the below we use the values $$M_{\pi^+} = 139.57018(35)~\mathrm{MeV}~,\qquad
M_{\pi^0} = 134.9766(6)~\mathrm{MeV}~.$$ The pion decay constant is measured in $\pi\to\mu\nu$ and the main uncertainty is the size of electromagnetic effects. $V_{ud}$ is known to sufficient precision from neutron and nuclear decays. We will adopt the value $$\label{eq:Fpi}
F_\pi = 92.2\pm0.1~\mathrm{MeV}$$ from the PDG [@Beringer:1900zz].

The next major input needed are the $\pi\pi$ scattering lengths and related quantities. The main theoretical underpinning of this are the Roy equations [@Roy:1971tc], a set of integral equations that the $\pi\pi$ scattering amplitude satisfies because of crossing and unitarity. These require as input two subtraction constants, or equivalently values of the scattering lengths $a_0^0$ and $a^2_0$, and phenomenological input for the higher waves and at short distances. A large (re)analysis was done in [@Ananthanarayan:2000ht]. This analysis confirmed a number of results from the 1970s and sharpened them. In [@Colangelo:2001df] the analysis was strengthened by two additional inputs: The scalar radius should have a value of about $0.6$ fm$^2$ and the [ChPT]{} series for the $\pi\pi$ amplitude was found to converge extremely well in the center of the Mandelstam triangle. This allowed to make a rather sharp prediction for the two subtraction constants if one assumes 