Minimal subtraction and momentum subtraction in quantum chromodynamics at two-loop order

The momentum-subtraction coupling constant ${\alpha }_{\mathrm{MOM}}$ yields consistently smaller one-loop corrections to many quantum-chromodynamics (QCD) processes than the minimal-subtraction couplings ${\alpha }_{\mathrm{MS}}$ and ${\alpha }_{\overline{\mathrm{MS}}}$. By shifting the renormalization scale $\mu$ of ${\alpha }_{\mathrm{MS}}\left(\mu \right)$, we obtain a minimal-subtraction coupling with the same small one-loop corrections. It is shown, by studying the effective charges of QCD, that at two-loop order this coupling constant will continue to yield corrections to physical quantities that are comparable to those obtained by momentum subtraction. We also introduce a momentum-subtraction scheme which treats the triple-gluon, quark, and ghost vertices equally at one-loop order and is more convenient for higher-order calculations than the MOM scheme.