Editing Beta function (physics) (section)

==Examples==
Beta functions are usually computed in some kind of approximation scheme. An example is [[Perturbation theory (quantum mechanics)|perturbation theory]], where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher [[Feynman graph|loop]] contributions, due to the number of loops in the corresponding [[Feynman graph]]s).

Here are some examples of beta functions computed in perturbation theory:

===Quantum electrodynamics===
{{Main|Quantum electrodynamics}}
The one-loop beta function in [[quantum electrodynamics]] (QED) is

*<math>\beta(e)=\frac{e^3}{12\pi^2}~,</math>
or, equivalently,
*<math>\beta(\alpha)=\frac{2\alpha^2}{3\pi}~,</math>

written in terms of the [[Fine-structure constant#In non-SI units|fine structure constant]] in natural units, {{math|''α'' {{=}}  ''e''<sup>2</sup>/4π}}.<ref>{{cite book |last1=Srednicki |first1=Mark Allen |title=Quantum field theory |date=2017 |publisher=Cambridge Univ. Press |location=Cambridge |isbn=978-0-521-86449-7 |page=446 |edition=13th printing}}</ref>

This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a [[Landau pole]]. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.

===Quantum chromodynamics===
{{Main|Quantum chromodynamics}}
The one-loop beta function in [[quantum chromodynamics]] with <math>n_f</math> [[Flavour (particle physics)#Quantum chromodynamics|flavours]] and <math>n_s</math> scalar colored bosons is
:<math>\beta(g)=-\left(11- \frac{n_s}{6} - \frac{2n_f}{3}\right)\frac{g^3}{16\pi^2}~,</math>

or
:<math>\beta(\alpha_s)=-\left(11- \frac{n_s}{6}-\frac{2n_f}{3}\right)\frac{\alpha_s^2}{2\pi}~,</math>

written in terms of  ''α<sub>s</sub>'' = <math>g^2/4\pi</math> .

Assuming ''n''<sub>''s''</sub>=0, if ''n''<sub>''f''</sub> ≤ 16,  the ensuing beta function  dictates that the coupling decreases with increasing energy scale, a phenomenon known as [[asymptotic freedom]]. Conversely, 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 <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>

===Standard Model Higgs–Yukawa Couplings===
{{Main|Infrared fixed point}}

In the [[Standard Model]], quarks and leptons have "[[Yukawa interaction|Yukawa coupling]]s" 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   ''[[Renormalization group|running]]''. The dynamics of Yukawa couplings of quarks are determined by the [[Exact renormalization group equation|renormalization group equation]]:

<math>\mu \frac{\partial}{\partial\mu} y  \approx \frac{y}{16\pi^2}\left(\frac{9}{2}y^2 - 8 g_3^2\right)</math>,

where <math>g_3</math> is the [[color charge|color]] [[gauge theory|gauge]] coupling (which is a function of  <math>\mu</math> and associated with [[asymptotic freedom]])  and <math>y</math> is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale <math>\mu</math>.

The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of [[Grand Unified Theory|grand unification]], <math> \mu \approx 10^{15} </math> GeV.  Therefore, the <math>y^2</math> term can be neglected in the above equation. Solving, we then find that <math>y</math>  is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, <math> \mu \approx 100 </math> GeV.

On the other hand, solutions to this equation for large initial values <math>y</math> cause the ''rhs'' to quickly approach smaller values as we descend in energy scale. The above equation then locks <math>y</math> to the QCD coupling <math>g_3</math>. This is known as the (infrared) quasi-fixed point of the renormalization group equation for the Yukawa coupling.<ref>{{cite journal|last1=Pendleton|first1=B.|last2=Ross|first2=G.G.|title=Mass and Mixing Angle Predictions from Infrared Fixed points|journal=Phys. Lett.|date=1981|volume=B98|issue=4 |page=291|doi=10.1016/0370-2693(81)90017-4|bibcode = 1981PhLB...98..291P }}</ref><ref>{{cite journal|last1=Hill|first1=C.T.|title=Quark and Lepton masses from Renormalization group fixed points|journal=Phys. Rev.|date=1981|volume=D24|issue=3 |page=691|doi=10.1103/PhysRevD.24.691|bibcode = 1981PhRvD..24..691H }}</ref> No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this quasi-fixed point value, and the corresponding quark mass is predicted.

===Minimal Supersymmetric Standard Model===
{{Main|Minimal Supersymmetric Standard Model#Gauge-Coupling Unification}}
Renomalization group studies in the Minimal Supersymmetric Standard Model (MSSM) of grand unification and the Higgs–Yukawa fixed points were very encouraging that the theory was on the right track. So far, however, no evidence of the predicted MSSM particles has emerged in experiment at the [[Large Hadron Collider]].