Some comments on calculations of the scalar radius of the pion and the chiral constant $\bar{l}_4$

Abstract

The pion scalar radius is given by $ =(6/π)\int_{4M^2_π}^\infty{\rm d}t δ_S(t)/t^2$, with $δ_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, $δ_S=δ_0$, $δ_0$ being the isoscalar, S-wave $ππ$ phase shift. Between $\bar{K}K$ threshold and $t^{1/2}\sim 1.5 {\rm GeV}$ I argued, in two previous letters, that one can approximate $δ_S\simδ_0$, because inelasticity is small, compared with the errors. This gives $ =0.75\pm0.07 {\rm fm}^2$ and the value $\bar{l}_4=5.4\pm0.5$ for the one-loop chiral perturbation theory constant, compared with the values given by Leutwyler and collaborators, $ =0.61\pm0.04 {\rm fm}^2$ and $\bar{l}_4=4.4\pm0.3$. At high energy, $t^{1/2}>1.5 {\rm GeV}$, I remarked that the value of $δ_S$ that follows from perturbative QCD agrees with my interpolation and disagrees with that of Leutwyler and collaborators. In a recent article, Caprini, Colangelo and Leutwyler claim that my estimate of the asymptotic phase $δ_S$ is incorrect as it neglects higher twist contributions. Here I remark that, when correctly calculated, higher twist contributions are likely negligible. I also show that chiral perturbation theory gives $\bar{l}_4=6.60\pm0.43$, compatible with my estimate but widely off the value $\bar{l}_4=4.4\pm0.3$ of Leutwyler and collaborators.