Gauge dependence of the renormalization-group functions in quantum chromodynamics

The gauge dependence of the renormalization-group (RG) functions in the momentum-space subtraction scheme is systematically studied in quantum chromodynamics. Fixed-gauge RG equations satisfied by physical quantities are found to be useful because we can treat the gauge parameter as a pure number in the solution. In massless quark theory, the first two coefficients of the $\beta$ function which appear in the fixed-gauge RG equation are found to be gauge independent and coincide with those of the minimal-subtraction $\beta$ function. Higher-order coefficients are gauge dependent and we calculate the third coefficient in a special momentum-subtraction scheme of Braaten and Leveille. In massive quark theory, even the first coefficients are in general gauge dependent. We present an algorithm to calculate these coefficients in an arbitrary momentum-space subtraction scheme.