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The pion scalar radius is given by $=(6/��)\int_{4M^2_��}^\infty{\rm d}s ��_S(s)/s^2$, with $��_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, $��_S=��_��$, $��_��$ being the isoscalar, S-wave $����$ phase shift. At high energy, $s>2 {\rm GeV}^2$, $��_S$ is given by perturbative QCD. In between I argued, in a previous letter, that one can interpolate $��_S\sim��_��$, because inelasticity is small, compared with the errors. This gives $=0.75\pm0.07 {\rm fm}^2$. Recently, Ananthanarayan, Caprini, Colangelo, Gasser and Leutwyler (ACCGL) have claimed that this is incorrect and one should have instead $��_S\simeq��_��-��$; then $=0.61\pm0.04 {\rm fm}^2$. Here I show that the ACCGL phase $��_S$ 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 $��_��(4m^2_K)$ different from what experiment indicates. If one uses the value for $��_��(4m^2_K)$ favoured by experiment, the ensuing phase $��_S$ is continuous, agrees with perturbative QCD expectations, and satisfies $��_S\simeq��_��$, thus confirming the correctness of my previous estimate, $=0.75\pm0.07 {\rm fm}^2$.
Version to be published in Phys. Letters. A few typos corrected. Plain YeX file. 5 figures
Nuclear and High Energy Physics, High Energy Physics - Phenomenology, High Energy Physics - Phenomenology (hep-ph), FOS: Physical sciences
Nuclear and High Energy Physics, High Energy Physics - Phenomenology, High Energy Physics - Phenomenology (hep-ph), FOS: Physical sciences
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