Lowest resonance in QCD from low-energy data

We worked out a generalization of $su(2)$ chiral perturbation theory, including a perturbative singlet scalar field. The approach suggests that the prediction for sensible low-energy observables converge faster towards their physical value. The physical mass and width of the scalar particle are obtained through a simultaneous analysis of the pion vector form factor and the $\gamma \gamma \to {\pi }^0{\pi }^0$ cross section. Both values are statistically consistent with the ones obtained by using Roy equations in $\pi -\pi$ scattering. In addition we find indications that the photon-photon-singlet coupling is quite small.

Figure 1

One-loop diagrams with scalar (dashed lines) contributing to the vector pion form factor.

Figure 2

One-loop diagrams with scalar (dashed lines) contributing to the $\gamma \gamma \to {\pi }^0{\pi }^0$ amplitude.

Figure 3

Pion form factor: The full line is obtained evaluating (8) at the central values of the parameters in (23). For comparison we plot the $\chi \mathrm{PT}$ result at $O(p^4)$, short-dashed line, with ${\overline{l}}_6=16.5$ [6]. If we decrease this value down to $\approx 15$ both curves, full and dashed, agree. The band covers the $1\sigma$ uncertainty around the values in (23). The long-dashed line is the $\chi \mathrm{PT}$ result at $O(p^6)$ and is shown for completeness.

Figure 4

Cross section for $\gamma \gamma \to {\pi }^0{\pi }^0$ integrated over $|\cos {\theta }^{*}|<0.8$ as a function of the $\pi \pi$ invariant mass $M_{\pi \pi }$: The full line is obtained evaluating (14) at the parameters (23) and its $1\sigma$ deviation is covered by the wider, yellow, band. The narrow pink band covers the $1\sigma$ deviation of (24).

Figure 5

Cross section for $\gamma \gamma \to {\pi }^0{\pi }^0$ integrated over $|\cos {\theta }^{*}|<0.8$ as a function of the $\pi \pi$ invariant mass $M_{\pi \pi }$: The full line is obtained evaluating (14) at (23). The green short-dashed line is the $O(p^4)$ $\chi \mathrm{PT}$ prediction [9] which does not depend on any low-energy constant. The long-dashed line is the $O(p^6)$ $\chi \mathrm{PT}$ result in [21]. The orange short-dashed curve shows the best fit performed using the data of this channel alone (21).