Large-$N_c$ naturalness in coupled-channel meson-meson scattering

The analysis of hadronic interactions with effective field theory techniques is complicated by the appearance of a large number of low-energy constants, which are usually fitted to data. On the other hand, the large-$N_c$ limit helps to impose natural short-distance constraints on these low-energy constants, providing a parameter reduction. A Bayesian interpretation of the expected $1/N_c$ accuracy allows for an easy and efficient implementation of these constraints, using an augmented ${\chi }^2$. We apply this approach to the analysis of meson-meson scattering, in conjunction with chiral perturbation theory to one loop and coupled-channel unitarity, and show that it helps to largely reduce the many existing ambiguities and simultaneously provide an acceptable description of the available phase shifts.

Figure 1

Meson-meson scattering process $A+B\to C+D$, with incoming momenta $k_1$, $k_2$ and outgoing momenta $k_3$, $k_4$. Dashed lines denote pseudoscalar mesons and the black dot their interaction.

Figure 2

Some $\mathcal{O}(p^8)$ loop contributions generated when coupled-channels unitarity is restored (Eqs. (29), (30) and (31)). They give rise to pathologies when estimated using anon-shell approximation. In the bottom panel, the line shows a possible source of a physical imaginary part, when the particles cut by it are put on the mass shell.

Figure 3

Results for meson-meson scattering observables as a function of the CM energy $\sqrt{s}$ (in MeV). We display the pseudodata, with their assumed uncertainties included in the fit (see Table 2). Solid lines represent fitted curves; dashed (blue) lines represent nonfitted curves.

Figure 4

Poles found in the SRS of the scalar-isoscalar elastic $\pi \pi$ amplitude. The $f_0(500)$ and $f_0(980)$ are clearly visible.

Figure 5

Resonance contributions to the pseudoscalar meson-meson scattering. The dotted lines denote pseudoscalar mesons, double lines the intermediate resonances, and the black dot their interaction. The first row contains the diagrams with intermediate vector and scalar resonances in the $s$-, $t$- and $u$-channel graphs. The pseudoscalar resonance contributions are shown in the second row.

Figure 6

Scalar and pseudoscalar resonance contributions to the pseudoscalar meson self-energy and the scalar one to the pion decay constant $f_{\pi }$.