. 6.23. Differential cross-section for charge-exchange scattering at 693 MeV/c. The data are from PS199; the curve from the PWA.</figcaption></figure>

<figure><img src="image_120.png" /><figcaption>Fig. 6.24. Nijmegen private communication: predictions for the $\bar{p}p \to \bar{n}n$ differential cross-section, multiplied by 0.955 and 0.975; the data are from PS206.</figcaption></figure>

It was already widely acknowledged (see, for instance, Section 6.1.2 and the discussion at the Archamps workshop [378]), that the charge-exchange differential cross-sections measurements at very low energy poses a challenge for every model. The Nijmegen group confirms this conclusion, though it is phrased differently: they basically wish to reject the PS173 data, which contribute too much to their $\chi^2$. They are, however, refrained from doing so completely, because there is no alternative data available below 300 MeV/c, and they adopt the philosophy that “imperfect data are perhaps better than no data at all”.

*Analysing power:* The Nijmegen PWA could fit very well also the $\bar{p}p \to \bar{p}p$ and $\bar{p}p \to \bar{n}n$ analysing power data, as can be seen in Figs. 6.25 and 6.26.

*π coupling:* As the PWA includes explicitly the possibility of exchanging a light particle, the analysis confirms the one-pion-exchange contribution, with a mass compatible with the measured mass for $\pi^\pm$. The coupling is also an output of the fit. Charge-exchange is particularly sensitive to the pion coupling.

From their analysis of $\bar{N}N$ data, the Nijmegen group published results for the $\pi NN$ coupling constant. In 1991, a PWA on the charge-exchange data alone (which included the first PS199 results of Ref. [111]) lead to a coupling constant $f_c^2 = 0.0751 \pm 11$ [322]. After the publication of all the PS199 analysing-power data in 1994, and using the full PWA93 $\bar{N}N$ data set, they found $f_c^2 = 0.0732 \pm 11$.

This result is considered by the authors as evidence for one-pion exchange in the $\bar{N}N$ interaction, and evidence for the G-parity rule.
<figure><img src="image_121.png" /><figcaption>Fig. 6.25. Analysing power in elastic scattering at 544, 679, 783, and 886 MeV/c. The data from the LEAR experiments PS172 and PS198 are compared with the PAW fits.</figcaption></figure>

### 6.4.4. Uniqueness of the solution of the PWA

As stressed in Section 3, one needs several spin measurements to determine completely a spin 1/2-spin 1/2 reaction. The Nijmegen PWA was based essentially on angular-distribution and analysing-power data. The depolarisation data for $\bar{p}p$ have enormous error bars. No accurate spin transfer or spin correlation is available.

As for the fits by the Paris group, one may address the question of the uniqueness of the solution, due to the lack of data. This is debated in Refs. [379,380]. Of course, the uniqueness is even less established for the earlier phase-shift analysis [374], done at a time when available data were even more scarce. The authors of the Nijmegen PWA have given a detailed answer. Let us quote them [370]: “A valid question is therefore: Can one do a PWA of the $\bar{p}p$ data, when there are essentially no “spin data”? The answer is yes! The proof that it can be done lies in the fact that we actually produced a $\bar{p}p$ PWA with a very good $\chi^2/N_d$. We have also checked this at length in our pp PWAs. We convinced ourselves that a pp PWA using only differential cross-section and analysing power data gives a pretty good solution. Of course, adding spin-transfer and spin-correlation data was
<figure><img src="image_122.png" /><figcaption>Fig. 6.26. Analysing power in charge-exchange scattering at 546, 656, 767, and 875 MeV/c. The data are from the LEAR experiment PS199. The curves are the PWA fits.</figcaption></figure>

helpful and tightened the error bands. However, most spin-transfer and spin-correlation data in the pp dataset actually did not give any additional information”.

Other invoked reasons were the knowledge of the long-range potential with $\pi$, $\rho$, $\omega$, $\varepsilon$ exchange, and the easy access to powerful computers.

In our opinion, the situation of PWA is comparable to that of optical potentials with detailed spin and isospin dependence. In this latter case, it was clearly shown (see Tables 6.1 and 6.2) that drastically different parameters lead to comparable fits.

## 6.5. Strangeness-exchange reactions

The results of the various runs of the PS185 experiment has motivated many phenomenological studies. We shall give below a brief account of some of the contributions. Clearly, some studies done independently and simultaneously are a little redundant, but equally valuable. So it is little arbitrary to present in more detail one rather than the other.