Before going into a discussion of the present knowledge of the scattering
length, we briefly introduce different notations existing in the literature,
which are relevant for the pionic hydrogen problem. With denoting the isospin
of the system, the scattering lengths can be represented by linear
combinations of the scattering lengths as
Instead of and sometimes the notations and
are used. The scattering lengths are related to the isospin-even
and isospin-odd scattering length, and , respectively, as follows
The present knowledge of the values for the scattering lengths , ,
is shown in Table 1.
The measurements of the hadronic shift in the atom with the relative accuracy of about 1% give the most precise value of the linear combination and can be used to constrain the phase shift analysis of the scattering data. On the other hand, the current accuracy of the widths measurements of about 10% is not sufficient for extracting a precise value of . This situation is in striking contrast with the theory where the isospin-odd scattering length is relatively better determined than the isospin-even scattering length , because large cancellations occur for the latter, so that the result is sensitive to certain theoretical parameters. A better experimental determination of the scattering length will be of big importance for reducing this theoretical uncertainty. Another problem that requires precision experimental data is related to the isospin symmetry breaking due to the mass difference of the up and the down quarks ( see  and references therein).