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 $<r_S^2>=(6/\pi ){\int }_{4M_{\pi }^2}^{\infty }\mathrm{d}t{\delta }_S(t)/t^2$, with ${\delta }_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, ${\delta }_S={\delta }_0$, ${\delta }_0$ being the isoscalar, S-wave $\pi \pi$ phase shift. Between $\bar{K}K$ threshold and $t^{1/2}\sim 1.5\mathrm{GeV}$ I argued, in two previous letters, that one can approximate ${\delta }_S\sim {\delta }_0$, because inelasticity is small, compared with the errors. This gives $<r_S^2>=0.75\pm 0.07{\mathrm{fm}}^2$ and the value ${\bar{l}}_4=5.4\pm 0.5$ for the one-loop chiral perturbation theory constant, compared with the values given by Leutwyler and collaborators, $<r_S^2>=0.61\pm 0.04{\mathrm{fm}}^2$ and ${\bar{l}}_4=4.4\pm 0.3$. At high energy, $t^{1/2}>1.5\mathrm{GeV}$, I remarked that the value of ${\delta }_S$ 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 ${\delta }_S$ 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 ${\bar{l}}_4=6.60\pm 0.43$, compatible with my estimate but widely off the value ${\bar{l}}_4=4.4\pm 0.3$ of Leutwyler and collaborators.

Comments: 6 pages; Plain TeX files