The S-Wave Pion-Nucleon Scattering Lengths from Pionic Atoms using Effective Field Theory

S.R. Beane, V. Bernard (Orsay), E. Epelbaum (Julich), Ulf-G. Meißner (Bonn), D.R. Phillips (Ohio University)

The pion-deuteron scattering length was computed to next-to-next-to-leading order in baryon chiral perturbation theory ($\chi$PT) [39]. We then formulated a modified power-counting which properly accounts for infrared enhancements engendered by the large size of the deuteron, as compared to the pion Compton wavelength. The new ingredient in this more practical power-counting is the observation that one may develop a hierarchy between the pion mass, $M_\pi$ and the deuteron binding momentum, $\gamma$. One can then pick out the pieces of the amplitude that become large in the formal limit in which $\gamma$ goes to zero and $M_\pi$ is held fixed [40]. This does not, of course, indicate that anything is amiss with baryon $\chi$PT power-counting. Baryon $\chi$PT should work just fine for $\pi$-d scattering in the threshold region, but high-order calculations are required if truly accurate results are desired. The advantage of the modified power-counting, and of EFT generally, is that it immediately isolates the large contributions, without requiring explicit calculations of matrix elements that end up being smaller than the theoretical error. We use the precise experimental value of the real part of the pion-deuteron scattering length determined from the decay of pionic deuterium, together with constraints on pion-nucleon scattering lengths from the decay of pionic hydrogen (See Fig. 5), to extract the isovector and isoscalar S-wave pion-nucleon scattering lengths, $a^-$ and $a^+$, respectively. We find $a^-=(0.0918 \pm 0.0013) M_\pi^{-1}$ and $a^+=(-0.0034 \pm 0.0007) M_\pi^{-1}$.

Figure 5: Plot of $a^+$ vs $-a^-$. The light shaded region and the dark band are from the experimental pionic-hydrogen width and shift, respectively, taken from Ref. [41]. The dotted line encompasses the constraints from $\pi$-N phase shift data and is taken from Ref. [42]. The dot is leading order $\chi$PT (current algebra). The two parallel bands are from the computed theoretical formulae evaluated with the NLO wavefunction with an ultraviolet cutoff of $500$ MeV (upper curve) and $600$ MeV (lower curve).