Beta function (physics): Difference between revisions

Revision as of 03:10, 21 November 2021

Latest revision as of 18:22, 9 June 2025

(26 intermediate revisions by 15 users not shown)

Line 4:

{{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]], ''μ'', of a given physical process described by [[quantum field theory]].

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

It is defined as

:: <math>\beta(g) = \mu \frac{\partial g}{\partial \mu} = \frac{\partial g}{\partial \logln(\mu)} ~,</math>

and, because of the underlying [[renormalization group]], it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''.

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 scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of beta function was first introduced by [[Ernst Stueckelberg]] and [[André Petermann]] in 1953,<ref>{{cite journal |author1-link=Ernst Stueckelberg |last1=Stueckelberg |first1=E.C.G. |author2-link=André Petermann |first2=A. |last2=Petermann |year=1953 |url=https://www.e-periodica.ch/cntmng?pid=hpa-001:1953:26::894 |title=La renormalisation des constants dans la théorie de quanta |journal=Helv. Phys. Acta |volume=26 |pages=499–520 |language=FR}}</ref> and independently postulated by [[Murray Gell-Mann]] and [[Francis E. Low]] in 1954.<ref>{{Cite journal |last=Fraser |first=James D. |date=2021-10-01 |title=The twin origins of renormalization group concepts |url=https://www.sciencedirect.com/science/article/pii/S0039368121001126 |journal=Studies in History and Philosophy of Science Part A |volume=89 |pages=114–128 |doi=10.1016/j.shpsa.2021.08.002 |issn=0039-3681}}</ref>

feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques.

== 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]].

Line 27 ⟶ 30:

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

or, equivalently,

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

written in terms of the [[Fine-structure constant#In non-SI units|fine structure constant]] in natural units, {{math|''α'' {{=}} ''e''²/4π}}.<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.

Line 41 ⟶ 44:

:<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_s'' = <math>g^2/4\pi</math> .

IfAssuming ''n''_''s''=0, if ''n''_''f'' ≤ 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-MillsYang–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 [[H.Hugh David Politzer|Politzer]],<ref>

{{cite journal

| author=H.David Politzer

Line 57 ⟶ 60:

| 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

Line 66 ⟶ 70:

| 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.[[Kurt Symanzik]] at a small meeting in Marseilles[[Marseille]] in June 1972, but he never published it.<ref>

{{cite journal

| author=G. 't Hooft

Line 76 ⟶ 81:

| 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-YukawaHiggs–Yukawa Couplingscouplings===

{{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.

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|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|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.

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.

The value of the quasi-fixed point is fairly precisely determined in the Standard Model, leading to a predicted [[top quark]] mass of 230 &nbsp;GeV.{{Citation needed|date=November 2021}} The observed top quark mass of 174 GeV is slightly lower than the standard model prediction by about 30% which suggests there may be more Higgs doublets beyond the single standard model Higgs boson.

===Minimal Supersymmetricsupersymmetric Standard Model===

{{Main|Minimal Supersymmetric Standard Model#Gauge-Coupling Unification}}

Renomalization group studies in the Minimalminimal Supersymmetricsupersymmetric Standard Model (MSSM) of grand unification and the Higgs-YukawaHiggs–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]].

==See also==

Line 105 ⟶ 110:

==References==

{{notelist}}

{{Reflist}}

Line 112 ⟶ 118:

* Zinn-Justin, Jean; ''Quantum Field Theory and Critical Phenomena,'' Oxford University Press (2002). Emphasis on the renormalization group and related topics.

{{Authority control}}

==External links==

[[Category:Quantum field theory]]

[[Category:Renormalization group]]

[[Category:Scaling symmetries]]