In this work, we perform the one-loop calculation of the scalar and pseudoscalar form factors in the framework of $U(3)$ chiral perturbation theory with explicit tree level exchanges of resonances. The meson-meson scattering calculation from Guo and Oller [Phys. Rev. D 84, 034005 (2011)] is extended as well. The spectral functions of the nonet scalar-scalar ($SS$) and pseudoscalar-pseudoscalar ($PP$) correlators are constructed by using the corresponding form factors. After fitting the unknown parameters to the scattering data, we discuss the resonance content of the resulting scattering amplitudes. We also study spectral-function sum rules in the $SS-SS$, $PP-PP$, and $SS-PP$ sectors as well as semilocal duality from scattering. The former relate the scalar and pseudoscalar spectra between themselves while the latter mainly connects the scalar spectrum with the vector one. Finally we investigate these items as a function of $N_C$ for $N_C>3$. All these results pose strong constraints on the scalar dynamics and spectroscopy that are discussed. They are successfully fulfilled by our meson-meson scattering amplitudes and spectral functions.

• Received 22 June 2012

DOI:https://doi.org/10.1103/PhysRevD.86.054006