Some comments on calculations of the scalar radius of the pion and the chiral constant Ī4

Abstract

The pion scalar radius is given by \( \left\langle {r_S^2 } \right\rangle = (6/\pi )\int {_{4M_\pi ^2 }^\infty } dt\delta _S (t)/t^2 \), with δ S the phase of the scalar form factor. Below \( \bar K \)K threshold, δ S = δ0, δ0 being the isoscalar, S-wave ππ phase shift. Between \( \bar K \)K threshold and t1/2 ∼ 1.5 GeV I argued, in two previous letters [Yndurain, Phys. Lett. B 578, 99 and (E) 586, 439 (2004); Phys. Lett. B 612, 245 (2005)], that one can approximate δ S ∼ δ0, because inelasticity is small, compared with the errors. This gives 〈r S 2 〉 = 0.75 ± 0.07 fm2 and the value \( \bar l \)4 = 5.4±0.5 for the one-loop chiral perturbation theory constant, compared with the values given by Leutwyler and collaborators, 〈r S 2 〉 = 0.61 ± 0.04 fm2 and \( \bar l \)4 = 4.4 ± 0.3. At high energy, t1/2 > 1.5GeV, I remarked that the value of δ 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 [Int. J. Mod. Phys. A 21, 954 (2006)] claim that my estimate of the asymptotic phase δ 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±0.43, compatible with my estimate but widely off the value \( \bar l \)4 = 4.4±0.3 of Leutwyler and collaborators. The results referred to have been published in F. J. Yndurain, The theory of quark and gluon interactions (Springer-Verlag, August 2006) p. 239 and pp. 336, 337. The full text of the contribution may be found, with full references, in hep-ph/0510317.