Isoscalar $S$-wave $\pi$-$N$ scattering length $a^{+}$ from $\pi$-deuteron scattering

We consider constraints on the isoscalar $S$-wave $\pi$-$N$ scattering length $a^{+}$ from $\pi$-deuteron scattering, to third order in small momenta and pion masses in chiral perturbation theory. To this order, the $\pi$-deuteron scattering length is determined by $a^{+}$ together with three-body corrections that involve no undetermined parameters. We extract a novel value for a combination of dimension two low-energy constants which is in agreement with previous determinations.