The scalar radius of the pion

Abstract

The pion scalar radius is given by 〈rS2〉=(6/π)∫4Mπ2∞dsδS(s)/s2, with δS the phase of the scalar form factor. Below K¯K threshold, δS=δπ, δπ being the isoscalar, S-wave ππ phase shift. At high energy, s>2 GeV2, δS is given by perturbative QCD. In between I argued, in a previous letter, that one can interpolate δS∼δπ, because inelasticity is small, compared with the errors. This gives 〈rS2〉=0.75±0.07 fm2. Recently, Ananthanarayan, Caprini, Colangelo, Gasser and Leutwyler (ACCGL) have claimed that this is incorrect and one should have instead δS≃δπ−π; then 〈rS2〉=0.61±0.04 fm2. Here I show that the ACCGL phase δS 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 δπ(4mK2) different from what experiment indicates. If one uses the value for δπ(4mK2) favoured by experiment, the ensuing phase δS is continuous, agrees with perturbative QCD expectations, and satisfies δS≃δπ, thus confirming the correctness of my previous estimate, 〈rS2〉=0.75±0.07 fm2.