Pionic hydrogen

We consider the problem of s-state energy shift according to the perturbation theory. Such analysis was performed for the pionic hydrogen in Ref. (Lyubovitskji and Rusetsky, 2000). Let Ho + Hc be the unperturbed Hamiltonian, whereas V is considered as a perturbation. The ground-state solution of the unperturbed Schrodinger equation in the center of mass (CM) system frame (E — Ho — Hc) To(0)) = 0, with E = M + m + E, is given by... 

In order to combine the results from pionic hydrogen and pionic deuterium, establishing a relation between And and ttN isoscalar and isovector lengths bo and b is the important problem. This relation can be expressed as... 

Gasser J. et al Ground-state energy of pionic hydrogen to one loop. Eur. Phys. J. C 26, 13-24 (2002). 

Schroder H. Ch. et al., The pion-nucleon scattering lengths from pionic hydrogen and deuterium, Eur. Phys. J. C 21, 473-488 (2001)... 

Deser S., Goldberger M. L., Baumann K. and Thirring W. Energy Level Displacements in Pi-Mesonic Atoms, Phys. Rev. 96, 774-776 (1954) Anagnostopoulos D. F. et al., Precision measurements in pionic hydrogen, Nucl. Phys. A 721, 849-852 (2003). 

Ivanov A. N. et al, On pionic hydrogen. Quantum field theoretic, relativistic covariant and model independent approach arXiv nucl-th/0306047. 

Sigg D. et al. Electromagnetic correction to the s-wave scattering lengths in pionic hydrogen, Nucl. Phys. A 609, 310, (1996)... 

The two sets may be combined by a relative measurement of pairs of transitions belonging to one set each. A first application of this new calibration method will be a precision determination of the hadronic shift and broadening of the ground state in pionic hydrogen [23]. Further possibilities are studies of... 

Its extension to heavy baryon chiral perturbation theory (HBCHPT) [3] allows to calculate many of the experimentally accessible processes in the meson nucleon sector. The check of the soundness of this approach requires high precision experiments. This resembles the situation in the development of QED during the last 50 years, where the measurement of the Lamb shift contributed much to the development of QED. In a comparable way the measurement of strong interaction shift and width in pionic hydrogen may be a key experiment in strong interaction physics at low energies. 

Here r is the Bohr radius of the pionic hydrogen atom with i b = 222.56 fm, Qo = 0.142 fm-1 is a kinematical factor and P=1.546 0.009 is the Panofsky ratio [6]. 6e and dr are electromagnetic corrections, which have recently been calculated with a potential model with an accuracy of about 0.5% [7], In a recent study the problem of the electromagnetic corrections is discussed and the potential model ansatz is critizised [8]. 

The shift and the width of the ground state in pionic hydrogen and deuterium have been determined in a series of experiments of the ETHZ-Neuchatel-PSI collaboration by measuring the 3-1 transition at 2886 eV with a reflection type crystal spectrometer [10]. An array of 6 cylindrically bent quartz crystals had been used in order to increase the statistics of the experiment. The pions were stopped in a cryogenic target inside a superconducting magnet (cyclotron trap I) and the X-rays were detected with CCD detectors developed at the University of Neuchatel. 

In conclusion it can be stated that the results from scattering data and atom experiments are still contradictory and therefore need further investigation. From the side of the atom experiments it should be clarified whether the shift and the width values of pionic hydrogen and deuterium are true strong interaction effects and are not spoiled by the interaction of the pionic atom with the surrounding molecules. In other words the shift and the width measurements for pionic hydrogen and deuterium should be extrapolated to zero pressure. In a second step state of the art electromagnetic corrections should be applied. 

Proposed Measurements of the Strong Interaction Shift and Width in Pionic Hydrogen... 

The planned measurements will be able to accumulate an intensity of more than 10000 events per transition enabling a determination of the transition energy with a statistical accuracy of better than 3 meV. For the 3-1 and 4-1 transitions in pionic hydrogen pionic oxygen and carbon transitions adjacent in energy are available as calibration lines, thus avoiding the systematic errors in the former experiment. In a first step the experiment will establish a result for the shift independent of pressure. In order to achieve this the position of the 3-1 and 4-1 lines will be measured as a function of pressure in the region between... 

Recent experiments determined the velocity state of the pionic hydrogen atom at the moment of the charge exchange reaction [23]. These results constrain the input parameters for the cascade calculations as well as the direct X-ray measurements from muonic hydrogen. The results of the cascade calculations can then be used to correct for the influence of the Doppler broadening. 

A successful X-ray spectroscopy of the quality required for the pionic hydrogen experiment is based on a narrow and well understood response function of the crystals. An energy calibration or an optimization can not be achieved with ffuorescence X-rays produced with X-ray tubes. Their width is an order of magnitude broader than the resolution of the crystals. The line shape is moreover influenced by poorly determined satellite lines. 

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