The pion–nucleon scattering lengths from pionic hydrogen and deuterium

This is the final publication of the ETH Zurich–Neuchâtel–PSI collaboration on the pionic hydrogen and deuterium precision X-ray experiments. We describe the recent hydrogen 3p–1s measurement, report on the determination of the Doppler effect correction to the transition line width, analyze the deuterium shift measurement and discuss implications of the combined hydrogen and deuterium results. From the pionic hydrogen 3p–1s transition experiments we obtain the strong-interaction energy level shift \(\varepsilon_{1s} = -7.108\pm0.013 (\mathrm{stat.})\pm0.034 (\mathrm{syst.})\) eV and the total decay width \(\Gamma_{1s} = 0.868\pm0.040 (\mathrm{stat.})\pm0.038 (\mathrm{syst.})\) eV of the \(1s\) state. Taking into account the electromagnetic corrections we find the hadronic \(\pi N\)s-wave scattering amplitude \(a_{\pi^{-}p\rightarrow\pi^{-}p} = 0.0883\pm0.0008 m_{\pi}^{-1}\) for elastic scattering and \(a_{\pi^{-}p\rightarrow\pi^{0}n} = -0.128\pm0.006 m_{\pi} ^{-1}\) for single charge exchange, respectively. We then combine the pionic hydrogen results with the 1s level shift measurement on pionic deuterium and test isospin symmetry of the strong interaction: our data are still compatible with isospin symmetry. The isoscalar and isovector \(\pi N\) scattering lengths (within the framework of isospin symmetry) are found to be \(b_0 = -0.0001^{+0.0009}_{-0.0021} m_{\pi}^{-1}\) and \(b_{1} = -0.0885^{+0.0010}_{-0.0021} m_{\pi} ^{-1}\), respectively. Using the GMO sum rule, we obtain from \(b_1\) a new value of the \(\pi N\) coupling constant (\(g_{\pi N} = 13.21_{-0.05}^{+0.11}\)) from which follows the Goldberger–Treiman discrepancy \(\Delta_{\mathrm{{GT}}} =0.027_{-0.008}^{+0.012}\). The new values of \(b_0\) and \(g_{\pi N}\) imply an increase of the nucleon sigma term by at least 9 MeV.