Light Mesons and Infrared Behavior of the Running Coupling Constant in QCD

A previous method for handling bound states in QCD is briefly revisited. Taking advantage of the Feynman-Schwinger representation for the iterated quark propagator in an external field, it is possible to give closed representations for certain appropriate (second order) two point and four point Green functions, H⁽²⁾(x - y) and H⁽⁴⁾(x₁, x₂, y₁, y₂), as path integrals on quark world lines. Then, starting from reasonable assumptions on the Wilson line correlators, a Bethe-Salpeter equation for H⁽⁴⁾ and a Dyson-Schwinger equation for H⁽²⁾ can be obtained, which are consistent with the Goldstone theorem in the chiral limit. Such equations are too complicate to be solved directly. However, a reduced Salpeter equation can be derived which is tractable and has been applied to a calculation of the meson spectrum. The results are in general good agreement with the data, but with the important exceptions of the light pseudo scalars (that are related to the breaking of the chiral symmetry). In this scenario two important improvements can be introduced: a) the fixed coupling constant can be replaced by a running coupling constant α_s(Q²) appropriately modified in the infrared region; b) the fixed mass in the reduced equation can be replaced for light quarks by an effective mass depending on the momentum of the particle, as suggested by the form of the DS equation. Then even the light pseudo scalar mesons can be made to agree with to their experimental value.