Editing Beta function (physics) (section)

===SU(N) Non-Abelian gauge theory===
While the (Yang–Mills) gauge group of QCD is <math>SU(3)</math>, and determines 3 colors, we can generalize to any number of colors, <math>N_c</math>, with a gauge group <math>G=SU(N_c)</math>. Then for this gauge group, with Dirac fermions in a [[Representations of Lie groups|representation]] <math>R_f</math> of <math>G</math> and with complex scalars in a representation <math>R_s</math>, the one-loop beta function is
:<math>\beta(g)=-\left(\frac{11}{3}C_2(G)-\frac{1}{3}n_sT(R_s)-\frac{4}{3}n_f T(R_f)\right)\frac{g^3}{16\pi^2}~,</math>

where <math>C_2(G)</math> is the [[Casimir invariant|quadratic Casimir]] of <math>G</math> and <math>T(R)</math> is another Casimir invariant defined by <math>Tr (T^a_RT^b_R) = T(R)\delta^{ab}</math> for generators <math>T^{a,b}_R</math> of the Lie algebra in the representation R. (For [[Weyl]] or [[Majorana fermions]], replace <math>4/3</math> by <math>2/3</math>, and for real scalars, replace <math>1/3</math> by <math>1/6</math>.) For gauge fields (''i.e.'' gluons), necessarily in the [[Adjoint representation of a Lie group|adjoint]] of <math>G</math>, <math>C_2(G) = N_c</math>; for fermions in the [[Fundamental representation|fundamental]] (or anti-fundamental) representation of <math>G</math>, <math>T(R) = 1/2</math>. Then for QCD, with <math>N_c = 3</math>, the above equation reduces to that listed for the quantum chromodynamics beta function.

This famous result was derived nearly simultaneously in 1973 by [[H. David Politzer|Politzer]],<ref>
{{cite journal
 | author=H.David Politzer
 | year=1973
 | title=Reliable Perturbative Results for Strong Interactions?
 | journal=Phys. Rev. Lett.
 | volume=30 | issue=26
 | pages=1346–1349
 | doi=10.1103/PhysRevLett.30.1346
 | url=http://inspirehep.net/record/81351?ln=en|bibcode = 1973PhRvL..30.1346P | doi-access=free
 }}</ref> [[David Gross|Gross]] and [[Frank Wilczek|Wilczek]],<ref>
{{cite journal
 | author=D.J. Gross and F. Wilczek
 | year=1973
 | title=Asymptotically Free Gauge Theories. 1
 | journal=Phys. Rev. D
 | volume=8 | issue=10
 | pages=3633–3652
 | doi=10.1103/PhysRevD.8.3633
 | url=http://inspirehep.net/record/81404 |bibcode = 1973PhRvD...8.3633G | doi-access=free
 }}.</ref>  for which the three were awarded the [[List of Nobel laureates in Physics|Nobel Prize in Physics]] in 2004.
Unbeknownst to these authors, [[Gerard 't Hooft|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.<ref>
{{cite journal
 | author=G. 't Hooft
 | year=1999
 | title= When was Asymptotic Freedom discovered?
 | journal=Nucl. Phys. B Proc. Suppl.
 | volume=74 | issue=1
 | pages=413–425
 | doi=10.1016/S0920-5632(99)00207-8
|arxiv = hep-th/9808154 |bibcode = 1999NuPhS..74..413T | s2cid=17360560
 }}</ref>