In this Chapter we describe the measurement of 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.
The Asterix experiment was carried out by stopping antiprotons in gaseous H at room temperature and pressure. A short description of the X-ray detector can be found in Sec. 2.4, a report on the performance of the full spectrometer in .
The large peak in Fig. 5.1 due to the Balmer- line is obviously not accompanied by a Lyman- line of similar strength. The spectrum evidences at the first glance the importance of annihilation from the 2P levels of protonium atoms: atoms in 2P states annihilate instead of radiating Lyman- X-rays. A fit to the data  shows some (weak) evidence for Lyman radiation with a fractional intensity of the K line of , compared to a total yield of the Balmer series of %. These data are used - together with Eq. (3.73) - to determine an average 2P strong interaction width of 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 . For a final state characterized by a partial width , the charges of the outgoing mesons and their four-momenta , the differential width for emission of a bremsstrahlung photon (after summation over all polarizations) can be written as
where is the four-momentum vector of the photon, and the solid angle element into which the photon is emitted. The photon energy distribution (5.1) exhibits the expected 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 angular momentum states; thus the direction of a primarily produced meson resonance is also correlated with the photon direction . But in the subsequent decay of the resonance, this weak correlation is washed out; angular correlations between, e.g., Balmer and charged particles can be neglected.
Equations (5.1,5.2) 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 and that of the nearest charged particle.
Figure 5.2 (upper panel) shows the angular distributions for with energies below (mostly originating from the atomic cascade). The angle is defined with respect to the direction of the higher-momentum particle; the angle is the polar angle, for 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. Figure 5.2 (lower panel) shows the angular distributions for above (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 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-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.
The fraction of annihilation in liquid H into all-neutral final states is given by the ratio of frequencies 
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 line is observed and thus the interpretation of the resulting spectrum is facilitated.
Results using 300 antiprotons were reported in . Figure 5.3a shows the spectrum obtained by stopping antiprotons and requiring absence of any charged particle in the final state . 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 are required. In Fig. 5.3b only events are considered with two observed in coincidence; the energy of the more energetic is plotted. The low-energy peak at originates from coincidences of a line belonging to the M series with a L line; note the absence of L. The broad peak at about stems from coincidences of an L with a K line. The apparent width of the peak () is much broader than the experimental resolution of indicating a sizable broadening due to strong interactions. When a cut is made to select events with one in the energy interval, the energy distribution of the second shows contributions from the full Balmer series.
The proper line-shape theory for broad lines (for which is not satisfied) is by no means trivial. To first order, the K 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
When the detection efficiency varies across the line, the Voigtian line shape has to be folded with it. The numerator in (5.4) is constant only when the transition matrix element for the radiative transition does not dependent on energy. This is, however, not the case. Electric dipole transitions prefer more energetic transitions suggesting that in (5.4) is proportional to . However, this integral would diverge. The situation was analysed by Ericson and Hambro  who developed a line-shape theory for broad lines. From the requirement of probability conservation, they deduced that the term contains an energy dependence
where and F is the hypergeometric function. In the proximity of the resonance energy , the relation 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 . Eq. (5.5) gives the best description of the data, with a central energy of the line at . The photon energy is shifted to a lower value than expected from QED ( is negative), the 1S level is shifted to a higher energy ( is positive). Thus a final result of
The yield of Balmer in all-neutral events is %. This is surprisingly close to the value % obtained with charged particles in the final state. Annihilation from S-states into all-neutral events proceeds dominantly only via the state (annihilation into any number of and requires positive -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 times larger from S-states. We deduce
The fraction of all-neutral events in which a K is emitted after a Balmer is detected is %. 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 -mesons decaying into .) 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 level.
The total number of L followed by annihilation into neutral particles is . The number of K followed by all-neutral annihilation is times smaller. This gives
Fig. 5.4 shows the mean time spent by antiprotons the scintillation counter at the entrance of the target and their annihilation. The time increases with the distance traveled by antiprotons before capture, but those stopping immediately after the scintillation counter T2 need
more time than those stopping in T2. The cascade time in solids is some ps only, hence the time difference corresponds to the time elapsed between capture in H and annihilation, at normal pressure and temperature,
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 .
The Obelix experiment  performed similar measurements using H 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 favors cascade times which are too short. Table 5.1 lists the final results.
In liquid H, the cascade time is too short to be measured. However, for the p system, the time can be deduced which elapses from the moment where the has a residual velocity of 0.004 c (where the sum of the measured decay momenta does not yet vanish) to nuclear absorption . 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.
The cold gas experiment used a variable H gas density over the range from 10 to 1/8 times STP. The first results were obtained using antiprotons with momenta of 300 and a Si(Li) detector . We discuss here the final results using Si(Li) detectors  and two gas-scintillation proportional detectors (GSPD) .
Figure 5.7 shows the low-energy part of the 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 H densities; they provide valuable information about the cascade processes which precede the emission of lines. The intensities are listed in Tables 5.4 and 5.5.
The high-energy part of the spectrum  is shown in Fig. 5.8 for various H 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 K 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 K line, they have narrow widths. The data were fitted with a polynomial background and a complex of eight lines corresponding to K , K , , 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 . The strong interaction parameters determined from the five data sets are mutually consistent and give average values of
In parallel to the data with the Si(Li) detectors, the GSPD was also used to detect the . The fit to the results, shown in Fig. 5.9, gives 
The Balmer series was not observed because of the need of rather thick windows. Thus no value was determined for . A search for the Lyman series of atoms was not successful and only upper limits (95% confidence level) for the yield of K X-rays per stopped antiproton were given: and at 0.25 and 0.92 , respectively .
Figure 5.10 shows expanded views of a high-statistics run using a target pressure of 30 mbar H (top) and D (bottom). The comparison of the two spectra evidences a peak at above 8 keV, the K line and a small contribution from K. No sign is seen from any K-line.
The authors of Ref.  tried to extract information on the energy splitting between the and 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.
More data were recorded than shown here. The K-lines were searched for with two Si(Li) detectors and an drift chamber. The three sets of data were combined into the final results
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.
Figure 5.11 shows the energy spectrum observed when antiprotons were stopped in the cyclotron trap operated at 20 mbar of H (a) and D (b) gas. The low-energy range demonstrates remarkable achievements in technology: individual lines are clearly identified, the L being the strongest one. Even the M series limit is observed, in H as a shoulder, in D as a peak.
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 and 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  to be the same for H and D. The difference of the spectra for H and D can, however, not be assigned to the Lyman series. There is an additional source of background of unknown origin, present only in the H data. This additional background was assigned to coherent interference between K-lines and bremsstrahlung, and then subtracted incoherently by the requirement that all three CCD’s gave compatible results on 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 lines from 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 CCD’s 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 is not fully explored. With this warning we quote their result on the hyperfine splitting:
The evidence for K lines from atoms is much weaker (Fig. 5.13b). There are several contaminant lines from gas impurities and from the target vessel; in particular the presence of a O(7-6) line at the proposed K energy is very unfortunate.
We caution the reader that in our view the identification of the observed structure with the K line from atoms is not unambiguously established. Also the yield of at a target pressure of 0.02 seems rather high. Batty, using cascade calculations, estimates the yield to be lower by one order of magnitude .
In parallel to the search for K X-rays, the line profile of the Balmer- 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 (CCD’s) 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, 2 2, and 2 levels. The data on exhibit a shoulder at the high energy side. Its relative intensity was determined by a two-component fit to be (9.5 0.9)%, in good agreement with the statistical population of 8.3% for the level. This level is expected to have a large strong-interaction shift. Hence, the high-energy tail is likely due to transitions. The main part of the line is attributed to the (unresolved) transitions to the three hyperfine levels, labeled ().
The mean energy of the D (P) transitions and the energy of the individual D 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 Tab. 3.3) and from potential models or using a single (broadened) Gaussian or Voigtian distribution. Both treatments lead to the same results on strong interaction parameters for the line. The relative intensities were always fixed to the statistical values. Under these assumptions the authors derived
From the intensity ratio (see Eq. 3.73) the authors deduce a spin-averaged value
The results on and on the direct crystal spectrometer measurement were combined assuming that yielding:
Using the same set up, data were taken also with D 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  were used with a common resolution given by the Ne calibration line. The fit did not reproduce the data. Much better agreement was obtained when the electromagnetic hyperfine splittings from  (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  and the hadronic shifts as given by  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:
The results on the strong interaction shift and width given in (5.10,5.11,5.12) and the mean value from from Table 5.2 are fully compatible even though we believe the errors to be sometimes underestimated. We give the linear average of the four measurements with a conservative estimate of the error:
Using the Trueman formula (3.90), we can relate these values to the complex S-wave scattering length
Similarly, we obtain the imaginary part of the P-wave scattering volume:
The ratio of the real to imaginary part of strong-interaction amplitude is read as
In Table 5.3 we summarize the results on strong interaction parameters. They will be compared to theoretical predictions in Chapter 6.
The experiments gave yields over a wide range of pressures. These are reproduced in Tables 5.4 and 5.5.
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 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 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.
Below keV the continues to loose 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.
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. Figure 5.15 shows the simulation of a capture process. The H atom is described by a classical system with a radius corresponding to the first Bohr orbit. Phases and 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 .
The cross section for protonium formation rises steeply for very low energies and is 2-3 Å for energies below 13.6 eV . Most antiprotons are captured when their energy is below the H ionisation energy; the principal quantum number is most often between 30 and 50 (sometimes even larger than 100) and the average orbital angular momentum about 20 . 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 and via
The “best” choice of the principal quantum number is then in the range
or . For the angular momentum states after capture, a statistical population seems plausible. Calculations show that the preferred distribution in has its maximum at about .
The rate for this effect is assumed to be given by the classical “size” of the atom and the collision frequency . eV is the dissociation energy of H molecules, is the density of hydrogen atoms, and is the protonium velocity.
The Auger effect is induced by the electric field seen by the p atom in the collision and is governed by the same matrix element 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 .
In collisions, peak electric field strengths of typically V/Å are experienced (for an impact parameter of 1.5 Å) for about 20 fs. The electron density integrated over the collision time is fs where is the electron density of H atoms at . Protonium atoms are neutral; hence they move along straight lines if the is attraction between the two collision partners is neglected. The path can be calculated using, e.g., the CTMC method . For an impact parameter of 1.5 Å, the minimal distance shrinks with increasing principal quantum number of the protonium atom. For , the minimal distance is by a factor 2 smaller than the impact parameter, for this effect is negligible. The straight-line approximation therefore underestimates the effect of the electric field and in particular the Stark mixing probability.
In a microscopic model, the coupled differential equation are integrated numerically and transition rates from any initial state to the other states are determined.
The cascade model starts at and assumes an initial distribution in and calculates the depopulation of these states until the residual population of 0.1% is reached. Each X-ray emission or annihilation from a S or P state is recorded. Thus the X-ray yields, the fraction of S and P state capture and the cascade time are determined. The intensities listed in Table 5.4 are shown in Fig. 5.17 and 5.18 and compared with the calculation.
The cascade of d atoms is very similar to that of protonium. Cascade calculations concentrated on the role of S- wave and P-wave capture . The yields are reproduced in Fig. 5.19.
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 is allowed from the state, into from the states and . The number of and events found in bubble chambers at BNL and CERN :et al.  found an unexpectedly large branching ratio for the reaction p annihilation , forbidden from S-state orbitals. After a long history of conflicting results, the Crystal Barrel Collaboration found a rate for 2 production of  for antiprotons stopping in liquid H, fully compatible with the old findings of Devons et al. . When compared to the rate, a P-wave fraction of 45% to p annihilation at rest in liquid H can be derived .
This large discrepancy is derived from two rare channels, with frequencies of about for and about for . It reflects the a large coupling to and a small coupling to from P states. With the measured rates for  and to  from P-states, the P-state contribution reduces to %, and there is no more conflict between the results derived from and from .
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 P-states for large is the same as for . This assumption is likely not true: in high- levels, Stark mixing of atomic states is very strong. The P level has a strong interaction width four times larger than the mean 2P width (compare (5.20) and (5.21)). High-n P levels can be repopulated after annihilation via Stark mixing collisions and the P levels have a larger chance to contribute to annihilation. There is practically no Stark mixing for n=2; atoms in the 2 P fine-structure levels annihilate and the P 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 % to % [363, 362]. This is a value compatible with most partial-wave analyses. Figure 5.20 shows the fraction of P-state annihilation as a function of H density.
Batty  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 . Figure 5.21 shows these factors as functions of the H density.
We notice a substantial increase of the contribution of the state with increasing density. This increase is responsible for the large branching ratio. In turn, this large branching ratio is only compatible with other determinations of the P-state capture rate, when we assign the majority of annihilations into 2 to the and not to the initial state.
The fraction of P-state annihilation in n annihilations is even more uncertain. From a comparison of p annihilation into in liquid H and D, the P-state capture fraction in D was estimated to % (after a cut on the proton momentum to ensure annihilation on a quasi-free nucleon) . Batty  estimated the P-state annihilation frequency from d annihilations into and into and derived a fraction % P-state capture. From cascade calculations he estimated this fraction to 40%. In summary, a P-state fraction of 30% for antiprotons stopping in liquid H seems realistic.