where formula_16 is the quadratic Casimir of formula_13 and formula_18 is another Casimir invariant defined by formula_19 for generators formula_20 of the Lie algebra in the representation R. (For Weyl or Majorana fermions, replace formula_21 by formula_22, and for real scalars, replace formula_23 by formula_24.) For gauge fields ("i.e." gluons), necessarily in the adjoint of formula_13, formula_26; for fermions in the fundamental (or anti-fundamental) representation of formula_13, formula_28. Then for QCD, with formula_29, the above equation reduces to that listed for the quantum chromodynamics beta function.