[Paper Review] Some comments on calculations of the scalar radius of the pion and the chiral constant $\bar{l}_4$

Félix Ynduráin|ArXiv.org|Oct 24, 2005

Particle physics theoretical and experimental studies1 references4 citations

TL;DR

This paper re-evaluates the pion scalar radius and the chiral perturbation theory constant $\bar{l}_4$ using unitarity and dispersion relations, arguing that the scalar phase shift $\delta_S$ can be approximated by the $\pi\pi$ $S$-wave phase shift $\delta_0$ below $1.5\,\text{GeV}$, leading to $\langle r^2_S \rangle = 0.75 \pm 0.07\,\text{fm}^2$ and $\bar{l}_4 = 5.4 \pm 0.5$. It challenges the value $\bar{l}_4 = 4.4 \pm 0.3$ from Leutwyler et al., showing that higher-twist contributions are negligible and that two-loop chiral perturbation theory yields $\bar{l}_4 = 6.60 \pm 0.43$, consistent with its estimate and incompatible with the lower value.

ABSTRACT

The pion scalar radius is given by $=(6/π)\int_{4M^2_π}^\infty{ m d}t δ_S(t)/t^2$, with $δ_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, $δ_S=δ_0$, $δ_0$ being the isoscalar, S-wave $ππ$ phase shift. Between $\bar{K}K$ threshold and $t^{1/2}\sim 1.5 { m GeV}$ I argued, in two previous letters, that one can approximate $δ_S\simδ_0$, because inelasticity is small, compared with the errors. This gives $=0.75\pm0.07 { m fm}^2$ and the value $\bar{l}_4=5.4\pm0.5$ for the one-loop chiral perturbation theory constant, compared with the values given by Leutwyler and collaborators, $=0.61\pm0.04 { m fm}^2$ and $\bar{l}_4=4.4\pm0.3$. At high energy, $t^{1/2}>1.5 { m GeV}$, 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 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 $\bar{l}_4=6.60\pm0.43$, compatible with my estimate but widely off the value $\bar{l}_4=4.4\pm0.3$ of Leutwyler and collaborators.

Motivation & Objective

• To re-evaluate the pion scalar radius and the chiral perturbation theory constant $\bar{l}_4$ using unitarity and dispersion relations.
• To challenge the value $\bar{l}_4 = 4.4 \pm 0.3$ reported by Leutwyler and collaborators.
• To argue that higher-twist contributions to the asymptotic phase of the scalar form factor are negligible.
• To show that two-loop chiral perturbation theory yields $\bar{l}_4 = 6.60 \pm 0.43$, compatible with the author's estimate but inconsistent with the lower value from Leutwyler et al.

Proposed method

• Uses the dispersion relation $\langle r^2_S \rangle = (6/\pi) \int_{4M_\pi^2}^\infty dt\, \delta_S(t)/t^2$ to compute the scalar radius.
• Approximates $\delta_S(t) \approx \delta_0(t)$ below $1.5\,\text{GeV}$, where inelasticity is small.
• Compares the asymptotic behavior of the scalar form factor with perturbative QCD predictions to validate the phase approximation.
• Performs a two-loop chiral perturbation theory fit to $\pi\pi$ scattering data, including constraints from low-energy theorems.
• Evaluates the impact of higher-twist contributions to the pion form factor and argues they are negligible when properly treated.
• Compares results with those from Caprini, Colangelo, and Leutwyler, showing inconsistency with their estimate of the asymptotic phase.

Experimental results

Research questions

• RQ1 Is the approximation $\delta_S(t) \approx \delta_0(t)$ valid below $1.5\,\text{GeV}$, given small inelasticity?
• RQ2 Are higher-twist contributions to the pion form factor's asymptotic phase significant or negligible?
• RQ3 Does two-loop chiral perturbation theory yield a value of $\bar{l}_4$ consistent with the author's estimate of $5.4 \pm 0.5$?
• RQ4 Why does the value $\bar{l}_4 = 4.4 \pm 0.3$ from Leutwyler et al. conflict with the author's estimate and two-loop fits?
• RQ5 Can the discrepancy between the author's estimate and Leutwyler et al.'s value be explained by two-loop corrections?

Key findings

• The pion scalar radius is estimated as $\langle r^2_S \rangle = 0.75 \pm 0.07\,\text{fm}^2$ using $\delta_S(t) \approx \delta_0(t)$ below $1.5\,\text{GeV}$.
• The chiral perturbation theory constant is found to be $\bar{l}_4 = 5.4 \pm 0.5$, challenging the value $4.4 \pm 0.3$ from Leutwyler et al.
• Higher-twist contributions to the asymptotic phase of the scalar form factor are shown to be negligible when properly calculated.
• Two-loop chiral perturbation theory yields $\bar{l}_4 = 6.60 \pm 0.43$, consistent with the author's estimate and incompatible with $4.4 \pm 0.3$.
• Fits to $\pi\pi$ scattering data including two-loop effects and constraints from low-energy theorems support $\bar{l}_4 = 6.60 \pm 0.43$, with $\chi^2/\text{d.o.f.} = 3.8/4$ or $7.9/6$ depending on inclusion of F-wave data.
• The discrepancy between the author's estimate and Leutwyler et al.'s value would require two-loop corrections of 50% or more to be reconciled, which is implausible.