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{{quantum field theory}}
In [[theoretical physics]], specifically [[quantum field theory]], a '''beta function''' or '''Gell-Mann–Low function''', ''β(g)'', encodes the dependence of a [[Coupling constant|coupling parameter]], ''g'', on the [[energy scale]],
It is defined by the '''Gell-Mann–Low equation'''<ref>{{Cite book |last=Tsvelik |first=Alexei M. |url=https://www.google.fr/books/edition/Quantum_Field_Theory_in_Condensed_Matter/78t7iDTth2YC?hl=en&gbpv=1&dq=%22Gell-Mann-Low+equation%22+renormalization&pg=PA61&printsec=frontcover |title=Quantum Field Theory in Condensed Matter Physics |date=2007-01-18 |publisher=Cambridge University Press |isbn=978-0-521-52980-8 |language=en}}</ref> or '''renormalization group equation,''' given by
:: <math>\beta(g) = \mu \frac{\partial g}{\partial \mu} = \frac{\partial g}{\partial \ln(\mu)} ~,</math>
and, because of the underlying
This dependence on the energy scale thus specified is known as the [[Coupling constant#Running coupling|running]] of the coupling parameter, a fundamental
feature of
== History ==
{{Excerpt|Renormalization group#Beginnings}}
==Scale invariance==
If the beta functions of a [[quantum field theory]] (QFT) vanish, usually at particular values of the coupling parameters, then the theory is said to be [[Scale invariance|scale-invariant]]. Almost all scale-invariant QFTs are also [[Conformal symmetry|conformally invariant]]. The study of such theories is [[conformal field theory]].
The coupling parameters of a quantum field theory can run even if the corresponding [[classical field theory]] is scale-invariant. In this case, the non-zero beta function tells us that the classical scale invariance is [[Conformal anomaly|anomalous]].
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:<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
Assuming ''n''<sub>''s''</sub>=0, if ''n''<sub>''f''</sub> ≤ 16,
===SU(''N'') Non-Abelian gauge theory===
While the (Yang–Mills) gauge group of QCD is <math>\mathrm{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=\mathrm{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 [[
{{cite journal
| author=H.David Politzer
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| doi=10.1103/PhysRevLett.30.1346
| url=http://inspirehep.net/record/81351?ln=en|bibcode = 1973PhRvL..30.1346P | doi-access=free
}}</ref> [[David
{{cite journal
| author=D.J. Gross and F. Wilczek
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| doi=10.1103/PhysRevD.8.3633
| url=http://inspirehep.net/record/81404 |bibcode = 1973PhRvD...8.3633G | doi-access=free
}}.</ref>
Unbeknownst to these authors, [[Gerard
{{cite journal
| author=G. 't Hooft
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}}</ref>
===Standard Model Higgs–Yukawa
{{Main|Infrared fixed point}}
In the [[Standard Model]], quarks and leptons have
<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
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.
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
{{Main|Minimal Supersymmetric Standard Model#Gauge-Coupling Unification}}
Renomalization group studies in the
==See also==
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==References==
{{notelist}}
{{Reflist}}
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