We study the pion mass dependence of the $\rho (770)$ and $f_0(600)$ masses and widths from one and two-loop unitarized chiral perturbation theory (ChPT). We show the consistency of one-loop calculations with lattice results for the $M_{\rho }$, $f_{\pi }$, and the isospin-2 scattering length $a_{20}$. Then, we develop and apply the modified inverse amplitude method formalism for two-loop ChPT. In contrast to the $f_0(600)$, the $\rho (770)$ is rather sensitive to the two-loop ChPT parameters—our main source of systematic uncertainty. We thus provide two-loop unitarized fits constrained by lattice information on $M_{\rho }$, $f_{\pi }$, by the $q\overline{q}$ leading $1/N_c$ behavior of the $\rho$ and by existing estimates of low-energy constants. These fits yield relatively stable predictions up to $m_{\pi }\simeq 300–350\mathrm{MeV}$ for the $\rho$ coupling and width as well as for all the $f_0(600)$ parameters. We confirm, to two loops, the weak $m_{\pi }$ dependence of the $\rho$ coupling and the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin relation, and the existence of two virtual $f_0(600)$ poles for sufficiently high $m_{\pi }$. At two loops one of these poles becomes a bound state when $m_{\pi }$ is somewhat larger than 300 MeV.

• Received 2 August 2010

DOI:https://doi.org/10.1103/PhysRevD.82.114002