Quark-Mass Dependence of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ρ</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>Mesons from Dispersion Relations and Chiral Perturbation Theory

Abstract

We use the one-loop chiral perturbation theory $\ensuremath{\pi}\ensuremath{\pi}$-scattering amplitude and dispersion theory in the form of the inverse amplitude method to study the quark-mass dependence of the two lightest resonances of the strong interactions, the ${f}_{0}(600)$ ($\ensuremath{\sigma}$) and the $\ensuremath{\rho}$ meson. As the main results, we find that the $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ coupling constant is almost quark mass independent and that the $\ensuremath{\rho}$ mass shows a smooth quark-mass dependence while that of the $\ensuremath{\sigma}$ shows a strong nonanalyticity. These findings are important for studies of the meson spectrum on the lattice.