$\rho$ meson, Bethe-Salpeter equation, and the far infrared

The Bethe-Salpeter equation in QCD connects the gauge-dependent gluon and quark degrees of freedom with the gauge-invariant properties of mesons. We study the $\rho$ meson mass and decay constant for various versions of the gauge-dependent input functions discussed in the literature, which start to differ generically below the hadronic scale and show qualitative different infrared behavior. We find that, once the gauge-dependent quark-gluon vertex is permitted to vary as well, the $\rho$ mass and decay constant is reproduced equally well for all forms investigated. A possible conclusion from this is that these $\rho$-meson properties are only sensitive to changes in the input at scales above a few hundred MeV.

Figure 1

The input running coupling (1). Here, and hereafter, SC denotes the scaling solution and DC the decoupling solutions. For more details, see 11.

Figure 2

The quark wave-function dressing (top) and mass function (bottom) for the four choices of parameters of the quark-gluon vertex fixed to pion properties, see Table .

Figure 3

The momentum dependence of the ansatz (4) for the parameter values $D$ and $\omega$ of Table  (top) and for the optimized values of Table  (bottom).