  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.
