The present proposal deals with a high precision study of strong interaction phenomena in the pionic hydrogen atom. It aims at concluding a series of experiments which have shown an increasing level of sophistication and precision [1,2,3,4].
The pionic hydrogen atom is dominated by the electromagnetic interaction
of its constituents. Their strong interaction is only effective in case the
wave functions of pions and protons significantly overlap, i.e. in the s-states,
which results in a broadening and shift of the s-states. The electromagnetic
binding energy of the 1s state is = eV whereas the strong
interaction shift of the ground state is about 7 eV (attractive
i.e. the state is stronger bound) and the width amounts
to about 1 eV only. The quantities which are finally extracted from the shift
and width measurement are the isospin scattering lengths in the pion-nucleon
system for both isospins. The relations of the measured quantities to the hadronic
scattering lengths describing the
processes respectively are given
by the Deser-type formulae [5,6]:
Here is the Bohr radius of the pionic hydrogen atom with , , are electromagnetic corrections, is a kinematical factor and is the Panofsky ratio . The resolution of a dedicated crystal spectrometer can reach a value of which permits a determination of shift and width with a relative precision of better than 1% provided enough statistics has been collected and the response function of the detection system is well known. The method is limited by the knowledge of the factors contributing to the Deser formulae and additional corrections which will be discussed later. The Panofsky ratio is known with an accuracy of about which is also the accuracy at which the electromagnetic corrections are known.
Some unique features of the exotic atom's method should be recalled:
On the theoretical side the description of the pion-nucleon system is considered to be a fundamental problem of QCD. The understanding of strong interaction in the confinement regime has advanced recently, as Chiral Perturbation Theory () was developed to perform calculations at low energies [11,12,13]. It offers the method to describe the pion-nucleon system quantitatively especially at low energy and allows to calculate certain combinations of scattering lengths with percent accuracy using all available experimental information.
The proposed experiment requires the cyclotron trap II at the E5 channel with the highest possible beam intensity. The X-rays emitted from the pionic hydrogen atom will be energy-analysed by a high resolution crystal spectroscopy using spherically bent Bragg crystals. In order to extract a line width with a relative accuracy of better than one percent the response function of the crystal spectrometer must be known with sufficient precision. Therefore the resolution and the response function of the Bragg crystals will be optimized and measured off beam with X-rays of single electron ions from an ECR ( Electron Cyclotron Resonance) source. The response function of the crystals will then be surveyed during measurement with X-rays of well defined line shape and energy from specially chosen pionic and muonic atoms which are not influenced by any broadening mechanism.
The required good knowledge of the response function alone is not sufficient to perform a successful determination of the width because the X-ray transitions will not only be broadened by strong interaction but also by Doppler effect. The pionic hydrogen atom changes its velocity during the cascade as its excitation energy can be transformed into kinetic energy (Coulomb deexcitation). The cross section for this process and therefore the development of the kinetic energy during the cascade is presently not well known. A possible way to assess the influence of Doppler broadening is to determine the kinetic energy distribution in muonic hydrogen which is a very similar system but not affected by strong interaction. The spectrum of the kinetic energies and hence the Doppler broadening here is rather similar to the pionic case as the reduced mass of the muonic hydrogen atom is only 21% smaller than the reduced mass of the pionic hydrogen atom. Compared to pions muons have a spin different from zero leading to a hyperfine structure of the atomic levels. The cascade processes leading to an acceleration of the exotic hydrogen atoms are not influenced by this fact as they occur at levels where the hyperfine splitting can be neglected. Therefore a high resolution spectroscopy of muonic hydrogen transitions allows one to determine the different contributions of the Coulomb deexcitation in this system. This knowledge can be transferred to pionic hydrogen and will serve as input for the fitting of the corresponding pionic hydrogen spectra.
The measurement of muonic hydrogen is in the present context considered as a necessary calibration. It can be stated, however, that a high resolution crystal spectrometry of muonic hydrogen unambiguously determines the kinetic energies developing during the cascade and thus will solve a long standing problem of the theory of cascade in exotic atoms.