The scalar radius of the pion

Abstract

The pion scalar radius is given by $<r^2_S>=(6/\pi)\int_{4M^2_\pi}^\infty{\rm d}s \delta_S(s)/s^2$, with $\delta_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, $\delta_S=\delta_\pi$, $\delta_\pi$ being the isoscalar, S-wave $\pi\pi$ phase shift. At high energy, $s>2 {\rm GeV}^2$, $\delta_S$ is given by perturbative QCD. In between I argued, in a previous letter, that one can interpolate $\delta_S\sim\delta_\pi$, because inelasticity is small, compared with the errors. This gives $<r^2_S>=0.75\pm0.07 {\rm fm}^2$. Recently, Ananthanarayan, Caprini, Colangelo, Gasser and Leutwyler (ACCGL) have claimed that this is incorrect and one should have instead $\delta_S\simeq\delta_\pi-\pi$; then $<r^2_S>=0.61\pm0.04 {\rm fm}^2$. Here I show that the ACCGL phase $\delta_S$ is pathological in that it is discontinuous for small inelasticity, does not coincide with what perturbative QCD suggests at high energy, and only occurs because these authors take a value for $\delta_\pi(4m^2_K)$ different from what experiment indicates. If one uses the value for $\delta_\pi(4m^2_K)$ favoured by experiment, the ensuing phase $\delta_S$ is continuous, agrees with perturbative QCD expectations, and satisfies $\delta_S\simeq\delta_\pi$, thus confirming the correctness of my previous estimate, $<r^2_S>=0.75\pm0.07 {\rm fm}^2$.