X-ray spectroscopy of the pionic deuterium atom

The pion-nucleon scattering lengths from pionic deuterium

“…In the experiment at PSI [1,2], one measures the energy of 3p − 1s X-ray transition, deducing the strong shift of the pionic deuterium in the 1s state and the πd scattering length from this measurement. At the first step, in order to obtain the strong shift, one has to subtract the so-called "electromagnetic shift" from the full measured value, where the former is calculated in the accuracy that matches the experimental precision.…”

“…In order to have a complete and transparent fieldtheoretical treatment of the pionic deuterium problem at all levels, we find it appropriate here to re-derive the expression for the full energy shift at order α 4 , α 3 (m d −m u ), and to check (at least, numerically) the results given in table 1 of Ref. [1].…”

“…[1]. Note that in this paper individual contributions are not specified explicitly, so the identification, which is given below and in table 1, is performed by analogy with the pionic hydrogen case [30].…”

“…In order to be able to compare with the existing results, our numerical calculations have been performed for the same values of the input parameters as in Ref. [1]. We take the deuteron binding energy to be ǫ = 2.22457 MeV, and the charge radii of the deuteron and of the pion are taken to be equal r 1/2 = 2.1303(66) fm, see [34] and references therein (the result changes slightly).…”

“…The Pionic Hydrogen collaboration at PSI [1,2] has performed high precision measurements of the strong interaction shift ε 1s and width Γ 1s of the 1s state of pionic deuterium from the 3p − 1s X-ray transition. The (complex) pion-deuteron scattering length was extracted from these measurements with the use of the leading-order Deser formula [3] −ε 1s + i Γ 1s 2 = 4E 1s r B a πd ,…”