Gauge-Independent W-Boson Partial Decay Widths
  We calculate the partial decay widths of the W boson at one loop in the standard model using the on-shell renormalization scheme endowed with a gauge-independent definition of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. We work in $R_\xi$ gauge and explicitly verify that the final expressions are independent of the gauge parameters. Furthermore, we establish the relationship between the on-shell and $\bar{\mathrm{MS}}$ definitions of the CKM matrix, both in its generic form and in the Wolfenstein parameterization. As a by-product of our analysis, we recover the beta function of the CKM matrix. 
B.A. Kniehl, F. Madricardo and M. Steinhauser