the coupling increases with decreasing energy scale. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory. SU(N) Non-Abelian gauge theory While the (Yang–Mills) gauge group of QCD is , and determines 3 colors, we can generalize to any number of colors, , with a gauge group . Then for this gauge group, with Dirac fermions in a representation of and with complex scalars in a representation , the one-loop beta function is where is the quadratic Casimir of and is another Casimir invariant defined by for generators of the Lie algebra in the representation R. (For Weyl or Majorana fermions, replace by , and for real scalars, replace by .) For gauge fields (i.e. gluons), necessarily in the adjoint of , ; for fermions in the fundamental (or anti-fundamental) representation of , . Then for QCD, with , the above equation reduces to that listed for the quantum chromodynamics beta function. This famous result was derived nearly simultaneously in 1973 by Politzer, Gross and Wilczek, for which the three were awarded the Nobel Prize in Physics in 2004. Unbeknownst to these authors, G. 't Hooft had announced the result in a comment following a talk by K. Symanzik at a small meeting in Marseilles in June 1972, but he never published it. Standard Model Higgs–Yukawa Couplings In the Standard Model, quarks and leptons have "Yukawa couplings" to the Higgs boson. These determine the mass of the particle. Most all of the quarks' and leptons' Yukawa couplings are small compared to the top quark's Yukawa coupling. These Yukawa couplings change their values depending on the energy scale at which they are measured, through running. The dynamics of Yukawa couplings of quarks are determined by the renormalization group equation: , where is the color gauge coupling (which is a function of and associated with asymptotic freedom) and is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale . The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification, GeV. Therefore, the term can be neglected in the above equation. Solving, we then find that is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, GeV. On the other hand, solutions to this equation for large initial