[ChPT]{}. The resulting values for the scattering lengths $a^0_0$ and $a^2_0$ have since been confirmed by other theoretical analyses using the same or similar methods [@DescotesGenon:2001tn; @GarciaMartin:2011cn]. There has since also been a large experimental effort to pin down these two scattering lengths by NA48/2 and others in both $K_{\ell4}$ [@Batley:2007zz] and $K\to 3\pi$ [@Batley:2005ax] decays and by DIRAC [@Adeva:2005pg]. The values we will use for the scattering lengths are those from [@Colangelo:2001df]: $$\label{eq:a0a2}
a^0_0 = 0.220\pm0.005\,,\qquad a^2_0= -0.0444\pm0.0010\,.$$

The pion scalar form factor is not directly measureable but it can be obtained from dispersion relations. The main part is given by the Omnés solution using the $\pi\pi$ S-wave from the analysis above but improvements are possible by including also other channels, first and foremost the $KK$ channel. The main conclusion from [@Donoghue:1990xh; @Moussallam:1999aq] is that $$\begin{aligned}
\label{eq:fs}
F_S^\pi(q^2) &=& F_S^\pi(0)\left(1+\frac{\langle
  r^2\rangle^\pi_S}{6}q^2+c^\pi_S q^4+\dots\right) 
\nonumber\\
\langle r^2\rangle^\pi_S &=& 0.61\pm0.04~\mathrm{fm}^2,
\qquad
c^\pi_S = 11\pm1~\mathrm{GeV}^{-4}~.\end{aligned}$$ A more recent discussion can be found in [@Ananthanarayan:2004xy].

The pion vector form factor can be measured directly as well as treated with dispersive methods. The direct fit to the low-energy data, which are dominated by [@Amendolia:1986wj], was done in [@Bijnens:1998fm]: $$\begin{aligned}
F_V^\pi(q^2) &=& F_V^\pi(0)\left(1+\frac{\langle
  r^2\rangle^\pi_V}{6}q^2+c^\pi_V q^4+\dots\right) 
\nonumber\\
