High-Precision Determination of the Pion-Nucleon $\sigma$ Term from Roy-Steiner Equations

We present a determination of the pion-nucleon ($\pi N$) $\sigma$ term ${\sigma }_{\pi N}$ based on the Cheng-Dashen low-energy theorem (LET), taking advantage of the recent high-precision data from pionic atoms to pin down the $\pi N$ scattering lengths as well as of constraints from analyticity, unitarity, and crossing symmetry in the form of Roy-Steiner equations to perform the extrapolation to the Cheng-Dashen point in a reliable manner. With isospin-violating corrections included both in the scattering lengths and the LET, we obtain ${\sigma }_{\pi N}=(59.1\pm 1.9\pm 3.0)\mathrm{MeV}=(59.1\pm 3.5)\mathrm{MeV}$, where the first error refers to uncertainties in the $\pi N$ amplitude and the second to the LET. Consequences for the scalar nucleon couplings relevant for the direct detection of dark matter are discussed.

Figure 1

Phase shifts ${\delta }_{l\pm }^{I_s}$ of the $s$-channel partial waves in degrees, obtained from the solution of the RS equations. The dashed line indicates our central solution, the bands the uncertainty estimate. The partial waves are labeled by the spectroscopic notation $L_{2I_{s}2J}$.