Inclusive hadronic distributions in jets and sub-jets with jet axis from color current

The hadronic $\kt$-spectrum and the gluon to quark average multiplicity ratio $r=N_g/N_q$ inside a high energy sub-jet are determined from a precise definition of the jet axis, including corrections of relative magnitude ${\cal O}(\sqrt{\alpha_s})$ with respect to the Modified Leading Logarithmic Approximation (MLLA), in the limiting spectrum approximation (assuming an infrared cut-off $Q_0 =\lqcd$). The results for the $\kt$-spectrum in the limiting spectrum approximation are found to be, after normalization, in impressive agreement with measurements by the CDF collaboration and the ratio has not been measured yet.