The scalar radius of the pion

Abstract

The pion scalar radius is given by $<r_S^2>=(6/\pi ){\int }_{4M_{\pi }^2}^{\infty }\mathrm{d}s{\delta }_S(s)/s^2$, 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, $s>2{\mathrm{GeV}}^2$, ${\delta }_S$ 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 $<r_S^2>=0.75\pm 0.07{\mathrm{fm}}^2$. 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 $<r_S^2>=0.61\pm 0.04{\mathrm{fm}}^2$. 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 ${\delta }_{\pi }(4m_K^2)$ different from what experiment indicates. If one uses the value for ${\delta }_{\pi }(4m_K^2)$ 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, $<r_S^2>=0.75\pm 0.07{\mathrm{fm}}^2$.

arXiv commentVersion to be published in Phys. Letters. A few typos corrected. Plain YeX file. 5 figures

References in corpus (7)

• ππscattering
• Meson-Meson interaction in a non-perturbative chiral approach
• N/D Description of Two Meson Amplitudes and Chiral Symmetry
• Meson-meson scattering within one loop Chiral Perturbation Theory and its unitarization
• Separation of S- wave pseudoscalar and pseudovector amplitudes in pi^- p---> pi^+ pi^- n reaction on polarized target
• Scalar form factors of light mesons
• The quadratic scalar radius of the pion and the mixed $\pi -K$ radius

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